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.

The nonlinear wavenumber is the inverse scale at which linear density contrast fluctuations become of order unity. For the one-dimensional convention $⟨\delta (k)\delta (k^{\prime })⟩=2\pi {\delta }^{(D)}(k+k^{\prime })P_L(k)$, a usual estimate is $k_{\mathrm{NL}}{P}_L(k_{\mathrm{NL}})/\pi \sim 1$. Loop momenta above this scale require the effective field theory of large-scale structure even when external modes are perturbative.

Gaussian initial conditions make every connected primordial correlation function above second order vanish. Wick theorem then reduces perturbative late-time correlation functions to sums of products of the linear power spectrum.

The continuity-equation kernel isIts symmetrization is ${\alpha }_s=1+({\mathbf{k}}_1\cdot {\mathbf{k}}_2)(k_1^{-2}+k_2^{-2})/2$.

The Euler-equation kernel is the symmetric function

The symmetrized kernel $F_n$ convolves $n$ linear density fields to produce the $n$th-order density perturbation. In an Einstein-de Sitter universe,

For plane-parallel pressureless matter in an Einstein-de Sitter universe, before shell crossing, the symmetrized growing-mode standard perturbation theory density kernel isIn particular $F_{2,s}=(q_1+q_2{)}^2/(2q_1{q}_2)$. An unsymmetrized representation of the one-dimensional beta mode-coupling kernel is $(q_1+q_2)/q_1$; its symmetric part is the expression above.

At fixed external momentum $\mathbf{k}$, mass and momentum conservation implyas $q\to \infty$. In particular, the constant term in the hard-momentum expansion must cancel.

For Gaussian initial conditions and the Einstein-de Sitter second-order standard perturbation theory density kernel, the unsmoothed density contrast obeysThere are three choices of the second-order field and two connected Wick contractions; the angular average of $F_2$ is $5/7+(2/7)/3=17/21$. Thus the reduced skewness is $6(17/21)=34/7$. Smoothing changes this result, and the unsmoothed calculation requires a finite variance.

For Gaussian initial conditions, the tree-level connected matter five-point function has perturbative-order partitionsThey correspond respectively to a four-valent star, an $F_3$--$F_2$ tree, and a chain of three $F_2$ vertices. Every topology contains four linear power spectra.

For Gaussian linear perturbations, the one-loop matter power spectrum iswhere $P_{22}$ correlates two second-order fields and $P_{13}$ correlates a linear field with a third-order field.

The $P_{22}$ contribution correlates two second-order density fields. For symmetrized kernels,

The $P_{13}$ contribution correlates a linear density field with a third-order field:Both orderings contribute $2P_{13}$ to the power spectrum.

At equal time, leading loop contributions from a very long-wavelength displacement cancel between perturbative diagrams. This is required by the equivalence principle, because a coherent long displacement merely translates short-scale structure.

A scale-free linear matter spectrum has $P(k)=A{k}^n$. Dimensional analysis then makes its one-loop $P_{22}$ contribution proportional to $A^2{k}^{2n+3}$ whenever the loop integral converges.

For a matter power spectrum $P(k)=A{k}^n$ at a fixed epoch, the Poisson equation gives potential Fourier amplitudes proportional to ${\delta }_k/k^2$, and the growing-mode peculiar velocity is proportional to ${\delta }_k\mathbf{k}/k^2$. With a spherical top-hat window function, their variances are proportional to $\int k^{n-2}W(kR{)}^{2}dk$ and $\int k^{n}W(kR{)}^{2}dk$, respectively. Rescaling $kR$ gives the displayed powers when the integrals converge: $1<n<5$ for the averaged potential and $-1<n<3$ for the bulk velocity. Absolute averaged potential has infrared sensitivity; band-limited fluctuations or potential differences can have different convergence requirements. The density variance converges only for $-3<n<1$, so these three absolute averages cannot all converge for one unbroken power law at every scale.

For an unsmoothed density spectrum $P(k)=A{k}^n$ and a scale-dependent smoothing window $\tilde{W}(kR)$, substituting $q=kR$ in the variance integral gives ${\sigma }_R^2\propto R^{-(n+3)}$, provided ${\int }_0^{\infty }{q}^{n+2}|\tilde{W}(q){|}^{2}dq$ converges. For a window tending to one at zero, infrared convergence requires $n>-3$; the ultraviolet condition depends on the window. A real-space spherical top-hat also requires $n<1$.

The symmetrized kernel $G_n$ convolves $n$ linear density fields to produce the $n$th-order peculiar-velocity divergence in cosmological standard perturbation theory.

For Gaussian initial conditions, the matter bispectrum through one loop is the sum of the tree diagram $B_{211}$ and the four one-loop topologies $B_{222}$, $B_{321}^I$, $B_{321}^{II}$, and $B_{411}$.

The connected correlation of one fourth-order and two first-order density contrasts contracts a pair of inputs at the fourth-order vertex into a loop. For each choice of the fourth-order external field there are twelve Wick contractions. In one dimension,For the one-dimensional cosmological density kernel, $F_4=-k_3^4/(24k_1{k}_2{q}^2)$ exactly. Its leading cutoff dependence is therefore proportional to $k_3^2{F}_2(k_1,k_2)P_L(k_1)P_L(k_2)\int dq{P}_L(q)/q^2$.

The connected correlation of three second-order density contrasts forms a triangle of three linear power spectra. It has eight Wick contractions. In one dimension, with $k_1+k_2+k_3=0$ and $P_L$ the power at the evaluation time,For the one-dimensional cosmological density kernel, its hard-loop integrand starts as $-k_1^2{k}_2^2{k}_3^2{P}_L(q{)}^3/q^6$.

For Gaussian initial conditions, the one-loop connected matter trispectrum contains the perturbative-order classes $T_{5111}$, $T_{4211}$, $T_{3311}$, $T_{3221}$, and $T_{2222}$. Each class includes every inequivalent external-leg assignment and Wick contraction.