Probing gravity and growth of structure with gravitational waves and galaxies’ peculiar velocity

The low-redshift velocity field is a unique probe of the growth of cosmic structure and gravity. We propose to use distances from gravitational wave (GW) detections, in conjunction with the redshifts of their host galaxies from wide field spectroscopic surveys (e.g., DESI, 4MOST, TAIPAN), to measure peculiar motions within the local Universe. Such measurement has the potential to constrain the growth rate $f{\sigma }_8$ and test gravity through determination of the gravitational growth index $\gamma$, complementing constraints from other peculiar velocity measurements. We find that binary neutron star mergers with associated counterpart at $z\lesssim 0.2$ that will be detected by the Einstein Telescope (ET) will be able to constrain $f{\sigma }_8$ to $\sim 3\%$ precision after 10 years of operations when combined with galaxy overdensities from DESI and TAIPAN. If a larger network of third generation GW detectors is available (e.g., including the Cosmic Explorer), the same constraints can be reached over a shorter timescale ($\sim 5\mathrm{years}$ for a 3 detectors network). The same events (plus information from their hosts’ redshifts) can constrain $\gamma$ to ${\sigma }_{\gamma }\lesssim 0.04$. This constraint is precise enough to discern general relativity from other popular gravity models at $3\sigma$. This constraint is improved to ${\sigma }_{\gamma }\sim 0.02–0.03$ when combined with galaxy overdensities. The potential of combining galaxies’ peculiar velocities with gravitational wave detections for cosmology highlights the need for extensive optical to near-infrared follow-up of nearby gravitational wave events, or exquisite GW localization, in the next decade.

Figure 1

Expected constraints on $f(z){\sigma }_8(z)$ using peculiar velocities measured using binary neutron star mergers from a 10 year 3G GW experiment (light blue errorbars with triangles) and in combination with DESI/TAIPAN (dark blue errorbars). DESI/TAIPAN RSD-only conservative expected constraints are shown by the green triangles, shifted to higher $z$ for visualization purposes. These are compared to constraints from a 10 year SN survey covering $2\pi$ of the sky that is able to recover a SN magnitude dispersion of ${\sigma }_M=0.08$ in the same redshift bins (red triangles, shifted to higher $z$ for visualization purposes). The results from a combination of such survey with DESI/TAIPAN RSDs is shown by the light orange triangles, and they are very similar to the SN-only case. Prospects for 100 BNSs from LIGO/Virgo (HLV) at design sensitivity are also reported (darker orange triangle), in combination with galaxy surveys. All the results represented by triangles are computed in this work. The remaining data points represent existing $f(z){\sigma }_8(z)$ measurements from 6dF [64], WiggleZ [65], SDSS-II LRG [66], SDSS-II Main Galaxy sample [67], BOSS [68], VIPERS [69] and eBOSS-CMASS [70]. We also report the forecast from [20] (black pentagon), who use a combination of GW and $\mathrm{PSC}z$ galaxy survey data. The darker purple line is the theoretical prediction for $f(z){\sigma }_8(z)$ in a Flat $\mathrm{\Lambda }\mathrm{CDM}$ Universe with $\gamma =0.55$ (GR), while the other curves show the theoretical prediction from $\gamma =0.42$ and $\gamma =0.68$ (which are the values predicted for $f(R)$ and DGP gravity, respectively).

Figure 2

Growth index uncertainty for different values of distance precision as a function of BNS volumetric rates integrated over time out to $z_{\max }<0.3$ (left panel), and as a function of the maximum redshift $z_{\max }$ out of which we consider GW events. The distance uncertainty ${\sigma }_{d_{*}}$ is the uncertainty at a reference distance $d_{*}$, here corresponding to $z_{*}=0.1$. The BNS volumetric rate considered for the specific estimates on the left, and all points on the right panel, is the maximum a posteriori value from the latest GW estimate, $1.09\times {10}^{-6}\times (h/0.679{)}^3{\mathrm{Mpc}}^{-3}{\mathrm{yr}}^{-1}$ [40]. On the left panel, also the low and high 90% CL limits in the rate are shown for an ideal 3G experiment (where the distance precision is 1% at $z=0.1$) after 5 years. The effect of the shift in number density from the high to the low bound of the 90% CL range is shown by the white line. The boxes represent limits within which we expect the constraint to fall for the various ET configurations studied in [27]. The white triangle shows our result using the [25] approximation for 1 ET. For comparison, we also show the constraints forecasted for DESI RSD and for an LSST–like SN peculiar velocity survey from [5] (lighter dashed lines). The distance precision for the ideal 3G experiment is also shown for reference on the right panel by the dashed line. The arrow shows possible constraints from the A+ LIGO/Virgo/KAGRA network, if improved viewing angle constraints can be derived from EM observations, reducing the distance uncertainty from order $\sim 20\%$ down to 10% or better.

Figure 3

Comparison between a SN (dashed lines) and a GW (solid lines) experiment to measure the growth index from peculiar velocities, for different distance uncertainties at $z=0.1$, as a function of the maximum distance reach of the SNe/GW mergers. In this idealized case, BNSs and SNe have the same value of $\frac{{\sigma }_v^2}{n}$ at $z=0.1$, implying that the reported ${\sigma }_d/d$ precision corresponds to a magnitude uncertainty at all redshifts ${\sigma }_M=\frac{5}{\ln (10)}\frac{{\sigma }_d}{d}$ for the SNe. Both SN and GW experiments cover 90% of the sky. Differences in the $\gamma$ constraints are driven by the different redshift dependence of the distance uncertainty measurement. The shaded region shows where constraints on $\gamma$ can discern between popular gravity models (${\sigma }_{\gamma }\lesssim 0.04$).