REVIEW
Stochastic Quantum Gravitational Inflation
General Relativity and Quantum Cosmology · Submitted 23 May 2005 · 35 references
Pith's verdict: Section 5 is a real methods contribution with a genuine two-loop check; Section 6 promises more than its toy model delivers, and the abstract should be softened accordingly.
T0 · AI review
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Referee report
The paper
The paper claims
Starobinskii's stochastic-inflation rules can be extended to theories with derivative interactions and constrained fields, the two features that separate gravity from the scalar models where the method was built.
The reading
What carries the argument
The reduction rules themselves, stated as (I) keep only the term with the smallest number of derivatives at each order in the field and (II) subtract a stochastic source from each time derivative of a linear term. Rule I is what makes the Hubble friction term $3H dot{phi}$ dominate the d'Alembertian during inflation, since the scale factor varies far faster than the field; rule II is what turns the resulting first-order equation into Starobinskii's Langevin equation $dot{phi} = f_phi - V'(phi)/3H$ with Gaussian white noise $langle f_phi(t) f_phi(t') rangle = (H^3/4pi^2) delta(t-t')$. The supporting device is the infrared-truncated Yang-Feldman equation, which shows where the rules come from and why the stochastic field commutes with its own time derivative — the free mode function is replaced by its constant long-wavelength limit $H/sqrt{2k^3}$, so the field becomes a commuting random variable rather than a quantum operator. The Fokker-Planck equation for the probability density follows, and in the quartic case its late-time stationary solution $varrho_infty(phi) propto exp[-(pi^2 lambda/9 H^4) phi^4]$ is the Starobinskii-Yokoyama non-perturbative result.
Extended reading notes
Core claim
The paper's claim is that two rules, applied to a field equation, recover the leading infrared logarithms of inflation: at each order in the field keep only the term with the fewest derivatives, and for each linear term subtract a stochastic source from every time derivative. The authors do not assert these rules abstractly; they derive them by infrared-truncating the Yang-Feldman equation, keeping only the constant term in the long-wavelength expansion of the mode functions and the leading term in the retarded Green's function, which reduces to $a'^3 G^{ret}_{IR} = (1/3H) theta(t-t') delta^3(x-x')$. They then check the rules in two models. In a theory with the derivative coupling $lambda sqrt{-g} g^{mu nu} A^2 partial_mu B partial_nu B$, the rules give $dot{A}_{IR} = f_A$ and $dot{B}_{IR} = f_B - (lambda/2) A_{IR}^2 dot{B}_{IR}$, whose exact operator solution has $B_{IR}$ attenuated by $1 + (lambda/2) A_{IR}^2$; expanding it reproduces the explicitly computed two-loop result $langle B^2 rangle = (H/2pi)^2 ln a - lambda H^4/(2^6 pi^4) ln^2 a$. In a model with a massless minimally coupled scalar $A$, a conformally coupled scalar $C$, and the coupling $-(1/2) kappa H^2 sqrt{-g} A^2 C$, the rules give $dot{A}_{IR} = f_A - (kappa H/3) A_{IR} C_{IR}$ with $C_{IR} = -(kappa/4) A_{IR}^2$, so that $A_{IR}$ obeys the same equation as a self-interacting scalar with an effective negative quartic coupling — a runaway. The paper closes with an argument that needs only the existence of a stochastic formulation: if back-reaction were negligible, the stochastic volume element $langle sqrt{-g} rangle = a^3 [1 - 4 pi G H^2 ln a]$ would shrink to zero at late times, so back-reaction cannot be ignored.
Load-bearing premise
The two rules are inferred from perturbative computations in scalar models and then extended to gravity by analogy; the authors state plainly that the rules should be re-derived model by model and that the gravitational case, with its derivative interactions and non-linearly related constrained metric components, is never worked out.
Editorial extensions
If this is right
- A mechanical recipe now exists for writing the stochastic Langevin equation of a theory whose interactions contain derivatives: differentiate the free field, keep the horizon-crossing delta-function term as the noise, and drop double derivatives of fields in favour of derivatives of the scale factor.
- In the derivative-coupling model, the field $B$ undergoes a spacetime-dependent field-strength renormalization, each set of modes being attenuated by $1 + (lambda/2) A_{IR}^2$ at horizon crossing, so its stochastic corrections die away rather than grow.
- In the constrained-field analogue, the conformally coupled field is slaved to the dynamical one, $C_{IR} = -(kappa/4) A_{IR}^2$, and the resulting effective quartic coupling is negative, which the authors expect to produce a runaway instability rather than a stable late-time distribution.
- The paper makes two concrete two-loop predictions that can be checked against explicit quantum field theory: $langle A_{IR}^2 rangle = (H^2/4pi^2) ln a + kappa^2 H^4/(2^5 3 pi^4) ln^3 a$ and $langle C_{IR}^2 rangle = 3 kappa^2 H^4/(2^8 pi^4) ln^2 a$.
- If a stochastic formulation of inflationary quantum gravity exists at all, then the stochastic volume element falls as $1 - 4pi G H^2 ln a$, so gravitational back-reaction must eventually become significant regardless of the details.
Reading between the lines
- The rules are validated only where the authors could also compute the answer by conventional perturbation theory, so the test is consistency rather than prediction; the genuinely new content is the transfer to gravity, which is asserted and not performed.
- The step $C_{IR} = -(kappa/4) A_{IR}^2$ relies on neglecting the variation of $A_{IR}^2$ against the exponentially growing scale factor inside a time integral; in gravity the constrained metric components are related to the dynamical ones non-locally as well as non-linearly, so this particular simplification has no obvious gravitational counterpart.
- The closing volume-element argument is a one-loop stochastic estimate, and it is worth noting that it establishes only that back-reaction is non-zero, not that it destroys de Sitter; the authors themselves say non-linear corrections might instead shift the expansion rate to a new constant.
- A natural next check is the two-loop computation the paper predicts for the constrained model, since agreement there would test the slaving approximation rather than merely the derivative-counting rules.
Original abstract
During inflation explicit perturbative computations of quantum field theories which contain massless, non-conformal fields exhibit secular effects that grow as powers of the logarithm of the inflationary scale factor. Starobinskii's technique of stochastic inflation not only reproduces the leading infrared logarithms at each order in perturbation theory, it can sometimes be summed to reveal what happens when inflation has proceeded so long that the large logarithms overwhelm even very small coupling constants. It is thus a cosmological analogue of what the renormalization group does for the ultraviolet logarithms of quantum field theory, and generalizing this technique to quantum gravity is a problem of great importance. There are two significant differences between gravity and the scalar models for which stochastic formulations have so far been given: derivative interactions and the presence of constrained fields. We use explicit perturbative computations in two simple scalar models to infer a set of rules for stochastically formulating theories with these features.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
During inflation explicit perturbative computations of quantum field theories which contain massless, non-conformal fields exhibit secular effects that grow as powers of the logarithm of the inflationary scale factor. Starobinskii's technique of stochastic inflation not only reproduces the leading infrared logarithms at each order in perturbation theory, it can sometimes be summed to reveal what happens when inflation has proceeded so long that the large logarithms overwhelm even very small coupling constants. It is thus a cosmological analogue of what the renormalization group does for the ultraviolet logarithms of quantum field theory, and generalizing this technique to quantum gravity is a problem of great importance. There are two significant differences between gravity and the scalar models for which stochastic formulations have so far been given: derivative interactions and the presence of constrained fields. We use explicit perturbative computations in two simple scalar models to infer a set of rules for stochastically formulating theories with these features.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | During inflation explicit perturbative computations of quantum field theories which contain massless, non-conformal fields exhibit secular effects that grow as powers of the logarithm of the inflationary scale factor. | background 1.00 | 0.08 | 0.04 | asserted 1.00 |
| s1 | Starobinskii's technique of stochastic inflation not only reproduces the leading infrared logarithms at each order in perturbation theory, it can sometimes be summed to reveal what happens when inflation has proceeded so long that the large logarithms overwhelm even very small coupling constants. | background 0.97 | 0.09 | 0.16 | asserted 0.72 |
| s2 | It is thus a cosmological analogue of what the renormalization group does for the ultraviolet logarithms of quantum field theory, and generalizing this technique to quantum gravity is a problem of great importance. | background 0.78 | 0.07 | 0.08 | asserted 1.00 |
| s3 | There are two significant differences between gravity and the scalar models for which stochastic formulations have so far been given: derivative interactions and the presence of constrained fields. | gap 0.84 | 0.07 | 0.05 | asserted 1.00 |
| s4 | We use explicit perturbative computations in two simple scalar models to infer a set of rules for stochastically formulating theories with these features. | contribution 0.56 | 0.94 | 0.23 | asserted 1.00 |
What was examined, and what was not
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References
How the literature uses it
102 arXiv papers cite this in 173 sentences
Papers whose main result relates to this one's
States the same result 1 paper
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We use explicit perturbative computations in two simple scalar models to infer a set of rules for stochastically formulating theories with these features.
A Leading Logarithm Approximation for Inflationary Quantum Field Theory · 1.00
Papers citing it · 117 sentences without a confident label
Contests it 2 papers
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Although Starobinsky's stochastic formalism [ref] solves the analogous problem for the large logarithms of scalar potential models [ref] , the derivative interactions of quantum gravity cause it to fail [ref] .
Unfinished Business in A Nonlinear Sigma Model on de Sitter Background · Introduction · contest 0.88
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This technique captures large logarithms that arise from the “tail” part of the propagator [ref] , but it fails for large logarithms which have an ultraviolet origin, as can sometimes happen with the derivative interactions of quantum gravity [ref] .
How Inflationary Gravitons Affect Gravitational Radiation · Epilogue · contest 0.74
Builds on it 2 papers
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We believe it may be possible to derive a non-perturbative resummation technique by extending the stochastic method which Starobinsky devised for the same purpose in scalar potential models [ref] .
Primordial Gravitational Waves Enhancement · Introduction · 0.95
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Here, we broadened this expression up to the next $λ^3$ order through the same reasoning as in [ref] ; see details in Appendix [ref].
IR finite correlation functions in de Sitter space, a smooth massless limit, and an autonomous equation · Starobinsky’s stochastic approach and its recurrent formula · 0.94
Uses its method or tool 10 papers
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This property of SdSET is essential for deriving the results in this paper.] (SdSET) [ref] to calculate corrections to Stochastic Inflation [ref] (see also [ref] ), which allows us to demonstrate that the onset of eternal inflation is incalculable when $Λ< f_π$.
A Tail of Eternal Inflation · Introduction · 0.99
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To evolve further requires a non-perturbative technique such as summing the series of leading infrared logarithms: [equation] Starobinskiĭ [ref] has developed a stochastic technique which exactly reproduces the leading infrared logarithms of scalar potential models [ref] , and which can be used to sum them whenever the scalar potential is bounded below [ref] .
A Phenomenological Model for the Early Universe · Introduction · 0.92
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There is a stochastic approach for investigating such a non-perturbative effect [ref] .
Non-linear sigma model in de Sitter space · Contribution from an interacting field : Non-perturbative effect · 0.94
Compares with it 3 papers
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Remarkably, by taking the time derivative of [ref], it precisely coincides [ref] with the local Langevin-like equation for the stochastic field in the original work [ref] .
From the Fokker-Planck equation to perturbative QFT's results in de Sitter space · Introduction · 0.80
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The leading contributions from infrared modes in slow-roll spacetimes (again for comoving observers) also vanish [ref] , but of course considering only long-wavelength modes can only account for a part of the complete result, which may or may not give the dominant contribution (the leading behaviour of scalar fields [ref] is correctly reproduced, but the treatment of gravitons is much more difficult [ref] ).
One-loop quantum gravitational backreaction on the local Hubble rate · Results · 0.85
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To expand [ref] up to the next $λ^3$ order, we obtained the same expressions in the spirit of [ref] in this part of the Appendix.
IR finite correlation functions in de Sitter space, a smooth massless limit, and an autonomous equation · Relation of Starobinsky's stochastic approach to perturbative QFT' series in the leading-logarithm approximation · 0.82
Cites it as support 2 papers
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The fact that (ultraviolet divergent) correlators of the full field $ϕ(x)$ agree, at leading logarithm order, with those of $φ(x)$ can be proved [ref] .
Resummations for Inflationary Quantum Gravity · Scalar Potential Models · 0.96
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eq:langevin faithfully reproduces the known resummed logarithmic divergences from QFT to leading order [ref] .
Multiple spectator condensates from inflation · Introduction · 0.96
Cites it as background 32 papers
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On the other hand, it has been suggested by many authors that the stochastic approach to inflation, originally pioneered by Starobinsky [ref] , functions as an infra-red regulator and leads to infra-red finite predictions [ref] .
A parton picture of de Sitter space during slow-roll inflation · Introduction · 0.80
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Early work focused on deriving a completely stochastic representation of the large logarithms induced by these models [ref] , and that approach has been extensively pursued by Kitamoto and Kitazawa [ref] .
Summing Inflationary Logarithms in Nonlinear Sigma Models · Introduction · 0.95
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The program to establish the correspondence between the stochastic approach and the results from perturbative quantum field theory in the leading logarithm approximation was launched by Woodard and collaborators [ref] ; see the recent one [ref] for the full reference list of further developments by that group and by many others [ref] .
From the Fokker-Planck equation to perturbative QFT's results in de Sitter space · Introduction · 0.95
Discussion
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