$k$-essence $f(R)$ Gravity Inflation. (arXiv:1902.03669v1 [gr-qc])

<a href="http://arxiv.org/find/gr-qc/1/au:+Nojiri_S/0/1/0/all/0/1">S. Nojiri</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Odintsov_S/0/1/0/all/0/1">S.D. Odintsov</a>, <a href="http://arxiv.org/find/gr-qc/1/au:+Oikonomou_V/0/1/0/all/0/1">V.K. Oikonomou</a>

In this work we study a modified version of $f(R)$ gravity in which higher

order kinetic terms of a scalar field are added in the action of vacuum $f(R)$

gravity. This type of theory is a type of $k$-essence $f(R)$ gravity, and it

belongs to the general class of $f(R,phi ,X)$ theories of gravity, where

$phi$ is a scalar field and

$X=frac{1}{2}partial^{mu}phipartial_{mu}phi$. We focus on the

inflationary phenomenology of the model, in the slow-roll approximation, and we

investigate whether viable inflationary evolutions can be realized in the

context of this theory. We use two approaches, firstly by imposing the

slow-roll conditions and by using a non-viable vacuum $f(R)$ gravity. As we

demonstrate, the spectral index of the primordial scalar perturbations and the

tensor-to-scalar ratio of the resulting theory can be compatible with the

latest observational data. In the second approach, we fix the functional form

of the Hubble rate as a function of the $e$-foldings number, and we modify

well-known vacuum $f(R)$ gravity reconstruction techniques, in order to find

the $k$-essence $f(R)$ gravity which can realize the given Hubble rate.

Accordingly, we calculate the slow-roll indices and the corresponding

observational indices, and we also provide general formulas of these quantities

in the slow-roll approximation. As we demonstrate, viability can be obtained in

this case too, however the result is strongly model dependent. In addition, we

discuss when ghosts can occur in the theory, and we investigate under which

conditions ghosts can be avoided by using a particular class of models.

Finally, we qualitatively discuss the existence of inflationary attractors for

the non-slow-roll theory, and we provide hints towards finding general de

Sitter attractors for the theory at hand.

In this work we study a modified version of $f(R)$ gravity in which higher

order kinetic terms of a scalar field are added in the action of vacuum $f(R)$

gravity. This type of theory is a type of $k$-essence $f(R)$ gravity, and it

belongs to the general class of $f(R,phi ,X)$ theories of gravity, where

$phi$ is a scalar field and

$X=frac{1}{2}partial^{mu}phipartial_{mu}phi$. We focus on the

inflationary phenomenology of the model, in the slow-roll approximation, and we

investigate whether viable inflationary evolutions can be realized in the

context of this theory. We use two approaches, firstly by imposing the

slow-roll conditions and by using a non-viable vacuum $f(R)$ gravity. As we

demonstrate, the spectral index of the primordial scalar perturbations and the

tensor-to-scalar ratio of the resulting theory can be compatible with the

latest observational data. In the second approach, we fix the functional form

of the Hubble rate as a function of the $e$-foldings number, and we modify

well-known vacuum $f(R)$ gravity reconstruction techniques, in order to find

the $k$-essence $f(R)$ gravity which can realize the given Hubble rate.

Accordingly, we calculate the slow-roll indices and the corresponding

observational indices, and we also provide general formulas of these quantities

in the slow-roll approximation. As we demonstrate, viability can be obtained in

this case too, however the result is strongly model dependent. In addition, we

discuss when ghosts can occur in the theory, and we investigate under which

conditions ghosts can be avoided by using a particular class of models.

Finally, we qualitatively discuss the existence of inflationary attractors for

the non-slow-roll theory, and we provide hints towards finding general de

Sitter attractors for the theory at hand.

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