Virial mass in warped DGP-inspired $\mathcal{L}(R)$ gravity

Abstract

A version of the virial theorem is derived in a brane-world scenario in the framework of a warped DGP model where the action on the brane is an arbitrary function of the Ricci scalar, $\mathcal{L}(R)$. The extra terms in the modified Einstein equations generate an equivalent mass term (geometrical mass), which give an effective contribution to the gravitational energy and offer viable explanation to account for the virial mass discrepancy in clusters of galaxies. We also obtain the radial velocity dispersion of galaxy clusters and show that it is compatible with the radial velocity dispersion profile of such clusters. Finally, we compare the result of the model with $\mathcal{L}(R)$ gravity theories.

Appendices

Appendix A: Field equations in four-dimensional $\mathcal{L}(R)$ gravity

In this appendix we derive the four-dimensional limit of Eq. (18). Substituting Λ(R), Σ(R), Θ(R) and Λ ₄(R) into Eq. (18), we obtain

(97)

Now we can recover the standard four-dimensional $\mathcal{L}(R)$ gravity from above equation in the limit κ ₅→0, while keeping the Newtonian gravitational constant $\kappa_{4}^{2} = \frac{1}{6}\kappa_{5}^{4}\lambda_{b}$ finite (Maeda et al. 2003)

(98)

where

(99)

neglecting the effective cosmological constant we can also rewrite equation (98) as

(100)

where $\mu^{2} \equiv\frac{1}{\kappa_{4}^{2}}$ and the curvature fluid energy-momentum tensor is defined as

(101)

which is exactly the four-dimensional field equation in $\mathcal{L}(R)$ gravity theories (Nojiri and Odintsov 2011; Capozziello and Laurentis 2011).

Appendix B: Virial theorem in $\mathcal{L}(R)$ gravity

Our aim in this appendix is to obtain the four-dimensional limit of the virial theorem in modified DGP model and to compare its results with pure $\mathcal{L}(R)$ gravity theories. From Eq. (49) the virial theorem in modified DGP model is given by

(102)

in the limit κ ₅→0, with keeping the Newtonian gravitational constant $\kappa_{4}^{2} = \frac{1}{6}\kappa_{5}^{4} \lambda_{b}$ finite, we obtain

(103)

neglecting the effective cosmological constant, Λ ₄, we can rewrite the above equation as

(104)

where $\mu^{2} \equiv\frac{1}{\kappa_{4}^{2}}$. As mentioned before we present $\frac{d\mathcal{L}}{dR}$ as $\frac{d\mathcal{L}}{dR} = 1+\varepsilon g'(R)$ where ε is a small quantity and g′(R) describes the modifications of the geometry due to the presence of the tensor Θ _μν . Therefore, we have

(105)

where

(106)

is the geometrical energy density. Equation (105) is exactly Eq. (17) in Boehmer et al. (2008). Also the resulting virial theorem from Eq. (105) is given by

(107)

where

(108)

(109)