Cosmography with the Einstein Telescope
CCosmography with the Einstein Telescope
B.S. Sathyaprakash, B.F. Schutz , and C. Van Den Broeck School of Physics and Astronomy, Cardiff University, 5, The Parade, Cardiff, UK, CF24 3AA Max Planck Institute for Gravitational Physics, Germany
Einstein Telescope (ET) is a 3rd generation gravitational-wave (GW) detector that is currentlyundergoing a design study. ET can detect millions of compact binary mergers up to redshifts 2-8. Asmall fraction of mergers might be observed in coincidence as gamma-ray bursts, helping to measureboth the luminosity distance and red-shift to the source. By fitting these measured values to acosmological model, it should be possible to accurately infer the dark energy equation-of-state, darkmatter and dark energy density parameters. ET could, therefore, herald a new era in cosmology.
PACS numbers: 04.30.Db, 04.25.Nx, 04.80.Nn, 95.55.Ym
The goal of modern cosmology is to measure the ge-ometrical and dynamical properties of the Universe byprojecting the observed parameters onto a cosmologicalmodel. The Universe has a lot of structure on smallscales, but on a scale of about 100 Mpc the distribu-tion of both baryonic (inferred from the electromagneticradiation they emit) and dark matter (inferred from largescale streaming motion of galaxies) components is quitesmooth. It is, therefore, quite natural to assume thatthe Universe is homogeneous and isotropic while describ-ing its large-scale properties. In such a model, the scalefactor a ( t ) , which essentially gives the proper distancebetween comoving coordinates, and curvature of spatialsections k, are the only quantities that are needed to fullycharacterize the properties of the Universe. The metricof a smooth homogeneous and isotropic spacetime is ds = − dt + a ( t ) dσ − kσ + σ (cid:0) dθ + sin θ dϕ (cid:1) , where t is the cosmic time-coordinate, ( σ, θ, ϕ ) are thecomoving spatial coordinates, and k is a parameter de-scribing the curvature of the t = const . spatial slices. k = 0 , ± , for flat, positively and negatively curvedslices, respectively. The evolution of a ( t ) depends on theparameter k, as well as the “matter” content of the Uni-verse. The latter could consist of radiation, baryons, darkmatter (DM), dark energy (DE), and everything else thatcontributes to the energy-momentum tensor.The Friedman equation, which is one of two Einsteinequations describing the dynamics of an isotropic and ho-mogeneous Universe, relates the cosmic scale factor a ( t )to the energy content of the Universe through H ( t ) = H (cid:20) ˆΩ M ( t ) − kH a + ˆΩ Λ ( t ) (cid:21) / , (1.1)where H ( t ) ≡ ˙ a ( t ) /a ( t ) is the Hubble parameter ( H = H ( t P ) being its value at the present epoch t P ), whileˆΩ M ( t ) and ˆΩ Λ ( t ) are the (dimensionless) energy densi-ties of the DM and DE, respectively. The above equa-tion has to be supplemented with the equation-of-stateof DM, assumed to be pressure-less fluid p = 0 [ ˆΩ M ( t ) =Ω M (1+ z ) , where Ω M = ˆΩ M ( t P )] and of DE, assumed to be of the form p = wρ Λ [ ˆΩ Λ ( t ) = Ω Λ (1 + z ) w ) , whereΩ Λ = Ω Λ ( t P )], with w = − H , Ω M , Ω Λ , w, k, . . . ) , which essentially determine thelarge-scale geometry and dynamics of the Universe. Inthe rest of this paper we shall assume that the spatialslices are flat (i.e., k = 0).Astronomers use “standard candles” to measure thegeometry of the Universe and the various cosmologicalparameters. A standard candle is a source whose in-trinsic luminosity L can be inferred from the observedproperties (such as the spectral content, time-variabilityof the flux of radiation, etc.). Since the observationsalso measure the apparent luminosity F , one can de-duce the luminosity distance D L to a standard candlefrom D L = (cid:112) L/ (4 πF ) . In addition, if the red-shift z tothe source is known then by observing a population ofsuch sources it will be possible to measure the variouscosmological parameters since the luminosity distance isrelated, when k = 0 , to the red-shift via D L = c (1 + z ) H (cid:90) z dz (cid:48) (cid:2) Ω M (1 + z (cid:48) ) + Ω Λ (1 + z (cid:48) ) w ) (cid:3) / . (1.2)There is no unique standard candle in astronomy thatworks on all distance scales. An astronomer, therefore,builds the distance scale by using several steps, each ofwhich works over a limited range of the distance. For in-stance, the method of parallax can determine distancesto a few kpc, Cepheid variables up to 10 Mpc, the Tully-Fisher relation works for several tens of Mpc, the D n - σ relation up to hundreds of Mpc and Type Ia supernovaeup to red-shifts of a few [1]. This way of building thedistance scale has been referred to as the cosmic distanceladder. For cosmography, a proper calibration of the dis-tance to high red-shift galaxies is based on the mutualagreement between different rungs of this ladder. It iscritical that each of the rungs is calibrated with as littlean error as possible.Cosmologists have long sought for standard candlesthat can work on large distance scales without being de-pendent on the lower rungs of cosmic distance ladder. In1986, one of us pointed out [2] that gravitational astron-omy can provide such a candle, or, more appropriately, a r X i v : . [ a s t r o - ph . C O ] J un Total mass (in M O. ) L u m i no s it y D i s t a n ce ( G p c ) Sky-ave. dist. Vs. Obs. M, n =0.25Sky-ave. dist. Vs phys. M, n=0.25 Sky-ave. dist. Vs Obs. M, n =0.10Sky-ave. dist. Vs phys. M, n =0.10 0.200.370.661.372.404.269.35 R e d s h i f t z Redshift ( z ) L u m i no s it y D i s t a n ce ( D L i n G p c ) FIG. 1: The left panel shows the range of the Einstein Telescope for inspiral signals from binaries as a function of the intrinsic (red solid line) and observed (blue dashed line) total mass. We assume that a source is visible if it produces an SNR of at least8 in ET. The right panel shows a realization of the source catalogue showing the measured luminosity distance (inferred fromGW observation of neutron star-black hole mergers) versus their red-shift (obtained by optical identification of the source).This catalogue is then fitted to a cosmological model. a standard siren , in the form of a chirping signal (i.e., asignal whose frequency increases as a function of time)from the coalescence of compact stars (i.e., neutron starsand black holes) in a binary. The basic reason for this isthat the gravitational-wave (GW) amplitude depends onthe ratio of a certain combination of the binary massesand the luminosity distance. For chirping signals GWobservations can measure both the amplitude of the sig-nal and the masses very accurately and hence infer theluminosity distance.Let us first recall in some detail how we might mea-sure the luminosity distance. We will first assume thatthe source is located close-by, i.e., its redshift z (cid:28) h αβ , which, in a suitable coordinate system andgauge, has only two independent components h + and h × ,h xx = − h yy = h + , h xy = h yx = h × , all other compo-nents being zero. A detector measures only a certainlinear combination of the two components, called the re-sponse h ( t ) given by h ( t ) = F + ( θ, ϕ, ψ ) h + ( t ) + F × ( θ, ϕ, ψ ) h × ( t ) , (1.3)where F + and F × are the detector antenna pattern func-tions, ψ is the polarization angle, and ( θ, ϕ ) are anglesdescribing the location of the source on the sky. The an-gles are all assumed to be constant for a transient sourcebut time-dependent for sources that last long enough sothat the Doppler modulation of the signal due to therelative motion of the source and detector cannot be ne-glected. For a coalescing binary consisting of two starsof masses m and m (total mass M ≡ m + m andsymmetric mass ratio ν ≡ m m /M ) and located at adistance D L , the GW amplitudes are given by h + ( t ) = 2 ν M / D − (1 + cos ( ι )) ω ( t − t ) / × cos[2Φ( t − t ; M, ν ) + Φ (cid:48) ] ,h × ( t ) = 4 ν M / D − cos( ι ) ω ( t − t ) / × sin[2Φ( t − t ; M, ν ) + Φ (cid:48) ] , (1.4)where ι is the angle of inclination of the binary’s orbitalangular momentum with the line-of-sight, ω ( t ) is the an-gular velocity of the equivalent one-body system around the binary’s centre-of-mass and Φ( t ; M, ν ) is the corre-sponding orbital phase. Parameters t and Φ (cid:48) are con-stants giving the epoch of merger and the orbital phaseof the binary at that epoch, respectively.The above expressions for h + and h × are the dominantterms in what is essentially a PN perturbative series. Wehave written down the expressions for a system consistingof non-spinning components on a quasi-circular orbit. Inreality, we cannot assume either to be true. Eccentric-ity might be negligible only in the case of stellar massbinaries expected to be observed by ground-based detec-tors, but both eccentricity and spins could be non-zero inthe case of merger of supermassive black holes expectedto be observed by the Laser Interferometer Space An-tenna (LISA) [3]. The argument below holds good towhatever order the amplitudes are written down and fornon-spinning objects on an eccentric orbit.Substituting the expressions given in Eq. (1.4) for h + and h × in Eq. (1.3), we get h ( t ) = ν M / D eff ω / cos[2Φ( t − t ; M, ν ) + Φ ] , (1.5) D eff ≡ D L (cid:2) F (1 + cos ( ι )) + 4 F × cos ( ι ) (cid:3) / , (1.6)Φ ≡ Φ (cid:48) + arctan (cid:20) − F × cos( ι ) F + (1 + cos ( ι )) (cid:21) . (1.7)Here D eff is the effective distance to the binary, whichis a combination of the true luminosity distance and theantenna pattern functions. Note that D eff ≥ D L . In thecase of non-spinning binaries on a quasi-circular orbit,therefore, the signal is characterized by nine parametersin all, (
M, ν, t , Φ , θ, ϕ, ψ, ι, D L ) . Since the phase Φ( t ) of the signal is known to a highorder in PN theory, one employs matched filtering to ex-tract the signal and in the process measures the two massparameters ( M, ν ) (parameters that completely deter-mine the phase evolution) and the two fiducial parame-ters ( t , Φ ) . In general, the response of a single inter-ferometer will not be sufficient to disentangle the lumi-nosity distance from the angular parameters. However,EM identification (i.e., electromagnetic, especially opti- W M W L w < W M > = 0.254< W L > = 0.739< w > = -0.96 s w = 0.18 s W L = 0.031 s W M = 0.045 < W M > = 0.260 s W M = 0.035< W L > = 0.736 s W L = 0.026< w > = -0.96 s w = 0.15 W M W L w W M w < W M > = 0.269< w > = -1.00 s W M = 0.025 s w = 0.076 < W M > = 0.268 s W M = 0.022 s w = 0.066< w > = -1.00 W M w FIG. 2: The plot on the left shows the distribution of errors in Ω M , Ω Λ and w, obtained by fitting 5,190 realizations of acatalogue of BNS merger events to a cosmological model of the type given in Eq. (1.2), with three free parameters. Thefractional 1- σ width of the distributions σ Ω M / Ω M , σ Ω Λ / Ω Λ , and σ w / | w | , are 18%, 4.2% and 18% (with weak lensing errors in D L , left panels) and 14%, 3.5% and 15% (if weak lensing errors can be corrected, right panels). The plot on the right is thesame, but assuming that Ω Λ is known to be Ω Λ = 0 .
73, and fitting the “data” to the model with two free parameters. Thefractional 1- σ widths in the distribution σ Ω M / Ω M and and σ w / | w | , are 9.4% and 7.6% (with weak lensing errors in D L , leftpanels) and 8.1% and 6.6% (if weak lensing errors can be corrected, right panels). cal, identification) of the source will determine the direc-tion to the source, still leaving three unknown parameters( ψ, ι, D L ). If the signal is a transient, as would be thecase in ground-based detectors, a network of three inter-ferometers will be required to measure all the unknownparameters and extract the luminosity distance.Although the inspiral signal from a compact binary isa standard siren, there is no way of inferring from it thered-shift to a source. The mappings M → (1 + z ) M , ω → ω/ (1 + z ) , and D L → (1 + z ) D L , in Eq. (1.4), leavethe signal invariant. Note that a source of total mass M at a red-shift z will simply appear to an observer tobe a binary of total mass (1 + z ) M . One must opticallyidentify the host galaxy to measure its red-shift. Thus,there is synergy in GW and EM observations which canmake precision cosmography possible, without the needto build a cosmic distance ladder.Over the next two decades GW interferometric de-tectors will provide a new tool for cosmology. Ad-vanced ground-based interferometers, operating around2015, are expected to detect ∼
40 binary neutron starmergers each year from within about 300 Mpc. Redshiftcould be measured to a (small) number of events asso-ciated with GRBs, thereby allowing an accurate deter-mination of the Hubble constant [4, 5]. Observation byLISA of extreme mass ratio inspirals could measure theHubble constant pretty accurately [6]. LISA will alsoobserve binary super-massive black hole mergers withSNRs ∼ few × ,
000 enabling the measurement of theDE equation-of-state to within several percents [7, 8].In the rest of this paper we will discuss how well itmight be possible to constrain cosmological parametersby GW observations of the inspiral signal of compact bi-naries by the Einstein Telescope (ET) — a third genera-tion GW interferometer that is currently under a designstudy [9]. ET is envisaged to be ten times more sensi-tive than the advanced ground-based detectors, coveringa frequency range of 1-10 Hz, posing new challenges in mitigating gravity gradient, thermal and quantum noise.The sky-position averaged distance up to which ETmight detect inspiral signals from coalescing binaries withan SNR of 8 is shown in Fig. 1. We plot the range both asa function of the intrinsic (red solid lines) and observed(blue dashed lines) total mass. A binary comprising two1 . M (cid:12) -neutron stars (BNS) can be observed from a red-shift of z (cid:39)
2, and that comprising a 1 . M (cid:12) -neutron starand a 10 M (cid:12) -black hole (NS-BH) from z (cid:39) ∼ several × for BNS and NS-BH.Such a large population of events to which luminosity dis-tances are known pretty accurately, would be very usefulfor measuring cosmological parameters. If, as suspected,BNS and NS-BH are progenitors of short-hard gamma-ray bursts (GRBs) [10], then it might be possible to makea coincident detection of a significant subset of the eventsin GW and EM windows and obtain both the luminositydistance to and red-shift of the source.Since GRBs are believed to be beamed with beamingangles of order 40 ◦ , we assume that only a small fraction( ∼ − ) of binary coalescences will have GRB or otherEM afterglows that will help us to locate the source onthe sky and measure its red-shift. Eventually, we willbe limited by the number of short-hard GRBs observedby detectors that might be operating at the time. As aconservative estimate, we assume that about 1 ,
000 BNSmergers will have EM counterparts over a three-year pe-riod. For definiteness we consider only BNS mergers andtake these to have component masses of (1 . , . M (cid:12) .How well would we measure cosmological parameterswith a catalogue of such sources? To answer this ques-tion we simulated 5,190 realizations of the catalogue con-taining 1,000 BNS coalescences with known red-shift andsky location, but the luminosity distance subject to sta-tistical 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-shiftrange 0 ≤ z ≤
2, distributed uniformly (i.e., with con-stant comoving number density) throughout this red-shift range. The luminosity distance to the source wascomputed by assuming an FRW cosmological model with H = 70 km s − Mpc − , Ω M = 0 .
27, Ω Λ = 0 .
73, and w = −
1, but the measured distance was drawn from aGaussian distribution whose width σ D L was determinedby the quadrature sum of the errors due to weak lens-ing and GW observation. Weak lensing error in D L was assumed to be 5% at z = 1 and linearly extrapo-lated to other red-shifts. GW observational error wasestimated from the covariance matrix C km of the five-dimensional parameter space of the unknown signal pa-rameters p k = ( M, ν, t , Φ , D L ): C km = Λ − km , Λ km = (cid:104) h k , h m (cid:105) , h k = ∂h∂p k . (1.8)Here the angular brackets denote the scalar product,which, for any two functions a ( t ) and b ( t ), is defined as (cid:104) a, b (cid:105) = 4 (cid:60) (cid:90) ∞ d fS h ( f ) A ( f ) B ∗ ( f ) (1.9)where A and B are the Fourier transforms of the func-tions a ( t ) and b ( t ), respectively, and S h ( f ) is the ETnoise power spectral density. Note that since GRBs areexpected to be strongly beamed, we did not take the an-gles ( ι, ψ ) associated with the unit normal to the planeof the inspiral as unknown variables. This assumptionis justified: even if the opening angle of a GRB beamis as large as 40 ◦ , the unit normal to the plane of theinspiral would still be confined to only 3% of the area ofa unit sphere. Averaging errors over ( ι, ψ ) with the con-straint ι < ◦ would then be little different from taking ι = 0 ◦ . We did, however, average the errors over the skyposition angles ( θ, φ ). We then fitted each realizationof the source catalogue to the cosmological model givenin Eq. (1.2), using the Levenberg-Marquardt algorithm[11, 12], in order to find a set of best fit parameters. Itturns out that a catalogue of 1,000 sources is not quiteenough for an accurate determination of all the parame-ters. However, assuming that H is known accurately, the algorithm gave the best fit parameters in (Ω M , Ω Λ , w ) foreach of the 5,190 realizations.The distribution P of the parameters obtained in thisway are shown in Fig. 2, where the vertical line is at thetrue value of the relevant parameter. The relative 1- σ errors in Ω Λ , Ω M and w, are 4.2%, 18% and 18% (withweak lensing) and 3.5%, 14% and 15% (with weak lensingerrors corrected). Although P ( w ) is quite symmetric, P (Ω M ) and P (Ω Λ ) are both skewed and their mean valuesare slightly off the true values. The medians, however,are coincident with the true values.In addition to H if Ω Λ is also known (or, equivalently,if Ω M + Ω Λ = 1), then one can estimate the pair (Ω M , w )more accurately, with the distributions as shown in Fig. 2with greatly reduced skewness, and 1- σ errors in Ω M and w, of 9.4% and 7.6% (with weak lensing) and 8.1% and6.6% (with lensing errors corrected). Finally, if w is theonly parameter unknown, it can be measured to an evengreater accuracy with 1- σ errors of 1.4% (with weak lens-ing) and 1.1% (with lensing errors corrected).The results of our simulation are quite encouraging butfurther work is needed to confirm the usefulness of GWstandard sirens in precision cosmology. Let us mentionsome that are currently being pursued. Spins of com-ponent stars can be legitimately neglected in the caseof neutron stars (and hence in BNS) but not for blackholes. The modulation in the signal caused by the spinof the black hole can improve parameter accuracies. Weassumed, for simplicity, that all our sources are BNS sys-tems with masses (1 . , . M (cid:12) . In reality, the cataloguewill consist of a range of NS and BH masses. A more re-alistic Monte Carlo simulation would draw binaries fromthe expected population rather than the same system,some of which (e.g. more massive systems) would leadto better, but others to worsened, parameter accuracies.The signal contains additional features, such as other har-monics of the orbital frequency than the second harmonicconsidered in this work and the merger and ringdown sig-nals. These are important for heavier systems and couldpotentially reduce the errors. These factors are currentlybeing taken into account to get a more reliable estimationof the usefulness of ET in precision cosmography. [1] W. L. Freedman, B. F. Madore, B. K. Gibson, L. Fer-rarese, D. D. Kelson, S. Sakai, J. R. Mould, R. C. Ken-nicutt, Jr., H. C. Ford, J. A. Graham, et al., Astrophys.J. , 47 (2001), arXiv:astro-ph/0012376.[2] B. Schutz, Nature , 310 (1986).[3] T. Apostolatos, C. Cutler, G. Sussman, and K. Thorne,Phys. Rev. D , 6274 (1994).[4] N. Dalal, D. E. Holz, S. A. Hughes, and B. Jain, Phys.Rev. D74 , 063006 (2006), astro-ph/0601275.[5] S. Nissanke, S. A. Hughes, D. E. Holz, N. Dalal, and J. L.Sievers (2009), 0904.1017.[6] C. L. MacLeod and C. J. Hogan, Phys. Rev.
D77 , 043512(2008), 0712.0618. [7] D. Holz and S. Hughes, Astrophys. J , 15 (2005),arXiv:astro-ph/0504616.[8] K. Arun, B. Iyer, B. Sathyaprakash, S. Sinha, andC. Van Den Broeck, Phys. Rev. D , 124002 (2007),arXiv:0707.3920.[9] Einstein telescope project , URL .[10] E. Nakar, Phys. Rept. , 166 (2007), astro-ph/0701748.[11] K. Levenberg, The Quarterly of Applied Mathematics ,164 (1944).[12] D. Marquardt, SIAM Journal on Applied Mathematics11