[Paper Review] Multicomponent perfect fluid with variable parameters in n Ricci-flat spaces

Jean‐Michel Alimi, V. R. Gavrilov+1 more|ArXiv.org|May 24, 2004

Geometric Analysis and Curvature Flows3 citations

TL;DR

This paper investigates a D-dimensional cosmological model with a multicomponent perfect fluid featuring variable barotropic parameters in n Ricci-flat spatial dimensions. Using exact Kasner-like solutions, it demonstrates that the model can describe anisotropic expansion with late-time isotropic acceleration in 4D and dynamical compactification of extra dimensions in (4+d)-D spacetime, offering a unified framework for cosmic acceleration and extra-dimensional stabilization without requiring exotic fields or cosmological constants.

ABSTRACT

D-dimensional cosmological model describing the evolution of a multicomponent perfect fluid with variable barotropic parameters in n Ricci-flat spaces is investigated. The equations of motion are integrated for the case, when each component possesses an isotropic pressure with respect to all spaces. Exact solutions are presented in the Kasner-like form. Some explicit examples are given: 4-dimensional model with an isotropic accelerated expansion at late times and (4+d)-dimensional model describing a compactification of extra dimensions.

Motivation & Objective

• To address the cosmic acceleration and dark energy problems by modeling a multicomponent perfect fluid with variable barotropic parameters in multidimensional spacetime.
• To investigate the dynamical compactification of extra dimensions in higher-dimensional cosmological models.
• To provide exact solutions in the Kasner-like form that describe both early-time anisotropic expansion and late-time isotropic acceleration.
• To explore the role of variable equation of state parameters in achieving stable compactification and accelerated expansion without introducing a cosmological constant or exotic scalar fields.

Proposed method

• The model employs a D-dimensional metric with n Ricci-flat factor spaces, each with its own scale factor and time-dependent exponent.
• The equations of motion are derived from a gravitational action with a multicomponent perfect fluid source, where each component has isotropic pressure across all spaces.
• The barotropic parameters are allowed to vary via a functional dependence on the Hubble-like variable γ₀, leading to a generalized 'second equation of state' for viscosity-like effects.
• Exact solutions are obtained under the assumption that the sum of fluid energy densities scales as e^(-2γ₀), mimicking stiff matter behavior.
• The solution takes a Kasner-like form, with exponents ε̃ⁱ determined by dimensionality and coupling parameters, satisfying modified Kasner constraints.
• The Einstein frame transformation is applied to analyze the effective expansion rate of the external 3D space, revealing decelerated expansion when internal dimensions contract.

Experimental results

Research questions

• RQ1 Can a multicomponent perfect fluid with variable barotropic parameters in n Ricci-flat spaces produce late-time isotropic acceleration in a 4D effective spacetime?
• RQ2 How does the dynamical evolution of scale factors in a (4+d)-dimensional model lead to the compactification of extra dimensions while the 3D space expands?
• RQ3 What conditions on the fluid parameters and coupling constants lead to stable solutions with contracting internal spaces and expanding external space?
• RQ4 How does the effective expansion rate of the external space change when transforming from the Jordan-Brans-Dicke frame to the Einstein frame?
• RQ5 What is the role of the generalized 'second equation of state' in enabling solutions that avoid the need for a cosmological constant or quintessence fields?

Key findings

• The model yields exact Kasner-like solutions for the scale factors, with exponents ε̃ⁱ given by a modified expression involving the dimensionality and coupling parameters.
• In the (4+d)-dimensional model, when the internal space contracts (ε̃² < 0), the external 3D space expands only with deceleration, as the effective exponent ȟ¹ < 1/3.
• For the external space to expand with acceleration, the internal space must expand or remain static, which contradicts the requirement for compactification.
• The solution satisfies modified Kasner constraints: ∑dᵢε̃ⁱ = 1 and ∑dᵢ(ε̃ⁱ)² = δ ≈ 1 when the coupling parameter A/⟨s,s⟩ is small.
• The condition for internal space contraction requires √[3(1+2/d)/(1+A/⟨s,s⟩)] > 1, which restricts the allowed range of A/⟨s,s⟩ to (0, 2+6/d).
• In the Einstein frame, the effective expansion exponent ȟ¹ is always less than the Jordan-Brans-Dicke frame exponent ε̃¹, confirming that acceleration is not achieved in the external space when internal dimensions contract.