One of the independent measures to the EM luminosity distance ${\rm d}_l^{\rm EM} (z)$ for a metric theory is through its relation with the geometric distance (the angular diameter distance) ${\rm d}_A(z)= {\rm d}_l^{\rm EM}(z)/(1+z)^2$, according to Etherington’s reciprocity theorem (Etherington 2007) or the distance duality relation. This relation is valid for EM probes if photon number is conserved, and photons propagate along null geodesics. The distance duality relation has been tested from several observations (Holanda, Lima & Ribeiro 2010; Holanda, Gonçalves & Alcaniz 2012; Liao et al. 2016) and will also be tested more stringently in the future (Liao et al. 2016; Arjona et al. 2020; Martinelli et al. 2020; Renzi et al. 2020). The angular diameter distance to any redshift z is related to the BAO scale θBAO in the matter correlation function by (Peebles & Yu 1970; Bond & Efstathiou 1984; Hu & Sugiyama 1996; Eisenstein & Hu 1998, 1997)
(4)$$\begin{eqnarray*}
\theta _{\rm BAO} (z)= \frac{r_{\rm s}}{(1+z){\rm d}_A(z)},
\end{eqnarray*}$$where, $r_{\rm s}= \int _{z_{\rm d}}^\infty {\rm d}z c_{\rm s}(z)/ H(z)$ is the sound horizon where zd denotes the drag redshift. As a result, we can relate the BAO scale with the EM luminosity distance ${\rm d}_l^{\rm EM}$ by the relation
(5)$$\begin{eqnarray*}
{\rm d}_l^{\rm EM} (z)= \frac{(1+z) r_{\rm s}}{\theta _{\rm BAO}(z)}.
\end{eqnarray*}$$Using equation (5) in equation (3), we can write
(6)$$\begin{eqnarray*}
{\rm d}^{\rm GW}_l(z)= \exp {\bigg (-\int {\rm d}z^{\prime } \frac{\gamma (z^{\prime })}{1+z^{\prime }}\bigg)}\frac{(1+z) r_{\rm s}}{\theta _{\rm BAO} (z)}.
\end{eqnarray*}$$This is the key equation of this paper. In this expression, the measurement of the term $\frac{r_{\rm s}}{\theta _{\rm BAO}(z)}$ comes from EM probes such as large-scale structure galaxy redshift surveys (Eisenstein et al. 2005; Dawson et al. 2013; Alam et al. 2017) and cosmic microwave background (CMB) (Spergel et al. 2003, 2007; Komatsu et al. 2011; Hinshaw et al. 2013; Planck Collaboration XIII 2016; Aghanim et al. 2020), and the measurement of $d_l^{GW}$ arises from the GW strain. We can write the above equation as
(7)$$\begin{eqnarray*}
{\rm d}^{\rm GW}_l(z)\theta _{\rm BAO}(z)= \exp {\bigg (-\int {\rm d}z^{\prime } \frac{\gamma (z^{\prime })}{1+z^{\prime }}\bigg)}(1+z) r_{\rm s}.
\end{eqnarray*}$$This relation shows that the product of the BAO angular scale θBAO(z) and the luminosity distance ${\rm d}^{\rm GW}_l(z)$ can measure the frictional term γ(z) as a function of redshift. So, the concordance between the EM geometric probes and the GW luminosity probes allows a way to test the theory of gravity. If the general theory of relativity is the correct theory of gravity, then the product between θBAO(z) and ${\rm d}^{\rm GW}_l(z)$ should vary with redshift as (1 + z). Any deviation from this scaling can be a signature of alternative theories of gravity. The quantities θBAO and ${\rm d}_l^{\rm GW}$ are measured from large-scale structure and GW data, and the value of rs depends on recombination physics and the sound speed in the baryon–photon fluid at the time of decoupling at redshift z ≈ 1100 (Silk 1968; Peebles & Yu 1970; Sunyaev & Zeldovich 1970; Hu, Sugiyama & Silk 1997). As a result, this relation is nearly model-independent, and can be written directly in terms of observables such as ${\rm d}_l^{\rm GW}(z)$, and θBAO(z).
Citation: Hu & Sugiyama 1996
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