Renormalized cosmological perturbation theory

We develop a new formalism to study nonlinear evolution in the growth of large-scale structure, by following the dynamics of gravitational clustering as it builds up in time. This approach is conveniently represented by Feynman diagrams constructed in terms of three objects: the initial conditions (e.g. perturbation spectrum), the vertex (describing nonlinearities) and the propagator (describing linear evolution). We show that loop corrections to the linear power spectrum organize themselves into two classes of diagrams: one corresponding to mode-coupling effects, the other to a renormalization of the propagator. Resummation of the latter gives rise to a quantity that measures the memory of perturbations to initial conditions as a function of scale. As a result of this, we show that a well-defined (renormalized) perturbation theory follows, in the sense that each term in the remaining mode-coupling series dominates at some characteristic scale and is subdominant otherwise. This is unlike standard perturbation theory, where different loop corrections can become of the same magnitude in the nonlinear regime. In companion papers we compare the resummation of the propagator with numerical simulations, and apply these results to the calculation of the nonlinear power spectrum. Remarkably, the expressions in renormalized perturbation theory can be written in a way that closely resembles the halo model.

Figure 1

Comparison between PT (left) and RPT (right) loop expansion for the nonlinear power spectrum in the Zel’dovich approximation. $P_{\mathrm{nl}}$ denotes the exact result for the nonlinear power spectrum, Eq. (3), whereas $P_{\mathrm{PT}}^{(\ell )}$ ($P_{\mathrm{RPT}}^{(\ell )}$) denotes the $\ell$-loop correction in PT (RPT). Dashed lines denote negative values. The resummation of the propagator involved in RPT leads to a well-defined perturbation expansion even in the nonlinear regime, unlike PT.

Figure 2

Graphical notation of the basic objects in the perturbative expansion, the initial field ${\varphi }_a$, the linear propagator $g_{ab}$, and the vertex ${\gamma }_{abc}^{(s)}$.

Figure 3

Diagrams up to order $n=4$ in the series expansion of $\Psi (\mathbf{k},\eta )$.

Figure 4

Diagrammatic notation for the initial power spectrum.

Figure 5

Diagrams for the correlation function $P_{ab}(\mathbf{k},\eta )$ up to two-loops (only 7 out of 29 two-loop diagrams are shown here). The dashed lines represent the points at which the two trees representing perturbative solutions to ${\Psi }_a$ and ${\Psi }_b$ have been glued together.

Figure 6

Diagrams for the propagator $G(k,\eta )$ up to two loops.

Figure 7

Diagrams corresponding to the renormalized vertex $\Gamma$, Eq. (41), up to one loop.

Figure 8

Diagrams for $\Sigma (\mathbf{k},s_1,s_2)$ up to two loops. The dashed lines enclose reducible fragments, see Sec. 4c.

Figure 9

All diagrams for $\Phi$ up to two loops. The dashed lines enclose reducible fragments for propagator and power spectrum renormalization. The dash-dotted lines enclose reducible fragments that lead to vertex renormalization, see Sec. 4c for a detailed discussion.

Figure 10

Integral Eqs. (45, 48) for the nonlinear propagator and the nonlinear power spectrum.

Figure 11

Resummed diagrams for the bispectrum $B_{abc}({\mathbf{k}}_1,{\mathbf{k}}_2,{\mathbf{k}}_3,\eta )$ up to one loop. All the remaining contributions are obtained by cyclic permutations of (${\mathbf{k}}_1,{\mathbf{k}}_2,{\mathbf{k}}_3$) on diagrams 2 through 6.

Figure 12

Renormalized series expansion for $\Sigma$ up to two loops.

Figure 13

Renormalized series expansion for $\Phi$ up to two loops.