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

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 $W(φ)=-1$ and $V(φ) = -Ξφ$, 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 $φ$ 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 $φ$ 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 $V(φ)$ 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 $W(φ)$ and a linear $V(φ)$ imply that $φ$ 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 $W(φ)$ and $V(φ)$, that can still make the theory reduce to Refracted Gravity in the weak-field limit, might alleviate these problems.

💡 Research Summary

In this paper the authors explore whether a single scalar‑tensor theory of gravity, belonging to the Brans–Dicke class, can simultaneously account for the dynamics of galaxies, galaxy clusters, the accelerated expansion of the Universe, and the formation of large‑scale structure without invoking any dark matter or dark energy components. The model is defined by two functions of the scalar field φ: the kinetic coupling W(φ) and the self‑interaction potential V(φ). The authors fix these to the simple forms

 W(φ)=−1, V(φ)=−Ξ φ with Ξ>0.

These choices are motivated by a previous work (Sanna et al. 2022) which showed that, in the weak‑field limit, the scalar field can be identified with twice the “gravitational permittivity” ε(ρ) that appears in Refracted Gravity (RG), a non‑relativistic modification of Poisson’s equation introduced to reproduce the observed rotation curves and mass‑to‑light relations of galaxies and clusters without dark matter. In this limit the modified Poisson equation reads

 ∇·(ε ∇Ψ)=4πG ρ,

and the identification φ=2ε reproduces exactly the RG equation.

Background cosmology
Assuming a spatially flat, matter‑dominated Friedmann‑Lemaître‑Robertson‑Walker (FLRW) background, the authors derive the modified Friedmann equations from the Brans–Dicke action with the above W and V. The scalar field contributes an effective energy density ρ_φ that behaves like a dark‑energy component. By solving the background equations analytically (for a pure matter‑dominated case) and numerically (including the scalar contribution), they find that the Hubble parameter H(z) matches current observational determinations (e.g., BAO, supernovae) when the constant Ξ is chosen of order the present‑day cosmological constant. Thus the model can reproduce the observed late‑time acceleration.

Linear perturbation theory
The core of the paper is the development of linear cosmological perturbation theory on top of the FLRW background. The metric perturbations are expressed in the Newtonian gauge with potentials Ψ and Φ, and the scalar field perturbation δφ is introduced. By linearising the field equations, the authors obtain a set of coupled equations for δ_m (the baryonic matter density contrast), Ψ, Φ, and δφ. A key result is the growth‑factor equation for the matter perturbations:

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