Standard perturbation theory in cosmology (SPT)

Standard perturbation theory expands the density contrast and velocity divergence of a pressureless irrotational fluid in powers of the linear density field.

Alpha mode-coupling kernel ( $α (k_{1}, k_{2})$ )

α (k_{1}, k_{2}) = \frac{( k _{1} + k _{2} ) \cdot k _{1}}{k _{1}^{2}} .

α_{s} = 1 + (k_{1} \cdot k_{2}) (k_{1}^{- 2} + k_{2}^{- 2}) /2

β (k_{1}, k_{2}) = \frac{∣ k _{1} + k _{2} ∣ ^{2} ( k _{1} \cdot k _{2} )}{2 k _{1}^{2} k _{2}^{2}} .

The symmetrized kernel

F_{n}

convolves

n

linear density fields to produce the

n

th-order density perturbation. In an Einstein-de Sitter universe,

F_{2} = \frac{5}{7} α_{s} + \frac{2}{7} β .

For Gaussian linear perturbations, the one-loop matter power spectrum is

P_{1 - loop} = a^{2} P + a^{4} (P_{22} + 2 P_{13}),

where

P_{22}

correlates two second-order fields and

P_{13}

correlates a linear field with a third-order field.

A scale-free linear matter spectrum has

P (k) = A k^{n}

. Dimensional analysis then makes its one-loop

P_{22}

contribution proportional to

A^{2} k^{2 n + 3}

whenever the loop integral converges.

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