Gogoi-Goswami f(R) gravity model

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Gogoi-Goswami f(R) gravity model is a dark energy f(R) gravity model which is a modification of general relativity.

History

The model was first introduced by D. J. Gogoi and U. D. Goswami to study the polarization modes of gravitational waves in f(R) gravity in metric formalism. In this theory, the Ricci scalar R of Einstein - Hilbert action is replaced by a function of R. Although there are a good number of f(R) gravity models, only a few can explain experimental results and can pass the observational constraints. Apart from this model, some other successful dark energy f(R) gravity models capable of explaining the drawbacks of general relativity up to a significant range are the Starobinsky model, Hu-Sawicki model and Tsujikawa model . There are five main requirements for an f(R) gravity model to describe the late-time dark energy problem. These are:

(1) A sufficient and suitable chameleon mechanism which allows $f(R)$ gravity to pass the constraints of local systems.

(2) A late-time stable de-Sitter solution.

(3) A positive effective gravitational coupling, which is obtained from $f'(R)>0$.

(4) A stable cosmological perturbation and positivity of the GWs for the scalar mode, which is obtained from $f''(R)>0$.

(5) An asymptotic behaviour to the $\Lambda$CDM model in the large curvature region.

The Gogoi-Goswami model can satisfy all these five requirements. The model is given by:

$f(R)=R-{\frac {\alpha }{\pi }}\,R_{c}\cot ^{-1}\!\left({\tfrac {R_{c}^{2}}{R^{2}}}\right)-\beta \,R_{c}\!\left[1-\exp \left({-\,{\tfrac {R}{R_{c}}}}\right)\right]$

where $\alpha$ and $\beta$ are two positive constants and $R_{c}$ is a characteristic curvature constant dimensionally equivalent to curvature scalar $R$. This model has two correction terms:

${\frac {\alpha }{\pi }}\,R_{c}\cot ^{-1}\!\left({\tfrac {R_{c}^{2}}{R^{2}}}\right)\;\;{\text{and}}\;\;\beta \,R_{c}\!\left[1-\exp \left({-\,{\tfrac {R}{R_{c}}}}\right)\right].$

The first correction factor is estimated by two parameters $\alpha$ and $R_{c}$. The second factor also has two model parameters $\beta$ and $R_{c}$ and it mimics the exponential f(R) gravity model.

Properties of the model

The action for the model is given by:

$S={\dfrac {1}{2\kappa }}\int d^{4}x{\sqrt {-g}}\left(R-{\frac {\alpha }{\pi }}\,R_{c}\cot ^{-1}\!\left({\tfrac {R_{c}^{2}}{R^{2}}}\right)-\beta \,R_{c}\!\left[1-\exp \left({-\,{\tfrac {R}{R_{c}}}}\right)\right]\right)+\int d^{4}x{\sqrt {-g}}\,{\mathcal {L}}_{m}\!\left[g^{\mu \nu },{\bar {\psi }}\,\right]\!$

here, ${\mathcal {L}}_{m}\!\left[g^{\mu \nu },{\bar {\psi }}\,\right]$ is the Lagrangian for a matter field ${\bar {\psi }}$ and $g_{\mu \nu }$ is the metric. The field equation can be obtained by the variation of the action with respect to the metric. The general f(R) gravity field equation is given by:

$f'(R)R_{\mu \nu }-{\dfrac {1}{2}}f(R)g_{\mu \nu }-\nabla _{\mu }\nabla _{\nu }f'(R)+g_{\mu \nu }\,\square f'(R)=\kappa ^{2}\,T_{\mu \nu }(g^{\mu \nu },{\bar {\psi }}),$

here $T_{\mu \nu }(g^{\mu \nu },{\bar {\psi }})={\dfrac {-\,2}{\sqrt {-g}}}{\dfrac {\delta \left({\sqrt {-g}}\,{\mathcal {L}}_{m}\!\left[g^{\mu \nu },{\bar {\psi }}\,\right]\right)}{\delta g^{\mu \nu }}}$ is the matter energy-momentum tensor and $f'(R)=\partial _{R}f(R)$. Using the form of f(R) in the above equation, one can obtain the field equations for this model. In Jordan frame, the square of the scalar field mass for this model is given by:

$m^{2}=\left[{\frac {R_{c}\exp \!\left(R/R_{c}\right)\left(\pi \left(R^{4}+R_{c}^{4}\right)^{2}-8\alpha R^{5}R_{c}^{3}\right)-\pi \beta (R+R_{c})\left(R^{4}+R_{c}^{4}\right)^{2}}{3\pi \beta \left(R^{4}+R_{c}^{4}\right)^{2}-6\alpha R_{c}^{4}\exp \!\left(R/R_{c}\right)\left(R_{c}^{4}-3R^{4}\right)}}\right]_{R\,=\,R_{0}}$

where $R_{0}$ is the background curvature. This expression shows that the mass of the scalar field is background curvature dependent. Apart from the background curvature, the mass of the scalar field also dependes on $\alpha$, $\beta$ and $R_{c}$. If the background curvature is zero i.e., in Minkowski space-time,

$m^{2}{\big |}_{R_{0}\,=\,0}={\frac {\pi (\beta -1)R_{c}}{6\alpha -3\pi \beta }}.$

It shows that the model can result in a massive scalar field even in the Minkowski space-time or at a far distance away from the matter source.

The model can easily pass the Solar system tests and has a wide feasible parametric space. Till now, the model has been constrained using GW170817 event and PSZ catalog.

See also

• Alternatives to general relativity
• General relativity
• f(R) gravity
• Inflation (cosmology)

References

This article "Gogoi-Goswami f(R) gravity model" is from Wikipedia. The list of its authors can be seen in its historical and/or the page Edithistory:Gogoi-Goswami f(R) gravity model. Articles copied from Draft Namespace on Wikipedia could be seen on the Draft Namespace of Wikipedia and not main one.

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