The focus of this Chapter is on describing the prospective sources of the gravitational wave universe accessible to present and future observations, from kHz, to mHz down to nano-Hz frequencies. The multi-frequency gravitational wave universe gives a deep view into the cosmos, inaccessible otherwise. It has as main actors core-collapsing massive stars, neutron stars, coalescing compact object binaries of different flavours and stellar origin, coalescing massive black hole binaries, extreme mass ratio inspirals, and possibly the very early universe itself. Here, we highlight the science aims and describe the gravitational wave signals expected from the sources and the information gathered in it. We show that the observation of gravitational wave sources will play a transformative role in our understanding of the processes ruling the formation and evolution of stars and black holes, galaxy clustering and evolution, the nature of the strong forces in neutron star interiors, and the most mysterious interaction of Nature: gravity. The discovery, by the LIGO Scientific Collaboration and Virgo Collaboration, of the first source of gravitational waves from the cosmos GW150914, and the superb technological achievement of the space mission LISA Pathfinder herald the beginning of the new phase of exploration of the universe.
Gravitational wave sources in the era of multi-frequency gravitational wave astronomy
Monica Colpi, Alberto Sesana
Dipartimento di Fisica G. Occhialini, Università degli Studi di Milano Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
School of Physics and Astronomy, The University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
to appear in the book
An Overview of Gravitational Waves:
Theory, Sources and Detection
edited by G. Auger and E. Plagnol (World Scientific, 2016)
- 1 Key science objectives of the multi-band gravitational wave astronomy
- 2 Prologue: GW150914 - the first cosmic source of gravitational waves
- 3 A new cosmic landscape
4 The electromagnetic universe
- 4.1 Neutron stars and stellar origin black holes in the realm of observations
- 4.2 Forming stellar origin compact binaries
- 4.3 Massive black holes in the realm of observations
- 4.4 The black hole desert
- 4.5 Formation of gravitational wave sources: the cosmological perspective
- 4.6 Massive black hole binary mergers across cosmic ages
- 5 The sources of the gravitational wave universe
- 6 Binaries as key sources of the gravitational universe
- 7 Waveforms and the Laws of Nature
- 8 Conclusion
1 Key science objectives of the multi-band gravitational wave astronomy
Gravitational wave sources have been anticipated and studied in the literature quite extensively during the last twenty years. These studies flourished in parallel to the building on Earth of the interferometric detectors Advanced LIGO and Virgo designed to explore the high frequency gravitational wave universe, and the proposal to construct in space a Laser Interferometer Space Antenna (LISA) to investigate lower frequency sources. The Pulsar Timing Array experiment (PTA) has been joining this world-wide effort by exploiting millisecond pulsars as high precision clocks to investigate the very low frequency domain.
With time, it has become clear that exploring the universe with gravitational waves from kHz to nano Hz makes it possible to discover new sources never anticipated before, and to provide complementary avenues for expanding our knowledge on the Laws of Nature, on Cosmology, and on the processes ruling the formation and evolution of compact objects as black holes within the realm of Relativistic Astrophysics and Galaxy Structure Formation. Exploring the universe with gravitational waves will help answering a number of fundamental questions in all these domains:
Laws of Nature
Is gravity in the strong field regime and dynamical sector as predicted by Einstein’s theory?
Are the properties of gravitational radiation as predicated by Einstein’s theory?
Does gravity couple to other dynamical fields, such as, massless or massive scalars?
Are (astrophysical) black holes described by the Kerr metric?
Are black holes hairless?
Are there naked singularities?
What is the behaviour of the short-range interaction at supra-nuclear densities?
What is the lowest energy state of baryonic matter at supra-nuclear densities?
Is gravitational collapse to a Kerr black hole unescapable?
Is the signal of coalescence from close pairs of neutron stars or/and black holes as predicted from Einstein’s theory?
Relativistic Astrophysics and Galaxy Structure Formation
What is the maximum mass of a neutron star and the minimum and maximum mass of a stellar origin black hole?
What is the mass function and redshift distribution of stellar origin neutron stars and black holes?
How do neutron star and black hole masses and spins evolve in relation to the environment and with cosmic epoch?
What is the physical mechanism behind supernovae and how asymmetric is their collapse?
How do stellar origin compact binaries form? Do they form in binary stars, or dynamically in dense star clusters or both?
Do black hole coalescence events of any flavour have an associated electromagnetic counterpart?
How can we identify the counterparts of neutron star binary mergers and of black hole-neutron star mergers? Are they related to known sources as short gamma-ray bursts (GRBs) and kilo-novae?
How many compact binaries of all flavours exist in the Milky Way and what do they tell us about the star formation history of our own galaxy?
Are ultra-compact white dwarf binaries the progenitors of Type Ia supernovae?
How do massive black hole form? Via accretion or/and aggregation of stellar origin black holes, or via the direct collapse of supermassive stars?
How do seed black holes grow to become giant through accretion and mergers, and how fast do they grow over cosmic time?
How often compact object binaries of the different flavours coalesce in galaxies and how does their coalescence rate evolve with redshift?
When did the first black holes form in pre-galactic halos, and what is their initial mass and spin distribution?
How do massive black holes pair in galaxy mergers and how fast to they coalesce?
What is the role of black hole mergers in galaxy formation?
Are massive black holes as light as inhabiting the cores of all dwarf galaxies?
What is the mass distribution of stellar remnants at the galactic centres and what is the role of mass segregation and relaxation in determining the nature of the stellar populations around the nuclear black holes in galaxies?
What is the merger rate of extreme mass ratio inspirals in galactic nuclei?
What is the architecture of the universe?
Using precise gravitationally calibrated distances and redshift measurements of coalescence events to what precision can we measure the Hubble flow?
What is the equation of state of dark energy, and to what precision can it be inferred from gravitational wave sources?
What can gravitational waves tell us about the physics beyond the Standard Model?
Can we measure or set bounds on cosmological gravitational wave backgrounds from the very early universe?
2 Prologue: GW150914 - the first cosmic source of gravitational waves
In the last years the detection of gravitational waves was perceived as imminent, following the rebuild of the LIGO interferometers at Hanford and Livingston , and of the interferometer Virgo in Pisa. Operating in the frequency interval between 10 Hz and 1000 Hz, Advanced LIGO and Virgo are designed to detect gravitational waves emitted by highly perturbed/deformed neutron stars, core collapsing massive stars and by the merger of pairs of neutron stars and stellar origin black holes. The rates of these events were so uncertain to prevent any definite prediction on the nature of the first signal, whether coming from neutron stars or black holes, or a combination of the two. But, on February 11th 2016, during the drafting of this chapter, from the LIGO Scientific Collaboration and The Virgo Collaboration came the announcement of 
the first direct detection of gravitational waves from a cosmic source;
the discovery of the most powerful astronomical event ever observed since the Big Bang;
the first detection of two stellar origin black holes coalescing in a single black hole, according to general relativity, and observed through the inspiral, merging and ringdown phases.
The discovery of this event, named GW150914, confirms, within statistical uncertainties and current precisionaaaWe remark that the current signal from GW150914 is not sufficient to exclude more exotic configurations than black holes as described in general relativity. Only precision observations of the late-time ringdown signal, where the differences in the quasinormal-mode spectrum eventually emerge, can be used to rule out exotic alternatives to black holes and to test quantum effects at the horizon scale. [3, 4]. ,
Einstein’s theory of gravity in the dynamical strong field regime, never tested before;
Einstein’s theory on the generation of gravitational radiation;
the unimpeded propagation of gravitational waves across the universe;
the existence of highly dynamical space-times that can form in the cosmos;
the existence and overall simplicity of black holes.
From an astrophysical perspective, the discovery of GW150914 provides
the first measure of the mass and spin of stellar black holes through gravity’s own messenger: the gravitational waves;
the first identification of ”heavy” stellar origin black holes with mass ;
the first definite proof of the existence of ”binary black holes”;
the most massive stellar origin black hole known to date, resulting from a merger product:
The signal from GW150914, lasting less than 0.45 seconds, is extraordinary simple . As shown in Figure 1, the signal sweeps upwards in amplitude and frequency of oscillation from 30 Hz to 250 Hz, with a peak gravitational wave strain of and significance of . The signal matches the waveform predicted by general relativity of the inspiral, merger, and ringdown of two black holes of and in the source-frame. The new black hole that formed has rest-frame mass and spin . The energy radiated in gravitational waves corresponds to , and to a peak luminosity of equivalent to . The source was observed with a matched filter signal-to-noise ratio of 23.7, and lies at a luminosity distance of Mpc, corresponding to a redshift (assuming the current -CDM cosmological model).
During the editing of this Chapter, the LIGO Scientific Collaboration and the Virgo Collaboration announced the discovery of a second signal: GW151226, detected at a significance greater than . Again a remarkable finding. The signal lasted in the LIGO frequency band for approximately 1 second, increasing in frequency and amplitude over about 55 cycles from 35 to 450 Hz (higher than that of the first event), and reached a peak gravitational strain of . The event detected with a signal-to-noise ratio of 13 is consistent again with the coalescence of two stellar origin black holes of and and a final mass of , in the rest frame of the source. The long lasting signal allowed the measure of the spin of one of the component black holes, which has spin parameter greater than 0.2. GW151226 is located at a distance of Mpc, corresponding to a redshift , similar to that of GW150914. While the high black hole masses in GW150914 lie in an almost unexplored interval, the masses in GW151226 are in line with those inferred in X-ray binaries . In we will discuss the repercussions of these findings on the origin and nature of stellar mass black holes.
In general, the signal expected from coalescing binaries of all flavours carries exquisite, and unique information on the masses and spins of the sources and of their internal structure: in the case of GW150914 and GW151226 the simplicity of two colliding black holes. Masses and spins will be the leitmotif of this Chapter on gravitational wave sources, as they are key parameters with which we will be describing the gravitational wave universe in a way complementary to that offered by electromagnetic observations, and often unaccessible otherwise. Key sources of the gravitational wave universe have been already presented in [7, 8, 9, 10] and will be described ahead in this Chapter. bbbRegretfully, the references have been limited to a minimum, due to the vastness of the topic. We will be mentioning main key and recent papers supplemented by specific reviews, which do include more extensive references.
3 A new cosmic landscape
The theory of general relativity by Einstein makes five key predictions: the existence of
an upper limit on the compactness of any self-gravitating object endowed of a surface (neutron stars being a chief example);
a maximum mass for stable, degenerate matter at nuclear densities, known as Oppenheimer Volkoff limit;
a maximum mass for (idealised) supermassive stars dominated by radiation pressure;
gravitational waves emitted by accelerated masses in non spherical motion.
The electromagnetic observations of the universe reveal the existence of
neutron stars, endowed by strong gravity, rapid rotation, intense magnetic fields, and high temperatures. Powered either by rotation or accretion or magnetic field dissipation, neutron stars are living in isolation or in binaries with stars or neutron stars as companions. They are ubiquitous and widespread in all the galaxies;
stellar origin black holes, powered by accretion and observed in a variety of X-ray binary systems with stars as companions. They are ubiquitous and widespread in all the galaxies;
active and quiescent supermassive black holes at the centres of galaxies;
galaxies with central supermassive black holes on their way to collide and merge;
an expanding universe changing and evolving on all scales, with gas fragmenting into stars inside dark matter halos (on the smallest scales), and galaxies embedded in dark matter halos assembling in galaxy’s clusters (on the largest scales);
an expanding universe at large, dominated by a dark energy component of unknown origin.
The combination of these two items set the frame for constructing a new cosmic landscape, that of the gravitational wave universe. By observing the universe with gravitational waves as messengers we will be able to answer to the deep questions outlined in . We now explore the content of each item, establishing connections among the different voices.
3.1 Black holes, neutron stars and supermassive stars as basic equilibrium objects
The simplest object to describe in nature is a black hole, representing the exact solution for the metric tensor of a point mass in otherwise empty space . The black hole solution found by Schwarzschild in 1916 describes the static, isotropic gravitational field generated by an uncharged point mass . The mass confined into a null volume, at , is surrounded by an event horizon, i.e. a boundary in spacetime defined with respect to the external universe inside of which events cannot affect any external observer. is not the baryonic mass only, but it includes the gravitational energy. Within the event horizon of a black hole all paths that electromagnetic waves could take are warped so as to fall farther into the black hole. In Schwarzschild coordinates, the event horizon, i.e. the surface of no return, appears as a critical spherical surface of radius , the Schwarzschild’s radius of the point mass . For the Sun the Schwarzschild radius of 2.95 km is deep in the solar interior, where Einstein’s equations in matter space exhibit no singularity. A key property of the Schwarzschild metric, which has no Newtonian analogue, is that below a radius, all circular orbits of massive particles are unstable (”isco” is acronym of innermost stable circular orbit). Massive particles moving on these geodesics are fated to cross the horizon, if subjected to an infinitesimal perturbation.
The Schwarzschild solution is a limiting case of a more general solution of the Einstein’s field equations found by Roy Kerr , which describes the spacetime metric of an axially-symmetric point mass surrounded by an event horizon, and describes an uncharged, rotating black hole with mass : here includes the negative contribution from the gravitational energy and the positive contribution by rotation, besides the matter load. The finding by Kerr is remarkable as it shows, contrary to Newtonian gravitation, that a mass endowed with rotation warps spacetime as the energy content from rotation becomes source of gravity itself, due to the non linearity of Einstein’s equation. Rotation is described by the spin vector where the norm of the vector is the spin parameter taking values between 0 and 1. Counterintuitively, the horizon of a Kerr black hole is smaller than and takes a simple expression: . Also depends on , and for , a test particle in co-rotation (counter-rotation) has placed at ().
Kerr black holes have become central for understanding the nature of singularities in general relativity, and a conjecture has been posed, known as cosmic censorship conjecture which asserts that no naked singularities form in Nature. In other words, it asserts that singularities (present in the classical description of gravity) are enclosed by a horizon so that information does not propagate into the rest of the universe, hidden from any observer at infinity by the event horizon of a black hole. For this reason the spin of a Kerr black hole is limited to values , with corresponding to a maximally rotating Kerr black hole (for a naked singularity would appear).
Black holes are fundamentally geometrical objects and a theorem, known as uniqueness theorem, states that Kerr black holes are the unique end-state of gravitational collapse [13, 14, 15]. This created the belief that all ”astrophysical” black holes (that form in Nature) are Kerr black holes, being the Kerr solution the only stationary solution of Einstein’s equation. The uniqueness theorem paved the way to a further conjecture, known as no-hair theorem. The no-hair theorem postulates that all black hole solutions of the Einstein-Maxwell equations of gravitation and electromagnetism in general relativity can be completely characterised by only three externally observable parameters: the mass , angular momentum , and electric charge. All other information (for which ”hair” is a metaphor) about the matter which formed a black hole or fall into it, disappears behind the black hole event horizon and is therefore permanently not accessible to external observers. A corollary of the no-hair conjecture asserts that the only deformations that black holes admit are those obtained by a change of mass and angular momentum as it occurs during the merger of two black holes, or during their substantial growth by accretion of in-falling matter. Any residual deformation is then radiated away by gravitational waves. cccWe do not consider here charged black holes and they are of no interest in astrophysics. Astrophysical black holes do not live in isolation and thus do not carry a charge. If charged, matter of opposite charge would fall in to obliterate any charge excess, making the black hole neutral.
While the theory of general relativity poses a limit on the angular momentum, no upper limit exists on the mass of a classical Kerr black hole. Only a lower bound exists, imposed by quantum mechanics natural units: the Planck mass below which a quantum description of gravity is desired . Astrophysical black holes are grouped in three classes or flavours possibly because of their different origin: the stellar black holes with masses in the interval , the (super-)massive black holes of , and the middleweight or intermediate mass black holes of . The boundaries of each interval are loosely defined and still arbitrary, as the physical mechanisms leading to the formation of massive black holes are uncertain. Detecting black holes of all flavours as gravitational wave sources will shed light into these mechanisms, and their potential connections.
Let us now proceed on considering the limit on the compactness of any astrophysical object endowed by a surface. A consequence of the Oppenheimer Volkoff equation for the equilibrium of a self-gravitating spherical body, of mass and radius is that its compactness defined as can not exceed a limiting value which holds for all stars, incompressible or not. An equilibrium object with can not exist with a finite surface. Note that the Schwarzschild radius (and in general ) violates this condition, implying the presence of a singularity in the interior solution of the Einstein’s equations.
Not only this condition poses a lower limit on the radius of any star, but paves the way to the idea that Kerr black holes inevitably form in nature, as soon as the condition is violated. Instability to collapse occurs when the pressure support against gravity, determined by the microphysical properties of matter, drops to the point that the total energy of the configuration is no longer a minimum . dddThe minimum here is computed with respect to all variations in the density profile that leave the number of particles unchanged, and unchanged and uniform the entropy per nucleon and chemical composition. The consequent loss of stability and evolution toward gravitational collapse occurs for a variety of reasons and here we highlight the most important.
In Newtonian gravity, gravitational collapse occurs when a stellar core supported by the degeneracy pressure of cold electrons becomes massive enough that electrons in their quantum states become ultra-relativistic, i.e. when their Fermi energy exceeds the electron rest mass energy . The reduced pressure support implied by this microphysical state transition occurs at the Chandrasekhar mass limit of (whose exact value depends on the chemical composition of the stellar core). Stellar evolution models show that when the iron core of a massive evolving star increases above core collapse ensues promptly. The dynamical contraction comes to a halt when the entire iron core at the Chandrasekhar mass limit has been transformed in a core of neutrons plus few exotic nuclei, at around or even above nuclear density . A new equilibrium, i.e. a neutron star endowed with a surface (of radius km) forms, supported by neutron degeneracy. The transformation of nuclear matter occurs following photodissociation of iron group nuclei and helium present in the core, and deleptonization of matter via weak interactions (mostly electron captures by protons) with the concomitant emission of neutrinos of all flavours. The universality of the process leads to the prediction that neutron stars at birth carry a mass close to the Chandrasekhar mass limit
The Oppenheimer Volkoff equation, which describes non-rotating neutron stars, does not admit stable equilibria above a maximum mass, , whose exact value depends on the details of the equation of state (EoS) of matter above nuclear density . If the Chandrasekhar mass limit refers to a Newtonian instability driven by microphysical processes (change in the degree of degeneracy) inside the star, the instability of neutron stars above the maximum mass limit is induced by general relativity only . What triggers the instability in a neutron star above the maximum mass is the pressure source term in the right hand side of the Oppenheimer Volkoff equation. The huge pressure (from microphysical processes) requested to counteract relativistic gravity acts as gravity source (before the degenerate neutrons become ultra-relativistic).
Rotation can contrast gravity, and uniformly rotating neutron stars can carry a mass higher than the corresponding static limit. The maximum mass of a uniformly rotating star is determined by the spin rate at which a fluid element at the equator moves on a geodesic so that any further speed-up would lead to mass shedding. This maximum mass can be determined numerically and is found to be at most larger than the non-rotating value . Only differentially rotating neutron stars can support significantly more mass than their non-rotating or uniformly rotating counterparts . Hyper-massive neutron stars are differentially rotating neutron stars with masses exceeding the maximum mass of a uniformly rotating neutron star. Because of the large angular momentum and shear, the hyper-massive neutron star is dynamically unstable to nonlinear instabilities leading to a bar mode deformation . In general, the collapse of the hyper-massive neutron star to a rotating black hole is temporarily prevented by its differential rotation, but a number of dissipative processes, such as magnetic fields, viscosity, or gravitational wave emission, will act so as to remove differential rotation. The hyper-massive neutron star can collapse directly to a black hole on the dynamical timescale. Alternatively, the star by loosing its differential rotation may evolve into a so-called supra-massive neutron star, i.e., an axisymmetric and uniformly rotating neutron star with mass exceeding the limit for non-rotating neutron stars (defined in the static limit). Magneto-rotational energy losses ultimately drive the star to collapse secularly to a black hole. In Nature, hyper-massive neutron stars likely form in the aftermath of a neutron star-neutron star merger that we will describe in .
According to many theoretical studies, there exists a range of values for the mass in the static limit, between and , due to current uncertainties in the behaviour of matter at supra-nuclear densities as shown in Figure 2. Quantum chromodynamics is expected to give a complete description of matter at the energy-densities of neutron stars. At present no unique model exists for describing the many-body ( baryons) nuclear interaction, understood as a residual coupling of the more fundamental interactions among quarks, and a phase transition to a free quark state may occur inside the star. A strange star made of strange quarks, representing the lowest energy state of matter at zero temperature, may also form in Nature .
A second example of a relativistic instability which conducts to the black hole concept is that of an equilibrium configuration dominated by radiation . This is called supermassive star, with hypothetical masses clustering around , but unknown dispersion.
Consider a supermassive star dominated by radiation, in convective equilibrium and with uniform chemical composition. Its mass is determined uniquely by the value of the photon entropy per baryon , where is the temperature of radiation and matter, the baryon number density and the radiation constant (with and the Boltzmann and Planck constants). Since a supermassive star has the structure of a Newtonian polytrope of index , supermassive stars can cover a wide mass spectrum, depending only on , and their mass is
independent on the central density . The interesting fact is that one can relate to the baryon load of the star. A simple calculation shows that , where is the matter to radiation pressure ratio. Arbitrarily large masses can be assembled for arbitrarily small values of , since . Since a polytropic relation connects the mass, radius and central density, one can compute the compactness parameter which turns out to be very small since supermassive stars are rarefied, loosely bound objects (for the radius is km) with low surface gravity owing to radiation pressure.
Although general relativity does not intervene in determining the overall structure of the star, it affects instead its stability, as a polytrope of is, in stellar structure, the trembling limit between stability and instability of the star, so it is necessary to take into account the small effect of matter pressure and general relativity which play little or no role in their structure calculation. A further important fact is that a supermassive star radiates energy at a rate very close to the Eddington limit (where is the mean baryon mass and the Thomson cross section), and therefore its total energy continues to decrease. Thus, the star evolves over time toward states or lower energy (more bound) and higher compaction. When plasma and general relativity effects are included in the stability analysis, one can show that the supermassive star loses its stability when its central density has reached a limiting value after having radiated away an energy erg, over its lifetime, a quite large value which is independent on the mass of the supermassive star. At the critical point of instability, the core temperature is large enough to ignite nuclear reactions which can affect the final fate. Once the supermassive star has reached the instability line through a progression of quasi-static equilibrium states, it can either explode or collapse to a black hole.
General relativity calculations  have shown that the nominal range of supermassive stars collapsing into a black hole lies between and . eeeThe maximum mass of a supermassive star is set by the comparison of two timescales: the thermal timescale and the timescale for the star to adjust to a new hydrostatic equilibrium, i.e. the dynamical timescale . If is shorter than the star can not any longer recover equilibrium, and rapid cooling leads the whole star to collapse. The thermal timescale at the boundary of stability equals at a mass . The collapse is homologous, with a velocity essentially linear with radius, and density profiles self-similar, although increasing in magnitude. Due to the homologous nature of the collapse, the entire mass moves inward coherently, crossing the event horizon in only a few light travel times . The concept of supermassive star has now evolved into the modern one of DCBH, acronym of a process which call for the Direct Collapse of a equilibrium structure as a supermassive star into a massive Black Hole [23, 24]. Supermassive stars, should they exist, are believed to grow via accretion onto an embryo of solar mass, in pristine, non-fragmenting gas clouds. In the context of structure formation models, they are viewed as equilibrium structures transiting through progressively more massive states (typically of ) that are conducive to stable episodes of hydrogen and helium burning. Later, they collapse into a black hole when crossing the general relativity instability limit either after exhaustion of the fuel or during the nuclear- burning phase . Ahead in this chapter we will return on this issue.
3.2 Gravitational wave sources: a first glimpse
Neutron stars and black holes are the most bound, lowest-energy states of self-gravitating matter known in the universe where gravity is in the strong field regime and the field is stationary. No processes can lower their energy state. When do neutron stars and black holes become sources of gravitational waves?
Gravity is the weakest interaction in nature but when high compactness combines with large scale, non-spherical coherent mass motions with velocity near the speed of light (as in a merger of two compact objects), then an immense luminosity can be emitted in gravitational waves, the luminosity scaling as . GW150914 is the first extraordinary example of a merger of two black holes moving at near 1/2 of the speed of light releasing a luminosity in excess of .
When perturbed out of equilibrium, during their formation or when colliding, neutron stars and black holes become often, and for a short time lapse, among the loudest sources of gravitational radiation in the universe. In particular, binary coalescences of compact objects are among the most powerful emitters that theory predicts. For binary coalescence we refer to the process of inspiral of two compact objects in a binary terminating with their merger into a new single unit, as is the case of GW150914 and GW151226.
In essence, the trait of a powerful gravitational wave source stems in its exquisite high degree of disequilibrium, leading to non-spherical dynamics under extreme conditions of compactness. Neutron stars, and black holes over a wide spectrum of masses are the protagonists of most of the violent events detectable by both current and next generations of interferometers, on Earth and in space.
The frequency of gravitational waves
Gravitational wave sources emit over a broad frequency range, and there is a close link between the frequency of the gravitational wave and mass and compactness of the source. The natural unit for is
where . In any self-bound system of mass and size , the natural frequency of oscillation, rotation, orbital revolution and dynamical collapse is of the order of
Since gravitational waves are emitted by accelerated, asymmetric mass motions, the frequency of the gravitational wave is expected to be close to the frequency of the source’s mass motions; , and in general as .
We focus now on the case of compact binary coalescences (CBCs). For black holes, whose horizon is between 2 and 1 (depending on the spin parameter ), the characteristic frequency of the wave near coalescence is , so that the total mass of the binary system determines the highest frequency of a coalescence signal. It is customary to introduce the frequency , equal to twice the Keplerian frequency of a test mass at the innermost stable circular orbit , as the characteristic frequency of a binary near coalescence. For stellar black holes with typical mass of
whereas for massive black holes
During the inspiral and merger sweeps upwards for up to . As neutron stars carry masses they can extend their gravitational wave emission at slightly higher frequencies kHz than stellar black holes.
According to the above relations, coalescing massive black holes are intrinsically low frequency sources, whereas coalescing stellar origin black holes and neutron stars combined in different arrangements are high frequency sources. In the case of a binary composed by a massive and a stellar black hole (denoted as extreme mass ratio inspiral, EMRI) the reference mass is that of the largest hole. Thus, EMRIs belong to the low frequency universe as the mass of the big black hole sets the frequency of emission.
In more detail, if is the minimum ( is the maximum) frequency of operation of an interferometer, equation (4) sets an upper (lower) limit on the mass of a binary that can be detected when nearing the final phase of plunge and coalescence. In the case of high frequency sources, this leads to
In the case of low-frequency sources
At nanoHz frequencies, the typical mass of a black hole binary near coalescence would be far in excess of . At these very low frequencies, its is possible to detect the signal from supermassive black holes of far from coalescence, i.e. at . The signal in this case is continuous and nearly monochromatic. By contrast, the signal from coalescing binaries is transient in nature and peaks during the latest phases of inspiral and plunge when the two objects reach separations comparable to their sizes: km for neutron stars, km for stellar origin black holes, and a few AU up to AU for massive black holes, depending in their mass. How this variety of coalescing binaries form in Nature is the subject of the following section. To this purpose we here overview key observational properties and key notions in the realm of current astronomical observations.
4 The electromagnetic universe
4.1 Neutron stars and stellar origin black holes in the realm of observations
Neutron stars are known to form in the aftermath of the gravitational collapse of massive stars () whose degenerate iron core is driven above the Chandrasekhar mass limit. The collapse releases erg. Most of the energy (99%) is emitted in neutrinos and only about erg into kinetic energy of the supernova explosion which is associated to the propagation of a shock wave, emerging when the infalling star’s envelope impacts on the dense neutronized core that settles into equilibrium: that of a young, hot neutron star. During shock break-out, the stellar envelope unbinds producing a luminous supernova . Crab with its remnants is a magnificent example of a successful supernova explosion. However, if the shock break-out is weaker, as for the case of heavier stars, a stellar black hole forms by fall back of part of the envelope onto the proto-neutron star driven above its maximum mass. At the extremes, direct collapse to a black hole can occur.
The mass of a compact object can be measured when it is a member of a binary system. At present, data from a variety of observations indicate that neutron stars likely show a bimodal, asymmetric distribution in their masses, with a low mass component centred around and dispersion , and a heavier component with a mass mean of and dispersion . The yellow strip in Figure 2 shows the range of neutron star masses observed in double neutron star binaries, and the red line shows the heaviest neutron star ever detected of  consistent with the expectation that neutron stars in ”binaries” experience re-cycling, i.e. a long-lived phase during which they accrete matter from the companion star. In this case, the mass of the compact object may not represent the mass at birth and gives information on the interaction of the neutron star with its companion. As shown in Figure 2, this finding already rules out the softest EoS for nuclear matter.
For stellar origin black holes, reliable dynamical mass measurements in low mass X-ray binaries are best described by a narrow mass distribution peaked around  with a clear divide between neutron stars and black holes, i.e. no remnants between and , often referred to as gap. Higher mass values are inferred in high mass X-ray binaries, and the mass of Cyg X-1, the first black hole discovered in X-rays is bound to values . The currently observed range of black hole masses is indicated in Figure 3 as lower grey strip, and the two black holes in GW151226 fall in the same range. But, the discovery of the two ”heavy” stellar black holes of and in GW150914 came as a surprise, though hypothesised by [30, 31] in their studies on binaries. Heavy stellar black holes, resulting from low metallicity progenitor stars, were considered earlier by  in the context of a class of sources known as Ultra Luminous X-ray sources, which often inhabit low metallicity galaxies.
The fate of massive stars and nature of the relic is a complex process to model. Studies by [34, 35, 36] show that the nature of the remnant depends to a large extent: (i) on the mass loss by stellar winds during the different evolutionary stages, driven by the opacity of the metals present in the star’s envelope and measured by the metallicity (defined as the logarithm in power of ten of the iron to hydrogen abundance ratio, and often expressed in units of the solar metallicity ); (ii) on rotation; (iii) on the strength of the shock break-out through the stellar envelope after core collapse and bounce; (iv) on the interplay between neutrino cooling and heating at the interface between the neutrino-sphere and the (stalled) shock; and (v) on the amount of fall-back material accreted onto the newly born hot neutron star, after formation of a reverse shock. These processes establish whether the collapse is delayed (lasting longer than 0.5 sec) or prompt (lasting less than 250 msec) and determine the value of the mass of the relic star.
Figure 3 shows the mass of the compact remnants as a function of the star’s initial mass, in the interval between [10, 100] for different values of the absolute metallicity , predicted by the models of  . The figure indicates that lower metallicity progenitor stars leave heavier relic black holes. The upper horizontal bands in the figure indicate the masses of the two black holes in GW150914 (with their uncertainties). Metallicities of the order of are necessary to form stellar origin black holes as heavy as those in GW150914, from stellar evolution models [30, 33, 37]. We further note that black holes with masses form in any metallicity environment. Thus the black holes in GW151226 do not pose constraints on the metallicity of the host galaxy [5, 38].
What is the shape and normalisation of the mass function of relic stars?
How far does the black hole mass function extend at high masses?
How can pairs of relativistic objects as those in GW150914 and GW151226 form in binaries and coalesce within the age of the universe?
Which are the astrophysical conditions for the rise of a substantial population of gravitational wave sources as GW150914?
4.2 Forming stellar origin compact binaries
Tutukov and Yungelson were the first to study the evolution of isolated massive binaries, and predicted the formation of merging binary compact objects of the different flavours [39, 40]. The formation of neutron star binaries (NS,NS) became the subject of intense studies soon after the discovery of PSR1913+16, the first binary pulsar for which we had evidence, albeit indirect, of the existence of gravitational waves [41, 42]. Formation models of neutron star-black hole (NS,BH*) and black hole-black hole (BH*,BH*) binaries have been developed in parallel despite lacking of any observational evidence. Compact binaries can form in the galactic field as outcome of stellar evolution in primordial binaries [43, 30, 31, 44]; in dense star clusters via stellar dynamical exchanges involving stars and black holes [45, 46, 47, 48]; or in more exotic environments as in the discs of active galactic nuclei .
Stellar population synthesis models are a powerful tool to establish how and in which fraction close pairs of compact objects can form in primordial binaries and coalesce within a Hubble time. The input parameters for starting a simulation are: (i) the shape of the underlying gravitational potential, (ii) the initial mass function (IMF) of massive stars on the zero-age main sequence, (iii) the metallicity of the parent gas cloud, (iv) the fraction of primordial binaries (and triplets), and (v) the distribution of the initial binary separation and eccentricity, which affect the degree of interaction of the two stars over their lifetime.
Stars lose mass via winds, but in binaries they can also donate their mass to the companion via mass exchange, as illustrated in Figure 4, which depicts the evolution of a binary system in a simplified way. Mass transfer occurs when the most massive star, which evolves first away from the main sequence, fills its Roche lobe. The pouring of mass on the companion star leads to a re-equilibration of the mass ratio and in general makes the less massive star the heaviest in the system as time evolves. After mass exchange, the star that evolves first becomes a Wolf-Rayet or a helium star (depending on the initial mass) that can go supernova. The supernova explosion can unbind the binary due to mass loss and recoil that accompany anisotropic core-collapse. Neutron stars are known to receive natal kicks at the time of their formation, with mean velocities of , so that the binary can break up. In fact, as many as of potential binaries may end up being disrupted after the first supernova explosion. This makes (NS,star) binaries very rare objects. Black holes which form either through fall back (with supernova display) or direct collapse, likely receive lower kicks but the three-dimensional distribution of their natal velocities is unknown . Thus, the weaker mass loss that may accompany their formation, and lower natal kicks may help a heavy binary to survive almost intact after the formation of the first compact object. Thus the rate of formation of (NS,star), (BH*,star) systems is not directly set by the shape of the IMF, as disruption mechanisms that break lighter binaries can limit the number of double neutron star systems that may form.
After birth of the first remnant, evolution continues, through a phase of Common Envelope evolution (depicted in Figure 4) when the second star becomes a giant and starts engulfing the companion remnant which spirals inwards via gas dynamical friction losing orbital angular momentum and energy, which is deposited as heat in the largely convective envelope. Then, the remnant star either merges plunging inside the dense core of the companion star, or lands on a close, very tight orbit after having ejected the entire envelope. In this last case, the core of the star evolves into a relic object and may go supernova, fall-back or direct collapse depending on its mass, so that binaries of all the three flavours can form. After common envelope and mass ejection, the tight binary that forms is less fragile against break up and can survive. If the two relic stars that managed to remain bound are sufficiently close (a few solar radii in separation), gravitational waves will drive the binary toward coalescence (as described in ), on timescales that may vary between a few Myr to Gyr or more. This avenue is affected by uncertainties on the common envelope evolution process, the kick distribution, and supernova modelling, so that the formation of compact binaries is a genuine statistical process and the rate of coalescences largely undetermined.
Within the scenario of primordial field binary formation, the effect of metallicity was anticipated by  who pointed out that heavy black hole binaries could form in low-metallicity environments. Figure 5 shows the distribution of (NS,NS), (NS,BH*), and (BH*,BH*) mergers as a function of the chirp mass of the binary (defined in the caption) from a population synthesis model and for two different metallicities. Depending on , whether it is solar or sub-solar, the expected number of compact binaries in all their arrangements changes dramatically, with black holes filling the high end of the mass distribution in the low metallicity channel. In the upper panel of Figure 5, we show the broad distribution of delay times as a function of the chirp mass. The delay time is defined as the time it takes a binary to coalesce since its formation as primordial stellar system. The distribution of delay times is broad, going from a Myr up to Myr. In general, population synthesis models suggest that the delay times follow a power-law distribution with slope , in the interval from 10 Myr up to Myr, i.e. a uniform distribution for logarithmic bin. In the context of primordial, isolated binaries, Belczynski et al.  find that the typical channel for the formation of GW150914 like binaries involves two very massive stars 40-100 formed in a low metallicity environment with that interacted once through stable mass transfer and once through a phase of common envelope evolution, with both black holes forming without no supernova display and low natal kick.
Recently and again in the context of primordial binaries, an alternative channel as been proposed for the origin of GW150914, named MOB (massive over-contact binary), which involves two very massive low-metallicity () stars in a tight binary which remains fully mixed due to their high spins induced by orbit synchronism driven from tides [44, 53, 50]. Rotation and tides transport the products of hydrogen burning throughout the stellar envelopes, enriching the entire star with helium and preventing the build-up of an internal chemical gradient. In this scenario there is no giant phase: both stars remain in stable contact filling their Roche lobes and eventually form two massive black holes, because the cores that collapse are massive.
Compact binaries can also form in dense, young star clusters or globular clusters and galactic nuclei, via dynamical processes . The high stellar densities, of the order of stars pc present in star’s clusters, favour the formation of black hole binaries via exchange interactions with other stars . In particular three body exchange interactions (BH*,star)+BH* (BH*,BH*)+star can lead to the built up of a population of massive compact object binaries . Being the heaviest objects in the cluster, these binaries mass segregate at the cluster centre on a timescale shorter than the two-body relaxation time, and continue to experience exchange encounters with other black holes that can further rearrange them in progressively heavier binaries, that can be as massive as GW150914 . Hardening due to scattering off stars can drive these binaries to coalesce within Gyr, and may also escape the parent cluster due to dynamical recoil [48, 47].
4.3 Massive black holes in the realm of observations
There is compelling evidence that besides stellar origin black holes, there exists a substantial population of supermassive black holes of that inhabit the centres of galaxies. This ”other flavour” is observed in two states: an active and a dormant state [55, 56].
Active supermassive black holes are accreting black holes at the centre of galaxies, which power the luminous, highly variable QSOs, and the less luminous Active Galactic Nuclei (AGN). The accretion paradigm states that outside the event horizon of a supermassive black hole, radiation is generated with high efficiency (%, higher than nuclear reactions) through the viscous dissipation of kinetic energy from gas orbiting deep in the gravitational potential of the hole. The energy escapes in the form of radiation, high velocity plasma outflows, and relativistic particles to produce luminosities of emitted over a wide spectrum and in 10% of the cases in the form of collimated radiation.
Dormant supermassive black holes appear ubiquitous in nearby bright galaxy spheroids. When dormant, their presence in inactive galaxies is revealed, albeit indirectly, through the measure of Doppler displacements in the spectral lines of stars and/or gas in the nuclear region of the galaxy. Often line-of-sight velocities show a Keplerian rise attributed to the presence of a point like gravitational potential dominating that of stars in the centre-most region of the galaxy. The Galactic Centre provides the most compelling evidence of a supermassive black hole. The Milky Way hosts a ”dark object” surrounded by a swarm of stars in Keplerian motion as close as The distance of the nearest star to the central object poses a lower limit on its compactness, found of the order of . No nuclear star cluster can remain in dynamical equilibrium at these densities, so that the black hole is the most simple and elegant hypothesis.
Figure 6 illustrates how broad is the mass distribution inferred from a sample of both dormant and active close-by supermassive black holes in galaxies of different morphology, kinematics and stellar masses . It extends from (the lightest black hole discovered in RGG118) to (the giant black hole in S50014+831). (Recently, the mass of S50014+813 as been revised downwards to : see e.g. ).fffThe mass of the black hole in bright, massive spheroids correlates with properties of the host galaxy in ellipticals and S/S0s with classical bulges. Two are the correlations, the , where is the stellar mass of the host galaxy, and a second (tighter) between and the velocity dispersion of the stars, measured far from the black hole . These correlations (often referred to as , and relations, the last shown as dashed or dotted lines in Figure 6: see  for details) state that bright galaxy spheroids with higher stellar velocity dispersions, i.e. with deeper gravitational potential wells, grow heavier black holes, and that brighter, more massive galaxies host more massive black holes. Despite being black holes tiny objects, with an influence gravitational radius extending out to pc (much smaller than the galaxy’s size of tens of kpc), black holes ”see” the galaxy they inhabit, and galaxies ”see” the central black hole they host. Consensus is rising that the relation is fossil evidence of a symbiotic co-evolution of black holes and bright spheroids. Most likely, the relation was established during the course of galaxy formation and assembly in episodes of self-regulated accretion and mergers when the black holes were active, creating a balance between accretion flows with their radiated power and gas at disposal for triggering both/either star formation and accretion. The correlation is poor instead when extended to a larger sample of galaxies types and galaxy masses, indicating that particularly in lower mass systems co-evolution never got to completion, or never started.
From the study of the kinematics of stars and gas in nearby galaxies, one can estimate the black hole local mass density: This mass density is remarkably close to the mass density increment that black holes experience over cosmic history (between ) due to efficient accretion [9, 55]. This last value is inferred considering that active black holes in galaxies contribute to the rise of the cosmic X-ray background resulting mostly from unresolved, obscured AGN of mass - . As the contribution to the local black hole mass density results from black holes in the same mass range, the close match between the two independent measures, and , indicates that radiatively efficient accretion () played a large part in the building of the mass of the 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. Thus, QSOs and AGN are believed to emerge from a population of seed black holes with masses in a range largely unconstrained (from up to ). This is because the mass of black holes increases sizeably due to accretion, over a relatively short -folding timescale yr compared to the age of the universe (where gives the luminosity in units of the Eddington luminosity, and the fraction of mass accreted by the hole in order to radiate a luminosity with efficiency ).
The spin of a black hole is also a key parameter in the context of gravitational wave astronomy, as together with the mass it can be inferred from the rich structure of the waveform (as illustrated ahead in this Chapter). Mass and spin are strongly coupled across the accretion history of a growing black hole. Spins determine directly the radiative efficiency, and thus also the rate at which the black hole mass is increasing. In radiatively efficient accretion discs, the efficiency varies from 0.057 (for ) to 0.15 (for ) and 0.43 (, for a maximally rotating black hole). Accretion on the other hand determines black hole’s spins since matter carries with it angular momentum (the angular momentum at the innermost stable circular orbit ). A non rotating black hole is spun-up to after increasing its mass by a factor , for prograde accretion. ggg Gas accretion from a geometrically thin disc limits the black-hole spin to , as photons emitted with angular momentum anti-parallel to the black hole spin are preferentially captured, having a larger cross section. In a magnetised, turbulent thick disc, the spin attains an equilibrium value . Conversely, a maximally rotating black hole is spun-down by retrograde accretion to , after growing by a factor .
The direction and norm of the black hole spin play a key role in the study of the spin-mass evolution of black holes. In a viscous accretion disc whose angular momentum is initially misaligned with the spin of the black hole, Lense-Thirring precession of the orbital plane, acting on the fluid elements, warps the disc forcing the gas close to the black hole to align (either parallel or anti-parallel) with the spin vector of the black hole. The timescale for warp propagation is very rapid and the warp extends out to rather large radii . Following conservation of total angular momentum, the black hole responds changing its spin direction . The spin starts precessing and the system evolves into a configuration of minimum energy where and are aligned and parallel, if . Black hole precession and alignment occur on a timescale shorter than the - folding accretion time scale (typically yr). If accretion tends to spin the black hole up after re-orienting the black hole spin. By contrast heavier black holes for which oppose more inertia and the spin direction does not suffer major re-orientations [64, 65].
Two limiting scenarios for the spin evolution have been proposed: Coherent accretion refers to accretion episodes from a geometrically thin disc lasting for a time longer than the black hole mass growth -folding time , bringing its spin up to its limiting value , and with parallel to . By contrast, chaotic accretion refers to a succession of accretion episodes that are incoherent, i.e. randomly oriented with . The black hole can then be spun-up or down, depending on the comparison between and . If accretion proceeds via uncorrelated episodes with co-rotating and counter-rotating material equally probable, the spin direction continues to change. 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 ( for ) than that of a co-rotating particle ( for ), and accordingly carries a larger orbital angular momentum. If chaotic accretion results in low spins .
The two scenarios of coherent and chaotic are at the extremes of a wide distribution of angular momenta for the accreting gas which is determined by the kinematic and dynamical properties of gas and stars at the galaxy’s centres which change over cosmic time as galaxies are not isolated systems.
At present the spin moduli of a handful (20) of AGN, hosted in low redshift late type galaxies, has been measured through the spectra of relativistically broadened iron lines, and are reported in Figure 7. The data points are then compared with a hybrid model by  which follows the joint evolution of the mass and the spin vector by precession and accretion, of a simulated population of growing black holes in late type (spiral) galaxies.
Mass and spin are directly encoded in the gravitational wave signal emitted during the merger of massive black holes, and mergers are detectable with space-borne detectors out to very large cosmological distances. Therefore, measuring the masses and spins of coalescing black holes over cosmic time will offer unprecedented details on how they have been evolving via repeated episodes of accretion and mergers.
4.4 The black hole desert
There is a black hole desert in the mass range between , the mass of the remnant black hole in GW150914 (the highest known as of today)
and the mass of the lightest supermassive black hole known at the centre of the dwarf galaxy RGG118, of , as depicted in Figure 8. Key questions arise that the new gravitational wave astronomy can answer:
Is the desert real, i.e. empty of middle sized black holes, or is the desert inhabited by black holes which we still do not detect?
If inhabited, is the desert populated by transition objects, resulting from the clustering/accretion of stellar black holes viewed as single building blocks?
Or is there a ”genetic divide” between stellar origin black holes and massive black holes growing from seeds of unknown origin?
What is the maximum mass for a stellar origin black hole?
Is there a gap in the mass function of stellar origin black holes induced by pair-instability supernovae?
(Here a maximum mass for a stellar origin black hole is intended not as fundamental mass limit (as in the case of neutron stars) but as a value related to the existence of a limit on the maximum mass of stars on the zero-age main sequence [67, 68, 69].)
To elaborate more on the above questions we notice that there is a conceptual distinction between stellar origin black holes and supermassive black holes: the first are the relic of the very massive stars that experienced stable and long lived episodes of nuclear burning. The second are possibly the relic of rare supermassive stars that may never experienced long-lasting phases of nuclear burning and which formed in rather extreme, isolated environments. There is also a ”morphological” distinction: stellar black holes (typically more than several millions per galaxy) are spread everywhere in all the galaxies of the universe, as stars are. In addition, they continue to form as stars do, in the galaxies. Instead, massive black holes (from the middleweight size to the giants) are found at the centres of galaxies (perhaps not in all), as single dark massive objects, and may have formed at early cosmic epochs or over a narrower interval of cosmic times, when the first galaxies were forming and assembling . If the desert is devoid of objects, this would unambiguously indicate that the physical conditions leading to the two flavours are distinct. Locating the dividing line is not easy.
Massive stars with sub-solar metallicities and masses between and explode as pair instability supernovae [34, 71, 69]. The pair instability is encountered when, late in the star’s life, a large amount of radiative energy is converted into electron-positron pairs which reduce the pressure support against gravity, cause rapid contraction of the core and trigger explosive burning of the CO core, ultimately leading to the disruption of the star. Thus, if the IMF is devoid of stars with mass in excess of , pair instability supernovae may actually limit the value of the maximum mass of a stellar origin black hole .
In the four panels of Figure 9, the mass of the relic black holes (red line) is plotted versus the initial mass of the progenitor star, for two selected values of the metallicity, and respectively, for both non-rotating stars and rapidly rotating stars (with velocity ). At sub-solar metallicities the maximum mass of the relic black hole coasts around values almost independently on rotation and on metallicity, provided is sub solar (). As shown in the bottom right panel, rapid rotation limits again and this occurs at lower progenitor masses . During core He burning, rotation driven mixing causes diffusion of matter from the He convective core into the surrounding radiative zones. Such an occurrence has the consequence of increasing the CO core mass at core He depletion and therefore of reducing the limiting initial mass that enters the pair instability regime. If the IMF is top heavy and contains stars in excess of , a gap between and (the He mass inside a 260 star ) should appear in the mass function of (single) stellar origin black holes, as anticipated in . Advanced LIGO and Virgo will likely help to shed light into this problem, if/when ”heavy” stellar binary black hole mergers will be further discovered.
In the logic of a ”genetic divide”, supermassive stars likely play a key role. The formation of radiation dominated equilibrium states with masses , evolving into a DCBH, i.e. a configuration collapsing directly into a black hole, requires rather extreme conditions to form, (i) namely pristine, metal free gas clouds (site of the forming supermassive star) irradiated by an intense flux of ultraviolet radiation to promote the dissociation of the main coolant, i.e. molecular hydrogen, and avoid fragmentation [23, 24, 25]; or/and (ii) the coherent collapse of massive gas clouds in major galaxy mergers. These scenarios would lead to a clean divide between the two black hole flavours, but Nature appears to be more complex and continuous.
There is indeed the possibility that the desert is filled of transition black holes. Processes of aggregation might have been in action, e.g. using as single building blocks stellar origin black holes which merge inside nuclear star clusters . Alternatively, episodes of supercritical accretion may drive stellar origin black holes to grow up to ”seed” sizes of when residing in gas rich disky galaxies. Photon trapping ensures that the momentum from outgoing radiation does not feed back to halt accretion which can continue until fuel exhaustion. In these cases, the desert zone would be filled by transition black holes with a broad range of masses between 100 and 1000 or more.
Runway collisions among massive stars, in young, dense nuclear star clusters at the centre of unstable proto-galactic discs [77, 78, 45] may also lead to middleweight black holes. The resulting runaway super-massive star, product of the multiple merger of stars (typically of mass ) may evolve into a quasi-star, i.e. a black hole surrounded by a massive accretion envelope , ultimately forming a black hole of intermediate mass. Detecting black hole coalescences of different flavours with both Earth and space-based detectors will enable us to shed light into this complex problem.
4.5 Formation of gravitational wave sources: the cosmological perspective
In this short paragraph, we show that the existence of close pairs of massive black holes fated to coalesce is a key, unescapable prediction of the process of clustering of cosmic structures, and that the formation of close binary systems comprising stellar origin black holes and neutron stars has a natural connection with the overall star formation history in the universe. The progenitor stars of GW150914 may have formed in a binary Gyr after the Big Bang (at ) in a metal poor environment, according to .
A plethora of observations show that today the energy content of our expanding universe is dominated by dark energy (68.3%), and by cold dark matter (CDM, 26.8%), with baryons contributing only at 4.9% level , and that the present spectrum of primordial density fluctuations contains more power at lower masses. At the earliest epoch, the universe was dominated by small scale density fluctuations. Regions with higher density grow in time to the point where they decouple from the Hubble flow and collapse and virialize, forming self-gravitating halos. The first dark matter halos that form grow bigger through mergers with other halos and accretion of surrounding matter along cosmic filaments. This is a bottom up process which leads to the hierarchical clustering of dark matter sub-structures and of the luminous components, the galaxies .
At present, most of the investigations of galaxies and of QSOs in the electromagnetic universe
feature the occurrence of three main epochs of evolution, along cosmic history :
The cosmic dawn which is the epoch extending from cosmic redshift when the universe was only a few Myr old to redshift , corresponding to Gyr. During this epoch, baryons in dark matter halos of begin to collapse and the first stars form and first seed black holes. Planck data indicate that between and 8.8 the universe completed the phase of cosmic re-ionisation of gas turning intergalactic neutral hydrogen into a hot tenuous plasma . At the limits of current capabilities, GRB 090423, the farthest long GRB observed (which signals the formation of a stellar origin black hole), exploded at , when the universe was 520 Myr old, and the most distant galaxy MACS0647-JD 420 and the most distant QSO ULAS J1120+0641 are found at and , 420 and 770 Myr after the big bang, respectively . These brightest sources are just probing the tip of an underlying distribution of fainter early objects for which little is known and which represent the building blocks of the largest structures.
The cosmic high noon follows, which is an epoch of critical transformations for galaxies, extending from to . Around , the luminous QSOs and the cosmic-integrated star formation rate have their peak. This is illustrated in Figure 10 where we show the cosmic-averaged star formation rate per unit comoving volume (in units of ) and the massive black hole accretion history (in the same units but enhanced by a factor 3,300 to help the comparison) as function of lookback time and redshift. Galaxies and seed black holes are expected to grow fast in this epoch which erases information of their properties at birth. In between redshift and 2, galaxies acquires about 50% of their mass, and widespread star formation can lead to the build up of populations of (NS,NS), (NS,BH*) and (BH*,BH*) fated to coalesce over cosmic time, and accessible to forthcoming and future observations.
The last epoch of cosmic fading traces a phase where star formation in galaxies, and QSO’s activity in galactic nuclei are both declining. It is a phase of slow evolution extending from to the present. Observations of galaxies and AGN give a description of a quieter universe where dormant supermassive black holes lurk at the centre of bright elliptical galaxies likely formed through galaxy mergers. Less massive (dwarf) galaxies in the near universe have undergone a quieter merger and accretion history than their brighter analogues (which formed earlier). They represent the closest analogue of lower mass high redshift dark matter halos from which galaxy assembly took off during cosmic dawn. Local, dwarf galaxies are the preferred site for the search of middleweight (or intermediate) black holes of . NGC 4359, a close-by bulgeless disky dwarf, houses in its centre a black hole of only . This indicates that nature provides a channel for the formation of middleweight black holes also in potential wells much shallower than that of the massive spheroids, and these galaxies are expected to host a class of gravitational waves sources, known as Extreme Mass Ratio Inspirals (EMRIs) that we will discuss ahead in this Chapter.
A number of important questions can be posed in the context of galaxy formation and evolution that the gravitational universe will try to answer:
When did the first black hole seeds form? Did they form only during cosmic dawn, i.e. over a limited interval of cosmic time?
How does the black hole mass and spin distribution evolve with cosmic time?
To what extent mergers affect the cosmic evolution of massive black holes?
4.6 Massive black hole binary mergers across cosmic ages
During cosmic dawn and high noon, the bottom-up assembly of galactic halos through galaxy mergers inevitably lead to the growth of an evolving population of
binary black holes in a mass range between . These are the target sources of the upcoming LISA-like observatory, in space.
When two galaxies with their dark matter halos merge, the time-varying gravitational field induced by the grand collision redistribute the
orbital energy of stars and gas discs in such a way that a new
galaxy with new morphology forms. At the same time, the black holes nested at the centres of the interacting galaxies have a long journey to travel before entering the phase of
gravitational driven inspiral . They experience four critical phases covering more than 10 orders of magnitude in dynamic range:
(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) when the stellar/gas mass enclosed in their relative orbit is comparable to the sum of
the black hole masses. In this phase, the two galaxies first sink by dynamical friction against the dark matter background to form a new
galaxy dragging the two black holes at the centre of the new system. Then, the black holes experience, as
individual massive particles, dynamical friction against the stars/gas and continue to spiral in and sink.
(2) The binary or hardening phase, when single stars scattering off the black holes extract tiny amount of their orbital energy and angular momentum. If present in large numbers, the binary continues to contract. In gas-rich galaxies, torques from a circum-binary gaseous disc surrounding the binary can also lead to hardening.
(3) The third phase of gravitational wave driven inspiral starts when the black holes get so close (typically at around or below pc) that they detach from their nearest environment, and gravitational waves dominate the loss of energy and angular momentum driving the binary to coalescence.
(4) Finally the new black hole that formed may experience a recoiling phase since gravitational waves carry away linear moment. Gravitational recoil velocities are between and . Thus the new black hole can either oscillate and sink back to the centre of the relic galaxy, or escape the galaxy. Only state-of-the-art numerical simulations can describe this long journey that begins at 10 kpc scales and ends when the two black hole coalesce, typically on scales of pc. The delay between the galaxy merger and black hole merger varies from Myr to many Gyrs . Figure 11 from  shows the three phases of a merger of two galaxies belonging to a group from a cosmological simulation (see the caption for details).
The merger of black holes in pristine dark matter halos is even more difficult to simulate as the dynamics is dominated by the gas and this requires use of self-consistent high resolution hydrodynamical cosmological simulations with rich input physics (chemistry network, cooling and radiative transport, turbulence and magnetic field dissipation) over a wide dynamical range. Preliminary studies indicate that the black hole dynamics is stochastic , implying a rather broad range of sinking timescales.
4.6.1 Reconstructing the cosmic evolution of massive black hole binary coalescences across the ages
Given a mass distribution of black hole seeds, a cosmological model for the growth and assembly of dark matter halos, and an accretion recipe, one can infer the merger rates of massive black holes. In Figure 12 we show the merger rate per redshift bins of black holes as a function of , for a variety of models of black hole seed formation, from Pop III stars to relic of supermassive stars collapsing as DCBH, computed using a Monte Carlo merger tree synthesis model within the extended Press and Schecter formalism for the assembly of galaxy halos . The uncertainties are large with merger rate excursions of about two orders of magnitude, ranging from ten to several hundreds events per year. Each halo had experienced few to few hundred mergers in its past life, placing mergers among the critical key mechanisms driving galaxy evolution.
5 The sources of the gravitational wave universe
Here we list the prospected sources of the gravitational wave universe, based on two criteria: the distinction between high and low frequency sources, and between short duration, transient and continuous sources. The concept of backgrounds is shortly introduced.
5.1 The high frequency gravitational universe
The sources of the high frequency universe, observed with ground-based detectors at frequencies between Hz and to a few 1000 Hz ( and Hz with the Einstein Telescope under design), can be grouped into four basic classes: compact binary coalescences, un-modelled bursts, continuous waves, and stochastic backgrounds[8, 7, 10]. These groups refer to different astrophysical settings and different algorithms for their detection.
Compact Binary Coalescences - CBCs refer to binaries hosting the relics of massive stars and comprise (NS,NS),(NS,BH*) and (BH*,BH*) binaries. CBCs are transient sources. They are detectable at the time of their coalescence, as they emit a sizeable fraction of their reduced-mass-energy, and have a modelled signal. (NS,NS) binaries are characterised by mass ratios (with ) close to , as observed in double neutron star binary systems, and a lower limit is . The mass ratio of (NS,BH*) and (BH*,BH*) binaries is less constrained, since we do not know the maximum mass of stellar black holes nor how they pair in binaries. GW150914 and GW151226 have mass ratio and , respectively [2, 5]. GW150914 and GW151226 are expected to be the first two of a rich population of CBCs of different flavours that will be observed in the forthcoming Advanced LIGO and Virgo science runs. In and we describe in depth CBCs and their expected signal.
Horizon luminosity distance - A key fact that makes binaries important sources is that the amplitude of the emission is calibrated just by a combination of the two masses (to leading orders). Given this, their detectability can be expressed in terms of the horizon luminosity distance for a detector, defined as the distance at which a detector measures a signal-to-noise ratio (SNR) of 8 for an optimally oriented (face-on) and optimally located binary. Figure 13 indicates this distance reach for Advanced LIGO in three of its design configurations. At present the distance reach for (NS,NS) binaries with Advanced LIGO and Virgo is Mpc, and Mpc for (BH*,BH*) binaries. At design sensitivity Advanced LIGO can detect neutron star binaries out to a distance of Mpc scales, and black hole binaries as GW150914 out to Gpc, as shown in the figure. Black holes of intermediate mass can also be detectable, if they form in binaries over this mass range. We remark that all these binaries would not be detectable otherwise (neutron star binaries are observed as pulsars only in our Milky Way). The third generation of Earth based detectors as ET will be able to detect (NS,NS) out to redshift corresponding to a distance of Gpc, and (BH*,BH*) as GW150914 out to a redshift (47 Gpc) allowing ET to explore binary populations at cosmological distances, at the end of the cosmic dawn and during high noon.
Expected coalescence rates - Prior to the discovery of GW150914, the rate of CBCs relied entirely on theoretical population synthesis models and dynamical models, and for (NS,NS) binaries on constraints derived from electromagnetic observations . The rates with their large uncertainties are in Figure 14. With the discovery of GW150914, the rates for (BH*,BH*) binaries now fall in the conservative range of and Mpc yr (we defer to  and arXiv:1606.04856v1).
Unmodelled Bursts refer to short-duration events caused by a sudden change of state in the source that do not have a near-universal waveforms. Here we outline their key features.
Core collapse supernovae (CCSNae) and hot remnants belong to this class [94, 95]. In CCSNae, among the most powerful explosions in the electromagnetic universe, the available energy reservoir of erg is set by the difference in gravitational binding energy between the pre-collapse iron core and the collapsed neutron star remnant. Much of this energy is initially stored as heat in the proto-neutron star and most of it () is released in the form of neutrinos, in kinetic energy of the explosion, is emitted in radiation across the electromagnetic spectrum, and an uncertain fraction is expected to be emitted in gravitational waves. Electromagnetic observations of CCSNae yield secondary observables, such as progenitor type and mass, explosion morphology and energy, and ejecta composition. By contrast, gravitational waves, much like neutrinos, are emitted from the innermost region (the core) of the CCSN and thus convey primary, direct live information on the dynamics of the core collapse and bounce. They potentially inform us not only on the general degree of asymmetry in the dynamics of the CCSN, but also more directly on the explosion mechanism, the structural and compositional evolution of the proto-neutron star, the rotation rate of the collapsed core, and the state of nuclear matter.
The violent dynamics in CCSNae and (possibly) in long GRBs (resulting from the collapse of rapidly rotating low-metallicity massive stars, dubbed as collapsars) is expected to give rise, if aspherical, to low amplitude bursts of gravitational waves with typical durations from milliseconds to seconds, over a wide frequency range, between 50-1000 Hz. The bursts have no universal features as gravitational wave emission is influenced by the stochastic dynamics driven by the richness in the input physics that accompany the infall of matter and its bounce. Many multi-dimensional processes may emit gravitational waves during core collapse and the subsequent post-bounce CCSN evolution. In we select a few mechanisms that lead to the emission of gravitational waves, and show the shape of the signal.
The proto-neutron star that forms at the end of a CCSN is a hot and rapidly evolving object. After the first tenths of seconds of the remnantÃs life, the lepton pressure in the interior decreases due to extensive neutrino losses, and the radius reduces to about 20-30 km. The subsequent evolution is Ãquasi-stationaryÃ, and can be described by a sequence of equilibrium configurations. In these states the hot star can display a rich spectrum of non-radial normal modes, which can excite emission of gravitational waves in narrow intervals around the characteristic frequency of the mode, and extending over times comparable to the damping timescale of the excited oscillation mode . CCSNae should be visible throughout the Milky Way with enhanced interferometric detector technology, while third generation observatories may be needed to explore events at a few Mpc, out to which the integrated CCSNae rate is yr. Detecting gravitational waves in coincidence with optical, X-ray, -ray radiation or neutrinos could give unprecedented insight into stellar collapse.
Pulsar glitches are also expected to fall in this category. Glitches are enigmatic spin-up events seen in (mainly) relatively young neutron stars like Crab and Vela. A glitch is a sudden increase (up to 1 part in ) in the rotational frequency of a pulsar. Following a glitch is a period of gradual recovery to a spin close to that observed before the glitch, due to braking provided by the emission of high energy particles and electromagnetic radiation. These gradual recovery periods have been observed to last from days to years. Currently, only multiple glitches of the Crab and Vela pulsars have been observed and studied extensively. The energy of these events is erg, i.e. of the order of , which set a benchmark energy level for the emission in gravitational waves by pulsars. These events are likely to be within reach of ET, but still too weak for Advanced LIGO and Virgo, and are observable only in the Galaxy.
Magnetar flares could be important sources of gravitational waves. Magnetars are associated to the high energy phenomena known as Soft Gamma Repeaters and Anomalous X-ray Pulsars. These sources host slowly spinning, isolated neutron stars endowed by ultra strong magnetic fields and whose emission is powered by the release of magnetic energy. On December 2004 a giant flare has been observed in SGR 1806-20 which released erg in high energy radiation, implying an internal magnetic field strength of Gauss. To explain this powerful emission, models require a substantial deformation of the neutron star in a direction non coincident with its spin axis. The newborn fast spinning magnetar may radiate for a few weeks gravitational waves at frequencies around a kHz, and may constitute a promising new class of gravitational wave emitters, visible once per year from galaxies in the Virgo cluster, out to a distance of 16 Mpc .
Asteroseismiology of neutron stars is (at least in principle) a promising avenue for studying neutron star interiors [96, 98]. Neutron stars have a rich oscillation spectrum associated to non-radial normal modes with frequencies in the kHz regime. They can be excited in different evolutionary phase, e.g. in rapidly and differentially rotating hot proto-neutron stars, or in old neutron stars recycled in binaries whose accreting layers are sites of repeated nuclear explosions that produce X-ray flares. In this last case, the rapid rise times of these instabilities may excite acoustic vibrations. If the rise time matches the period of a mode, than a substantial fraction of the energy released can be channeled into mechanical vibrations and a large fraction of this energy could be carried away by gravitational waves, when other mode-damping mechanisms (e.g. viscosity) are less efficient.
Continuous wave source are generally persistent sources which produce signals of roughly constant amplitude and frequency, i.e. varying relatively slowly over the observation time. A number of mechanisms may cause the neutron star to emit a continuous signal. These include deformations generated either by strains in the star’s crust or by intense magnetic fields, precession, and long-lived oscillation modes of the fluid interior. Target sources for this type of emission are the rotation powered neutron stars in the Milky Way .
More than 2000 radio pulsars have been detected for which the sky location and frequency evolution have been accurately measured. Among them, several tens have spin frequencies greater than 20 Hz so that they are in the Advanced LIGO and Virgo bandwidth reach. In the search of the gravitational wave signal, pulsars are assumed to be triaxial stars emitting gravitational waves at precisely twice their observed spin frequencies (i.e. the emission mechanism is an quadrupole), with the wave phase-locked with the electromagnetic signal. No signal has been reported so far from targeted pulsars. This null result can therefore be interpreted as upper limit on the strength of the gravitational wave emission, and thus as upper limit on the level of asymmetry seeded in the star’s equilibrium structure. Theoretical modelling of bumpy neutron stars has mainly focused on establishing what the largest possible neutron star mountain would be . Expressing this in terms of a (quadrupole) ellipticity, detailed modelling of crustal strains suggest where is the crustal breaking strain. State-of-the-art calculations indicate that solid phases may also be present at high densities, allowing the construction of stars with larger deformations. The magnetic field also tends to deform the star. For typical pulsar field strengths the deformation is , but it can be larger by a factor if the core is super-conducting with critical field strength of G .
An observational milestone was reached recently, when the LIGO and Virgo data analysis from the first nine months of the S5 science run was carried on to beat the Crab pulsar spin-down limit. It was found that no more than of the spin-down energy was being emitted in the gravitational wave channel. This limit already indicates that Crab is not a maximally strained quark star for which larger ellipticities are allowed [100, 99, 98].
Stochastic backgrounds can arise from a large population of weak sources, so that there are many comparable strength signals with overlapping frequencies in each resolvable frequency bin. In the high frequency regime, the likely sources would be a population of inspiralling binaries at much greater distances than the resolvable CBCs. With the LIGO detection of GW150914 there might exist a population of ”heavy” binary black holes with mass above contributing to the stochastic gravitational wave background at a level higher than previously expected from CBCs .
A stochastic background can also arise from the primordial gravitational waves produced at the inflationary epoch. The standard cosmological model places this background at even lower levels than the expected foreground from unresolved binaries, but alternative models can produce strong cosmological backgrounds in different frequency bands. Consequently, non-detections can place meaningful constraints on alternative cosmological models.
5.2 The low frequency gravitational universe
The milli-Hz frequency range (between 0.1 mHz and 0.1Hz) will be probed by space based interferometers such as eLISA [102, 9]hhheLISA or ’evolving’ LISA is, at the time of writing, a descoped version of the original LISA mission, to be flown by ESA in the early 2030s. For the purpose of our general discussion, there is no need to differenciate between the details of the two concepts. Detailed eLISA desings under consideration can be found in . and is usually referred to as the low frequency universe. This is expected to be by far the richest window in terms of number, loudness, distance reach and diversity of sources, including massive black hole coalescences (MBHCs), extreme mass ratio inspirals (EMRIs), galactic and extragalactic binaries of stellar mass compact objects, and more.
Massive black hole coalescences (MBHCs) are binaries resulting from the collision and merger of galaxies, and are detected at the time of their coalescence. Figure 15 shows that eLISA will observe signals coming from MBHCs in the mass range between and , with typical binary mass ratios , out to redshift (if they already exist) corresponding to a luminosity distance of Gpc and an age of the universe of 180 Myr. Overlaid to constant signal-to-noise ratio contours are mass-redshift evolutionary pathways ending with the formation of a supermassive black hole representing (i) an analogue of SgrA (the black hole at our Galactic centre); (ii) a typical quasar at ; and (iii) two distant quasars at . White dots mark merger events and highlight the fact that any massive black hole we observe in bright galaxies today has grown cutting through the eLISA sensitivity band. The forthcoming LISA-like observatory will therefore provide an highly complete census of MBHCs throughout the universe.
Expected rates - The expected rate of MBHCs is weakly constrained as it depends on the occupation fraction of (seed) black holes in haloes as a function of redshift, on their mass distribution (as depicted in Figure 12), on their accretion history, and on the pairing and hardening efficiency inside the new galaxy that has formed [89, 103, 85]. Cosmological simulations of the galaxy assembly anchored to estimates of the local galaxy-merger-rate predict a few to few-hundred coalescences per year . Mergers are inevitable in a hierarchical universe, and whatever is the route to the massive black hole build-up, eLISA will provide a unique window to test MBHCs. MBHCs pinpoint places where galaxy mergers occur and in the eLISA band-width they inform us on the evolution of massive black holes in the low mass end of their distribution extending down to the desert zone.
Physics and astrophysics with precision gravitational wave measurements - In virtue of the extremely high signal-to-noise ratio of most of the events, MBHC parameters will be extracted with exquisite precision . Individual redshifted masses can be measured with an error of , on both components. Even more interestingly, the spins of two massive black holes can be determined to an absolute uncertainty down to 0.01 in the best cases. This is a critical measurement, because the efficiency of accretion and mass growth of MBHs strongly depends on their spins which are currently difficult to determine through electromagnetic observations  (see ).
The distinctive high signal-to-noise ratio of MBHCs will allow black hole ”spectroscopy” i.e., the direct measure of several frequencies and damping times associated to the quasi-normal modes present in the ringdown signal of the newborn massive black hole [105, 106]. This will make it possible to carry on direct precision tests of the no-hair theorem. Violations of general relativity predictions may indicate new physics or the presence of exotic dark objects such as, e.g. boson stars that carry a surface . The comparison between spectroscopy measurements from the LIGO-Virgo data on (BH*,BH*) coalescences and those from LISA  on MBHCs mapping the heaviest holes will be of enormous value: the proof of the universality of black holes over a mass range of more than six orders of magnitude.
Constraining the massive black hole cosmic history - While individual MBHC measurements will allow exquisite tests of general relativity and will probe several distinctive feature of massive black hole physics, information on the astrophysical evolution is encoded in the statistical properties of the observed population. As first illustrated in , observations of multiple MBHCs can be combined together to learn about their formation and cosmic evolution. In particular the mass distribution of the ensemble of observed events encodes precious information about the nature of the first seeds, whereas the spin distribution will constrain the primary mode of accretion that grows them to become supermassive .
Cosmography - Another peculiar property of MBHCs is that their luminosity distance can be directly measured as it is encoded in the gravitational wave signal, and its estimate does not involve cross-calibrations of successive distance indicators at different scales (as the distance ladder in the electromagnetic universe) since the gravitational wave luminosity of MBHCs is determined by gravitational physics, only. Thus MBHCs are standard sirens (we defer to for an exact definition). A LISA like interferometer can provide the distance to the source to a stunning few percent accuracy. If an electromagnetic counterpart to the MBHC event can be observed [109, 110], it will make it possible to reconstruct the luminosity distance versus redshift relation, as shown in Figure 16 offering the possibility of measuring the Hubble parameter at the level of , and of inferring bounds on the dark matter and dark energy content of the universe .
Extreme mass ratio inspirals (EMRIs) describe the inspiral and possibly the plunge of stellar mass compact objects into a massive black hole at the centre of a galaxy [111, 9]. EMRIs still fall in the class of ”binaries” despite their small mass ratio .
EMRI flavours and expected rate - Massive black holes in galactic nuclei are surrounded by a swarm of stars and compact objects. The densities can be as high as stars pc. In such extreme environments, stars are easily deflected on very low angular momentum orbits, owing to repeated, distant stellar encounters and thus can enter the massive black hole sphere of influence. The fate of main sequence stars on such ”plunging” orbits is to be tidally disrupted . But, compact objects such as neutron stars, stellar black holes and white dwarfs (for central black holes of ) can be captured in extremely eccentric orbits, with periastron of , avoiding disruption. Their orbit will then slowly circularise because of gravitational wave emission and the slow inspiral can in principle lead to observable EMRI signals. In general, stellar black holes are expected to dominate the observed rate for a LISA-like detector. This is because dynamical mass segregation tends to concentrate the heavier compact stars nearer the massive black hole [113, 114, 115], and because black hole EMRIs have higher signal-to-noise ratio, and so can be seen out to a much larger distance, typically of few Gpc (). Their expected rate is uncertain due to the currently poor knowledge on the low mass end of the massive black hole mass function in galaxies (at ) and to the large uncertainties on the properties of typical compact object distributions in galactic nuclei. In general, a Milky Way type massive black hole is expected to form an EMRI every 10 Myr, implying a detection rate for a LISA-like mission in the figure of hundreds per year [116, 9, 117]. There is, however, at least a factor of 100 uncertainty on this number.
Astrophysics and fundamental physics with EMRIs - High rates imply large astrophysical payouts following detection. The number and mass distribution of EMRIs will inform us about the unconstrained low end of the mass function of massive black holes and on the dynamics of compact objects in the dense environment of galactic nuclei on scales that are impossible to probe otherwise. The requirement of matching hundreds of thousands of cycles to dig out the signal from the data stream, implies that detections will automatically come with exquisite parameter estimation . Figure 17 shows that the mass of the two black holes and the spin of the massive black hole can be determined generally to better than a part in ten thousand, a precision that is unprecedented in astronomical measurements. This will make it possible to perform massive black hole population studies on a sample of relatively low redshift, quiescent black holes, complementary to the higher redshift, merging systems seen as MBHCs. EMRIs ensure that the inspiralling object essentially acts as a test particle in the background space-time of the central massive black hole. As such, the hundreds of thousands of wave cycles collected at the detector encode a very precise mapping of the stationary spacetime metric of the central massive black hole, providing the ultimate test of its Kerr nature, complementary to the ringdown one possible with MBHCs. As shown in Figure 17 deviations as small as 0.1% from the Kerr mass-quadrupole moment will be detectable for typical EMRIs, pushing testing of spacetime metric to a whole new level.
Continuous sources comprise double white dwarfs in binaries (WD,WD), and possibly (NS,NS) and (BH,BH), in the Milky Way emitting a nearly monochromatic signal, preferentially located at the low frequency end of the eLISA sensitivity interval . A number of (WD,WD) binaries are already known to emit a nearly monochromatic signal in the eLISA band since they have been discovered in the electromagnetic window, and are known as verification binaries. The discovery of new (up to a few thousand for a two-year mission) ultra-compact binaries with orbital periods below one hour and typically 5 to 10 minutes, determined from the periodicity of the gravitational wave, is one of the main objectives of a LISA-like mission. For a number of systems it is possible to measure the first time derivative of the frequency, and thus determine a combination of the masses of the two component stars that can be used to distinguish white dwarf, neutron star and black hole binaries. This will give precious insight on the distribution of the binaries in their different arrangements and flavours, present in the thin and thick discs of our Galaxy as well as in the halo and inside globular clusters. The highest signal-to-noise-ratio systems will allow us to study the complex physics of white dwarf interactions in binaries and to establish how systems survive as interacting binaries. We recall that (WD,WD) binaries are considered to be potential progenitors of Type Ia supernovae [118, 9].
Galactic foreground describes the signal coming from an unresolved population of million compact binaries emitting each a nearly monochromatic gravitational wave, which are confined in the thick disc of the Milky Way, preferentially (WD,WD) binaries which create a confusion-limited noise at frequencies below a few mHz, as illustrated in Figure 18. Its average level is comparable to the instrument noise, but due to its strong modulation during the year (by more than a factor of two) it can be detected. The overall strength can be used to learn about the distribution of the sources in the Milky Way.
Cosmological background refers to the signal(s) coming from the primordial universe. The frequency band of a LISA-like detector corresponds to 0.1 to 100 TeV energy scales in the early universe, at which new physics is expected to become visible. We defer to  for a authoritative description of the potential sources of the primordial universe.
5.3 The very low frequency gravitational universe
Moving further down in frequency, we enter the very low frequency universe probed by Pulsar Timing Array experiments EPTA , PPTA , NANOGrav  and the International PTA (IPTA), which are especially sensitive in a window extending from Hz to Hz. Arrays of millisecond pulsars can be used to detect correlated signals such caused by passing gravitational waves. The dominant contribution at these frequencies is expected to come from supermassive black hole binaries (SMBHBs) in their slow inspiral phase, month or years prior to merging .
Background from supermassive black hole binaries refers to the incoherent superposition of signals coming from a large number of SMBHBs of forming in massive galaxy mergers out to redshift which gives rise to a confusion-limited foreground, and on top of which particularly bright or nearby sources might be individually resolved. The main traits of the background are described at the end of this Chapter, but we can anticipate the obvious payout of a PTA detection.
The background from SMBHBs informs us of the existence of a vast population of sub-pc (to be precise, sub-0.01pc) SMBHBs expected to rise according to the current cosmological model of galaxy assembly, and for which we have only indirect evidence . Recently, a number of galaxies in the verge of merging has been discovered in large surveys, each galaxy harbouring an active supermassive black hole . But, these mergers are in their early stage of pairing, as galaxies are observed interacting on scales of several kpcs. The detection of this foreground can provide a measure of the efficiency of the pairing and hardening of these SMBHBs on pc and sub-pc scales. This will enable us to distinguish the role of stellar and/or gas dynamics in removing energy and angular momentum from the binary (overcoming the last parsec problem, i.e. the possible stalling of the binary due its weak coupling with the environment . In particular, from the shape and amplitude of the signal we will learn whether binaries are eccentric or circular in their approach to coalescence, possibly constraining the efficiency of the mutual coupling with stars or/and gas. Identification and sky localisation of individual sources, will also open the possibility of identifying their electromagnetic counterpart, making multi-messenger studies of SMBHBs possible. In we describe in more detail the background detectable by PTA.
The unknown is a universe hosting totally unexpected sources over the whole multi-frequency gravitational wave sky. History shows that every time a new window became accessible to electromagnetic observations we discovered sources that were never anticipated.
6 Binaries as key sources of the gravitational universe
In this section, we introduce shortly key concepts required to identify the main traits of binaries as astrophysical sources of gravitational waves, and defer to the book by M. Maggiore, Gravitational Waves, and Poisson and Will, Gravity, for a comprehensive overview.
In Newtonian gravity, two point masses in a binary move on circular or elliptical orbits around the common centre of mass. The motion is periodic with constant Keplerian frequency , where is the semi-major axis of the relative orbit and the total mass of the binary of components and , respectively. In general relativity, binary systems emit gravitational waves which radiate away orbital energy and angular momentum. In the case of circular binaries the gravitational wave, which tracks the large scale motion, is monochromatic with frequency equal to . To compensate the radiative energy losses, binaries back-react gradually hardening, i.e. decreasing their semi-major axis and increasing the orbital frequency . The emission is weak initially and a phase of nearly adiabatic contraction, lasting hundred to thousand million years, anticipates the phase of inspiral, merger and ringdown, which produce a detectable signal.
The inspiral refers to the phase when the two binary components can still be considered as structureless and their dynamics (both conservative and dissipative) can be described by Post Newtonian (PN) theory. In this phase, which is the longer lasting, the signal, called chirp, has a characteristic shape, with both the amplitude and frequency of the wave slowly sweeping to higher values. This phase is crucial in obtaining first estimates of the binary system’s parameters most of which can be extracted by matching the observed signal on general relativity predictions. When the binary companions are spinning, the signal is modulated by spin-orbit and spin-spin couplings, and this modulation encodes in addition to the masses, orbit inclination, distance and sky location, also the spins of the two interacting bodies.
The merger refers to the phase of ”very late inspiral” and coalescence (no longer described within the PN formalism). Moving at around one third of the speed of light, the two bodies experience extreme gravitational fields so that their dynamics and signal can be described only in the realm of Numerical Relativity (NR). The merger signal lasts for a shorter time (milliseconds for stellar origin black holes, minutes for massive black holes) compared to the inspiral, and in this phase finite-size effects become important for neutron star mergers, as the stars carry a surface. NR simulations which account for the full non-linear structure of the Einstein’s equation are highly successful in tracing the dynamics and the gravitational wave radiation.
The ringdown refers to the phase when the coalescence end-product relaxes to a new stationary equilibrium solution of the Einstein field equations: a new black hole for (BH*,BH*) and (NS,BH*) mergers or a hot hyper-massive or supra-massive neutron star or a black hole for the case of (NS,NS) mergers. Likewise MBHCs and EMRIs end with the formation of a new black hole. The emitted radiation can be computed using Perturbation Theory and it consists of a superposition of quasi-normal modes of the compact object that forms. These modes carry a unique signature that depend only on the mass and spin in the case of black holes [105, 106].
The merger and ringdown parts of the signal last for a short duration, yet they carry tremendous luminosity. Their inclusion in a matched filter search for binary systems dramatically increases the distance reach and the accuracy at which the masses and spins can be measured. The access to the latest stages gives precious insight into the structure of neutron stars and the EoS at supra-nuclear densities; and in the case of black holes the possibility of testing gravity in the genuinely strong-field dynamical sector, and possibly prove the ”no-hair” conjecture [105, 128]. The emission of gravitational waves from a binary is a continuous process and phenomenological models for the merger dynamics have been developed, the most remarkable among the various approaches being the Effective One Body (EOB) theory and the Phenom models which permits a continuous description of the three phases, as predicted by general relativity [129, 130, 131, 132], including also tidal effects in the case of (NS,NS) coalescences .
6.1 Description of the inspiral
Binaries are irreversibly driven to coalescence, and the reference frequency of the gravitational wave at the time of coalescence is
representing twice the Keplerian frequency of a test mass orbiting around a non-spinning binary black hole of mass (seen as single unit) at the innermost stable circular orbit Neutron stars are so compact that their equilibrium radii are smaller than for many EoSs and typical masses , so that represents a reference frequency for coalescing compact objects in general.
Radiated energy and angular momentum - back reaction
In the inspiral phase and to leading order, the power radiated by a circular binary (averaged over a orbital period) is
where is the frequency of the gravitational wave emitted, the unitless frequency, and
is the so called chirp mass expressed either in terms of the symmetric mass ratio, (equal to for an equal mass binary), or of the reduced mass . The luminosity near coalescence () does not depend upon the mass of the coalescing objects, but on the symmetric mass ratio only, approaching the value The independence on is just a consequence of the fact that [energy/time] is equivalent to [mass/time], and time is equivalent to mass in units. For a short time lapse, this huge gravitational wave luminosity is far in excess of the electromagnetic luminosity of the entire universe (when ).
We then remark that merging black holes of stellar origin of emit the same luminosity as merging black holes of or , for a given as the two fundamental constants and fix the scale uniquely.
The orbital angular momentum from a binary is radiated away at a orbit-averaged rate
in the direction of . When the binary nears coalescence, whose value depends on and .
Binaries with non zero eccentricity and equal semi-major axis , lose energy and angular momentum at a higher rate, as during closest approach when the mutual interaction is strongest, radiation is emitted more effectively. The two rates are enhanced by a factor in (9), and in (11), with respect to a circular binary. In the case of eccentric binaries, the signal carries a dependence on the eccentricity and the emission spectrum is far richer than for a circular binary as more harmonics of the fundamental frequency enter the expression, with .
The emission of gravitational waves costs energy, and the source of radiation is the orbital energy of the binary, given by according to the virial theorem (computed to lowest order assuming Newtonian dynamics). Likewise, angular momentum is radiated away at the expense of the orbital angular momentum where denotes its direction.
The inspiral can be represented as a sequence of quasi-closed orbits where both the semi-major axis and eccentricity vary with time. During adiabatic contraction, and according to (9) and (11), one can prove that energy is extracted more rapidly than angular momentum, and binaries become more and more circular so that only little or null eccentricity is left at the time of coalescence. (Notice however that this is not necessarily true for EMRIs as in their inspiral they can retain significant eccentricity up to the innermost circular orbit.)
The total energy of the binary decreases adiabatically at a rate equal to . The binary hardens, the semi-major axis decreases, and the gravitational wave frequency increases at a rate
Equation (12) is derived setting with inferred using the expression of the binary’s Newtonian energy given few lines above. Equation (12) shows that to leading order the frequency evolution of the gravitational wave emitted by a circular binary is determined uniquely by the chirp mass . The evolution of is slow initially and it progresses faster and faster with time, given the rapid dependence of on the frequency itself. The solution to (12)
describes the rise in the frequency of the gravitational wave emitted by the binary when chirping, where gives the epoch of merger. At , the frequency of the wave formally diverges, but a non diverging cut-off frequency is found when the system evolves into the relativistic state and the two masses merge.
Figure 1 of shows (bottom row) the spectacular chirp observed in GW150914 , i.e. the increase in frequency during binary inspiral, and the convergence of a finite value at merger as a new black hole has formed.
According to (13), a binary observed at a frequency takes a time to coalesce equal to
which is a steep function of the frequency .
The late inspiral, merger and ringdown phases have a very short duration. In terms of the dimensionless frequency this time is