[Paper Review] Cosmological perturbations in the presence of a solid with positive pressure

V. Balek, Matej Škovran|arXiv (Cornell University)|Jan 27, 2014

Cosmology and Gravitation Theories6 references3 citations

TL;DR

This paper investigates cosmological scalar perturbations in a universe containing a solid with positive pressure, focusing on radiation-like (w = 1/3) and stiff (w > 1/3) solids. It shows that large-scale perturbations are suppressed if the shear stress-to-energy density ratio ξ is positive, and enhanced if ξ is negative; only very small |ξ| ≲ 10⁻⁵ yields a flat spectrum consistent with observations, requiring radiation to dominate long enough before recombination to match data.

ABSTRACT

Evolution of scalar perturbations in a universe containing solid matter with positive pressure is studied. Solution for pure solid is found and matched with solution for ideal fluid, including the case when the pressure to energy density ratio $w$ has a jump. Two classes of solutions are explored in detail, solutions with radiation-like solid ($w = 1/3$) and solutions with stiff solid ($w > 1/3$) appearing in a universe filled with radiation. For radiation-like solid, an almost flat spectrum of large-scale perturbations is obtained only if the shear stress to energy density ratio $ξ$ is close to zero, $|ξ| \lesssim 10^{-5}$. For a solid with stiff equation of state, large-scale perturbations are enhanced for $ξ$ negative and suppressed for $ξ$ positive. If the solid dominated the dynamics of the universe long enough, perturbations could end up suppressed as much as by several orders of magnitude, and in order that the inclination of the large-scale spectrum is consistent with observations, radiation must have prevailed over the solid long enough before recombination. In Newtonian gauge, corrections to metric and energy density are typically much greater than 1 in the first period after the shear stress appears, but the linearized theory is still applicable because the corrections stay small when one uses the proper-time comoving gauge.

Motivation & Objective

• To study the evolution of scalar cosmological perturbations in a universe containing a solid with positive pressure, particularly in the radiation-dominated era.
• To determine how the shear modulus and equation of state (w) of the solid affect the power spectrum of large-scale perturbations.
• To assess whether a solid with positive w can reproduce the observed nearly scale-invariant spectrum of CMB anisotropies without conflicting with data.
• To explore the conditions under which a solid with positive w can be consistent with cosmological observations, especially regarding the amplitude and tilt of the primordial power spectrum.

Proposed method

• The paper derives the energy-momentum tensor for a solid using a Lagrangian formulation with internal coordinates, distinguishing between shear and bulk moduli.
• It uses the comoving gauge to linearize perturbations and derives the evolution equations for metric and energy density corrections in the presence of shear stress.
• The analysis includes matching solutions for pure solid and ideal fluid components, particularly when w exhibits a discontinuity.
• It employs the Newtonian gauge to compute corrections to the metric and energy density, showing they remain small despite large relative corrections.
• The paper uses the traceless part of the metric perturbation to isolate shear effects and relates them to the shear modulus μ and energy density ρ₊.
• It computes sound speeds for longitudinal and transverse modes in the solid using μ and λ (Lamé parameters), assuming adiabatic perturbations with constant entropy per particle.

Experimental results

Research questions

• RQ1 How do scalar perturbations evolve in a radiation-dominated universe containing a solid with positive pressure, particularly for w = 1/3 and w > 1/3?
• RQ2 What constraints does the observed nearly scale-invariant CMB power spectrum impose on the shear stress-to-energy density ratio ξ in a solid?
• RQ3 Can a solid with positive w reproduce the observed amplitude and tilt of cosmological perturbations without requiring fine-tuning of initial conditions?
• RQ4 How does the presence of a solid affect the Silk damping of small-scale anisotropies, especially if the solid has viscosity?
• RQ5 Under what conditions can a solid with positive w be consistent with observations, particularly when radiation dominates before recombination?

Key findings

• For a radiation-like solid (w = 1/3), the large-scale perturbation spectrum is nearly flat only if the shear stress-to-energy density ratio satisfies |ξ| ≲ 10⁻⁵.
• For a stiff solid (w > 1/3), large-scale perturbations are suppressed when ξ > 0 and enhanced when ξ < 0, with suppression potentially reaching several orders of magnitude if the solid dominates long enough.
• If the solid dominates the dynamics for a significant period, the initial amplitude of perturbations must be increased to match observations, indicating a need for non-standard initial conditions.
• The linearized theory remains valid despite metric and energy density corrections exceeding 1 in the first period after shear stress appears, due to the use of the proper-time comoving gauge.
• The ratio δp/δρ equals dp/dρ, confirming consistency with the sound speed defined via c_S₀² = dp/dρ in the fluid limit.
• The paper confirms that adiabatic perturbations with constant entropy per particle are sufficient to derive the main results, and that entropy perturbations would introduce additional terms proportional to δS in δρ and δp.