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Cosmological perturbation theory

Theory of the evolution of cosmological structure

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Due to the gauge invariance of general relativity, the correct formulation of cosmological perturbation theory is subtle. In particular, when describing an inhomogeneous spacetime, there are often no preferred coordinate choices. There are currently two distinct approaches to perturbation theory in classical general relativity:

  • gauge-invariant perturbation theory based on foliating a spacetime with hyper-surfaces, and
  • 1+3 covariant gauge-invariant perturbation theory based on threading a spacetime with frames.

Newtonian perturbation theory

Focusing on the effect of matter on structure formation in the hydrodynamical fluid regime is useful because dark matter has dominated structure growth for most of the universe's history. This regime is on sub-Hubble scales <H−1, (where H is the Hubble parameter), whereupon spacetime can be taken to be flat, and the effects of general relativity can be neglected. But these scales are above a cut-off, such that perturbations in pressure and density are sufficiently linear δP,δρ≪1. Assume low pressure P≪ρ, so that radiative effects can be ignored, and low speeds u≪c, making relativistic effects negligible.

The first governing equation follows from matter conservation, namely, the continuity equation

∂ρ∂t+3Hρ+1a⋅∇(ρv)=0,

where a is the scale factor and v is the peculiar velocity. All variables are evaluated at time t and the divergence ∇ is in comoving coordinates. The second equation is the Euler equation

ρdu→dt=ρ(∂∂t+1av⋅∇)u=−1a∇P−1aρ∇Φ,

where Φ is the gravitational potential. It is a consequence of momentum conservation. In the Newtonian limit, the potential obeys Poisson's equation for gravity

1a2∇2Φ=4πGρ.

The equations so far are fully nonlinear, and can be hard to interpret intuitively. It is therefore useful to consider a perturbative expansion and examine each order separately,

ρ=ρ¯(1+δ),u=Hax+v,P=P¯+δP,Φ=Φ¯+δΦ.

where x is a comoving coordinate.

At linear order, the continuity equation becomes

δ˙=−1aθ,

where θ≡∇⋅v is the velocity divergence. And the linear Euler equation is

ρ¯(v˙+Hv)=−1a∇δP−1aρ¯∇δΦ.

Combining the continuity equation, Euler, and Poisson equations yields a simple master equation governing evolution

(∂2∂2t+2H∂∂t−cs21a∇2−4πGρ¯)δ=0,

where the speed of sound is defined to be cs2≡δP/ρ¯δ, giving a closure relation. This master equation admits wave solutions in δ(x,t), describing how matter fluctuations grow over time due to a combination of competing effects: the fluctuation's self-gravity, pressure forces, the universe's expansion, and the background gravitational field.

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