Other coarse-graining functions also exist, introducing colored noise [88], which gives rise to non-Markovian memory effects.

We retain only the zero modes, ${\varphi }_s\simeq {\varphi }_{s0}+\mathcal{O}(k)$, and regulate the comoving volume $(2\pi {)}^3{\delta }^{(3)}(0)$ as $\mathrm{\Omega }$. For convenience, we drop the subscript 0 and work in a unit comoving volume, $\mathrm{\Omega }=1$, so that the zero modes are canonically normalized.

This structure is known as the open EFT of cosmology [61, 62, 63, 91, 92].

This is the Markovian approximation in the language of open quantum systems, referring to the time locality of the density matrix evolution [1, 4]. We emphasize that this is conceptually different from the use of “(non)Markov” in Refs. [54, 99] for Langevin dynamics, where it denotes that the amplitude of the white noise is history dependent.

See Ref. [102] for a derivation of the leading FP equation from holographic renormalization, where late-time evolution is interpreted as boundary renormalization [103]. As in EFTs more generally, this renormalization implements the resummation of IR logarithms [32].

To match with Ref. [24], we need to include the renormalization that generates an effective mass $v_r(\varphi )=v(\varphi )+m_{\mathrm{eff}}^2{\varphi }^2$ and then drop higher order terms in $\lambda ,\lambda m_{\mathrm{eff}}^2,\cdots$.

See also Ref. [107] for a discussion of the static patch.