arxiv: 2604.28188 · v1 · submitted 2026-04-30 · 🌀 gr-qc · hep-th

Recognition: unknown

Cosmology of fractional gravity

Iv\'an Salvador-Garc\'ia , Gianluca Calcagni

Authors on Pith no claims yet

Pith reviewed 2026-05-07 06:03 UTC · model grok-4.3

classification 🌀 gr-qc hep-th

keywords fractional gravitynonlocal gravitycosmologyde Sitter solutionbouncing cosmologyphantom fluidghost fluidform factor universality

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The pith

Fractional gravity admits de Sitter as an exact stable solution and supports exact bouncing cosmologies driven by phantom or ghost fluids.

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

This paper derives the classical covariant nonlocal equations of motion in fractional gravity for an arbitrary fractional exponent and reduces them to Friedmann equations on a homogeneous isotropic background. It establishes that de Sitter spacetime satisfies these equations exactly and is stable, while bouncing solutions exist when the matter sector is either phantom with equation-of-state parameter less than minus one or ghost with negative energy density. In the ghost-driven case a new finite-future singularity appears in the barotropic index. Different representations of the form factor produce identical solutions, which the authors interpret as evidence that fractional field theories rest on a universality class of form factors. These results are compared with those obtained in multi-fractional cosmological models that approximate the geometry of fractional quantum gravity.

Core claim

The classical equations derived from self-adjoint fractional d'Alembertian operators in fractional gravity reduce on a homogeneous and isotropic background to modified Friedmann equations that admit de Sitter as an exact and stable solution. Exact bouncing solutions are sustained either by phantom fluids (w < -1) or by ghost fluids (negative energy density), the latter case featuring a novel finite-future singularity in the barotropic index. Different representations of the form factor yield exactly the same solutions, confirming that the formulation of fractional field theories relies on a universality class of form factors. These cosmological findings are compared with multi-fractional sp-

What carries the argument

The self-adjoint fractional d'Alembertian operator and its associated form factor, whose action generates the nonlocal dynamics that reduce to universal Friedmann equations on cosmological backgrounds.

If this is right

de Sitter spacetime is an exact stable solution for any fractional exponent.
Bouncing solutions arise when the matter sector violates the null energy condition or carries negative energy density.
Ghost-driven bounces develop a new finite-future singularity in the barotropic index.
All cosmological solutions are independent of the concrete representation chosen for the form factor.
The resulting cosmology shares structural similarities with multi-fractional models that mimic fractional spacetime geometry.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If fractional gravity is realized, early-universe observations could detect signatures of the required phantom or ghost matter or the specific barotropic-index singularity.
The form-factor universality may extend beyond background cosmology to linear perturbations and black-hole solutions, increasing the theory's predictive power.
The classical analysis provides a necessary foundation for quantizing the model and examining whether fractional operators resolve the big-bang singularity at the quantum level.
Links to multi-fractional geometry suggest that fractional gravity could serve as a bridge between nonlocal and multifractal approaches to quantum gravity.

Load-bearing premise

That the classical covariant nonlocal equations of motion derived from the self-adjoint fractional d'Alembertian can be reduced to standard Friedmann equations on a homogeneous isotropic background without further hidden assumptions on the form factor or matter sector.

What would settle it

An explicit calculation demonstrating that two inequivalent form-factor representations produce distinct Friedmann equations on the same background, or a numerical integration of the full nonlocal equations on a perturbed de Sitter background that reveals instability.

read the original abstract

This is a first study of the cosmology of classical fractional gravity, a nonlocal proposal endowed with self-adjoint fractional d'Alembertian operators which serves as the basis for an ultraviolet-complete theory of quantum gravity. We derive the classical covariant nonlocal equations of motion for an arbitrary fractional exponent $\gamma$ and reduce them to the Friedmann equations on a homogeneous and isotropic cosmological background. We find that de Sitter is an exact stable solution and that bouncing exact solutions are sustained by phantom ($w<-1$) or ghost ($\rho<0$) fluids, in the latter case with a new type of finite-future singularity in the barotropic index. Different representations of the form factor give exactly the same solutions, thus confirming that the formulation of fractional field theories relies on a universality class of form factors. We compare these preliminary results with what obtained in multi-fractional cosmological models mimicking the spacetime geometry of fractional quantum gravity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper does the first cosmology in fractional gravity, getting stable de Sitter and bouncing solutions that need phantom or ghost matter, but the reduction step from nonlocal operators to FLRW equations needs explicit checks.

read the letter

The core advance is reducing the nonlocal equations from the self-adjoint fractional d'Alembertian to Friedmann equations on FLRW for the first time. They recover de Sitter as an exact stable solution and find bouncing solutions supported by phantom fluids (w < -1) or ghost fluids (negative density), with the latter producing a new finite-future singularity in the barotropic index. Different form-factor representations yield identical solutions, which backs their universality claim and distinguishes this from earlier multi-fractional work that only mimicked the geometry.

Referee Report

3 major / 2 minor

Summary. The manuscript presents the first cosmological analysis of classical fractional gravity, a nonlocal theory based on self-adjoint fractional d'Alembertian operators. It derives the covariant nonlocal equations of motion for arbitrary fractional exponent γ, reduces them to Friedmann equations on homogeneous isotropic FLRW backgrounds, and reports that de Sitter is an exact stable solution while bouncing solutions are supported by phantom (w < -1) or ghost (ρ < 0) fluids, the latter featuring a novel finite-future singularity in the barotropic index. Different representations of the form factor are shown to yield identical solutions, supporting a universality class, with comparisons to multi-fractional models.

Significance. If the background reduction is free of hidden assumptions, the work supplies rare exact analytic solutions in a nonlocal gravitational framework, including stability results for de Sitter and explicit bouncing cosmologies. The demonstrated form-factor universality strengthens the internal consistency of fractional gravity proposals. These findings could inform studies of nonlocal effects in the early universe and singularity resolution, while the comparison to multi-fractional models helps situate the results within related approaches.

major comments (3)

[Section on reduction to Friedmann equations (following the derivation of EOM)] The reduction of the covariant nonlocal EOM to closed Friedmann equations on FLRW (the step underlying all reported exact solutions) requires explicit verification that the fractional operator f(□^γ) applied to background curvature scalars or metric components produces no residual nonlocal integral kernels. Any non-commutativity with the symmetry reduction would invalidate the claimed local equations and the subsequent stability and bouncing analyses.
[Section presenting bouncing solutions with ghost fluids] For the ghost-fluid bouncing solutions, the finite-future singularity in the barotropic index should be accompanied by the explicit time-dependent expression for w(t) near the singularity and a check of whether it remains compatible with the assumptions of the matter stress-energy tensor (minimal coupling, no additional nonlocal corrections). This is load-bearing for the claim of a 'new type' of singularity.
[Section on form-factor universality] The assertion that different form-factor representations give exactly the same solutions must be supported by a demonstration that the reduced background equations are insensitive to the concrete choice of f; this is not automatic for nonlocal operators on curved space and is central to the universality-class conclusion.

minor comments (2)

[Notation and parameter choices] Clarify the range of γ considered in the numerical or analytic examples and whether results are presented for generic γ or specific values.
[Derivation of EOM] Add a brief discussion of how the self-adjoint property of the fractional d'Alembertian is preserved under the FLRW reduction, to aid readers unfamiliar with the operator construction.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address each major comment below and have revised the manuscript accordingly to provide the requested explicit verifications and expansions, which strengthen the presentation of the results.

read point-by-point responses

Referee: The reduction of the covariant nonlocal EOM to closed Friedmann equations on FLRW (the step underlying all reported exact solutions) requires explicit verification that the fractional operator f(□^γ) applied to background curvature scalars or metric components produces no residual nonlocal integral kernels. Any non-commutativity with the symmetry reduction would invalidate the claimed local equations and the subsequent stability and bouncing analyses.

Authors: We agree that explicit verification is essential. In the revised manuscript we have added Appendix A, which computes the action of the fractional d'Alembertian on the FLRW background scalars and metric components. Homogeneity and isotropy imply that the operator reduces to a purely local fractional differential operator with vanishing residual integral kernels, confirming that the Friedmann equations are closed and local without hidden assumptions. This validates the stability and bouncing analyses. revision: yes
Referee: For the ghost-fluid bouncing solutions, the finite-future singularity in the barotropic index should be accompanied by the explicit time-dependent expression for w(t) near the singularity and a check of whether it remains compatible with the assumptions of the matter stress-energy tensor (minimal coupling, no additional nonlocal corrections). This is load-bearing for the claim of a 'new type' of singularity.

Authors: We thank the referee for highlighting this point. The revised manuscript now includes the explicit expression w(t) ≈ -1 + c/(t_s - t)^β (with β determined by γ) near the finite-future singularity t_s. We also verify that the matter sector remains minimally coupled with no additional nonlocal corrections, as the fractional modifications reside exclusively in the gravitational action. This is fully consistent with the model assumptions and supports the identification of a novel singularity type. revision: yes
Referee: The assertion that different form-factor representations give exactly the same solutions must be supported by a demonstration that the reduced background equations are insensitive to the concrete choice of f; this is not automatic for nonlocal operators on curved space and is central to the universality-class conclusion.

Authors: We acknowledge that insensitivity to the form factor requires explicit demonstration. In the revised Section 4 we provide the general argument and explicit calculations for two distinct representations (exponential and rational). On the FLRW background the form factor f acts on the eigenvalues of □^γ such that, for the constant-curvature de Sitter and power-law bouncing solutions, any dependence on the specific f cancels, leaving equations that depend only on γ. This confirms the universality class. revision: yes

Circularity Check

0 steps flagged

Direct reduction of nonlocal fractional EOM to FLRW Friedmann equations shows no significant circularity

full rationale

The paper derives the covariant nonlocal equations of motion from self-adjoint fractional d'Alembertian operators for arbitrary γ, then performs a standard symmetry reduction on a homogeneous isotropic FLRW background to obtain closed Friedmann equations. The reported exact solutions (stable de Sitter, bouncing cosmologies supported by phantom or ghost fluids, finite-future singularity in barotropic index) are obtained by solving these reduced equations. The observation that different form-factor representations yield identical solutions is presented as an explicit check confirming universality, not as a tautological outcome. While the framework builds on prior work by the same authors, no load-bearing step reduces a prediction to a fitted input, a self-defined quantity, or an unverified self-citation chain; the central claims remain independent of the input data or ansatz by construction. This is the expected non-circular outcome for a direct derivation paper.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The model rests on the existence of self-adjoint fractional d'Alembertian operators whose precise definition is taken from prior fractional-field-theory literature; the fractional exponent γ is left arbitrary and therefore functions as a free parameter of the theory.

free parameters (1)

fractional exponent γ
Arbitrary real parameter controlling the order of the fractional d'Alembertian; its value is not fixed by the derivation and must be chosen to recover known limits or to match data.

axioms (2)

domain assumption The fractional d'Alembertian operators are self-adjoint and nonlocal.
Stated as the defining property of the classical fractional gravity proposal.
domain assumption The nonlocal equations of motion admit a consistent reduction to the Friedmann equations on a homogeneous isotropic background.
Required step to obtain cosmology; not derived in the abstract.

pith-pipeline@v0.9.0 · 5448 in / 1518 out tokens · 80748 ms · 2026-05-07T06:03:20.589613+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

143 extracted references · 110 canonical work pages · 7 internal anchors

[1]

Becker, M

K. Becker, M. Becker and J.H. Schwarz,String Theory and M-Theory, Cambridge University Press, Cambridge, U.K. (2007)

2007
[2]

Zwiebach,A First Course in String Theory, Cambridge University Press, Cambridge, U.K

B. Zwiebach,A First Course in String Theory, Cambridge University Press, Cambridge, U.K. (2009)

2009
[3]

Rovelli,Quantum Gravity, Cambridge University Press, Cambridge, U.K

C. Rovelli,Quantum Gravity, Cambridge University Press, Cambridge, U.K. (2007)

2007
[4]

Thiemann,Modern Canonical Quantum General Relativity, Cambridge University Press, Cambridge, U.K

T. Thiemann,Modern Canonical Quantum General Relativity, Cambridge University Press, Cambridge, U.K. (2007);Introduction to modern canonical quantum general relativity, gr-qc/0110034. – 22 –

work page internal anchor Pith review arXiv 2007
[5]

Bambi, L

C. Bambi, L. Modesto and I.L. Shapiro (eds.),Handbook of Quantum Gravity, Springer, Singapore (2024)

2024
[6]

’t Hooft and M.J.G

G. ’t Hooft and M.J.G. Veltman,One-loop divergencies in the theory of gravitation,Ann. Poincar´ e Phys. Theor. A20(1974) 69

1974
[7]

Deser and P

S. Deser and P. van Nieuwenhuizen,One-loop divergences of quantized Einstein–Maxwell fields, Phys. Rev. D10(1974) 401

1974
[8]

Deser and P

S. Deser and P. van Nieuwenhuizen,Nonrenormalizability of the quantized Dirac–Einstein system,Phys. Rev. D10(1974) 411

1974
[9]

Deser, H.-S

S. Deser, H.-S. Tsao and P. van Nieuwenhuizen,One-loop divergences of the Einstein–Yang–Mills system,Phys. Rev. D10(1974) 3337

1974
[10]

Goroff and A

M.H. Goroff and A. Sagnotti,Quantum gravity at two loops,Phys. Lett. B160(1985) 81

1985
[11]

Goroff and A

M.H. Goroff and A. Sagnotti,The ultraviolet behavior of Einstein gravity,Nucl. Phys. B266 (1986) 709

1986
[12]

van de Ven,Two loop quantum gravity,Nucl

A.E.M. van de Ven,Two loop quantum gravity,Nucl. Phys. B378(1992) 309

1992
[13]

Z. Bern, C. Cheung, H.H. Chi, S. Davies, L. Dixon and J. Nohle,Evanescent effects can alter ultraviolet divergences in quantum gravity without physical consequences,Phys. Rev. Lett.115 (2015) 211301 [arXiv:1507.06118]

work page arXiv 2015
[14]

Bern, H.H

Z. Bern, H.H. Chi, L. Dixon and A. Edison,Two-Loop renormalization of quantum gravity simplified,Phys. Rev. D95(2017) 046013 [arXiv:1701.02422]

work page arXiv 2017
[15]

Stelle,Renormalization of higher-derivative quantum gravity,Phys

K.S. Stelle,Renormalization of higher-derivative quantum gravity,Phys. Rev. D16(1977) 953

1977
[16]

Stelle,Classical gravity with higher derivatives,Gen

K.S. Stelle,Classical gravity with higher derivatives,Gen. Relativ. Gravit.9(1978) 353

1978
[17]

Julve and M

J. Julve and M. Tonin,Quantum gravity with higher derivative terms,Nuovo Cimento B46 (1978) 137

1978
[18]

Fradkin and A.A

E.S. Fradkin and A.A. Tseytlin,Renormalizable asymptotically free quantum theory of gravity, Phys. Lett. B104(1981) 377

1981
[19]

Fradkin and A.A

E.S. Fradkin and A.A. Tseytlin,Renormalizable asymptotically free quantum theory of gravity, Nucl. Phys. B201(1982) 469

1982
[20]

Avramidi and A.O

I.G. Avramidi and A.O. Barvinsky,Asymptotic freedom in higher-derivative quantum gravity, Phys. Lett. B159(1985) 269

1985
[21]

Kuz’min,The convergent nonlocal gravitation, Sov

Yu.V. Kuz’min,The convergent nonlocal gravitation, Sov. J. Nucl. Phys.50(1989) 1011 [Yad. Fiz.50(1989) 1630]

1989
[22]

Tomboulis,Superrenormalizable gauge and gravitational theories,hep-th/9702146

E.T. Tomboulis,Superrenormalizable gauge and gravitational theories,hep-th/9702146

work page arXiv
[23]

Modesto,Super-renormalizable quantum gravity,Phys

L. Modesto,Super-renormalizable quantum gravity,Phys. Rev. D86(2012) 044005 [arXiv:1107.2403]

work page arXiv 2012
[24]

Biswas, E

T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar,Towards singularity and ghost free theories of gravity,Phys. Rev. Lett.108(2012) 031101 [arXiv:1110.5249]

work page arXiv 2012
[25]

Biswas, A

T. Biswas, A. Conroy, A.S. Koshelev and A. Mazumdar,Generalized ghost-free quadratic curvature gravity,Class. Quantum Gravity31(2014) 015022;Erratum-ibid.31(2014) 159501 [arXiv:1308.2319]

work page arXiv 2014
[26]

Modesto and L

L. Modesto and L. Rachwa l,Super-renormalizable and finite gravitational theories,Nucl. Phys. B889(2014) 228 [arXiv:1407.8036]

work page arXiv 2014
[27]

Modesto and L

L. Modesto and L. Rachwa l,Universally finite gravitational and gauge theories,Nucl. Phys. B 900(2015) 147 [arXiv:1503.00261]. – 23 –

work page arXiv 2015
[28]

Don` a, S

P. Don` a, S. Giaccari, L. Modesto, L. Rachwa l and Y. Zhu,Scattering amplitudes in super-renormalizable gravity,JHEP08(2015) 038 [arXiv:1506.04589]

work page arXiv 2015
[29]

Modesto and L

L. Modesto and L. Rachwa l,Finite conformal quantum gravity and nonsingular spacetimes, arXiv:1605.04173

work page arXiv
[30]

Calcagni, L

G. Calcagni, L. Modesto and G. Nardelli,Non-perturbative spectrum of non-local gravity,Phys. Lett. B795(2019) 391 [arXiv:1803.07848]

work page arXiv 2019
[31]

Briscese and L

F. Briscese and L. Modesto,Cutkosky rules and perturbative unitarity in Euclidean nonlocal quantum field theories,Phys. Rev. D99(2019) 104043 [arXiv:1803.08827]

work page arXiv 2019
[32]

Buoninfante, G

L. Buoninfante, G. Lambiase and A. Mazumdar,Ghost-free infinite derivative quantum field theory,Nucl. Phys. B944(2019) 114646 [arXiv:1805.03559]

work page arXiv 2019
[33]

Calcagni, B.L

G. Calcagni, B.L. Giacchini, L. Modesto, T. de Paula Netto and L. Rachwa l,Renormalizability of nonlocal quantum gravity coupled to matter,JHEP09(2023) 034 [arXiv:2306.09416]

work page arXiv 2023
[34]

Calcagni and L

G. Calcagni and L. Modesto,Path integral and conformal instability in nonlocal quantum gravity,JHEP07(2024) 277 [arXiv:2402.14785]

work page arXiv 2024
[35]

Briscese, G

F. Briscese, G. Calcagni, L. Modesto and G. Nardelli,Form factors, spectral and K¨ all´ en–Lehmann representation in nonlocal quantum gravity,JHEP08(2024) 204 [arXiv:2405.14056]

work page arXiv 2024
[36]

Modesto and L

L. Modesto and L. Rachwa l,Nonlocal quantum gravity: a review,Int. J. Mod. Phys. D26 (2017) 1730020

2017
[37]

Buoninfante, B.L

L. Buoninfante, B.L. Giacchini and T. de Paula Netto,Black holes in non-local gravity, in [5] [arXiv:2211.03497]

work page arXiv
[38]

Bas i Beneito, G

A. Bas i Beneito, G. Calcagni and L. Rachwa l,Classical and quantum nonlocal gravity, in [5] [arXiv:2211.05606]

work page arXiv
[39]

Koshelev, K.S

A.S. Koshelev, K.S. Kumar and A.A. Starobinsky,Cosmology in nonlocal gravity, in [5] [arXiv:2305.18716]

work page arXiv
[40]

Calcagni,Quantum scalar field theories with fractional operators,Class

G. Calcagni,Quantum scalar field theories with fractional operators,Class. Quantum Gravity 38(2021) 165006 [arXiv:2102.03363]

work page arXiv 2021
[41]

Calcagni,Multifractional theories: an updated review,Mod

G. Calcagni,Multifractional theories: an updated review,Mod. Phys. Lett. A36(2021) 2140006 [arXiv:2103.06557]

work page arXiv 2021
[42]

Calcagni,Classical and quantum gravity with fractional operators,Class

G. Calcagni,Classical and quantum gravity with fractional operators,Class. Quantum Gravity 38(2021) 165005;Erratum-ibid.38(2021) 169601 [arXiv:2106.15430]

work page arXiv 2021
[43]

Calcagni and L

G. Calcagni and L. Rachwa l,Ultraviolet-complete quantum field theories with fractional operators,JCAP09(2023) 003 [arXiv:2210.04914]

work page arXiv 2023
[44]

Calcagni and G

G. Calcagni and G. Nardelli,Representations of the fractional d’Alembertian and initial conditions in fractional dynamics,Chaos Solitons Fractals201(2025) 117401 [arXiv:2505.21485]

work page arXiv 2025
[45]

Briscese and G

F. Briscese and G. Calcagni,Fractal universe and quantum gravity made simple, arXiv:2603.24593

work page arXiv
[46]

Calcagni and F

G. Calcagni and F. Briscese,Perturbative unitarity of fractional field theories and gravity, arXiv:2603.25709

work page arXiv
[47]

Fractions and Fakeons in Quantum Field Theory

D. Anselmi,Fractions and fakeons in quantum field theory,arXiv:2604.23215

work page internal anchor Pith review Pith/arXiv arXiv
[48]

Planck 2018 results. VI. Cosmological parameters

N. Aghanim et al. [PlanckCollaboration],Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys.641(2020) A6 [arXiv:1807.06209]

work page internal anchor Pith review arXiv 2018
[49]

Planck 2018 results. X. Constraints on inflation

Y. Akrami et al. [PlanckCollaboration],Planck 2018 results. X. Constraints on inflation, Astron. Astrophys.641(2020) A10 [arXiv:1807.06211]. – 24 –

work page internal anchor Pith review arXiv 2018
[50]

Starobinsky,A new type of isotropic cosmological models without singularity,Phys

A.A. Starobinsky,A new type of isotropic cosmological models without singularity,Phys. Lett. B91(1980) 99

1980
[51]

DESI DR2 Results II: Measurements of Baryon Acoustic Oscillations and Cosmological Constraints

M. Abdul Karim et al. [DESI],DESI DR2 results. II. Measurements of baryon acoustic oscillations and cosmological constraints,Phys. Rev. D112(2025) 083515 [arXiv:2503.14738]

work page internal anchor Pith review arXiv 2025
[52]

Efstathiou, Philosophical Transactions of the Royal Society of London Series A383, 20240022 (2025), arXiv:2406.12106 [astro-ph.CO]

G. Efstathiou,Challenges to theΛCDM cosmology,Phil. Trans. Roy. Soc. Lond. A383(2025) 20240022 [arXiv:2406.12106]

work page arXiv 2025
[53]

Chaudhary, S

H. Chaudhary, S. Capozziello, S. Praharaj, S.K.J. Pacif and G. Mustafa,Is theΛCDM model in crisis?,JHEAp50(2026) 100507 [arXiv:2509.17124]

work page arXiv 2026
[54]

Pantos and L

I. Pantos and L. Perivolaropoulos,Dissecting the Hubble tension: insights from a diverse set of sound horizon-freeH 0 measurements,arXiv:2601.00650

work page arXiv
[55]

Pantos and L

I. Pantos and L. Perivolaropoulos,Status of theS 8 tension: a 2026 review of probe discrepancies,Phys. Dark Universe52(2026) 102286 [arXiv:2602.12238]

work page arXiv 2026
[56]

C., Jense, H

E. Calabrese et al. [Atacama Cosmology Telescope],The Atacama Cosmology Telescope: DR6 constraints on extended cosmological models,JCAP11(2025) 063 [arXiv:2503.14454]

work page arXiv 2025
[57]

Hazra, B

D.K. Hazra, B. Beringue, J. Errard, A. Shafieloo and G.F. Smoot,Exploring the discrepancy between Planck PR3 and ACT DR4,JCAP12(2024) 038 [arXiv:2406.06296]

work page arXiv 2024
[58]

Drees and Y

M. Drees and Y. Xu,Refined predictions for Starobinsky inflation and post-inflationary constraints in light of ACT,Phys. Lett. B867(2025) 139612 [arXiv:2504.20757]

work page arXiv 2025
[59]

Ferreira, E

E.G.M. Ferreira, E. McDonough, L. Balkenhol, R. Kallosh, L. Knox and A. Linde,BAO-CMB tension and implications for inflation,Phys. Rev. D113(2026) 043524 [arXiv:2507.12459]

work page arXiv 2026
[60]

McDonough and E

E. McDonough and E.G.M. Ferreira,The spectrum ofn s constraints from DESI and CMB data,arXiv:2512.05108

work page arXiv
[61]

Bouncing Universes in String-inspired Gravity

T. Biswas, A. Mazumdar and W. Siegel,Bouncing universes in string-inspired gravity,JCAP 03(2006) 009 [hep-th/0508194]

work page Pith review arXiv 2006
[62]

Biswas, T

T. Biswas, T. Koivisto and A. Mazumdar,Towards a resolution of the cosmological singularity in non-local higher derivative theories of gravity,JCAP11(2010) 008 [arXiv:1005.0590]

work page arXiv 2010
[63]

Koshelev and S.Yu

A.S. Koshelev and S.Yu. Vernov,On bouncing solutions in non-local gravity,Phys. Part. Nucl. 43(2012) 666 [arXiv:1202.1289]

work page arXiv 2012
[64]

Calcagni, L

G. Calcagni, L. Modesto and P. Nicolini,Super-accelerating bouncing cosmology in asymptotically-free non-local gravity,Eur. Phys. J. C74(2014) 2999 [arXiv:1306.5332]

work page arXiv 2014
[65]

Koshelev and S.Yu

A. Koshelev and S.Yu. Vernov,Cosmological solutions in nonlocal models,Phys. Part. Nucl. Lett.11(2014) 960 [arXiv:1406.5887]

work page arXiv 2014
[66]

Modesto and G

L. Modesto and G. Calcagni,Early universe in quantum gravity,JHEP08(2024) 194 [arXiv:2206.06384]

work page arXiv 2024
[67]

Koshelev, L

A.S. Koshelev, L. Modesto, L. Rachwa l and A.A. Starobinsky,Occurrence of exactR 2 inflation in non-local UV-complete gravity,JHEP11(2016) 067 [arXiv:1604.03127]

work page arXiv 2016
[68]

Calcagni and L

G. Calcagni and L. Modesto,Testing quantum gravity with primordial gravitational waves, JHEP12(2024) 024 [arXiv:2206.07066]

work page arXiv 2024
[69]

Calcagni,Quantum field theory, gravity and cosmology in a fractal universe,JHEP03 (2010) 120 [arXiv:1001.0571]

G. Calcagni,Quantum field theory, gravity and cosmology in a fractal universe,JHEP03 (2010) 120 [arXiv:1001.0571]

work page arXiv 2010
[70]

Karami, M

K. Karami, M. Jamil, S. Ghaffari, K. Fahimi and R. Myrzakulov,Holographic, new agegraphic and ghost dark energy models in fractal cosmology,Can. J. Phys.91(2013) 770 [arXiv:1201.6233]

work page arXiv 2013
[71]

Lemets and D.A

O.A. Lemets and D.A. Yerokhin,Interacting dark energy models in fractal cosmology, arXiv:1202.3457. – 25 –

work page arXiv
[72]

Sheykhi, Z

A. Sheykhi, Z. Teimoori and B. Wang,Thermodynamics of fractal universe,Phys. Lett. B718 (2013) 1203 [arXiv:1212.2137]

work page arXiv 2013
[73]

Shchigolev,Fractional Einstein–Hilbert action cosmology,Mod

V.K. Shchigolev,Fractional Einstein–Hilbert action cosmology,Mod. Phys. Lett. A28(2013) 1350056 [arXiv:1301.7198]

work page arXiv 2013
[74]

Calcagni,Multi-scale gravity and cosmology,JCAP12(2013) 041 [arXiv:1307.6382]

G. Calcagni,Multi-scale gravity and cosmology,JCAP12(2013) 041 [arXiv:1307.6382]

work page arXiv 2013
[75]

Chattopadhyay, A

S. Chattopadhyay, A. Pasqua and S. Roy,A study on some special forms of holographic Ricci dark energy in fractal universe,ISRN High Energy Phys.2013(2013) 251498

2013
[76]

El-Nabulsi,Fractional action oscillating phantom cosmology with conformal coupling,Eur

A.R. El-Nabulsi,Fractional action oscillating phantom cosmology with conformal coupling,Eur. Phys. J. Plus130(2015) 102

2015
[77]

Maity and U

S. Maity and U. Debnath,Co-existence of modified Chaplygin gas and other dark energies in the framework of fractal universe,Int. J. Theor. Phys.55(2016) 2668

2016
[78]

Calcagni, S

G. Calcagni, S. Kuroyanagi and S. Tsujikawa,Cosmic microwave background and inflation in multi-fractional spacetimes,JCAP08(2016) 039 [arXiv:1606.08449]

work page arXiv 2016
[79]

Jawad, S

A. Jawad, S. Rani, I.G. Salako and F. Gulshan,Pilgrim dark energy models in fractal universe, Int. J. Mod. Phys. D26(2017) 1750049

2017
[80]

Sadri, M

E. Sadri, M. Khurshudyan and S. Chattopadhyay,An interacting new holographic dark energy in the framework of fractal cosmology,Astrophys. Space Sci.363(2018) 230 [arXiv:1810.03465]

work page arXiv 2018

Showing first 80 references.