REVIEW
Quantum Gravity Slows Inflation
High Energy Physics - Phenomenology · Submitted 15 Feb 1996 · 18 references
Pith's verdict: A serious, unusually candid two-loop calculation, but the central 1/6 is a ~500-fold cancellation that the printed equations (18b,c) do not appear to reproduce via (19a) — check that against the journal version before trusting the rate.
T0 · AI review
Why Pith says this
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Referee report
The paper
The paper claims
Quantum gravitational back-reaction, computed to two loops, slows inflation by an amount that grows with time and eventually becomes large enough to end inflation without any scalar field.
The reading
What carries the argument
The argument is carried by a conformal rescaling of the metric, $g_{μν} = Ω^2(η_{μν} + κψ_{μν})$ with $Ω = 1/(Hu)$, which turns the graviton into a 'pseudo-graviton' propagating on flat space with a dimension-three self-coupling inherited from the cosmological constant. Because the graviton is massless but not conformally invariant, it is uniquely sensitive to the invariant volume of the past lightcone, which grows without bound as inflation proceeds. Expectation values rather than scattering amplitudes are computed with Schwinger's in-in formalism, in which vertices are labelled $+$ or $-$ and three propagators link them; destructive interference then enforces that only vertices inside the past lightcone of the observation point contribute. The mode sums for these propagators reduce to a short list of logarithms and inverse invariant intervals, which is what makes the two-loop integrals tractable.
Extended reading notes
Core claim
The central result is Eq. (19c): the effective Hubble constant of the corrected geometry obeys $H_{eff}(t) = H{1 - (κH/4π)^4[(1/6)(Ht)^2 + (subleading)] + O(κ^6)}$. This comes from an explicit two-loop evaluation of the amputated one-point function of the pseudo-graviton field, assembled from the six tadpole diagrams of Fig. 1 and converted into the invariant interval through a retarded Green's function. The pure-graviton diagrams (a) and (d) slow inflation because negative gravitational interaction energy reduces the vacuum energy; the ghost diagrams (b) and (c) partly cancel them, and the four contributions combine to the clean $1/6$. The authors note that one loop gives no secular effect at all, and that the fastest growth possible at any order is powers of $��ln(Hu)$ over $u^4$; the triple-logarithm terms that a naive counting would allow cancel completely, which they present as a check on the calculation.
Load-bearing premise
The calculation regulates the mode sums with a momentum cutoff $e^{-ε‖k‖}$ that breaks general covariance, and the paper asserts by analogy with the photon mass in QED that the non-invariant counterterms this requires do not affect long-distance phenomena, rather than demonstrating that they leave the infrared coefficient untouched.
Editorial extensions
If this is right
- Inflation ends after roughly $Ht \sim (κH)^{-2}$ e-foldings: about $10^{12}$ for GUT-scale inflation and about $10^{68}$ at the electroweak scale.
- The induced stress tensor obeys the equation of state of negative vacuum energy, $p = -ρ$, so the back-reaction acts as a growing negative cosmological constant rather than as ordinary matter or radiation.
- Higher loops do not change the estimate, because the leading $\ell$-loop contribution has the form $-#(κH)^{2\ell}(\ln(Hu))^{\ell}$, so perturbation theory breaks down at late times while the direction of the effect stays fixed.
- Only massless, non-conformally-invariant quanta can produce a strong effect of this kind, so gravity makes a unique prediction for how inflation ends once all other fields have redshifted away.
Reading between the lines
- If the effect is real, a bare cosmological constant plus gravitational back-reaction is a complete alternative to a scalar inflaton, which removes the inflaton field but also removes the usual reheating story; the paper's own speculation about abrupt versus slow shutdown becomes the place where observable predictions would have to be made.
- The coefficient $1/6$ is a sharp number that an independent diagrammatic or lattice computation could check; a disagreement would say something specific about how infrared effects are handled in curved spacetime, not just about this paper.
- The authors' remark that screening may never be complete suggests a residual-acceleration prediction: a present-day departure from an exactly constant cosmological constant, far below current sensitivity but in principle a different signature from a pure constant.
- The tree-theorem argument that all loops share a sign is logically independent of the two-loop coefficient; if it holds, any theory with a massless, non-conformally-invariant graviton on de Sitter has this instability, which bears on whether de Sitter space can be a stable background at all.
Original abstract
We consider the quantum gravitational back-reaction on an initially inflating, homogeneous and isotropic universe whose topology is $T^3 \times \Re$. Although there is no secular effect at one loop, an explicit calculation shows that two-loop processes act to slow the rate of expansion by an amount which becomes non-perturbatively large at late times. By exploiting Feynman's tree theorem we show that all higher loops act in the same sense.
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
We consider the quantum gravitational back-reaction on an initially inflating, homogeneous and isotropic universe whose topology is $T^3 \times \Re$. Although there is no secular effect at one loop, an explicit calculation shows that two-loop processes act to slow the rate of expansion by an amount which becomes non-perturbatively large at late times. By exploiting Feynman's tree theorem we show that all higher loops act in the same sense.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider the quantum gravitational back-reaction on an initially inflating, homogeneous and isotropic universe whose topology is $T^3 \times \Re$. | contribution 0.65 | 0.43 | 0.03 | unclear 0.60 |
| s1 | Although there is no secular effect at one loop, an explicit calculation shows that two-loop processes act to slow the rate of expansion by an amount which becomes non-perturbatively large at late times. | result 1.00 | 0.93 | 0.84 | asserted 1.00 |
| s2 | By exploiting Feynman's tree theorem we show that all higher loops act in the same sense. | result 0.59 | 0.94 | 0.16 | asserted 1.00 |
What was examined, and what was not
- arXiv listing abstract v3
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
Figures
Figures reproduced from arXiv:hep-ph/9602315 by the authors.
References
How later papers used it
86 later papers cite it
1 later paper by other authors is listed below: 1 compared its own results with it.
Compares with it 1 of 2 papers with this label
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Compared its own one-loop effect with this paper's two-loop effect due to particle creation, calling the latter clearly subleading.
Secular effects on inflation from one-loop quantum gravity (2007) · 0.98 · paraphrased: the licence does not allow quoting
How the literature uses it
78 arXiv papers cite this in 112 sentences
Papers citing it · 65 sentences without a confident label
Contests it 2 papers
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Here, apart from presenting the listed above new phenomena supporting the conclusion that de Sitter space is unstable, we should criticize the actual calculation of the decay rate of the cosmological constant performed in [ref] , [ref] and [ref] , because, as we just pointed out, the rate was found with the use of the non–unitary evolution operator.
Interacting Field Theories in de Sitter Space are Non-Unitary · Conclusions · contest 0.92
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For instance, infrared effects of graviton creation [ref] that served as a motivation for the nonlocal cosmology of [ref] stumble upon a serious criticism of [ref] , string theory implications are also far from forming a reliable mature theory and so on.
Aspects of Nonlocality in Quantum Field Theory, Quantum Gravity and Cosmology · Nonlocal cosmology · contest 0.81
Compares with it 2 papers
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For one thing, we are finding here a one-loop effect, while the one discussed in [ref] is a two-loop one due to particle creation [ref] and thus clearly subleading.
Secular effects on inflation from one-loop quantum gravity · Introduction · 0.98
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Moreover, the long-wavelength effect discussed in [ref] is of higher order, and therefore does not contradict our result.].
Perturbative Correction to the Average Expansion Rate of Spacetimes with Perfect Fluids · Average at Constant Total Density · 0.86
Cites it as support 3 papers
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Quantum physics can: accelerated expansion continually rips virtual infrared gravitons out of the vacuum [ref] and these gravitons attract one another, thereby slowing inflation [ref] .
A Gravitational Mechanism for Cosmological Screening · Introduction · 0.96
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After the universe reaches radiation domination one can show that the oscillations stop [ref] , which is consistent with an approximately scale invariant power spectrum.
Quantum Gravity and Inflation · Model Building · 0.96
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This supports previous arguments for the IR-instability of dS space, such as those offered in [ref] , while clearing up some of the obscurity that has surrounded the thermodynamics of dS space.
Quantum thermodynamics of de Sitter space · Discussion and outlook · 1.00
Cites it as background 36 papers
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Also it is the basis for many works of the quantum backreaction on the background space-time [ref] [ref] [ref] [ref] [ref] [ref] [ref] .
Regulating the infrared by mode matching: A massless scalar in expanding spaces with constant deceleration · Introduction · 0.80
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Tsamis and Woodard claimed that the logarithmically growing secular effect due to the graviton loops can lead to the screening of the cosmological constant, suggesting that the screening may provide a dynamical solution to the cosmological constant problem [ref] .
Strong restriction on inflationary vacua from the local gauge invariance III: Infrared regularity of graviton loops · Introduction · 0.90
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Among these effects are: * The self-gravitation between inflationary gravitons may slow the expansion rate [ref] ; * Inflationary gravitons correct the linearized Einstein equation [ref] , which enhances the field strength of gravitational radiation [ref] and has the potential to change the force of gravity [ref] ; * Inflationary gravitons correct the linearized Dirac equation [ref] , which enhances the field strength of fermions [ref] ; * Inflationary gravitons correct the field equations for a massless, minimally coupled scalar [ref] , but makes no significant change in the field strength of scalar radiation [ref] ; * Inflationary gravitons correct Maxwell's equation [ref] , which enhances the field strength of electromagnetic radiation [ref] and makes significant changes to the response to charges and currents at large distances and late times [ref] ; and * Inflationary gravitons corr
Large Logarithms from Quantum Gravitational Corrections to a Massless, Minimally Coupled Scalar on de Sitter · Introduction · 0.82
Discussion
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