A critical look at stochastic inflation
The stochastic approach aims at describing the long-wavelength part of quantum fields during inflation by a classical stochastic theory. It is usually formulated in terms of Langevin equations, giving rise to a Fokker-Planck equation for the probability distribution function of the fields, and possibly their momenta. The link between these two descriptions is ambiguous in general, as it depends on an implicit discretisation procedure, the two prominent ones being the Itô and Stratonovich prescriptions. Here we show that the requirement of general covariance under field redefinitions is verified only in the latter case, however at the expense of introducing spurious ‘frame’ dependences. This feature disappears when there is only one source of stochasticity, like in slow-roll single-field inflation, but manifests itself when taking into account the full phase space, or in the presence of multiple fields. Despite these difficulties, we use physical arguments to write down a covariant Fokker-Planck equation that describes the diffusion of light scalar fields in non-linear sigma models in the overdamped limit. We apply it to test scalar fields in de Sitter space and show that some statistical properties of a class of two-field models with derivative interactions can be reproduced by using a correspondence with a single-field model endowed with an effective potential. We also present explicit results in a simple extension of the single-field theory to a hyperbolic field space geometry.
Introduction.— The theory of cosmological inflation can boast from a spectacular success to explain the patterns observed in the Cosmic Microwave Background anisotropies Ade et al. (2016a, b). According to it, cosmological fluctuations originate from microscopic quantum fluctuations stretched to cosmological scales. This paradigm is therefore deeply rooted both in quantum field theory and in general relativity. However, in its conventional formulation, the geometry of spacetime and the fields active during inflation are split between an homogeneous background, obeying classical equations of motion, and spatial fluctuations, which are treated quantum-mechanically Weinberg (2008); Peter and Uzan (2009).
In the absence of a full theory of quantum gravity, this practical approximation scheme is justified in situations in which the classical behaviour dominates the dynamics. Conceptually, it is not entirely satisfactory though, and moreover, it is expected to break down in the presence of very light scalar fields.
The stochastic formalism aims at addressing this issue by deriving a classical effective theory for the coarse-grained super-Hubble part of the quantum fields driving inflation Starobinsky (1986); Starobinsky and Yokoyama (1994). The expansion of the universe results in a continuous flow of initially sub-Hubble modes joining the super-Hubble sector, which gives rises in this framework to a stochastic dynamics. Stochastic inflation is at the heart of the concepts of eternal chaotic inflation and the multiverse Linde (1986a, b); Goncharov et al. (1987), with possibly far-reaching consequences. Moreover, as the production of primordial black holes during inflation necessitates very flat potentials, in which quantum diffusion effects are non-negligible, it is also necessary to take it into account in the context of the identification of the nature and origin of dark matter Pattison et al. (2017); Kawasaki and Tada (2016); Biagetti et al. (2018); Ezquiaga and
García-Bellido (2018). Scrutinising the theoretical grounds of stochastic inflation is therefore conceptually and practically very important. In this short paper, we summarise the salient features of a detailed investigation carried out in companion papers Pinol et al. (2018a, b), highlighting in particular a conceptual issue of stochastic inflation that has been hitherto unnoticed, related to the Itô versus Stratonovich dilemna in statistical physics van Kampen (1981).
Itô versus Stratonovich.— It is well known that stochastic differential equations (SDE) are not completely defined in general unless they are accompanied with a prescription to make sense of their stochasticity. Different types of consistent stochastic calculus have been defined by mathematicians, chief amongst them Itô and Stratonovich calculus Ito (1944); Stratonovich (1966), that physically correspond to different discretisation procedures for stochastic processes111We concentrate on these two well known discretisations, but one can easily consider more general -discretisatons Pinol et al. (2018b), where , recovering Itô calculus for and the Stratonovich one for . (see e.g. Risken and Haken (1989); Kampen (2007)). It will prove useful to present them in a rather general way. For this, let us consider an arbitrary number of random variables , which depend on time , and that obey the generic Langevin equations
where are independent normalised Gaussian white noises, verifying , and whose numbers need not be the same as the number of ’s. Both the deterministic and stochastic parts and are in general functions of the fields and of , although we will omit to mention the time-dependence for simplicity. In a discretised version, the physical ambiguity comes from whether the strengths of the random kicks are determined by the amplitudes of the noises at a time immediately before the kicks — this is the Itô choice — or, motivated by the fact that white noises are idealisations of random processes with finite correlation time, whether they are determined by the average of the noises amplitudes over the duration of the kick — this is the Stratonovich choice. The two procedures are physically distinct, and the corresponding probability distribution functions (PDF) verify the distinct Fokker-Planck (FP) equation
where the so-called drift is given by in the Itô prescription, and by in the Stratonovich one, and where the diffusion coefficient reads in both cases. The two frameworks differ in the presence of multiplicative noises, i.e. when the depend on the random variables , which results in the noise-induced drift in the Stratonovich prescription, in addition to the deterministic drift. The Stratonovich FP operator can be rewritten as , so that the choice between the two prescriptions is reminiscent of the factor ordering problem in quantum mechanics. Mathematically, a given set of SDE can always be rewritten in either of the two prescriptions by changing the expression of in the Langevin equations (1), but physically the deterministic part of the evolution is often specified, so that other physical considerations should be taken into account to fix the ambiguity.
In the context of inflation, it has been argued that the Itô prescription should be preferred to respect causality Fujita et al. (2014); Tokuda and Tanaka (2018), although the Stratonovich choice is in no way problematic in this respect. On the other hand, the Stratonovich prescription has been advocated by the fact that the white noises should be treated as a limit of colored noises when the smooth decomposition between short and long-wavelength modes becomes sharp Mezhlumian and Starobinsky (1991).
It has been also argued that the choice between the two prescriptions exceeds the accuracy of the stochastic approach Starobinsky (private
communication); Vennin and Starobinsky (2015). We will show that while this latter point is indeed true in a certain sense for slow-roll single-field inflation, the Itô versus Stratonovich dilemna strikes back in a non-trivial manner as soon as more than one degree of freedom is involved, in the full phase space or when multiple fields are taken into account.
Slow-roll single-field inflation.— We begin by considering the simple framework of slow-roll single-field inflation, with the corresponding Langevin equation Starobinsky (1986)
where . Here, is the coarse-grained scalar field, denotes its potential, the deterministic drift describes the overdamped, slow-roll, behaviour, and the amplitude of the noise term is given by the size of super-Hubble fluctuations of a light scalar field (we will discuss below the origin and regime of validity of Eq. (3) and consider more general situations). It is now well understood that the number of e-folds of expansion , where is the scale factor, should be used as the time variable in stochastic inflation, in order to agree with cosmological perturbation theory and to warrant the agreement with perturbative quantum field theory Finelli et al. (2009, 2010); Vennin and Starobinsky (2015).
Note that there is a considerable conceptual and technical difference between test and non-test scalar fields. If a test scalar field in a fixed spacetime geometry is considered,
the instances of in (3) should be replaced by the deterministically determined Hubble scale , which is constant in the most studied case of de Sitter spacetime.
The stochastic formalism can then be seen as a powerful method to resum infrared divergences of a light quantum scalar field in de Sitter spacetime.
In addition, in the case of a scalar field driving inflation, the Langevin equation (3) has been used to compute various observable quantities like the power spectrum of the curvature perturbation, using the techniques of first passage time analysis, recovering results of conventional cosmological perturbation theory in a suitable classical limit Vennin and Starobinsky (2015).
Using the same techniques, it is easy to show that the Itô or Stratonovich prescriptions lead in this context to the same results to an excellent accuracy.
Let be the number of e-folds of inflation that is realised by starting from the initial value , from which the distribution of primordial density perturbations can be easily computed with the stochastic- formalism Fujita et al. (2013, 2014); Vennin and Starobinsky (2015); Assadullahi et al. (2016). The PDF of this stochastic variable is equivalently defined by its Fourier transform, the characteristic function .
From the Fokker-Planck equation,
one can show that satisfies , where indicates the adjoint operator. Whereas
, one has , with the same boundary conditions, and where is the dimensionless potential in Planck units. As for consistently working in the classical gravity regime, one can see that in the regime of validity of stochastic inflation, the Itô or Stratonovich prescriptions for slow-roll single-field inflation lead to the same results, both when the classical result of standard perturbation theory is recovered, and when stochastic effects are large. This can be easily confirmed by deriving recursive differential equations between the moments of , and computing numerically all related observables like the power spectrum and the scalar spectral index Pinol et al. (2018b).
General covariance.— The physical criterion that we use to shed new light on the conceptual issues of stochastic inflation is the one of its covariance under general coordinate transformations. The conundrum can be stated rather concisely: on one hand, Itô calculus does not verify the standard chain rule, and the corresponding Fokker-Planck equation does not respect general covariance under field redefinitions. On the other hand, Stratonovich calculus does respect general covariance, but at the expense of introducing spurious ‘frame’ dependences.222Itô calculus has nothing particular in this respect, as only Stratonovich calculus respects general covariance in the general class of -discretisations.
To present the issue in a simple but rather general form, let us consider the generalisation of (3) to non-linear sigma models, with Lagrangian , assuming for simplicity that all properly normalised fields are effectively light. If one uses the slow-roll classical equation of motion to determine the drift, such a generalisation reads
where are stochastic noises, and . A crucial point is that any derivation of this equation or generalisations thereof — be it from a heuristic approach at the level of the equations of motion, or from the more elaborate perspective of deriving a coarse-grained action within the Schwinger-Keldysh formalism Pinol et al. (2018a) — does not result in Langevin equations of the form (1), but only determines the correlation functions of the noises, i.e. determines the diffusion matrix , and not a set of ‘square-roots’ , which can differ by arbitrary field-dependent rotations . For instance, in a generalised slow-roll approximation, one obtains
but no prescription to write down as a weighted sum of independent white noises. The correlations (5) can be realised by writing with , for any set of vielbeins of the field space metric. This arbitrariness is innocuous in Itô calculus, as the precise expression of does not alter the Fokker-Planck equation (2), that depends only on the deterministic drift and on the diffusion matrix. It does matter in the Stratonovich interpretation however, as the field space derivatives of enter into the FP equation.
Now comes the issue of general covariance: physical quantities should not depend on arbitrary field redefinitions that one can perform at the level of the action. For non-linear sigma models studied here, the fields are indeed merely coordinates on the field space manifold of metric . To discuss the covariance of the FP equation, it is convenient to express it in terms of the rescaled PDF , which is a scalar under general field redefinitions , and in terms of which any sensible stochastic theory should be manifestly covariant. Using the above equations, one obtains that this required property is not satisfied in the Itô interpretation. This can be traced back to the general fact that in Itô calculus, the drift in the Langevin equation (1) does not transform as a vector under redefinitions , whereas the drift in Eq. (4) does classically transform as a vector under field space redefinitions (see Pinol et al. (2018b) for details). If one uses the Stratonovich interpretation however, one obtains the manifestly covariant result
where field space indices are raised with the inverse metric, and denotes the covariant derivative with respect to the field space metric. As announced, through their covariant derivatives in the second line, this comes however at the expense of a spurious dependence on the arbitrary choice of vielbeins, which can result in qualitatively very different results Pinol et al. (2018b). Note also that for a general metric, no privileged frame exists that would make this effect disappear. We remark of course that the potential or the field space curvature in general does introduce special directions in field space, however only in the mass matrix of fluctuations, and hence not under the current assumption that all properly normalised field fluctuations are light. Moreover, had we considered a more general situation in which the right-hand side of Eq. (5) is different, this would not have changed the fact that the sub-Hubble physics dictates only , and that the Stratonovich interpretation, necessary to maintain general covariance, forces us to introduce an artificial structure that does not drop out from physical observables without further modifications, be it in a curved or flat field space.
It is also interesting to use this general point of view to shed different light on our discussion of single-field slow-roll inflation. If one allows field redefinition in that case, at the somewhat artificial expense of not having a standard kinetic term, different ‘coordinates’ would yield different results in the Itô interpretation, because of its lack of general covariance. Such a difficulty disappears in the Stratonovich one however, as in one dimension, there is no ambiguity in defining the square-root up to an irrelevant sign.
First principle ‘derivation’ in phase space.— An attempt to derive the stochastic theory from first principles in a general setup enables us to spell out various assumptions behind such a description, which is important in order to delineate its regime of validity, and shows that the difficulties described above hold true in more complete descriptions. Leaving details to Pinol et al. (2018a), using the Schwinger-Keldysh formalism, one derives there a coarse-grained action for the long-wavelength part of fields in non-linear sigma models, taking into account the full phase space using the Hamiltonian language, as well as the coupling to gravity, unifying and generalising previous works Morikawa (1990); Moss and Rigopoulos (2017); Grain and Vennin (2017); Tokuda and Tanaka (2018). Assuming that an effective classical description emerges, this gives rises to the following set of equations:
where represent the conjugate momentum of (divided by ), , , , and where the two-point correlation functions of the stochastic noises can be computed in principle Pinol et al. (2018a). This system of equations is not of the type (1), for several reasons. The dynamics is strictly speaking non-Markovian Gautier and Serreau (2012), as the statistical properties of the noises do not depend only on the current location in phase space, but on the entire past history for each realisation. Indeed, the physics of the small scale fluctuations is dictated by the coarse-grained dynamics, which itself depends on all the stochastic kicks it has received. The stochastic noises are also in general non-Gaussian — a feature that would give rise not to a FP but to a more general Kramers-Moyal equation Risken and Haken (1989); Riotto and Sloth (2011) — and colored, owing to the smooth splitting between the long wavelength modes and the short ones that are integrated out Matarrese et al. (2004); Liguori et al. (2004); Winitzki and Vilenkin (2000). Even when the assumptions of a Markovian dynamics sourced by Gaussian white noises are adequate, the subtlety pointed out above is still present: one needs to introduce by hand ‘square-roots’ of the correlation matrix of the noises, and , which transform as field-space vectors and covectors under field redefinitions (see Pinol et al. (2018b) for details). They do not appear in the Itô FP equation, which however does not respect the criterion of general covariance, but explicitly appear in the covariant Stratonovich FP equation:
where the phase space PDF is a proper scalar under field redefinitions, is the number of fields, and is a covariant derivative in phase space. In addition to the arbitrariness that we point out in the diffusion term, note that the first line of (8) provides a neat reformulation of the deterministic evolution in non-linear sigma-models. It has a similar formal structure as a covariant Boltzmann equation in curved spacetime Straumann (2013), and can be useful beyond stochastic inflation, for instance to study
attractor behaviour and the sensitivity to initial conditions. In Ref. Pinol et al. (2018b), we also discuss the relationship between the full phase space description here and the overdamped limit in the previous section, and in particular how the latter can be approximately derived from the former in a suitable limit.
Stochastic diffusion in curved field space.— The various limitations of stochastic inflation that we have described should not diminish its successes, notably to reproduce non-trivial results from quantum field theory in curved spacetime Tsamis and Woodard (2005); Finelli et al. (2009); Garbrecht et al. (2014, 2015). In the current lack of a more fundamental understanding, in the following we take the pragmatic point of view that the dependence on the arbitrary choice of vielbeins in (6) is an artefact coming from the formulation of the problem in terms of SDE, which constitutes an intermediate step towards the final physical result, which is the computation of the PDF. Hence, we dismiss the last term in (6), which is technically equivalent to adding a suitable noise-induced term to the deterministic drift in (4). Considering test scalar fields in de Sitter space for simplicity in the following, we thus take
as our covariant equation that describes the corresponding diffusion of light scalar fields in a curved field space in the overdamped limit. Given that Eq. (9) is the simplest multifield covariantisation of the corresponding single-field equation, one can wonder whether field space curvature invariants can also enter into this equation Vilenkin (1999). To answer this question, one can use our knowledge of the relevant microphysics, i.e. of quantum fluctuations of fields in non-linear sigma models. The Riemann curvature of the field space does enter at quadratic order in the action, however only in the effective mass matrix of the fluctuations. While in general this can have important consequences like the geometrical destabilisation of inflation Renaux-Petel and Turzyński (2016); Renaux-Petel et al. (2017); Garcia-Saenz et al. (2018), we deduce that no curvature invariant should enter into the description of effectively light fields, and that Eq. (9) is thus sufficient for this purpose.
Let us now apply it to 2-field models with field space metric . For a generic potential, one can easily derive from (9) evolution equations for arbitrary correlation functions, including for :
For sum separable potentials, this system of equations has the interesting feature of involving only . When and the non-canonical function are polynomials in , it can thus be solved iteratively, starting from any initial distribution of , and independently of the one of . Let us concentrate on the interesting and simple case of an hyperbolic geometry, with , and a quartic potential . Formally Taylor-expanding the correlation functions as a function of the number of e-folds, , one obtains the recursive relations , from which all the correlation functions can be deduced to arbitrary order . With initial conditions for , one finds for instance (see Pinol et al. (2018b) for more results)
Compared to the single-field case, the field space geometry breaks the symmetry of , hence the appearance of non-zero correlations of odd powers of . One can also see that in addition to the series expansion in coming from the potential, the field space curvature comes with an expansion in .
This model, as a simple generalisation of the well studied single-field case, offers an interesting playground for studying stochastic effects in curved field space, and it would be interesting to confront these predictions to quantum field theory computations, which are comparably much more involved.
Like in the single-field case, at late time, truncating the series to any finite order is misleading, but Eqs. (9)–(10) enable one to understand the asymptotic behaviour of the system. For a generic and for sum-separable potentials, one can see from Eq. (10) that the statistics of can indeed be computed as in a single-field model with an effective potential , with equilibrium distribution . Of course, this stationary distribution need not exist in general, i.e. when it is not normalisable, for instance in the previous example with , which describes a simple Brownian motion around the linearly evolving shifted value . Eventually note that the consistency of our approach requires that its results should be such that all canonically normalised field fluctuations are light around the mean trajectory, which depends on the precise choice of potential and the initial distribution of both fields.
Conclusions.— The stochastic approach aims at describing the long-wavelength part of quantum fields, in inflationary or de Sitter spacetimes, by a classical stochastic theory. As any effective theory, it comes with its regime of validity and its limitations, several of which that have already been emphasised in the literature. Here we highlighted an important conceptual issue of the stochastic approach that has not been previously pointed out, although similar difficulties are well identified in statistical physics (see e.g. van Kampen (1986); Graham (1985); Aron et al. (2016); Cugliandolo and Lecomte (2017)): the Itô interpretation of the Langevin equations does not respect general covariance under field redefinitions, while the Stratonovich one does, but at the expense of introducing spurious ‘frame’ dependences. This feature holds in the overdamped description in terms of fields only (a description of the Einstein-Smoluchowski’s type in the statistical physics’ language), but also in a more complete phase space (Kramers) description. Despite these limitations, we used physical arguments to write down a manifestly covariant and physically motivated Fokker-Planck equation, with the aim of describing the quantum diffusion and the late-time behaviour of effectively light fields in a curved field space. We applied it to test scalar fields in de Sitter space and showed that for a certain class of two-field models with derivative interactions, some statistical properties can be derived using a correspondence with a single-field model endowed with an effective potential. We also studied a simple extension of the single-field theory to a hyperbolic field space geometry, making predictions that would be interesting to compare to first principles quantum field theory computations. Eventually, it would be interesting to study whether the ambiguities that we have described can be circumvented by using discretisation schemes that are more complex than simple -discretisations Cugliandolo et al. (2018), by using nonperturbative renormalisation group techniques (see e.g. Guilleux and Serreau (2015); Prokopec and Rigopoulos (2017)), or by using the tools of open quantum systems and directly working with the quantum density matrix (see e.g. Burgess et al. (2015, 2016); Collins et al. (2017)).
Acknowledgements. We are grateful to Camille Aron, Guillaume Faye, Vivien Lecomte, Martin Lemoine, Jérôme Martin, Cyril Pitrou, Gerasimos Rigopoulos, Julien Serreau, Alexei A. Starobinsky and Vincent Vennin for useful discussions. S.RP is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 758792, project GEODESI). Y. Tada was supported by grants from Région Île de France and Grand-in-Aid for JSPS Research Fellow (JP18J01992).
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