Modified gravitational-wave propagation and standard sirens

Studies of dark energy at advanced gravitational-wave (GW) interferometers normally focus on the dark energy equation of state $w_{\mathrm{DE}}(z)$. However, modified gravity theories that predict a nontrivial dark energy equation of state generically also predict deviations from general relativity in the propagation of GWs across cosmological distances, even in theories where the speed of gravity is equal to $c$. We find that, in generic modified gravity models, the effect of modified GW propagation dominates over that of $w_{\mathrm{DE}}(z)$, making modified GW propagation a crucial observable for dark energy studies with standard sirens. We present a convenient parametrization of the effect in terms of two parameters $({\mathrm{\Xi }}_0,n)$, analogue to the $(w_0,w_a)$ parametrization of the dark energy equation of state, and we give a limit from the LIGO/Virgo measurement of $H_0$ with the neutron star binary GW170817. We then perform a Markov chain Monte Carlo analysis to estimate the sensitivity of the Einstein Telescope (ET) to the cosmological parameters, including $({\mathrm{\Xi }}_0,n)$, both using only standard sirens, and combining them with other cosmological data sets. In particular, the Hubble parameter can be measured with an accuracy better than 1% already using only standard sirens while, when combining ET with current $\mathrm{CMB}+\mathrm{BAO}+\mathrm{SNe}$ data, ${\mathrm{\Xi }}_0$ can be measured to 0.8%. We discuss the predictions for modified GW propagation of a specific nonlocal modification of gravity, recently developed by our group, and we show that they are within the reach of ET. Modified GW propagation also affects the GW transfer function, and therefore the tensor contribution to the ISW effect.

Figure 1

The DE EoS in the RR nonlocal model with $u_0=0$ (blue solid line). By comparison we also show the $(w_0,w_a)$ parametrization with the values $w_0=-1.15$ and $w_a=0.09$ (magenta dashed line), obtained by fitting the function $w_{\mathrm{DE}}(a)$ to the parametrization (5) near $a=1$.

Figure 2

The function $\delta (z)$ in the RR nonlocal model (blue solid line) and the fit given by Eq. (24).

Figure 3

The ratio $d_L^{\mathrm{gw}}(z)/d_L^{\mathrm{em}}(z)$ in the RR nonlocal model, as a function of redshift (blue solid line) and the fit given by Eq. (22).

Figure 4

The relative difference $\mathrm{\Delta }{d}_L/d_L$ between $w\mathrm{CDM}$ with $w=-1.1$ and $\mathrm{\Lambda }\mathrm{CDM}$. Green, dot-dashed curve: using the same values of ${\mathrm{\Omega }}_M$, $H_0$ for the two models. Magenta, dashed curve: using in each model its own best-fit values of ${\mathrm{\Omega }}_M$, $H_0$.

Figure 5

The relative difference $\mathrm{\Delta }{d}_L/d_L$ between $w\mathrm{CDM}$ with $w=-0.9$ and $\mathrm{\Lambda }\mathrm{CDM}$. Green, dot-dashed curve: using the same values of ${\mathrm{\Omega }}_M$, $H_0$ for the two models. Magenta, dashed curve: using in each model its own best-fit values of ${\mathrm{\Omega }}_M$, $H_0$.

Figure 6

The relative difference $\mathrm{\Delta }{d}_L/d_L$ between the nonlocal RR model and $\mathrm{\Lambda }\mathrm{CDM}$. Green, dot-dashed curve: using the same values of ${\mathrm{\Omega }}_M$, $H_0$ for the two models. Magenta, dashed curve: using in each model its own best-fit values of ${\mathrm{\Omega }}_M$, $H_0$. Blue solid curve: the relative difference using, for the RR model, the GW luminosity distance (and the respective best-fit values of ${\mathrm{\Omega }}_M$, $H_0$ for the two models).

Figure 7

A sample of 1000 sources distributed in redshift according to Eq. (76), and scattered in $d_L(z)$ according to the ET error estimate (74). The black and red curves show the theoretical prediction for $d_L(z)$ and the $1\sigma$ ET error, respectively. The cosmological model assumed is $\mathrm{\Lambda }\mathrm{CDM}$ with ${\mathrm{\Omega }}_M=0.3087$ and $H_0=67.64$.

Figure 8

The distribution of sources as a function of redshift.

Figure 9

The $1\sigma$ and $2\sigma$ contours of the two-dimensional likelihood in the $({\mathrm{\Omega }}_M,H_0)$ plane in $\mathrm{\Lambda }\mathrm{CDM}$, with the contribution from $\mathrm{CMB}+\text{BAO}+\mathrm{SNe}$ (red), the contribution from $10^3$ standard sirens at ET (gray) and the overall combined contours (blue).

Figure 10

The $1\sigma$ and $2\sigma$ contours of the two-dimensional likelihood in the $({\mathrm{\Omega }}_M,w_0)$ plane in $w\mathrm{CDM}$, with the contribution from $\mathrm{CMB}+\text{BAO}+\mathrm{SNe}$ (red), the contribution from $10^3$ standard sirens at ET (gray) and the overall combined contours (blue).

Figure 11

The two-dimensional likelihood in the $(w_0,w_a)$ plane, with the combined contribution from $\mathrm{CMB}+\text{BAO}+\mathrm{SNe}$ (red), the contribution from $10^3$ standard sirens at ET (gray), and the total combined result (blue).

Figure 12

The two-dimensional likelihood in the $({\mathrm{\Xi }}_0,w_0)$ plane, with the combined contribution from $\mathrm{CMB}+\text{BAO}+\mathrm{SNe}$ (red), the contribution from $10^3$ standard sirens at ET (gray), and the total combined result (blue).

Figure 13

The two-dimensional likelihood in the $({\mathrm{\Xi }}_0,H_0)$ plane, with the combined contribution from $\mathrm{CMB}+\text{BAO}+\mathrm{SNe}$ (red), the contribution from $10^3$ standard sirens (gray), and the total combined result (blue).

Figure 14

The two-dimensional likelihood in the $({\mathrm{\Xi }}_0,{\mathrm{\Omega }}_M)$ plane, with the combined contribution from $\mathrm{CMB}+\text{BAO}+\mathrm{SNe}$ (red), the contribution from $10^3$ standard sirens (gray), and the total combined result (blue).

Figure 15

The two-dimensional likelihood in the $(n,H_0)$ plane, from $10^3$ standard sirens at ET.

Figure 16

Average and standard deviation of $\mathrm{\Delta }{\chi }^2={\chi }_{\mathrm{RR}}^2-{\chi }_{\mathrm{\Lambda }\mathrm{CDM}}^2$. The horizontal line corresponds to the threshold value $\mathrm{\Delta }{\chi }^2=6$.

Figure 17

As in Fig. 16, restricting to sources with redshift $0.07<z<0.7$.

Figure 18

As in Fig. 16, restricting to sources with $0.7<z<2$.

Figure 19

As in Fig. 16, using the electromagnetic luminosity distance of the RR model.

Figure 20

The DE EoS in the RR nonlocal model with $u_0=250$.

Figure 21

The ratio $d_L^{\mathrm{gw}}(z)/d_L^{\mathrm{em}}(z)$ in the RR nonlocal model with $u_0=250$, as a function of redshift (blue solid line) compared to the fitting function (22) with $n=2.3$ (dashed).

Figure 22

As in Fig. 16, for $u_0=250$.