Quantum scale of inflation and particle physics of the early universe

A.O.Barvinsky; A.Yu.Kamenshchik

REVIEW

Quantum scale of inflation and particle physics of the early universe

A.O.Barvinsky, A.Yu.Kamenshchik

General Relativity and Quantum Cosmology · Submitted 29 Apr 1994 · 28 references

Pith's verdict: A real derived constraint (A ≤ 1.3) sitting on a one-loop Z ≈ 10¹¹ whose higher-loop corrections are never estimated, with the headline numbers largely inherited from COBE normalization.

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arXiv IDgr-qc/9404062 (version 2)

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CitationA.O.Barvinsky, A.Yu.Kamenshchik. "Quantum scale of inflation and particle physics of the early universe." arXiv:gr-qc/9404062 (1994). https://pith.science/paper/SVZLLAJH

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Editor's note

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Worth your time if you care where the inflationary scale comes from, but read the numbers as a consistency check, not a prediction. What is new here is not the suppression mechanism — that is their own earlier work [8–11] — but its application to the Bardeen–Bond–Salopek nonminimal inflaton with large negative ξ. Extremizing (3.4) gives a clean peak at φ_I² = 2|I₁|/(Z+2), and the algebra from (4.2) through (4.3) is internally consistent: N = 8π²/A, so N ≥ 60 becomes A ≤ 1.3. That is a genuine, checkable constraint on the particle content, and it is useful even to someone who does not buy the whole framework. Now the soft spots. First, the headline numbers are mostly inputs dressed as outputs. H ≈ 10⁻⁵ m_P follows from λ/ξ² ≈ 10⁻¹⁰, which §4 takes from COBE; N ≈ 60 follows from picking A ≈ 0.76 inside the allowed window. So the 'good correspondence with observation' of [abstract:s2] is a consistency check. What is actually derived is φ_I ≈ 0.03 m_P, and the fact that it lands sub-Planckian and inside the slow-roll domain is not nothing. Second, and this is where I would press hardest: Z ≈ 10¹¹ is a one-loop number generated by ξ²/λ ≈ 10¹⁰, not by a small coupling, and the paper gives no estimate of Z₂/Z₁. The §5 argument that large |ξ| might improve the loop expansion is about the effective gravitational constant — the graviton sector — not the matter loops that build Z. The claim that the result is 'robust against inclusion of multi-loop corrections' is stated as a hope, and if Z₂ is not suppressed then φ_I, σ, and the A bound all move. Third, minor but real: eq. (3.2), on which everything else rests, is cited to [20], listed as in preparation. I would want that pinned down before leaning on it. The supersymmetry conclusion is presented as a suggestion and reads like one; A ≤ 1.3 is a wide window, and any model with a near-cancellation between boson and fermion contributions fits. Bottom line: a real mechanism, a real derived constraint, and honest about its own asymptotic limitations. Send it to referees.

Referee report

Major objections (4)

[§4, Eq. (4.3); Abstract] The abstract and §4 present H(φ_I) ≈ 10^-5 m_P as a parameter of the peak 'found' by the calculation, but Eq. (4.3) gives H(φ_I) = √(λ/(12|ξ|)) φ_I, and with φ_I = m_P√(8π(1+δ)/(|ξ|A)) this becomes H = m_P√(2πλ(1+δ)/(3ξ²A)). Since §4 fixes λ/ξ² ≈ 10^-10 from COBE, H ≈ 10^-5 m_P is a re-expression of the COBE normalization, not an independent output of the peak condition. The genuinely derived quantities are φ_I (given ξ and A) and N = 8π²/A. The paper should say explicitly that H is a consistency check, and soften the abstract's 'the quantum scale of inflation is calculated' accordingly.
[§3, Eqs. (3.3)-(3.4); §5] The peak location φ_I² = 2|I_1|/(Z+2) and the bound A ≤ 1.3 both depend on Z ≈ 10^11, which is computed only at one loop, Z = 6ξ²A/λ + O(ξ) in Eq. (4.2). Z is large not because a coupling is large but because of the ratio ξ²/λ ≈ 10^10, so the usual loop-counting suppression is not automatic. §5 offers only 'a hope' that the result is robust against multi-loop corrections. The paper should either estimate the leading two-loop contribution to Z at φ_I or state explicitly that the peak parameters are conditional on one-loop dominance. The separate concern about the 1/φ² truncation of I(φ) is less serious: the neglected I_2/φ⁴ term is smaller than I_1/φ² by m_P²/(16π|ξ|φ_I²) ≈ 10^-3 at the peak, so the 'strictly valid only as φ→∞' caveat in §3 is not numerically damaging.
[§3, Eq. (3.1); §4, Eq. (4.2)] Eq. (3.1) says Γ_loop accumulates the quantum effects of all physical fields g = (φ, f), but the sum in Eq. (4.2) runs only over the matter fields χ, A, ψ, and the text states that 'gravitons and inflaton field do not contribute to A.' Please state explicitly whether the inflaton fluctuation is included in Z. If it is, its mass term m²_φ = 3λφ² gives 9λ²φ⁴/(12H⁴) = 108ξ², which corresponds to A_inflaton = 18λ ≈ 0.9 for λ = 0.05, comparable to the bound A ≤ 1.3. That would tighten the allowed range for the matter sector and shift the numbers in Eq. (4.3).
[§4, Eq. (4.4); §5; Abstract] The bound 0 < A ≤ 1.3 is an O(1) constraint, not a fine-tuning. The lower bound is not stated in the paper but follows from requiring φ_I < m_P, which gives A > 8π/|ξ| ≈ 1.3×10^-3 for |ξ| = 2×10^4; the allowed window therefore spans about three orders of magnitude. The claim in §5 that 'it is only supersymmetry that can constrain the values ... so as to provide a subtle balance' and the abstract's 'likely to be enforced by the (quasi) supersymmetric nature' are stronger than this calculation supports. The authors should either quantify the cancellation required (e.g., how close the boson and fermion sums must be) or soften the inference.

Minor objections (7)

[§4, Eq. (4.3); Abstract] N = 8π²/A, so the abstract's N ≈ 60 corresponds to A ≈ 1.3, the upper edge of the allowed range. For A ≈ 1, N ≈ 80, and for smaller A, N is larger. State N as a bound N ≥ 60 rather than as a value.
[§4] For the quoted parameters, σ/φ_I = √I_1/((Z+2)φ_I) ≈ 3×10^-6, not 10^-5. The claimed correspondence with the COBE amplitude 10^-5 is therefore a factor-of-three statement; please check the arithmetic.
[§3, Eq. (3.2)] The derivation of Eq. (3.2) is attributed to ref. [20], listed as 'in preparation.' Please point to the published [9] (Phys. Reports 230, 1993) or make the derivation self-contained.
[§5] The dismissal of the no-boundary state (eternal inflation) as 'physically unacceptable' is a preference, not a derivation; please state the observational or theoretical basis for preferring the tunnelling state.
[§2, Eq. (2.1)] The sign conventions for ρ_NB and ρ_T should be stated explicitly (I = -3m_P⁴/(8U) < 0), since the flatness at large φ depends on it.
[§5] The 'third quantization' paragraph is speculative and could be shortened or moved to a note.
[References and text] There are several typographical errors: 'reckognized' → 'recognized' (twice), 'intermidiate' → 'intermediate', 'complimentary' → 'complementary', 'exeedingly' → 'exceedingly', 'LAP' → 'LEP' in ref. [26], 'B¨ ohm' → 'Böhm', 'Eath' → 'D'Eath', 'M arciano' → 'Marciano', 'Uni versity' → 'University', 'Bol shaya' → 'Bolshaya'. The boldface H versus italic H notation is also confusing and should be defined clearly at first use.

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The paper

The paper claims

The scale at which inflation ran is not a free parameter in this account: a one-loop quantum-gravity correction turns the flat, unnormalizable distribution of inflationary universes into a sharp peak that lands on the observed inflationarye

The reading

Inflation theory normally takes the starting value of the inflaton field, and hence the energy scale of inflation, as a free input. Quantum cosmology is supposed to supply it, since the universe's quantum state gives a probability distribution over inflationary histories, but at tree level that distribution is flat at large field values and cannot be normalized, which undermines the semiclassical reasoning inflation rests on. The authors show that adding the one-loop effective action of all matter fields, evaluated on the de Sitter instanton, multiplies the distribution by a power of the inflaton field that both normalizes it and produces a sharp peak. For a nonminimally coupled inflaton with large negative coupling $|xi|$ and the tunnelling wavefunction, that peak sits at a Hubble constant of about $10^{-5}m_P$, with relative width about $10^{-5}$ and about 60 e-foldings, provided one combination $A$ of the Higgs, gauge and Yukawa couplings lies between zero and 1.3. The authors argue that only a (quasi) supersymmetric matter sector would naturally deliver that balance.

What carries the argument

The carrier of the argument is the anomalous scaling $Z$: the integrated second DeWitt coefficient of all quantum fields on the four-sphere of radius $1/H$, appearing as the coefficient of $ln(H/mu)$ in the one-loop Euclidean effective action. Because $phi^2/H^2 = 12|xi|/lambda$ on the instanton, the quartic-in-mass contributions give $Z_{1- m loop} = 6xi^2 A/lambda + O(xi)$ with $A = rac{1}{2lambda}(sum_chi lambda_chi^2 + 6^{-1}sum_A g_A^4 - 6^{-1}sum_psi f_psi^4)$. This single number does three jobs at once: it normalizes the distribution, it sets the peak position and width through $phi_I^2 = 2|I_1|/(Z+2)$, and it fixes the number of e-foldings as $8pi^2/A$, independent of $xi$ and $lambda$.

Extended reading notes

Core claim

The central claim is that the quantum gravitational scale of inflation is calculable, and that it comes out in the right range. For a scalar field with a large negative constant of nonminimal coupling to curvature, the tree-level no-boundary and tunnelling distributions $rho sim e^{mp I(phi)}$ are flat in the slow-roll domain and diverge as $phi o infty$. Replacing the squared wavefunction with the diagonal of the density matrix after tracing out all other fields, and including the one-loop effective action $Gamma_{ m loop}simeq Zln(H/mu)$ on the de Sitter instanton, changes the distribution to $rho simeq exp[pm 3m_P^4/8U],phi^{-Z-2}$. For $Z>-1$ this is normalizable and has a sharp maximum at $phi_I$ with $phi_I^2 = 2|I_1|/(Z+2)$ and dispersion $sigma^2 = |I_1|(Z+2)^2$. Choosing the tunnelling state, which requires $delta>-1$ and so a finite inflationary stage, and taking $Z$ dominated by its one-loop value $Z simeq 6xi^2 A/lambda$, the peak parameters become $phi_I = m_Psqrt{8pi(1+delta)/(|xi|A)}$, $H(phi_I)=sqrt{|xi|},sigma$, and $N(phi_I)=8pi^2/A$. With $|xi|sim 2imes10^4$, $lambdasimeq 0.05$ and $Asimeq 1$ this gives $phi_Isimeq 0.03,m_P$, $sigmasimeq 10^{-7}m_P$, $Hsimeq 10^{-5}m_P$ and $Nsimeq 60$, the standard inflationary range, on the single condition $0<A le 1.3$.

Load-bearing premise

The one-loop form $Gamma_{ m loop}simeq Zln(H/mu)$, derived in the limit of large inflaton field, is assumed to still govern the distribution at the finite peak value $phi_Isimeq 0.03,m_P$ where $Zsim 10^{11}$; the paper itself says this equation is strictly valid only as $phi o infty$ and is used at intermediate scales as a qualitatively good approximation.

Editorial extensions

If this is right

If the argument holds, the initial value of the inflaton is not a free parameter but is fixed by the particle content of the theory through the single combination $A$ of quartic Higgs, gauge and Yukawa couplings.
The tunnelling wavefunction is favoured over the no-boundary one: only the tunnelling peak yields a finite inflationary stage with about 60 e-foldings, while the no-boundary peak corresponds to eternal inflation on the negative slope of the potential.
The relative width of the peak, about $10^{-5}$, numerically matches the observed level of density perturbations, though the paper notes that stochastic noise of the same order is superimposed on it during inflation.
The requirement $0<A le 1.3$ constrains the matter sector, and the authors argue a quasi-supersymmetric spectrum is what would satisfy it, tying early-universe cosmology to the unification of the gauge couplings.
The same loop mechanism that normalizes the distribution suppresses over-Planckian energy scales, which is what makes the semiclassical expansion self-consistent in the first place.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The bound $A le 1.3$ is a narrow numerical window on a boson-minus-fermion combination of quartic couplings, so the peak can be read as a cosmological handle on the superpartner spectrum rather than as a generic consistency condition.
Because $N = 8pi^2/A$ is independent of $xi$ and $lambda$, the duration of inflation is fixed entirely by the matter sector; a future measurement pinning $N$ more tightly than 60 would translate directly into a bound on $A$.
The paper's own gap, the low-$phi$ region where the distribution blows up and slow roll fails, is where a third-quantized or baby-universe treatment would have to take over; the near-zero-probability desert between that region and the peak is what keeps the two domains separable.
If the peak width is genuinely of order $10^{-5}$ in $H$, the quantum spread of the Hubble constant at nucleation is comparable to the observed amplitude of scalar perturbations, which hints that the observed spectrum may carry an imprint of the nucleation distribution and not only of the inflationary dynamics.

Original abstract

The quantum gravitational scale of inflation is calculated by finding a sharp probability peak in the distribution function of chaotic inflationary cosmologies driven by a scalar field with large negative constant $\xi$ of nonminimal interaction. In the case of the no-boundary state of the universe this peak corresponds to the eternal inflation, while for the tunnelling quantum state it generates a standard inflationary scenario. The sub-Planckian parameters of this peak (the mean value of the corresponding Hubble constant ${\mbox{\boldmath $H$}}\simeq 10^{-5}m_P$, its quantum width $\Delta{\mbox{\boldmath $H$}}/{\mbox{\boldmath $H$}}\simeq 10^{-5}$ and the number of inflationary e-foldings ${\mbox{\boldmath $N$}}\simeq 60$) are found to be in good correspondence with the observational status of inflation theory, provided the coupling constants of the theory are constrained by a condition which is likely to be enforced by the (quasi) supersymmetric nature of the sub-Planckian particle physics model.

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arXiv listing abstract v2. Results and the main claim are underlined; show every role.

The quantum gravitational scale of inflation is calculated by finding a sharp probability peak in the distribution function of chaotic inflationary cosmologies driven by a scalar field with large negative constant $\xi$ of nonminimal interaction. In the case of the no-boundary state of the universe this peak corresponds to the eternal inflation, while for the tunnelling quantum state it generates a standard inflationary scenario. The sub-Planckian parameters of this peak (the mean value of the corresponding Hubble constant ${\mbox{\boldmath $H$}}\simeq 10^{-5}m_P$, its quantum width $\Delta{\mbox{\boldmath $H$}}/{\mbox{\boldmath $H$}}\simeq 10^{-5}$ and the number of inflationary e-foldings ${\mbox{\boldmath $N$}}\simeq 60$) are found to be in good correspondence with the observational status of inflation theory, provided the coupling constants of the theory are constrained by a condition which is likely to be enforced by the (quasi) supersymmetric nature of the sub-Planckian particle physics model.

background contribution method result interpretation main claim

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#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The quantum gravitational scale of inflation is calculated by finding a sharp probability peak in the distribution function of chaotic inflationary cosmologies driven by a scalar field with large negative constant $\xi$ of nonminimal interaction.	method 0.50	0.87	0.38	asserted 1.00
s1	In the case of the no-boundary state of the universe this peak corresponds to the eternal inflation, while for the tunnelling quantum state it generates a standard inflationary scenario.	interpretation 0.72	0.58	0.11	asserted 1.00
s2	The sub-Planckian parameters of this peak (the mean value of the corresponding Hubble constant ${\mbox{\boldmath $H$}}\simeq 10^{-5}m_P$, its quantum width $\Delta{\mbox{\boldmath $H$}}/{\mbox{\boldmath $H$}}\simeq 10^{-5}$ and the number of inflationary e-foldings ${\mbox{\boldmath $N$}}\simeq 60$) are found to be in good correspondence with the observational status of inflation theory, provided the coupling constants of the theory are constrained by a condition which is likely to be enforced by the (quasi) supersymmetric nature of the sub-Planckian particle physics model.	result 0.92	0.93	0.66	asserted 0.76

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.96.

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sources read

arXiv listing abstract v2

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full text
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levels run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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States the same result 2 papers

The sub-Planckian parameters of this peak (the mean value of the corresponding Hubble constant ${\mbox{\boldmath $H$}}\simeq 10^{-5}m_P$, its quantum width $\Delta{\mbox{\boldmath $H$}}/{\mbox{\boldmath $H$}}\simeq 10^{-5}$ and the number of inflationary e-foldings ${\mbox{\boldmath $N$}}\simeq 60$) are found to be in good correspondence with the observational status of inflation theory, provided the coupling constants of the theory are constrained by a condition which is likely to be enforced by the (quasi) supersymmetric nature of the sub-Planckian particle physics model.

Quantum Origin of the Early Universe and the Energy Scale of Inflation · 0.98
The sub-Planckian parameters of this peak (the mean value of the corresponding Hubble constant $H\simeq 10^{-5}m_P$, its quantum width $\Delta H/H\simeq 10^{-5}$ and the number of inflationary e-foldings $N\geq 60$) are found to be in good correspondence with the observational status of inflation theory, provided the coupling constants of the theory are constrained by a condition which is likely to be enforced by the (quasi) supersymmetric nature of the sub-Planckian particle physics model.

Quantum origin of the early inflationary Universe · 0.98

Papers citing it · 18 sentences without a confident label

Uses its method or tool 1 paper

In our investigations we apply the dynamical systems methods in exploring stationary states represented by critical points in the phase space as well as their stability [ref] .

Twister quintessence scenario · Introduction · 0.93

Cites it as support 1 paper

It has been studied that the inflationary models that are in conflict with observation in the standard cold inflationary framework, can nonetheless satisfy the observational constraints in the modified gravity framework [ref] .

Inflationary Dynamics of Mutated Hilltop Inflation in Einstein-Gauss-Bonnet Gravity Under New Slow-Roll Approximations with Generalised Reheating · Introduction · 0.98

Cites it as background 19 papers

In this respect, recently the possibility of producing the inflation stage by the Higgs boson with a non-minimal coupling to the gravitation has been studied [ref] .

Decoupling of Higgs boson from the inflationary stage of Universe evolution · Introduction · 0.96
This model was proposed in 1984 in Ref. [ref] and developed in Refs. [ref] .

More accurate slow-roll approximations for inflation in scalar-tensor theories · Introduction · 0.98
Inflationary models with the Ricci scalar multiplied by a function of the scalar field are being intensively studied in cosmology [ref] .

Cosmological attractor inflation from the RG-improved Higgs sector of finite gauge theory · Introduction · 0.83

Discussion

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