Consistent effective theory of long-wavelength cosmological perturbations

Effective field theory provides a perturbative framework to study the evolution of cosmological large-scale structure. We investigate the underpinnings of this approach, and suggest new ways to compute correlation functions of cosmological observables. We find that, in contrast with quantum field theory, the appropriate effective theory of classical cosmological perturbations involves interactions that are nonlocal in time. We describe an alternative to the usual approach of smoothing the perturbations, based on a path-integral formulation of the renormalization group equations. This technique allows for improved handling of short-distance modes that are perturbatively generated by long-distance interactions.

Figure 1

Diagrammatic representation of the solution for ${\varphi }_{(1)}^i(\tau )$, ${\varphi }_{(2)}^i(\tau )$, and ${\varphi }_{(3)}^i(\tau )$, as given by Eqs. (3.3), (3.4), and (3.5). Notation is as follows: vertical solid lines are associated with a Green function $G_j^i$. Vertices represent the interaction $M_{jk}^i$, and the position of each vertex is integrated over time. A solid line emerging from the bottom horizontal dotted line represents an initial condition ${\varphi }_{\text{in}}^j$, and the line reaching the upper horizontal dotted line is the quantity being calculated.

Figure 2

Diagrammatic representation of the correlator $⟨{\varphi }_{(1)}^i{\varphi }_{(3)}^j⟩$, as expressed in Eq. (3.9). Here we have explicitly indicated the quantities associated with each line and vertex. The bottom brackets represent contraction of the two lines, which is carried out by summing with the linear power spectrum. Momentum in the loop labeled with indices $l,m,n,o$ is integrated over. The other possible contraction, linking ${\varphi }_{(1)}^o$ and ${\varphi }_{(1)}^p$, vanishes in the theory of LSS.

Figure 3

Diagrammatic representation of the Taylor expansion for ${\varphi }_{\mathrm{S}}^i$ considered as a functional of ${\varphi }_{\mathrm{L}}^j$, as expressed in Eq. (3.16). Dashed lines represent (arbitrary numbers of) the field ${\varphi }_{\mathrm{S}}$, and the NL blob stands for nonlinear interactions. In the second diagram we see the effects of the background field ${\varphi }_{\mathrm{L}}$, thought of as an external source.

Figure 4

Diagrammatic representation of the solution for $\mathrm{\Delta }{\varphi }_{\mathrm{L}(2)}^i$, as expressed in Eq. (3.21). Dashed lines are the short-wavelength field ${\varphi }_{\mathrm{S}0}$, while solid lines are the long-wavelength field ${\varphi }_{\mathrm{L}}$. As in Fig. 3, NL blobs represent nonlinear interactions.

Figure 5

Diagrammatic representation of the renormalization group equation (4.22). The curved bracket at the bottom represents a contraction with a factor of $d{P}^{ij}/d\mathrm{\Lambda }$. The second graph on the right stands for a sum over various ways to distribute and contract the incoming lines.