[Paper Review] Self-interaction in a cosmic dark fluid: The four-kernel rheological extension of the equations of state

Alexander B. Balakin, Alexei S. Ilin|arXiv (Cornell University)|Mar 30, 2022

Cosmology and Gravitation Theories54 references5 citations

TL;DR

This paper proposes a four-kernel rheological model for self-interaction in a cosmic dark fluid, extending equations of state using Volterra-type integral operators to describe fading memory effects. The model yields an exactly integrable cosmological framework with explicit solutions for scale factor, Hubble parameter, and acceleration, revealing new behaviors including super-exponential expansion, symmetric bounce, and quasi-periodic dynamics, with classification into Big Rip, Little Rip, and Pseudo Rip scenarios based on asymptotic analysis.

ABSTRACT

We establish a new self-consistent model of coupling between the cosmic dark energy and dark matter in the framework of the rheological approach, which is based on the representation of the equations of state in terms of integral operators of the Volterra-type. We elaborate the so-called four-kernel model, in the framework of which both the dark energy and dark matter pressures are presented by two integrals containing the energy densities of the dark energy and dark matter. For the Volterra operators, the kernels of which are associated with the effects of fading memory, the corresponding isotropic homogeneous cosmological model is shown to be exactly integrable. We consider the classification of the model exact solutions, based on the analysis of roots of the characteristic polynomial associated with the key equation of the presented model. The scalars of the pressure and energy-density of the dark energy and dark matter, the Hubble function and acceleration parameter are presented explicitly as the functions of the dimensionless scale factor. The scale factor as the function of the cosmological time is found in quadratures and is described analytically, qualitatively and numerically. Asymptotic analysis allowed us to classify the models with respect to behavior typical for the Big Rip, Little Rip and Pseudo Rip (de Sitter type). Two intriguing exact cosmological solutions are discussed, which describe the super-exponential expansion and the symmetric bounce. New solutions are presented, which correspond to the quasi-periodic behavior of the state functions of the dark fluid and of the geometric characteristics of the Universe.

Motivation & Objective

• To develop a self-consistent model of dark energy and dark matter coupling using a rheological approach based on integral equations of motion.
• To extend the equations of state for dark energy and dark matter using Volterra-type integral operators with fading memory kernels.
• To derive and solve a key integro-differential equation of Euler-type order up to six, enabling exact cosmological solutions.
• To classify exact solutions based on asymptotic behavior, identifying models corresponding to Big Rip, Little Rip, and Pseudo Rip cosmologies.
• To present explicit analytical and numerical solutions for the scale factor, Hubble parameter, and acceleration parameter as functions of the dimensionless scale factor.

Proposed method

• Formalism is built on isotropic homogeneous spacetime, with equations of state for dark energy and dark matter expressed as two Volterra integrals each, involving four fading memory kernels.
• The model uses Volterra integral operators with multiplicative kernels to represent time-delayed responses, modeling nonlocality in time via memory effects.
• The balance equations and gravitational field equations are combined to derive a key linear differential equation of Euler type for the dark energy density.
• The key equation is derived in multiple orders (2nd to 6th) depending on the completeness of coupling parameters, with auxiliary coefficients defined in appendices.
• Exact solutions are obtained via characteristic polynomial analysis, and the scale factor is found in quadrature form.
• Asymptotic analysis of solutions classifies cosmological behavior into Big Rip, Little Rip, and Pseudo Rip types based on divergence or boundedness of scale factor and Hubble parameter.

Experimental results

Research questions

• RQ1 How can self-interaction in the dark fluid be modeled using nonlocal, memory-dependent equations of state?
• RQ2 What exact cosmological solutions emerge from a four-kernel Volterra-type extension of the dark energy and dark matter equations of state?
• RQ3 How do the asymptotic behaviors of the scale factor and Hubble parameter classify the model into Big Rip, Little Rip, or Pseudo Rip scenarios?
• RQ4 Can the model support super-exponential expansion or symmetric bounce dynamics?
• RQ5 What role do the characteristic roots of the key Euler-type differential equation play in classifying the model's cosmological evolution?

Key findings

• The model yields an exactly integrable cosmological framework, with the scale factor expressible in quadrature form and analytically solvable via characteristic polynomial roots.
• Explicit expressions are derived for the Hubble parameter, acceleration parameter, and energy densities of dark energy and dark matter as functions of the dimensionless scale factor.
• The model admits a new exact solution describing super-exponential expansion, distinct from standard ΛCDM or phantom models.
• A symmetric bounce solution is identified, where the scale factor reaches a minimum and expands symmetrically in both time directions.
• Quasi-periodic behavior is found in state functions and geometric characteristics, indicating oscillatory dynamics in the dark fluid's evolution.
• The model classifies into Big Rip, Little Rip, and Pseudo Rip scenarios based on the asymptotic behavior of the scale factor and Hubble parameter, with the classification determined by the roots of the characteristic polynomial.