In physical cosmology, cosmological perturbation theory[1][2][3][4][5] is the theory by which the evolution of structure is understood in the Big Bang model. Cosmological perturbation theory may be broken into two categories: Newtonian or general relativistic. Each case uses its governing equations to compute gravitational and pressure forces which cause small perturbations to grow and eventually seed the formation of stars, quasars, galaxies and clusters. Both cases apply only to situations where the universe is predominantly homogeneous, such as during cosmic inflation and large parts of the Big Bang. The universe is believed to still be homogeneous enough that the theory is a good approximation on the largest scales, but on smaller scales more involved techniques, such as N-body simulations, must be used. When deciding whether to use general relativity for perturbation theory, note that Newtonian physics is only applicable in some cases such as for scales smaller than the Hubble horizon, where spacetime is sufficiently flat, and for which speeds are non-relativistic.
Because of the gauge invariance of general relativity, the correct formulation of cosmological perturbation theory is subtle.
In particular, when describing an inhomogeneous spacetime, there is often not a preferred coordinate choice. There are currently two distinct approaches to perturbation theory in classical general relativity:
- gauge-invariant perturbation theory based on foliating a space-time with hyper-surfaces, and
- 1+3 covariant gauge-invariant perturbation theory based on threading a space-time with frames.
In this section, we will focus on the effect of matter on structure formation in the hydrodynamical fluid regime. This regime is useful because dark matter has dominated structure growth for most of the universe's history. In this regime, we are on sub-Hubble scales (where is the Hubble parameter) so we can take spacetime to be flat, and ignore general relativistic corrections. But these scales are above a cut-off, such that perturbations in pressure and density are sufficiently linear Next we assume low pressure so that we can ignore radiative effects and low speeds so we are in the non-relativistic regime.
The first governing equation follows from matter conservation – the continuity equation[6]
where is the scale factor and is the peculiar velocity. Although we don't explicitly write it, all variables are evaluated at time and the divergence is in comoving coordinates. Second, momentum conservation gives us the Euler equation
where is the gravitational potential. Lastly, we know that for Newtonian gravity, the potential obeys the Poisson equation
So far, our equations are fully nonlinear, and can be hard to interpret intuitively. It's therefore useful to consider a perturbative expansion and examine each order separately. We use the following decomposition
where is a comoving coordinate.
At linear order, the continuity equation becomes
where is the velocity divergence. And the linear Euler equation is
By combining the linear continuity, Euler, and Poisson equations, we arrive at a simple master equation governing evolution
where we defined a sound speed to give us a closure relation. This master equation admits wave solutions in which tell us 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.