Linear perturbation theory and structure formation in a Brans-Dicke theory of gravity without dark matter

TL;DR

This work tests a covariant Brans–Dicke gravity with $W(\varphi)=-1$ and $V(\varphi)=-\Xi\varphi$ as a unified description of dark matter and dark energy in a baryon-only universe. The authors derive the background cosmology, perform linear perturbation analysis, and compute the growth of structure and lensing implications, finding that while the background can fit $H(z)$, the growth of perturbations is significantly delayed (growth occurs at $z<1$) due to the $H^{-1}(z)$-scaled source term, and lensing is suppressed because $\nabla^2\Phi=0$ while $\nabla^2\Psi$ tracks $\delta\rho$ with a factor $2/\varphi$. The quasi-static approximation confirms these features and reveals parameter degeneracies, with best fits favoring $\varphi_0>1$ and $\Omega_{\Xi 0}\sim0.5$, but the model fails to reproduce the observed timing of structure formation. Overall, the study shows that this particular BD+RG realization cannot account for large-scale structure formation, suggesting that different choices of $W(\varphi)$ and $V(\varphi)$ (still reducing to RG in the weak-field limit) are needed to reconcile background expansion, growth history, and lensing.

Abstract

We investigate the formation of the large-scale cosmic structure in a scalar-tensor theory of gravity belonging to the class of the Brans--Dicke theories. The universe contains baryonic matter alone and neither dark matter nor dark energy. The two arbitrary functions of the scalar field characterizing the kinetic term and the self-interaction potential are set to $W(\varphi)=-1$ and $V(\varphi) = -Ξ\varphi$, respectively, with $Ξ$ a positive constant. In the weak-field limit, the theory reduces to Refracted Gravity, a non-relativistic theory whose modified Poisson equation contains the scalar field $\varphi$ that provides the gravitational boost required to describe the dynamics of galaxies and galaxy clusters without dark matter. In a flat, matter-dominated, homogeneous and isotropic universe the same scalar field $\varphi$ drives the accelerated expansion of the universe and describes the observed redshift evolution of the Hubble-Lemaître parameter $H(z)$. However, in the equation of the growth factor of the linear perturbation theory, the form of $V(\varphi)$ makes the coefficient of the source of the gravitational field proportional to $H^{-1}(z)$; therefore the gravitational field is strongly suppressed at early times and structure formation is delayed to redshift $z< 1$, in disagreement with the observation of formed galaxies at much larger redshifts. In addition, the form of $W(\varphi)$ and a linear $V(\varphi)$ imply that $\varphi$ generates twice the gravitational boost on massive particles than on photons, with possible observable consequences on the gravitational lensing phenomenon. It remains to be investigated whether different choices of $W(\varphi)$ and $V(\varphi)$, that can still make the theory reduce to Refracted Gravity in the weak-field limit, might alleviate these problems.

Figures (5)

• Figure 1: Comparison between the evolution of $H(a)$ (left) and $\varphi(a)/\varphi_0$ (right) derived analytically in a radiation-free universe and numerically in a universe containing both matter and radiation. The bottom panels show the percentage difference between the analytical and numerical solutions. The two solutions agree with each other on the entire redshit range except for $H(a)$ at $z \gtrsim 100$.
• Figure 2: Redshift evolution of the Hubble-Lemaı tre parameter $H(z)$ for different values of $(\varphi_0, \, \Omega_{\Xi 0})$ (solid curves). The solid points with error bars show the observational estimates from Yu:2017iju.
• Figure 3: Examples of the MCMC chains. We show six sets of four panels each. Each set depicts the evolution of a single chain, including the burn-in set. The upper and middle sets of panels show the pair of free parameters $(\varphi_0, \Omega_{\Xi 0})$, with the Hubble-Lemaı tre parameter $H_0$ added for a consistency check. The prior for the value of $\varphi_0$ is set uniformly in the range $[0, 100]$. In the panels with the red curves, the initial value for $\varphi_0$ is chosen uniformly in the range $0<\varphi_0<1$, whereas in the middle sets (panels with the blue curves) this range was $1<\varphi_0<100$. The two lower sets with the green curves show the pair $(\Omega_0, \Omega_{\Xi 0})$.
• Figure 4: Posterior distributions of the free parameters $\Omega_0$, $\Omega_{\Xi 0}$ and $H_0$ from the MCMC analysis. The median and the 0.16 and 0.84 quantiles are reported for each distribution (orange lines and dashed blue lines, respectively). In the $(\Omega_0, \Omega_{\Xi 0})$ parameter space, the chains preferentially populate the region adjacent to the upper limit of $\Omega_{\Xi 0}$, with the preferred value for $\Omega_0$ consistent with zero at $2\sigma$.
• Figure 5: Evolution of the density contrast $\delta_{\textbf{k}}$ (top panel) and scalar field perturbations $\delta\varphi_{\textbf{k}}$ (bottom panel) as a function of redshift, for $k = 1 \, \text{Mpc}^{-1}$. Different curves assume different values for the parameter $\varphi_0$; the other free parameter, $\Omega_{\Xi 0}$, was fixed to the upper limit of Eq. (\ref{['omega_xi condition']}), that was shown to be the best-fit value for the background evolution, as explained in Section \ref{['Numerical background solution and $H(z)$ fit']}. The dashed red line in the top panel represents the $\Lambda$CDM prediction.