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

Probing the theory of with gravitational lensing of gravitational 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, France 2 Sorbonne Universites, Institut Lagrange de Paris, 98 bis Boulevard Arago, 75014 Paris, France 3 Center for Computational , Flatiron Institute, 162 5th Avenue, 10010, New York, NY, USA 4 The Johns Hopkins University, Department of & , 3400 N. Charles Street, Baltimore, MD 21218, USA 5 Beecroft Institute for and Particle Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK

23 March 2020

ABSTRACT The cross-correlation of gravitational strain with upcoming galaxy surveys probe theories of gravity in a new way. This method enables testing the theory of gravity by combining the effects from both gravitational lensing of gravitational waves and the propagation of gravitational waves in . We find that within 10 years, the combination of the Advanced-LIGO and VIRGO detector networks with planned galaxy surveys should detect weak gravitational lensing of gravitational waves in the low (z < 0.5). With the next generation gravitational wave experi- ments such as Voyager, LISA, Cosmic-Explorer and Einstein , we can extend this test of the theory of gravity to larger by exploiting the synergies between electromagnetic wave and gravitational wave probes. Key words: gravitational wave, large-scale structure of Universe, Weak lensing

1 INTRODUCTION Cryogenic Gravitational wave Telescope) (Akutsu et al. 2019) and LIGO-India (Unnikrishnan 2013), we can mea- The Lambda-Cold Dark (LCDM) model of cosmol- sure the gravitational wave signals from binary neutron ogy matches well all of the best available data sets probing (NS-NS), stellar- binary black holes (BH-BH), black cosmic background (CMB), galaxy distributions, hole-neutron (BH-NS) systems over the range and supernovae (Aghanim et al. 2018; Alam et al. 2017b; 10 − 1000 Hz and up to a redshift of about one. Along Abbott et al. 2018, 2019; Jones et al. 2018). However 95% with the ground-based gravitational wave detectors, - of the content of this model is composed of dark and based gravitational wave experiment such as LISA (Laser that is not yet understood. Several theoretical Interferometer Space Antenna) Klein et al.(2016); Amaro- models aid our understanding of this unknown part of the Seoane et al.(2017) are going to measure the gravitational Universe and lead to observable predictions which can be ex- wave signal emitted from supermassive binary black holes plored using electromagnetic probes of the Universe. With in the frequency range ∼ 10−4 − 10−1 Hz. In the future, the discovery of gravitational waves Abbott et al.(2016a, with ground-based gravitational wave experiments such as 2017b), a new observational probe has opened up which is

arXiv:1908.08951v2 [astro-ph.CO] 20 Mar 2020 Voyager, Cosmic Explorer Abbott et al.(2017a) and the capable of bringing new insights to our studies of the Uni- Einstein Telescope1, we will be able to probe gravitational verse. Study of the imprints of the dark matter and late-time wave sources up to a redshift z = 8 and z = 100 respectively cosmic acceleration on gravitational waves will provide new Sathyaprakash et al.(2019). This exquisite multi-frequency avenues for understanding these unknowns. probe of the Universe is going to provide an avenue towards With the ongoing ground-based gravitational wave ob- understanding the history of the Universe up to high redshift servatories such as Advanced-LIGO (Laser Interferometer and unveil the true of and dark matter. Gravitational-Wave Observatory) (Abbott et al. 2016c), In this paper, we explore the imprints of the cosmic Virgo ()(Acernese et al. 2015), and with density field on astrophysical gravitational waves and fore- the upcoming observatories such as KAGRA (Large-scale casts for measuring them using cross-correlations with the galaxy density field and galaxy weak lensing. Intervening ? [email protected] † [email protected] ‡ [email protected] 1 http://www.et-gw.eu

c 2019 The Authors 2 Mukherjee, Wandelt, & Silk cosmic structures induce distortions, namely magnifications tional weak lensing due to the distribution of the density and demagnifications in the gravitational wave signal when fluctuations in the Universe and its effect on cosmological gravitational waves propagate through spacetime. According observables such as the galaxy field and gravitational waves. to the theory of , these distortions should be universal for gravitational wave signals of any frequency for a fixed density field. These imprints should exhibit cor- 2.1 Weak lensing of galaxies relations with other probes of the cosmic density field such The observed distribution of the galaxies can be translated as galaxy surveys, as both of the distortions have a common into a density field as a function of sky position and redshift source of origin. of the source as We estimate the feasibility of measuring the lensing of ρ (ˆn, z) the gravitational wave signals emitted from astrophysical δ(ˆn, z) = m − 1, (1) sources in the frequency band 10−1000 Hz and 10−4 −10−1 ρ¯m(z)

Hz by cross-correlating with the upcoming large-scale struc- where ρm(ˆn, z) is the matter density at a positionn ˆ at the ture surveys such as DESI (Dark Energy Spectroscopic redshift z andρ ¯m denotes the mean density at z. The den- Instrument) Aghamousa et al.(2016), EUCLID Refregier sity field of the galaxies is assumed to be a biased tracer of et al.(2010), LSST (The Large Synoptic Survey Telescope) the underlying dark matter density field which can be writ- LSST Collaboration et al.(2009), SPHEREx Dore ten in Fourier space in terms of the scale-dependent bias et al.(2018a), WFIRST (Wide Field Survey Tele- bg(k) as δg(k) = bg(k)δDM (k). In the Fourier domain, we scope)(Dore et al. 2018b), etc. which also probes the cosmic define the galaxy spectrum in terms of the dark mat- 2 density field. The cross-correlation of the gravitational wave ter power spectrum by the relation Pg(k) = bg(k) PDM (k), ∗ 0 signal with the cosmic density field is going to validate the where the power spectrum is defined as hδX0 (k)δX (k )i = 3 ~ ~ 0 theory of gravity, nature of dark energy and the impact of (2π) PX0X (k)δD(k − k )(δD denotes the Dirac delta func- dark matter on gravitational wave with a high SNR over tion). a large range of redshift as discussed in Sec.5. The auto- The inhomogeneous distribution of matter between correlation of the GW sources can also be used as a probe of galaxies and the observer causes distortions in galaxy the lensing signal. However this is going to be important in shapes. The lensing convergence field can be written in terms the regime of large numbers of GW sources with high statis- of the projected line of sight matter density as tical significance such as from the 2 and Z zs 3 Ω H2(1 + z)χ(z) Cosmic Explorer (Abbott et al. 2017a). κ (ˆn) = dz m 0 g 2 cH(z) The paper is organized as follows. In Sec.2, we discuss 0 (2) Z ∞ 0 0 the effect of lensing on the galaxy distribution and on grav- 0 dngal(z ) (χ(z ) − χ(z)) dz 0 0 δ(χ(z)ˆn, z), itational waves. In Sec.3, we discuss the implication of the z dz (χ(z )) gravitational lensing of gravitational waves in validating the 0 dngal(z ) theory of gravity and the of cosmology. In where zs is the redshift of the source plane and dz0 Sec.4 and Sec.5, we obtain the estimator for measuring the is the normalized redshift distribution of galaxies. The con- signal and the forecast of measuring this from the upcoming vergence field is related to the cosmic shear by the relation gravitational wave observatories and LSS missions. Finally, Kaiser(1998) in Sec.6, we discuss the conclusions of this and its 2 2 ~ k1 − k2 ~ k1k2 ~ future scope. κg(k) = 2 2 γ1(k) + 2 2 γ2(k), (3) k1 + k2 k1 + k2

where, γ1 and γ2 are the cosmic shears in the Fourier space with ~k = (k1, k2). From the observables such as the galaxy 2 WEAK LENSING distributions and its shape, the lensing field can be inferred The general predicts universal null from the data of the upcoming galaxy surveys (Aghamousa geodesics for both gravitational wave and electromagnetic et al. 2016; Refregier et al. 2010; LSST Science Collaboration wave, that can be written in terms of the Friedmann- et al. 2009; Dore et al. 2018b,a). Lemaitre-Robertson-Walker (FLRW) metric with line ele- ments given by ds2 = −dt2 + a(t)2(dx2 + dy2 + dz2), where 2.2 Weak lensing of gravitational wave a(t) is the scale factor, is set to one at present and decreases to zero in the past. However, in the presence of density The strain of the gravitational waves from coalescing bi- fluctuations, both electromagnetic and gravitational waves naries can be written in terms of the redshifted propagate along the perturbed FLRW metric which can be Mz = (1+z)M as (Hawking & Israel 1987; Cutler & Flana- written as ds2 = −(1 + 2Φ)dt2 + a(t)2(1 + 2Ψ)(dx2 + dy2 + gan 1994; Poisson & Will 1995; Maggiore 2008) 2 dz ). One of the inevitable effects of matter perturbations r 5 G5/6M2(f M )−7/6 on both electromagnetic and gravitational waves is gravita- z z z iφz (4) h(fz) = Q(angles) 3/2 2/3 e , tional lensing which can lead to magnification, spatial de- 24 c π dL 3/5 flections and time delays in the arrival times of the signal (m1m2) where M = 1/5 is the true chirp mass in terms (Kaiser & Squires 1993; Kaiser 1998; Bartelmann & Schnei- (m1+m2) der 2001). In this paper, we discuss the aspect of gravita- of the of the individual binaries m1 and m2 and dL is the luminosity distance to the binaries which can be written for the standard LCDM model of cosmology as R z dz0 2 d = (c(1 + z)/H ) √ . G and c are the L 0 0 0 3 http://www.et-gw.eu Ωm(1+z ) +(1−Ωm)

MNRAS 000,1–16 (2019) Weak lensing of gravitational waves 3 and of respectively. Prop- Gravitational lensing of gravitational waves is going to agation of the gravitational wave in the presence of a matter probe both of these aspects of the theory of gravity from the distribution leads to magnification in the strain of gravita- observations. Several alternative theories of gravity consider tional waves in the geometric optics limit which can be writ- deviations in Eq. (8)(Cardoso et al. 2003; Saltas et al. 2014; ten as 3 (Takahashi 2006; Laguna et al. 2010; Cutler & Holz Nishizawa 2018) and Eq. (9)(Carroll et al. 2006; Bean et al. 2009; Camera & Nishizawa 2013; Bertacca et al. 2018) 2007; Hu & Sawicki 2007; Schmidt 2008; Silvestri et al. 2013; Baker et al. 2015; Lombriser & Taylor 2016; Lombriser & ˜ h(ˆn, fz) = h(fz)[1 + κgw(ˆn)], (5) Lima 2017; Saltas et al. 2014; Baker & Bull 2015; Slosar where κgw(ˆn) is the lensing convergence and can be written et al. 2019) such as (i) g(k, z) 6= 0, (ii) the modified Poisson in terms of the density fluctuations δ(χ(z)ˆn, z) along the line Equation, (iii) running of the mass which acts like of sight by the relation a frictional term leading to damping of the gravitational strain more than the usual damping due to the luminosity Z zs 2 3 ΩmH0 (1 + z)χ(z) distance, (iv) mass of the (v) speed of propagation κgw(ˆn) = dz 2 cH(z) of gravitational waves and (vi) the anisotropic stress term as 0 (6) Z ∞ dn (z0) (χ(z0) − χ(z)) a source term in the gravitational equation dz0 gw δ(χ(z)ˆn, z), 0 0 Eq. (8). A joint analysis of all of these observational aspects z dz (χ(z )) can be performed to test the validity of the general theory 0 dngw(z ) where dz0 is the normalized redshift distribution of the of relativity and cosmic acceleration using the framework gravitational wave sources. The redshift distribution of the discussed in this paper. GW sources can be determined from the electromagnetic counterparts 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 the theory of general relativity and the standard model of cos- An avenue to study gravitational wave propagation in mology, in the absence of anisotropic stress, metric pertur- the presence of matter perturbations is to cross-correlate bations are related by Φ = −Ψ, and are related to the matter the gravitational wave strain with the cosmic density field perturbation by the relation probed by other avenues such as the galaxy field, CMB and line-intensity mapping. In this paper, we explore the ∇2Φ = 4πGa2ρ δ, (7) m cross-correlation of the galaxy field with the gravitational where G is the gravitational constant and ρm is the matter wave strain from astrophysical gravitational wave sources density. which are going to be measured from upcoming gravita- tional wave observatories such as Advanced-LIGO Abbott et al.(2017a), Virgo Acernese et al.(2015), KAGRA Akutsu et al.(2019), LIGO-India Unnikrishnan(2013) and LISA 3 PROBE OF GENERAL THEORY OF Klein et al.(2016); Amaro-Seoane et al.(2017). The joint RELATIVITY AND DARK ENERGY study of the gravitational wave data along with the data Gravitational waves bring a new window to validate the gen- from large-scale structure and CMB missions will probe all eral theory of relativity and as the observable signatures (listed above) due to modifications in correct explanation of the theory of gravity and cosmic ac- the theory of gravity and dark energy by measuring the dis- celeration. Propagation of gravitational waves in the pres- tortion in the gravitational wave strain. ence of cosmic structures brings two avenues to explore the Universe: (i) the propagation of gravitational wave in the spacetime is defined by a perturbed FLRW metric Previous studies have discussed the effect of gravita- µ µν tional lensing of gravitational waves on measuring cosmolog- DµD hαβ + 2Rµανβ h = 0. (8) ical parameters using gravitational wave events only Taka- where Dµ is the covariant derivative and Rµανβ is the cur- hashi(2006); Laguna et al.(2010); Cutler & Holz(2009); vature tensor; and Camera & Nishizawa(2013); Bertacca et al.(2018); Con- (ii) For a perfect fluid energy- tensor, the evo- gedo & Taylor(2019). The measurement of the luminosity lution of the metric perturbations Φ and Ψ and its rela- distance using gravitational waves can also probe the cur- tionship with the matter perturbation can be written at all vature of the Universe (Congedo & Taylor 2019). In this cosmological epochs by the relation Hu & Sawicki(2007) work, we examine the cross-correlation of the gravitational wave strain with electromagnetic probes. This approach in- Ψ(k, z) + Φ(k, z) g(k, z) ≡ =0, troduces two new aspects. First, we can study the synergy Ψ(k, z) − Φ(k, z) (9) between the electromagnetic sector and the gravitational 1 2 2 ∇ (Ψ(k, z) − Φ(k, z)) =4πGa ρmδ, wave sector. Different theories of gravity lead to modifica- 2 tions in Eq. (8) and Eq. (9), and a joint estimation of both of these aspects is necessary to constrain these models. Sec- 3 The effects of matter perturbations on the GW signal apart ond, the cross-correlation study with large samples of data from the effect from lensing are explored in previous studies (La- from galaxy surveys makes it possible to detect the lensing guna et al. 2010) signal from noise-dominated gravitational wave data.

MNRAS 000,1–16 (2019) 4 Mukherjee, Wandelt, & Silk

4 MEASURING THE LENSING OF THE 4.2 Estimator GRAVITATIONAL WAVE USING The observed gravitational wave signal depends upon both CROSS-CORRELATIONS WITH GALAXY gravitational wave strain from the source and the effect from SURVEYS the gravitational wave propagation, which can be expressed 4.1 Theoretical signal according to Eq. (5). The observed DL from the gravita- tional wave strain can be written as The propagation of the electromagnetic waves and gravi- tational waves along the same perturbed geodesics brings dL(ˆn) DL(ˆn) = + gw(ˆn), an inevitable correlation between the galaxy ellipticity and 1 + κ(ˆn) (13) 2 distortion in the gravitational wave strain according to the = dL(ˆn)(1 − κ(ˆn)) + gw(ˆn) + O(κ ), theory of general relativity and the standard LCDM model of cosmology. The gravitational wave strain gets lensed by where gw is the observational error from the gravita- tional wave observation with zero mean and dL = (c(1 + the foreground galaxies present between the gravitational 0 z )/H ) R zs √ dz . In the weak lensing limit wave source and us. The theoretical power spectrum of the s 0 0 0 3 Ωm(1+z ) +(1−Ωm) gravitational wave lensing in the spherical harmonic basis κgw << 1, which is used to obtain the second equation in 4 can be written as Eq. 13. The luminosity distance can be inferred indepen- j dently of the chirp mass using the relation derived by Schutz κgwκgal K K Cl ≡h(κgw)lm(κg)l0m0 iδll0 δmm0 (1986) Z dz H(z)  j  −1 2 = 2 Wκgw (χ(z))Wκg (χ(z)) 1 dν/dt πMz χ c DL ∝ , where τ ≡ ∝ , ¯ 2 11/3 8/3  h(t)τν ν (πMz) ν × P ((l + 1/2)/χ(z))(χ(z)) , 2/3 δ ¯ Mz(πνzMz) (10) and h(t) ∝ , j dL κgwδ K K Cl ≡h(κgw)lmδl0m0 iδll0 δmm0 (14) Z  dz H(z) j where τ is the time-scale related to the change of the fre- = bg Wκgw (χ(z))Wδ (χ(z)) χ2 c quency of the gravitational wave signal and h¯ is the gravi-  tational wave strain averaged over detectors and source ori- × P ((l + 1/2)/χ(z))(χ(z)) , δ entations. However, the source inclination angle is impor- tant to estimate the luminosity distance accurately (Nis- where W i ,W i and W i are the kernels for gravitational κgw κg δ sanke et al. 2010). By using the two states of wave lensing, galaxy lensing and the galaxy field respectively the gravitational wave signal h+ and h×, the degeneracy be- for the tomographic bins indicated by the bin index j in tween the source inclination angle and luminosity distance redshift, which can be defined as can be lifted Nissanke et al.(2010). 1 Z ∞ dn (z0) (χ(z0) − χ(z)) Eq. (13) shows that there exists a multiplicative bias to W (χ(z)) = dz0 gw/gal , κgw/κgal H(z) dz0 (χ(z0)) κgw from dL . So, in order to remove this bias, we need to z make an estimate of the true luminosity distance for each dn W j (χ(z)) = (z ). sources. The estimate of true luminosity distance to the δ dz j (11) gravitational wave sources can be estimated using the red- shift (zs) of the source (from the EM counterparts (Nissanke In this analysis, we have taken a linear bias model with et al. 2013)) and best-fit cosmological parameters (obtained the value of the bias parameter as bg = 1.6 (Anderson et al. using the data from other cosmological probes such as CMB, es 2012; Alam et al. 2017a; Desjacques et al. 2018) and the supernovae, etc.) in the relation dL = dL(1+s), where s is non-linear dark matter power spectrum from the numerical the error in the measurement of true dL due to redshift error code CLASS (Blas et al. 2011; Lesgourgues 2011). The power and the error in best-fit cosmological parameters. Then the es spectra for galaxy lensing and gravitational wave lensing, estimator for κgw using the the ratio of DL and dL is κ κj j C gw gal and Cκgwδ , are plotted in Fig.1 as a function of l l ˆ DL(ˆn) the redshift of the gravitational wave sources. The redshift DL(ˆn) ≡ 1 − es dL (ˆn) distribution of the gravitational wave sources is taken as gw (15) dngw (1 − κgw(ˆn)) + (ˆn) dl dz (zs) = δ(z − zs) and the distribution of the galaxies is = 1 − . taken according to Models 1 + s(ˆn)

dN 2 3/2 For s << 1, we can write the above equation as (zs) =zs exp(−(zs/z0) ) Model-I, dz ˆ gw  2  (12) DL(ˆn) = κgw(ˆn) − (ˆn) + s(ˆn) − s(ˆn)κgw(ˆn). (16) dN (zs/z0) dl (zs) = exp(−zs/z0) Model-II. dz (2z0) This is feasible for the gravitational wave sources with elec- The signal of the cross-correlation gets stronger at high red- tromagnetic counterparts, the first case we will discuss. For shift due to more overlap between the gravitational wave those gravitational wave sources that do not have electro- lensing kernel and the galaxy kernel. magnetic counterparts, eq. 15 is the estimator of the lensing signal, which produces a multiplicative bias in estimation of the lensing potential κgw. We will return to this case below 4 We have used the Limber approximation (k = (l + 1/2)/χ(z)). and 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 case with 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 analysis with ∆z = 0.2 for LIGO and ∆z = 0.5 for LISA.

4.2.1 Gravitational wave events with electromagnetic 4.2.2 Gravitational wave events without electromagnetic counter part counterpart

For the case when s << 1 the corresponding estima- Returning now to the case of gravitational wave sources tor for gravitational wave lensing and galaxy lensing cross- without electromagnetic counterparts, the error term s ∼ 1. correlations in real space can be written as5 The corresponding cross-correlation signal can be written as κ (ˆn0)κ (ˆn) ˆκgwκg 0 g gw ˆκgwκg 0 ˆ C (ˆn.nˆ ) = h i, C (ˆn.nˆ ) = hDL(ˆn)κg(ˆn)i, 1 + s(ˆn) 0 (( = hκ (ˆn)κ (ˆn )i + h ((ˆn()κ((ˆn)i  1  gw g (s g = Cκg κgw (ˆn.nˆ0) ,  0 −hs(ˆn)ihκgw(ˆn)κg(ˆn )i, 1 + s(ˆn) (17) 0 (18) ˆκgwδ 0 ˆ κ δ 0 κgw(ˆn)δ(ˆn ) C (ˆn.nˆ ) = hDL(ˆn)δ(ˆn)i, Cˆ gw (ˆn.nˆ ) = h i, 0  1 + s(ˆn) = hκgw(ˆn)δ(ˆn )i + hs(ˆn)δ(ˆn)i   1   0 κgwδ 0 −hs(ˆn)ihκgw(ˆn)δ(ˆn )i. = C (ˆn.nˆ ) , 1 + s(ˆn) The estimated cross-correlation signal can have a multi- Here, the term  is not correlated with the lensing and den- s plicative bias if h i 6= 0 for sources without EM counter- sity field of the galaxy δ and κ and hence goes to zero on s g gw parts. However, for a large number of gravitational wave averaging over a large number of sources. As a result, the sources, all that is required to remove this bias is N (z), estimator of the cross-correlation remains unbiased by any gw the redshift distribution of the gravitational wave sources. error in the luminosity distance measurement for the gravi- Using the clustering approach, redshifts of the gravitational tational wave sources with electromagnetic counterpart. wave sources can be obtained using the methods described in (Menard et al. 2013; Oguri 2016; Mukherjee & Wandelt 2018). These estimators in the spherical harmonic basis 6

6 Any field X(ˆn) can be expressed in the spherical harmonics 5 P The quantities with hat denote the estimator of the signal basis as X(ˆn) = lm XlmYlm(ˆn)

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ˆκgwκg 1 P can be expressed as Cl = 2l+1 m(κgw)lm(κg)lm the Universe will be probed by future missions such as DESI ˆκGW δ 1 P Aghamousa et al.(2016), EUCLID Refregier et al.(2010), and Cl = 2l+1 m(κgw)lmδlm. The real space cross- correlation estimator given in Eq. 17 is related to the spher- LSST LSST Science Collaboration et al.(2009), SPHEREx ical harmonic space cross-correlation estimator by the rela- Dore et al.(2018a), WFIRST (Dore et al. 2018b), having tion a wide sky coverage and access to galaxies up to high red-   shift ( about z ∼ 3). For the large-scale structure surveys, ˆ 0 X 2l + 1 0 ˆXX0 CXX 0 (ˆn.nˆ ) = Pl(ˆn.nˆ )Cl , (19) we consider the survey setup with two different normalized 4π l redshift distributions according to Eq. 12 with the value of 0 where, Pl(ˆn.nˆ ) are the Legendre polynomials. z0 = 0.2 and z0 = 0.5. We have considered two different sets 2 The variance of the cross-correlation signal can be writ- of number density of galaxies as ng = 0.5 per arcmin for 2 ten in terms of the available overlapping sky fraction of both z0 = 0.2 and 45 per arcmin for z0 = 0.5 in this analysis. the missions fsky as The redshift distribution of the large scale structure surveys overlaps with the the redshift distribution of the gravita- 1  gw−X 2 κgwκgw DD XX XX tional wave sources. (σl ) = (Cl + Nl )(Cl + Nl ) fsky(2l + 1) For gravitational waves, we have considered the set-up  κ X for both ground-based and space-based gravitational wave + (C gw )2 , l experiments such as Advanced-LIGO and LISA. For the (20) gravitational wave detectors, we have considered the cur- 7 XX 2 rent instrument noise for Advanced-LIGO and LISA as where, X ∈ g, κg, Nl = σX /nX corresponds to the shape shown in Fig.3. The LISA noise curves are calculated using noise (σ = 0.3) and the galaxy (σ = 1) for X = 8 κg g the online tool with the instrument specification provided κg and g respectively. nX denotes the number density of DD in Table1(Klein et al. 2016). sources per sr. Nl is the variance related to the estimation The currently ongoing ground-based gravitational wave of the gravitational wave lensing signal, which can be written observatories such as Advanced-LIGO Abbott et al.(2017a) as and VIRGO Acernese et al.(2015), and the future detectors  2 2  4π σd σ 2 2 such as KAGRA Akutsu et al.(2019), LIGO-India Unnikr- DD l b l θmin/8 ln 2 (21) Nl = 2 + 2 e , Ngw dl dl ishnan(2013) will be joining these in the coming decade. These experiments will be capable of reaching up to a red- where Ngw is the number of gravitational wave sources, θmin is the sky localization area of the gravitational wave sources, shift of z ∼ 1 and are going to detect the gravitational 2 wave primarily from stellar origin binary mergers σdl is the error in the luminosity distance due to the detector 2 (BH-BH), black hole mergers (BH-NS) and bi- noise and σb is the error due to the error in the gravitational wave source redshift and uncertain values of cosmological nary neutron star mergers (NS-NS). The exact rates parameters. of these binary mergers are not yet known and are predicted The joint estimation of all the effects from the prop- from the current observation of the gravitational wave events agation of the gravitational wave signal can be written in Abbott et al.(2016b, 2017b). an unified frame work by combining the observable from We have taken the number of the sources of gravi- the gravitational wave and the cosmic probes such as the tational waves as a free parameter for each of these bi- nary species. Among these sources, BH-NSs and NS-NSs galaxy surveys δg and κg as are expected to have an electromagnetic counterpart (Bhat-  ˆ  DL tacharya et al. 2019; Nissanke et al. 2013) and hence the O =  δˆ  (22) redshift to these sources can be measured. We have consid- κˆg ered the spectroscopic measurement of the source redshift and the corresponding covariance matrix can be expressed for the electromagnetic counterpart from BH-NS and NS- as NS. Sources such as BH-NSs and NS-NSs with electromag- netic counterparts can have an accurate estimation of the  DLDL κgwδ κgwκg  Cl Cl Cl sky localization (less than one arcsec), which makes it pos- C ≡ hOO†i =  κgwδ δδ δκg  .  Cl Cl Cl  sible to improve the cross-correlation signal with the LSS κgwκg δκg κg κg Cl Cl Cl surveys. For the stellar mass BH-BHs, we do not expect any electromagnetic counterpart and as a result, the redshift of Here, the terms in bold indicates the new cosmological the source cannot be measured. For BH-BHs we have consid- probes which will be available from the cross-correlation ered that the redshift of the source is completely unknown. of the gravitational wave signal with the galaxy field. The For BH-BHs without any electromagnetic counterparts, we cross-correlation of the galaxy surveys with the stochastic will be relying only on measurements from the gravitational gravitational wave signal is another probe to study the un- wave signal and as a result, the sky localization error is going resolved sources (Mukherjee & Silk 2019). have relatively large errors compared to the case of BH-NS and NS-NS. For the combination of five detectors, we can 5 FORECAST FOR THE MEASUREMENT OF LENSING OF GRAVITATIONAL WAVE 7 https://dcc.ligo.org/cgi-bin/DocDB/ShowDocument? The measurement of the galaxy lensing/gravitational wave .submit=Identifier&docid=T1800044&version=5 lensing and galaxy/gravitational wave lensing can be ex- 8 http://www.srl.caltech.edu/~shane/sensitivity/ plored from upcoming missions. The large-scale structure of 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 latest projected noise. The LISA noise is plotted for one year of integration time with the LISA instrument specifications given in Table1.

Table 1. Specifications of LISA instrument properties Amaro- tectors such as LISA (Klein et al. 2016; Amaro-Seoane et al. Seoane et al.(2017) is used with 1 year of integration time to 2017; Smith & Caldwell 2019) will also be operational in produce Fig.3 using the online tool Larson et al.(2000, 2002). the next decade and will be able to reach to very high redshifts z ∼ 20. These experiments are going to explore supermassive BH-BH systems over a wide range of masses ∼ 103 − 107 M . Such sources are going to be observed in Specifications Values the frequency band 10−4 − 10−1 Hz for time-scales of days Armlength 2.5 × 109 m to years before mergers of the binaries. The event rates for these sources are largely unknown and are mainly based on Optics diameter 0.3 m studies made from simulations Micic et al.(2007). For the 7 1064 nm heavier supermassive BH-BH systems such as 10 M , the event rates are going to be less than the case for the interme- Laser power 2.0 W 4 diate supermassive BH-BH systems of mass 10 M , accord- Optical train efficiency 0.3 ing to our current understanding of the formation of these binaries. We consider three different event rates for unit red- Acceleration noise 3 × 10−15 √m s2 Hz shift bins, 50, 100 and 300 in this analysis for LISA. For the Position noise 2 × 10−9 √m supermassive binary BHs, we can expect to see electromag- Hz netic counterparts over a wide of electromagnetic frequency Sensitivity noise floor Position spectrum (Giacomazzo et al. 2012; Haiman 2018; Palenzuela et al. 2010; Farris et al. 2015; et al. 2014; Armitage & Astrophysical Noise Natarajan 2002). This is going to be useful for obtaining the redshift of the gravitational wave sources by the electromag- netic follow up surveys. In the future, the next-generation gravitational wave expect about 10 sq. deg sky localization error in the sources observatories will include Voyager 9, which is going to of the gravitational wave (Nissanke et al. 2011; Pankow et al. achieve an instrument noise of ∼ 10−24 Hz−1/2. There are 2018). In this analysis, we have taken the angular resolution also proposals for ground-based observatories with larger θ (or equivalently the maximum value of l, lmax ∝ 1/∆θ) as the free parameter and obtained the results for a few different choices. 9 https://dcc.ligo.org/public/0142/T1700231/003/ Along with the ground-based detectors, space-based de- 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 κgwκg Cl 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 = 31.5 M . (c) The BH-BH sources are considered for equal mass-ratios with Mtotal = 60 M . 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)

κgwδ Figure 5. The cumulative SNR for the measurement of the cross-correlation between gravitational wave lensing and galaxy field Cl 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 a 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 = 31.5 M . (c) The BH-BH sources are considered for equal mass-ratio with Mtotal = 60 M . 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) Weak lensing of gravitational waves 11 baselines such as the Einstein Telescope (Hild et al. 2011) identified as individual events for redshifts below 0.5 and we and Cosmic Explorer (Abbott et al. 2017a). These obser- considered the low redshift galaxy survey as shown by the vatories are capable of exploring stellar-mass BH-BHs, BH- red coloured curve in Fig.2 to estimate the cross-correlation NSs and NS-NSs up to a redshift about z ∼ 20 for the signal between galaxy lensing and gravitational wave lens- frequency range 10−1000 Hz. These observatories are going ing. The measurability of the signal remains weak at nearly κGW κg to have orders-of-magnitude improvement in the instrument all the values of redshift for Cl with lmax = 3000. But noise over LIGO. As a result, our proposed methodology κGW δ the signal from Cl can be detected with high SNR for is going to be an exquisite probe of the Universe for better redshift z > 0.35 from the sources of the gravitational wave understanding gravity, dark matter and dark energy by com- which will be detected by Advanced-LIGO-like detectors. bining the probes from gravitational wave and electromag- BH-NS systems with the lower (or higher) mass of the black netic wave. In future work, we will estimate the prospects hole can be identified as individual events only up to a lower of this approach for the next-generation gravitational wave (or higher) redshift, and the corresponding detection thresh- detectors. old will vary. The BH-NS systems are the best scenario for The signal-to-noise ratio for the measurement of the studying the cross-correlation signal as this can have an elec- cross-correlation of the gravitational wave lensing by cross- tromagnetic counterpart and can also be detected up to a correlating with the galaxy-lensing (X = κGW ) and with high redshift than for the NS-NS systems. the galaxy density field X = δ can be estimated using For the BH-BH systems, with Mtotal = 60 M , gravita- κ X C GW 2 Plmax l tional wave events can be detected up to a redshift z = 1. (S/N) = κ X 2 . Using the gravitational wave l (σ GW ) These sources are possibly not going to have electromag- detector noise, we estimate the error in the luminosity dis- netic counterparts and as a result, the redshifts of the ob- tance σ using a Fisher analysis by considering the two dl jects will remain unknown. We have considered the redshift polarization states of the gravitational wave signal, of the GW sources to be unknown to obtain the forecast Z ∞ X 1 2 ∂h+,× 1 ∂h+,× for BBHs. Along with the absence of the redshifts, these FDlDl = 4f d ln f 2 . (23) 2 ∂Dl |hn(f)| ∂Dl +,× 0 sources are going to have poor sky localization ∼ 10 sq. de- gree even with the five detector network (LIGO-H, LIGO-L, The corresponding Cramer-Rao bound on the parameters Virgo, KAGRA, LIGO-India) (Pankow et al. 2018). Using q −1 is obtained as σD = (F ). The noise due to lensing l DlDl a minimum sky localization error of 10 sq. degree and with causes an error in the estimation of the luminosity distance. for the case with unknown redshift, we estimate the mea- κGW κg κGW δ This can be written in terms of the fitting form as given by surability of Cl and Cl for a galaxy survey shown Hirata et al.(2010). The error in the redshift σz = C(1 + z) by the black curve in Fig.2 (which reaches up to redshift of the gravitational wave sources leads to an additional error z = 1). The measurability of the signal remains marginal es ∂dl in the estimation of the dl as σs = ∂z σz. We have taken for 5 years of observation time with the current estimate of C = 0.03 for the cases with photo-metric measurements of the LIGO events. The estimates of the mean signal can also the redshift, and C = 0 for spectroscopic measurements of be biased for a lesser number of gravitational wave samples, the redshift. due to the unavailability of the true luminosity distance to In Fig.4 and Fig.5, we show the cumulative SNR for the GW source (as shown in Eq. 16). For a large number κGW κg κGW δ the measurement of Cl and Cl signals for binary of the gravitational wave sources, clustering redshift of the neutron star, black hole neutron star systems and binary gravitational wave source can be inferred by using the spa- black holes respectively. For binary neutron stars, current tial correlations (Menard et al. 2013; Oguri 2016; Mukherjee gravitational wave detectors will be able to detect individ- & Wandelt 2018). ual objects with high SNR for redshifts z < 0.2. As a re- Detection of more events is going to improve the pro- sult, we need to consider galaxy surveys which are going to jected SNR. In these estimates of the SNR for NS-NSs, probe the low redshift Universe such as DESI (Aghamousa BH-NSs, and BH-BHs, we have only considered the inspiral et al. 2016). Using the galaxy distribution shown by the red phase of the gravitational wave signal. Further improvement colours in Fig.2, we estimate the cumulative SNR up to red- of the SNR is possible if we include merger and the ringdown shift z = 0.2 for lmax = 3000 and assuming spectroscopic phase of the gravitational wave strain. This will be consid- measurements of the redshifts of the binary neutron stars. ered in a future analysis including the complete waveform Detection is not possible for the currently considered event of the gravitational wave signal. κGW κg rate within five years of observation time from Cl . But For the space-based gravitational wave detectors such as κGW δ for Cl , we can expect a cumulative SNR of 3 detection of LISA, the gravitational wave events are going to be detected the signal up to redshift z = 0.2 for the highest event rates up to a redshift about z ∼ 20 Amaro-Seoane et al.(2017). and the best sky-fraction scenario. For the redshift range This enables us to probe the cosmological signal up to a 0.3 > z > 0.2, LIGO will be detecting several very low SNR high redshift by cross-correlating with galaxy surveys which gravitational wave events. The cross-correlation signal with reach up to a high redshift. Using the redshift distribution of the low SNR gravitational wave events (which may not be galaxies shown by the blue curve in Fig.2 which mimics the considered as a detected event) and if the electromagnetic surveys such as LSST, we estimate the SNR for the measura- κGW κg κGW δ counterpart of these sources can be identified, then more bility of Cl and Cl for different masses of the BH- κGW δ 4 7 than a 3-σ measurement is possible only from Cl . How- BHs ranging from 10 − 10 . In this analysis, we have con- ever, this scenario is extremely rare and the detection of the sidered a maximum time period of one year of inspiral phase signal beyond z > 0.2 is unlikely. before the merger. The merger and the ringdown phase for For the BH-NS systems, we have estimated the SNR for these sources are not considered in the analysis. For the MNS = 1.5 M and MBH = 30 M . These sources can be LISA 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 lensing κgwκg 5 6 Cl from LISA for an observation time of 4 years for equal mass binary black holes with individual masses (a) 10 , (b) 10 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 κgwδ 5 6 Cl from LISA for an observation time of 4 years for equal mass binary black holes with individual masses (a) 10 , (b) 10 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) 14 Mukherjee, Wandelt, & Silk ror of σz/(1 + z) = 0.03 in the determination of the redshift tational wave observatories makes it possible to study these of gravitational wave sources from the electromagnetic coun- aspects and opens a new window in observational cosmology. terpart. Measurement of the electromagnetic counterpart is The astrophysical gravitational wave signals which are emit- also going to be useful to reduce the sky localization error ted from compact objects of different types, such as white of the source of the gravitational wave and hence improves dwarfs, neutron stars and black holes, span a vast range of the detection of the cross-correlation signal. For two differ- masses, gravitational wave and cosmological red- ent values of lmax, lmax = 200 and lmax = 1000, we estimate shifts. All of these sources are transients and are going to be κGW κg κGW δ the SNR for the measurement of Cl and Cl signal frequent. As a result, they can probe the underlying theory for four years of observation time and show the correspond- of gravity and the statistical properties of the cosmological ing estimates in Fig.6 and Fig.7 respectively. As can be perturbations present in the Universe multiple times from seen from these figures, the discovery space from synergy different sources. As a result, all the above-mentioned as- between LISA and LSST is promising for each mass win- pects of fundamental physics can be studied using transient dow of BH-BHs. These SNRs are going to improve with astrophysical gravitational wave sources. the inclusion of the merger and the ring down phase of the In this paper, we have explored one of the important gravitational wave signal. Further improvement in the SNR effects of matter perturbations, the weak lensing of the grav- is possible with the increase in the LISA observation time itational wave strain, and propose an avenue to explore to ten years, increase in the number of gravitational wave this signal using cross-correlations with other cosmological event rates and also with galaxy surveys which can go up probes such as galaxy surveys. The measurement of the lens- to higher redshift. By combining all the high redshift probes ing signal leads to a measurement of the of the such as EUCLID Refregier et al.(2010), LSST LSST Science gravitational potential over a wide range of cosmic redshifts Collaboration et al.(2009), SPHEREx Dore et al.(2018a) and also probes the connection of the matter perturbation and WFIRST Dore et al.(2018b), the detectability of the with the gravitational potential. The joint study of cosmic signal is going to improve significantly. At redshifts higher density field and gravitational wave propagation in space- than z > 3, the temperature and polarization anisotropies time promises to be a powerful avenue towards a better un- of the CMB is going to be a powerful probe to measure the derstanding of the theory of gravity and the dark sector lensing signal to even higher redshift as shown by (Mukher- (dark matter and dark energy) of the Universe. jee et al. 2019). Using the cross-correlation between CMB- Several alternative theories of gravity and models of lensing and gravitational wave lensing, we are going to have dark energy can be distinguished from the evolution of the higher SNR measurements from redshift z > 3 than possi- cosmic structures (Carroll et al. 2006; Bean et al. 2007; Hu & ble from the galaxy surveys with the redshift distributions Sawicki 2007; Schmidt 2008; Silvestri et al. 2013; Baker et al. considered in this analysis (Fig.2). Complementary infor- 2015; Baker & Bull 2015; Slosar et al. 2019) and also from mation from cross-correlation with intensity mapping of the the propagation of gravitational wave in spacetime (Car- 21-cm line is another avenue to be explored in future work. doso et al. 2003; Saltas et al. 2014; Nishizawa 2018). As a result, the studies of the cross-correlations between the gravitational wave signal and cosmic structures should re- sult in a more comprehensive understanding of the theory 6 CONCLUSIONS of gravity and dark energy. The imprint of the lensing po- The general theory of relativity provides a unique solution tential is also a direct probe of the dark matter distribu- to the propagation of gravitational waves which depends tion in the Universe, and measurement of this via gravi- upon the underlying spacetime metric. Gravitational waves tational wave is going to explore the gravitational effects of according to the theory of general relativity are massless dark matter on gravitational waves. The measurement of the and should propagate along null geodesics with the speed of gravitational lensing signal from gravitational waves with a light. As a result, in the framework of the standard model frequency ranging from 10−4 to 103 Hz will probe the uni- of cosmology, gravitational wave should propagate through versality of gravity on a wide range of energy scales and lead the perturbed FLRW metric where the perturbations arise to a direct probe of the . Finally, this due to the spatial fluctuations in the matter density in every new avenue is also going to illuminate several hitherto un- position. As a result of this metric perturbation, the gravi- explored windows which can be studied from the synergies tational wave strain is distorted. between electromagnetic waves and gravitational waves. Study of these distortions in the gravitational wave Ground based gravitational wave detectors such as strain while propagating through the perturbed metric can LIGO-US Abbott et al.(2017a) , Virgo Acernese et al. be a direct observational probe of several aspects of funda- (2015), KAGRA Akutsu et al.(2019), LIGO-India Unnikr- mental physics, which include (i) the validation of general ishnan(2013) and space based gravitational wave detec- relativity from the propagation of gravitational wave sig- tor such as LISA Klein et al.(2016); Amaro-Seoane et al. nals and from the connection between metric perturbations (2017) are going to probe a large range of cosmic redshift and matter perturbations, (ii) properties of dark energy and 0 < z . 20 with gravitational wave sources such as NS- its effect on gravitational waves, (iii) gravitational effects of NS, BH-NS, stellar mass BH-BH and supermassive BH-BH. dark matter on gravitational waves and evolution with cos- By cross-correlating with the data from upcoming galaxy mic redshift, (iv) testing the equivalence principle from the surveys such as DESI (Aghamousa et al. 2016), EUCLID multi-frequency character of the gravitational wave signal, (Refregier et al. 2010), LSST (LSST Science Collaboration (v) equivalence between the electromagnetic sector and the et al. 2009), SPHEREx (Dore et al. 2018a), WFIRST (Dore gravitational sector over the cosmic time-scale. et al. 2018b), we will be able to obtain a high SNR detec- The coming decade with ongoing and upcoming gravi- 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 with ACKNOWLEDGEMENT electromagnetic counterparts such NS-NS, BH-NS and su- S. M. would like to thank Karim Benabed, Luc Blanchet, per massive BH-BHs. For stellar mass black hole binaries, Sukanta Bose, Neal Dalal, Guillaume Faye, Zoltan Haiman, we do not expect electromagnetic counterparts, and hence David Spergel, and Samaya Nissanke for useful discussions. poorly determined source redshifts. As a result, the measur- The results of this analysis are obtained using the Hori- ability of the weak lensing signal is going to be marginal zon cluster hosted by Institut d’Astrophysique de Paris. We from stellar mass BH-BHs. thank Stephane Rouberol for smoothly running the Horizon cluster. The Flatiron Institute is supported by the Simons We find that the most promising avenue with Advanced- Foundation. The work of SM and BDW is also supported LIGO will be the cross-correlation of lensed black hole neu- by the Labex ILP (reference ANR-10-LABX-63) part of the tron star mergers and the galaxy overdensity CκGW δ. The l Idex SUPER, and received financial state aid managed by contributions from binary neutron stars and binary black the Agence Nationale de la Recherche, as part of the pro- holes are marginal and only improve if the event rates are gramme Investissements d’avenir under the reference ANR- larger than considered in this analysis. However, with fu- 11-IDEX-0004-02. The work of BDW is also supported by ture surveys such as Cosmic Explorer and Einstein Tele- the grant ANR-16-CE23-0002. BDW thanks the CCPP at scope, we can explore the spatial cross-correlation of binary NYU for hospitality during the completion of this work. In black holes and galaxies in order to obtain the clustering this analysis, we have used the following packages: IPython redshift (Menard et al. 2013) of the sources (Oguri 2016). (P´erez& Granger 2007), Matplotlib (Hunter 2007), NumPy This can result in an improvement in the correlation sig- (van der Walt et al. 2011), and SciPy (Jones et al. 01). nal. By fitting the shape of the correlation function between the spatial position of binary black holes and underlying galaxies, an accurate estimate of the source redshift is pos- sible (Mukherjee & Wandelt 2018) for a large sample of bi- REFERENCES nary black holes. The gravitational wave sources from LISA are going to provide a high SNR measurement from both Abbott B. P., et al., 2016a, Phys. Rev. Lett., 116, 061102 CκGW κg and CκGW δ over a vast range of gravitational wave Abbott B. P., et al., 2016b, Astrophys. J., 833, L1 l l Abbott B. P., et al., 2016c, Phys. 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