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Doing science with eLISA:

[1cm] Astrophysics and cosmology in the millihertz regime

[3cm]

Pau Amaro-Seoane, Sofiane Aoudia, Stanislav Babak, Pierre Binétruy, Emanuele Berti, Alejandro Bohé, Chiara Caprini, Monica Colpi, Neil J. Cornish, Karsten Danzmann, Jean-François Dufaux, Jonathan Gair, Oliver Jennrich, Philippe Jetzer, Antoine Klein, Ryan N. Lang, Alberto Lobo, Tyson Littenberg, Sean T. McWilliams, Gijs Nelemans, Antoine Petiteau, Edward K. Porter, Bernard F. Schutz, Alberto Sesana, Robin Stebbins, Tim Sumner, Michele Vallisneri, Stefano Vitale, Marta Volonteri, and Henry Ward





Max Planck Institut für Gravitationsphysik (Albert-Einstein-Institut), Germany

APC, Univ. Paris Diderot, CNRS/IN2P3, CEA/Irfu, Obs. de Paris, Sorbonne Paris Cité, France

Department of Physics and Astronomy, The University of Mississippi, University, MS 38677, USA

Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena CA 91125, USA

UPMC-CNRS, UMR7095, Institut d’Astrophysique de Paris, F-75014, Paris, France

Institut de Physique Théorique, CEA, IPhT, CNRS, URA 2306, F-91191Gif/Yvette Cedex, France

University of Milano Bicocca, Milano, I-20100, Italy

Department of Physics, Montana State University, Bozeman, MT 59717, USA

Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, UK

ESA, Keplerlaan 1, 2200 AG Noordwijk, The Netherlands

Institute of Theoretical Physics University of Zürich, Winterthurerstr. 190, 8057 Zürich Switzerland

Washington University in St. Louis, One Brookings Drive, St. Louis, MO 63130, USA

Institut de Ciències de l’Espai (CSIC-IEEC), Campus UAB, Torre C-5, parells, planta, ES-08193, Bellaterra, Barcelona, Spain

Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742

Gravitational Astrophysics Laboratory, NASA Goddard Spaceflight Center, 8800 Greenbelt Rd., Greenbelt, MD 20771, USA

Department of Physics, Princeton University, Princeton, NJ 08544, USA

Department of Astrophysics, Radboud University Nijmegen, The Netherlands

Institute for Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium

Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands

NASA Liason, NASA GSFC, USA

Imperial College, UK

Jet Propulsion Laboratory, California Inst. of Technology, Pasadena, CA 91109, USA

University of Trento, Department of Physics, I38050 Povo, Trento, Italy

Institut d’Astrophysique de Paris, 98bis Boulevard Arago, 75014 Paris, France

Astronomy Department, University of Michigan, Ann Arbor, MI 48109, USA

Institute for Gravitational Research, Department of Physics & Astronomy Kelvin Building, University of Glasgow, Glasgow

Contents

Abstract

This document introduces the exciting and fundamentally new science and astronomy that the European New Gravitational Wave Observatory (NGO) mission (derived from the previous LISA proposal) will deliver. The mission (which we will refer to by its informal name “eLISA”) will survey for the first time the low-frequency gravitational wave band (about  mHz to 1 Hz), with sufficient sensitivity to detect interesting individual astrophysical sources out to . The measurements described here will address the basic scientific goals that have been captured in ESA’s “New Gravitational Wave Observatory Science Requirements Document”; they are presented here so that the wider scientific community can have access to them. The eLISA mission will discover and study a variety of cosmic events and systems with high sensitivity: coalescences of massive black holes binaries, brought together by galaxy mergers; mergers of earlier, less-massive black holes during the epoch of hierarchical galaxy and black-hole growth; stellar-mass black holes and compact stars in orbits just skimming the horizons of massive black holes in galactic nuclei of the present era; extremely compact white dwarf binaries in our Galaxy, a rich source of information about binary evolution and about future Type Ia supernovae; and possibly most interesting of all, the uncertain and unpredicted sources, for example relics of inflation and of the symmetry-breaking epoch directly after the Big Bang. eLISA’s measurements will allow detailed studies of these signals with high signal-to-noise ratio, addressing most of the key scientific questions raised by ESA’s Cosmic Vision programme in the areas of astrophysics and cosmology. They will also provide stringent tests of general relativity in the strong-field dynamical regime, which cannot be probed in any other way. This document not only describes the science but also gives an overview on the mission design and orbits. LISA’s heritage in the eLISA design will be clear to those familiar with the previous proposal, as will its incorporation of key elements of hardware from the LISA Pathfinder mission, scheduled for launch by ESA in 2014. But eLISA is fundamentally a new mission, one that will pioneer the completely new science of low-frequency gravitational wave astronomy.

1 Introduction

Our view of the Universe has changed dramatically over the past century. Less than a hundred years ago our own Galaxy, the Milky Way, was believed to be our own island-Universe. The discovery of hundreds of billion galaxies like our own, and of billion luminous sources such as Quasi Stellar Objects (QSOs), changed our perception of the cosmic landscape. New astronomical objects were discovered with the advent of radio and X-ray Astronomy. Relativistic binaries composed of compact stars (such as white dwarfs or neutron stars) and stellar-mass black holes are among these sources of electromagnetic radiation. According to the accretion paradigm, supermassive black holes at galactic centres are the simplest explanation for the power emitted by distant, luminous QSOs, but a conclusive test of this hypothesis is still lacking.

The remarkable discovery of the recession of galaxies and of the fossil microwave background radiation, almost contemporary to the discovery of X-ray sources, has further led to the emergence of a cosmological paradigm, the Big Bang, that has revolutionized our description of the Universe. We now know that our Universe had a beginning and that its luminous components (in particular, galaxies and QSOs) evolve jointly and in concordance with the evolution of the underlying dark matter permeating the Universe.

According to General Relativity, black holes and compact binaries are expected to be powerful sources of gravitational waves. Rather than “seeing” electromagnetic radiation, as all of astronomy has done until present, eLISA will “hear” the vibrations of the fabric of spacetime itself, emitted coherently by macroscopic bodies. Studying these signals will convey rich new information about the behaviour, the structure and the history of the Universe, and it will clarify several issues in fundamental physics.

Gravitational waves travel undisturbed through spacetime, and when observed they offer a new and uniquely powerful way to probe the very distant Universe, from the extremely early Big Bang to the early epoch of galaxy and black hole seed formation. This may allow us to address deep questions. What powered the Big Bang? How did galaxies and their black holes form and evolve? What is the structure of spacetime around the massive objects we believe to be black holes? What is the nature of the mysterious dark matter and dark energy accelerating the expansion of the Universe?

eLISA is a space-based mission designed to measure gravitational radiation over a broad band at frequencies ranging between mHz and Hz. In this frequency band the Universe is richly populated by strong sources of gravitational waves. For binary systems the characteristic gravitational-wave frequency is twice the Keplerian orbital frequency, which in turn is proportional to , where is the total mass of the binary and its semi-major axis. In the eLISA frequency band, gravitational waves are produced by close binaries of stellar-mass objects with orbital periods of a few to several minutes. Massive black hole binaries with and mass ratio on the verge of coalescing have orbital frequencies sweeping to higher and higher values, until the binary separation becomes as small as the scale of the event horizon . Finally, eLISA could observe binaries comprising a massive black hole and a stellar-mass compact object (e.g., a stellar-mass black hole) skimming the horizon of the larger black hole before being captured: these systems are commonly referred to as extreme mass ratio inspirals (EMRIs). Furthermore, a stochastic background in the eLISA frequency band can be generated by less conventional sources, such as phase transitions in the very early Universe and/or cosmic strings.

This note is a comprehensive survey of the eLISA science case. We consider all the relevant astrophysical and cosmological gravitational wave sources and explore eLISA detection performances in terms of sensitivity, SNR distributions, and parameter estimation.

In Section 2 we will briefly describe the mission concept and the basic design of the instrument, introducing the eLISA sensitivity curve that will be used throughout the study.

Our survey of the eLISA science case will start in Section 3 by exploring the nearest observable sources of gravitational waves, i.e. compact stellar-mass binaries in the Milky Way. eLISA will study the gravitational wave signals from thousands of stellar-mass close binaries in the Galaxy and will give information on the extreme endpoints of stellar evolution. eLISA will provide distances and detailed orbital and mass parameters for hundreds of these binaries. This is a rich trove of information for mapping and reconstructing the history of stars in the Galaxy, and it can reveal details of the tidal and non-gravitational influences on the binary evolution associated with the internal physics of the compact remnants themselves.

Then we will summarize the science objectives that are relevant for the astrophysics of black holes (Section 4). Current electromagnetic observations are probing only the tip of the black hole mass distribution in the Universe, targeting black holes with large masses, between and . Conversely, eLISA will be able to detect the gravitational waves emitted by black hole binaries with total mass (in the source rest frame) as small as and up to , out to a redshift as large as . eLISA will detect fiducial sources out to redshift , and so it will explore almost all the mass-redshift parameter space relevant for addressing scientific questions on the evolution of the black hole population. Redshifted masses will be measured to an unprecedented accuracy, up to the 0.1 – 1 % level, whereas absolute errors in the spin determination are expected to be in the range 0.01 to 0.1, allowing us to reconstruct the cosmic evolution of massive black holes. Black holes are expected to transit into the mass interval to which eLISA is sensitive along the course of their cosmic evolution. Thus, eLISA will map and mark the loci where galaxies form and cluster, using black holes as clean tracers of their assembly.

eLISA will also bring a new revolutionary perspective to the study of galactic nuclei, as shown in Section 5. Orbits of stellar objects captured by the massive black hole at the galactic centre evolve by gravitational radiation. By capturing their signal, eLISA will offer the deepest view of nearby galactic nuclei, exploring regions that are invisible to electromagnetic techniques. eLISA will probe the dynamics of compact objects in the space-time of a Kerr black hole, providing information on the space density of those objects.

In Section 6 we address key questions concerning the nature of spacetime and gravity. GR has been extensively tested in the weak field regime, both in the solar system and via binary pulsar observations. eLISA will provide a unique opportunity to probe GR in the strong field limit. eLISA will observe the coalescence of massive black hole binaries moving at speeds close to the speed of light, and enable us to test the dynamics of curved spacetime when gravitational fields are strong. By observing a large number of orbital cycles during the last few years of the inspiral of a stellar mass object into a massive black hole, eLISA will allow us to measure precisely the parameters of the central object (including its quadrupole moment) in the near Universe. Any deviations in the orbital motion from GR predictions will leave an imprint in the gravitational wave phase. Thus, measurements of the mass, spin and quadrupole moment of the central object will allow us to check the Kerr nature of the central massive object, and to test for the first time the black hole hypothesis.

Lastly, as we describe in Section 7, eLISA will probe new physics and cosmology with gravitational waves, and search for unforeseen sources of gravitational waves. The eLISA frequency band in the relativistic early Universe corresponds to horizon scales where phase transitions or extra dimensions may have caused catastrophic, explosive bubble growth and efficient gravitational wave production. eLISA will be capable of detecting a stochastic background from such events from about 100 GeV to about 1000 TeV, if gravitational waves in the eLISA band were produced with sufficient efficiency.

In closing the note, we will present a summary of the science objectives of the eLISA mission. We summarize the key theoretical and observational goals of the eLISA science case in Section 8. There, in a schematic bullet-point form, we enumerate the scientific goals related to each class of gravitational wave sources and the observational performance of eLISA in achieving such goals. This could serve as a compact summary of eLISA science, as well as a reference point for other space-based gravitational wave detector proposals.

2 Description of the mission

eLISA is a European-led variant of LISA that can be launched before 2022 . The basic principle of gravitational wave detection for eLISA is the same as for LISA: it is a laser interferometer designed to detect the passage of a gravitational wave by measuring the time-varying changes of optical pathlength between free-falling masses. Many design and technological developments were migrated from LISA, however there are some substantial differences.

The two measurement arms are defined by three spacecraft orbiting the Sun in a triangular configuration (see figure 1). A key feature of the eLISA concept is a set of three orbits that maintain a near-equilateral triangular formation with an armlength , without the need for station-keeping. Depending on the initial conditions of the spacecraft, the formation can be kept in an almost constant distance to the Earth or be allowed to slowly drift away to about m, the outer limit for communication purposes. A very attractive feature of the eLISA orbits is the almost constant sun-angle of 30 degrees with respect to the normal to the top of the spacecraft, thereby resulting in an extremely stable thermal environment, minimizing the thermal disturbances on the spacecraft.

Figure 1: The eLISA orbits: The constellation is shown trailing the Earth by about 20 degrees (or ) and is inclined by 60 degrees with respect to the ecliptic. The trailing angle will vary over the course of the mission duration from 10 degrees to 25 degrees. The separation between the spacecraft is m.

One of the three spacecraft serves as the “central hub” and defines the apex of a “V”. Two other, simpler spacecraft are positioned at the ends of the V-shaped constellation. The central spacecraft houses two free-falling “test masses” that define the endpoint of the two interferometer arms. The other spacecraft contain one test mass each, defining two more endpoints (see figure 2). Each spacecraft accommodates the interferometry equipment for measuring changes in the arm length. For practical reasons, this measurement is broken up into three distinct parts (see figure 3): the measurement between the spacecrafts, i.e. between the optical benches that are fixed to each spacecraft, and the measurement between each of the test masses and its respective optical bench. Those measurements are recombined in a way that allows us to reconstruct the distance between the test masses which is insensitive to the noise in the position of the spacecraft with respect to the test masses.

Figure 2: The constellation of the three eLISA spacecraft constitutes the science instrument. The central spacecraft harbors two send/receive laser ranging terminals, while the end spacecraft has one each. The laser in the end spacecraft is phase-locked to the incoming laser light. The blue dots indicate where interferometric measurements are taken. The sketch leaves out the test mass interferometers for clarity.
Figure 3: Partition of the eLISA measurement. Each measurement between two test masses is broken up into three different measurements: two between the respective test mass and the spacecraft and one between the two spacecraft (S/C). As the noise in the measurement is dominated by the shot noise in the S/C-S/C measurement, the noise penalty for the partitioning of the measurement is negligible. The blue (solid) dots indicate where the interferometric measurements are taken.

A second key feature of the eLISA concept is that the test masses are protected from disturbances as much as possible by a careful design and the "drag-free" operation. To establish the drag-free operation, a housing around the test mass senses the relative position of test mass and spacecraft, and a control system commands the spacecraft thrusters to follow the free-falling mass. Drag-free operation reduces the time-varying disturbances to the test masses caused by force gradients arising in a spacecraft that is moving with respect to the test masses. The requirements on the power spectral density of the residual acceleration of the test mass is

(1)

or

(2)

where is the frequency.

The third key feature, the distance measuring system, is a continuous interferometric laser ranging scheme, similar to that used for radar-tracking of spacecraft. The direct reflection of laser light, such as in a normal Michelson interferometer, is not feasible due to the large distance between the spacecrafts. Therefore, lasers at the ends of each arm operate in a "transponder" mode. A laser beam is sent out from the central spacecraft to an end spacecraft. The laser in the end spacecraft is then phase-locked to the incoming beam thus returning a high-power phase replica. The returned beam is received by the central spacecraft and its phase is in turn compared to the phase of the local laser. A similar scheme is employed for the second arm. In addition, the phases of the two lasers serving the two arms are compared within the central spacecraft. The combined set of phase measurements together with some auxiliary modulation allows to determine the relative optical path changes with simultaneous suppression of the laser frequency noise and clock noise below the secondary (acceleration and displacement) noise. The displacement noise has two components: the shot noise, with a required power spectral density

(3)

and the other (combined) measurement noise with a required power spectral density

(4)

According to the requirements, eLISA achieves the strain noise amplitude spectral density (often called sensitivity) showed in figure 4 which can be analytically approximate as , where :

(5)

This allows to detect a strain of about in a 2-year measurement with an SNR of 1 (displacement sensitivity of over a path length of ). The feasible reduction of disturbances on test masses and the displacement sensitivities achievable by the laser ranging system yield a useful measurement frequency bandwidth from Hz to 1 Hz (the requirement is Hz to 1 Hz; the goal is Hz to 1 Hz).

Figure 4: Sensitivity of eLISA (averaged over all sky locations and polarisations) versus frequency: the solid red curve is obtained numerically using the simulator LISACode 2.0 (Petiteau et al., 2008) and the dashed blue curve is the analytic approximation based on equation 5. For a reference, we also depict the sensitivity curve of LISA (dotted, green curve).

3 Ultra-Compact Binaries

1 Overview

The most numerous sources in the low-frequency gravitational wave band are ultra-compact binary stars: double stars in which two compact objects, such as white dwarfs and neutron stars, orbit each other with short periods. They have relatively weak gravitational wave signals in comparison to massive black hole binaries, but are numerous in the Galaxy and even the Solar neighbourhood.

Several thousand systems are expected to be detected individually, with their parameters determined to high precision, while the combined signals of the millions of compact binaries in the eLISA band will form a foreground signal. This is in contrast to less than 50 ultra-compact binaries known today. The number of detections will allow for detailed study of the entire WD binary population. In particular, the most numerous sources are double white dwarfs, which are one of the candidate progenitors of type Ia supernovae and related peculiar supernovae. eLISA will determine the merger rate of these binaries. The detailed knowledge of the ultra-compact binary population also constrains the formation of these binaries and thus many preceding phases in binary evolution. This has a strong bearing on our understanding of many high-energy phenomena in the Universe, such as supernova explosions, gamma-ray bursts and X-ray sources, as they share parts of the evolution history of the binaries detectable by eLISA.

As many of the Galactic sources are rather close (within a few kpc), they will be detectable at high SNR (often larger than ), allowing detailed studies of individual binaries. For many hundreds, the frequency and phase evolution can be studied, enabling the study of the physics of tides and mass transfer in unprecedented detail. The extreme conditions of short orbital periods, strong gravitational fields and high mass-transfer rates are unique in astrophysics.

The information provided by eLISA will be different from what can be deduced by electromagnetic observations. In particular, eLISA’s capability to determine distances and inclinations, as well as the fact that the gravitational wave signals are unaffected by interstellar dust, provide significant advantages over other detection techniques. Compared to Gaia, eLISA will observe a quite different population. Gravitational wave observations allow us to determine the distances to binaries that are right in the Galactic centre rather than to those close to the Sun. The distance determinations will make it possible to map the distribution of many compact binaries in the Galaxy, providing a new method to study Galactic structure. The inclination determinations allow the study of binary formation by comparing the average angular momentum of the binaries to that of the Galaxy. Electromagnetic observations and gravitational wave observations are complementary to one another; dedicated complementary observing programs as well as public data releases will allow simultaneous and follow-up electromagnetic observations of binaries identified by eLISA.

A number of guaranteed detectable sources are known to date from electromagnetic observations. Some of these can be used to verify instrument performance by looking for a gravitational signal at twice the orbital period and comparing the signal with expectations. In addition, once eLISA has detected several nearby binaries and determined their sky position they can be observed optically, thus providing an additional quantitative check on instrument sensitivity.

2 Instrument verification

Figure 5: Artist impression of a detached double white dwarf binary (left) and an interacting binary in which a neutron star accretes material from a white dwarf donor. The Earth is shown to set the scale. Courtesy BinSim by Rob Hynes.

There are currently about 50 know ultra-compact binaries. They come in two flavours: systems in which the two stars are well apart, called detached binaries, and systems in which the two stars are so close together that mass is flowing from one star to the other, called interacting binaries (see figure 5).

A subset of the known ultra-compact binaries have been recognised as instrument verification sources, as they should be detected in a few weeks to months and thus can be used to verify the performance of the instrument (Stroeer and Vecchio, 2006). The most promising verification binaries, shown as green squares in figure 6, are the shortest-period interacting binaries HM Cnc (RX J0806.3+1527), V407 Vul, ES Cet and the recently discovered 12 minute period detached system SDSS J0651+28 (Brown et al., 2011), whose lightcurve is shown in figure 8. For a decade it has remained unclear if the measured periods of HM Cnc and V407 Vul were actually orbital periods, but recent results from the Keck telescope on HM Cnc (Roelofs et al., 2010) show conclusively that this system has an orbital period of 5.4 minutes. As V407 Vul has almost identical properties, this implies that this also really is a binary with an orbital period of 9.5 minutes. As the signal from the verification binaries is essentially monochromatic with a well known frequency within the eLISA mission time, astrophysical effects such as those discussed in section 4 will not hamper their detection. As more and more wide field and synoptical surveys are completed, the number of ultra-compact binaries is gradually increasing and is expected to continue to do so in the future. Already several new binaries have been found in the SDSS and the PTF (Rau et al., 2010, Levitan et al., 2011) while surveys such as Pan-Starrs, the EGAPS and in the future LSST will also find new systems. However, most of the systems found so far have relatively long orbital periods (longer than about 30 minutes). Two pilot surveys in principle capable of finding ultra-compact binaries with periods less than 30 minutes are underway or will start soon: the RATS (Barclay et al., 2011) and the OmegaWhite survey.

Figure 6: Strain amplitude spectral density versus frequency for the verification binaries and the brightest binaries,expected from a simulated Galactic population of ultra-compact binaries. The solid line shows the sensitivity of eLISA. Verification binaries are indicated as green squares with their names indicated, blue squares are other known binaries. Strongest 100 simulated binaries are shown in red, strongest 1000 as black dots. The integration time for the binaries is two years. Based on Roelofs et al. (2006, 2010), Brown et al. (2011) for the known binaries and Nelemans et al. (2004) for the simulation.

Interacting ultra-compact binaries with neutron star accretors are strong X-ray sources and new discoveries are expected, both through the continued monitoring of the sky to search for X-ray transients with RXTE, MAXI and other satellites, as well as through dedicated X-ray and optical surveys of the Galactic bulge that are currently happening (Jonker et al., 2011). With these developments we expect that several tens of verification sources should be available for eLISA, allowing detailed tests of the performance of the instrument.

3 eLISA as a workhorse: thousands of new binaries

Ultra-compact binaries will completely dominate the number of source detections by eLISA. Current estimates suggest the numbers of resolved compact binaries that will be detected by eLISA to be in the thousands (Webbink, 2010). We provide a visual impression in figure 6 by showing the 100 (red dots) and 1000 (black dots) strongest binaries from a MonteCarlo realization of the galaxy compact white dwarf binary population. The shortest period systems will be the most numerous, the majority having periods between 5 and 10 minutes. eLISA will revolutionise our knowledge of such a population, especially given that only two of the known fifty sources have periods less than ten minutes. As these systems are relatively short lived and faint, there is no hope to detect them in significant numbers by any other means than via gravitational radiation, as there are only several thousand expected to exist in the whole Galaxy. Their detection will allow us to test different models for the common-envelope phase, a significant uncertainty in our understanding of binary evolution and many high-energy phenomena. The internal statistical accuracy delivered by the sheer number of detected sources will ensure that the common-envelope phase will be put to the most critical test expected in the midterm future. The same population can be used to constrain models for type Ia supernovae and peculiar supernovae, as well as the formation of ultra-compact binaries in globular clusters.

The outcome of the common envelope phase

Only a minority of the stars in the Universe are single, leaving the majority to be part of a binary, a triple or a higher-order system. On the order of half of the binaries formed with sufficiently small orbital separation, so that the stars will interact during the evolution of the components into giants or super giants. Especially for low-mass stars, the majority of interactions are unstable and will lead to runaway mass transfer. Based on the observed short orbital periods of binaries that have passed this stage it is argued that somehow the companion of the giant ends up inside the giant’s outer layers. During that common envelope phase, (dynamical) friction reduces the velocity of the companion, leading to orbital shrinkage and transfer of angular momentum from the orbit into the envelope of the giant. Along with angular momentum, orbital energy is deposited in the envelope, whose matter is then unbound from the giant’s core, leading to a very compact binary consisting of the core of the giant and the original companion (Paczynski, 1976).

Virtually all compact binaries and most of the systems giving rise to high-energy phenomena (such as X-ray binaries, relativistic binary pulsars and possibly gamma-ray bursts) have experienced at least one common-envelope phase. Given the importance of this phase in high-energy astrophysics, our understanding of the physics and our ability to predict the outcome of the common-envelope phase are poor. Theoretical progress to understand the phase from first physical principles is slow (e.g. Taam and Sandquist, 2000, Taam and Ricker, 2010) and the standard formalism described above has been challenged by observational tests (Nelemans and Tout, 2005, De Marco et al., 2011). Comparison of the parameters of the thousands of binaries detected by eLISA with model predictions will provide a direct test of the different proposed outcomes of the common-envelope phase and our understanding of the preceding binary evolution in general.

3.1 Type Ia supernovae and sub-luminous supernovae

Type Ia supernovae have been the heralds of a new paradigm in Cosmology: cosmic acceleration (Riess et al., 1998, Perlmutter and Riess, 1999) for which the 2011 Nobel Prize in Physics was awarded. However, there are different scenarios proposed for the progenitors of SN Ia. One is the merger of two (carbon-oxygen) white dwarfs that are brought together via gravitational wave radiation (Pakmor et al., 2010) which is exactly the population eLISA will be probing. By determining the number of systems in the Galaxy and their period distribution, the rate at which they will merge will be measured. By comparing that to the inferred SNIa rate for an Sbc galaxy, the viability of this progenitor scenario will be determined. The significant efforts in the past decade to find more supernovae and the advent of wide field optical surveys have revealed a host of new types of supernovae (Perets et al., 2010, Kasliwal et al., 2010, Sullivan et al., 2011, Perets et al., 2011). Some of these have been suggested to originate in the interaction between two white dwarfs at very short periods, again exactly the population to which eLISA is sensitive (Perets et al., 2010, Waldman et al., 2011).

Formation of ultra-compact binaries in globular clusters

Globular clusters have a strong overabundance of bright X-ray sources per unit mass compared to the field, probably due to dynamical interactions. Many of these have turned out to be so-called ultra-compact X-ray binaries, in which a neutron star accreted material from a white dwarf companion is a very compact orbit, exactly the type of sources that eLISA may see. However, it is not clear if the same enhancement will operate for the much more numerous white dwarf binaries. The angular resolution that can be achieved with eLISA is such that globular clusters can be resolved, so that the cluster sources can be distinguished from the Galactic disc sources. This enables eLISA to determine the number of ultra-compact binaries in globular clusters and thus to provide a direct test of the overabundance of white dwarfs binaries in globular clusters. That in turn can be used to test models for dynamical interactions in clusters.

The foreground of Galactic gravitational waves

Figure 7: Level of the Galactic gravitational wave signal as a function of time. Black is the total signal, the red after removal of the resolved binaries. The yearly variation of the Galactic foreground is clearly seen. Based on the Ruiter et al. (2009) Galactic model.

At frequencies below a few mHz the number of sources in the Galaxy is so large ( to , see e.g. Yu and Jeffery, 2010, Ruiter et al., 2010) that only a small percentage, the brightest sources, will be individually detected. The vast majority will form an unresolved foreground signal in the detector, which is quite different from and much stronger than any diffuse extragalactic background (Farmer and Phinney, 2003).

This foreground is often described as an additional noise component, which is misleading for two reasons. The first is that there is a lot of astrophysical information in the foreground. The overall level of the foreground is a measure of the total number of ultra-compact binaries, which gives valuable information given the current uncertainty levels in the normalisation of the population models. The spectral shape of the foreground also contains information about the homogeneity of the sample, as simple models of a steady state with one type of binary predict a very distinct shape. In addition, the geometrical distribution of the sources can be detected by eLISA.

Due to the concentration of sources in the Galactic centre and the inhomogeneity of the eLISA antenna pattern, the foreground is strongly modulated over the course of a year (see figure 7), with time periods in which the foreground is more than a factor two lower than during other periods (Edlund et al., 2005). The characteristics of the modulation can be used to learn about the distribution of the sources in the Galaxy, as the different Galactic components (thin disk, thick disk, halo) contribute differently to the modulation and their respective amplitude can be used, for example, to set upper limits on the halo population (e.g. Ruiter et al., 2009).

4 Studying the astrophysics of compact binaries using eLISA

Although the effect of gravitational radiation on the orbit will dominate the evolution of the binaries detected by eLISA, additional physical processes will cause strong deviations from the simple point-mass approximation. The two most important interactions that occur are tides – when at least one of the stars in a binary system is not in co-rotation with the orbital motion or when the orbit is eccentric – and mass transfer. Because many binaries will be easily detected, these interactions do not hamper their discovery, but instead will allow tests of the physics underlying these deviations. By providing a completely complementary approach, gravitational wave measurements are optimal to the study of short period systems, in contrast to the current bias towards bright electromagnetic systems and events.

Physics of tidal interaction

Figure 8: Lightcurve of SDSS J0651+28, folded on the 12 minute orbital period. Except for the two eclipses at phase and , the sinusoidal variation due to the tidal distortion of the primary white dwarf. From Brown et al. (2011)

eLISA measurements of individual short-period binaries will give a wealth of information on the physics of tides and the stability of the mass transfer. For detached systems with little or no interaction, the frequency evolution is well understood as that of two point masses. The strain amplitude , the frequency and its derivatives then are connected by

(6)
(7)
(8)

where is the chirp mass, are the masses of the binary constituents and is the distance. Thus the measurement of , , provides chirp mass and distance; the additional measurement of gives a direct test of the dominance of gravitational wave radiation in the frequency evolution. Tidal interaction between white dwarfs in detached systems before the onset of mass transfer will give rise to distinct deviations of the frequency evolution as compared to systems with no or little tidal interaction. The strength of the tidal interaction is virtually unknown, with estimates ranging over many orders of magnitude (Marsh et al., 2004), although the high temperature of the white dwarf in the recently discovered 12 min double white dwarf may suggest efficient tidal heating (Piro, 2011). Knowledge of the strength of the tides is not only important for understanding the physics of tides in general and of white dwarf interiors, it also has important consequences for the tidal heating (and possibly optical observability) of eLISA sources and for the stability of mass transfer between white dwarfs (Racine et al., 2007, Willems et al., 2010, Marsh, 2011, Fuller and Lai, 2011).

In globular clusters, dynamical interactions may produce eccentric double white dwarf systems, which can be used to constrain white dwarf properties and masses (Valsecchi et al., 2011).

Physics of mass-transfer stability

Detached ultra-compact binaries will evolve to shorter and shorter periods due to the angular momentum loss through gravitational wave radiation. At sufficiently short orbital periods (a few minutes) one of the stars becomes larger than its Roche lobe – the equipotential surface that crosses the minimum of the potential between the two stars – and material leaks out of the potential well of one star upon the other star. Depending on the difference between the change of the radius of this star and the Roche lobe upon mass transfer, there may be positive or negative feedback, leading to either limited, stable mass transfer, or a runaway mass-transfer instability.

For double white dwarfs and white dwarf-neutron star binaries the stability of the ensuing mass transfer has important consequences, for the number of detectable sources, as well as for a number of open astrophysical questions. The stable systems will form interacting binaries, AM CVn systems or ultra-compact X-ray binaries, that can be detected through their gravitational wave emissions. eLISA will detect a number of detached double white dwarfs and AM CVn systems that are so close to the onset of mass transfer that the stability of the mass transfer can be tested directly by comparing their numbers. In addition, eLISA will detect several ultra-compact X-ray binaries at the very early stages of mass transfer, providing a test of the mass transfer stability in these systems as well (Marsh, 2011).

For AM CVn systems, a major uncertainty in the mass-transfer stability is again the tidal interaction between the two white dwarfs. Most likely the mass transfer will proceed via the direct impact configuration: due to the proximity of the two stars, the mass transfer stream lands directly on the surface of the accreting white dwarf, rather than wrapping around the accreting stars and interacting with itself to form a flat accretion disk in the plane of the orbit (Webbink, 1984, Marsh and Steeghs, 2002). The stability of the mass transfer depends critically on the tidal interaction between the two white dwarfs (Marsh et al., 2004): In the absence of any tidal interaction, there will be additional angular momentum loss from the orbit due to the transfer of angular momentum from the orbit to the accreting star which will consequently spin up. This is different from cases where the accretion is via a disc for which most of the angular momentum generally is stored in the disc and eventually via very efficient tidal interaction put back into the orbit. Efficient tidal coupling between the accreting star and the companion has the ability to return the angular momentum back to the orbit (see Racine et al., 2007, D’Souza et al., 2006), thus reducing the magnitude of the spin-up. The difference between efficient and inefficient tidal coupling is rather dramatic: the fraction of double white dwarfs estimated to survive the onset of mass transfer can drop from about  % to  % (Nelemans et al., 2001) depending on assumptions about the tidal coupling. This difference is easily measurable with eLISA. Short-term variations in the secular evolution of the systems experiencing mass transfer will change the frequency evolution, but are likely to be rare and will not prevent the detection of these systems (Stroeer and Nelemans, 2009).

For ultra-compact X-ray binaries (see e.g. figure 9), the stability issue is completely different. At the onset, the mass transfer is orders of magnitude above the Eddington limit for a neutron star (the mass transfer rate at which the potential energy liberated in the accretion can couple to the infalling gas to blow it away). For normal stars and white dwarfs, this would likely lead to a complete merger of the system, but the enormous amount of energy liberated when matter is falling into the very deep potential well of a neutron star allows matter to be dumped on it at rates up to a thousand times the Eddington limit if the white dwarf has a low mass (see Yungelson et al., 2002). This allows the formation of ultra-compact X-ray binaries from white dwarf-neutron star pairs. eLISA will unambiguously test this prediction by detecting several tens of ultra-compact X-ray binaries with periods between 5 and 20 minutes.

Double white dwarf mergers

The 80% to 99.8% of the double white dwarfs that experience run-away mass transfer and merger give rise to quite spectacular phenomena. Mergers of double white dwarfs have been proposed as progenitors of single subdwarf O and B stars, R Corona Borealis stars and maybe all massive white dwarfs (e.g. Webbink, 1984). In addition, the merger of a sufficiently massive double white dwarf can be a trigger for type Ia supernova events (see Pakmor et al., 2010). Alternatively, if the merger does not lead to an explosion, a (rapidly spinning) neutron star will be formed. This is one possible way to form isolated millisecond radio pulsars as well as magnetars, which have been proposed as sites for short gamma-ray bursts (e.g. Levan et al., 2006).

Although it is not expected that eLISA will witness the actual merger of a double white dwarf as the event rate in our Galaxy is too low, it will certainly detect the shortest-period binaries known, expected at a period of about two minutes, and give an extremely good estimate of their merger rate. In addition, if the actual merger takes many orbits as recently found in simulations (Dan et al., 2011), eLISA may observe them directly.

By measuring (chirp) masses and coalescence times, eLISA will directly determine the merger rate for double white dwarfs with different masses, which can then be compared with the rates and population of their possible descendants determined by other means (Stroeer et al., 2011).

Neutron star and black hole binaries

Figure 9: Imprint of the 40 min orbital period on the arrival times of the X-ray pulsations in the ultra-compact X-ray binary XTE J0929-314. From Galloway et al. (2002).

The current observational and theoretical estimates of the formation rate of neutron star binaries are highly uncertain and predict several tens of neutron star binaries to be detected by eLISA (e.g. Nelemans et al., 2001, Belczynski et al., 2010). The number of ultra-compact stellar-mass black hole binaries in the Galaxy is even more uncertain (e.g. Belczynski et al., 2002); furthermore, these binaries are likely to be detectable only through their gravitational wave emission as they are electromagnetically quiet.

eLISA will thus constrain the formation rate estimates and the numbers of neutron star binaries and ultra-compact stellar mass black hole binaries. As these systems can be seen throughout the Galaxy, the samples for all these populations will be complete at the shortest periods. Thus the sample will be independent of selection effects, such as those present in radio pulsar surveys and X-ray surveys, that pick up only transient X-ray sources. In addition, by the time eLISA will fly, Advanced LIGO and Virgo will likely have detected a number of double neutron star mergers from far away galaxies, so these measurements together will test our ability to extrapolate our population models from our own galaxy to the rest of the Universe.

A special situation might arise for the case of millisecond X-ray pulsars, in ultra-compact X-ray binaries. In the last decade, observations of X-ray pulsations from many ultra-compact X-ray binaries have enabled astrophysicists to determine the rotation rate of the neutron star in the binary using the NASA mission RXTE (Wijnands, 2010). As had been expected on theoretical grounds, neutron stars are spinning rapidly (several hundred times per second) due to the angular momentum gained from infalling matter. The measurements give credence to the idea that these rapidly spinning neutron stars observed as millisecond radio pulsars are descendants of accreting neutron stars in binary systems (e.g. Bhattacharya and van den Heuvel, 1991). However, the exact role of ultra-compact binaries in the formation of these pulsars has yet to be established. The distribution of spin periods discovered in X-ray binaries suggests additional neutron star angular momentum loss on top of the plasma physics interaction between the accretion and magnetic field of the spinning neutron stars (Chakrabarty et al., 2003) which could be due to strong gravitational wave emission (Bildsten (1998); but see Watts et al. (2008) and Patruno et al. (2011)). In that case, ultra-compact X-ray binaries might be the only sources that could be studied simultaneously with eLISA and ground based detectors, with eLISA detecting the orbital period and the ground based detector detecting the neutron star spin period.

5 Studies of galactic structure with eLISA

Figure 10: Spitzer GLIMPSE model of the Milky Way, showing bulge, bar and spiral arms. The resolved binaries are expected to trace the old stellar populations of the Milky Way. Courtesy NASA/JPL-Caltech/R. Hurt (SSC/Caltech)

One of the major capabilities of eLISA is that it will determine distances for hundreds of compact binaries by measuring their (see equation 7). The ability of eLISA to determine distances depends on the mission lifetime, as larger life times lead to more accurate measurements. The directional dependence of the Galactic foreground as well as the directional accuracy for the resolved systems allow a statistical assessment of the contributions of the different Galactic components, such as the Galactic bulge (with its bar), the thin and thick disc and the Galactic halo (for a realistic MW model, see in figure 10).

The Galactic center is one of the most interesting areas of the Galaxy, with a central massive black hole surrounded by a dense assembly of stars with intriguing properties. Dynamical effects, in particular mass segregation, will lead to many interactions close to the central black hole so that wide binaries will become tighter or will be disrupted (for a review see Alexander, 2005). This likely leads to an increase in the number of ultra-compact binaries as well as the possibility of EMRIs (see 2). eLISA will put much more stringent constraints on these populations than current observations (see e.g. Roelofs et al., 2007), which are limited by the electromagnetic faintness of the sources, or theoretical predictions, which are limited by our current understanding of the processes leading to compact binary formation. Distance determinations to the many ultra-compact binaries around the Galactic centre will allow for an independent distance determination.

The level and shape of the double white dwarf foreground as well as the distribution of resolved sources will provide information on the scale height of the ultra-compact binary population (Benacquista and Holley-Bockelmann, 2006) in the disc of the Galaxy.

The distribution of sources in the Galactic halo will be significantly different from the other Galactic components. In principle the halo population is expected to be much smaller than the rest of the Galaxy (Ruiter et al., 2009, Yu and Jeffery, 2010), but it might be enhanced as the formation and evolution of binaries in the halo may have been quite different. Such old and metal-poor population can be studied locally only in globular clusters, where the formation and evolution of binaries is generally completely altered by dynamical effects. Two of the known AMCVn systems may belong to the halo. They have very low metal abundances and have anomalous velocities. If true this implies that a large number of AMCVn stars are in the halo, maybe as many as in the rest of the Galaxy.

The eLISA directional sensitivity will immediately pick up any strong halo population if it exists.

Finally, for many of the resolved sources the eLISA measurements will also provide an accurate estimate of their orbital inclination. For the first time, this will give hints on the dynamics of the formation of binaries from interstellar clouds, because the angular momentum vectors of the binaries is related (in a statistical way) to the overall angular momentum of the Galaxy.

4 Astrophysical Black Holes

1 Overview

Astrophysical black holes appear to come in nature into two flavours: the “stellar mass” black holes of to approximately resulting from the core collapse of very massive stars, and the “supermassive” black holes of that, according to the accretion paradigm, power the luminous QSO. The former light up the X-ray sky, albeit only in our neighbourhood, as stellar mass black holes fade below detection limits outside our local group. The latter are detected as active nuclei, over the whole cosmic time accessible to our current telescopes. Electromagnetic evidence of black holes in the mass range is less common, due to the intrinsic difficulty of detecting such faint sources in external galaxies. However, it is in this mass interval, elusive to electromagnetic observations, that the history of supermassive black hole growth is imprinted.

Supermassive black holes inhabit bright galaxies, and are ubiquitous in our low-redshift Universe. The discovery of close correlations between the mass of the supermassive black hole with key properties of the host has led to the notion that black holes form and evolve in symbiosis with their galaxy host. In agreement with the current paradigm of hierarchical formation of galactic structures and with limits imposed by the cosmic X-ray background light, astrophysical black holes are believed to emerge from a population of seed black holes with masses in the range , customarily called intermediate mass black holes. The mass and spin of these black holes change sizably in these interactions as they evolve over cosmic time through intermittent phases of copious accretion and merging with other black holes in galactic halos. In a galactic merger, the black holes that inhabit the two colliding galaxies spiral in under the action of dynamical friction, and pair on sub-galactic scales forming a Keplerian binary: binary black holes thus appear as the inescapable outcome of galaxy assembly. When two massive black holes coalesce, they become one of the loudest sources of gravitational waves in the Universe.

eLISA is expected to target coalescing binaries of during the epoch of widespread cosmic star formation and up to , and to capture the signal of a coalescing binary of beyond the era of the earliest known QSO (). Gravitational waveforms carry information on the spins of the black holes that eLISA will measure with exquisite precision, providing a diagnostic of the mechanism of black hole growth. The detection of coalescing black holes not only will shed light into the phases of black hole growth and QSO evolution, but will pierce deep into the hierarchical process of galaxy formation.

2 Black holes in the realm of the observations

Figure 11: The correlation between the black hole mass and the luminosity of the host galaxy’s stellar bulge (left), and host galaxy’s bulge velocity dispersion (right) for all detections in galaxies near enough for current instruments to resolve the region in which the black hole mass dominates the dynamics (adapted from Gültekin et al., 2009)

Dormant and active supermassive black holes

QSOs are active nuclei so luminous that they often outshine their galaxy host. They are sources of electromagnetic energy, with radiation emitted across the spectrum, almost equally, from X-rays to the far-infrared, and in a fraction of cases, from rays to radio waves. Their variability on short timescales revealed that the emitting region is compact, only a few light hours across.

There is now scientific consensus that the electromagnetic power from QSO and from the less luminous AGN results from accretion onto a supermassive black hole of (Salpeter, 1964, Zel’dovich and Novikov, 1964, Krolik, 1999). Escaping energy in the form of radiation, high velocity plasma outflows, and ultra relativistic jets can be generated with high efficiency (, higher than nuclear reactions) just outside the event horizon, through viscous stresses on parcels of gas orbiting in the gravitational potential of the black hole. The accretion paradigm has thus been, and still is, at the heart of the hypothesis of black holes as being “real” sources in our cosmic landscape. eLISA will offer the new perspective of revealing these black holes as powerful sources of gravitational waves, probing the smallest volumes of the large scale Universe.

Massive black holes are tiny objects compared to their host galaxies. The event horizon of a Kerr black hole of mass scales as , and it is far smaller than the optical radius of the galaxy host: . The distance out to which a black affects the kinematic of stars and gas (the gravitational influence radius), , is also small compared to the optical radius of the host, (where is the velocity dispersion of the stars of the galactic bulge).

For a long time, QSO and more generally the less luminous AGN phenomena were understood as caused by a process exclusively confined to the nuclear region of the host. This picture of disjoint black hole and galaxy evolution changed with the advent of the HST (Ferrarese and Ford, 2005).

Observations of almost all bright galaxy spheroids in the near universe reveal that the velocities of stars and gas start to rise in a Keplerian fashion at their centres, highlighting the presence of a dark point-mass which dominates the central gravitational potential. The same observations provide the mass of this dark object, hypothesised to be a quiescent black hole. The proximity of these galaxies to Earth allowed for a full optical characterisation of the host, and this ultimately led to the discovery of tight correlations – depicted in figure 11, from Gültekin et al. (2009) – between the black hole mass and the optical luminosity and velocity dispersion of the stars measured far from the black hole (Gültekin et al., 2009, Ferrarese and Merritt, 2000, Gebhardt et al., 2000, Tremaine et al., 2002, Graham et al., 2011). The relations state that galaxy spheroids with higher stellar velocity dispersions, i.e. with deeper gravitational potential wells and higher stellar masses and luminosities, host heavier central black holes with little dispersion in the correlation. Thus more massive galaxies grow more massive black holes: the black hole sees the galaxy that it inhabits, and the galaxy sees the black hole at its centre despite its small influence radius (Magorrian et al., 1998, Marconi and Hunt, 2003, Häring and Rix, 2004).

Consensus is rising that the relation of figure 11 is fossil evidence of a co-evolution of black holes and galaxies. The relation may have been established along the course of galactic mergers and in episodes of self-regulated accretion (Mihos and Hernquist, 1996, Di Matteo et al., 2005, Hopkins et al., 2006, 2008, 2009, Croton et al., 2006, Somerville et al., 2008, Johansson et al., 2009, Lamastra et al., 2010). However, the origin of the relation (Silk and Rees, 1998, King, 2003, Wyithe and Loeb, 2003, Ciotti et al., 2010), and its evolution at look-back times is still unclear (Robertson et al., 2006, Peng, 2007, Treu et al., 2007, Woo et al., 2008). The similarity between the evolution, over cosmic time, of the luminosity density of QSOs and the global star formation rate (Boyle and Terlevich, 1998, Kauffmann and Haehnelt, 2000) points to the presence of a symbiotic growth, which is still under study (Schawinski et al., 2010, 2011).

The census of black holes, from the study of the kinematics of stars and gas in nearby galaxies, has further led to the estimate of the black hole local mass density: (Aller and Richstone, 2002, Marconi et al., 2004, Lauer et al., 2007, Tundo et al., 2007). Whether this mass density traces the initial conditions, i.e. the mass since birth, obtained at most by rearranging individual masses via coalescences, or the mass acquired via major episodes of accretion in active AGN phases can only be inferred using additional information: that resulting from the AGN demographics and from studies of the X-ray cosmic background.

Two arguments provide information about how much of the black hole growth occurred through accretion of gas, in phases when the black hole is active as AGN. The first is the existence of a limiting luminosity for an accreting black hole, corresponding to when the radiation pressure force equals gravity. Above this limit material that would be responsible for the emission can not fall onto the black hole, as it is pushed away. This limit is the Eddington luminosity ( and are the Thomson cross section and proton mass). The AGN luminosity is normally a fraction of the Eddington luminosity, since as soon as approaches the radiation pressure force against gravity self-regulates the accretion flow to , providing also a lower bound on . The second argument is that “light is mass”, i.e. that any light output from accretion (at a luminosity level ) increases the black hole’s mass at a rate , where is the rest-mass accreted per unit time and the accretion efficiency, i.e. how much of the accreted mass is converted into radiation. Accordingly, the black hole’s mass increases exponentially in relation to the self-regulated flow, with an -folding time . For – typical of radiatively efficient accretion onto a non-rapidly rotating black hole (Shapiro and Teukolsky, 1979) – and , this timescale is short (about  %) compared to the age of the Universe, indicating that black holes can enhance their mass via accretion by orders of magnitude.

Active black holes in galaxies are known to contribute to the rise of a cosmic X-ray background resulting mostly from unresolved and obscured AGN of mass , in the redshift interval (Merloni, 2004). As energy from accretion is equivalent to mass, the X-ray light present in the background mirrors the increment experienced by the black holes over cosmic history due to accretion. This mass-density increment is found to be (Marconi et al., 2004, Yu and Tremaine, 2002, Soltan, 1982).

As the contribution to the local (zero redshift) black hole mass density results from black holes of comparable mass , the close match between the two independent measures, and , indicates that radiatively efficient accretion () played a large part in the building of supermassive black holes in galaxies, from redshift to now. It further indicates that information residing in the initial mass distribution of the, albeit unknown, black hole seed population is erased during events of copious accretion, along the course of cosmic evolution.

Massive black holes in the cosmological framework

Figure 12: A state-of-art hydrodynamical simulation by Di Matteo et al. (2008) visualising the cosmic evolution of the baryonic density field and of their embedded black holes, in the CDM cosmology. Each panel shows the same region of space ( Mpc on a side) at different redshift, as labelled. The circles mark the positions of the black holes, with a size that encodes the mass, as indicated in the top left panel (numerical force resolution limits the lowest black hole mass to ). The projected baryonic density field is colour-coded with brightness proportional to the logarithm of the gas surface density. The images show that the black holes emerge in halos starting at high redshift (as early as ) and later grow by accretion driven by gas inflows that accompany the hierarchical build-up of ever larger halos through merging. As the simulation evolves the number of black holes rapidly increases and larger halos host increasingly larger black holes. No black holes as massive as are present in the simulated box because they are extremely rare.
Figure 13: A cartoon of the merger-tree history for the assembly of a galaxy and its central black holes. Time increases along the arrow. Here the final galaxy is assembled through the merger of twenty smaller galaxies housing three seed black holes, and four coalescences of binary black holes.

These key findings hint in favour of the existence, at any redshift, of an underlying population of black holes of a smaller variety, with masses of , that grew in mass along cosmic history inside their galaxies, through episodes of merging and accretion. The evolution of black holes mimics closely that of their host galaxies within the currently favoured cosmological paradigm: a Universe dominated by cold dark matter (CDM).

Observations show that the mass content of the Universe is dominated by CDM, with baryons contributing only at a  % level to the CDM, and that the spectrum of primordial density perturbations contains more power at lower masses (Mo et al., 2010). Thus, at the earliest epoch, the Universe was dominated by small density perturbations. Regions with higher density grow in time, to the point where they decouple from the Hubble flow and collapse and virialise, forming self gravitating halos. The first objects that collapse under their own self-gravity are small halos that grow bigger through mergers with other halos and accretion of surrounding matter. This is a bottom up path, and the process is known as hierarchical clustering. As halos cluster and merge to build larger ones, baryons follow the CDM halo potential well and, similarly, black holes form and evolve in the same bottom-up fashion (White and Rees, 1978, Haiman and Loeb, 1998, Haehnelt et al., 1998, Wyithe and Loeb, 2002, Volonteri et al., 2003).

State-of-the-art hydrodynamical cosmological simulations (Di Matteo et al., 2008) illustrate (figure 12) where and when the massive black holes form and how they are connected with the evolving background baryonic density field. As illustrated in figure 12 and as inferred in statistical models based on the extended Press-Schechter (EPS) formalism, most of the black holes transit into the mass interval for which eLISA is sensitive during their cosmic evolution (Volonteri et al., 2003). Figure 13 sketches and simplifies conceptually the complex net terminating with the formation of a bright galaxy at zero redshift, highlighting the sites where black holes form, cluster within halos, pair with other black holes, and eventually coalesce.

Black holes in the sensitivity window of eLISA

Is there any observational evidence of black holes of this variety in the Universe that may be observed by eLISA? The Milky Way hosts in its bulge a black hole of (Ghez et al., 2005, Gillessen et al., 2009), providing an example of a black hole that does not fall into the population that can be traced by luminous QSO. Black holes in the mass range are now increasingly found in low mass spiral galaxies and dwarfs with and without a bulge (Greene and Ho, 2004, Barth et al., 2004, Greene et al., 2008, Jiang et al., 2011a, Kuo et al., 2011, Xiao et al., 2011, Jiang et al., 2011b) and evidence exists that some of these low mass black holes of cohabit nuclear star clusters (Ferrarese et al., 2006, Wehner and Harris, 2006, Barth et al., 2009, Seth et al., 2008, Graham and Spitler, 2009, Bekki and Graham, 2010).

Dwarf galaxies in the galactic field are believed to undergo a quieter merger and accretion history than their brighter analogues. They may represent the closest example of low mass halos from which galaxy assembly took off. Late type dwarfs are thus the preferred site for the search of pristine black holes (Volonteri and Natarajan, 2009). NGC 4359, a close-by bulgeless, disky dwarf houses in its centre a black hole of only (Peterson et al., 2005). This key discovery shows that nature provides a channel to black hole formation also in potential wells much shallower than that of the massive spheroids.

These middleweight mass black holes are numerous at high redshifts (Di Matteo et al., 2008), but are invisible with today instrumentation, given their low intrinsic luminosity and far-out distance. Furthermore, they become invisible to electromagnetic observations near as close to this redshift the intergalactic medium becomes opaque to their light, due to intervening absorption of the neutral hydrogen (Fan et al., 2006a, Miralda-Escude, 1998). ULAS J1120+0641 holds the record of being the further distant known QSO, at redshift , and hosts a bright, very massive black hole of (Mortlock et al., 2011). Its light was emitted before the end of the reionisation, i.e. before the theoretically predicted transition of the interstellar medium from an electrically neutral to an ionised state (Fan et al., 2006a).

3 Galaxy mergers and black hole coalescence

Figure 14: The different stages of the merger between two identical Milky-Way-like gas-rich disc galaxies (from Mayer et al., 2007). The panels show the density maps of the gas component in logarithmic scale, with brighter colours for higher densities. The four panels to the left show the large-scale evolution at different times. The boxes are  kpc on a side (top) and  kpc on a side (bottom). During the interaction tidal forces tear the galactic disks apart, generating spectacular tidal tails and plumes. The panels to the right show a zoom in of the very last stage of the merger, about 100 million years before the two cores have fully coalesced (upper panel), and 2 million years after the merger (middle panel), when a massive, rotating nuclear gaseous disc embedded in a series of large-scale ring-like structures has formed. The boxes are now  kpc on a side. The two bottom panels, with a grey colour scale, show the detail of the inner  pc of the middle panel; a massive nuclear disc, shown edge-on (left) and face-on (right), forms in the aftermath of the merger (of ). The two black holes continue to sink inside the disc and form a Keplerian binary; they are shown in the face-on image.

A grand collision between two galaxies of comparable mass (called major merger) is not destructive event, but rather a transformation, as the two galaxies, after merging, form a new galaxy with a new morphology. Individual stars do not collide during the merger, as they are tiny compared to the distances between them. The two galaxies pass through each other and complex, time-varying gravitational interactions redistribute the energy of each star in such a way that a new bound galaxy forms. Gas clouds instead collide along the course of the merger: new stars form, and streams of gas flow in the nuclear region of the newly forming galaxy. The massive black holes in the grand collision behave like stars. A key question for the eLISA science case is: do black holes coalesce as their galaxies merge?

The fate of black holes in merging galaxies can only be traced using numerical simulations at the limits of current numerical resolution. Not only isolated black holes are tiny, but also binary black holes are. They form a tight binary system within a galaxy when the mass in stars enclosed in the binary orbit becomes negligible compared to the total mass of the binary , and their Keplerian velocity exceeds the velocity of the stars, . This occurs when their relative separation decays below about , i.e. when . Binary black holes on the verge of coalescing within less than a Hubble time are even smaller, as they touch when their separation is of the size of the event horizon. The timescale for coalescence by gravitational waves only is a sensitive function of the binary separation, scaling as (Peters, 1964). Therefore, gravitational waves guide the inspiral only when is less than a critical value (using scaling relations) that is of  pc –  pc for a circular binary in the eLISA mass interval. Typical orbital periods at are of a few years to tens of years, and the hole’s relative velocities are as high as  km/s –  km/s.

Black holes have to travel a distance from  kpc –  kpc down to  pc –  pc, before entering the gravitational wave inspiral regime, in a galaxy. Given the huge dynamical range, different physical mechanisms are guiding their sinking (Colpi and Dotti, 2011). We can distinguish four phases for the dynamics of black holes on their way to and after merging:

  1. the pairing phase, when the black holes pair on galactic scales following the dynamics of the galaxies they inhabit until they form a Keplerian binary (on pc scales);

  2. the binary phase, when the Keplerian binary continues to harden at the centre of the galaxy remnant;

  3. the gravitational wave phase, when black hole inspiral is dominated by loss of energy and angular momentum by gravitational waves; and finally

  4. the recoiling phase, when the now single black hole either oscillates or escapes the galaxy following gravitational recoil.

In major merger of galaxies the black holes pair under the action of dynamical friction against the dark matter background, that brakes the disc/bulge which they inhabit (Begelman et al., 1980, Chandrasekhar, 1943, Ostriker, 1999, Colpi et al., 1999). Pairing occurs on the typical timescale of a galactic merger of a few billion years. A few million years after the new galaxy has formed, a Keplerian binary forms on the scale of  pc –  pc, under the action of dynamical friction by stars and gas (Mayer et al., 2007). Figure 14 shows the evolution of the two gas-discs in merging galaxies similar to the Milky Way. The galaxies host a central black hole, and the black holes end forming a Keplerian binary embedded in a massive nuclear disc (Mayer et al., 2007). The subsequent hardening of the binary orbit (phase II) is controlled by the inflow of stars from larger radii, and by the gas rotating in a circum-binary disc (Merritt and Milosavljević, 2005, Colpi and Dotti, 2011, Escala et al., 2004, Dotti et al., 2006, 2007, 2009b, Callegari et al., 2009). In gas rich environments, and for black holes of mass smaller than about , gas-dynamical torques on the binary suffice to drive the system down to the gravitational wave inspiral domain (Cuadra et al., 2009, Hayasaki et al., 2008, Hayasaki and Okazaki, 2009, MacFadyen and Milosavljević, 2008, Armitage and Natarajan, 2005, Ivanov et al., 1999, Gould and Rix, 2000) if the gas does not fragment in stars (Lodato et al., 2009).

Stars are ubiquitous, and in stellar bulges the black holes lose orbital energy and angular momentum by ejecting stars that scatter individually off the black holes (Quinlan, 1996, Milosavljević and Merritt, 2001, Makino and Funato, 2004, Merritt and Poon, 2004, Berczik et al., 2005, Merritt and Milosavljević, 2005, Merritt, 2006, Merritt et al., 2007, Sesana et al., 2006, 2007a, 2008a, Perets and Alexander, 2008). These stars approach the binary from nearly radial orbit, and shrink the binary down to the gravitational wave phase, if they are present in sufficient number to carry away the energy for the binary to decay down to . These stars, ejected with high velocities, are lost by the galaxy, and the timescale of sinking of the binary depends on the rate at which new stars are supplied from far-out distances. Self-consistent high resolution direct N-body simulations (Khan et al., 2011, Preto et al., 2011, Berczik et al., 2006) indicate that the stellar potential of the remnant galaxy retains, in response to the anisotropy of the merger, a sufficiently high degree of rotation and triaxiality to guarantee a large reservoir of stars on centrophilic orbits that can interact with the black holes down to the transit from the binary phase to the gravitational wave phase. This seems to be a universal process. When coalescence occurs, the merger remnant “recoils”because of the anisotropic emission of gravitational waves (Baker et al., 2008), moving away from the gravitational centre of the galaxy. The kicked black hole, may return after a few oscillations down to the nuclear regions of the host galaxy, or escape the galaxy depending on the magnitude of the kick (Gualandris and Merritt, 2008, Blecha and Loeb, 2008, Blecha et al., 2011, Guedes et al., 2011).

4 Dual, binary and recoiling AGN in the cosmic landscape

Figure 15: Active black holes in colliding galaxies. Arp 299 (upper left panel) is the interacting system resulting from the collision of two gas-rich spirals, and hosts a dual AGN, i.e. two black holes “active” during the pairing phase. The accreting black holes are visible in the X-rays and are located at the optical centres of the two galaxies, at a separation of  kpc(Ballo et al., 2004). X-ray view of NGC6240 (lower left panel) an ultra luminous infrared galaxy considered to be a merger in a well advanced phase (Komossa et al., 2003). X-ray observations with the Chandra Observatory let to the discovery of two strong hard X-ray unresolved sources embedded in the diluted soft X-ray emission (red) of a starburst. The dual AGN are at a separation of  pc. Composite X-ray (blue)/radio (pink) image of the galaxy cluster Abell 400 (upper right panel) showing radio jets immersed in a vast cloud of multimillion degree X-ray emitting gas that pervades the cluster. The jets emanate from the vicinity of two supermassive black holes (a dual radio-loud AGN) housed in two elliptical galaxies in the very early stage of merging. Composite optical and X-ray image of NGC 3393 (lower right panel), a spiral galaxy with no evident signs of interaction. In its nucleus, two active black holes have been discovered at a separation of only  pc (Fabbiano et al., 2011). The closeness of the black holes embedded in the bulge, provide a hitherto missing observational point to the study of galaxy-black hole evolution: the phase when the black holes are close to forming a Keplerian binary. The regular spiral morphology and predominantly old circum-nuclear stellar population of this galaxy, indicates that a merger of a dwarf with a large spiral led to the formation of the binary (Callegari et al., 2009, 2011).

Surprisingly, the closest example of an imminent merger is in our Local Group. Andromeda (M31) along with a handful of lesser galaxies does not follow Hubble’s law of cosmic expansion: it is falling toward us at a speed of about  km/s. M31 is a member of a group of galaxies, including the Milky Way, that form a gravitationally bound system, the Local Group. M31 and the Milky Way each house a massive black hole (van der Marel et al., 1994) and are on a collision course, with a merger possibly before the Sun expands into a red giant ( billion years) (Cox and Loeb, 2008). Observations are now revealing the presence of many colliding galaxies in the Universe, and in a number of cases two active black holes are visible through their X-ray or radio emission.

The existence of binary AGN, i.e. of two active black holes bound in a Keplerian fashion, is still debatable at the observational level, as they are rare objects (Volonteri et al., 2009). Two cases deserve attention. The first case is 0402+379, a radio source in an elliptical galaxy showing two compact flat-spectrum radio nuclei, only  pc apart (Rodriguez et al., 2006, 2009). The second case is OJ 287, a source displaying a periodic variability of 12 years (Valtonen et al., 2006, 2008, 2010, 2011) that has been interpreted as being a Keplerian binary with evidence of orbital decay by emission of gravitational waves. A number of sub-parsec binary black hole candidates have been proposed (Tsalmantza et al., 2011, Eracleous et al., 2011) based on the recognition that gas clouds orbiting one/two black hole(s) can leave an imprint in the optical spectra of the AGN (Montuori et al., 2011, Shen and Loeb, 2010, Bogdanović et al., 2008, Boroson and Lauer, 2009, Shields et al., 2009, Decarli et al., 2010, Barrows et al., 2011). Follow-up observations will be necessary to assess their true nature.

Recoiling AGN, i.e. recoiling black holes observed in an active phase (Loeb, 2007, Devecchi et al., 2009, O’Leary and Loeb, 2009, Merritt et al., 2009b), have been searched recently, and there has been a claim of a discovery (Komossa et al., 2008), even though alternative interpretations are also viable (Dotti et al., 2009a, Bogdanović et al., 2009). Two spatially off-set AGN have been found in deep surveys with kinematic properties that are consistent with being two recoiling black holes (Jonker et al., 2010, Civano et al., 2010).

The most remarkable, albeit indirect, evidence of coalescence events is found in bright elliptical galaxies that are believed to be product of mergers. Bright elliptical galaxies show light deficits (cores) in their surface brightness profiles, i.e. lack of stars in their nuclei (Kormendy and Bender, 2009), and this missing light correlates with the mass of the central black hole (Merritt, 2006, Merritt et al., 2007). Thus, cores are evidence of binary black holes scouring out the nuclear stars via three-body scattering (Merritt et al., 2007) or even via post-merger relaxation following a kick (Boylan-Kolchin et al., 2004, Gualandris and Merritt, 2008, Guedes et al., 2009). Lastly, mergers change the black hole’s spin directions due to conservation of angular momentum. Reorientation of the black hole spin following coalescence is now believed to be at the heart of X-shaped radio galaxies, where an old jet coexists with a new jet of different orientation (Merritt and Ekers, 2002, Liu, 2004). This would be again a sign of a fully accomplished coalescence event.

5 Seed black holes

Models of hierarchical structure formation predict that galaxy sized dark matter halos start to become common at redshift (Mo et al., 2010). This is the beginning of the nonlinear phase of density fluctuations in the Universe, and hence also the epoch of baryonic collapse leading to star and galaxy formation. Different populations of seed black holes have been proposed in the range (Volonteri, 2010).

Small mass seeds () may result from the core collapse of the first generation of massive stars (Pop III) that form from unstable metal-free gas clouds, at and in halos of (Haiman et al., 1996, Tegmark et al., 1997, Abel et al., 2002, Omukai and Palla, 2001, Bromm et al., 2002, Ripamonti et al., 2002, Omukai and Palla, 2003). Pop III stars as massive as or larger collapse into a black hole of similar mass after only about  Myr (Heger et al., 2003, Madau and Rees, 2001). The formation of Pop III stars remains a poorly understood process (Zinnecker and Yorke, 2007) and the maximum mass reached by individual stars is unknown (see, e.g., Clark et al., 2011).

Large seeds form in heavier halos (of ) from the collapse of unstable gaseous discs of . This route, ending with the formation of a very massive quasi-star, assumes that fragmentation is suppressed possibly by turbulence and by an intense ultraviolet background light, in an environment of low metallicity (Haehnelt and Rees, 1993, Loeb and Rasio, 1994, Bromm and Loeb, 2003, Koushiappas et al., 2004, Begelman et al., 2006, Lodato and Natarajan, 2006, Dotan et al., 2011, Dijkstra et al., 2008). Collapsing clouds can have significant angular momentum (Bullock et al., 2001), and thus additional momentum transport is required for the self-gravitating disc to form a supermassive star (Shlosman et al., 1989, Begelman and Shlosman, 2009). The very massive quasi-star of about , burns hydrogen and helium in its core and once formed, the star collapses into a black hole when metallicity is below some threshold, as the alternative is its entire explosion (Montero et al., 2011, Shibata and Shapiro, 2002). Recently, other formation routes have been explored for the large seeds (). They comprise the formation of a massive star via stellar runaway collisions in young dense star clusters (Portegies Zwart and McMillan, 2002, Gürkan et al., 2004, Devecchi and Volonteri, 2009), and the relativistic collapse of stellar mass black holes in nuclear star clusters invaded by a large inflow of gas (Davies et al., 2011). Lastly, much heavier seeds resulting from direct collapse of nuclear gas in a gas-rich galactic merger have been proposed as the origin of black holes (Mayer et al., 2010).

The subsequent step is to follow the evolution of the black hole seeds according to the growth of the halos they inhabit, and the mode of accretion (Volonteri and Begelman, 2010, Devecchi and Volonteri, 2009, Volonteri et al., 2008). This is an inherently model dependent process. Observations of nearby dormant black holes and of AGN at higher redshift help in constraining their evolution, but theoretical models still have some unconstrained degrees of freedom.

6 Evolving massive black hole spins via coalescence and accretion events

Astrophysical black holes are fully described by the mass and angular momentum , referred to as spin. The modulus of is usually specified in terms of the dimensionless spin parameter defined so that For a specified mass , a black hole described by GR cannot have , without showing a naked singularity (and this is forbidden by the Cosmic Censorship conjecture).

Both coalescences and accretion change , (or ) and the orientation of in a significant manner (Berti and Volonteri, 2008, Volonteri et al., 2007).

Spins in black hole coalescences

With the advent of numerical relativity, it became possible to accurately determine the evolution of the initial spins of the black holes to the final spin of the remnant black hole in a merger event (Pretorius, 2005, Baker et al., 2006, Campanelli et al., 2006, Rezzolla et al., 2008a, Centrella et al., 2010).

Numerical relativity simulations for equal mass, non spinning black holes find a spin (Scheel et al., 2009) for the merged black hole, resulting from the angular momentum of the orbit. Extrapolation of black hole coalescences with large initial spins (larger than approximately ) exactly aligned with the orbital angular momentum find a final (Marronetti et al., 2008). When mergers occur with retro- and pro-grade orbits equally distributed, as it is expected in the case of astrophysical black holes (Hughes and Blandford, 2003), the average spin of the merger remnant is about , close to the expectation for non spinning holes (Berti and Volonteri, 2008).

For almost any configuration of spins and mass ratio, the emission pattern of the gravitational wave is anisotropic, leading to a gravitational recoil (Campanelli et al., 2007, González et al., 2007, Lousto and Zlochower, 2011b).

Numerical studies show that initially nonspinning black holes or binaries with spins aligned with the orbital angular momentum are recoiling with a velocity below about  km/s. By contrast, the recoil is dramatically larger, up to approximately  km/s, for binaries of comparable mass and black holes with large spins in peculiar non-aligned configurations (Lousto and Zlochower, 2011a). Thus, unexpectedly, spins (regulated by coalescence and accretion) affect the retention fraction of black holes in galactic halos, and this has consequences on the overall evolution of black holes in galaxies (Volonteri et al., 2010, Schnittman, 2007, Schnittman and Buonanno, 2007, Kesden et al., 2010).

Spins and black hole accretion

The evolution of mass and spin of astrophysical black holes are strongly correlated, also when considering accretion. Spins determine directly the radiative efficiency , and so the rate at which mass is increasing. In radiatively efficient accretion discs (Shakura and Sunyaev, 1973) varies from (for ) to (for ) and (for ). Accretion on the other hand determines black hole spins since matter carries with it angular momentum (the angular momentum at the innermost stable circular orbit of a Kerr black hole). A non-rotating black hole is spun-up to after increasing its mass by a factor , for prograde accretion (Bardeen, 1970). Conversely, a maximally rotating black hole is spun-down by retrograde accretion to , after growing by a factor .

Accretion imposes limits on the black hole spin. Gas accretion from a geometrically thin disc limits the black-hole spin to , as photons emitted by the disc and with angular momentum anti-parallel to the black hole spin are preferentially captured, having a larger cross section, limiting its rotation (Thorne, 1974). The inclusion of a jet, as studied in magneto-electrodynamic simulations, reduces this limit to (Gammie et al., 2004), and changes in the accretion geometry produce a similar effect (Popham and Gammie, 1998).

How black holes are fed from the large scale down to the hole’s influence radius () is presently unknown, and the spin is sensitive to the way gas is accreted with time (Volonteri et al., 2007). Two limiting modes of accretion can occur. Coherent accretion refers to accretion from a geometrically thin disc, lasting longer than a few black hole mass growth -folding times. During coherent accretion the black hole can more than double its mass, bringing its spin up to the limit imposed by basic physics, either or . By contrast, chaotic accretion refers to a succession of accretion episodes that are incoherent, i.e. randomly oriented. The black hole can then spin-up or down, and spin-down occurs when counter-rotating material is accreted, i.e. when the angular momentum of the disc is strongly misaligned with respect to (i.e. ). If accretion proceeds via short-lived, uncorrelated episodes with co-rotating and counter-rotating material equally probable, spins tend to be small (King and Pringle, 2006, 2007, Moderski et al., 1998): counter-rotating material spins the black hole down more than co-rotating material spins it up, as the innermost stable orbit of a counter-rotating test particle is located at a larger radius than that of a co-rotating particle, and accordingly carries a larger orbital angular momentum.

The direction of the black hole spin is also an important element in the study of black holes. In a viscous accretion disc that is misaligned with the spin of the black hole, Lense-Thirring precession of the orbital plane of fluid elements warps the disc, forcing the gas close to the black hole to align (either parallel or anti-parallel) with the spin of the black hole. Warping is a rapid process that causes alignment of the disc out to , depending on (Bardeen and Petterson, 1975). Following conservation of total angular momentum, the black hole responds to the warping through precession and alignment, due to dissipation in the disc (Scheuer and Feiler, 1996, Perego et al., 2009) evolving into a configuration of minimum energy where the black hole and disc are aligned (King et al., 2005). This process is short ( yrs) compared to the typical accretion time scale, allowing astrophysical black holes to evolve into a quasi-aligned spin-orbit configuration prior to coalescence (Dotti et al., 2010).

According to these theoretical findings, masses and spins evolve dramatically following coalescence and accretion events. The spin offers the best diagnostics on whether the black holes prior to coalescence have experienced either coherent or chaotic accretion episodes. Both, mass and spin, are directly encoded into the gravitational waves emitted during the the merger process. eLISA will measure the masses and spins of the black holes prior to coalescence, offering unprecedented details on how black hole binaries have been evolving along of the course of the galactic merger and along cosmic history.

7 Cosmological massive black hole merger rate

As eLISA creates a new exploratory window on the evolution of black holes, covering a mass and redshift range that is out of the reach of current (and planned) instruments, its expected detection rate is observationally unconstrained. Today we can probe dormant black holes down to masses of about (Magorrian et al., 1998, Xiao et al., 2011) in the local Universe only, and their massive (i.e. heavier than ) active counterparts out to redshift (Fan et al., 2006b, 2004, 2003, 2001). Any estimate of the eLISA detection rate necessarily has to rely on extrapolations based on theoretical models matching the properties of the observable black hole population.

Observationally, the black hole merger rate can be inferred only at relatively low redshift, by counting the fraction of close pairs in deep galaxy surveys. Given a galaxy density per co-moving megaparsec cube , a fraction of close pairs , and a characteristic merger timescale , the merger rate density of galaxies (number of mergers per year per co-moving megaparsec cube), is given by .

Estimates of have been produced by several groups in the last decade (Patton et al., 2002, Lin et al., 2004, Bell et al., 2006, De Propris et al., 2007, Lin et al., 2008, de Ravel et al., 2009, Xu et al., 2011), using deep spectroscopic galaxy surveys like COMBO, COSMOS and DEEP2. Surveys are obviously flux limited, and usually an absolute magnitude cutoff (which translate into a stellar mass lower limit) is applied to obtain a complete sample of galaxy pairs across a range of redshifts. The galaxy merger rate is therefore fairly well constrained only at redshift for galaxies with stellar mass larger than approximately . From compilation of all the measurement (Xu et al., 2011), typical average massive galaxy merger rates at lie in the range . By applying the black hole-host relations (Gültekin et al., 2009), the galaxy stellar mass cutoff is converted into a lower limit to the hosted black hole mass. Assuming a black hole occupation fraction of one (appropriate for massive galaxies) and integrating over the appropriate co-moving cosmological volume, this translates into an observational estimate of the massive black hole merger rate for and .

These estimates can be compared to the rate predicted by Monte Carlo merger trees (Volonteri et al., 2003) based on the EPS formalism (Press and Schechter, 1974, Lacey and Cole, 1993, Sheth and Tormen, 1999), which are used to reconstruct the black hole assembly, and thus to infer eLISA detection rates, in the CDM cosmology. The evolutionary path (outlined in the previous sections) can be traced back to very high redshift () with high resolution via numerical EPS Monte Carlo realisations of the merger hierarchy. Sesana et al. (2008b) carried a detailed comparison of the merger rate predicted by such models in the and range with those inferred by galaxy pair counting, finding a generally broad consistency within a factor of .

On the theoretical side, massive black hole merger rates can be computed from semi-analytic galaxy formation models coupled to massive N-body simulations tracing the cosmological evolution of dark matter halos (De Lucia et al., 2006, Bertone et al., 2007, Guo et al., 2011), such as the Millennium Run (Springel et al., 2005). Such models are generally bound to the limiting resolution of the underlying N-body simulations, and are therefore complete only for galaxy masses larger than approximately . In a companion study, Sesana et al. (2009) also showed that the merging black hole mass functions predicted by EPS based merger trees is in excellent agreement with those extracted by semi-analytic galaxy formation model in the mass range , where semi-analytic models can be considered complete.

Merger rates obtained by EPS merger trees are therefore firmly anchored to low redshift observations and to theoretical galaxy formation models. Nevertheless, the lack of observations in the mass range of interest for eLISA leaves significant room for modelling, and theoretical astrophysicists have developed a large variety of massive black holes formation scenarios that are compatible with observational constraints (Volonteri et al., 2003, Koushiappas et al., 2004, Begelman et al., 2006, Lodato and Natarajan, 2006). The predicted coalescence rate in the eLISA window depends on the peculiar details of the models, ranging from a handful up to few hundred events per year (Haehnelt, 1994, Wyithe and Loeb, 2003, Sesana et al., 2004, Enoki et al., 2004, Sesana et al., 2005, Rhook and Wyithe, 2005, Koushiappas and Zentner, 2006, Sesana et al., 2007b). A recent compilation, encompassing a wide variety of assembly history models, can be found in (Sesana et al., 2011).

8 Massive black hole binaries as gravitational waves sources: what can eLISA discover?

In the eLISA window of detectability massive black hole binary coalescence is a three-step process comprising the inspiral, merger, and ring-down (Flanagan and Hughes, 1998). The inspiral stage is a relatively slow, adiabatic process in which the black holes spiral together on nearly circular orbits. The black holes have a separation wide enough so that they can be treated analytically as point particles through the Post Newtonian (PN) expansion of their binding energy and radiated flux (Blanchet, 2006). The inspiral is followed by the dynamical coalescence, in which the black holes plunge and merge together, forming a highly distorted, perturbed remnant. At the end of the inspiral, the black hole velocities approach . At this stage the PN approximation breaks down, and the system can only be described by a numerical solution of the Einstein equations. The distorted remnant settles into a stationary Kerr black hole as it rings down, by emitting gravitational radiation. This latter stage can be, again, modelled analytically in terms of black hole perturbation theory. At the end of the ring-down the final black hole is left in a quiescent state, with no residual structure besides its Kerr spacetime geometry.

In recent years there has been a major effort in constructing accurate waveforms inclusive of the inspiral merger and ring-down phases (Pretorius, 2005, Campanelli et al., 2006, Baker et al., 2006). Even a few orbital cycles of the full waveform are computationally very demanding. “Complete” waveforms can be designed by stitching together analytical PN waveforms for the early inspiral with a (semi)phenomenologically described merger and ring-down phase (Santamaría et al., 2010, Pan et al., 2011, Damour et al., 2011), calibrated against available numerical data. In the following estimations we will mostly employ phenomenological waveforms constructed in frequency domain, as described in Santamaría et al. (2010). Self-consistent waveforms of this type (the so called PhenomC waveforms) are available for non-spinning binaries and for binaries with aligned spins. In the case of binaries with misaligned spins, we use “hybrid” waveforms obtained by stitching precessing PN waveforms for the inspiral with PhenomC waveforms for the merger/ring-down. This stitching is performed by projecting the orbital angular momentum and individual spins onto the angular momentum of the distorted black hole after merger. Given a waveform model, a first measure of the eLISA performance is the SNR of a binary merger with parameters in the relevant astrophysical range.

Detector performance

Figure 16: Constant-contour levels of the sky and polarisation angle-averaged signal-to-noise ratio (SNR) for equal mass non-spinning binaries as a function of their total mass and cosmological redshift . The total mass is measured in the rest frame of the source. The SNR is computed using PhenomC waveforms (Santamaría et al., 2010), which are inclusive of the three phases of black hole coalescence (in jargon: inspiral, merger, and ring-down, as described in the text).
Figure 17: The figure shows constant-contour levels of the sky and polarisation angle averaged signal-to-noise ratio (SNR) for non-spinning binaries, at cosmological redshift (left panel) and (right panel) in the plane. Here is the total mass of the binary in the source rest frame, and is the mass ratio. The SNR is computed from the full non spinning PhenomC waveform inclusive of inspiral, merger and ring-down, as in figure 16.

Figure 16 shows eLISA SNRs for equal mass, non-spinning coalescing binaries. Here we use PhenomC waveforms and we compute the SNR as a function of the rest-frame total binary mass and of the redshift , averaging over all possible source sky locations and wave polarisation, for two-year observations. The plot highlights the exquisite capabilities of the instrument in covering almost all the mass-redshift parameter space relevant to massive black hole astrophysics. It is of importance to emphasise that current electromagnetic observations are probing only the tip of the massive black hole distribution in the Universe. Our current knowledge of massive black holes is bound to instrument flux limits, probing only the mass range at . Conversely, eLISA will be able to detect the gravitational waves emitted by sources with total mass (in the source rest frame) as small as at cosmological distances inaccessible to any other astrophysical probe. A binary with total mass in the interval can be detected out to a redshift as remote as with a . By contrast, a binary as massive as a few can be detected with high SNR in our local Universe (). Binaries with total mass between can be detected with a , between These intervals in mass and redshift can be considered as optimal for a deep and extensive census of the black hole population in the Universe.

Figure 17 shows constant-contour levels of the SNR expected from binaries with different mass ratios (defined as , where is the mass of the less massive black hole in the binary) located at redshift and . The plots show the SNR reduction that occurs with decreasing , as unequal mass binaries have lower strain amplitudes than equal mass binaries. They also show how SNR decreases with increasing redshift, and thus with increasing luminosity distance. Notice however that even at , binaries in the mass range with mass ratio can be detected with .

Parameter estimation

Figure 18: SNR distribution as a function of cosmological redshift, computed using the inspiral, merger and ring-down waveform PhenomC for spinning binaries (Santamaría et al., 2010). The solid line corresponds to the mean value, and the grey area to the distribution corresponding to 10th and 90th percentile of SNR distribution. These results are based on a catalogue of 15360 sources obtained combining 25 realisations of each of the four fiducial massive black hole evolution models. For reference, the dashed-blue line indicates the sky-averaged SNR, for one year of integration, computed for an equal mass coalescing binary of .
Figure 19: Parameter estimation accuracy evaluated on a source catalogue, obtained combining ten Monte Carlo realisations of the coalescing massive black hole binary population, predicted by the four SE-SC-LE-LC models. The top panels show the distributions of the fractional errors in the estimation of the redshifted masses of the primary (left) and secondary (right) black hole. The middle panels show the absolute error distributions on the measurement of the primary (left) and secondary (right) black hole spin, while the bottom panels show the fractional error distribution on the luminosity distance , and the sky location accuracy (in deg). Errors are evaluated considering full waveforms.

Figures 16 and 17 describe the detectability of single events, and for these individual events, it is possible to extract information on the physical parameters of the source. Waveforms carry information on the redshifted mass (the mass measured at the detector is times the mass at the source location) and on the spin of the individual black holes prior to coalescence. The measure of the mass and spin is of importance in astrophysics. Except for the Galactic centre (Ghez et al., 2005, Gillessen et al., 2009), the mass of astrophysical black holes is estimated with uncertainties ranging from  % to a factor of about , depending on the technique used and the type of source. As far as spin is concerned, its measure is only indirect, and it is derived through modelling of the spectrum, or of the shape of emission lines, mainly by fitting the skewed relativistic K iron line. There are few notable examples, but uncertainties are still large. By contrast, spins leave a distinctive peculiar imprint in the waveform.

In section 5 and 6 we explored different routes to seed black hole formation and to their subsequent assembly and growth through mergers and accretion episodes. Different physically motivated assumptions lead to different black hole evolution scenarios, and, as we highlighted above, the lack of observational constraints allowed theoretical astrophysicists to develop a large variety of massive black holes formation scenarios. To assess the astrophysical impact of eLISA, we simulate observations assuming a fiducial set of four cosmological black hole evolution scenarios: SE refers to a model where the seeds have small (S) mass about (from Pop III stars only) and accretion is coherent, i.e. resulting from extended (E) accretion episodes: SC refers to a model where seeds are small but accretion is chaotic (C), i.e., resulting from uncorrelated episodes; and finally, LE and LC refer to models where the seed population is heavy (L stands for large seeds of ) and accretion is extended and chaotic, respectively. The models are almost the same used in previous studies by the LISA Parameter Estimation Task Force (Arun et al., 2009). The only difference is that in the extended accretion model, spins are not assumed to be perfectly aligned to the binary orbital angular momentum. The angles of misalignment relative to the orbit are drawn randomly in the range 0 to 20 degree, consistent with the finding of recent hydrodynamical simulations of binaries forming in wet mergers (Dotti et al., 2010). These models encompass a broad range of plausible massive black hole evolution scenarios, and we use them as a testbed for eLISA capabilities in a fiducial astrophysical context. Each massive black hole binary, coalescing at redshift , is characterised by the (rest frame) total mass (with and the mass of the primary and secondary black hole), mass ratio , spin vectors and ; spin magnitudes are denoted by and . The orientations of the spins are drawn as described above for the extended (E) accretion models, and completely random for the chaotic (C) accretion models. Here we generate several Monte Carlo realisations of each model and we sum up all the generated sources in a single “average” catalogue (we will consider models separately in the next section). Catalogues are generated by selecting , , , , according to the distribution predicted by the individual models, and by randomising other source parameters (sky location, polarisation, inclination, initial phase, coalescence time) according to the appropriate distribution.

Figure 18 shows the average source SNR as a function of the source redshift. According to the simulated models, eLISA will detect sources with out to . Note that the astrophysical capabilities of eLISA are not limited by the detector design, but by the population of astrophysical sources. If there were a coalescing black hole binary of out to redshift , eLISA would reveal such a source. Our models, and accordingly our SNR distribution, do not have such an event.

Figure 19 shows error distributions in the source parameter estimation, for all the events in the combined catalogue. We used a hybrid approach of joining inspiral with PhenomC waveforms, as described above, to evaluate uncertainties based on the Fisher matrix approximation.

The figure illustrates the importance and power of including the full waveform modelling, comprising all the stages of the binary merger, in order to reach a high level of precision measurements. It is found that individual black hole redshifted masses can be measured with unprecedented precision, i.e. with an error of  % –  %, on both components. No other astrophysical tool has the capability of reaching a comparable accuracy. As far as spins are concerned, the analysis shows that the spin of the primary massive black hole can be measured with an exquisite accuracy, to a absolute uncertainty. This precision in the measure mirrors the fact that the primary black hole leaves a bigger imprint in the waveform. The measurement is more problematic for that can be either determined to an accuracy of , or remain completely undetermined, depending on the source mass ratio and spin amplitude. We emphasise that the spin measure is a neat, direct measurement, that does not involve complex, often degenerate, multi-parametric fits of high energy emission processes.

The source luminosity distance error has a wide spread, usually ranging from to only few percent. Note that this is a direct measurement of the luminosity distance to the source, which, again, cannot be directly obtained (for cosmological objects) at any comparable accuracy level by any other astrophysical means. eLISA is a full sky monitor, and the localisation of the source in the sky is also encoded in the waveform pattern. Sky location accuracy is typically estimated in the range square degrees.

9 Reconstructing the massive black hole cosmic history through eLISA observations

eLISA will be an observatory. The goal is not only to detect sources, but also to extract valuable astrophysical information from the observations. While measurements for individual systems are interesting and potentially very useful for making strong-field tests of GR (see section 2), it is the properties of the set of massive black hole binary mergers that are observed which will carry the most information for astrophysics. Gravitational wave observations of multiple binary mergers may be used together to learn about their formation and evolution through cosmic history.

As any observatory, eLISA will observe a set of signals. After signal extraction and data analysis, these observations will provide a catalogue of coalescing binaries, with measurements of several properties of the sources (masses, mass ratio, spins, distances, etc) and estimated errors. The interesting questions to ask are the following: can we discriminate among different massive black hole formation and evolution scenarios on the basis of gravitational wave observations alone? Given a set of observed binary coalescences, what information can be extracted about the underlying population? For example, will gravitational wave observations alone tell us something about the mass spectrum of the seed black holes at high redshift, that are inaccessible to conventional electromagnetic observations, or about the poorly understood physics of accretion? These questions were extensively tackled in (Sesana et al., 2011) in the context of LISA.

Selection among a discrete set of models

First we consider a discrete set of models. As argued above, in the general picture of massive black hole cosmic evolution, the population is shaped by the seeding process and the accretion history. The four models we study here are the SE, SC, LE, and LC models introduced in the previous section (Arun et al., 2009). As a first step, we test here if eLISA observations will provide enough information to enable us to discriminate between those models, assuming that the Universe is well described by one of them.

Each model predicts a theoretical distribution of coalescing massive black hole binaries. A given dataset of observed events can be compared to a given model by computing the likelihood that the observed dataset is a realisation of model . When testing a dataset against a pair of models and , we assign probability to model , and probability to model . The probabilities and are a measure of the relative confidence we have in model and , given an observation . Once eLISA data is available, each model comparison will yield this single number, , which is our confidence that model is correct. Since the eLISA data set is not currently available, we can only work out how likely it is that we will achieve a certain confidence with future eLISA observations.

We therefore generate independent realisations of the population of coalescing massive black hole binaries in the Universe predicted by each of the four models. We then simulate gravitational wave observations by producing datasets of observed events (including measurement errors), which we statistically compare to the theoretical models. We consider only sources that are observed with SNR larger than eight in the detector. We set a confidence threshold of , and we count what fraction of the realisations of model yield a confidence when compared to an alternative model . We repeat this procedure for every pair of models. For simplicity, in modelling gravitational wave observations, we focus on circular, non-spinning binaries; therefore, each coalescing black hole binary in the population is characterised by only three intrinsic parameters – redshift , mass , and mass ratio – and we compare the theoretical trivariate distribution in these parameters predicted by the models to the observed values in the dataset . In terms of gravitational waveform modelling, our analysis can therefore be considered extremely conservative.

Results are shown in the left-hand panel of table 1, for a one year observation. The vast majority of the pair comparisons yield a  % confidence in the true model for almost all the realisations — we can perfectly discriminate among different models. Similarly, we can always rule out the alternative (false) model at a  % confidence level. Noticeable exceptions are the comparisons of models LE to LC and SE to SC, i.e., among models differing by accretion mode only. This is because the accretion mode (efficient versus chaotic) particularly affects the spin distribution of the coalescing systems, which was not considered here. To extend this work, we added to our analysis the distribution of the merger remnant spins , and compared the theoretical distribution predicted by the models to the observed values (including determination errors once again). The spin of the remnant can be reasonably determined in about  % of the cases only; nevertheless, by adding this information, we are able to almost perfectly discriminate between the LE and LC and the SE and SC models, as shown in the right hand panel of table 1.

Without spins With spins
SE SC LE LC SE SC LE LC
(l) SE 0.48 0.99 0.99 SE 0.96 0.99 0.99
SC 0.53 1.00 1.00 SC 0.13 1.00 1.00
LE 0.01 0.01 0.79 LE 0.01 0.01 0.97
LC 0.02 0.02 0.22 LC 0.02 0.02 0.06
Table 1: Summary of all possible comparisons of the pure models. Results are for one year of observation with eLISA. We take a fixed confidence level of . The numbers in the upper-right half of each table show the fraction of realisations in which the row model will be chosen at more than this confidence level when the row model is true. The numbers in the lower-left half of each table show the fraction of realisations in which the row model cannot be ruled out at that confidence level when the column model is true. In the left table we consider the trivariate , , and distribution of observed events; in the right table we also include the observed distribution of remnant spins, .

Constrains on parametric models

Figure 20: Likelihood distribution of the mixing fraction , for a particular realisation of the model . The true mixing parameter, marked by a dashed vertical line, was .

In the preceding section we demonstrated the potential of eLISA to discriminate among a discrete set of “pure” models given a priori. However, the true massive black hole population in the Universe will probably result from a mixing of known physical processes, or even from a completely unexplored physical mechanism. A meaningful way to study this problem is to construct parametric models that depend on a set of key physical parameters, , describing, for instance, the seed mass function and redshift distribution, the accretion efficiency etc. and to investigate the potential of eLISA to constrain these parameters. Such a parametric family of models is not available at the moment, but we can carry out a similar exercise by mixing two of our pure models, and , to produce a model in which the number of events of a particular type is given by [A], where is the number of events of that type predicted by model , is the corresponding number predicted by model and is the “mixing fraction”. In this case we generate datasets from a mixed model with a certain unknown , and we estimate the parameter by computing the likelihood that the data is drawn from a mixed distribution, as a function of . A specific example is shown in figure 20. Here the underlying model is [SE][LE], with . eLISA observations will allow us to pin-down the correct value of the mixing parameter with an uncertainty of . More complex examples of multi-model mixing can be found in (Sesana et al., 2011). Although highly idealised, this exercise demonstrate the potential of eLISA observations to constrain the physics and astrophysics of massive black hole along their entire cosmic history, in a mass and redshift range inaccessible to conventional electromagnetic observations.

5 Extreme mass ratio inspirals and astrophysics of dense stellar systems

1 The Galactic Centre: a unique laboratory

The discovery, in the local Universe, of dark, massive objects lurking at the centres of nearly all bright galaxies is one of the key findings of modern-day astronomy, the most spectacular being the case of the dark object in our own Galaxy (Schödel et al., 2003, Ghez et al., 2003, Eisenhauer et al., 2005, Ghez et al., 2005, 2008, Gillessen et al., 2009). The nucleus of the Milky Way is one hundred times closer to Earth than the nearest large external galaxy Andromeda, and one hundred thousand times closer than the nearest QSO. Due to its proximity, it is the only nucleus in the Universe that can be studied and imaged in great detail. The central few parsecs of the Milky Way house gas cloud complexes in both neutral and hot phases, a dense luminous star cluster, and a faint radio source SgrA of extreme compactness ( light minutes across). Observations, using diffraction-limited imaging and spectroscopy in the near-infrared, have been able to probe the densest region of the star cluster and measure the stellar dynamics of more than two hundred stars within a few light days of the dynamic centre. The latter is coincident, to within  arcsec, with the compact radio source SgrA. The stellar velocities increase toward SgrA with a Kepler law, implying the presence of a central dark mass (Gillessen et al., 2009). This technique has also led to the discovery of nearly thirty young stars that orbit the innermost region: the so called S0 (or S) stars. These young stars are seen to move on Keplerian orbits, with S02 (or S2) the showcase star orbiting the putative black hole on a highly eccentric () orbit with a period of  years. The periapsis of this orbit requires a lower limit on the density of the dark point-like mass concentration of more than pc (Maoz, 1998). Additionally, a lower limit of more than pc can be inferred from the compactness of the radio source (Genzel et al., 2010). These limits provide compelling evidence that the dark point-mass at SgrA is a black hole. A cluster of dark stars of this mass and density (e.g. composed of neutron stars, stellar black holes or sub-stellar entities such as brown dwarfs, planets and rocks) can not remain in stable equilibrium for longer than  years (Maoz, 1998, Genzel et al., 2000, 2006), and the only remaining, albeit improbable, hypothesis is a concentration of heavy bosons (a boson star, (Colpi et al., 1986)) or of hyperlight black holes (Maoz, 1998, ,). Overall, the measurements at the Galactic Centre are consistent with a system composed of a massive black hole and an extended close-to-isotropic star cluster, with the young S0 (or S) stars the only population showing a collective rotation pattern in their orbits (Genzel et al., 2010).

2 Extreme Mass Ratio Inspirals in galactic nuclei

Can we probe the nearest environs of a massive black hole other than the Galactic Centre? Massive black holes are surrounded by a variety of stellar populations, and among them are compact stars (stellar black holes, neutron stars and white dwarfs). White dwarfs, neutron stars, and stellar-mass black holes all share the property that they reach the last stable orbit around the central massive black hole before they are tidally disrupted. A compact star can either plunge directly toward the event horizon of the massive black hole, or gradually spiral in and fall into the hole, emitting gravitational waves. The latter process is the one of primary interest for eLISA. Gravitational waves produced by inspirals of stellar-mass compact objects into massive black holes are observable by eLISA. The mass of the compact object is typically of the order of a few solar masses, while the mass of the central black holes detectable by eLISA is from to . Because the mass ratio for these binaries is typically around , these sources are commonly referred to as EMRI.

The extreme mass ratio ensures that the inspiralling object essentially acts as a test particle in the background space-time of the central massive black hole. EMRI detections thus provide the best means to probe the environment of an astrophysical black hole and its stellar surroundings. White dwarfs, neutron stars, and stellar-mass black holes can all in principle lead to observable EMRI signals. However, stellar-mass black holes, being more massive, are expected to dominate the observed rate for eLISA, for two reasons: mass segregation tends to concentrate the heavier compact stars nearer the massive black hole, and black hole inspirals have higher signal-to-noise, and so can be seen within a much larger volume.

Three different mechanisms for the production of EMRI have been explored in the literature. The oldest and best-understood mechanism is the diffusion of stars in angular-momentum space, due to two-body scattering. Compact stars in the inner  pc will sometimes diffuse onto very high eccentricity orbits, such that gravitational radiation will then shrink the orbit’s semi-major axis and eventually drive the compact star into the massive black hole. Important physical effects setting the overall rate for this mechanism are mass segregation, which concentrates the more massive stellar-mass black holes () close to the central black hole, and resonant relaxation, which increases the rate of orbit diffusion in phase-space (Hopman and Alexander, 2006b): the orbits of stars close to a massive black hole are nearly Keplerian ellipses, and these orbits exert long term torques on each other, which can modify the angular momentum distribution of the stars and enhance the rate of EMRI formation (Gürkan and Hopman, 2007). However, subtle relativistic effects can reduce the estimated rates from relaxation processes (Merritt et al., 2011). In addition to the two-body scattering mechanism, other proposed channels for EMRI are tidal disruption of binaries that pass close to the central black hole (Miller et al., 2005), and creation of massive stars (and their rapid evolution into black holes) in the accretion discs surrounding the central massive black hole (Levin, 2007). Tidal break up of incoming stellar binaries may already have been seen in the Milky Way following the remarkable discovery of a number of so-called hypervelocity stars observed escaping from our Galaxy (e.g., Brown et al., 2009). They are believed to be the outcome of an ejection following the break-up of two bound stars by the tidal field of SgrA. All these mechanisms give specific predictions on the eccentricity and inclination of EMRI events that can be extracted from the gravitational wave signal (Miller et al., 2005).

When the orbiting object is close enough (within a few horizon radii from the large black hole) gravitational radiation dominates energy losses from the system, and the semimajor axis of the orbit shrinks. Radiation is emitted over hundreds of thousand of orbits as the object inspirals to the point where it is swallowed by the central massive black hole. Over short periods of time, the emitted radiation can be thought of as a snapshot that contains detailed information about the physical parameters of the binary. The detection of the emitted gravitational wave signal will give us very detailed information about the orbit, the mass, and spin of the massive black hole as well as the mass of the test object (Hopman, 2009, Gair et al., 2010, Preto and Amaro-Seoane, 2010).

The measurement of even a few EMRI will give astrophysicists a totally new and different way of probing dense stellar systems determining the mechanisms that shape stellar dynamics in the galactic nuclei and will allow us to recover information on the emitting system with a precision which is not only unprecedented in the history of astrophysics, but beyond that of any other technique (Amaro-Seoane et al., 2007, Porter, 2009, Babak et al., 2010).

3 A probe of galactic dynamics

The centre-most part of the stellar spheroid, i.e. the galactic nucleus, constitutes an extreme environment in terms of stellar dynamics. With stellar densities higher than pc and relative velocities exceeding  km/s, collisional processes (i.e.  collective gravitational encounters among stars) are important in shaping the density profiles of stars. The mutual influence between the massive black hole and the stellar system occurs thanks to various mechanisms. Some are global, like the capture of stars via collisional relaxation, or accretion of gas lost by stars through stellar evolution, or adiabatic adaptation of stellar orbits to the increasing mass of the black hole. Others involve a very close interaction, like the tidal disruption of a star or the formation of an EMRI.

The distribution of stars around a massive black hole is a classical problem in stellar dynamics (Bahcall and Wolf, 1976, 1977), and of importance for EMRI is the distribution of stellar black holes. Objects more massive than the average star, such as stellar black holes, tend to segregate at the centre of the stellar distribution in the attempt to reach, through long-distance gravitational encounters, equipartition of kinetic energy. A dense, strongly mass-segregated cusp of stellar black hole is expected to form near a massive black hole, and such a cusp plays a critical role in the generation of EMRI. The problem of the presence of a dark cusp has been addressed, for the Galactic Centre, by different authors, from a semi-analytical and numerical standpoint (Sigurdsson and Rees, 1997, Miralda-Escudé and Gould, 2000, Freitag et al., 2006a, b, Hopman and Alexander, 2006a). A population of stellar black holes can leave an imprint on the dynamics of the S0 (or S) stars at the Galactic Centre, inducing a Newtonian retrograde precession on their orbits (Mouawad et al., 2005). Current data are not sufficient to provide evidence of such deviations from Keplerian orbits, so that the existence of a population of stellar black holes is yet to be confirmed (Gillessen et al., 2009, Merritt et al., 2009a).

4 A probe of the masses of stellar and massive black holes

It is very difficult to measure the mass of black holes, both of the massive and stellar variety. In the case of massive black holes, methods based on following the innermost kinematics are difficult for low-mass black holes in the range . These black holes have low intrinsic luminosities even when they are active, making detection hard. Performing dynamical measurements at these masses through stellar kinematics requires extremely high spatial resolution. Nowadays with adaptive optics we could optimistically hope to get a handful of measurements through stellar kinematics about  kpc away, although future  m –  m telescopes can reach out to the Virgo cluster ( Mpc). Exquisite gas-dynamical measurements are possible for only a handful of active black holes using water megamaser spots in a Keplerian circumnuclear disk (Kuo et al., 2011). Still, the black hole in the centre of our own galaxy lies in this range, and placing constraints on the mass function of low-mass black holes has key astrophysical implications. Observations show that the masses of black holes correlate with the mass, luminosity and the stellar velocity dispersion of the host (Gültekin et al., 2009). These correlations imply that black holes evolve along with their hosts throughout cosmic time. One unanswered question is whether this symbiosis extends down to the lowest galaxy and black hole masses due to changes in the accretion properties (Mathur and Grupe, 2005), dynamical effects (Volonteri et al., 2007), or cosmic bias (Volonteri and Natarajan, 2009). eLISA will discover the population of massive black holes in galaxies smaller than the Milky Way, that are difficult to access using other observational techniques, and provide insights on the co-evolution of black holes and their hosts.

Difficulties, albeit of a different nature, exist in measuring the masses and mass distribution of stellar black holes. Stellar black holes are observed as accreting X-ray sources in binaries. According to stellar evolution, black holes result from the core collapse of very massive stars, and their mass is predicted to be in excess of the maximum mass of a neutron star, which is still not fully constrained. Depending on the state of nuclear matter, this limit varies from about to about (Shapiro and Teukolsky, 1986). The maximum mass of a stellar black hole is not constrained theoretically, and is known to depend sensitively on the metallicity of the progenitor star. The masses of stellar black holes are inferred using Kepler’s third law, or through spectral analysis of the emission from the hole’s accretion disc. These techniques can be used only for black holes in a binary system. Current measurements of the black hole mass indicate a range for stellar black holes from about up to , but uncertainties in the estimate can be as large as a factor of two (Orosz, 2003). In addition, stellar black holes in interacting binaries are a very small and probably strongly biased fraction of the total stellar black hole population. They are formed from stars that have lost their hydrogen mantle due to mass transfer (and thus formed in a different way than the vast majority of stellar black holes. eLISA will measure the mass of the stellar black holes again with unprecedented precision providing invaluable insight on the process of star formation in the dense nuclei of galaxies, where conditions appear extreme.

5 Detecting extreme mass ratio inspirals with eLISA

Figure 21: Left: The maximum detectable redshift (or horizon) for EMRIs assuming a two year mission lifetime, and a . The red curve is computed using the analytic waveform model and has a maximum redshift of for black hole masses at the source rest-frame of . The green and blue curves are sky-averaged horizons, computed using the Teukolsky waveform model, for two different values of the spin of the central black holes, (green) and (blue).
Right: The distribution of maximum EMRI SNRs versus redshift for eLISA. The dashed horizontal line denotes the SNR threshold of 20.
Figure 22: The distribution of errors from a Markov Chain Monte Carlo analysis for a source at with an SNR of 25. The plot shows the error distributions for the central black holes mass and spin , the mass of the compact object and the eccentricity at plunge .

EMRI are compact stars moving on relativistic orbits around a massive black hole. As the compact object spends most of its time in the strong field regime, its orbit is very complex and difficult to model. While not fully realistic, a set of phenomenological waveforms have been developed (Barack and Cutler, 2004), the AK waveforms, which fully capture the complexity of the model. These waveforms are defined by a 14 dimensional parameter set, of which the most physically relevant are the masses of the central black hole and of the compact object, and respectively, the spin of the massive black hole , the eccentricity of the orbit at plunge, , the sky position of the source with respect to the detector, and the luminosity distance to the source, . In addition to these approximate models, more accurate EMRI waveform models have been computed using black hole perturbation theory, in which the inspiraling object is regarded as a small perturbation to the background spacetime of the large black hole. The perturbation theory framework was first outlined in (Teukolsky, 1973) and gave rise to the Teukolsky equation. However, the solution of this equation is computationally expensive, and results have only recently been obtained for a selection of generic orbits (Drasco and Hughes, 2006). Nonetheless, results have been fully tabulated for certain restricted types of orbit. For the calculations described here we will use data for circular-equatorial orbits (Finn and Thorne, 2000, Gair, 2009a). We can use both models to compute the maximum detectable redshift, or the horizon for EMRI detection, as a function of mass.

To calculate the detection limit of EMRI for eLISA using the AK waveforms, we must perform a Monte Carlo simulation over the waveform parameters. We explore the mass range . As not much is known about the distribution of spins or eccentricities for EMRI, we consider uniform distributions for the spins in the range , and for eccentricities at plunge in the interval . We fix the mass of the inspiraling body to to represent the inspiral of a stellar black hole, as these are expected to dominate the event rate (Gair et al., 2004). The detection horizon for neutron star and white dwarf inspirals is significantly less than for black holes. The final assumption required is to set a threshold of detection. While a SNR threshold of was thought to be justified in the past, advances in search algorithms have recently demonstrated that EMRI with SNR about is sufficient for detection (Cornish, 2011, Gair et al., 2008, Babak et al., 2010), allowing us to assume an SNR threshold of in this analysis. Assuming a mission lifetime of two years, and plunge times between , a large scale Monte Carlo simulation was run over all 14 parameters. In figure 21 (left) we plot the maximum detectable redshift (also referred to as horizon) as a function of intrinsic mass of the massive black hole. Systems with intrinsic mass in the range from are detectable in the local Universe at redshift of , while systems in the range from should be detectable by eLISA to , corresponding to a co-moving volume of about  Gpc.

Figure 21 (left) also shows the maximum detectable redshift as a function of the mass of the central massive black hole, computed for circular-equatorial inspirals using the Teukolsky equation for the same masses of the inspiralling compact object and massive black holes. This curve shows the sky-averaged horizon, i.e., the maximum redshift at which the SNR averaged over inclinations and orientations of the EMRI system reaches the threshold value of . Tabulated Teukolsky results are only available for selected values of the spin of the central black hole, so we show the horizon assuming all the central black holes have either spin or . The Teukolsky horizon appears significantly lower than the AK horizon, but this is a result of the sky-averaging approximation – the sky averaged SNR is expected to be a factor of about lower than the SNR of an “optimally oriented” binary. The AK horizon was computed using a Monte Carlo simulation over orientations and sky locations for the source and will therefore approach the value for an optimally-oriented binary. The difference between the sky-averaged Teukolsky horizon and the AK horizon is therefore consistent with the expected level of difference. The maximum horizon for the Teukolsky curves is at a similar value for the mass of the central black hole as the AK results – somewhat lower for and higher for , as we would expect since inspirals into more rapidly spinning black holes emit radiation at higher frequencies, which shifts the peak sensitivity to higher masses. For the same reason, we see that the eLISA horizon is at a higher redshift for more rapidly spinning central black holes.

In figure 21 (right) we plot the distribution of maximum SNR as a function of redshift for the Monte Carlo simulation performed using the AK waveforms. A nearby EMRI will be detectable with SNR of many tens, with SNRs of 30 being available out to . EMRI can be detected with an SNR of 20 up to , up to a volume of about  Gpc, encompassing the last 6 billion years of the Universe.

EMRIs are the most complex sources to model and to search for. However, if they can be detected, this complexity will allow us to estimate the parameters of the system with great accuracy (Cornish, 2011, Gair et al., 2008, Babak et al., 2010). For an EMRI detected with a certain SNR, the parameter estimation accuracy does not strongly depend on the detector configuration, since any detected EMRI will be observed for many waveform cycles. For this reason the parameter estimation accuracy achievable with eLISA is essentially the same as reported in the published LISA literature (Barack and Cutler, 2004, Huerta and Gair, 2009). For any EMRI observed with SNR above the detection threshold of , we expect to measure the mass and spin of the central massive black hole with a precision to better than a part in . This is illustrated in Figure 22, that shows the results from a Markov Chain Monte Carlo analysis (Cornish and Porter, 2006) of a source at with . The plots show the distribution of errors for a particular source that would be recovered by analysing the data from the detector. Results are shown for the mass and spin of central black hole, the mass of the stellar black hole, and the eccentricity at plunge . Our analysis also shows that the luminosity distance to the source is determined with an accuracy of less than  % and the source sky location can be determined to around square degrees. While the SNR is quite low for this source, the accuracy in the estimation of parameters is very good.

6 Estimating the event rates of extreme mass ratio inspirals for eLISA

We can use the horizon distances described in the preceding section to compute the likely number of EMRI events that eLISA will detect, if we make further assumptions about the EMRI occurring in the Universe. This depends on the black hole population and on the rate at which EMRI occur around massive black holes with particular properties. The latter is poorly known, and we will use results from Hopman (2009) and Amaro-Seoane and Preto (2011) for the rate of inspirals involving black holes. The rate is found to scale with the central black hole mass, , as . We do not consider neutron star and white dwarf inspirals in these rate estimates as the expected number of detections with eLISA is less than one in both cases, due to the considerably reduced horizon distance for these events. We therefore fix the mass of the inspiraling body at , as in the previous section.

To model the black hole population, we take the mass function of black holes to be in the intrinsic mass range . Using the assumption that there is no evolution in the black hole mass function, we sampled sources from a uniform distribution in co-moving volume. These assumptions are consistent with the mass function derived from the observed galaxy luminosity function using the relation, and excluding Sc-Sd galaxies (Aller and Richstone, 2002, Gair et al., 2004, Gair, 2009a). For the results using the AK waveform model, we choose the spin of the central object uniformly in the range , the eccentricity of the orbit at plunge uniformly in the range and all angles to be uniform or uniform in cosine as appropriate. For the Teukolsky based results we do not need to specify the angles, as we use a sky and orientation averaged sensitivity, and we do not specify the eccentricity or inclination as the orbits are all circular and equatorial (although we assume equal numbers of prograde and retrograde inspirals). As before, the Teukolsky results are available for fixed values of the spin only, so we estimate the event rate assuming that all the black holes have spin , or .

It is important also to correctly randomise over the plunge time of the EMRI. For the AK calculation, we choose the plunge time uniformly in , with time measured relative to the start of the eLISA observation and assuming an eLISA lifetime of 2 years. Although sufficiently nearby events with plunge times greater than 5 years in principle could be detected, it was found that such events contribute less than one event to the total event rate. For the Teukolsky calculation, we evaluated the observable lifetime for every event, which is the amount of time during the inspiral that eLISA could start to observe that will allow sufficient SNR to be accumulated over the mission lifetime to allow a detection (Gair, 2009a).

In table 2 we give the results of this calculation for different waveform models and black hole spins. The predicted number of events depends on the assumptions about the waveform model and the spin of the black holes, but it is in the range of events in two years. The number of events predicted for the AK model is higher because the presence of eccentricity in the system tends to increase the amount of energy radiated in the eLISA band. The analytic kludge estimates include randomisation over the black hole spin, orbital eccentricity and inclination, so the true detection rate is likely to be closer to this number, although this depends on the unknown astrophysical distribution of EMRI parameters. Even with as few as 10 events, Gair et al. (2010) show that the slope of the mass function of massive black holes in the mass range can be determined to a precision of about , which is the current level of observational uncertainty.

Teukolsky
Waveform model with black hole spin Analytic Kludge
Number of events 30 35 55 50
Table 2: Estimated number of EMRI events detectable by eLISA. The first three columns shows the results computed using the Teukolsky waveform model, assuming all black holes have fixed spin of , or . The last column shows results computed using the analytic kludge waveform model.

7 Black hole coalescence events in star clusters

In closing this section on astrophysical black holes, we explore briefly the possibility that an instrument like eLISA will detect coalescences between low-mass massive black holes (called also intermediate-mass black holes) in the mass range . These coalescence events do not result from the assembly of dark matter halos, but rather they are local coalescence events occurring in star clusters under extreme (and largely unexplored) astrophysical conditions. Given the tiny radius of gravitational influence (about  pc) of such light black holes on the surrounding dense stellar environment, their detection is extremely difficult, and their existence has never been confirmed, though evidence has been claimed in a number of globular clusters (see Miller and Colbert, 2004, Miller, 2009, and references therein).

An intermediate-mass black hole may form in a young cluster if the most massive stars sink to the cluster’s centre due to mass segregation before they evolve and explode. There, they start to physically collide. The most massive star gains more and more mass and forms a runaway star that may collapse to form an intermediate-mass black hole (Portegies Zwart and McMillan, 2000, Gürkan et al., 2004, Portegies Zwart et al., 2004, Freitag et al., 2006a, c). Intermediate-mass black holes can be observed by eLISA via the inspiral of compact objects such as stellar mass black holes (Konstantinidis et al., 2011), or when they form a binary. The formation of an intermediate-mass black hole binary can occur via star-cluster star-cluster collisions like those found in the Antennæ galaxy (Amaro-Seoane and Freitag, 2006a, Amaro-Seoane et al., 2010a), or via formation in situ (Gürkan et al., 2006).

eLISA will observe intermediate-mass black hole binaries with out to a few Gpc (Santamaría et al., 2010), and it will detect stellar-mass black holes plunging into an intermediate-mass black hole in a massive star cluster in the local Universe (Konstantinidis et al., 2011). Event rates are hard to predict due to large uncertainties in the dynamical formation of intermediate mass black holes in star clusters, but we may observe as many as a few events per year (Amaro-Seoane and Freitag, 2006a). The detection of even a single event would have great importance for astrophysics, probing the existence of black holes in this unexplored mass range and shedding light on the complex dynamics of dense stellar clusters.

6 Confronting General Relativity with Precision Measurements
of Strong Gravity

1 Setting the stage

GR is a theory of gravity in which gravitational fields are manifested as curvature of spacetime. GR has no adjustable parameters other than Newton’s gravitational constant, and it makes solid, specific predictions. Any test can therefore potentially be fatal to its viability, and any failure of GR can point the way to new physics. Confronting GR with experimental measurements, particularly in the strong gravitational field regime, is therefore an essential enterprise. In fact, despite its great successes, we know that GR cannot be the final word on gravity, since it is a classical theory that necessarily breaks down at the Planck scale. As yet there is no complete, quantum theory of gravity, and gravitation is not unified with the other fundamental forces. Under such a premise, several stress tests of GR have been proposed, each of them potentially fatal to the theory, however all of them involve low energies and length-scales much larger than the Planck scale.

Although so far GR has passed all the tests to which it has been subjected (Will, 2006), most of these tests were set in the weak-field regime, in which the parameter is much smaller than one. Here is the typical velocity of the orbiting bodies, their total mass, and their typical separation.

For the tests of GR that have been carried out in our Solar System, expected second-order GR corrections to the Newtonian dynamics are of the order , and so to date it has been sufficient to expand GR equations to the first Post-Newtonian (PN) order. Solar System tests are completely consistent with GR to this order of approximation.

Binary pulsars, which are essentially very stable and accurate clocks with typical orbital velocities (), are excellent laboratories for precision tests of GR (Lorimer, 2008). Current observations of several binary pulsars are perfectly consistent with the GR predictions, with orbits again calculated to the first PN order. Observations of the first binary pulsar to be discovered, PSR 1913+16, also provided the first astrophysical evidence for gravitational radiation, a 2.5-PN-order effect. Loss of energy due to gravitational-wave emission (radiation reaction) causes the binary orbit to shrink slowly; its measured period derivative agrees with GR predictions to within  %, consistent with measurement error bars (Weisberg and Taylor, 2005). Another double pulsar system, SR J0737-3039 A and B, allows additional tests of GR that were not available prior to its discovery (Kramer et al., 2006). In that system, the orbital period derivative is consistent with GR at the  % level, and the Shapiro delay agrees to within  % with the predictions of GR (Kramer and Wex, 2009).

However, the gravitational fields responsible for the orbital motion in known binary pulsars are not much stronger than those in the Solar System: the semimajor axis of the orbit of PSR 1913+16 is about . Such weak fields limit the ability of binary pulsars to probe nonlinear GR dynamics. They do provide important tests of strong-field static gravity, as the redshift at the surface of a neutron star is of order .

Figure 23: Gravitational wave signal for the final few orbits, plunge, merger and ringdown of a coalescing binary. The total mass of system , mass ratio , spin magnitudes , misalignment between spins and orbital angular momentum few degrees, the distance to the source . The inset shows the signal on a larger data span.

eLISA observations of coalescing massive black hole binaries, or of stellar-mass compact objects spiralling into massive black holes, will allow us to confront GR with precision measurements of physical regimes and phenomena that are not accessible through Solar System or binary pulsar measurements. The merger of comparable-mass black hole binaries produces an enormously powerful burst of gravitational radiation, which eLISA will be able to measure with amplitude SNR as high as a few hundred, even at cosmological distances.

In the months prior to merger, eLISA will detect the gravitational waves emitted during the binary inspiral; from that inspiral waveform, the masses and spins of the two black holes can be determined to high accuracy. Given these physical parameters, numerical relativity will predict very accurately the shape of the merger waveform, and this can be compared directly with observations, providing an ideal test of pure GR in a highly dynamical, strong-field regime.

Stellar-mass compact objects spiralling into massive black holes will provide a qualitatively different test, but an equally exquisite one. The compact object travels on a near-geodesic of the spacetime of the massive black hole. As it spirals in, its emitted radiation effectively maps out the spacetime surrounding the massive black hole. Because the inspiralling body is so small compared to the central black hole, the inspiral time is long and eLISA will typically be able to observe of order cycles of inspiral waveform, all of which are emitted as the compact object spirals from 10 horizon radii down to a few horizon radii. Encoded in these waves is an extremely high precision map of the spacetime metric just outside the central black hole. Better opportunities than these for confronting GR with actual strong-field observations could hardly be hoped for.

The LIGO and Virgo detectors should come online around 2015, and their sensitivity is large enough that they should routinely observe stellar mass black hole coalescences, where the binary components are of roughly comparable mass. However, even the brightest black hole mergers that LIGO and Virgo should observe will still have an amplitude SNR about 10 to 100 times smaller than the brightest massive black hole coalescences that eLISA will observe. The precision with which eLISA can measure the merger and ringdown waveforms will correspondingly be better by the same factor when compared to ground-based detectors. The situation is similar for the EMRI described in the previous section: while ground-based detectors may detect binaries with mass ratios of about (e.g., a neutron star spiralling into a black hole), in observations lasting approximately cycles, the precision with which the spacetime can be mapped in such cases is at least two orders of magnitude worse than what is achievable with eLISA’s EMRI sources. Thus eLISA will test our understanding of gravity in the most extreme conditions of strong and dynamical fields, and with a precision that is two orders of magnitude better than that achievable from the ground.

GR has been extraordinarily fruitful in correctly predicting new physical effects, including gravitational lensing, the gravitational redshift, black holes and gravitational waves. GR also provided the overall framework for modern cosmology, including the expansion of the Universe. However, our current understanding of the nonlinear, strong gravity regime of GR is quite limited. Exploring gravitational fields in the dynamical, strong-field regime could reveal new objects that are unexpected, but perfectly consistent with GR, or even show violations of GR.

The best opportunity for making such discoveries is with an instrument of high sensitivity. Ground-based detectors like LIGO and Virgo will almost certainly always have to detect signals by extracting them from deep in the instrumental noise, and they will therefore depend on prior predictions of waveforms. eLISA, on the other hand, will have enough sensitivity that many signals will show themselves well above noise; unexpected signals are much easier to recognize with such an instrument.

2 Testing strong-field gravity: The inspiral, merger, and ringdown of massive black hole binaries

eLISA’s strongest sources are expected to be coalescing black hole binaries where the components have roughly comparable masses, . Their signal at coalescence will be visible by eye in the data stream, standing out well above the noise, as illustrated in figure 23.

As discussed in section 8, black hole binary coalescence can be schematically decomposed into three stages (inspiral, merger, and ringdown), all of which will be observable by eLISA for a typical source. The inspiral stage is a relatively slow, adiabatic process, well described by the analytic PN approximation. The inspiral is followed by the dynamical merger of the two black holes, that form a single, highly distorted black hole remnant. Close to merger, the black hole velocities approach and the PN approximation breaks down, so the waveform must be computed by solving the full Einstein equations via advanced numerical techniques. The distorted remnant black hole settles down into a stationary rotating solution of Einstein’s equations (a Kerr black hole) by emitting gravitational radiation. This is the so called “ringdown” phase, where the gravitational wave signal is a superposition of damped exponentials QNM, and therefore similar to the sound of a ringing bell.

While numerical relativity is required to understand the gravitational radiation emitted during merger, the post-merger evolution – i.e., the black hole “quasinormal ringing” – can be modelled using black hole perturbation theory. The final outcome of the ringdown is the Kerr geometry, with a stationary spacetime metric that is determined uniquely by its mass and spin, as required by the black hole“no-hair” theorem.

For equal-mass black hole binaries with total mass in the range , where is the cosmological redshift of the source, the inspiral SNR and post-inspiral (merger plus ring-down) SNR are within an order of magnitude of each other. From a typical eLISA observation of the inspiral part of the signal, it will be possible to determine the physical parameters of the binary to extremely high accuracy. Using these parameters, numerical relativity can predict very precisely the merger and ringdown waves. Measurements of the individual masses and spins will allow us to predict the mass and the spin of the remnant black hole (Rezzolla et al., 2008b), which can be directly tested against the corresponding parameters extracted from the ringdown. The merger and ringdown waveforms will typically have an SNR of for binary black holes with total mass at , so an extremely clean comparison will be possible between the observed waveforms and the predictions of GR.

The inspiral stage: comparing inspiral rate with predictions of General Relativity

With orbital velocities typically in the range , most of the inspiral stage can be well described using high-order PN expansions of the Einstein equations. The inspiral waveform is a chirp: a sinusoid that increases in frequency and amplitude as the black holes spiral together. Depending on the source parameters, eLISA will be able to observe the final stages of the inspiral, for up to one year in some favourable cases. To give a practical reference, when the gravitational-wave frequency sweeps past  mHz, the time remaining until merger is approximately

(9)

where, as above, is the total mass of the binary and is the symmetric mass ratio. eLISA will observe the last gravitational wave inspiral cycles, depending on the total mass and distance of the source. Since the inspiral signal is quite well understood theoretically, matched filtering can be used to recognise these inspirals up to a year before the final merger, at a time when the total SNR is still small. Moreover, as the total SNR in the inspiral is quite large in many cases, and such signals are long lived, matched filtering based on the inspiral waveform alone can determine the system parameters to very high accuracy. Both masses can be determined to within a fractional error of about , and the spin of the primary black hole can be measured to an accuracy of  % or better.

The nonlinear structure of GR (and possible deviations from GR) could be encoded in a phenomenological way by considering hypothetical modifications of the gravitational wave amplitude and phasing, as proposed by different authors (Arun et al., 2006, Yunes and Pretorius, 2009). The relatively large strength of the inspiral gravitational wave signal will allow a sensitive test of GR by comparing the rate of the observed inspiral (phase evolution) to predictions of the PN approximation to GR (Mishra et al., 2010, Cornish et al., 2011, Huwyler et al., 2011, Li et al., 2011).

The merger stage: spectacular bursts

The inspiral is followed by a dynamical merger that produces a burst of gravitational waves. This is a brief event, comprising a few cycles lasting about , yet very energetic: during the merger the gravitational wave luminosity is , emitting more power than all the stars in the observable Universe. The final merger of massive binaries occurs in the very strong-field, highly nonlinear and highly dynamical regime of GR, and is the strongest gravitational wave source that eLISA is expected to see. eLISA will be able to see the merger of two black hole beyond redshift , and for mergers of two black hole at the SNR will be about . As mentioned above, eLISA observations of the inspiral yield a good measurement of the masses and spins of the black holes. With these in hand, numerical relativity will make a very specific prediction for the merger and ringdown radiation from the system. Comparison with the waveform that eLISA actually observes will allow us to confront the predictions of GR with an ultra-high precision measurement in the fully nonlinear and dynamical regime of strong gravity for the first time.

The ringdown stage: black hole spectroscopy

Although numerical relativity waveforms from colliding holes naturally include the ringdown waves, these waves are also well understood analytically. GR predicts, as a consequence of the “no-hair” theorem, that every excited black hole emits gravitational waves until it reaches a time-independent state characterised entirely by its mass and spin. These ringdown waves consist of a set of superposed black hole QNM waves with exponentially damped sinusoidal time dependence, plus a far weaker “tail”. The modes are strongly damped as their energy is radiated away to infinity, so the final ringdown stage is brief, lasting only a few cycles.

The QNM of Kerr black hole can be computed using perturbation theory: the spacetime metric is written as the Kerr metric plus a small perturbation, and Einstein’s equations are expanded to first-order in that perturbation. The solutions can be decomposed into a sum of damped exponentials with complex eigenfrequencies (Chandrasekhar and Detweiler, 1975) that can be computed to essentially arbitrary accuracy (Leaver, 1985).

While there are infinitely many modes (corresponding to the angular order and overtone number of the perturbation from the stationary state), the lowest-order modes are the most readily excited and the least strongly damped, so in practice only a few modes are likely to be observed.

The frequencies and damping times of these ringdown QNM (tabulated in Berti et al., 2009) are completely determined by the mass and the spin of the remnant black hole.

A data analysis strategy based on multi-mode searches will be necessary for an accurate estimation of the mass and spin of the final black hole (Berti et al., 2006, 2007). Furthermore, if we can measure at least two different QNM in a ringdown signal, the ringdown radiation itself will provide a strong-field test of the hypothesis that the central massive objects in galactic nuclei are indeed Kerr black holes. The reason is that a two-mode signal contains four parameters (the frequencies and damping times of each mode), which must all be consistent with the same mass and spin values (Dreyer et al., 2004). Just like we can identify chemical elements via spectroscopic measurements, we can uniquely identify a black hole (determine its mass and spin) from the spectrum of its ringdown radiation.

If GR is correct but the observed radiation is emitted from a different source (exotic proposals include boson stars and gravastars, among others), the spectrum would most certainly be inconsistent with the QNM spectrum of Kerr black holes in GR (Yoshida et al., 1994, Berti and Cardoso, 2006, Chirenti and Rezzolla, 2007, Pani et al., 2009). The same should occur if GR does not correctly describe gravity in the extremes of strong fields and dynamical spacetimes. The fact that black hole oscillations should produce different radiation spectra in different theories of gravity is true in general (Barausse and Sotiriou, 2008), and the spectrum was studied in some specific extensions of GR, such as Einstein-dilaton-Gauss-Bonnet gravity (Pani and Cardoso, 2009). The possibility of testing the no-hair theorem with QNM depends on the accuracy with which frequencies and damping times can be measured, which in turn depends on the SNR of the ringdown signal. As shown in (Berti, 2006, Berti et al., 2007), SNR larger than should be sufficient to identify the presence of a second mode and use it for tests of the no-hair theorem. This is only marginally achievable with advanced Earth-based detectors, but SNR of this order should be the norm for the black hole mergers detectable by eLISA. Furthermore, recent work showed that multi-mode ringdown waveforms could encode information on parameters of the binary before merger, such as the binary’s mass ratio (Kamaretsos et al., 2011), and this would provide further consistency checks on the strong-field dynamics of general relativity.

3 Extreme mass ratio inspirals: precision probes of Kerr spacetime

EMRI are expected to be very clean astrophysical systems, except perhaps in the few percent of galaxies containing accreting massive black holes, where interactions with the accretion disk could possibly affect the EMRI dynamics. Over timescales of the order of a day, the orbits of the smaller body are essentially geodesics in the spacetime of the massive black hole. On longer timescales, the loss of energy and angular momentum due to gravitational-wave emission causes the smaller body to spiral in; i.e., the geodesic’s “constants” of motion change slowly over time. Over a typical eLISA observation time (years), EMRI orbits are highly relativistic (radius smaller than 10 Schwarzschild radii) and display extreme forms of periastron and orbital plane precession due to the dragging of inertial frames by the massive black hole’s spin. Figure 24 shows two sample waveforms, corresponding to short stretches of time.

Figure 24: Segments of generic EMRI waveforms (Drasco and Hughes, 2006). These are the plus-polarised waves produced by a test mass orbiting a black hole that is spinning at 90 % of the maximal rate allowed by general relativity, a distance from the observer. Top panel: Slightly eccentric and inclined retrograde orbit modestly far from the horizon. Bottom panel: Highly eccentric and inclined prograde orbit much closer to the horizon. The amplitude modulation visible in the top panel is mostly due to Lense-Thirring precession of the orbital plane. The bottom panel’s more eccentric orbit produces sharp spikes at each pericentre passage.

Given the large amount of gravitational wave cycles collected in a typical EMRI observation (about ), a fit of the observed gravitational waves to theoretically calculated templates will be very sensitive to small changes in the physical parameters of the system. As mentioned above, this sensitivity makes the search computationally challenging, but it allows an extremely accurate determination of the source parameters, once an EMRI signal is identified. Assuming that GR is correct and the central massive object is a black hole, eLISA should be able to determine the mass and spin of the massive black hole to fractional accuracy of about for gravitational wave signals with an SNR of 20 (Barack and Cutler, 2004).

This level of precision suggests that we can use EMRI as a highly precise observational test of the “Kerr-ness” of the central massive object. That is, if we do not assume that the larger object is a black hole, we can use gravitational waves from an EMRI to map the spacetime of that object. The spacetime outside a stationary axisymmetric object is fully determined by its mass moments and current multipole moments . Since these moments fully characterise the spacetime, the orbits of the smaller object and the gravitational waves it emits are determined by the multipolar structure of the spacetime. By observing these gravitational waves with eLISA we can therefore precisely characterise the spacetime of the central object. Extracting the moments from the EMRI waves is analogous to geodesy, in which the distribution of mass in the Earth is determined by studying the orbits of satellites. Black hole geodesy, also known as holiodesy, is very powerful because Kerr black holes have a very special multipolar structure. A Kerr black hole with mass and spin parameter (in units with ) has multipole moments given by

(10)

Thus, , , and , and similarly for all other multipole moments; they are all completely determined by the first two moments, the black hole mass and spin. This is nothing more than the black hole “no-hair” theorem: the properties of a black hole are entirely determined by its mass and spin.

For inspiraling trajectories that are slightly eccentric and slightly non-equatorial, in principle all the multipole moments are redundantly encoded in the emitted gravitational waves (Ryan, 1995), through the time-evolution of the three fundamental frequencies of the orbit: the fundamental frequencies associated with the , , and motions (Drasco and Hughes, 2004), or, equivalently, the radial frequency and the two precession frequencies.

The mass quadrupole moment of a Kerr black hole can be measured to within for signals with an SNR of 30 (Barack and Cutler, 2004), At the same time and will be estimated to an accuracy of .

Any inconsistency with the Kerr multipole structure could signal a failure of GR, the discovery of a new type of compact object, or a surprisingly strong perturbation from some other material or object. For a review of the different hypotheses regarding the nature of the central object see (Sopuerta, 2010, Babak et al., 2011).

Other tests of the Kerr nature of the central massive object have also been proposed. EMRI signals can be used to distinguish definitively between a central massive black hole and a boson star (Kesden et al., 2005). In the black hole case the gravitational wave signal “shuts off” shortly after the inspiraling body reaches the last stable orbit (and then plunges through the event horizon), while for a massive boson star, the signal does not fade, and its frequency derivative changes sign, as the body enters the boson star and spirals toward its centre. Similarly, if the central object’s horizon is replaced by some kind of membrane (this is the case for the so-called gravastars) the orbital radiation produced by the orbiting body could resonantly excite the QNM of the gravastar, with characteristic signatures in the gravitational wave energy spectrum that would be detectable by eLISA (Pani et al., 2009).

Other studies within GR considered axisymmetric solutions of the Einstein field equations for which the multipole moments can differ from the Kerr metric, such as the Manko-Novikov solution. These studies revealed ergodic orbital motion in some parts of the parameter space (Gair, 2009b) as a result of the loss of the third integral of motion. A similar study suggested that the inspiralling body could experience an extended resonance in the orbital evolution when the ratio of intrinsic frequencies of the system is a rational number (Lukes-Gerakopoulos et al., 2010). If detected, these features would be a robust signature of a deviation from the Kerr metric.

These and similar studies of “bumpy” Kerr black holes – spacetime metrics with a multipolar stucture that deviates from the Kerr spacetime by some “tunable” amount (Ryan, 1995, Collins and Hughes, 2004, Glampedakis and Babak, 2006, Hughes, 2006, Vigeland et al., 2011, Vigeland and Hughes, 2010) – focussed on understanding whether the best fit to eLISA data is consistent with the Kerr solution within general relativity. However, an even more exciting prospect is that modifications in EMRI waveforms might arise because the true theory of gravity is in fact different from GR. For example, black holes in dynamical Chern-Simons theory (a parity-violating, quantum-gravity inspired extension of GR) deviate from Kerr black holes in the fourth multipole moment . This affects geodesic motion, and therefore the phasing of the gravitational wave signal (Sopuerta and Yunes, 2009, Pani et al., 2011).

Gravitational wave observations of black hole-black hole binaries cannot discriminate between GR and scalar-tensor theories of gravity. The reason is that black holes do not support scalar fields; i.e., they have no scalar hair. However, eLISA could place interesting bounds on scalar-tensor theories using observations of neutron stars spiralling into massive black holes (Berti et al., 2005, Yagi and Tanaka, 2010). These limits will be competitive with – but probably not much more stringent than – Solar System and binary pulsar measurements (Esposito-Farèse, 2004).

Finally, eLISA observations of compact binaries could provide interesting bounds on Randall-Sundrum inspired braneworld models (McWilliams, 2010, Yagi et al., 2011). A general framework to describe deviations from GR in different alternative theories and their imprint on the gravitational wave signal from EMRI can be found in (Gair and Yunes, 2011).

Most high-energy modifications to GR predict the existence of light scalar fields (axions). If such scalar fields exist, as pointed out long ago by Detweiler and others (Detweiler, 1980), rotating black holes could undergo a superradiant “black hole bomb” instability for some values of their spin parameter. Depending on the mass of axions, string-theory motivated “string axiverse” scenarios predict that stable black holes cannot exist in certain regions of the mass/angular momentum plane (Arvanitaki and Dubovsky, 2011). Furthermore, this superradiant instability could produce a surprising result: close to the resonances corresponding to a superradiant instability the EMRI would stop, and the orbiting body would float around the central black hole. These “floating orbits” (for which the net gravitational energy loss at infinity is entirely provided by the black hole’s rotational energy) are potentially observable by eLISA, and they could provide a smoking gun of high-energy deviations from general relativity (Cardoso et al., 2011, Yunes et al., 2011).

In conclusion we remark that, if GR must be modified, the “true” theory of gravity should lead to similar deviations in all observed EMRI. For this reason, statistical studies of EMRI to test GR would alleviate possible disturbances that may cause deviations in individual systems, such as interactions with an accretion disk (Barausse et al., 2007, Barausse and Rezzolla, 2008, Kocsis et al., 2011), perturbations due to a second nearby black hole (Yunes et al., 2011) or by a near-by star, which could allow us to investigate different models of how stars distribute around a massive black hole (Amaro-Seoane et al., 2012).

4 Intermediate mass ratio binaries

A loud gravitational wave source for eLISA would be the IMRI of binaries comprising a middleweight (or equivalently intermediate-mass) black hole, with mass in the range of a few times to a few times , along with either a massive black hole () or a solar-mass black hole. Currently there is no fully convincing evidence for the existence of intermediate-mass black holes, primarily due to the enormous observational difficulties of resolving the central region of dwarf galaxies and/or globular clusters, the two most likely places where they might reside. eLISA is one of the most promising observatories for discovering these middleweight black holes.

The strength of the gravitational wave signal from an IMRI lies between that of massive black hole binaries and EMRI, and the signal itself carries features of both limiting types, including a relatively fast frequency evolution and comparable contribution of several harmonics to the total strength of the signal. According to the proposed eLISA sensitivity, IMRI could be seen up to redshift . There are good reasons to expect that IMRI orbits may have measurable eccentricity (Amaro-Seoane and Freitag, 2006b, Amaro-Seoane, 2006, Amaro-Seoane et al., 2009, 2010b, Amaro-Seoane and Santamaría, 2010, Sesana, 2010). It may also be possible in some cases to observe the gravitational spin-spin coupling between the two black holes (equivalent to the Lense-Thirring effect). The precision in the measurements of the source parameters will lie between that of EMRI and comparable-mass binaries.

5 The mass of the graviton

In GR, gravitational waves travel with the speed of light and the graviton is hence massless. Alternative theories with a massive graviton predict an additional frequency-dependent phase shift of the observed waveform. The dominant effect can be expressed at 1-PN order, and would change the PN coefficient in the stationary-phase approximation to the Fourier transform of the waveform as follows:

(11)

where is again the symmetric mass ratio. This term alters the time of arrival of waves of different frequencies, causing a dispersion, and a corresponding modulation in the phase of the signal that depends on the Compton wavelength and the distance to the binary. Hence, by tracking the phase of the inspiral waves, eLISA should set bounds in the range on the graviton Compton wavelength (Berti et al., 2011, Huwyler et al., 2011), improving current Solar System bound on the graviton mass, () by several orders of magnitude.

Statistical observations of an ensemble of black hole coalescence events could be used to yield stringent constraints on other theories whose deviations from GR are parametrized by a set of global parameters: examples considered so far in the literature include theories with an evolving gravitational constant (Yunes et al., 2010), massive Brans-Dicke theories (Alsing et al., 2011) and Lorentz-violating modifications of GR (Mirshekari et al., 2011).

7 Cosmology

1 New physics and the early Universe

Gravitational waves penetrate all of cosmic history, which allows eLISA to explore scales, epochs, and new physical effects not accessible in any other way (see figure 25). Indeed a detectable gravitational wave background in the eLISA band is predicted by a number of new physical ideas for early cosmological evolution (Maggiore, 2000, Hogan, 2006). Two important mechanisms for generating stochastic backgrounds are phase transitions in the early Universe and cosmic strings.

Gravitational waves produced after the Big Bang form a fossile radiation: expansion prevents them from reaching thermal equilibrium with the other components because of the weakness of the gravitational interaction. Important information on the first instants of the Universe is thus imprinted in these relics and can be decoded. The mechanical effect of expansion is simply to redshift the corresponding frequency. Assuming that the wavelength is set by the apparent horizon size at the time of production (when the temperature of the Universe is ), the redshifted frequency is

(12)

Thus, the eLISA frequency band of about  mHz to  mHz today corresponds to the horizon at and beyond the Terascale frontier of fundamental physics. This allows eLISA to probe bulk motions at times about seconds after the Big Bang, a period not directly accessible with any other technique. Taking a typical broad spectrum into account, eLISA has the sensitivity to detect cosmological backgrounds caused by new physics active in the range of energy from  TeV to  TeV, if more than a modest fraction of about of the energy density is converted to gravitational radiation at the time of production.

Various sources of gravitational wave background of cosmological origin are presented in detail in Binétruy et al. (2012). Here we will only briefly summarize the main mechanisms leading to the potentially observable backgrounds.

Figure 25: The observed (redshifted) frequency of wave-generating phenomena is shown as a function of cosmic scale factor , with the present epoch at the right. The redshifted Hubble rate (horizon scale) is shown in black for a standard Grand Unified Theory (GUT) and a lower temperature Terascale (TeV) inflationary cosmology. Blue regions are accessible to electromagnetic (EM) observations: the Universe since recombination (right box) and cosmic microwave background (CMB) fluctuations (left box). The red bar shows the range of cosmic history accessible through eLISA from processes within the horizon up to about  TeV.

A standard example of new physics is a first-order phase transition resulting in bubble nucleation and growth, and subsequent bubble collisions and turbulence. Phase transitions also often lead to the formation of one-dimensional topological defects known as cosmic strings. Among possible topological defects, cosmic strings are unique from a cosmological point of view because, whereas their energy density should grow with the expansion, they interact and form loops which decay into gravitational waves. Thus cosmic strings tend to form networks with a typical scaling behaviour, losing energy mainly through gravitational radiation with a very broad and uniquely identifiable spectrum. Besides topological defects, cosmic strings could also find their origin among the fundamental objects of string theory, the theory that is aiming at providing a unified framework for all particles and forces of nature. Indeed, although fundamental strings were devised as submicroscopic objects, it has been progressively realized (Copeland et al., 2004) that some of these strings could be stretched to astronomical size by the cosmic expansion. eLISA will be our most sensitive probe for these objects by several orders of magnitude and so offers the possibility of detecting direct evidence of fundamental strings.

In order to distinguish backgrounds of gravitational waves from those waves emitted by point sources, it is essential to make use of the successive positions of eLISA around the Sun, and thus to wait a sufficient amount of time (of the order of a few months). It is more difficult to disentangle an isotropic cosmological (or astrophysical) background from an instrumental one, all the more because the eLISA “Mother-Daughter” configuration, providing only two measurement arms, does not allow to use Sagnac calibration (Hogan and Bender, 2001). Luckily, in the case of phase transitions as well as cosmic strings, the spectral dependence of the signal is well predicted and may allow to distinguish cosmological backgrounds as long as they lie above the eLISA sensitivity curve.

First-order cosmological phase transitions: Bulk motion from bubble nucleation, cavitation, collisions, turbulence

Abundant evidence suggests that the physical vacuum was not always in its current state, but once had a significantly higher free energy. This idea is fundamental and general: it underlies symmetry breaking in theories such as the Standard Model and its supersymmetric extensions, and cosmological models including almost all versions of inflation. Common to all these schemes is the feature that a cold, nearly uniform free energy contained in the original (false) vacuum is liberated in a phase transition to a final (true) vacuum, and eventually converted into thermal energy of radiation and hot plasma.

In many theories beyond the Standard Model, the conversion between vacuum states corresponds to a first-order phase transition. In an expanding Universe this leads to a cataclysmic process. After supercooling below the critical temperature for the transition, a thermal or quantum jump across an energy barrier leads to the formation of bubbles of the new phase. The bubbles rapidly expand and collide. The internal energy is thus converted to organised flows of mass-energy, whose bulk kinetic energy eventually dissipates via turbulence and finally thermalises. The initial bubble collision and subsequent turbulent cascade lead to relativistic flows and acceleration of matter that radiate gravitational waves on a scale not far below the horizon scale (Witten, 1984, Hogan, 1986, Kamionkowski et al., 1994, Huber and Konstandin, 2008, Caprini et al., 2009).

The gravitational wave energy density typically depends on two parameters: is the duration of the transition in Hubble units and is the fraction of energy density available in the source (false vacuum, relativistic motion). Typically , where is the the fraction of radiation energy today, and the fraction of vacuum energy which is converted into bulk kinetic energy during the phase transition. Strong first order phase transitions are obtained for but, in the context of specific models, increasing may increase as well.

Dynamics of warped sub-millimetre extra dimensions

Superstring theory provides examples of strong first order phase transitions in the Terascale region. It requires, for mathematical consistency, several extra dimensions. The sizes of these dimensions, their shapes, and how they are stabilised are yet to be determined. If they exist, gravity can penetrate into them, so they must be small or warped – with a size below the sub-millimetre scale limit set by direct laboratory tests of the gravitational inverse-square law. The scales probed by Standard Model particles and fields are much smaller than this, but fields other than gravity might be confined to a 3-dimensional subspace or (mem)brane plunged in the higher dimensional space.

Since the the Hubble length at the Terascale is about a millimetre, the current threshold where possible new effects of extra dimensions might appear happens to be about the same for experimantal gravity in the laboratory as for the cosmological regime accessible to eLISA. It is even possible that new properties of gravity on this scale are related to cosmic dark energy, whose energy density is about in particle physics units.

The dynamics associated with the stabilisation of extra dimensions at a certain size or warp radius might introduce a source of free internal energy released coherently on a mesoscopic, i.e. sub-millimetre to nanometre scale, leading to a detectable background