[Paper Review] AdS non-linear curvature-squared and curvature-quartic multidimensional (D=8) gravitational models with stabilized extra dimensions

Tamerlan Saidov, Alexander Zhuk|ArXiv.org|Apr 19, 2006

Cosmology and Gravitation Theories5 references3 citations

TL;DR

This paper investigates an 8-dimensional gravitational model with nonlinear curvature terms $ R + R^2 + R^4 $, showing that for specific parameter regions, the extra dimensions are stabilized via a freezing mechanism in a negative minimum of the effective potential. The resulting four-dimensional spacetime is asymptotically AdS with a negative cosmological constant, achieved through analytical treatment of the effective potential in critical dimension $ D=8 $.

ABSTRACT

We investigate $D$-dimensional gravitational model with curvature-quadratic and curvature-quartic correction terms: $R+R^2+R^4$. It is assumed that the corresponding higher dimensional spacetime manifold undergos a spontaneous compactification to a manifold with warped product structure. Special attention is paid to the stability of the extra-dimensional factor space for a model with critical dimension D=8. It is shown that for certain parameter regions the model allows for a freezing stabilization of this space. The effective four-dimensional cosmological constant is negative and the external four-dimensional spacetime is asymptotically AdS.

Motivation & Objective

• To investigate the stabilization of extra dimensions in multidimensional gravity models with nonlinear curvature invariants.
• To analyze the behavior of the effective potential for the internal space scale factor in a critical dimension $ D=8 $.
• To determine parameter regions where the extra dimensions are frozen at a stable minimum.
• To establish the conditions under which the effective four-dimensional cosmological constant becomes negative.
• To extend previous $ f(R) $-gravity models to include $ R^4 $ terms and analyze their impact on moduli stabilization.

Proposed method

• Transform the non-linear $ f(R) = R + R^2 + R^4 $ gravity model into an equivalent linear gravity model coupled to a scalar field via conformal transformation.
• Derive the effective potential $ U_{\text{eff}}(\hat{\beta}^1, \phi) $ for the internal space volume modulus and scalar field.
• Use the critical dimension $ D=8 $, defined as $ D=2N $ for $ N=4 $, to reduce the degree of the extremum condition equation and simplify analysis.
• Apply the extremum condition $ Df - 2\bar{R}f' = 0 $ to locate minima of the effective potential.
• Analyze asymptotic behavior of the potential at $ \phi \to \infty $ to assess stability and avoid pathological behavior.
• Visualize the effective potential using contour plots and 3D forms for specific parameter sets $ \alpha=1, \gamma=1, \Lambda_8 = -0.1 $.

Experimental results

Research questions

• RQ1 Can the $ R + R^2 + R^4 $ gravitational model in $ D=8 $ dimensions stabilize the extra dimensions through a minimum of the effective potential?
• RQ2 What parameter regions in the $ R^4 $-dominated model allow for a freezing stabilization of the internal space?
• RQ3 Does the effective four-dimensional cosmological constant remain negative in the stabilized configuration?
• RQ4 How does the critical dimension $ D=8 $ simplify the analysis of extremum conditions for the effective potential?
• RQ5 What is the asymptotic behavior of the effective potential at large scalar field values, and does it indicate stability or instability?

Key findings

• For certain parameter regions, the model allows for a freezing stabilization of the extra dimensions at a negative minimum of the effective potential.
• The effective four-dimensional spacetime is asymptotically AdS, with a negative cosmological constant arising from the stabilization mechanism.
• The critical dimension $ D=8 $ reduces the degree of the extremum equation from 4 to 3, simplifying the analytical treatment of the potential's extrema.
• The effective potential $ U_{\text{eff}}(\hat{\beta}^1, \phi) $ reaches a global minimum at $ \hat{\beta}^1 = 0 $, $ \phi \approx -2.45 $, for the chosen parameters.
• Asymptotic analysis shows that for $ D=8 $, the potential exhibits stable behavior at large $ \phi $, avoiding catastrophic instabilities seen in supercritical dimensions.
• Graphical visualizations confirm the existence of a well-defined minimum in the effective potential, supporting the stability of the compactified internal space.