Cosmology in nonlinear multidimensional gravity and the Casimir effect

Abstract

We study the possible cosmological models in Kaluza-Klein-type multidimensional gravity with a curvature-nonlinear Lagrangian and a spherical extra space, taking into account the Casimir energy. First, we find a minimum of the effective potential of extra dimensions, leading to a physically reasonable value of the effective cosmological constant in our 4D spacetime. In this model, the huge Casimir energy density is compensated by a fine-tuned contribution of the curvature-nonlinear terms in the original action. Second, we present a viable model with slowly evolving extra dimensions and power-law inflation in our space-time. In both models, the results formulated in Einstein and Jordan frames are compared.

Local Regions with Expanding Extra Dimensions

“…In [25,26], it has been shown that the models with higher derivatives could lead to stationary solutions. The analysis was based on the model where the initial action was taken in the form [27,28]…”

“…The dot above the variables stays for the time derivative. The last term is related to the Casimir effect, with k C of the order 10 −3 -10 −4 ; see [27,28] and references therein.…”

On Cosmology in Nonlinear Multidimensional Gravity with Multiple Factor Spaces

“…In accord with the above-said, we seek a minimum of W ignoring the Casimir contribution. If we additionally assume m = n, the expressions (15) and (16) are symmetric with respect to x and y , and it makes sense to seek a minimum of W on the line x = y , which substantially simplifies the process. It turns out that the case n = 3 is degenerate because, instead of two parameters C 1 and C 2 , both W and K depend on the single combination C 1 + C 2 .…”

“…As in [15,16], we begin with a sufficiently general D -dimensional gravitational action, then, under suitable assumptions on the space-time geometry, follow a dimensional reduction and a transition to the Einstein conformal frame. After that, we demonstrate the existence of such sets of the initial parameters that provide a minimum of the effective potential at a physically reasonable length scale, and it is also shown that the kinetic term of the effective scalar fields is positive-definite, hence these minima really describe stable stationary configurations.…”

“…In this paper we continue our study of cosmological models in the framework of nonlinear multidimensional gravity, taking into account possible contributions of quantum vacuum effects. In [15,16], the space-time geometry was chosen in the extended Kaluza-Klein form, M 4 × S n , where M 4 is the observed weakly curved 4D space-time, and S n is an n-dimensional sphere of sufficiently small size to be invisible by modern instruments. Now we extend the study to more complex geometries containing a number (at least two) of internal factor spaces.…”