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 of gravitational waves and galaxy surveys Suvodip Mukherjee1;2?, Benjamin D. Wandelt1;2;3 y & Joseph Silk1;2;4;5 z 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 Astrophysics, Flatiron Institute, 162 5th Avenue, 10010, New York, NY, USA 4 The Johns Hopkins University, Department of Physics & Astronomy, 3400 N. Charles Street, Baltimore, MD 21218, USA 5 Beecroft Institute for Cosmology and Particle Astrophysics, University of Oxford, Keble Road, Oxford OX1 3RH, UK 23 March 2020 ABSTRACT The cross-correlation of gravitational wave 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 spacetime. 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 redshift Universe (z < 0:5). With the next generation gravitational wave experi- ments such as Voyager, LISA, Cosmic-Explorer and Einstein Telescope, we can extend this test of the theory of gravity to larger redshifts 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 Matter (LCDM) model of cosmol- sure the gravitational wave signals from binary neutron stars ogy matches well all of the best available data sets probing (NS-NS), stellar-mass binary black holes (BH-BH), black cosmic microwave background (CMB), galaxy distributions, hole-neutron star (BH-NS) systems over the frequency 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, space- of the content of this model is composed of dark energy and based gravitational wave experiment such as LISA (Laser dark matter 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 nature of dark energy and dark matter. Gravitational-Wave Observatory) (Abbott et al. 2016c), In this paper, we explore the imprints of the cosmic Virgo (Virgo interferometer)(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] y [email protected] z [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 general relativity, 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 (^n; z) the gravitational wave signals emitted from astrophysical δ(^n; 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(^n; z) is the matter density at a positionn ^ 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 Science Collaboration et al.(2009), SPHEREx Dore ten in Fourier space in terms of the scale-dependent bias et al.(2018a), WFIRST (Wide Field Infrared 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 power 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 Einstein Telescope 2 and Z zs 3 Ω H2(1 + z)χ(z) Cosmic Explorer (Abbott et al. 2017a). κ (^n) = dz m 0 g 2 cH(z) The paper is organized as follows. In Sec.2, we discuss 0 (2) Z 1 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)^n; 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 standard model 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 work 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 theory of relativity 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 chirp mass 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 ).