An Unfinished Collaboration with A. A. Starobinsky

TL;DR

This paper summarizes early-stage results from a collaboration attempting to apply stochastic methods to study infrared quantum-gravity effects during cosmic inflation, highlighting ongoing research efforts.

Contribution

Introduces an initial exploration of stochastic techniques in quantum gravity within an inflationary universe context, based on unfinished collaborative work.

Findings

Preliminary insights into IR quantum-gravity effects

Application of stochastic approach to inflationary spacetime

Foundation for future research in quantum cosmology

Abstract

The present text summarizes some of the results obtained in 2008 during the initial stages of a collaboration with Starobinsky which remained unfinished. The collaboration was an attempt to apply the stochastic approach to infrared (IR) quantum-gravity effects in an inflationary spacetime.

Equations

$$d{s}^2=-d{t}^2+e^{2Ht}{\hat{g}}_{ij}(\mathbf{x},t)d{x}^{i}d{x}^j,$$

$${\hat{g}}_{ij}(\mathbf{x},t)=a_{ij}(\mathbf{x})+b_{ij}(\mathbf{x})e^{-2Ht}+c_{ij}(\mathbf{x})e^{-3Ht}+\cdots .$$

$${\hat{g}}_{ij}(\mathbf{x},t)={\delta }_{ij}+h_{ij}(\mathbf{x},t),$$

$$h_{ij}(\mathbf{x},t)=\sum _e\kappa \int \frac{d^{3}k}{(2\pi {)}^{\frac{3}{2}}}{e}_{ij}{a}_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{x}}\frac{{\chi }_{\mathbf{k}}(\eta )}{a(\eta )}+h.c.$$

$$a=e^{Ht}=-\frac{1}{H\eta }.$$

$${\chi }_{\mathbf{k}}^{\prime \prime }(\eta )+\left(k^2-\frac{a^{\prime \prime }}{a}\right){\chi }_{\mathbf{k}}(\eta )=0,$$

$${\chi }_{\mathbf{k}}(\eta )=\frac{e^{-ik\eta }}{\sqrt{2k}}\left(1-\frac{i}{k\eta }\right).$$

$$[a_{\mathbf{k}},a_{{\mathbf{k}}^{\prime }}^{†}]=\delta (\mathbf{k}-{\mathbf{k}}^{\prime }),$$

$$⟨0|h_{ij}^2(\mathbf{x},t)|0⟩=2{\kappa }^2{\left(\frac{H}{2\pi }\right)}^2{\int }_{k_{\min }}^{k_{\max }}\frac{dk}{k}(1+(k\eta {)}^2).$$

$$⟨0|h_{ij}^2(\mathbf{x},t)|0{⟩}_{\mathrm{IR}}\approx 2{\kappa }^2{\left(\frac{H}{2\pi }\right)}^2\ln \frac{a(\eta )}{a({\eta }_0)}=2{\kappa }^2{\left(\frac{H}{2\pi }\right)}^{2}H(t-t_0).$$

$$\varphi (\mathbf{x},t)=\bar{\varphi }(\mathbf{x},t)+\int \frac{d^{3}k}{(2\pi {)}^{\frac{3}{2}}}\theta (k-ϵa(t)H)\left[a_{\mathbf{k}}{\varphi }_{\mathbf{k}}(t)e^{i\mathbf{k}\cdot \mathbf{x}}+h.c.\right].$$

$$f(\mathbf{x},t)\equiv ϵa(t)H^2\int \frac{d^{3}k}{(2\pi {)}^{\frac{3}{2}}}\delta (k-ϵa(t)H)\left[a_{\mathbf{k}}{\varphi }_{\mathbf{k}}(t)e^{i\mathbf{k}\cdot \mathbf{x}}+h.c.\right].$$

$$\frac{\partial \bar{\varphi }(\mathbf{x},t)}{\partial t}=-\frac{1}{3H}{V}^{\prime }(\bar{\varphi })+f(\mathbf{x},t),$$

$${\hat{g}}^{is}{\partial }_t{\hat{g}}_{sj}=2{\hat{K}}_j^i,$$

$${\partial }_t{\hat{K}}_j^i+3H{\hat{K}}_j^i+H\hat{K}{\delta }_j^i+\hat{K}{\hat{K}}_j^i+a^{-2}{\hat{R}}_j^i=0.$$

$${\hat{g}}_{ij}(\mathbf{x},t)={\bar{g}}_{ij}(\mathbf{x},t)+h_{ij}^{\mathrm{UV}}(\mathbf{x},t),$$

$$h_{ij}^{\mathrm{UV}}(\mathbf{x},t)=\sum _e\kappa \int \frac{d^{3}k}{(2\pi {)}^{\frac{3}{2}}}\theta (k-ϵa(t)H)\left[e_{ij}{a}_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{x}}\frac{{\chi }_{\mathbf{k}}(\eta )}{a(\eta )}+h.c.\right].$$

$$\frac{\partial {\bar{g}}_{ij}(\mathbf{x},t)}{\partial t}\approx f_{ij}(\mathbf{x},t),$$

$$f_{ij}(\mathbf{x},t)\equiv ϵ\kappa a(t)H^2\sum _e\int \frac{d^{3}k}{(2\pi {)}^{\frac{3}{2}}}\delta (k-ϵa(t)H)\left[e_{ij}{a}_{\mathbf{k}}{e}^{i\mathbf{k}\cdot \mathbf{x}}\frac{{\chi }_{\mathbf{k}}(\eta )}{a(\eta )}+h.c.\right].$$

$$⟨f_{ij}(t)f_{kl}(t^{\prime })⟩\propto ({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}-\frac{2}{3}{\delta }_{ij}{\delta }_{kl}){\left(\frac{\kappa H}{2\pi }\right)}^{2}H\delta (t-t^{\prime }).$$

$$d{s}^2=\mathrm{Tr}[(\bar{g}{)}^{-1}d\bar{g}{]}^2={\bar{g}}^{ik}{\bar{g}}^{jl}d{\bar{g}}_{ij}d{\bar{g}}_{kl},$$

$${\bar{g}}_{ij}(t)=\sum _{a=1}^3{N}_i^a{e}^{2{\beta }_a}{N}_j^a;\mathrm{with}{\beta }_1+{\beta }_2+{\beta }_3=0.$$

$$d{X}^{\mu }(T)=e_a^{\mu }(X)d{w}^a(T)-\frac{1}{2}{G}^{\alpha \beta }{\Gamma }_{\alpha \beta }^{\mu }dT,$$

$$\frac{\partial \rho (X)}{\partial T}=\frac{1}{2}{\Delta }_G\rho (X),$$

$$\frac{\partial \rho (\bar{g})}{\partial T}=\frac{1}{2}{\Delta }_S\rho (\bar{g}),$$

$$T\sim {\left(\frac{\kappa H}{2\pi }\right)}^{2}Ht.$$

$${\bar{g}}_{ij}(T)=\sum _{a=1}^3{N}_i^a(T)e^{2{\beta }_a(T)}{N}_j^a(T),$$

$$2{\beta }_1(T)$$

$$2{\beta }_2(T)$$

$$2{\beta }_3(T)$$

$$N_2^1(T)-N_2^1(0)\sim {\int }_0^T\exp ({\beta }_2(T^{\prime })-{\beta }_1(T^{\prime }))d{w}_{12}(T^{\prime }).$$

$${\beta }_2(T)-{\beta }_1(T)={\beta }_2(0)-{\beta }_1(0)-\frac{1}{4}T+\frac{1}{2}{\bar{w}}_2(T)-\frac{1}{2}{\bar{w}}_1(T)\approx -\frac{1}{4}T+O(T^{\frac{1}{2}}).$$

Full text

1 Introduction

My dear friend Alexei Alexander Starobinsky (henceforth Alyosha) visited the Institut des Hautes Etudes Scientifiques (IHES) in October 2008. During this visit, we started a collaboration on the impact of infrared quantum gravity effects on the global structure of de Sitter (or inflationary) spacetimes. I had always hoped that Alyosha would come back at IHES to allow us to continue our discussions, and to hopefully finalize the preliminary results we had reached in October 2008. Alas this did not happen. Though I had the pleasure of meeting again Alyosha in person during some later visits of mine in Moscow (notably during the 2019 conference dedicated to the 100th anniversary of Khalatnikov), and though Alyosha kindly agreed to speak (remotely) during the 2021 “Damour Fest” at IHES, we were both dragged away from our collaborative project by other scientific pursuits, and we never finalized our collaboration. As an homage to him, and as a token of my admiration for all the deep and beautiful ideas he contributed to physics, the present text will present the gist of our discussions, as they were left at the time (together with some additional considerations on Brownian motion in the space of 3-by-3 matrices). Hopefully, the approach we pursued (crucially based on Alyosha’s stochastic approach to inflation) has still some value, in spite of the many developments that have since happened (and are still happening) in the study of infrared effects in de Sitter spacetimes. Before entering into details, let me mention that the stochastic approach to inflation is the idea, pioneered in Ref. Starobinsky:1982ee (see also Vilenkin:1983xq ; Linde:1986fd , as well as Starobinsky:1986fx ; Starobinsky:1994bd ), that the infrared part of a quantum fluctuating field during inflation may be considered as a classical (i.e., $c$-number) space-dependent stochastic field satisfying a local (i.e., without spatial derivatives) evolution equation where the classical evolution is completed by an additional Langevin-like fluctuating term (coming from the higher-frequency quantum fluctuations). For recent reviews of the stochastic approach to inflation see Ref. Cruces:2022imf ; Woodard:2025cez . For entries into the still growing literature on infrared effects in de Sitter (or inflationary) spacetimes, see Polyakov:2012uc ; Anderson:2017hts ; Akhmedov:2019cfd ; Akhmedov:2019esv ; Cohen:2020php ; Miao:2024shs ; Woodard:2025smz . The last two references, of which I became aware while writing the present account, have close links with the aim of my unfinished collaboration with Alyosha. I leave to the future to decide whether they make it irrelevant by superseding it.

2 First starting point

The first starting point of our discussions was the following classic result of Alyosha. In 1983, Alyosha derived the asymptotic structure, at large times, of the general solution of Einstein’s equations in presence of a (positive) cosmological constant $\Lambda=3H^{2}$ Starobinsky:1982mr (in $3+1$ dimensions, and when neglecting other types of stress-energy tensor), namely (in synchronous gauge)

[TABLE]

where the rescaled spatial metric $\hat{g}_{ij}({\bf x},t)$ admits an expansion in powers of $e^{-Ht}$ of the form

[TABLE]

One can further impose the unimodularity condition $\det a_{ij}({\bf x})=1$. The latter expanded solution111In an anti-de-Sitter context, it would be called a Fefferman-Graham expansion. is general in that it contains four physical arbitrary functions of three variables: there are two arbitrary functions in the unimodular spatial metric $a_{ij}({\bf x})$ (considered modulo 3-diffeomorphisms); the function $b_{ij}({\bf x})$ is determined by the Ricci tensor of $a_{ij}({\bf x})$ (namely $H^{2}b_{ij}({\bf x})=R_{ij}[a]-\frac{1}{4}R[a]a_{ij}$); and the other two arbitrary functions of three variables are contained in $c_{ij}({\bf x})$ (which must satisfy $c_{i}^{i}=0$ and $\nabla^{j}c_{ij}=0$). [The latter constraints use $a_{ij}({\bf x})$ as background metric.] Terms beyond $c_{ij}({\bf x})$ in the expansion (2) are fully determined by $a_{ij}({\bf x})$ and $c_{ij}({\bf x})$.

The expansion (2) is valid in many spatial coordinate systems. E.g., one recovers the global de Sitter metric with $\hat{g}_{ij}({\bf x},t)dx^{i}dx^{j}=H^{-2}e^{-2Ht}\cosh^{2}Htd\Omega_{3}^{2}$ (where $d\Omega_{3}^{2}$ is a round 3-sphere metric), which leads to $a_{ij}({\bf x})dx^{i}dx^{j}=\frac{1}{4}H^{-2}d\Omega_{3}^{2}$ and $b_{ij}({\bf x})=2a_{ij}({\bf x})$. Alternatively, the Poincaré-patch case corresponds to $a_{ij}({\bf x})=\delta_{ij}$ and $b_{ij}({\bf x})=0$.

The crucial point here is that though this general solution exhibits a phenomenon of local isotropization, dubbed the cosmic no-hair behavior (with an exponentially vanishing Weyl tensor, $C_{\mu\nu\kappa\lambda}C^{\mu\nu\kappa\lambda}\propto e^{-6Ht}$), its global spatial geometry is conformal, at large times, to an arbitrary (unimodular) 3-metric $a_{ij}({\bf x})=\lim_{t\to\infty}\hat{g}_{ij}({\bf x},t)$. The conformal spatial geometry defined by $a_{ij}({\bf x})$ is globally inhomogeneous and anisotropic. It can be arbitrarily different from the round sphere obtained for a global de Sitter spacetime. On the other hand, it is time-independent.

3 Second starting point

As just reviewed, in a classical context the large-time limit of the conformal spatial metric exists and defines a globally inhomogeneous and anisotropic, but time-independent geometry. The second basic starting point of my collaboration with Alyosha was to extend his stochastic approach to inflation to an approximate description of the evolution of $\hat{g}_{ij}({\bf x},t)$ in a quantum context, and to argue that a suitably coarse-grained version of the quantum $\hat{g}_{ij}({\bf x},t)$ would define a stochastically evolving (unimodular) 3-metric $a_{ij}({\bf x},t)$ becoming exponentially anisotropic as $t\to+\infty$.

Let us first recall another classic result of Alyosha, concerning the production of “relict gravitational radiation” from quantum gravitational inflationary fluctuations Starobinsky:1979ty . Linearized quantum gravitational fluctuations in a de Sitter background, i.e. small fluctuations in the (conformally rescaled) spatial metric of the type,

[TABLE]

can be Fourier-decomposed as (with $\kappa\equiv\sqrt{32\pi G}$)

[TABLE]

Here, we assumed for simplicity that we work in a Poincaré patch, and we introduced the conformal time $\eta$, varying between $-\infty$ and $0^{-}$, and such that

[TABLE]

In Eq. (4), $e_{ij}$ labels two different normalized ($e_{ij}e_{ij}=1$) transverse-traceless polarization tensors, and the corresponding mode functions222For simplicity we leave implicit the extra label, say $e=+,\times$, on the mode functions, and on the annihilation operators. $\chi_{\bf k}(\eta)$ satisfy the equation (with ${}^{\prime}=d/d\eta$)

[TABLE]

whose solution (corresponding to the Bunch-Davies vacuum) is

[TABLE]

Finally, $a_{\bf k}$ is a corresponding quantum annihilation operator satisfying the canonical commutation relation

[TABLE]

and defining the Bunch-Davies vacuum through $a_{\bf k}|0\rangle=0$.

A first way to see the physics associated with the quantum gravitational field (4) is to compute the VEV of $h_{ij}^{2}({\bf x},t)$, namely

[TABLE]

The latter integral is both UV and IR divergent. The UV (power-law) divergence is the usual local-Minkowski divergence of the two polarization modes of the gravitational field (with $h_{ij}(L)\sim\kappa/L$ over a small (sub-Hubble) spatial length $L\ll H^{-1}$). On the other hand, the IR (logarithmic) divergence involves the lower bound $k_{\rm min}=Ha(\eta_{0})$ corresponding to the Hubble scale of some initial time, $\eta_{0}$. When integrated over the interval of momenta $k_{\rm min}<k<Ha(\eta)$ (where the upper bound $Ha(\eta)$ corresponds to the Hubble scale at the considered current time, $\eta$), the contribution of the IR logarithm then gives (using $a(\eta)=e^{Ht}$)

[TABLE]

We see on the latter equation that the IR part of the quantum gravitational fluctuations leads to a linear-in-time wandering of the squared metric fluctuation. We recognize a Brownian motion of $h_{ij}(t)$ due to quantum modes coming from the UV, being redshifted by the expansion, and passing through the Hubble scale at time $t$. Essentially, Eq. (10) says that each one of the two polarization modes contributes a random amplitude $\pm\kappa\frac{H}{2\pi}$, during each time interval $\Delta t=H^{-1}$.

4 Towards a stochastic approach to super-Hubble quasi-classical IR fluctuations of the de Sitter metric

Alyosha’s stochastic approach to quantum fluctuating light scalar fields (with mass $m\ll H$) during inflation consists in representing the Heisenberg operator of the quantum field $\phi$ as Starobinsky:1982ee ; Starobinsky:1986fx ; Starobinsky:1994bd

[TABLE]

Here $\bar{\phi}({\bf x},t)$ is a coarse-grained version of $\phi({\bf x},t)$, i.e. averaged over a constant physical volume larger (given some $\epsilon<1$) than a Hubble volume. The second term in Eq. (11), which involves a step function $\theta(x)$ and annihilation/creation operators $a_{\bf k},a^{\dagger}_{\bf k}$, represents the short-wavelength counterpart of $\phi({\bf x},t)$. The presence of the step function means that quantum modes of wave number $k$ coming from the UV get redshifted by the expansion, $k^{\rm phys}=\frac{k}{a(t)}$, and thereby continuously transit from the quantum high-frequency part of $\phi({\bf x},t)$ to the low-frequency part $\bar{\phi}({\bf x},t)$ as $k^{\rm phys}=\frac{k}{a(t)}$ becomes smaller than $\epsilon H$. The basic idea is to approximate the low-frequency part $\bar{\phi}({\bf x},t)$ by a classical stochastic field which can wander to large values in field space, contrary to the high-frequency quantum part, which is treated as staying “small” (after suitably subtracting UV divergences). When taking the time derivative of Eq. (11), the derivative of the step function generates a term in $\partial\phi({\bf x},t)/\partial t$ of the form $-f({\bf x},t)$, where

[TABLE]

The term $+f({\bf x},t)$ then must appear as an additional term in the time derivative of the quasi-classical low-frequency field $\bar{\phi}({\bf x},t)$. The main technical feature of the stochastic approach is then to consider the term $+f({\bf x},t)$ as a Langevin-type stochastic force driving (in addition to classical terms) the evolution of $\bar{\phi}({\bf x},t)$. E.g., in the case of a scalar field with self-interacting potential $V(\phi)$, one gets the approximate first-order evolution equation

[TABLE]

where $\bar{\phi}({\bf x},t)$ is treated as a classical stochastic variable, and $f({\bf x},t)$ as a random Gaussian noise, having the statistical properties defined by the quantum averages of $a_{\bf k}$ and $a^{\dagger}_{\bf k}$. Refs. Starobinsky:1994bd ; Woodard:2025cez have shown that this stochastic approach does correctly describe IR effects in various field models. See also Ref. Kiefer:1998qe for a discussion of why the low-frequency quantum fluctuations can be treated as classical stochastic variables.

The aim of our unfinished collaboration was to use this stochastic approach to derive an approximate evolution of the low-frequency part (say $\bar{g}_{ij}({\bf x},t)$) of the conformal metric $\hat{g}_{ij}({\bf x},t)$. We explored several possibilities for defining $\bar{g}_{ij}({\bf x},t)$ and for writing down its Langevin-type evolution equation. Let me mention one of them.

If we start from the $3+1$ decomposition of Einstein’s equations (with $\Lambda$) in a synchronous (or Gauss) gauge, and rewrite them in terms of $\hat{g}_{ij}({\bf x},t)\equiv a^{-2}g_{ij}({\bf x},t)$, with $a(t)=e^{Ht}$, they read (with $\hat{K}\equiv\hat{K}^{s}_{\;s}$)

[TABLE]

Let us start, around some spatial point, and at some time, with a local value of the metric of the type

[TABLE]

where the initial value of the low-frequency part is the flat Euclidean metric, $\bar{g}_{ij}({\bf x},t_{0})=\delta_{ij}$, while the high-frequency part reads

[TABLE]

Taking the time derivative of the step function in $h^{\rm UV}_{ij}({\bf x},t)$ suggests that the low-frequency part will approximately evolve as

[TABLE]

where the Langevin force $f_{ij}({\bf x},t)$ (induced by the time derivative of $h^{\rm UV}_{ij}({\bf x},t)$) reads

[TABLE]

The trace, $\delta^{ij}f_{ij}$, of this Langevin force vanishes, and its two-point correlation has (when considering for simplicity a fixed spatial point333Alyosha told me that he expected the (physical) spatial correlation length $R_{c}$ to be (asymptotically) of (logarithmic) order $\ln(HR_{c})\sim(\kappa H)^{-1}$.) the following structure:

[TABLE]

When going beyond the approximation of linear stochastic perturbations of $\bar{g}_{ij}({\bf x},t)$ around an Euclidean metric $\delta_{ij}$, described by Eq. (17), it seems consistent to define the nonlinear stochastic evolution of $\bar{g}_{ij}({\bf x},t)$ by requiring that $\bar{g}_{ij}({\bf x},t)$ stays unimodular, and satisfies a geometrically-defined first-order stochastic evolution equation in the space, say $S$, of unimodular, positive-definite symmetric 3-by-3 matrices.

Let us recall that the space of unimodular, positive-definite symmetric 3-by-3 matrices $\bar{g}$ is a 5-dimensional Riemannian space, endowed with the metric

[TABLE]

and that it is equivalent to the symmetric space $S\equiv SL(3)/SO(3)$. The equivalence between these two 5-dimensional spaces is made particularly clear by parametrizing the matrix $\bar{g}=(\bar{g}_{ij}({\bf x},t))$ by means of an Iwasawa decomposition $KAN$ of the group $SL(3)$. Namely, a generic group element $\gamma\in SL(3)$ can be uniquely decomposed as the product $kan$ of three 3-by-3 matrices, with: $k\in SO(3)$; $a$ being diagonal, positive, and unimodular; and with $n$ being upper triangular (with ones on the diagonal). The coset $S\equiv SL(3)/SO(3)$ is described by the $an$ part, which is in one-to-one correspondence with $\bar{g}=\gamma^{t}\gamma=n^{t}a^{t}an=n^{t}An$ ($t$ denoting a transpose, and $A=a^{t}a$ being diagonal, positive and unimodular). More explicitly, we can write at each spatial point ${\bf x}$

[TABLE]

Here, $N^{a}_{\;\>i}$ denotes the components of the upper triangular matrix $n$, while $A=a^{t}a={\rm diag}[e^{2\beta_{a}}]$. The explicit expression of the metric $ds^{2}$ in the Iwasawa coordinates $\beta_{a}$, $N^{a}_{\;\>i}$ can be found in Henneaux:1981su .

The infinitesimal neighbourhood of $\bar{g}_{ij}=\delta_{ij}$, say $\bar{g}_{ij}=\delta_{ij}+\epsilon f_{ij}$ (with $f_{ij}$ symmetric and traceless), is parametrized by $\beta_{a}\to\epsilon\beta_{a}$ and $N^{a}_{\;\;i}=\delta_{ai}+\epsilon n_{ai}$, with $n_{ai}$ strictly upper triangular (i.e. with $n_{12},n_{13},n_{23}$ as only non-zero elements). The correspondence between $f_{ij}$ and $\beta_{a},n_{ai}$ is $f_{aa}=2\beta_{a}$, $f_{12}=n_{12}(=f_{21}),f_{13}=n_{13},f_{23}=n_{23}$. The Langevin term (18) in the evolution of $\bar{g}_{ij}(t)$ near $\delta_{ij}$ then defines a corresponding Langevin term in the evolution of the coordinates, $\beta_{a}$, $n_{12},n_{13},n_{23}$, of the coset element $an$ near the identity.

The issue is then to define, at the nonlinear level (when $\bar{g}$ starts wandering away from the identity), a definite stochastic evolution of $\bar{g}$ over the curved manifold $S$. The classic work of Itô Ito1950 has shown how to define stochastic differential equations on a curved manifold in a geometrically covariant way, i.e. independently of the choice of coordinates on the considered manifold. In the case of the coset space $S$, the five numbers $\beta_{1}$, $\beta_{2}$, $n_{12},n_{13},n_{23}$ parametrizing $\bar{g}$ are particular coordinates. One wants to consider a first-order stochastic differential system that is equivariant under the choice of parametrization of the elements of $S$. Indeed, one might have used coordinates on $S$ based, e.g., on a Gauss decomposition of the matrix $\bar{g}_{ij}$, as used, e.g., in Damour:2011yk .

Itô found that a geometrically covariant stochastic differential equation for the motion of the coordinates $X^{\mu}$ of a point on a Riemannian manifold (with metric tensor $G_{\mu\nu}(X)$ in the coordinates $X^{\mu}$) must involve an additional (non-covariant) drift vector. The most natural Brownian motion equation for $X^{\mu}(T)$ (with a suitable scaled time variable $T$) reads

[TABLE]

where $e^{\mu}_{a}(X)$ is a local orthonormal frame (“repère mobile”, satisfying $G_{\mu\nu}e^{\mu}_{a}e^{\nu}_{b}=\delta_{ab}$), $dw^{a}(T)$ a normalized white noise in the tangent space ($\langle dw^{a}(T)dw^{b}(T^{\prime})\rangle=\delta^{ab}\delta(T-T^{\prime})$) , and $\Gamma^{\mu}_{\alpha\beta}$ the Christoffel coefficients of $G_{\mu\nu}(X)$. The diffusion (or Fokker-Planck) equation describing the evolution of the (one-point) probability distribution function (pdf) $\rho(X)$ then reads

[TABLE]

where $\Delta_{G}$ denotes the Laplace-Beltrami operator defined by the metric $G_{\mu\nu}(x)$.

It was argued above that the IR quantum fluctuations of the spatial metric $\bar{g}_{ij}({\bf x},t)$, when starting from the flat metric $\delta_{ij}$, satisfy $\frac{\partial\bar{g}_{ij}(t)}{\partial t}\approx f_{ij}(t)$ with a white noise $f_{ij}$ defined by Eq. (18). Viewing $\bar{g}_{ij}({\bf x},t)$, at each spatial point ${\bf x}$, as a point $X^{\mu}$ in the 5-dimensional symmetric space $S\equiv SL(3)/SO(3)$ (endowed with the metric (20)), it seems natural to assume that, at the nonlinear level (when $\bar{g}$ starts wandering away from the identity), the stochastic wandering of $\bar{g}_{ij}({\bf x},t)$ over the 5-dimensional symmetric space $S$ will be described by the simple geometric Brownian motion (22), with associated diffusion equation

[TABLE]

for the one-point pdf of $\bar{g}$ on $S\equiv SL(3)/SO(3)$. Here $\Delta_{S}$ denotes the Laplace-Beltrami operator on $S$, endowed with the metric (20). [We also wondered whether in Eq. (24) $\Delta_{S}$ should be augmented by a scalar-curvature term, say $\Delta_{S}+\xi R_{S}$ with $\xi=O(1)$.] The heat equation (24) involves a suitably rescaled (dimensionless) time variable,

[TABLE]

While writing up the results of my discussions with Alyosha, I became aware of the large mathematical literature discussing Brownian motions on symmetric spaces. For entries in the literature see notably Refs.Malliavin1974 and Anker1992 .

A remarkable fact is that the long-time behavior of a Brownian motion on a non-compact symmetric space (such as $S$) exhibits special features linked to the negative sectional curvature of such spaces. Namely, when using an Iwasawa parametrization, say $\bar{g}=n^{t}a^{t}an=n^{t}An$, heat diffusion in symmetric spaces $G/K$ of rank $\ell\geq 2$ asymptotically (for large times $T$) proceeds in the following way: (i) on the one hand, the off-diagonal component $n(T)$ (parametrized by $N^{a}_{\;\;i}$ in Eq. (21)), tends to some (random) finite limit as $T\to+\infty$; while, (ii) on the other hand, the logarithm $\alpha=\ln A$ of the diagonal component (equal to $\alpha={\rm diag}(2\beta^{a})$ in the notation of Eq. (21)) asymptotically proceeds in a fixed direction (Weyl vector) in the linear Cartan space, and with a fixed, finite speed.

In our case, $S=G/K$ with $G=SL(3,R)$ and $K=SO(3)$, the rank (i.e. the number of independent $\beta$’s) is $\ell=2$. Actually, in this case, one can explicitly write down the solution of the stochastic differential equation satisfied by the five Iwasawa coordinates $N^{a}_{\;\;i}$ and $\beta_{a}$ (with the constraint $\beta_{1}+\beta_{2}+\beta_{3}=0$), in terms of six Wiener processes $w_{I}(T)$, $I=1,2,\cdots,6$, as was done in the recent Ref. Matsumoto2022 . [One must subtract from the first three Wiener processes in Section 3.2 of Matsumoto2022 their trace $\frac{1}{3}(w_{1}+w_{2}+w_{3})$, i.e. replace $w_{a}$ by $\bar{w}_{a}=w_{a}-\frac{1}{3}(w_{1}+w_{2}+w_{3})$]. This leads to a result of the following form for the asymptotic (large $T$) stochastic evolution of the metric $\bar{g}_{ij}(T)$ (considered at each given spatial point $\bf x$):

[TABLE]

where the three diagonal (logarithmic) components are given by simple explicit expressions involving Wiener processes (i.e. Brownian motions), namely

[TABLE]

while the off-diagonal components are given by stochastic integrals (involving three more independent Wiener processes $w_{12}(T),w_{13}(T),w_{23}(T)$) of the type

[TABLE]

The latter stochastic integrals converge to finite (random) limits as $T\to+\infty$. The existence of a finite limit follows from the fact that the exponents entering the differential of $N^{a}_{\;\>i}(T)$ are decreasing approximately linearly in $T$; e.g. (recalling that Brownian motions $w_{a}(T)$ behave as $O(T^{\frac{1}{2}})$ when $T\to+\infty$)

[TABLE]

We note in passing that the asymptotic (large $T$) stochastic evolution of the metric $\bar{g}_{ij}(T)$ is reminiscent of the Belinsky-Khalatnikov-Lifshitz Belinsky:1970ew dynamics of the spatial metric $g_{ij}(\tau)$ (considered at a fixed spatial point) near a cosmological singularity. Indeed, when parametrizing small cosmic proper times $t\to 0$ by $\tau=\int\frac{dt}{\sqrt{g}}\sim-\ln t\to+\infty$, and using an Iwasawa parametrization, the off-diagonal components $N^{a}_{\;\>i}(\tau)$ of the metric have finite limits as $\tau\to+\infty$, while the diagonal components of the metric, $a_{1}^{2}=e^{2\beta_{1}}$, $a_{2}^{2}=e^{2\beta_{2}}$, $a_{3}^{2}=e^{2\beta_{3}}$, have a stochastic behavior made of an infinite succession of Kasnerlike periods during which the $\beta_{a}$’s evolve approximately linearly with $\tau$, say $\beta_{a}\approx\beta_{a}^{0}+c_{a}\tau$, with velocities $c_{a}=O(1)$, see, e.g., Damour:2002et ; Damour:2007nb .

When going back from the heat-diffusion of the rescaled metric $\bar{g}_{ij}$ over $S$ with respect to the rescaled time $T$, Eq. (25), to the stochastic evolution of the original spatial metric $g_{ij}({\bf x},t)=e^{2Ht}\hat{g}_{ij}({\bf x},t)$ in synchronous gauge, the final result is that the spatial metric at each spatial point would stochastically wander in the following approximate way

[TABLE]

where we recall that $T\sim\left(\frac{\kappa H}{2\pi}\right)^{2}\,Ht$.

This quantum asymptotic behavior differs from the classical cosmic-no-hair asymptotic behavior of Eqs. (1), (2):

[TABLE]

in that the time-independent limiting classical rescaled (unimodular) metric $a_{ij}({\bf x})$ is replaced by a (unimodular) metric whose diagonal elements (in an Iwasawa parametrization) keep evolving in an exponential way. This corresponds to a quantum violation of the cosmic-no-hair behavior (the rescaled metric becoming more and more anisotropic in the future). However, we note that the anisotropic deviations from a uniformly expanding de Sitter metric are fractionally small. Indeed, we have an anisotropic Hubble expansion for the diagonal components $a_{a}^{2}$ of the type $a_{a}\approx\exp H_{a}t$, where $a=1,2,3$ and

[TABLE]

with $c_{a}\propto(+\frac{1}{4},0,-\frac{1}{4})$ (modulo an overall factor of order unity). Here the dimensionless (one-loop) factor $\left(\frac{\kappa H}{2\pi}\right)^{2}$ (which is proportional to $G$) must be assumed to be small in the present treatment. Note also that, in the approximation used here, we have no indication of the presence of a quantum back reaction slowing down the exponential inflation (as was suggested by several authors, notably by Polyakov Polyakov:1982ug , and by Tsamis and Woodard Tsamis:1996qq ). This is not unexpected as our approximation is a resummed version of one-loop effects, while the perturbative computations of Ref. Tsamis:1996qq (see also Miao:2024shs ; Woodard:2025smz ) found a slowing down of inflation at the two-loop level, i.e., $H_{\rm effective}\sim H\left[1-c\left(\frac{\kappa H}{2\pi}\right)^{4}(Ht)^{2}\right]$ with a positive coefficient $c=O(1)$.

I leave to future work the task444I also leave to future work the task of putting on a firmer ground the sketchy derivation given above, with precise numerical estimates of the various $O(1)$ factors that entered it. of comparing the results sketched above to the recent leading-logarithm quantum gravity results of Ref. Miao:2024shs . This task poses the delicate problem of defining observable quantities that could be compared in both approaches (and that are physically meaningful within a quantum cosmological spacetime).

Summarizing: a possibly too naive, and certainly preliminary, application of Starobinsky’s stochastic approach to IR quantum-gravity effects in an inflationary spacetime suggests a picture where the low-frequency part of the metric (coarse-grained on Hubble scales) undergoes (modulo the overall exponential expansion factor $e^{2Ht}$) a Brownian motion in the 5-dimensional symmetric space $SL(3)/SO(3)$. This leads to a quantum violation of the cosmic-no-hair behavior in which the rescaled metric $\bar{g}_{ij}({\bf x},t)$ does not tend to a finite limit in the asymptotic future, but becomes more and more anisotropic in a way reminiscent of the Belinsky-Khalatnikov-Lifshitz behavior near a singularity.

Acknowledgements.

It is a pleasure to thank Sasha Polyakov and Richard Woodard for enlightening discussions about IR effects in de Sitter spacetimes, and Yvain Bruned and Laurence Field for informative exchanges about stochastic differential equations. The present work was partly supported by the “2021 Balzan Prize for Gravitation: Physical and Astrophysical Aspects”.