# The large scale structure in the 3D luminosity-distance space and its cosmological applications

###### Abstract

Future gravitational wave (GW) observations are capable of detecting millions of compact star binary mergers in extragalactic galaxies, with luminosity-distance () measurement accuracy and better than arcminute positioning accuracy. This will open a new window of the large scale structure (LSS) of the universe, in the 3D luminosity-distance space (LDS), instead of the 3D redshift space of galaxy spectroscopic surveys. The baryon acoustic oscillation and the AP test encoded in the LDS LSS constrain the - (comoving angular diameter distance) relation and therefore the expansion history of the universe. Peculiar velocity induces the LDS distortion, analogous to the redshift space distortion, and allows for a new structure growth measure . When the distance duality is enforced (), the LDS LSS by itself determines the redshift to level accuracy, and alleviates the need of spectroscopic follow-up of GW events.But a more valuable application is to test the distance duality to level accuracy, in combination with conventional BAO and supernovae measurements. This will put stringent constraints on modified gravity models in which the gravitational wave deviates from the electromagnetic wave . All these applications require no spectroscopic follow-ups.

###### pacs:

98.80.-k; 98.80.Es; 98.80.Bp; 95.36.+xIntroduction.—
Discoveries of gravitational wave (GW) produced by black hole
(BH)/neutron star (NS)-BH/NS mergers
(Abbott
et al., 2016a, 2017a, 2017b, 2017c)
have opened the era of
gravitational wave astronomy. These GW events can serve as standard
sirens to measure cosmological distance from first principles
(Schutz, 1986; Abbott
et al., 2017d) and therefore avoid
various systematics associated with traditional
methods. It will then have profound impact on
cosmology. However, to fulfill this potential, usually it requires
spectroscopic follow-ups to determine redshifts of their host
galaxies or electromagnetic counterparts. This will be challenging, for
the third generation GW experiments such as the Big Bang Observer
(BBO, Cutler and Harms (2006); Cutler and Holz (2009)), and the
Einstein Telescope^{1}^{1}1https://tds.virgo-gw.eu/?call_file=ET0106C10.pdf, which will detect millions of
these GW events. Various alternatives have been proposed to
circumvent this stringent need of spectroscopic follow-ups Namikawa et al. (2016); Oguri (2016); Nair et al. (2018); Mukherjee and Wandelt (2018).

We point out a new possibility to circumvent this challenge. These GW events are hosted by galaxies and are therefore tracers of the large scale structure (LSS). With arcminute positioning accuracy and level accuracy in the luminosity distance determination achievable by BBO, we are able to map the 3D large scale structure in the luminosity-distance space (LDS). It is analogous to the redshift space LSS mapped by the conventional spectroscopic redshift surveys of galaxies (). Therefore it also contains valuable information of baryon acoustic oscillation (BAO), both across the sky and along the line of sight. As BAO in the redshift space measures the comoving angular diameter distance and at given redshift bins, BAO in LDS measures and at given bins. Here is the comoving radial distance. Both the - relation and the - relation constrain cosmology (Fig. 1), without the need of redshift. Furthermore, both and can be converted into cosmological redshift through the distance duality relation . Similar to the redshift space distortion (RSD), peculiar velocity also induces the luminosity-distance space distortion (LDSD). This will enable a new measure of structure growth rate , which differs from measured in RSD by a redshift dependent factor.

The luminosity-distance space LSS.— Each GW event provides a 3D position (, ). With millions of them, arcminute positioning accuracy, and accuracy in , we are able to measure the number density fluctuation over effectively the entire cosmic volume. This LSS is statistically anisotropic, since differs from its cosmological value ,

(1) |

Here is the lensing convergence, describing the effect of
gravitational lensing magnification. This effect is a highly valuable
source of cosmological information
(e.g. Cutler and Holz (2009)). is the physical
peculiar velocity (Hui and Greene, 2006) and is the
line of sight unit vector. If an object is moving
away from us (), photons/GWs take longer time to reach us and suffer
more cosmic dimming. At , and . Naively one would think the lensing effect
overwhelms the peculiar velocity effect. This is indeed the case if
we can subtract with cosmological redshift from spectroscopic follow-up. However, what affects the LDS LSS is the gradient of and along
the line of sight. Under the distance observer approximation and up to
leading order, . Since is lack of variation along the line of
sight, its contribution is sub-dominant comparing to the velocity
gradient contribution^{2}^{2}2The area amplification of lensing
adds an extra term to . However, it is
orders of magnitude smaller than the intrinsic clustering
(e.g. Yang and Zhang (2011)). . The LDS power spectrum then resembles
the Kaiser Kaiser (1987) plus Finger of God formula in RSD,

(2) |

Here and . () is the wavevector perpendicular (parallel) to the line of sight. is the density bias of GW host galaxies. describes the FOG effect. There are two majo differences to RSD. First,

(3) |

It differs from in RSD by a redshift dependent factor. This arises from the different effects of peculiar velocity on the luminosity distance (, and on redshift (). The prefactor in Eq. 3 is zero at and increases with . It becomes larger than unity at , where the peculiar velocity induced distortion is larger in LDS than in redshift space. The second difference is that the factor shown up in FOG should be replaced by .

Cosmological applications.— Now we proceed to constraints on , and using the LDS power spectrum measurement. Assuming Gaussian distribution in the power spectrum measurement errors, the Fisher matrix is

(4) |

The sum is over bins. Instead of directly fitting , and , we fit their ratios () with respect to the fiducial cosmology, along with . Namely . () scales the pair separation perpendicular (parallel) to the line of sight. Under such scaling,

(5) |

Statistical error in the power spectrum measurement is

(6) |

is the number of independent Fourier modes in the bin, proportional to the survey volume . () is the window function parallel(perpendicular) to the line of sight, due to statistical errors in the measurement and angular positioning.

We adopt the fiducial cosmology as the CDM cosmology with , , , , and . We are targeting at BBO or experiments of comparable capability. BBO has a positioning accuracy better than 1 arc-minute for all NS/BH-NS/BH mergers in the horizon Cutler and Holz (2009). Since we are only interested at large scale (), to excellent approximation. In contrast, and the distance measurement error has a significant effect. For typical and , the induced damping is significant at . This limites the power spectrum measurement to the linear regime. On one hand, it reduces the constraining power. On the other hand, it simplifies the theoretical modeling, and allows us to neglect the FOG term in Eq. 2. is the average number density of GW events in the survey volume. The local NS-NS merger rate is constrained to Abbott et al. (2017c). The BH-BH merger rate is a factor of smaller Abbott et al. (2016b). Therefore is dominated by NS-NS mergers. For the evolution of NS-NS merger rate, we adopt the model in Cutler and Harms (2006); Cutler and Holz (2009). For the bestfit , the total number of GW events per year is at respectively.

We find that the luminosity-distance space LSS is capable of constraining , and in multiple bins to a few percent accuracy (Fig. 2). These estimations adopt years and . Since the power spectrum measurement error is shot noise dominated, the statistical errors roughly scale as . But their dependence on is more complicated. Fig. 2 shows the cases of , within the reach of BBO capability. has major impact on cosmology, by significantly affecting the number of accessible Fourier modes. For which may be achieved by BH-BH merger observations of BBO or NS-NS mergers observations of more advanced experiments, cosmological constraints can be significantly improved, especially for and .

These constraints alone are able to constrain dark energy, demonstrated in Fig. 1. One way to under its constraining power is that, when the distance duality holds (), the - relation is equivalent to the more familiar - relation in the supernovae cosmology. It indeed contains valuable information of dark energy. However, due to lower number density and larger error in the measurement, these constraints are significantly worse than what will be achieved by stage IV redshift surveys such as DESI DESI Collaboration et al. (2016) and Euclid Amendola et al. (2016).

Nevertheless, these measurements are unique in constraining modified gravity (MG) models. In these models, GW propagation may differ from electromagnetic wave propagation and . This has been proposed and been applied to constrain gravity (e.g. Deffayet and Menou (2007); Pardo et al. (2018)). There are two degrees of freedom to modify the GW propagation equation Belgacem et al. (2018). One allows for deviation between the GW speed and the speed of light. However, GW170817 Abbott et al. (2017c) has constrained the relative difference to be within Abbott et al. (2017e), and ruled out a large fraction of MG models (e.g. Amendola et al. (2018)). In contrast, the other degree of freedom is essentially unconstrained. This is to modify the friction term in the GW propagation equation. Belgacem et al. (2018) parametrizes this modification as . To avoid confusion of with the commonly used LSS symbol, we adopt a different notation . leads to

(7) |

Usually we expect no deviation from GR in the early epoch (). A simple parameterization satisfying this condition is . Under this parametrization, .

Combining the - and/or - measurements from electromagnetic wave telescopes, and the - measurements here, we can measure . Combining the - and - measurements can also constrain this ratio. BBO can measure this ratio and constrain to percent level accuracy (Fig. 3). It will then be sensitive to MG models such as the RR model with correction in the action Maggiore (2014); Belgacem et al. (2018). Since this test of gravity is on the tensor part of space-time metric, it is highly complementary to tests on the scalar part. The statistical error here is dominated by the GW observations. will allow for better than accuracy in , and longer observations can further help.

Further applications.— We point out that future GW experiments will map LSS in a new space, namely the luminosity-distance space (LDS), through the luminosity-distance determined using NS/BH-NS/BH mergers. We present a proof of concept study on its major LSS patterns (BAO and LDSD), and list a few cosmological applications (constraining dark energy, determining cosmological redshift and probing gravity). It has other applications. One is to probe the primordial non-Gaussianity. Another is to probe the horizon scale gravitational potential, since it alters the luminosity distance and generate a relativistic correction to the number density distribution of GW events. Both require the LSS measurement near the horizon scale. The LDS LSS is in particular suitable since it naturally covers the whole sky and can extend to . Furthermore, the LDS LSS is free of all systematics associated with dust extinction, star confusion, masks and survey boundaries, due to the transparency of GWs. This will also make it advantageous in probing horizon scale LSS.

Including the cross correlation with the redshift space LSS overlapping in the survey volume, its power in constraining cosmology can be significantly enhanced. First, it will enable more accurate redshift determination, in a way independent of galaxy clustering modelling and different to existing proposals Oguri (2016); Nair et al. (2018); Mukherjee and Wandelt (2018). Since galaxy surveys have higher number density, the constraint on (and ) will be improved by than what shown in Fig. 3. This point will be addressed in a companion paper. Combining the LSS in the two spaces will also reduce cosmic variance in constraining primordial non-Gaussianity, gravitational potential and peculiar velocity, following the cosmic variance cancellation technique McDonald and Seljak (2009). Furthermore, cross correlations beween the luminosity-distance space and redshift space are also valuable for studies of stellar evolution and galaxy formation, such as constraining the NS/BH-galaxy relation. Notice that all these applications only require the overlap of GW observations and galaxy observations in cosmic volume. No spectroscopic follow-ups of GW events are required at all. Given the advance of DESI, Euclid, SKA and even more advance surveys Dodelson et al. (2016), this requirement will be automatically satisfied. Given these potentials, we recommend more comprehensive studies of LSS in the luminosity-distance space.

Finally we address that the above proposal does not invalidate the usefulness of spectroscopic redshift follow-ups. With spectroscopic redshifts, the lensing field can be measured to high accuracy Cutler and Holz (2009). The velocity field, instead of the velocity gradient causing LDSD, can be determined as well Zhang and Chen (2008). Therefore massive spectroscopic follow-ups of GW events, although highly challenging, will be highly desirable as well.

Acknowledgement.— This work was supported by the National Science Foundation of China (11621303, 11433001, 11653003, 11320101002), and National Basic Research Program of China (2015CB85701).

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