MNRAS 000, 1–16 (2019) Preprint 23 March 2020 Compiled using MNRAS LATEX style file v3.0

Probing the theory of gravity with gravitational lensing ofgravitational waves and galaxy surveys

Suvodip Mukherjee1,2?, Benjamin D. Wandelt1,2,3 † & Joseph Silk1,2,4,5 ‡1 Institut d’Astrophysique de Paris, 98bis Boulevard Arago, 75014 Paris, France2 Sorbonne Universites, Institut Lagrange de Paris, 98 bis Boulevard Arago, 75014 Paris, France3 Center for Computational Astrophysics, Flatiron Institute, 162 5th Avenue, 10010, New York, NY, USA4 The Johns Hopkins University, Department of Physics & Astronomy, 3400 N. Charles Street, Baltimore, MD 21218, USA5 Beecroft Institute for Cosmology and Particle Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK

23 March 2020

ABSTRACTThe cross-correlation of gravitational wave strain with upcoming galaxy surveys probetheories of gravity in a new way. This method enables testing the theory of gravityby combining the effects from both gravitational lensing of gravitational waves andthe propagation of gravitational waves in spacetime. We find that within 10 years,the combination of the Advanced-LIGO and VIRGO detector networks with plannedgalaxy surveys should detect weak gravitational lensing of gravitational waves in thelow redshift Universe (z < 0.5). With the next generation gravitational wave experi-ments such as Voyager, LISA, Cosmic-Explorer and Einstein Telescope, we can extendthis test of the theory of gravity to larger redshifts by exploiting the synergies betweenelectromagnetic wave and gravitational wave probes.

Key words: gravitational wave, large-scale structure of Universe, Weak lensing

1 INTRODUCTION

The Lambda-Cold Dark Matter (LCDM) model of cosmol-ogy matches well all of the best available data sets probingcosmic microwave background (CMB), galaxy distributions,and supernovae (Aghanim et al. 2018; Alam et al. 2017b;Abbott et al. 2018, 2019; Jones et al. 2018). However 95%of the content of this model is composed of dark energy anddark matter that is not yet understood. Several theoreticalmodels aid our understanding of this unknown part of theUniverse and lead to observable predictions which can be ex-plored using electromagnetic probes of the Universe. Withthe discovery of gravitational waves Abbott et al. (2016a,2017b), a new observational probe has opened up which iscapable of bringing new insights to our studies of the Uni-verse. Study of the imprints of the dark matter and late-timecosmic acceleration on gravitational waves will provide newavenues for understanding these unknowns.

With the ongoing ground-based gravitational wave ob-servatories such as Advanced-LIGO (Laser InterferometerGravitational-Wave Observatory) (Abbott et al. 2016c),Virgo (Virgo interferometer)(Acernese et al. 2015), and withthe upcoming observatories such as KAGRA (Large-scale

? mukherje@iap.fr† wandelt@iap.fr‡ silk@iap.fr

Cryogenic Gravitational wave Telescope) (Akutsu et al.2019) and LIGO-India (Unnikrishnan 2013), we can mea-sure the gravitational wave signals from binary neutron stars(NS-NS), stellar-mass binary black holes (BH-BH), blackhole-neutron star (BH-NS) systems over the frequency range10 − 1000 Hz and up to a redshift of about one. Alongwith the ground-based gravitational wave detectors, space-based gravitational wave experiment such as LISA (LaserInterferometer Space Antenna) Klein et al. (2016); Amaro-Seoane et al. (2017) are going to measure the gravitationalwave signal emitted from supermassive binary black holesin the frequency range ∼ 10−4 − 10−1 Hz. In the future,with ground-based gravitational wave experiments such asVoyager, Cosmic Explorer Abbott et al. (2017a) and theEinstein Telescope1, we will be able to probe gravitationalwave sources up to a redshift z = 8 and z = 100 respectivelySathyaprakash et al. (2019). This exquisite multi-frequencyprobe of the Universe is going to provide an avenue towardsunderstanding the history of the Universe up to high redshiftand unveil the true nature of dark energy and dark matter.

In this paper, we explore the imprints of the cosmicdensity field on astrophysical gravitational waves and fore-casts for measuring them using cross-correlations with thegalaxy density field and galaxy weak lensing. Intervening

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2 Mukherjee, Wandelt, & Silk

cosmic structures induce distortions, namely magnificationsand demagnifications in the gravitational wave signal whengravitational waves propagate through spacetime. Accordingto the theory of general relativity, these distortions shouldbe universal for gravitational wave signals of any frequencyfor a fixed density field. These imprints should exhibit cor-relations with other probes of the cosmic density field suchas galaxy surveys, as both of the distortions have a commonsource of origin.

We estimate the feasibility of measuring the lensing ofthe gravitational wave signals emitted from astrophysicalsources in the frequency band 10−1000 Hz and 10−4−10−1

Hz by cross-correlating with the upcoming large-scale struc-ture surveys such as DESI (Dark Energy SpectroscopicInstrument) Aghamousa et al. (2016), EUCLID Refregieret al. (2010), LSST (The Large Synoptic Survey Telescope)LSST Science Collaboration et al. (2009), SPHEREx Doreet al. (2018a), WFIRST (Wide Field Infrared Survey Tele-scope)(Dore et al. 2018b), etc. which also probes the cosmicdensity field. The cross-correlation of the gravitational wavesignal with the cosmic density field is going to validate thetheory of gravity, nature of dark energy and the impact ofdark matter on gravitational wave with a high SNR overa large range of redshift as discussed in Sec. 5. The auto-correlation of the GW sources can also be used as a probe ofthe lensing signal. However this is going to be important inthe regime of large numbers of GW sources with high statis-tical significance such as from the Einstein Telescope 2 andCosmic Explorer (Abbott et al. 2017a).

The paper is organized as follows. In Sec. 2, we discussthe effect of lensing on the galaxy distribution and on grav-itational waves. In Sec. 3, we discuss the implication of thegravitational lensing of gravitational waves in validating thetheory of gravity and the standard model of cosmology. InSec. 4 and Sec. 5, we obtain the estimator for measuring thesignal and the forecast of measuring this from the upcominggravitational wave observatories and LSS missions. Finally,in Sec. 6, we discuss the conclusions of this work and itsfuture scope.

2 WEAK LENSING

The general theory of relativity predicts universal nullgeodesics for both gravitational wave and electromagneticwave, that can be written in terms of the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric with line ele-ments given by ds2 = −dt2 + a(t)2(dx2 + dy2 + dz2), wherea(t) is the scale factor, is set to one at present and decreasesto zero in the past. However, in the presence of densityfluctuations, both electromagnetic and gravitational wavespropagate along the perturbed FLRW metric which can bewritten as ds2 = −(1 + 2Φ)dt2 + a(t)2(1 + 2Ψ)(dx2 + dy2 +dz2). One of the inevitable effects of matter perturbationson both electromagnetic and gravitational waves is gravita-tional lensing which can lead to magnification, spatial de-flections and time delays in the arrival times of the signal(Kaiser & Squires 1993; Kaiser 1998; Bartelmann & Schnei-der 2001). In this paper, we discuss the aspect of gravita-

2 http://www.et-gw.eu

tional weak lensing due to the distribution of the densityfluctuations in the Universe and its effect on cosmologicalobservables such as the galaxy field and gravitational waves.

2.1 Weak lensing of galaxies

The observed distribution of the galaxies can be translatedinto a density field as a function of sky position and redshiftof the source as

δ(n, z) =ρm(n, z)

ρm(z)− 1, (1)

where ρm(n, z) is the matter density at a position n at theredshift z and ρm denotes the mean density at z. The den-sity field of the galaxies is assumed to be a biased tracer ofthe underlying dark matter density field which can be writ-ten in Fourier space in terms of the scale-dependent biasbg(k) as δg(k) = bg(k)δDM (k). In the Fourier domain, wedefine the galaxy power spectrum in terms of the dark mat-ter power spectrum by the relation Pg(k) = bg(k)2PDM (k),where the power spectrum is defined as 〈δX′(k)δ∗X(k′)〉 =

(2π)3PX′X(k)δD(~k − ~k′) (δD denotes the Dirac delta func-tion).

The inhomogeneous distribution of matter betweengalaxies and the observer causes distortions in galaxyshapes. The lensing convergence field can be written in termsof the projected line of sight matter density as

κg(n) =

∫ zs

0

dz3

2

ΩmH20 (1 + z)χ(z)

cH(z)∫ ∞z

dz′dngal(z

′)

dz′(χ(z′)− χ(z))

(χ(z′))δ(χ(z)n, z),

(2)

where zs is the redshift of the source plane anddngal(z

′)dz′

is the normalized redshift distribution of galaxies. The con-vergence field is related to the cosmic shear by the relationKaiser (1998)

κg(~k) =k2

1 − k22

k21 + k2

2

γ1(~k) +k1k2

k21 + k2

2

γ2(~k), (3)

where, γ1 and γ2 are the cosmic shears in the Fourier spacewith ~k = (k1, k2). From the observables such as the galaxydistributions and its shape, the lensing field can be inferredfrom the data of the upcoming galaxy surveys (Aghamousaet al. 2016; Refregier et al. 2010; LSST Science Collaborationet al. 2009; Dore et al. 2018b,a).

2.2 Weak lensing of gravitational wave

The strain of the gravitational waves from coalescing bi-naries can be written in terms of the redshifted chirp massMz = (1+z)M as (Hawking & Israel 1987; Cutler & Flana-gan 1994; Poisson & Will 1995; Maggiore 2008)

h(fz) = Q(angles)

√5

24

G5/6M2z(fzMz)

−7/6

c3/2π2/3dLeiφz , (4)

where M = (m1m2)3/5

(m1+m2)1/5is the true chirp mass in terms

of the masses of the individual binaries m1 and m2 anddL is the luminosity distance to the binaries which canbe written for the standard LCDM model of cosmology asdL = (c(1 + z)/H0)

∫ z0

dz′√Ωm(1+z′)3+(1−Ωm)

. G and c are the

MNRAS 000, 1–16 (2019)

Weak lensing of gravitational waves 3

gravitational constant and speed of light respectively. Prop-agation of the gravitational wave in the presence of a matterdistribution leads to magnification in the strain of gravita-tional waves in the geometric optics limit which can be writ-ten as 3 (Takahashi 2006; Laguna et al. 2010; Cutler & Holz2009; Camera & Nishizawa 2013; Bertacca et al. 2018)

h(n, fz) = h(fz)[1 + κgw(n)], (5)

where κgw(n) is the lensing convergence and can be writtenin terms of the density fluctuations δ(χ(z)n, z) along the lineof sight by the relation

κgw(n) =

∫ zs

0

dz3

2

ΩmH20 (1 + z)χ(z)

cH(z)∫ ∞z

dz′dngw(z′)

dz′(χ(z′)− χ(z))

(χ(z′))δ(χ(z)n, z),

(6)

wheredngw(z′)

dz′ is the normalized redshift distribution of thegravitational wave sources. The redshift distribution of theGW sources can be determined from the electromagneticcounterparts or else can also be estimated by using the clus-tering of the GW source positions with the galaxy catalogs(Oguri 2016; Mukherjee & Wandelt 2018). According to thetheory of general relativity and the standard model of cos-mology, in the absence of anisotropic stress, metric pertur-bations are related by Φ = −Ψ, and are related to the matterperturbation by the relation

∇2Φ = 4πGa2ρmδ, (7)

where G is the gravitational constant and ρm is the matterdensity.

3 PROBE OF GENERAL THEORY OFRELATIVITY AND DARK ENERGY

Gravitational waves bring a new window to validate the gen-eral theory of relativity and cosmological constant as thecorrect explanation of the theory of gravity and cosmic ac-celeration. Propagation of gravitational waves in the pres-ence of cosmic structures brings two avenues to explore theUniverse:

(i) the propagation of gravitational wave in the spacetimeis defined by a perturbed FLRW metric

DµDµhαβ + 2Rµανβhµν = 0. (8)

where Dµ is the covariant derivative and Rµανβ is the cur-vature tensor; and

(ii) For a perfect fluid energy-momentum tensor, the evo-lution of the metric perturbations Φ and Ψ and its rela-tionship with the matter perturbation can be written at allcosmological epochs by the relation Hu & Sawicki (2007)

g(k, z) ≡ Ψ(k, z) + Φ(k, z)

Ψ(k, z)− Φ(k, z)=0,

1

2∇2(Ψ(k, z)− Φ(k, z)) =4πGa2ρmδ,

(9)

3 The effects of matter perturbations on the GW signal apartfrom the effect from lensing are explored in previous studies (La-

guna et al. 2010)

Gravitational lensing of gravitational waves is going toprobe both of these aspects of the theory of gravity from theobservations. Several alternative theories of gravity considerdeviations in Eq. (8) (Cardoso et al. 2003; Saltas et al. 2014;Nishizawa 2018) and Eq. (9) (Carroll et al. 2006; Bean et al.2007; Hu & Sawicki 2007; Schmidt 2008; Silvestri et al. 2013;Baker et al. 2015; Lombriser & Taylor 2016; Lombriser &Lima 2017; Saltas et al. 2014; Baker & Bull 2015; Slosaret al. 2019) such as (i) g(k, z) 6= 0, (ii) the modified PoissonEquation, (iii) running of the Planck mass which acts likea frictional term leading to damping of the gravitationalstrain more than the usual damping due to the luminositydistance, (iv) mass of the graviton (v) speed of propagationof gravitational waves and (vi) the anisotropic stress term asa source term in the gravitational wave propagation equationEq. (8). A joint analysis of all of these observational aspectscan be performed to test the validity of the general theoryof relativity and cosmic acceleration using the frameworkdiscussed in this paper.

An avenue to study gravitational wave propagation inthe presence of matter perturbations is to cross-correlatethe gravitational wave strain with the cosmic density fieldprobed by other avenues such as the galaxy field, CMBand line-intensity mapping. In this paper, we explore thecross-correlation of the galaxy field with the gravitationalwave strain from astrophysical gravitational wave sourceswhich are going to be measured from upcoming gravita-tional wave observatories such as Advanced-LIGO Abbottet al. (2017a), Virgo Acernese et al. (2015), KAGRA Akutsuet al. (2019), LIGO-India Unnikrishnan (2013) and LISAKlein et al. (2016); Amaro-Seoane et al. (2017). The jointstudy of the gravitational wave data along with the datafrom large-scale structure and CMB missions will probe allobservable signatures (listed above) due to modifications inthe theory of gravity and dark energy by measuring the dis-tortion in the gravitational wave strain.

Previous studies have discussed the effect of gravita-tional lensing of gravitational waves on measuring cosmolog-ical parameters using gravitational wave events only Taka-hashi (2006); Laguna et al. (2010); Cutler & Holz (2009);Camera & Nishizawa (2013); Bertacca et al. (2018); Con-gedo & Taylor (2019). The measurement of the luminositydistance using gravitational waves can also probe the cur-vature of the Universe (Congedo & Taylor 2019). In thiswork, we examine the cross-correlation of the gravitationalwave strain with electromagnetic probes. This approach in-troduces two new aspects. First, we can study the synergybetween the electromagnetic sector and the gravitationalwave sector. Different theories of gravity lead to modifica-tions in Eq. (8) and Eq. (9), and a joint estimation of bothof these aspects is necessary to constrain these models. Sec-ond, the cross-correlation study with large samples of datafrom galaxy surveys makes it possible to detect the lensingsignal from noise-dominated gravitational wave data.

MNRAS 000, 1–16 (2019)

4 Mukherjee, Wandelt, & Silk

4 MEASURING THE LENSING OF THEGRAVITATIONAL WAVE USINGCROSS-CORRELATIONS WITH GALAXYSURVEYS

4.1 Theoretical signal

The propagation of the electromagnetic waves and gravi-tational waves along the same perturbed geodesics bringsan inevitable correlation between the galaxy ellipticity anddistortion in the gravitational wave strain according to thetheory of general relativity and the standard LCDM modelof cosmology. The gravitational wave strain gets lensed bythe foreground galaxies present between the gravitationalwave source and us. The theoretical power spectrum of thegravitational wave lensing in the spherical harmonic basiscan be written as 4

Cκgwκ

jgal

l ≡〈(κgw)lm(κg)l′m′〉δKll′δKmm′

=

∫dz

χ2

H(z)

c

[Wκgw (χ(z))W j

κg (χ(z))

× Pδ((l + 1/2)/χ(z))(χ(z))

],

Cκgwδ

j

l ≡〈(κgw)lmδl′m′〉δKll′δKmm′

=

∫dz

χ2

H(z)

cbg

[Wκgw (χ(z))W j

δ (χ(z))

× Pδ((l + 1/2)/χ(z))(χ(z))

],

(10)

where W iκgw , W

iκg and W i

δ are the kernels for gravitationalwave lensing, galaxy lensing and the galaxy field respectivelyfor the tomographic bins indicated by the bin index j inredshift, which can be defined as

Wκgw/κgal(χ(z)) =1

H(z)

∫ ∞z

dz′dngw/gal(z

′)

dz′(χ(z′)− χ(z))

(χ(z′)),

W jδ (χ(z)) =

dn

dz(zj).

(11)

In this analysis, we have taken a linear bias model withthe value of the bias parameter as bg = 1.6 (Anderson et al.2012; Alam et al. 2017a; Desjacques et al. 2018) and thenon-linear dark matter power spectrum from the numericalcode CLASS (Blas et al. 2011; Lesgourgues 2011). The powerspectra for galaxy lensing and gravitational wave lensing,

Cκgwκ

jgal

l and Cκgwδ

j

l , are plotted in Fig. 1 as a function ofthe redshift of the gravitational wave sources. The redshiftdistribution of the gravitational wave sources is taken asdngwdz

(zs) = δ(z − zs) and the distribution of the galaxies istaken according to Models

dN

dz(zs) =z2

s exp(−(zs/z0)3/2) Model-I,

dN

dz(zs) =

((zs/z0)2

(2z0)

)exp(−zs/z0) Model-II.

(12)

The signal of the cross-correlation gets stronger at high red-shift due to more overlap between the gravitational wavelensing kernel and the galaxy kernel.

4 We have used the Limber approximation (k = (l+ 1/2)/χ(z)).

4.2 Estimator

The observed gravitational wave signal depends upon bothgravitational wave strain from the source and the effect fromthe gravitational wave propagation, which can be expressedaccording to Eq. (5). The observed DL from the gravita-tional wave strain can be written as

DL(n) =dL(n)

1 + κ(n)+ εgw(n),

= dL(n)(1− κ(n)) + εgw(n) +O(κ2),

(13)

where εgw is the observational error from the gravita-tional wave observation with zero mean and dL = (c(1 +

zs)/H0)∫ zs

0dz′√

Ωm(1+z′)3+(1−Ωm). In the weak lensing limit

κgw << 1, which is used to obtain the second equation inEq. 13. The luminosity distance can be inferred indepen-dently of the chirp mass using the relation derived by Schutz(1986)

DL ∝1

h(t)τν2, where τ ≡

(dν/dt

ν

)−1

∝ πM2z

(πMz)11/3ν8/3,

and h(t) ∝ Mz(πνzMz)2/3

dL,

(14)

where τ is the time-scale related to the change of the fre-quency of the gravitational wave signal and h is the gravi-tational wave strain averaged over detectors and source ori-entations. However, the source inclination angle is impor-tant to estimate the luminosity distance accurately (Nis-sanke et al. 2010). By using the two polarization states ofthe gravitational wave signal h+ and h×, the degeneracy be-tween the source inclination angle and luminosity distancecan be lifted Nissanke et al. (2010).

Eq. (13) shows that there exists a multiplicative bias toκgw from dL . So, in order to remove this bias, we need tomake an estimate of the true luminosity distance for eachsources. The estimate of true luminosity distance to thegravitational wave sources can be estimated using the red-shift (zs) of the source (from the EM counterparts (Nissankeet al. 2013)) and best-fit cosmological parameters (obtainedusing the data from other cosmological probes such as CMB,supernovae, etc.) in the relation desL = dL(1+εs), where εs isthe error in the measurement of true dL due to redshift errorand the error in best-fit cosmological parameters. Then theestimator for κgw using the the ratio of DL and desL is

DL(n) ≡ 1− DL(n)

desL (n)

= 1−(1− κgw(n)) +

εgwdl

(n)

1 + εs(n).

(15)

For εs << 1, we can write the above equation as

DL(n) = κgw(n)− εgwdl

(n) + εs(n)− εs(n)κgw(n). (16)

This is feasible for the gravitational wave sources with elec-tromagnetic counterparts, the first case we will discuss. Forthose gravitational wave sources that do not have electro-magnetic counterparts, eq. 15 is the estimator of the lensingsignal, which produces a multiplicative bias in estimation ofthe lensing potential κgw. We will return to this case belowand discuss how to remove the bias.

MNRAS 000, 1–16 (2019)

Weak lensing of gravitational waves 5

(a)

(b)

Figure 1. We show the correlation for (a) the LIGO sources and (b) the LISA sources between κgw with κg and δ for different tomographic

redshift bins indicated by the zbin index given by (zgw, zg/κg ). The redshift range is taken from [0, 1] and [0, 3] for the LIGO and LISA

sources with the bin width ∆z = 0.2 and ∆z = 0.5 respectively. For the galaxy survey, we have taken the redshift distribution for (a)Model-I and (b) Model-II with z0 = 0.5 and for gravitational wave we assume a single source plane dngw/dz(zs) = δD(z − zs).

MNRAS 000, 1–16 (2019)

6 Mukherjee, Wandelt, & Silk

Figure 2. We show the normalized distribution of the galaxy surveys for Model-I and Model-II given in Eq. 12. We have used the casewith Model-I and z0 = 0.2 for the cross-correlation with the NS-NS and BH-NS sources which can be observed by LIGO. For the BH-BH

sources from LIGO, we have used Model-I with z0 = 0.5. For the LISA gravitational wave sources, we have studied the cross-correlation

signal for the galaxy distribution shown by Model-II with z0 = 0.5. The colored bands shows the tomographic bins chosen in this analysiswith ∆z = 0.2 for LIGO and ∆z = 0.5 for LISA.

4.2.1 Gravitational wave events with electromagneticcounter part

For the case when εs << 1 the corresponding estima-tor for gravitational wave lensing and galaxy lensing cross-correlations in real space can be written as5

Cκgwκg (n.n′) = 〈DL(n)κg(n)〉,= 〈κgw(n)κg(n

′)〉+((((((〈εs(n)κg(n)〉

−〈εs(n)〉〈κgw(n)κg(n′)〉,

Cκgwδ(n.n′) = 〈DL(n)δ(n)〉,

= 〈κgw(n)δ(n′)〉+〈εs(n)δ(n)〉

−〈εs(n)〉〈κgw(n)δ(n′)〉.

(17)

Here, the term εs is not correlated with the lensing and den-sity field of the galaxy δg and κgw and hence goes to zero onaveraging over a large number of sources. As a result, theestimator of the cross-correlation remains unbiased by anyerror in the luminosity distance measurement for the gravi-tational wave sources with electromagnetic counterpart.

5 The quantities with hat denote the estimator of the signal

4.2.2 Gravitational wave events without electromagneticcounterpart

Returning now to the case of gravitational wave sourceswithout electromagnetic counterparts, the error term εs ∼ 1.The corresponding cross-correlation signal can be written as

Cκgwκg (n.n′) = 〈κg(n′)κgw(n)

1 + εs(n)〉,

= Cκgκgw (n.n′)

⟨1

1 + εs(n)

⟩,

Cκgwδ(n.n′) = 〈κgw(n)δ(n′)

1 + εs(n)〉,

= Cκgwδ(n.n′)

⟨1

1 + εs(n)

⟩,

(18)

The estimated cross-correlation signal can have a multi-plicative bias if 〈εs〉 6= 0 for sources without EM counter-parts. However, for a large number of gravitational wavesources, all that is required to remove this bias is Ngw(z),the redshift distribution of the gravitational wave sources.Using the clustering approach, redshifts of the gravitationalwave sources can be obtained using the methods describedin (Menard et al. 2013; Oguri 2016; Mukherjee & Wandelt2018).

These estimators in the spherical harmonic basis 6

6 Any field X(n) can be expressed in the spherical harmonics

basis as X(n) =∑lmXlmYlm(n)

MNRAS 000, 1–16 (2019)

Weak lensing of gravitational waves 7

can be expressed as Cκgwκgl = 1

2l+1

∑m(κgw)lm(κg)lm

and CκGW δl = 1

2l+1

∑m(κgw)lmδlm. The real space cross-

correlation estimator given in Eq. 17 is related to the spher-ical harmonic space cross-correlation estimator by the rela-tion

CXX ′(n.n′) =∑l

(2l + 1

4π

)Pl(n.n

′)CXX′

l , (19)

where, Pl(n.n′) are the Legendre polynomials.

The variance of the cross-correlation signal can be writ-ten in terms of the available overlapping sky fraction of boththe missions fsky as

(σgw−Xl )2 =1

fsky(2l + 1)

((C

κgwκgw

l +NDDl )(CXXl +NXXl )

+ (CκgwX

l )2

),

(20)

where, X ∈ g, κg, NXXl = σ2

X/nX corresponds to the shapenoise (σκg = 0.3) and the galaxy shot noise (σg = 1) for X =κg and g respectively. nX denotes the number density ofsources per sr. NDDl is the variance related to the estimationof the gravitational wave lensing signal, which can be writtenas

NDDl =4π

Ngw

(σ2dl

d2l

+σ2b

d2l

)el

2θ2min/8 ln 2, (21)

where Ngw is the number of gravitational wave sources, θminis the sky localization area of the gravitational wave sources,σ2dl

is the error in the luminosity distance due to the detectornoise and σ2

b is the error due to the error in the gravitationalwave source redshift and uncertain values of cosmologicalparameters.

The joint estimation of all the effects from the prop-agation of the gravitational wave signal can be written inan unified frame work by combining the observable fromthe gravitational wave and the cosmic probes such as thegalaxy surveys δg and κg as

O =

DLδκg

(22)

and the corresponding covariance matrix can be expressedas

C ≡ 〈OO†〉 =

CDLDLl C

κgwδ

l Cκgwκg

l

Cκgwδ

l Cδδl Cδκgl

Cκgwκg

l Cδκgl C

κgκgl

.

Here, the terms in bold indicates the new cosmologicalprobes which will be available from the cross-correlationof the gravitational wave signal with the galaxy field. Thecross-correlation of the galaxy surveys with the stochasticgravitational wave signal is another probe to study the un-resolved sources (Mukherjee & Silk 2019).

5 FORECAST FOR THE MEASUREMENT OFLENSING OF GRAVITATIONAL WAVE

The measurement of the galaxy lensing/gravitational wavelensing and galaxy/gravitational wave lensing can be ex-plored from upcoming missions. The large-scale structure of

the Universe will be probed by future missions such as DESIAghamousa et al. (2016), EUCLID Refregier et al. (2010),LSST LSST Science Collaboration et al. (2009), SPHERExDore et al. (2018a), WFIRST (Dore et al. 2018b), havinga wide sky coverage and access to galaxies up to high red-shift ( about z ∼ 3). For the large-scale structure surveys,we consider the survey setup with two different normalizedredshift distributions according to Eq. 12 with the value ofz0 = 0.2 and z0 = 0.5. We have considered two different setsof number density of galaxies as ng = 0.5 per arcmin2 forz0 = 0.2 and 45 per arcmin2 for z0 = 0.5 in this analysis.The redshift distribution of the large scale structure surveysoverlaps with the the redshift distribution of the gravita-tional wave sources.

For gravitational waves, we have considered the set-upfor both ground-based and space-based gravitational waveexperiments such as Advanced-LIGO and LISA. For thegravitational wave detectors, we have considered the cur-rent instrument noise for Advanced-LIGO 7 and LISA asshown in Fig. 3. The LISA noise curves are calculated usingthe online tool 8 with the instrument specification providedin Table 1 (Klein et al. 2016).

The currently ongoing ground-based gravitational waveobservatories such as Advanced-LIGO Abbott et al. (2017a)and VIRGO Acernese et al. (2015), and the future detectorssuch as KAGRA Akutsu et al. (2019), LIGO-India Unnikr-ishnan (2013) will be joining these in the coming decade.These experiments will be capable of reaching up to a red-shift of z ∼ 1 and are going to detect the gravitationalwave primarily from stellar origin binary black hole mergers(BH-BH), black hole neutron star mergers (BH-NS) and bi-nary neutron star mergers (NS-NS). The exact event ratesof these binary mergers are not yet known and are predictedfrom the current observation of the gravitational wave eventsAbbott et al. (2016b, 2017b).

We have taken the number of the sources of gravi-tational waves as a free parameter for each of these bi-nary species. Among these sources, BH-NSs and NS-NSsare expected to have an electromagnetic counterpart (Bhat-tacharya et al. 2019; Nissanke et al. 2013) and hence theredshift to these sources can be measured. We have consid-ered the spectroscopic measurement of the source redshiftfor the electromagnetic counterpart from BH-NS and NS-NS. Sources such as BH-NSs and NS-NSs with electromag-netic counterparts can have an accurate estimation of thesky localization (less than one arcsec), which makes it pos-sible to improve the cross-correlation signal with the LSSsurveys. For the stellar mass BH-BHs, we do not expect anyelectromagnetic counterpart and as a result, the redshift ofthe source cannot be measured. For BH-BHs we have consid-ered that the redshift of the source is completely unknown.For BH-BHs without any electromagnetic counterparts, wewill be relying only on measurements from the gravitationalwave signal and as a result, the sky localization error is goinghave relatively large errors compared to the case of BH-NSand NS-NS. For the combination of five detectors, we can

7 https://dcc.ligo.org/cgi-bin/DocDB/ShowDocument?

.submit=Identifier&docid=T1800044&version=58 http://www.srl.caltech.edu/~shane/sensitivity/

MakeCurve.html

MNRAS 000, 1–16 (2019)

8 Mukherjee, Wandelt, & Silk

Figure 3. We plot the gravitational wave strain noise for Advanced-LIGO and LISA. The Advanced-LIGO noise is from the latestprojected noise. The LISA noise is plotted for one year of integration time with the LISA instrument specifications given in Table 1.

Table 1. Specifications of LISA instrument properties Amaro-

Seoane et al. (2017) is used with 1 year of integration time toproduce Fig. 3 using the online tool Larson et al. (2000, 2002).

Specifications Values

Armlength 2.5× 109 m

Optics diameter 0.3 m

Wavelength 1064 nm

Laser power 2.0 W

Optical train efficiency 0.3

Acceleration noise 3× 10−15 m

s2√Hz

Position noise 2× 10−9 m√Hz

Sensitivity noise floor Position

Astrophysical Noise White dwarf

expect about 10 sq. deg sky localization error in the sourcesof the gravitational wave (Nissanke et al. 2011; Pankow et al.2018). In this analysis, we have taken the angular resolutionθ (or equivalently the maximum value of l, lmax ∝ 1/∆θ)as the free parameter and obtained the results for a fewdifferent choices.

Along with the ground-based detectors, space-based de-

tectors such as LISA (Klein et al. 2016; Amaro-Seoane et al.2017; Smith & Caldwell 2019) will also be operational inthe next decade and will be able to reach to very highredshifts z ∼ 20. These experiments are going to exploresupermassive BH-BH systems over a wide range of masses∼ 103 − 107 M. Such sources are going to be observed inthe frequency band 10−4 − 10−1 Hz for time-scales of daysto years before mergers of the binaries. The event rates forthese sources are largely unknown and are mainly based onstudies made from simulations Micic et al. (2007). For theheavier supermassive BH-BH systems such as 107 M, theevent rates are going to be less than the case for the interme-diate supermassive BH-BH systems of mass 104 M, accord-ing to our current understanding of the formation of thesebinaries. We consider three different event rates for unit red-shift bins, 50, 100 and 300 in this analysis for LISA. For thesupermassive binary BHs, we can expect to see electromag-netic counterparts over a wide of electromagnetic frequencyspectrum (Giacomazzo et al. 2012; Haiman 2018; Palenzuelaet al. 2010; Farris et al. 2015; Gold et al. 2014; Armitage &Natarajan 2002). This is going to be useful for obtaining theredshift of the gravitational wave sources by the electromag-netic follow up surveys.

In the future, the next-generation gravitational waveobservatories will include Voyager 9, which is going toachieve an instrument noise of ∼ 10−24 Hz−1/2. There arealso proposals for ground-based observatories with larger

9 https://dcc.ligo.org/public/0142/T1700231/003/

T1700231-v3.pdf

MNRAS 000, 1–16 (2019)

Weak lensing of gravitational waves 9

(a)

(b)

(c)

Figure 4. The cumulative SNR for the measurement of the cross-correlation between gravitational wave lensing and galaxy lensing

Cκgwκgl for an observation time of 5 years with the instrument noise of Advanced-LIGO. (a) For the NS-NS, the detection threshold of

the gravitational wave sources above redshift of 0.2 is going to be less than 10 − σ and hence will not be identified as a LIGO event.(b) For BH-NS, we have considered here the case for a mass ratio of 20 with Mtotal = 31.5M. (c) The BH-BH sources are considered

for equal mass-ratios with Mtotal = 60M. The region shaded in pink indicates below 3 − σ detection of the cosmological signal. The

dotted-line indicates the upper and lower limits of the event rates, solid-line indicates the mean value of the event rates.

MNRAS 000, 1–16 (2019)

10 Mukherjee, Wandelt, & Silk

(a)

(b)

(c)

Figure 5. The cumulative SNR for the measurement of the cross-correlation between gravitational wave lensing and galaxy field Cκgwδ

lfor an observation time of 5 years with the instrument noise of Advanced-LIGO. (a) For the NS-NS, the detection threshold of thegravitational wave sources above a redshift of 0.2 is going to be less than 10−σ and hence will not be identified as a LIGO event. (b) ForBH-NS, we have considered here the case for a mass ratio of 20 with Mtotal = 31.5M. (c) The BH-BH sources are considered for equal

mass-ratio with Mtotal = 60M. The region shaded in pink indicates below 3− σ detection of the cosmological signal. The dotted-lineindicates the upper and lower limits of the event rates, solid-line indicates the mean value of the event rates.

MNRAS 000, 1–16 (2019)

Weak lensing of gravitational waves 11

baselines such as the Einstein Telescope (Hild et al. 2011)and Cosmic Explorer (Abbott et al. 2017a). These obser-vatories are capable of exploring stellar-mass BH-BHs, BH-NSs and NS-NSs up to a redshift about z ∼ 20 for thefrequency range 10−1000 Hz. These observatories are goingto have orders-of-magnitude improvement in the instrumentnoise over LIGO. As a result, our proposed methodologyis going to be an exquisite probe of the Universe for betterunderstanding gravity, dark matter and dark energy by com-bining the probes from gravitational wave and electromag-netic wave. In future work, we will estimate the prospectsof this approach for the next-generation gravitational wavedetectors.

The signal-to-noise ratio for the measurement of thecross-correlation of the gravitational wave lensing by cross-correlating with the galaxy-lensing (X = κGW ) and withthe galaxy density field X = δ can be estimated using

(S/N)2 =∑lmaxl

CκGWX

l

(σκGWX )2. Using the gravitational wave

detector noise, we estimate the error in the luminosity dis-tance σdl using a Fisher analysis by considering the twopolarization states of the gravitational wave signal,

FDlDl =∑+,×

1

2

∫ ∞0

4f2d ln f∂h+,×

∂Dl

1

|hn(f)|2∂h+,×

∂Dl. (23)

The corresponding Cramer-Rao bound on the parameters

is obtained as σDl =√

(F−1DlDl

). The noise due to lensing

causes an error in the estimation of the luminosity distance.This can be written in terms of the fitting form as given byHirata et al. (2010). The error in the redshift σz = C(1 + z)of the gravitational wave sources leads to an additional errorin the estimation of the desl as σs = ∂dl

∂zσz. We have taken

C = 0.03 for the cases with photo-metric measurements ofthe redshift, and C = 0 for spectroscopic measurements ofthe redshift.

In Fig. 4 and Fig. 5, we show the cumulative SNR forthe measurement of C

κGW κgl and CκGW δ

l signals for binaryneutron star, black hole neutron star systems and binaryblack holes respectively. For binary neutron stars, currentgravitational wave detectors will be able to detect individ-ual objects with high SNR for redshifts z < 0.2. As a re-sult, we need to consider galaxy surveys which are going toprobe the low redshift Universe such as DESI (Aghamousaet al. 2016). Using the galaxy distribution shown by the redcolours in Fig. 2, we estimate the cumulative SNR up to red-shift z = 0.2 for lmax = 3000 and assuming spectroscopicmeasurements of the redshifts of the binary neutron stars.Detection is not possible for the currently considered eventrate within five years of observation time from C

κGW κgl . But

for CκGW δl , we can expect a cumulative SNR of 3 detection of

the signal up to redshift z = 0.2 for the highest event ratesand the best sky-fraction scenario. For the redshift range0.3 > z > 0.2, LIGO will be detecting several very low SNRgravitational wave events. The cross-correlation signal withthe low SNR gravitational wave events (which may not beconsidered as a detected event) and if the electromagneticcounterpart of these sources can be identified, then morethan a 3-σ measurement is possible only from CκGW δ

l . How-ever, this scenario is extremely rare and the detection of thesignal beyond z > 0.2 is unlikely.

For the BH-NS systems, we have estimated the SNR forMNS = 1.5M and MBH = 30M. These sources can be

identified as individual events for redshifts below 0.5 and weconsidered the low redshift galaxy survey as shown by thered coloured curve in Fig. 2 to estimate the cross-correlationsignal between galaxy lensing and gravitational wave lens-ing. The measurability of the signal remains weak at nearlyall the values of redshift for C

κGW κgl with lmax = 3000. But

the signal from CκGW δl can be detected with high SNR for

redshift z > 0.35 from the sources of the gravitational wavewhich will be detected by Advanced-LIGO-like detectors.BH-NS systems with the lower (or higher) mass of the blackhole can be identified as individual events only up to a lower(or higher) redshift, and the corresponding detection thresh-old will vary. The BH-NS systems are the best scenario forstudying the cross-correlation signal as this can have an elec-tromagnetic counterpart and can also be detected up to ahigh redshift than for the NS-NS systems.

For the BH-BH systems, with Mtotal = 60M, gravita-tional wave events can be detected up to a redshift z = 1.These sources are possibly not going to have electromag-netic counterparts and as a result, the redshifts of the ob-jects will remain unknown. We have considered the redshiftof the GW sources to be unknown to obtain the forecastfor BBHs. Along with the absence of the redshifts, thesesources are going to have poor sky localization ∼ 10 sq. de-gree even with the five detector network (LIGO-H, LIGO-L,Virgo, KAGRA, LIGO-India) (Pankow et al. 2018). Usinga minimum sky localization error of 10 sq. degree and withfor the case with unknown redshift, we estimate the mea-surability of C

κGW κgl and CκGW δ

l for a galaxy survey shownby the black curve in Fig. 2 (which reaches up to redshiftz = 1). The measurability of the signal remains marginalfor 5 years of observation time with the current estimate ofthe LIGO events. The estimates of the mean signal can alsobe biased for a lesser number of gravitational wave samples,due to the unavailability of the true luminosity distance tothe GW source (as shown in Eq. 16). For a large numberof the gravitational wave sources, clustering redshift of thegravitational wave source can be inferred by using the spa-tial correlations (Menard et al. 2013; Oguri 2016; Mukherjee& Wandelt 2018).

Detection of more events is going to improve the pro-jected SNR. In these estimates of the SNR for NS-NSs,BH-NSs, and BH-BHs, we have only considered the inspiralphase of the gravitational wave signal. Further improvementof the SNR is possible if we include merger and the ringdownphase of the gravitational wave strain. This will be consid-ered in a future analysis including the complete waveformof the gravitational wave signal.

For the space-based gravitational wave detectors such asLISA, the gravitational wave events are going to be detectedup to a redshift about z ∼ 20 Amaro-Seoane et al. (2017).This enables us to probe the cosmological signal up to ahigh redshift by cross-correlating with galaxy surveys whichreach up to a high redshift. Using the redshift distribution ofgalaxies shown by the blue curve in Fig. 2 which mimics thesurveys such as LSST, we estimate the SNR for the measura-bility of C

κGW κgl and CκGW δ

l for different masses of the BH-BHs ranging from 104 − 107. In this analysis, we have con-sidered a maximum time period of one year of inspiral phasebefore the merger. The merger and the ringdown phase forthese sources are not considered in the analysis. For theLISA BH-BH sources, we have considered a photometric er-

MNRAS 000, 1–16 (2019)

12 Mukherjee, Wandelt, & Silk

(a)

(b)

(c)

Figure 6. The cumulative SNR for the measurement of the cross-correlation between the gravitational wave lensing and galaxy lensingCκgwκgl from LISA for an observation time of 4 years for equal mass binary black holes with individual masses (a) 105, (b) 106 and (c)

107. The region shaded in pink indicates below 3-σ detection of the cosmological signal. The projected SNRs are obtained for two sky

localization errors (i) ∆θ = 1 deg (shown in blue) and (ii) ∆θ = 0.2 deg (shown in purple).

MNRAS 000, 1–16 (2019)

Weak lensing of gravitational waves 13

(a)

(b)

(c)

Figure 7. The cumulative SNR for the measurement of the cross-correlation between the gravitational wave lensing and galaxy field

Cκgwδ

l from LISA for an observation time of 4 years for equal mass binary black holes with individual masses (a) 105, (b) 106 and (c)107. The region shaded in pink indicates below 3-σ detection of the cosmological signal. The projected SNRs are obtained for two skylocalization errors (i) ∆θ = 1 deg (shown in blue) and (ii) ∆θ = 0.2 deg (shown in purple).

MNRAS 000, 1–16 (2019)

14 Mukherjee, Wandelt, & Silk

ror of σz/(1 + z) = 0.03 in the determination of the redshiftof gravitational wave sources from the electromagnetic coun-terpart. Measurement of the electromagnetic counterpart isalso going to be useful to reduce the sky localization errorof the source of the gravitational wave and hence improvesthe detection of the cross-correlation signal. For two differ-ent values of lmax, lmax = 200 and lmax = 1000, we estimatethe SNR for the measurement of C

κGW κgl and CκGW δ

l signalfor four years of observation time and show the correspond-ing estimates in Fig. 6 and Fig. 7 respectively. As can beseen from these figures, the discovery space from synergybetween LISA and LSST is promising for each mass win-dow of BH-BHs. These SNRs are going to improve withthe inclusion of the merger and the ring down phase of thegravitational wave signal. Further improvement in the SNRis possible with the increase in the LISA observation timeto ten years, increase in the number of gravitational waveevent rates and also with galaxy surveys which can go upto higher redshift. By combining all the high redshift probessuch as EUCLID Refregier et al. (2010), LSST LSST ScienceCollaboration et al. (2009), SPHEREx Dore et al. (2018a)and WFIRST Dore et al. (2018b), the detectability of thesignal is going to improve significantly. At redshifts higherthan z > 3, the temperature and polarization anisotropiesof the CMB is going to be a powerful probe to measure thelensing signal to even higher redshift as shown by (Mukher-jee et al. 2019). Using the cross-correlation between CMB-lensing and gravitational wave lensing, we are going to havehigher SNR measurements from redshift z > 3 than possi-ble from the galaxy surveys with the redshift distributionsconsidered in this analysis (Fig. 2). Complementary infor-mation from cross-correlation with intensity mapping of the21-cm line is another avenue to be explored in future work.

6 CONCLUSIONS

The general theory of relativity provides a unique solutionto the propagation of gravitational waves which dependsupon the underlying spacetime metric. Gravitational wavesaccording to the theory of general relativity are masslessand should propagate along null geodesics with the speed oflight. As a result, in the framework of the standard modelof cosmology, gravitational wave should propagate throughthe perturbed FLRW metric where the perturbations arisedue to the spatial fluctuations in the matter density in everyposition. As a result of this metric perturbation, the gravi-tational wave strain is distorted.

Study of these distortions in the gravitational wavestrain while propagating through the perturbed metric canbe a direct observational probe of several aspects of funda-mental physics, which include (i) the validation of generalrelativity from the propagation of gravitational wave sig-nals and from the connection between metric perturbationsand matter perturbations, (ii) properties of dark energy andits effect on gravitational waves, (iii) gravitational effects ofdark matter on gravitational waves and evolution with cos-mic redshift, (iv) testing the equivalence principle from themulti-frequency character of the gravitational wave signal,(v) equivalence between the electromagnetic sector and thegravitational sector over the cosmic time-scale.

The coming decade with ongoing and upcoming gravi-

tational wave observatories makes it possible to study theseaspects and opens a new window in observational cosmology.The astrophysical gravitational wave signals which are emit-ted from compact objects of different types, such as whitedwarfs, neutron stars and black holes, span a vast range ofmasses, gravitational wave frequencies and cosmological red-shifts. All of these sources are transients and are going to befrequent. As a result, they can probe the underlying theoryof gravity and the statistical properties of the cosmologicalperturbations present in the Universe multiple times fromdifferent sources. As a result, all the above-mentioned as-pects of fundamental physics can be studied using transientastrophysical gravitational wave sources.

In this paper, we have explored one of the importanteffects of matter perturbations, the weak lensing of the grav-itational wave strain, and propose an avenue to explorethis signal using cross-correlations with other cosmologicalprobes such as galaxy surveys. The measurement of the lens-ing signal leads to a measurement of the evolution of thegravitational potential over a wide range of cosmic redshiftsand also probes the connection of the matter perturbationwith the gravitational potential. The joint study of cosmicdensity field and gravitational wave propagation in space-time promises to be a powerful avenue towards a better un-derstanding of the theory of gravity and the dark sector(dark matter and dark energy) of the Universe.

Several alternative theories of gravity and models ofdark energy can be distinguished from the evolution of thecosmic structures (Carroll et al. 2006; Bean et al. 2007; Hu &Sawicki 2007; Schmidt 2008; Silvestri et al. 2013; Baker et al.2015; Baker & Bull 2015; Slosar et al. 2019) and also fromthe propagation of gravitational wave in spacetime (Car-doso et al. 2003; Saltas et al. 2014; Nishizawa 2018). Asa result, the studies of the cross-correlations between thegravitational wave signal and cosmic structures should re-sult in a more comprehensive understanding of the theoryof gravity and dark energy. The imprint of the lensing po-tential is also a direct probe of the dark matter distribu-tion in the Universe, and measurement of this via gravi-tational wave is going to explore the gravitational effects ofdark matter on gravitational waves. The measurement of thegravitational lensing signal from gravitational waves with afrequency ranging from 10−4 to 103 Hz will probe the uni-versality of gravity on a wide range of energy scales and leadto a direct probe of the equivalence principle. Finally, thisnew avenue is also going to illuminate several hitherto un-explored windows which can be studied from the synergiesbetween electromagnetic waves and gravitational waves.

Ground based gravitational wave detectors such asLIGO-US Abbott et al. (2017a) , Virgo Acernese et al.(2015), KAGRA Akutsu et al. (2019), LIGO-India Unnikr-ishnan (2013) and space based gravitational wave detec-tor such as LISA Klein et al. (2016); Amaro-Seoane et al.(2017) are going to probe a large range of cosmic redshift0 < z . 20 with gravitational wave sources such as NS-NS, BH-NS, stellar mass BH-BH and supermassive BH-BH.By cross-correlating with the data from upcoming galaxysurveys such as DESI (Aghamousa et al. 2016), EUCLID(Refregier et al. 2010), LSST (LSST Science Collaborationet al. 2009), SPHEREx (Dore et al. 2018a), WFIRST (Doreet al. 2018b), we will be able to obtain a high SNR detec-tion of the weak lensing signal with these surveys, as shown

MNRAS 000, 1–16 (2019)

Weak lensing of gravitational waves 15

in Fig. 4-7. The measurability is best for the sources withelectromagnetic counterparts such NS-NS, BH-NS and su-per massive BH-BHs. For stellar mass black hole binaries,we do not expect electromagnetic counterparts, and hencepoorly determined source redshifts. As a result, the measur-ability of the weak lensing signal is going to be marginalfrom stellar mass BH-BHs.

We find that the most promising avenue with Advanced-LIGO will be the cross-correlation of lensed black hole neu-tron star mergers and the galaxy overdensity CκGW δ

l . Thecontributions from binary neutron stars and binary blackholes are marginal and only improve if the event rates arelarger than considered in this analysis. However, with fu-ture surveys such as Cosmic Explorer and Einstein Tele-scope, we can explore the spatial cross-correlation of binaryblack holes and galaxies in order to obtain the clusteringredshift (Menard et al. 2013) of the sources (Oguri 2016).This can result in an improvement in the correlation sig-nal. By fitting the shape of the correlation function betweenthe spatial position of binary black holes and underlyinggalaxies, an accurate estimate of the source redshift is pos-sible (Mukherjee & Wandelt 2018) for a large sample of bi-nary black holes. The gravitational wave sources from LISAare going to provide a high SNR measurement from bothCκGW κgl and CκGW δ

l over a vast range of gravitational wavesource masses 104 M − 107 M by cross-correlating withthe large scale structure surveys with the access to high cos-mological redshift. In future work, we will present a jointBayesian framework for the combined analysis of the gravi-tational wave data and cosmological data. For this, we usethe full gravitational wave waveforms including the inspiral,merger and the ring-down phases, instead of only the inspi-ral phase considered in this analysis. Inclusion of the mergerand the ringdown phases is going to improve the SNR pre-sented in this paper.

In summary, we point out a new window for the emerg-ing paradigm of multi-messenger observational cosmologywith transient sources. This window will exploit both grav-itational wave and electromagnetic waves in order to testthe propagation and lensing of gravitational waves, poten-tially leading to high precision tests of gravity on cosmolog-ical scales. In addition to testing gravity in new ways, weexpect this probe to be useful for developing an improvedunderstanding of the standard model of cosmology. All theelectromagnetic observational probes of the Universe pointto additional matter content beyond baryonic matter, en-tirely inferred through its gravitational effects. The cause ofthese effects is associated in the standard model of cosmol-ogy with a cold dark matter component, which is neitherunderstood theoretically, nor directly detected in particlephysics experiments. We show in this paper that gravita-tional waves will bring a new and independent window toconfirm the existence of dark matter in the Universe and ex-ploit the gravitational effect of dark matter on gravitationalwaves as a new and complementary probe on its physicalnature. The detailed potential for constraints on the theo-retical models from this new observational window, enabledby advances in gravitational wave detector capabilities anda new generation of galaxy redshift surveys, remain to bedetermined.

ACKNOWLEDGEMENT

S. M. would like to thank Karim Benabed, Luc Blanchet,Sukanta Bose, Neal Dalal, Guillaume Faye, Zoltan Haiman,David Spergel, and Samaya Nissanke for useful discussions.The results of this analysis are obtained using the Hori-zon cluster hosted by Institut d’Astrophysique de Paris. Wethank Stephane Rouberol for smoothly running the Horizoncluster. The Flatiron Institute is supported by the SimonsFoundation. The work of SM and BDW is also supportedby the Labex ILP (reference ANR-10-LABX-63) part of theIdex SUPER, and received financial state aid managed bythe Agence Nationale de la Recherche, as part of the pro-gramme Investissements d’avenir under the reference ANR-11-IDEX-0004-02. The work of BDW is also supported bythe grant ANR-16-CE23-0002. BDW thanks the CCPP atNYU for hospitality during the completion of this work. Inthis analysis, we have used the following packages: IPython(Perez & Granger 2007), Matplotlib (Hunter 2007), NumPy(van der Walt et al. 2011), and SciPy (Jones et al. 01 ).

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