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  Energy density of dust in f(R) gravity

+ 1 like - 0 dislike

Usually during the calculation of the Einstein-Hilbert action, we consider the integral of the variation of the Ricci tensor to be zero as the variation of it, at the boundary was assumed to be zero, as:



Now, if we are to continue calculating this integral with the vanishing boundary, we obtain the following expression for the variation of the Ricci tensor:
Using the above result in the action, we obtain:
Now, under the vacuum conditions, $\Lambda=0$, and  $T_{\mu{\nu}}=0$, we obtain a value for the Ricci scalar as:
Now, in the case of Dust (in co-moving coordinates), $T_{\mu{\nu}}=\rho{g_{0\mu}g_{0\nu}}$, we obtain the following:
This yields:
But we know that the density in the dust model has the following form:
This implies:

If we replace $1/G$ with a scalar field $\phi$, which varies from one place to another, then we obtain the following form:
Note: the D'alembertian for a scalar field is: $\Box=\frac{1}{\sqrt{|g|}}\partial_{\mu}(\sqrt{|g|}\partial^{\mu})$, hence, $\kappa=G/c^{4}=1/(c^{4}\phi)$.
From Brans-Dicke theory:$$\Box{\phi}=\frac{8\pi{T}}{(3+2\omega)}$$
where $\omega$ is the dimensionless Dicke coupling constant.
This yields: >$$\Lambda\approx\frac{4\pi{T}}{\phi(3+2\omega)}$$

Under strong coupling, i.e. $\omega>-1.5$, $\Lambda$ is positive ($\Lambda\approx\frac{2\pi{\rho}}{3\phi}$ when $\omega=1.5$). Under weak coupling, i.e. $\omega<-1.5$, $\Lambda$ is negative. Is my derivation correct? What can be the possible implications of this? I am aware that in f(R) gravity, in Einstein field equations when linearized on a de Sitter background gives rise to wave equations for the de Sitter covariant field $\phi_{\mu{\nu}}$: $$(\Box-2\Lambda)\phi\approx{8\pi{\kappa}T_{\mu{\nu}}}$$

When $\Lambda$ is negative, we can observe that the equation takes the form of Klein-Gordon equations for a massive spin-zero field.

asked Jun 20, 2017 in Theoretical Physics by Naveen (85 points) [ revision history ]
recategorized Jun 20, 2017 by Dilaton

Somehow it is not clear to me what do you mean by expressions such as $R = 3 \Box$. On the left-hand-side I see a scalar and on the right a differential operator. Are you assuming a space of test functions or tensors? What is it and what is its meaning?

I am not sure about that part too, but it seems to make sense when I use it in Einstein field equations when linearized on a de Sitter background, which gives rise to wave equations for the de Sitter covariant field ϕμν. ​Also the d'alembertian would change its form for a scalar field as when I replace (1/G) with a scalar field ϕ​.

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