The fifth force, propagated by the scalar degree of freedom, affects the Poisson equations associated to the Newtonian potential Φ, as well as the relativistic one, Ψ, according to (Kobayashi, Watanabe & Yamauchi 2015; Crisostomi & Koyama 2018; Dima & Vernizzi 2018),
(1)$$\begin{eqnarray*}
\frac{\text{d} \Phi (r)}{\text{d}r} = \frac{G M(r)}{r^2} \left[1+\frac{3}{4}Y_1\left(\frac{\rho (r)}{\bar{\rho }(r)}\right)\left(2+\frac{\text{d}\ln \rho }{\text{d}\ln r}\right)\right],
\end{eqnarray*}
$$(2)$$\begin{eqnarray*}
\frac{\text{d} \Psi (r)}{\text{d}r} =\frac{G M(r)}{r^2}\left[1-\frac{15}{4}Y_2\left(\frac{\rho (r)}{\bar{\rho }(r)}\right)\right].
\end{eqnarray*}
$$In the above equations, we have assumed spherical symmetry. $\bar{\rho }(r)$ is the (spatially) average density at radius r from the centre of the galaxy cluster, and Y1, Y2 correspond to the dimensionless fifth-force couplings. Finally, G is the Newton’s constant. Although the dynamics of member galaxies in the cluster is governed by the potential Φ, lensing is sourced by the combination
(3)$$\begin{eqnarray*}
\frac{\mathrm{ d}}{\mathrm{ d}r} \Phi _{\rm {lens}} = \frac{1}{2}\frac{\mathrm{ d}}{\mathrm{ d}r}(\Phi + \Psi).
\end{eqnarray*}
$$Therefore, kinematical observations allow for contraints on Y1, while lensing constrains both Y1 and Y2. The right-hand side of above equation can be expressed in terms of the density profile ρ(r) according to the relevant equations for Φ and Ψ above. The dominant source of pressureless matter density in the cluster comes from dark matter, which density we choose to model with a Navarro-Frenk-White (NFW) of Navarro, Frenk & White (1997) profile as
(4)$$\begin{eqnarray*}
\rho (r)=\frac{\rho _\text{s}}{r/r_\text{s}(1+r/r_\text{s})^2},
\end{eqnarray*}
$$with ρs is a characteristic density and rs the radius at which the logarithmic derivative of the density profile takes the value −2. The NFW profile has been shown to provide an overall good agreement with observations and simulations over a broad range of scales in GR (e.g. Biviano et al. 2013; Umetsu et al. 2016; Peirani et al. 2017) and in MG (e.g. Lombriser et al. 2012a; Wilcox et al. 2016). Moreover, the GR analyses with lensing and internal kinematics of both clusters indicate that the total mass profile is well fitted by the NFW model (Biviano et al. 2013; Umetsu et al. 2016; Caminha et al. 2017; Sartoris et al. 2020). Under the assumption of a NFW profile, we can re-write the equation for the potential Φ in an effective way as
(5)$$\begin{eqnarray*}
\frac{\text{d}\Phi }{\text{d}r} \equiv \frac{G M_{\text{dyn}}}{r^2}=\frac{G}{r^2}\left[ M_{\rm {NFW}}(r)+M_1(r)\right],
\end{eqnarray*}
$$which serves as a definition of the dynamical mass Mdyn. Notice that, G here is still Newton’s constant as measure in the Solar system. The fifth-force contribution M1 is defined in terms of the NFW parameters as
(6)$$\begin{eqnarray*}
M_1(r)= M_{200}\frac{Y_1}{4}\frac{r^2(r_\text{s}-r)}{(r_\text{s}+r)^3}\times [\ln (1+c)- c/(1+c)]^{-1}.
\end{eqnarray*}
$$where c = r/rs is the concentration and M200 is the mass of a sphere of radius r200 enclosing an average density 200 times the critical density of the universe at that redshift. In a similar fashion, the relevant expression for the lensing mass can be found by computing
(7)$$\begin{eqnarray*}
M_{\text{lens}}(r) =\frac{r^2}{2G}\left[\frac{\text{d}\Psi }{\text{d}r}+\frac{\text{d}\Phi }{\text{d}r}\right].
\end{eqnarray*}
$$$$\begin{eqnarray*}
M_{\text{lens}}=M_{\text{NFW}}+\frac{r^2M_{200}\left[Y_1(r_\text{s}-r)-5Y_2(r_\text{s}+r)\right]}{4[\log (1+c_{200})-c_{200}/(1+c_{200})]}\frac{1}{(r_\text{s}+r)^{3}},
\end{eqnarray*}
$$which can be effectively re-expressed in terms of the dynamical mass as
(8)$$\begin{eqnarray*}
M_{\text{lens}} \equiv M_{\text{dyn}}+M_2,
\end{eqnarray*}
$$with M2 the contribution from the fifth force defined through
(9)$$\begin{eqnarray*}
M_2=\frac{r^2M_{200}}{8(r_\text{s}+r)^{3}}\frac{Y_1(r-r_\text{s})-5Y_2(r_\text{s}+r)}{[\ln (1+c)-c/(1+c)]}.
\end{eqnarray*}
$$In view of the above equations, it is important to emphasize again that, although the fifth force effect enters the dynamical mass only through the coupling Y1, the lensing mass is affected by both Y1 and Y2. This is expected, since lensing is sourced by the combination of the two potentials Φ and Ψ, equation (3). Note also that, with gravitational lensing observations, one reconstructs the projected surface mass density profile Σ(R), where R is the projected radius from the cluster centre. We refer to e.g. Umetsu (2020) for an explicit discussion of the physics and mathematical framework.
Citation: Umetsu et al. 2016
Citation function: Similarities
Citation: Umetsu et al. 2016
Citation function: Similarities