Аннотация:
Many significant gravitational phenomena have been predicted and
discovered by General Relativity (GR), despite of all the successes
and many nice theoretical properties, GR is not complete theory of
gravity. One of actual approaches towards more complete theory of
gravity is its nonlocal modification.
We consider nonlocal modification of the Einstein theory of gravity in
framework of the pseudo-Riemannian geometry. The nonlocal term has the
form $\,\mathcal{H}(R)\,\mathcal{F} (\Box)\,\mathcal{G}(R)$, where
$\mathcal{H}$ and $\mathcal{G}$ are differentiable functions of the
scalar curvature $R$, and $\mathcal{F} (\Box) = \sum_{n=0}^{\infty}\,
f_n\,\Box^n$, where $f_n$ are analytic functions of the d’Alembert
operator $\Box$. Our motivation to modify gravity in an analytic
nonlocal way comes mainly from string theory, in particular from string
field theory and $p$-adic string theory.
Using calculus of variations we derived the corresponding equations of
motion. The variation of action is induced by variation of the metric
tensor $\,g_{\mu\nu}.$ We consider several models of the above mentioned
type, as well as the case when the scalar curvature is constant.
Moreover, we consider space-time perturbations of the de Sitter space.
It was shown that gravitational waves are described in the class of
nonlocal models $\,\mathcal{H}(R)\,\mathcal{F} (\Box)\,\mathcal{G}(R),$
with respect to Minkowski metric by the same equations as in general
relativity.
Recently, we deal with the cases where:
- $\mathcal{H}(R)=\mathcal{G}(R)=
\displaystyle{R-4\,\Lambda},$ and
- $\mathcal{H}(R)=\mathcal{G}(R)=
\displaystyle{\sqrt{R-2\,\Lambda}}.$
Specially, we paid our attention to the case (2) with scaling factor of
the form $a(t)=A\,t^{\frac{2}{3}}\,e^{\frac{\Lambda}{14}\,t^2}$, and we
find some new cosmological solutions, and we test validity of obtained
solutions with experimental data.
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