The role of cosmological constant in f(R, G) gravity

Abhik Kumar Sanyal; Chandramouli Sarkar

REVIEW 2 major objections 5 minor 27 references

The role of cosmological constant in f(R, G) gravity

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Canonical quantization of Einstein-Hilbert gravity with a Gauss-Bonnet-squared term is contradictory unless a cosmological constant is added, making vacuum energy essential in the early universe.

desk verdict The claimed necessity of a cosmological constant rests on a Hamilton–Jacobi differentiation artifact; the phase-space construction is solid but the main conclusion is unsupported. read the letter →

arxiv 1908.05680 v1 pith:6DQOESY5 submitted 2019-08-15 gr-qc hep-th

Abhik Kumar Sanyal , Chandramouli Sarkar This is my paper · ORCID

classification gr-qchep-th PACS 04.60.-m04.50.Kd98.80.Qc95.36.+x

keywords cosmologicalconstantGauss-Bonnetsquaredgravityf(RG)canonicalquantizationphase-spacestructuresemiclassicalapproximationHamilton-JacobiequationdeSitterinflation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper attempts to establish that the action $\int(\alpha R+\gamma\mathcal{G}^2)\sqrt{-g}\,d^4x$, with $\mathcal{G}$ the Gauss-Bonnet invariant, cannot be consistently quantized in a flat cosmological model without a cosmological constant. Constraint quantization produces a Schrödinger-like equation whose effective potential gives de Sitter inflation only if the coupling $\gamma$ is negative, opposite to the sign required by the classical and slow-roll solutions. The on-shell Hamilton-Jacobi function also fails to satisfy the Hamilton-Jacobi equation. Adding $\Lambda$ removes both failures and yields a semiclassical wavefunction peaked around the classical de Sitter trajectory; the authors read this as evidence that vacuum energy in the form of $\Lambda$ is required already in the very early universe.

What carries the argument

The machinery is the constrained Hamiltonian obtained by treating $z=a^2$ and $x=\dot z/N$ as the configuration variables (with $N$ the lapse), so that the Gauss-Bonnet-squared action becomes a constrained system whose quantization proceeds through a Schrödinger-like equation with internal time $\sigma=z^{11/2}$. Two equations carry the contradiction: the extremum condition $\partial V_e/\partial x=0$ on the effective potential $V_e$, which the paper requires to reproduce the classical de Sitter solution; and the zeroth-order Hamilton-Jacobi equation obtained by inserting $\Psi=\Psi_0 e^{iS/\hbar}$ into the quantized equation. These algebraic conditions convert the consistency of the quantum theory into sign conditions on the coupling $\gamma$ and into a demand that the on-shell action equal a solution of the Hamilton-Jacobi equation, and it is precisely these conditions that fail without $\Lambda$ and hold with it.

What would settle it

Solve the $\Lambda=0$ zeroth-order Hamilton-Jacobi equation (47a) for a real solution $S_0$ without first imposing the classical trajectory; if one exists with positive $\gamma$ and preserved probability conservation, the claimed semiclassical contradiction would not force the cosmological constant.

Watch

Extended reading notes

Core claim

The paper's central claim is that the phase-space and canonical quantization of the higher-order action are the place where the need for the cosmological constant shows up. Classically the vacuum de Sitter solution and the slow-roll inflationary solution both require $\gamma>0$. In the quantized theory, extremizing the effective potential with respect to the auxiliary velocity $x$ produces the same de Sitter form only when $\gamma<0$, and the Hamilton-Jacobi function $S_0$ computed from the classical solution does not solve the Hamilton-Jacobi equation. With $\Lambda$ added to the action, the extremum condition no longer flips the sign of $\gamma$, and $S_0$ satisfies the Hamilton-Jacobi equation under a relation that ties $\lambda$, $\alpha$, and $\gamma$, giving a semiclassical wavefunction sharply peaked on the classical inflationary trajectory. The authors conclude that geometry alone cannot be fundamental in the very early universe; at least the vacuum energy of other fields, $\Lambda$, must be present.

Load-bearing premise

The load-bearing assumption is that the special point of the quantum effective potential must reproduce the classical de Sitter expansion; if that match is not required, the sign flip of the coupling is not necessarily a contradiction.

Editorial extensions

If this is right

With $\Lambda$ in the action, the semiclassical wavefunction is strongly peaked about the classical inflationary solutions, so the quantized theory has a well-defined classical limit in the early universe.
Without $\Lambda$, probability conservation forces the operator-ordering index to $n=-5$ and holds only for flat spatial sections; with $\Lambda$ the same condition persists, restricting the quantum theory to the $k=0$ sector.
The absence of a power-law radiation-dominated solution is not cured by $\Lambda$, so the G² action alone cannot supply the full post-inflationary history; additional curvature scalars or matter are needed.
The classical vacuum de Sitter solution is recovered in the limit $\Lambda\to0$, so the consistent quantized theory reduces to the original action in that limit while retaining the same phase-space structure.
The consistency condition relating $\lambda$, $\alpha$, and $\gamma$ means the cosmological constant is not freely chosen at the quantum level but tied to the inflationary scale and the coupling constants.

Reading between the lines

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

If the same consistency test were applied to other higher-curvature invariants, the recurrence of a sign flip or a broken Hamilton-Jacobi equation would suggest that vacuum energy is a generic prerequisite for quantum cosmology in higher-order gravity; the paper does not run those cases.
Reading $\Lambda$ as the sum of zero-point energies of all fields turns the paper's consistency condition into a quantitative bridge to the measured vacuum energy, but the paper makes no numerical estimate.
A different operator-ordering prescription than the one used in the quantization step could change the $n=-5$ conclusion and therefore deserves a dedicated check before the necessity claim is adopted broadly.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know before reading further: the paper's central claim—that the Einstein–Hilbert action with a G^2 term requires a cosmological constant to make consistent quantum gravity—rests on a differentiation slip. When you keep x and z independent in the Hamilton–Jacobi function S0, Eq. (47a) is satisfied identically on the classical solution. The reported contradiction is an artifact of substituting x = 2λz before differentiating. The reader's stress-test note is right; I checked it.

What is genuinely useful here is the phase-space construction. The authors carefully apply Dirac's algorithm to the Einstein–Hilbert plus Gauss–Bonnet squared action, obtain a Hamiltonian in the z (scale-factor-squared) and x (velocity) variables, and verify that it reproduces the field equations. The observation that the probability interpretation forces k = 0 and n = −5 is a concrete, checkable result. The classical de Sitter and slow-roll solutions are also derived cleanly.

The soft spots, in order of severity:

1. The Hamilton–Jacobi “pathology” (Section 3.4) is the load-bearing one. S0 = ∫px dx + ∫pz dz is computed with the on-shell relation x = 2λz inserted too early. The correct partial derivatives with respect to independent x and z satisfy the HJ equation; the paper checks a total derivative along the trajectory instead. This removes the main mathematical support for Λ.

2. The sign-of-gamma pathology (Section 3.3) is also unconvincing. The authors require the effective potential's extremum to reproduce the classical de Sitter solution, but nothing in canonical quantum cosmology forces that. The sign flip is an assumption, not a derivation.

3. The radiation-era no-power-law claim is asserted without proof. It might be true, but it is not established.

4. The Λ resolution is a fix-up, not a prediction. Adding Λ precisely to remove the two pathologies and then concluding Λ is necessary is close to assuming the conclusion.

So: the paper is worth a serious referee because the phase-space technique is legitimate and the HJ subtlety is instructive, but in its current form the central conclusion is not supported. I would suggest rejecting with an invitation to resubmit after correcting the differentiation and either dropping or properly justifying the extremum assumption. A reader working on canonical quantization of higher-order gravity would get value from the construction, but should be warned against taking the Λ conclusion at face value.

Referee Report

2 major / 5 minor

Summary. The paper studies the action A=∫(αR+γG^2)√-g d^4x in a flat Robertson-Walker spacetime. It constructs a phase-space structure via Dirac constraint analysis, performs canonical quantization, and carries out a WKB/semiclassical approximation around the vacuum de-Sitter solution. The manuscript claims three pathologies: absence of a power-law solution in the radiation-dominated era, a sign flip of the coupling γ when the effective potential is extremized to obtain a de-Sitter solution, and failure of the Hamilton-Jacobi function S0 to satisfy the leading-order Hamilton-Jacobi equation. It then adds a cosmological constant and claims these pathologies are resolved, concluding that Λ is necessary in the very early universe.

Significance. If the central claim were correct, it would be a striking result: canonical quantization of Gauss-Bonnet-squared gravity would single out a cosmological constant as indispensable. The paper does contain a useful Dirac constraint analysis and a classical consistency check of the resulting Hamiltonian, and the equality of the on-shell action with the Hamilton-Jacobi function at zeroth order is a nice check. However, the two load-bearing pathologies behind the headline conclusion—the effective-potential sign flip and the Hamilton-Jacobi failure—rest on an unjustified assumption and on a differentiation error, respectively. The surviving content is a technical canonical-quantization study whose main interpretive conclusion is not supported.

major comments (2)

[§3.4, Eqs. (47a)–(49)] The claimed Hamilton-Jacobi contradiction is an artifact of differentiating an on-shell, z-only expression. Equation (49) is obtained after substituting the classical relation x=2λz into the integrals of Eq. (48), but Eq. (47a) is a partial differential equation in two independent variables. One must use ∂S0/∂x and ∂S0/∂z at fixed (x,z). For S0(x,z)=12γ(2λ)^(11/2)x^(3/2)−4αλz^(3/2)−(6912/7)γλ^7z^(3/2), the derivatives reproduce exactly the momenta px and pz of Eq. (48). Substituting these into the left-hand side of Eq. (47a), with x=2λz and α=96γλ^6, gives an identically vanishing result. The statement that Eq. (47a) is not satisfied is therefore false, and the 'third pathology' disappears once the partial derivatives are taken correctly.
[§3.3, Eq. (41)] The second pathology rests on the unstated assumption that the extremum of the effective potential V_e with respect to the auxiliary variable x must reproduce the classical de-Sitter solution. No physical principle in minisuperspace quantum cosmology requires the stationary point of V_e to coincide with the classical trajectory; x is a velocity-type variable introduced via ˙z=Nx, and the classical solution is a solution of the Hamiltonian constraint, not necessarily a stationary point of V_e. The paper itself later concedes in §4 that 'classical solution is not expected to match at the extremum of the potential.' Without this assumption, the sign of γ at ∂V_e/∂x=0 is not evidence of an inconsistency, and the conclusion that Λ is required to cure this sign flip is unsupported.

minor comments (5)

[§2.1] The assertion that the field equations admit no power-law solution in the radiation-dominated era is made without a supporting calculation. If this claim is retained, a demonstration should be provided; if not, it should be removed or downgraded.
[§3.2] The statement that the operator-ordering index n=−5, being different from n=−1 in R^2 gravity, constitutes 'a clear contradiction' is not justified. Different higher-order theories are not required to yield the same operator ordering, so this should be presented as an observation rather than a pathology.
[Eq. (48)] The integrals in Eq. (48) are written with integration constants omitted, and the two sides display different functional arguments (for instance, ∫px dx is written with z^{3/2} even though px is a function of x). The use of the on-shell relation x=2λz should be made explicit at this step.
[Throughout] There are numerous typographical errors and garbled equations (e.g., 'Ind ia', 's quared', and the typesetting around Eq. (41)). The extremization condition in Eq. (41) is particularly hard to verify as printed and should be corrected.
[Abstract/Conclusion] The overall argument is close to circular: Λ is introduced in §4 as a remedy for the pathologies and then concluded to be necessary. Since the pathologies are not established, the conclusion that Λ must be present in the very early universe does not follow.

Circularity Check

2 steps flagged · score 7.0 of 10

The claimed need for a cosmological constant is a self-imposed cure: Lambda is added ad hoc to remove pathologies, and one of those pathologies (the Hamilton-Jacobi failure) is an artifact of differentiating after substituting x=2lambda z.

self definitional [Section 4 and Section 5, Eqs. (51), (61), and Conclusion]
"We have therefore improvised the action under the addition of a cosmological constant term to observe that such pathologies are removed, leading to mathematical consistency of the theory. This proves the very importance of considering the presence of cosmological constant, which is essentially the sum of zero point energies of all quantum fields, available in the very early universe."

The modified action (51) is chosen by hand to include Lambda, and the consistency conditions in Section 4 (e.g., Eq. (61), Lambda = 3 lambda^2 + 4320 gamma lambda^8/alpha) are imposed precisely so that the effective-potential extremum no longer requires a sign flip of gamma and so that the Hamilton-Jacobi check can be declared consistent. The paper then treats this imposed consistency as evidence that Lambda is necessary. That is the conclusion restating the input: Lambda was added to cure the pathologies, so observing that the pathologies are cured proves only that the added term does what it was added to do. No independent observable or externally fixed value of Lambda is predicted, so the central claim reduces to the construction.
other [Section 3.4, Eqs. (47a)-(50)]
"Substituting gamma = alpha/(96 lambda^6) in view of relation (8) both in (49) and (50), one finally finds A0 = S0 = 12/7 alpha lambda z^(3/2), and the classical on-shell action matches exactly with the Hamilton-Jacobi function. Alas! Equation (47a) is not satisfied for the form of S0 so obtained."

The 'form of S0' in Eq. (49) is obtained from Eq. (48) after the classical relation x = 2 lambda z has been substituted into the momentum integrals, making S0 a function of z alone. Checking Eq. (47a) with this z-only S0 sets S0,x = 0, which is not the partial derivative required for a Hamilton-Jacobi function of independent configuration-space variables (x,z). If S0 is instead kept as S0(x,z) = 12 gamma (2 lambda)^(11/2) x^(3/2) - 4 alpha lambda z^(3/2) - (54/7) gamma (2 lambda)^7 z^(3/2), then S0,x = px and S0,z = pz, and substituting into Eq. (47a) together with x = 2 lambda z and alpha = 96 gamma lambda^6 gives zero identically.

full rationale

The phase-space construction and the classical checks in Section 3.1 are self-contained: the Hamiltonian is derived from the action and correctly reproduces the classical field equations, so that part is not circular. The circularity enters in the argument for Lambda. The effective-potential extremum is required, without independent justification, to reproduce the classical de-Sitter solution; when the coupling constant comes out negative this is called a contradiction, and Lambda is then chosen in Eq. (61) to make the sign positive. That is a fitted input presented as a necessity. More seriously, the claimed Hamilton-Jacobi failure is an artifact: S0 in Eq. (49) is written only in z after substituting x = 2 lambda z, so checking Eq. (47a) with S0,x = 0 is not the required partial-derivative check. If S0 is kept as a function of independent (x,z), Eq. (47a) is identically satisfied on the classical solution with alpha = 96 gamma lambda^6. The Lambda-dependent consistency condition is therefore a cure for a contradiction that does not exist. Finally, the conclusion explicitly says the action was 'improvised' by adding Lambda so that pathologies are removed, and then takes that removal as proof that Lambda is important. The central claim thus reduces to the input of the modified action. Score 7 reflects that the classical Hamiltonian work is independent, but the paper's principal conclusion is forced by construction and by an erroneous differentiation step.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the validity of the Dirac quantization of this higher-order action, on the assumption that effective-potential extremization should match classical solutions, and on the WKB framework. The addition of Lambda is a free adjustment. No new physical entities are introduced.

free parameters (4)

gamma (G^2 coupling) = positive for classical solutions; no numerical value
Coupling of the Gauss-Bonnet squared term; appears in the de-Sitter solution (8) and slow-roll solution (14); its sign is the subject of the claimed contradiction.
Lambda (cosmological constant) = determined by consistency conditions (56) and (61); not predicted
Introduced ad hoc in Section 4 to remove pathologies; expressed in terms of lambda, alpha, and gamma; no independent prediction is made.
lambda (de-Sitter expansion rate) = constrained by (65) in terms of alpha and gamma
Parameter of the exponential solution; used throughout the on-shell computations.
n (operator ordering index) = -5
Chosen to make the probability continuity hold for k=0; the difference from n=-1 is called a contradiction in Section 3.3.

assumptions (4)

standard math Dirac's constraint algorithm for constrained Hamiltonian systems is applicable to this higher-order action.
Used in Section 3.1 to construct the phase-space structure of the G^2 action.
domain assumption Canonical quantization of the minisuperspace model, with sigma as internal time, is a valid approach.
The paper quantizes the reduced Hamiltonian (24) to a Schrodinger-like equation with sigma = a^11 as time.
ad hoc to paper The extremum of the effective potential must yield the classical de-Sitter solution.
Section 3.3; this assumption underlies the claimed sign-of-gamma contradiction.
domain assumption The WKB semiclassical approximation with the on-shell action should satisfy the Hamilton-Jacobi equation (47a).
Section 3.4; the paper treats the failure of this condition as a pathology.

how reviews work

0 comments

Cite this review

Pith. "Pith review of The role of cosmological constant in f(R, G) gravity." pith.science (2026). https://pith.science/paper/6DQOESY5

@misc{pith2026190805680,
  author       = {Pith},
  title        = {Pith review of: The role of cosmological constant in f(R, G) gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6DQOESY5}},
  note         = {Machine review of arXiv:1908.05680}
}

read the original abstract

Einstein-Hilbert action is supplemented by Gauss-Bonnet squared term, its phase-space structure is constructed and canonical quantization is performed. Resolution of a contradiction that emerges in the process, requires the presence of other fields at least in the form of vacuum energy-density, commonly known as the cosmological constant. This reveals the very importance of the presence of other fields at least in the form of cosmological constant, in the very early universe.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 23 canonical work pages

[1]

Parker, Particle creation in expanding universes, Ph ys

L. Parker, Particle creation in expanding universes, Ph ys. Rev. Lett. 21, 562 (1968)

work page 1968
[2]

Parker, Quantized ﬁelds and particle creation in expa nding universes, I

L. Parker, Quantized ﬁelds and particle creation in expa nding universes, I. Phys. Rev. 183, 1057 (1969)

work page 1969
[3]

Parker, Quantized ﬁelds and particle creation in expa nding universes, II

L. Parker, Quantized ﬁelds and particle creation in expa nding universes, II. Phys. Rev. D 3, 346 (1971)

work page 1971
[4]

Papastamatiou and L

J. Papastamatiou and L. Parker, Asymmetric creation of m atter and antimatter in the expanding universe, Phys Rev D 19, 2283 (1979)

work page 1979
[5]

N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge: Cambridge University Press (1982)

work page 1982
[6]

S. A. Fulling, Aspects of Quantum Field Theory in Curved S pacetime, Cambridge: Cambridge University Press (1989). 11

work page 1989
[7]

V. F. Mukhanov and S. Winitzki, Introduction to Quantum F ields in Gravity, Cambridge: Cambridge University Press (2007)

work page 2007
[8]

A. K. Sanyal, S. Debnath, S. Ruz, Canonical formulation o f the curvature-squared action in the presence of a lapse function, Class. Quantum Grav. 29, 215007 (2012), arXiv:1108.5869 [gr-qc]

work page Pith review arXiv 2012

Show all 27 references

[9]

S. Ruz, R. Mandal, S. Debnath and A. K. Sanyal, Resolving t he issue of branched Hamiltonian in modiﬁed Lanczos-Lovelock gravity, Gen Relativ Gravit 48, 86 (2016)

2016
[10]

Debnath, S

S. Debnath, S. Ruz, R. Mandal and A. K. Sanyal, History of cosmic evolution with modiﬁed Gauss-Bonnet- dilatonic coupled term, Eur. Phys. J. C. 77, 318 (2017), arXi v:1608.04669v1 [gr-qc]

2017 arXiv
[11]

Mandal and A

R. Mandal and A. K. Sanyal, Equivalent and inequivalent canonical structures of higher order theories of gravity, Phys. Rev. D 96, 084025 (2017), arXiv:1709.05201v 3 [gr-qc]

2017 arXiv
[12]

Mandal, C

R. Mandal, C. Sarkar and A. K. Sanyal, Early universe wit h modiﬁed scalar-tensor theory of gravity, JHEP, 05, 078 (2018), arXiv:1801.04056v2 [hep-th]

2018 arXiv
[13]

A. K. Sanyal, Degenerate Hamiltonian operator in highe r-order canonical gravity - the problem and a remedy, arXiv:1807.02769 [gr-qc]

arXiv
[14]

C. G. Callan, E. J. Martinec, M. J. Perry and D. Friedan, S trings in Background Fields, Nucl. Phys. B 262 593 (1985)

1985
[15]

D. J. Gross, J.H. Sloan, The quartic eﬀective action for the heterotic string, Nucl. Phys. B 291, 41 (1987)

1987
[16]

R. R. Metsaev, A. A. Tseytlin, Two-loop β -function for the generalized bosonic sigma model, Phys. Le tt. B 191, 354 (1987)

1987
[17]

M. C. Bento, O. Bertolami, Cosmological Solutions of Hi gher-Curvature String Eﬀective Theories with Dilatons, Phys. Lett. B 368, 198 (1996), arXiv:gr-qc/95030 57

1996
[18]

Antoniadis, E

I. Antoniadis, E. Gava, K.S. Narain, Moduli Correction s to Gauge and Gravitational Couplings in four dimensional Superstrings, Nucl. Phys. B 383, 93 (1992), arX iv:hep-th/9204030

1992 arXiv
[19]

Antoniadis, J

I. Antoniadis, J. Rizos, K. Tamvakis, Singularity-fre e cosmological solutions of the superstring eﬀective action, Nucl. Phys. B 415, 497 (1994)

1994
[20]

Deser, String-Generated Gravity Mode ls, Phys

D.G Boulware, S. Deser, String-Generated Gravity Mode ls, Phys. Rev. Lett. 55, 2656 (1985)

1985
[21]

Nojiri and S

S. Nojiri and S. D. and Odintsov, Modiﬁed GaussBonnet th eory as gravitational alternative for dark energy, Phys. Lett. B 631, 1 (2005)

2005
[22]

A. V. Astashenok, S. Cappozziello and S. D, Odintsov, Ex treme neutron stars from Extended Theories of Gravity, JCAP: 2015 (2015) arXiv:1408.3856 [gr-qc]

2015 arXiv
[23]

Abbas, D

G. Abbas, D. Momeni, M. Aamir Ali, R. Myrzakulov and S. Qa isar, Anisotropic Compact Stars in f (G) Gravity, Astrophys.Space Sci. 357, 158 (2015), arXiv:1501 .00427 [gr-qc]

2015
[24]

Nojiri, S

S. Nojiri, S. D. Odintsov, and P. V. Tretyakov, From inﬂa tion to dark energy in the non-minimal modiﬁed gravity, Prog. Theor. Phys. Suppl. 172, 81 (2008)

2008
[25]

Bamba, S

K. Bamba, S. D. Odintsov, L. Sebastiani and S. Zerbini, F inite-time future singularities in modiﬁed GaussBonnet and f (R, G) gravity and singularity avoidance, Eur. Phys. J. C67:295 ( 2010) arXiv:0911.4390 [hep-th]

2010 arXiv
[26]

Capozziello, M

S. Capozziello, M. Francaviglia and A. N. Makar, Higher -Order Gauss-Bonnet Cosmology, Astrophys.Space Sci. 349 603 (2014), arXiv:1304.5440v3 [gr-qc]

2014 arXiv
[27]

De Laurentis, M

M. De Laurentis, M. Paolella, S. Capozziello, Cosmolog ical inﬂation in F(R, G ) gravity, Phys. Rev. D 91, 083531 (2015), arXiv:1503.04659 [gr-qc] 12

2015 arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Parker, Particle creation in expanding universes, Ph ys

L. Parker, Particle creation in expanding universes, Ph ys. Rev. Lett. 21, 562 (1968)

work page 1968

[2] [2]

Parker, Quantized ﬁelds and particle creation in expa nding universes, I

L. Parker, Quantized ﬁelds and particle creation in expa nding universes, I. Phys. Rev. 183, 1057 (1969)

work page 1969

[3] [3]

Parker, Quantized ﬁelds and particle creation in expa nding universes, II

L. Parker, Quantized ﬁelds and particle creation in expa nding universes, II. Phys. Rev. D 3, 346 (1971)

work page 1971

[4] [4]

Papastamatiou and L

J. Papastamatiou and L. Parker, Asymmetric creation of m atter and antimatter in the expanding universe, Phys Rev D 19, 2283 (1979)

work page 1979

[5] [5]

N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge: Cambridge University Press (1982)

work page 1982

[6] [6]

S. A. Fulling, Aspects of Quantum Field Theory in Curved S pacetime, Cambridge: Cambridge University Press (1989). 11

work page 1989

[7] [7]

V. F. Mukhanov and S. Winitzki, Introduction to Quantum F ields in Gravity, Cambridge: Cambridge University Press (2007)

work page 2007

[8] [8]

A. K. Sanyal, S. Debnath, S. Ruz, Canonical formulation o f the curvature-squared action in the presence of a lapse function, Class. Quantum Grav. 29, 215007 (2012), arXiv:1108.5869 [gr-qc]

work page Pith review arXiv 2012