Multi-fluid cosmology in Einstein gravity: analytical solutions

Abstract

We review analytical solutions of the Einstein equations which are expressed in terms of elementary functions and describe Friedmann-Lemaître-Robertson-Walker universes sourced by multiple (real or effective) perfect fluids with constant equations of state. Effective fluids include spatial curvature, the cosmological constant, and scalar fields. We provide a description with unified notation, explicit and parametric forms of the solutions, and relations between different expressions present in the literature. Interesting solutions from a modern point of view include interacting fluids and scalar fields. Old solutions, integrability conditions, and solution methods keep being rediscovered, which motivates a review with modern eyes.

Cosmological analogies for geophysical flows, Lagrangians, and new analogue gravity systems

“…for 0 ≤ x ≤ x 0 . The scale factor of the analogous spatially flat expanding universe reduces to the well-known power-law a(t) a * (t − t 0 ) 1/3 (with a * a positive constant) caused by a stiff fluid or a free scalar field [28].…”

Cosmological analogies for geophysical flows, Lagrangians, and new analogue gravity systems

“…for 0 ≤ x ≤ x 0 . The scale factor of the analogous spatially flat expanding universe reduces to the well-known powerlaw a(t) ≃ a * (t − t 0 ) 1/3 (with a * a positive constant) caused by a stiff fluid or a free scalar field [28].…”

When can we compute analytically lookback time, age of the universe, and luminosity distance?

“…Equivalently, it happens when the integral expressing lookback time, age, or luminosity distance is of a special form contemplated by the Chebyshev theorem of integration [1,2]. The truncation of the hypergeometric series, or the equivalent Chebyshev theorem, were used in the 1960s [3,4,5,6], and were recently rediscovered [7], to derive two-and three-fluid (or effective fluid) analytical solutions of the Einstein-Friedmann equations (see [8] for a review). a e-mail: sjose21@ubishops.ca b e-mail: alexandre.leblanc3@usherbrooke.ca c e-mail: vfaraoni@ubishops.ca When the matter content of the FLRW universe consists of at most three non-interacting fluids or effective fluids (which includes spatial curvature and/or the cosmological constant Λ , if present), and assuming that the equations of state of these fluids are of the form P = wρ with w constant and rational, the situations in which lookback time, age, and luminosity distance are analytical and simple are classified by means of the Chebyshev theorem [1,2].…”