Cosmography with the Einstein Telescope
Einstein Telescope (ET) is a 3rd generation gravitational-wave (GW) detector that is currently undergoing a design study. ET can detect millions of compact binary mergers up to redshifts 2-8. A small fraction of mergers might be observed in coincidence as gamma-ray bursts, helping to measure both the luminosity distance and red-shift to the source. By fitting these measured values to a cosmological model, it should be possible to accurately infer the dark energy equation-of-state, dark matter and dark energy density parameters. ET could, therefore, herald a new era in cosmology.
pacs:04.30.Db, 04.25.Nx, 04.80.Nn, 95.55.Ym
The goal of modern cosmology is to measure the geometrical and dynamical properties of the Universe by projecting the observed parameters onto a cosmological model. The Universe has a lot of structure on small scales, but on a scale of about 100 Mpc the distribution of both baryonic (inferred from the electromagnetic radiation they emit) and dark matter (inferred from large scale streaming motion of galaxies) components is quite smooth. It is, therefore, quite natural to assume that the Universe is homogeneous and isotropic while describing its large-scale properties. In such a model, the scale factor which essentially gives the proper distance between comoving coordinates, and curvature of spatial sections are the only quantities that are needed to fully characterize the properties of the Universe. The metric of a smooth homogeneous and isotropic spacetime is
where is the cosmic time-coordinate, are the comoving spatial coordinates, and is a parameter describing the curvature of the spatial slices. for flat, positively and negatively curved slices, respectively. The evolution of depends on the parameter as well as the “matter” content of the Universe. The latter could consist of radiation, baryons, dark matter (DM), dark energy (DE), and everything else that contributes to the energy-momentum tensor.
The Friedman equation, which is one of two Einstein equations describing the dynamics of an isotropic and homogeneous Universe, relates the cosmic scale factor to the energy content of the Universe through
where is the Hubble parameter ( being its value at the present epoch ), while and are the (dimensionless) energy densities of the DM and DE, respectively. The above equation has to be supplemented with the equation-of-state of DM, assumed to be pressure-less fluid [ where ] and of DE, assumed to be of the form [ where ], with corresponding to a cosmological constant. The goal of cosmography is to measure which essentially determine the large-scale geometry and dynamics of the Universe. In the rest of this paper we shall assume that the spatial slices are flat (i.e., ).
Astronomers use “standard candles” to measure the geometry of the Universe and the various cosmological parameters. A standard candle is a source whose intrinsic luminosity can be inferred from the observed properties (such as the spectral content, time-variability of the flux of radiation, etc.). Since the observations also measure the apparent luminosity , one can deduce the luminosity distance to a standard candle from In addition, if the red-shift to the source is known then by observing a population of such sources it will be possible to measure the various cosmological parameters since the luminosity distance is related, when to the red-shift via
There is no unique standard candle in astronomy that works on all distance scales. An astronomer, therefore, builds the distance scale by using several steps, each of which works over a limited range of the distance. For instance, the method of parallax can determine distances to a few kpc, Cepheid variables up to Mpc, the Tully-Fisher relation works for several tens of Mpc, the - relation up to hundreds of Mpc and Type Ia supernovae up to red-shifts of a few Freedman et al. (2001). This way of building the distance scale has been referred to as the cosmic distance ladder. For cosmography, a proper calibration of the distance to high red-shift galaxies is based on the mutual agreement between different rungs of this ladder. It is critical that each of the rungs is calibrated with as little an error as possible.
Cosmologists have long sought for standard candles that can work on large distance scales without being dependent on the lower rungs of cosmic distance ladder. In 1986, one of us pointed out Schutz (1986) that gravitational astronomy can provide such a candle, or, more appropriately, a standard siren, in the form of a chirping signal (i.e., a signal whose frequency increases as a function of time) from the coalescence of compact stars (i.e., neutron stars and black holes) in a binary. The basic reason for this is that the gravitational-wave (GW) amplitude depends on the ratio of a certain combination of the binary masses and the luminosity distance. For chirping signals GW observations can measure both the amplitude of the signal and the masses very accurately and hence infer the luminosity distance.
Let us first recall in some detail how we might measure the luminosity distance. We will first assume that the source is located close-by, i.e., its redshift , although we will later relax this condition.
Gravitational waves are described by a second rank tensor , which, in a suitable coordinate system and gauge, has only two independent components and , all other components being zero. A detector measures only a certain linear combination of the two components, called the response given by
where and are the detector antenna pattern functions, is the polarization angle, and are angles describing the location of the source on the sky. The angles are all assumed to be constant for a transient source but time-dependent for sources that last long enough so that the Doppler modulation of the signal due to the relative motion of the source and detector cannot be neglected. For a coalescing binary consisting of two stars of masses and (total mass and symmetric mass ratio ) and located at a distance , the GW amplitudes are given by
where is the angle of inclination of the binary’s orbital angular momentum with the line-of-sight, is the angular velocity of the equivalent one-body system around the binary’s centre-of-mass and is the corresponding orbital phase. Parameters and are constants giving the epoch of merger and the orbital phase of the binary at that epoch, respectively.
The above expressions for and are the dominant terms in what is essentially a PN perturbative series. We have written down the expressions for a system consisting of non-spinning components on a quasi-circular orbit. In reality, we cannot assume either to be true. Eccentricity might be negligible only in the case of stellar mass binaries expected to be observed by ground-based detectors, but both eccentricity and spins could be non-zero in the case of merger of supermassive black holes expected to be observed by the Laser Interferometer Space Antenna (LISA) Apostolatos et al. (1994). The argument below holds good to whatever order the amplitudes are written down and for non-spinning objects on an eccentric orbit.
Here is the effective distance to the binary, which is a combination of the true luminosity distance and the antenna pattern functions. Note that In the case of non-spinning binaries on a quasi-circular orbit, therefore, the signal is characterized by nine parameters in all,
Since the phase of the signal is known to a high order in PN theory, one employs matched filtering to extract the signal and in the process measures the two mass parameters (parameters that completely determine the phase evolution) and the two fiducial parameters In general, the response of a single interferometer will not be sufficient to disentangle the luminosity distance from the angular parameters. However, EM identification (i.e., electromagnetic, especially optical, identification) of the source will determine the direction to the source, still leaving three unknown parameters . If the signal is a transient, as would be the case in ground-based detectors, a network of three interferometers will be required to measure all the unknown parameters and extract the luminosity distance.
Although the inspiral signal from a compact binary is a standard siren, there is no way of inferring from it the red-shift to a source. The mappings , and in Eq. (4), leave the signal invariant. Note that a source of total mass at a red-shift will simply appear to an observer to be a binary of total mass . One must optically identify the host galaxy to measure its red-shift. Thus, there is synergy in GW and EM observations which can make precision cosmography possible, without the need to build a cosmic distance ladder.
Over the next two decades GW interferometric detectors will provide a new tool for cosmology. Advanced ground-based interferometers, operating around 2015, are expected to detect binary neutron star mergers each year from within about 300 Mpc. Redshift could be measured to a (small) number of events associated with GRBs, thereby allowing an accurate determination of the Hubble constant Dalal et al. (2006); Nissanke et al. (2009). Observation by LISA of extreme mass ratio inspirals could measure the Hubble constant pretty accurately MacLeod and Hogan (2008). LISA will also observe binary super-massive black hole mergers with SNRs enabling the measurement of the DE equation-of-state to within several percents Holz and Hughes (2005); Arun et al. (2007).
In the rest of this paper we will discuss how well it might be possible to constrain cosmological parameters by GW observations of the inspiral signal of compact binaries by the Einstein Telescope (ET) — a third generation GW interferometer that is currently under a design study ET (). ET is envisaged to be ten times more sensitive than the advanced ground-based detectors, covering a frequency range of 1- Hz, posing new challenges in mitigating gravity gradient, thermal and quantum noise.
The sky-position averaged distance up to which ET might detect inspiral signals from coalescing binaries with an SNR of 8 is shown in Fig. 1. We plot the range both as a function of the intrinsic (red solid lines) and observed (blue dashed lines) total mass. A binary comprising two -neutron stars (BNS) can be observed from a red-shift of , and that comprising a -neutron star and a -black hole (NS-BH) from .
The expected rate of coalescences per year within the horizon of ET is for BNS and NS-BH. Such a large population of events to which luminosity distances are known pretty accurately, would be very useful for measuring cosmological parameters. If, as suspected, BNS and NS-BH are progenitors of short-hard gamma-ray bursts (GRBs) Nakar (2007), then it might be possible to make a coincident detection of a significant subset of the events in GW and EM windows and obtain both the luminosity distance to and red-shift of the source.
Since GRBs are believed to be beamed with beaming angles of order , we assume that only a small fraction () of binary coalescences will have GRB or other EM afterglows that will help us to locate the source on the sky and measure its red-shift. Eventually, we will be limited by the number of short-hard GRBs observed by detectors that might be operating at the time. As a conservative estimate, we assume that about BNS mergers will have EM counterparts over a three-year period. For definiteness we consider only BNS mergers and take these to have component masses of .
How well would we measure cosmological parameters with a catalogue of such sources? To answer this question we simulated 5,190 realizations of the catalogue containing 1,000 BNS coalescences with known red-shift and sky location, but the luminosity distance subject to statistical errors from GW observation and weak lensing. One such realization is shown in Fig. 1 (right panel). We assumed that the sources were all in the red-shift range , distributed uniformly (i.e., with constant comoving number density) throughout this red-shift range. The luminosity distance to the source was computed by assuming an FRW cosmological model with , , , and , but the measured distance was drawn from a Gaussian distribution whose width was determined by the quadrature sum of the errors due to weak lensing and GW observation. Weak lensing error in was assumed to be 5% at and linearly extrapolated to other red-shifts. GW observational error was estimated from the covariance matrix of the five-dimensional parameter space of the unknown signal parameters :
Here the angular brackets denote the scalar product, which, for any two functions and , is defined as
where and are the Fourier transforms of the functions and , respectively, and is the ET noise power spectral density. Note that since GRBs are expected to be strongly beamed, we did not take the angles associated with the unit normal to the plane of the inspiral as unknown variables. This assumption is justified: even if the opening angle of a GRB beam is as large as , the unit normal to the plane of the inspiral would still be confined to only 3% of the area of a unit sphere. Averaging errors over with the constraint would then be little different from taking . We did, however, average the errors over the sky position angles . We then fitted each realization of the source catalogue to the cosmological model given in Eq. (2), using the Levenberg-Marquardt algorithm Levenberg (1944); Marquardt (1963), in order to find a set of best fit parameters. It turns out that a catalogue of 1,000 sources is not quite enough for an accurate determination of all the parameters. However, assuming that is known accurately, the algorithm gave the best fit parameters in for each of the 5,190 realizations.
The distribution of the parameters obtained in this way are shown in Fig. 2, where the vertical line is at the true value of the relevant parameter. The relative 1- errors in and are 4.2%, 18% and 18% (with weak lensing) and 3.5%, 14% and 15% (with weak lensing errors corrected). Although is quite symmetric, and are both skewed and their mean values are slightly off the true values. The medians, however, are coincident with the true values.
In addition to if is also known (or, equivalently, if ), then one can estimate the pair more accurately, with the distributions as shown in Fig. 2 with greatly reduced skewness, and 1- errors in and of 9.4% and 7.6% (with weak lensing) and 8.1% and 6.6% (with lensing errors corrected). Finally, if is the only parameter unknown, it can be measured to an even greater accuracy with 1- errors of 1.4% (with weak lensing) and 1.1% (with lensing errors corrected).
The results of our simulation are quite encouraging but further work is needed to confirm the usefulness of GW standard sirens in precision cosmology. Let us mention some that are currently being pursued. Spins of component stars can be legitimately neglected in the case of neutron stars (and hence in BNS) but not for black holes. The modulation in the signal caused by the spin of the black hole can improve parameter accuracies. We assumed, for simplicity, that all our sources are BNS systems with masses In reality, the catalogue will consist of a range of NS and BH masses. A more realistic Monte Carlo simulation would draw binaries from the expected population rather than the same system, some of which (e.g. more massive systems) would lead to better, but others to worsened, parameter accuracies. The signal contains additional features, such as other harmonics of the orbital frequency than the second harmonic considered in this work and the merger and ringdown signals. These are important for heavier systems and could potentially reduce the errors. These factors are currently being taken into account to get a more reliable estimation of the usefulness of ET in precision cosmography.
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