We show that quantum-induced marginal deformations of the Starobinsky
gravitational action of the form $R^{2(1 -\alpha)}$, with $R$ the Ricci scalar
and $\alpha$ a positive parameter, smaller than one half, can account for the
recent experimental observations by BICEP2 of primordial tensor modes. We also
suggest natural microscopic (non) gravitational sources of these corrections
and demonstrate that they lead generally to a nonzero and positive $\alpha$.
Furthermore we argue, that within this framework, the tensor modes probe
theories of grand unification with a large scalar field content.Comment: 5 pages, 1 figure, 2 column

Codello, A.

Joergensen, J.

Sannino, F.

Svendsen, O.

English

arXiv

Alessandro Codello

Jakob Joergensen

Francesco Sannino

Ole Svendsen

Crossref

Journal of High Energy Physics

Marginally deformed Starobinsky gravity

We show that quantum-induced marginal deformations of the Starobinsky gravitational action of the form $R^{2(1 -\alpha)}$, with $R$ the Ricci scalar and $\alpha$ a positive parameter, smaller than one half, can account for the recent experimental observations by BICEP2 of primordial tensor modes. We also suggest natural microscopic (non) gravitational sources of these corrections and demonstrate that they lead generally to a nonzero and positive $\alpha$. Furthermore we argue, that within this framework, the tensor modes probe theories of grand unification with a large scalar field content

Sannino, Francesco

Syddansk Universitets Forskerportal

       University of Southern DenmarkMarginally Deformed Starobinsky GravityCodello, A.; Joergensen, J.; Sannino, Francesco; Svendsen, O.Published in:Journal of High Energy PhysicsDOI:10.1007/JHEP02(2015)050Publication date:2015Citation for pulished version (APA):Codello, A., Joergensen, J., Sannino, F., & Svendsen, O. (2015). Marginally Deformed Starobinsky Gravity.Journal of High Energy Physics, 2015, Article 50. https://doi.org/10.1007/JHEP02(2015)050Go to publication entry in University of Southern Denmark's Research PortalTerms of useThis work is brought to you by the University of Southern Denmark.Unless otherwise specified it has been shared according to the terms for self-archiving.If no other license is stated, these terms apply:            • You may download this work for personal use only.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying this open access versionIf you believe that this document breaches copyright please contact us providing details and we will investigate your claim.Please direct all enquiries to puresupport@bib.sdu.dkDownload date: 25. Oct. 2025Marginally Deformed Starobinsky GravityAlessandro Codello,∗ Jakob Joergensen,† Francesco Sannino,‡ and Ole Svendsen§CP3-Origins & the Danish IAS University of Southern Denmark, Campusvej 55, DK-5230 Odense M, DenmarkWe show that quantum-induced marginal deformations of the Starobinsky gravitational action ofthe form R2(1−α), with R the Ricci scalar and α a positive parameter smaller than one half, can generatesizable primordial tensor modes. We also suggest natural microscopic sources of these correctionsand demonstrate that they generally lead to a nonzero and positive α. Furthermore we argue, thatwithin this framework, the scalar spectral index and tensor modes probe theories of grand unificationincluding theories not testable at the electroweak scale.Preprint: CP3-Origins-2014-13 DNRF90 and DIAS-2014-13The fundamental origin of the inflationaryparadigm is a central problem in cosmology [1–4]. The simplest models of inflation typicallyintroduce new scalar degrees of freedom.An intriguing possibility is that gravity itselfis directly responsible for the inflationary pe-riod of the universe. This requires one to gobeyond the time-honored Einstein action, forexample by adding a R2-term as in the Starobin-sky model [1]. This approach is highly naturalsince it enables gravity itself to drive inflationwithout resorting to the introduction of new adhoc scalar fields. The Starobinsky model in iso-lation predicts a nearly vanishing ratio of ten-sor to scalar modes (r), a result challenged byBICEP2 results [5]. However, independentlyon the validity of the BICEP2 result [6], it isof fundamental importance to know the mod-ifications on the Starobinsky model stemmingfrom the embedding of a matter theory of par-ticle physics in the gravitational theory.According to [7] cosmology can be usedqualitatively to establish the quantization ofgravity. In fact, by combining cosmological ob-servations with an effective field theory (EFT)treatment of gravity [8, 9] one can start esti-mating the parameters entering gravity’s effec-tive action. An actual discovery of primordialtensor modes can therefore be used to deter-mine these parameters at the inflationary scale,which may turn out to be close to the grandunification energy scale.To lowest order, the effective action for grav-ity can be parametrized asS =∫d4x√−g−M2p2 R + h0R2 + h1R3++c0C2 + e0E + ...],where Mp is the Planck mass. Beyond anexpansion in the Ricci scalar R, we formallyincluded the Weyl conformal tensor C2 andthe Euler four dimensional topological termE. However we can drop E since it is a totalderivative. Furthermore when gravity is quan-tized around the Friedmann Lemaitre Robert-son Walker metric the Weyl terms are sub–leading since the geometry is conformally flat[11]. We are left with an f (R) form of theEFT. Higher powers of R, C2 and E are nat-urally suppressed by the Planck mass scale.If inflation occurs at energy scales much be-low the Planck scale the EFT is accurate. Wemust, however, take into account also marginaldeformations including, for example, logarith-mic corrections to the action above. Because ofthe similarity between the EFT description ofgravity and the chiral Lagrangian for QuantumChromo Dynamics we expect the quantum-induced logarithmic corrections to play a fun-damental role for a coherent understanding oflow energy gravitational dynamics at the in-flationary scale. This is exactly what happensin hadronic processes involving pions at lowenergies.We encode these ideas in a simple f (R) formof the gravitational action formulated in theJordan frame:SJ =∫d4x√−g−M2p2 R + hM4αp R2(1−α) .(1)We assume that α is a real parameterwith 2|α| < 1 and h is a dimension-less parameter. We linearize the actionvia SJ =∫d4x√−g[f (y) + f ′(y)(R − y)]withf (R) = −M2p2R + hM4αp R2(1−α). The equation ofmotion for y implies R = y provided f ′′(y) doesnot vanish. Introducing the conformal modeψ = − f ′(y) with V(ψ) = −y(ψ)ψ− f (y(ψ)) we ar-rive, after having introduced the mass dimen-arXiv:1404.3558v3  [hep-ph]  10 Feb 20152sion one real scalar field φ via 2ψ −M2P = ξφ2at:SJ =∫d4x√−g−M2p + ξφ22 R − V(φ) , (2)with V(φ) = λφ4(φMp)4γ, α = γ/(1 + 2γ) andh1+2γ =(ξ41+2γ1+γ)2(1+γ) 1λ(1+2γ) . This providesthe explicit relation between (1) and the ef-fective quantum-corrected non-minimally cou-pled scalar field theory used in [12]. Here wecan simply set the non-minimal coupling valueof ξ associated to the φ2R term to unity. Thisis valid since this action is equivalent to (1),where only the two independent parameters hand α appear. In this way h depends only on γand λ. We will retain the explicit dependenceon ξ to ease the comparison with the resultsobtained in [12].An important difference with respect to thenon-minimally coupled theory of [12] is thepresence, already in the Jordan frame, of a ki-netic term for φ. In the f (R) framework thekinetic term for φ is absent in the Jordan framebut it is, however, generated via the followingconformal transformation of the metric:gµν → Ω(φ)2gµν, Ω(φ)2 = 1 +ξφ2M2p. (3)This transformation allows us to rewrite boththeories in terms of a propagating scalar fieldminimally coupled to ordinary Einstein grav-ity. This is the Einstein frame. Here the theoryreadsSE =∫d4x√−g−M2p2 R + 12 gµν∂µχ∂νχ −U(χ) , U(χ) ≡ (Ω−4V) (φ (χ)) . (4)The canonically normalized scalar field χ is re-lated to φ via12(dχdφ)2=M2p(σM2p + (σ + 3ξ) ξφ2)(M2p + ξφ2)2 . (5)The map from the Jordan frame of f (R) gravityto the Einstein frame with a canonically nor-malized field is obtained for σ = 0 and agreeswith other derivations [13]. The case σ = 1corresponds to a φ with an initial kinetic termin the Jordan frame. Because of this σ depen-dence it is, in principle, possible to disentanglethe two original models. To obtain an explicitrelation between χ and φ we assume that in-flation occurs at large values of the the scalarfield, i.e. φ Mp√ξyieldingχ ' κMp log √ξφMp , κ = √2σξ + 6 . (6)For large values of the non-minimal couplingξ it is not possible to differentiate between thetwo values of σ. The Einstein frame potentialtakes the formU (χ) =λM4pξ2(1 + exp[−2χκMp])−2︸                         ︷︷                         ︸φ4-Inflationξ−2γ exp[4γχκMp]︸             ︷︷             ︸Corrections from γ. (7)The underbraced ’φ4-Inflation’-term refers tothe potential one would obtain by settingγ = 0,that is, non-minimally coupled φ4-Inflation.This limit corresponds to the Starobinskymodel [1].The Starobinsky potential is recovered forα = 0. For 0 < α < 0.5 the potential growsexponentially and, as shown in [12], it leads toa successful inflationary model with nonzeroprimordial tensor modes. On the other hand3FIG. 1: Constraints on the parameter α bydirect comparison with the combined fit ofBICEP2 and Planck data provided in [5].forαnegative the potential is unable to produceenough e-folds. In Fig. 1 generic modificationsof the Starobinsky model are confronted withBICEP2 and PLANCK data. We observe thatcosmology may constrain the deformation pa-rameter α, and as we will show shortly, α holdsinformation regarding the generic particle con-tent embedded in this gravity model of infla-tion.Different extensions of the original Starobin-sky model including, for example, new degreesof freedom, have been investigated [14]. We be-lieve that our approach is a minimal one which,as we shall see, also leads to relevant testablephenomenological implications.We now argue that these marginal deforma-tions, needed from a purely phenomenologi-cal standpoint, arise naturally within a field-theoretical approach to quantum gravity. Togain insight we start by expanding (1) in pow-ers of α and writeSJ '∫d4x√−g−M2p2 R + hR21 − 2α log  RM2p + O(α2) .(8)The logarithmic term is reminiscent of whatone would obtain via trace-log evaluations ofquantum corrections. There are several pos-sible sources for these corrections. They mayarise, for example, by integrating out matterfields, or they can arise directly from gravityloops. To sum-up the entire series of logarith-mic corrections, and hence recover the R2(1−α),we expect that a renormalization group im-proved computation is needed. This stronglysuggests that one can determine α if a morefundamental theory were at our disposal. Inthe absence of a full theory of quantum grav-ity we start here by comparing different pre-dictions for the coefficient of the logarithmicterm in (8) stemming out from: i) Integratingout minimally coupled non-interacting NS realscalar fields [15] (only non–conformal invari-ant matter contributes); ii) gravity correctionsvia the effective field theory (EFT) approach[8–10]; iii) gravity corrections within higherderivative gravity (HDG) [16]. The action (8)naturally arises after a direct computation ofthe quantum corrections stemming from the Trlog ∆ operator. This is deduced via heat kernelmethods [17], where ∆ is the laplacian aris-ing as the Hessian of the minimally coupledscalar action, the Einstein-Hilbert action in theeffective field theory case or the HDG action.The R2 log R/µ2 term emerges upon evaluatingthe operator on S4. For dimensional reasons,further corrections can be parametrized by anh(R/µ2) R2 term, where h is now a function ofR/µ2, with µ the renormalization scale. Theexplicit computations via heat kernel methodsshows [17] that leading order quantum fluctua-tions indeed induce a logarithmic form for h asin (8). This fact alone immediately shows thelink between the exponent α and the coefficientof the beta function of the coupling of the R2term, as a scale derivative with respect to themass scale in (8) shows. But we can give a bet-ter argument noticing that, because h dependson the ratio R/µ2, we have 2R∂Rh = −µ∂µh andone can determine the R dependence once thebeta function, with respect to µ, of h is known.Non–local R2 log(−/µ2) quantum correctionscan also be derived in a similar way [8]. Cos-mological applications of the non-local termshave recently been studied in [18]; other kindsof non-local actions, not generated by quan-tum loops, have also been considered in [19]4and [20]. Here we are interested in the cos-mological role of non-analytical terms whichare present in the effective quantum gravityaction. Our analysis favors the idea that non-analytic terms are more significant for inflationthan non-local ones. To the lowest order thebeta function is µ∂µh = C(4π)2 with C a constantdepending on the source of quantum correc-tions considered. After an RG improvement,the equation for h readsR ∂Rh = −C2(4π)2h . (9)The improvement is related to the appearanceof a factor h(R/µ2) on the right hand-side of theequation above. If one sets h(R/µ2) = 1, on theright-hand side, we only obtain the first loga-rithmic correction of (8). Using (9) we constructthe log–resummed solutionh(R) = h(R0)( RR0)− C2(4π)2. (10)Here R0 = µ20 is a given renormalization scale.We therefore have α = C4(4π)2 and the constanth in (1) is h(R0). If C > 0 this would naturallylead to a positive α. An explicit evaluation ofC gives [15, 16]:C =NS72minimally coupled scalars,C =14EFT gravity, (11)C =536HDG.Remarkably we deduce a positive exponent re-gardless of the underlying theory used to de-termine the associated quantum corrections tothe gravitation action. Massive particles (weconsider scalars of mass m for simplicity) leadto the beta function µ∂µh = C(4π)2 (1 + m2/µ2)−1[16]. When the renormalisation scale is taken tobe Planck’s mass the effect of the mass term isnegligible. Smaller renormalisation scales gen-erally tend to reduce the value of C and thus ofα, but in particular they don’t affect it’s sign.From (11) we deduce that quantum gravita-tional contributions can account, at most, fora 3% increase in r as compared to the originalStarobinsky model. Therefore any larger valueof r can only be generated by adding matter cor-rections. This, in turn, can be used to constrainparticle physics models minimally coupled tof (R) gravity. Furthermore, as it is evident formFig 1, for small r the spectral index (ns) de-pends sensitively on the particular value of α.We can therefore provide the following generalconstraint at the one sigma confidence level onthe number of scalar fields minimally coupledto f (R) gravityNS ≤ 85 . (12)The corresponding r values cannot exceed0.007. To exemplify the power of our resultswe now compare (12) with popular modelsof grand unification (GUT) such as minimalSU(5) that features 34 scalars and (non)minimalSO(10) featuring (297) 109 scalars. It is clearthat only models with a low content of scalarsare preferred by current experiments.Values of r around and above 0.2 can beachieved at two sigma confidence level onlyby allowing for the presence of thousandsof scalars. This corresponds to the upperpart of Fig. 1. Here one might hope to usenon-minimal models of supersymmetric GUTswhich would otherwise be physically excludedwithin the paradigm investigated here.We have pointed out that non-analytic termsare presents in the effective quantum actionand can have a role in cosmology. We have alsochecked that higher order terms of the type R3cannot lead to sizable nonzero tensor modes.Therefore we arrive at the general conclusion,that if inflation is driven by an f (R) theory ofgravity, a natural form for this function is themarginally deformed Starobinsky action pro-vided in (1) with a positive α whose size isrelated to the microscopic theory dictating thetrace-log quantum corrections. This form canbe tested by current and future experimentalresults and constitutes a natural generalizationof the original Starobinsky action.Work supported by the Danish National Re-search Foundation DNRF:90 grant.∗ codello@cp3-origins.net† joergensen@cp3-origins.net‡ sannino@cp3-origins.net§ svendsen@cp3-origins.net[1] A. A. Starobinsky, JETP Lett. 30, 682 (1979)[Pisma Zh. Eksp. Teor. Fiz. 30, 719 (1979)]; idem;Phys. Lett. B 91, 99 (1980).[2] V. F. Mukhanov and G. V. Chibisov, JETP Lett.33, 532 (1981) [Pisma Zh. Eksp. Teor. Fiz. 33, 549(1981)].[3] A. H. Guth, Phys. Rev. D 23, 347 (1981).[4] A. D. Linde, Phys. Lett. B 108, 389 (1982).[5] P. A. R. Ade et al. [BICEP2 Collaboration],arXiv:1403.3985 [astro-ph.CO].5[6] R. Flauger, J. C. Hill and D. N. Spergel,arXiv:1405.7351 [astro-ph.CO].[7] L. M. Krauss and F. Wilczek, Phys. Rev. D 89(2014) 047501 [arXiv:1309.5343 [hep-th]].[8] J. F. Donoghue, Phys. Rev. D 50 (1994) 3874 [gr-qc/9405057].[9] C. P. Burgess, arXiv:0912.4295 [gr-qc].[10] J. F. Donoghue, AIP Conf. Proc. 1483 (2012) 73[arXiv:1209.3511 [gr-qc]].[11] M. Iihoshi, S. V. Ketov and A. Morishita, Prog.Theor. Phys. 118 (2007) 475 [hep-th/0702139].[12] J. Joergensen, F. Sannino and O. Svendsen,arXiv:1403.3289 [hep-ph]. To appear in Phys.Rev. D.[13] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys.82 (2010) 451 [arXiv:0805.1726 [gr-qc]].[14] S. Ferrara, A. Kehagias and A. Riotto,arXiv:1403.5531 [hep-th].[15] A. Codello, New J. Phys. 14 (2012) 015009[arXiv:1108.1908 [gr-qc]].[16] A. Codello, R. Percacci and C. Rahmede, An-nals Phys. 324 (2009) 414 [arXiv:0805.2909 [hep-th]].[17] I. G. Avramidi, Lect. Notes Phys. M 64 (2000) 1.[18] J. F. Donoghue and B. K. El-Menoufi, Phys. Rev.D 89 (2014) 104062 [arXiv:1402.3252 [gr-qc]].[19] S. Deser and R. P. Woodard, Phys. Rev. Lett. 99(2007) 111301 [arXiv:0706.2151 [astro-ph]].[20] M. Maggiore and M. Mancarella, Phys. Rev. D90 (2014) 023005 [arXiv:1402.0448 [hep-th]].

Marginally Deformed Starobinsky Gravity

       University of Southern DenmarkMarginally Deformed Starobinsky GravityCodello, A.; Joergensen, J.; Sannino, Francesco; Svendsen, O.Published in:Journal of High Energy PhysicsDOI:10.1007/JHEP02(2015)050Publication date:2015Citation for pulished version (APA):Codello, A., Joergensen, J., Sannino, F., & Svendsen, O. (2015). Marginally Deformed Starobinsky Gravity.Journal of High Energy Physics, 2015, Article 50. https://doi.org/10.1007/JHEP02(2015)050Go to publication entry in University of Southern Denmark's Research PortalTerms of useThis work is brought to you by the University of Southern Denmark.Unless otherwise specified it has been shared according to the terms for self-archiving.If no other license is stated, these terms apply:            • You may download this work for personal use only.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying this open access versionIf you believe that this document breaches copyright please contact us providing details and we will investigate your claim.Please direct all enquiries to puresupport@bib.sdu.dkDownload date: 20. Oct. 2025Marginally Deformed Starobinsky GravityAlessandro Codello,∗ Jakob Joergensen,† Francesco Sannino,‡ and Ole Svendsen§CP3-Origins & the Danish IAS University of Southern Denmark, Campusvej 55, DK-5230 Odense M, DenmarkWe show that quantum-induced marginal deformations of the Starobinsky gravitational action ofthe form R2(1−α), with R the Ricci scalar and α a positive parameter smaller than one half, can generatesizable primordial tensor modes. We also suggest natural microscopic sources of these correctionsand demonstrate that they generally lead to a nonzero and positive α. Furthermore we argue, thatwithin this framework, the scalar spectral index and tensor modes probe theories of grand unificationincluding theories not testable at the electroweak scale.Preprint: CP3-Origins-2014-13 DNRF90 and DIAS-2014-13The fundamental origin of the inflationaryparadigm is a central problem in cosmology [1–4]. The simplest models of inflation typicallyintroduce new scalar degrees of freedom.An intriguing possibility is that gravity itselfis directly responsible for the inflationary pe-riod of the universe. This requires one to gobeyond the time-honored Einstein action, forexample by adding a R2-term as in the Starobin-sky model [1]. This approach is highly naturalsince it enables gravity itself to drive inflationwithout resorting to the introduction of new adhoc scalar fields. The Starobinsky model in iso-lation predicts a nearly vanishing ratio of ten-sor to scalar modes (r), a result challenged byBICEP2 results [5]. However, independentlyon the validity of the BICEP2 result [6], it isof fundamental importance to know the mod-ifications on the Starobinsky model stemmingfrom the embedding of a matter theory of par-ticle physics in the gravitational theory.According to [7] cosmology can be usedqualitatively to establish the quantization ofgravity. In fact, by combining cosmological ob-servations with an effective field theory (EFT)treatment of gravity [8, 9] one can start esti-mating the parameters entering gravity’s effec-tive action. An actual discovery of primordialtensor modes can therefore be used to deter-mine these parameters at the inflationary scale,which may turn out to be close to the grandunification energy scale.To lowest order, the effective action for grav-ity can be parametrized asS =∫d4x√−g−M2p2 R + h0R2 + h1R3++c0C2 + e0E + ...],where Mp is the Planck mass. Beyond anexpansion in the Ricci scalar R, we formallyincluded the Weyl conformal tensor C2 andthe Euler four dimensional topological termE. However we can drop E since it is a totalderivative. Furthermore when gravity is quan-tized around the Friedmann Lemaitre Robert-son Walker metric the Weyl terms are sub–leading since the geometry is conformally flat[11]. We are left with an f (R) form of theEFT. Higher powers of R, C2 and E are nat-urally suppressed by the Planck mass scale.If inflation occurs at energy scales much be-low the Planck scale the EFT is accurate. Wemust, however, take into account also marginaldeformations including, for example, logarith-mic corrections to the action above. Because ofthe similarity between the EFT description ofgravity and the chiral Lagrangian for QuantumChromo Dynamics we expect the quantum-induced logarithmic corrections to play a fun-damental role for a coherent understanding oflow energy gravitational dynamics at the in-flationary scale. This is exactly what happensin hadronic processes involving pions at lowenergies.We encode these ideas in a simple f (R) formof the gravitational action formulated in theJordan frame:SJ =∫d4x√−g−M2p2 R + hM4αp R2(1−α) .(1)We assume that α is a real parameterwith 2|α| < 1 and h is a dimension-less parameter. We linearize the actionvia SJ =∫d4x√−g[f (y) + f ′(y)(R − y)]withf (R) = −M2p2R + hM4αp R2(1−α). The equation ofmotion for y implies R = y provided f ′′(y) doesnot vanish. Introducing the conformal modeψ = − f ′(y) with V(ψ) = −y(ψ)ψ− f (y(ψ)) we ar-rive, after having introduced the mass dimen-arXiv:1404.3558v3  [hep-ph]  10 Feb 20152sion one real scalar field φ via 2ψ −M2P = ξφ2at:SJ =∫d4x√−g−M2p + ξφ22 R − V(φ) , (2)with V(φ) = λφ4(φMp)4γ, α = γ/(1 + 2γ) andh1+2γ =(ξ41+2γ1+γ)2(1+γ) 1λ(1+2γ) . This providesthe explicit relation between (1) and the ef-fective quantum-corrected non-minimally cou-pled scalar field theory used in [12]. Here wecan simply set the non-minimal coupling valueof ξ associated to the φ2R term to unity. Thisis valid since this action is equivalent to (1),where only the two independent parameters hand α appear. In this way h depends only on γand λ. We will retain the explicit dependenceon ξ to ease the comparison with the resultsobtained in [12].An important difference with respect to thenon-minimally coupled theory of [12] is thepresence, already in the Jordan frame, of a ki-netic term for φ. In the f (R) framework thekinetic term for φ is absent in the Jordan framebut it is, however, generated via the followingconformal transformation of the metric:gµν → Ω(φ)2gµν, Ω(φ)2 = 1 +ξφ2M2p. (3)This transformation allows us to rewrite boththeories in terms of a propagating scalar fieldminimally coupled to ordinary Einstein grav-ity. This is the Einstein frame. Here the theoryreadsSE =∫d4x√−g−M2p2 R + 12 gµν∂µχ∂νχ −U(χ) , U(χ) ≡ (Ω−4V) (φ (χ)) . (4)The canonically normalized scalar field χ is re-lated to φ via12(dχdφ)2=M2p(σM2p + (σ + 3ξ) ξφ2)(M2p + ξφ2)2 . (5)The map from the Jordan frame of f (R) gravityto the Einstein frame with a canonically nor-malized field is obtained for σ = 0 and agreeswith other derivations [13]. The case σ = 1corresponds to a φ with an initial kinetic termin the Jordan frame. Because of this σ depen-dence it is, in principle, possible to disentanglethe two original models. To obtain an explicitrelation between χ and φ we assume that in-flation occurs at large values of the the scalarfield, i.e. φ Mp√ξyieldingχ ' κMp log √ξφMp , κ = √2σξ + 6 . (6)For large values of the non-minimal couplingξ it is not possible to differentiate between thetwo values of σ. The Einstein frame potentialtakes the formU (χ) =λM4pξ2(1 + exp[−2χκMp])−2︸                         ︷︷                         ︸φ4-Inflationξ−2γ exp[4γχκMp]︸             ︷︷             ︸Corrections from γ. (7)The underbraced ’φ4-Inflation’-term refers tothe potential one would obtain by settingγ = 0,that is, non-minimally coupled φ4-Inflation.This limit corresponds to the Starobinskymodel [1].The Starobinsky potential is recovered forα = 0. For 0 < α < 0.5 the potential growsexponentially and, as shown in [12], it leads toa successful inflationary model with nonzeroprimordial tensor modes. On the other hand3FIG. 1: Constraints on the parameter α bydirect comparison with the combined fit ofBICEP2 and Planck data provided in [5].forαnegative the potential is unable to produceenough e-folds. In Fig. 1 generic modificationsof the Starobinsky model are confronted withBICEP2 and PLANCK data. We observe thatcosmology may constrain the deformation pa-rameter α, and as we will show shortly, α holdsinformation regarding the generic particle con-tent embedded in this gravity model of infla-tion.Different extensions of the original Starobin-sky model including, for example, new degreesof freedom, have been investigated [14]. We be-lieve that our approach is a minimal one which,as we shall see, also leads to relevant testablephenomenological implications.We now argue that these marginal deforma-tions, needed from a purely phenomenologi-cal standpoint, arise naturally within a field-theoretical approach to quantum gravity. Togain insight we start by expanding (1) in pow-ers of α and writeSJ '∫d4x√−g−M2p2 R + hR21 − 2α log  RM2p + O(α2) .(8)The logarithmic term is reminiscent of whatone would obtain via trace-log evaluations ofquantum corrections. There are several pos-sible sources for these corrections. They mayarise, for example, by integrating out matterfields, or they can arise directly from gravityloops. To sum-up the entire series of logarith-mic corrections, and hence recover the R2(1−α),we expect that a renormalization group im-proved computation is needed. This stronglysuggests that one can determine α if a morefundamental theory were at our disposal. Inthe absence of a full theory of quantum grav-ity we start here by comparing different pre-dictions for the coefficient of the logarithmicterm in (8) stemming out from: i) Integratingout minimally coupled non-interacting NS realscalar fields [15] (only non–conformal invari-ant matter contributes); ii) gravity correctionsvia the effective field theory (EFT) approach[8–10]; iii) gravity corrections within higherderivative gravity (HDG) [16]. The action (8)naturally arises after a direct computation ofthe quantum corrections stemming from the Trlog ∆ operator. This is deduced via heat kernelmethods [17], where ∆ is the laplacian aris-ing as the Hessian of the minimally coupledscalar action, the Einstein-Hilbert action in theeffective field theory case or the HDG action.The R2 log R/µ2 term emerges upon evaluatingthe operator on S4. For dimensional reasons,further corrections can be parametrized by anh(R/µ2) R2 term, where h is now a function ofR/µ2, with µ the renormalization scale. Theexplicit computations via heat kernel methodsshows [17] that leading order quantum fluctua-tions indeed induce a logarithmic form for h asin (8). This fact alone immediately shows thelink between the exponent α and the coefficientof the beta function of the coupling of the R2term, as a scale derivative with respect to themass scale in (8) shows. But we can give a bet-ter argument noticing that, because h dependson the ratio R/µ2, we have 2R∂Rh = −µ∂µh andone can determine the R dependence once thebeta function, with respect to µ, of h is known.Non–local R2 log(−/µ2) quantum correctionscan also be derived in a similar way [8]. Cos-mological applications of the non-local termshave recently been studied in [18]; other kindsof non-local actions, not generated by quan-tum loops, have also been considered in [19]4and [20]. Here we are interested in the cos-mological role of non-analytical terms whichare present in the effective quantum gravityaction. Our analysis favors the idea that non-analytic terms are more significant for inflationthan non-local ones. To the lowest order thebeta function is µ∂µh = C(4π)2 with C a constantdepending on the source of quantum correc-tions considered. After an RG improvement,the equation for h readsR ∂Rh = −C2(4π)2h . (9)The improvement is related to the appearanceof a factor h(R/µ2) on the right hand-side of theequation above. If one sets h(R/µ2) = 1, on theright-hand side, we only obtain the first loga-rithmic correction of (8). Using (9) we constructthe log–resummed solutionh(R) = h(R0)( RR0)− C2(4π)2. (10)Here R0 = µ20 is a given renormalization scale.We therefore have α = C4(4π)2 and the constanth in (1) is h(R0). If C > 0 this would naturallylead to a positive α. An explicit evaluation ofC gives [15, 16]:C =NS72minimally coupled scalars,C =14EFT gravity, (11)C =536HDG.Remarkably we deduce a positive exponent re-gardless of the underlying theory used to de-termine the associated quantum corrections tothe gravitation action. Massive particles (weconsider scalars of mass m for simplicity) leadto the beta function µ∂µh = C(4π)2 (1 + m2/µ2)−1[16]. When the renormalisation scale is taken tobe Planck’s mass the effect of the mass term isnegligible. Smaller renormalisation scales gen-erally tend to reduce the value of C and thus ofα, but in particular they don’t affect it’s sign.From (11) we deduce that quantum gravita-tional contributions can account, at most, fora 3% increase in r as compared to the originalStarobinsky model. Therefore any larger valueof r can only be generated by adding matter cor-rections. This, in turn, can be used to constrainparticle physics models minimally coupled tof (R) gravity. Furthermore, as it is evident formFig 1, for small r the spectral index (ns) de-pends sensitively on the particular value of α.We can therefore provide the following generalconstraint at the one sigma confidence level onthe number of scalar fields minimally coupledto f (R) gravityNS ≤ 85 . (12)The corresponding r values cannot exceed0.007. To exemplify the power of our resultswe now compare (12) with popular modelsof grand unification (GUT) such as minimalSU(5) that features 34 scalars and (non)minimalSO(10) featuring (297) 109 scalars. It is clearthat only models with a low content of scalarsare preferred by current experiments.Values of r around and above 0.2 can beachieved at two sigma confidence level onlyby allowing for the presence of thousandsof scalars. This corresponds to the upperpart of Fig. 1. Here one might hope to usenon-minimal models of supersymmetric GUTswhich would otherwise be physically excludedwithin the paradigm investigated here.We have pointed out that non-analytic termsare presents in the effective quantum actionand can have a role in cosmology. We have alsochecked that higher order terms of the type R3cannot lead to sizable nonzero tensor modes.Therefore we arrive at the general conclusion,that if inflation is driven by an f (R) theory ofgravity, a natural form for this function is themarginally deformed Starobinsky action pro-vided in (1) with a positive α whose size isrelated to the microscopic theory dictating thetrace-log quantum corrections. This form canbe tested by current and future experimentalresults and constitutes a natural generalizationof the original Starobinsky action.Work supported by the Danish National Re-search Foundation DNRF:90 grant.∗ codello@cp3-origins.net† joergensen@cp3-origins.net‡ sannino@cp3-origins.net§ svendsen@cp3-origins.net[1] A. A. Starobinsky, JETP Lett. 30, 682 (1979)[Pisma Zh. Eksp. Teor. Fiz. 30, 719 (1979)]; idem;Phys. Lett. B 91, 99 (1980).[2] V. F. Mukhanov and G. V. Chibisov, JETP Lett.33, 532 (1981) [Pisma Zh. Eksp. Teor. Fiz. 33, 549(1981)].[3] A. H. Guth, Phys. Rev. D 23, 347 (1981).[4] A. D. Linde, Phys. Lett. B 108, 389 (1982).[5] P. A. R. Ade et al. [BICEP2 Collaboration],arXiv:1403.3985 [astro-ph.CO].5[6] R. Flauger, J. C. Hill and D. N. Spergel,arXiv:1405.7351 [astro-ph.CO].[7] L. M. Krauss and F. Wilczek, Phys. Rev. D 89(2014) 047501 [arXiv:1309.5343 [hep-th]].[8] J. F. Donoghue, Phys. Rev. D 50 (1994) 3874 [gr-qc/9405057].[9] C. P. Burgess, arXiv:0912.4295 [gr-qc].[10] J. F. Donoghue, AIP Conf. Proc. 1483 (2012) 73[arXiv:1209.3511 [gr-qc]].[11] M. Iihoshi, S. V. Ketov and A. Morishita, Prog.Theor. Phys. 118 (2007) 475 [hep-th/0702139].[12] J. Joergensen, F. Sannino and O. Svendsen,arXiv:1403.3289 [hep-ph]. To appear in Phys.Rev. D.[13] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys.82 (2010) 451 [arXiv:0805.1726 [gr-qc]].[14] S. Ferrara, A. Kehagias and A. Riotto,arXiv:1403.5531 [hep-th].[15] A. Codello, New J. Phys. 14 (2012) 015009[arXiv:1108.1908 [gr-qc]].[16] A. Codello, R. Percacci and C. Rahmede, An-nals Phys. 324 (2009) 414 [arXiv:0805.2909 [hep-th]].[17] I. G. Avramidi, Lect. Notes Phys. M 64 (2000) 1.[18] J. F. Donoghue and B. K. El-Menoufi, Phys. Rev.D 89 (2014) 104062 [arXiv:1402.3252 [gr-qc]].[19] S. Deser and R. P. Woodard, Phys. Rev. Lett. 99(2007) 111301 [arXiv:0706.2151 [astro-ph]].[20] M. Maggiore and M. Mancarella, Phys. Rev. D90 (2014) 023005 [arXiv:1402.0448 [hep-th]].

Springer - Publisher Connector

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Published for SISSA by Springer
Received: November 25, 2014
Accepted: January 19, 2015
Published: February 9, 2015
Marginally deformed Starobinsky gravity
Alessandro Codello, Jakob Joergensen, Francesco Sannino and Ole Svendsen
CP3-Origins & the Danish IAS, University of Southern Denmark,
Campusvej 55, DK-5230 Odense M, Denmark
E-mail: codello@cp3-origins.net, joergensen@cp3-origins.net,
sannino@cp3-origins.net, svendsen@cp3-origins.net
Abstract: We show that quantum-induced marginal deformations of the Starobinsky
gravitational action of the form R2(1−α), with R the Ricci scalar and α a positive parameter
smaller than one half, can generate sizable primordial tensor modes. We also suggest
natural microscopic sources of these corrections and demonstrate that they generally lead
to a nonzero and positive α. Furthermore we argue, that within this framework, the scalar
spectral index and tensor modes probe theories of grand unification including theories not
testable at the electroweak scale.
Keywords: Cosmology of Theories beyond the SM, Models of Quantum Gravity
ArXiv ePrint: 1404.3558
Open Access, c© The Authors.
Article funded by SCOAP3.
doi:10.1007/JHEP02(2015)050
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The fundamental origin of the inflationary paradigm is a central problem in
cosmology [1–5]. The simplest models of inflation typically introduce new scalar degrees
of freedom.
An intriguing possibility is that gravity itself is directly responsible for the inflationary
period of the universe. This requires one to go beyond the time-honored Einstein action, for
example by adding a R2-term as in the Starobinsky model [1, 2]. This approach is highly
natural since it enables gravity itself to drive inflation without resorting to the introduction
of new ad hoc scalar fields. The Starobinsky model in isolation predicts a nearly vanishing
ratio of tensor to scalar modes (r), a result challenged by BICEP2 results [6]. However,
independently on the validity of the BICEP2 result [7], it is of fundamental importance
to know the modifications on the Starobinsky model stemming from the embedding of a
matter theory of particle physics in the gravitational theory.
According to [8] cosmology can be used qualitatively to establish the quantization
of gravity. In fact, by combining cosmological observations with an effective field theory
(EFT) treatment of gravity [9, 10] one can start estimating the parameters entering grav-
ity’s effective action. An actual discovery of primordial tensor modes can therefore be used
to determine these parameters at the inflationary scale, which may turn out to be close to
the grand unification energy scale.
To lowest order, the effective action for gravity can be parametrized as
S =
∫
d4x
√−g
[
− M
2
p
2
R+ h0R
2 + h1R
3 + c0C
2 + e0E + . . .
]
,
where Mp is the Planck mass. Beyond an expansion in the Ricci scalar R, we formally
included the Weyl conformal tensor C2 and the Euler four dimensional topological term
E. However we can drop E since it is a total derivative. Furthermore when gravity is
quantized around the Friedmann Lemaitre Robertson Walker metric the Weyl terms are
sub-leading since the geometry is conformally flat [12]. We are left with an f(R) form of the
EFT. Higher powers of R, C2 and E are naturally suppressed by the Planck mass scale. If
inflation occurs at energy scales much below the Planck scale the EFT is accurate. We must,
however, take into account also marginal deformations including, for example, logarithmic
corrections to the action above. Because of the similarity between the EFT description of
gravity and the chiral Lagrangian for Quantum Chromo Dynamics we expect the quantum-
induced logarithmic corrections to play a fundamental role for a coherent understanding of
low energy gravitational dynamics at the inflationary scale. This is exactly what happens
in hadronic processes involving pions at low energies.
We encode these ideas in a simple f(R) form of the gravitational action formulated in
the Jordan frame:
SJ =
∫
d4x
√−g
[
−M
2
p
2
R+ hM4αp R
2(1−α)
]
. (1)
We assume that α is a real parameter with 2|α| < 1 and h is a dimensionless
parameter. We linearize the action via SJ =
∫
d4x
√−g [f(y) + f ′(y)(R− y)] with
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f(R) = −M2p2 R+ hM4αp R2(1−α). The equation of motion for y implies R = y provided
f ′′(y) does not vanish. Introducing the conformal mode ψ = −f ′(y) with V (ψ) =
−y(ψ)ψ − f(y(ψ)) we arrive, after having introduced the mass dimension one real scalar
field φ via 2ψ −M2P = ξφ2 at:
SJ =
∫
d4x
√−g
[
−M
2
p + ξφ
2
2
R− V (φ)
]
, (2)
with V (φ) = λφ4
(
φ
Mp
)4γ
, α = γ/(1 + 2γ) and h1+2γ =
(
ξ
4
1+2γ
1+γ
)2(1+γ)
1
λ(1+2γ) . This pro-
vides the explicit relation between (1) and the effective quantum-corrected non-minimally
coupled scalar field theory used in [13]. Here we can simply set the non-minimal coupling
value of ξ associated to the φ2R term to unity. This is valid since this action is equivalent
to (1), where only the two independent parameters h and α appear. In this way h depends
only on γ and λ. We will retain the explicit dependence on ξ to ease the comparison with
the results obtained in [13].
An important difference with respect to the non-minimally coupled theory of [13] is
the presence, already in the Jordan frame, of a kinetic term for φ. In the f(R) framework
the kinetic term for φ is absent in the Jordan frame but it is, however, generated via the
following conformal transformation of the metric:
gµν → Ω(φ)2gµν , Ω(φ)2 = 1 + ξφ
2
M2p
. (3)
This transformation allows us to rewrite both theories in terms of a propagating scalar
field minimally coupled to ordinary Einstein gravity. This is the Einstein frame. Here the
theory reads
SE =
∫
d4x
√−g
[
−M
2
p
2
R+
1
2
gµν∂µχ∂νχ− U(χ)
]
, U(χ) ≡ (Ω−4V ) (φ (χ)) . (4)
The canonically normalized scalar field χ is related to φ via
1
2
(
dχ
dφ
)2
=
M2p
(
σM2p + (σ + 3ξ) ξφ
2
)(
M2p + ξφ
2
)2 . (5)
The map from the Jordan frame of f(R) gravity to the Einstein frame with a canonically
normalized field is obtained for σ = 0 and agrees with other derivations [14]. The case
σ = 1 corresponds to a φ with an initial kinetic term in the Jordan frame. Because of this
σ dependence it is, in principle, possible to disentangle the two original models. To obtain
an explicit relation between χ and φ we assume that inflation occurs at large values of the
the scalar field, i.e. φ Mp√
ξ
yielding
χ ' κMp log
(√
ξφ
Mp
)
, κ =
√
2σ
ξ
+ 6 . (6)
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Figure 1. Constraints on the parameter α by direct comparison with the combined fit of BICEP2
and Planck data provided in [6].
For large values of the non-minimal coupling ξ it is not possible to differentiate between
the two values of σ. The Einstein frame potential takes the form
U (χ) =
λM4p
ξ2
(
1 + exp
[−2χ
κMp
])−2
︸ ︷︷ ︸
φ4-Inflation
ξ−2γ exp
[
4γχ
κMp
]
︸ ︷︷ ︸
Corrections from γ
. (7)
The underbraced ‘φ4-Inflation’-term refers to the potential one would obtain by setting γ =
0, that is, non-minimally coupled φ4-Inflation. This limit corresponds to the Starobinsky
model [1, 2].
The Starobinsky potential is recovered for α = 0. For 0 < α < 0.5 the potential
grows exponentially and, as shown in [13], it leads to a successful inflationary model with
nonzero primordial tensor modes. On the other hand for α negative the potential is unable
to produce enough e-folds. In figure 1 generic modifications of the Starobinsky model are
confronted with BICEP2 and PLANCK data. We observe that cosmology may constrain
the deformation parameter α, and as we will show shortly, α holds information regarding
the generic particle content embedded in this gravity model of inflation.
Different extensions of the original Starobinsky model including, for example, new
degrees of freedom, have been investigated [15]. We believe that our approach is a minimal
one which, as we shall see, also leads to relevant testable phenomenological implications.
We now argue that these marginal deformations, needed from a purely phenomenolog-
ical standpoint, arise naturally within a field-theoretical approach to quantum gravity. To
gain insight we start by expanding (1) in powers of α and write
SJ '
∫
d4x
√−g
{
−M
2
p
2
R+ hR2
[
1− 2α log
(
R
M2p
)]
+O(α2)
}
. (8)
The logarithmic term is reminiscent of what one would obtain via trace-log evaluations
of quantum corrections. There are several possible sources for these corrections. They
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may arise, for example, by integrating out matter fields, or they can arise directly from
gravity loops. To sum-up the entire series of logarithmic corrections, and hence recover the
R2(1−α), we expect that a renormalization group improved computation is needed. This
strongly suggests that one can determine α if a more fundamental theory were at our
disposal. In the absence of a full theory of quantum gravity we start here by comparing
different predictions for the coefficient of the logarithmic term in (8) stemming out from:
i) Integrating out minimally coupled non-interacting NS real scalar fields [16] (only non-
conformal invariant matter contributes); ii) gravity corrections via the effective field theory
(EFT) approach [9–11]; iii) gravity corrections within higher derivative gravity (HDG) [17].
The action (8) naturally arises after a direct computation of the quantum corrections
stemming from the Tr log ∆ operator. This is deduced via heat kernel methods [18],
where ∆ is the laplacian arising as the Hessian of the minimally coupled scalar action,
the Einstein-Hilbert action in the effective field theory case or the HDG action. The
R2 logR/µ2 term emerges upon evaluating the operator on S4. For dimensional reasons,
further corrections can be parametrized by an h(R/µ2)R2 term, where h is now a function
of R/µ2, with µ the renormalization scale. The explicit computations via heat kernel
methods shows [18] that leading order quantum fluctuations indeed induce a logarithmic
form for h as in (8). This fact alone immediately shows the link between the exponent α and
the coefficient of the beta function of the coupling of the R2 term, as a scale derivative with
respect to the mass scale in (8) shows. But we can give a better argument noticing that,
because h depends on the ratio R/µ2, we have 2R∂Rh = −µ∂µh and one can determine
the R dependence once the beta function, with respect to µ, of h is known. Non-local
R2 log(−/µ2) quantum corrections can also be derived in a similar way [9]. Cosmological
applications of the non-local terms have recently been studied in [19]; other kinds of non-
local actions, not generated by quantum loops, have also been considered in [20] and [21].
Here we are interested in the cosmological role of non-analytical terms which are present
in the effective quantum gravity action. Our analysis favors the idea that non-analytic
terms are more significant for inflation than non-local ones. To the lowest order the beta
function is µ∂µh =
C
(4pi)2
with C a constant depending on the source of quantum corrections
considered. After an RG improvement, the equation for h reads
R∂Rh = − C
2(4pi)2
h . (9)
The improvement is related to the appearance of a factor h(R/µ2) on the right hand-side
of the equation above. If one sets h(R/µ2) = 1, on the right-hand side, we only obtain the
first logarithmic correction of (8). Using (9) we construct the log-resummed solution
h(R) = h(R0)
(
R
R0
)− C
2(4pi)2
. (10)
Here R0 = µ
2
0 is a given renormalization scale. We therefore have α =
C
4(4pi)2
and the
constant h in (1) is h(R0). If C > 0 this would naturally lead to a positive α. An explicit
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evaluation of C gives [16, 17]:
C =
NS
72
minimally coupled scalars,
C =
1
4
EFT gravity, (11)
C =
5
36
HDG.
Remarkably we deduce a positive exponent regardless of the underlying theory used to
determine the associated quantum corrections to the gravitation action. Massive particles
(we consider scalars of mass m for simplicity) lead to the beta function µ∂µh =
C
(4pi)2
(1 +
m2/µ2)−1 [17]. When the renormalisation scale is taken to be Planck’s mass the effect of
the mass term is negligible. Smaller renormalisation scales generally tend to reduce the
value of C and thus of α, but in particular they don’t affect it’s sign.
From (11) we deduce that quantum gravitational contributions can account, at most,
for a 3% increase in r as compared to the original Starobinsky model. Therefore any larger
value of r can only be generated by adding matter corrections. This, in turn, can be used
to constrain particle physics models minimally coupled to f(R) gravity. Furthermore, as
it is evident form figure 1, for small r the spectral index (ns) depends sensitively on the
particular value of α. We can therefore provide the following general constraint at the one
sigma confidence level on the number of scalar fields minimally coupled to f(R) gravity
NS ≤ 85 . (12)
The corresponding r values cannot exceed 0.007. To exemplify the power of our results we
now compare (12) with popular models of grand unification (GUT) such as minimal SU(5)
that features 34 scalars and (non)minimal SO(10) featuring (297) 109 scalars. It is clear
that only models with a low content of scalars are preferred by current experiments.
Values of r around and above 0.2 can be achieved at two sigma confidence level only
by allowing for the presence of thousands of scalars. This corresponds to the upper part of
figure 1. Here one might hope to use non-minimal models of supersymmetric GUTs which
would otherwise be physically excluded within the paradigm investigated here.
We have pointed out that non-analytic terms are presents in the effective quantum
action and can have a role in cosmology. We have also checked that higher order terms of
the type R3 cannot lead to sizable nonzero tensor modes. Therefore we arrive at the general
conclusion, that if inflation is driven by an f(R) theory of gravity, a natural form for this
function is the marginally deformed Starobinsky action provided in (1) with a positive α
whose size is related to the microscopic theory dictating the trace-log quantum corrections.
This form can be tested by current and future experimental results and constitutes a natural
generalization of the original Starobinsky action.
Acknowledgments
Work supported by the Danish National Research Foundation DNRF:90 grant.
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Open Access. This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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       University of Southern DenmarkMarginally Deformed Starobinsky GravityCodello, A.; Joergensen, J.; Sannino, Francesco; Svendsen, O.Published in:Journal of High Energy PhysicsDOI:10.1007/JHEP02(2015)050Publication date:2015Citation for pulished version (APA):Codello, A., Joergensen, J., Sannino, F., & Svendsen, O. (2015). Marginally Deformed Starobinsky Gravity.Journal of High Energy Physics, 2015, Article 50. https://doi.org/10.1007/JHEP02(2015)050Go to publication entry in University of Southern Denmark's Research PortalTerms of useThis work is brought to you by the University of Southern Denmark.Unless otherwise specified it has been shared according to the terms for self-archiving.If no other license is stated, these terms apply:            • You may download this work for personal use only.            • You may not further distribute the material or use it for any profit-making activity or commercial gain            • You may freely distribute the URL identifying this open access versionIf you believe that this document breaches copyright please contact us providing details and we will investigate your claim.Please direct all enquiries to puresupport@bib.sdu.dkDownload date: 22. Oct. 2025Marginally Deformed Starobinsky GravityAlessandro Codello,∗ Jakob Joergensen,† Francesco Sannino,‡ and Ole Svendsen§CP3-Origins & the Danish IAS University of Southern Denmark, Campusvej 55, DK-5230 Odense M, DenmarkWe show that quantum-induced marginal deformations of the Starobinsky gravitational action ofthe form R2(1−α), with R the Ricci scalar and α a positive parameter smaller than one half, can generatesizable primordial tensor modes. We also suggest natural microscopic sources of these correctionsand demonstrate that they generally lead to a nonzero and positive α. Furthermore we argue, thatwithin this framework, the scalar spectral index and tensor modes probe theories of grand unificationincluding theories not testable at the electroweak scale.Preprint: CP3-Origins-2014-13 DNRF90 and DIAS-2014-13The fundamental origin of the inflationaryparadigm is a central problem in cosmology [1–4]. The simplest models of inflation typicallyintroduce new scalar degrees of freedom.An intriguing possibility is that gravity itselfis directly responsible for the inflationary pe-riod of the universe. This requires one to gobeyond the time-honored Einstein action, forexample by adding a R2-term as in the Starobin-sky model [1]. This approach is highly naturalsince it enables gravity itself to drive inflationwithout resorting to the introduction of new adhoc scalar fields. The Starobinsky model in iso-lation predicts a nearly vanishing ratio of ten-sor to scalar modes (r), a result challenged byBICEP2 results [5]. However, independentlyon the validity of the BICEP2 result [6], it isof fundamental importance to know the mod-ifications on the Starobinsky model stemmingfrom the embedding of a matter theory of par-ticle physics in the gravitational theory.According to [7] cosmology can be usedqualitatively to establish the quantization ofgravity. In fact, by combining cosmological ob-servations with an effective field theory (EFT)treatment of gravity [8, 9] one can start esti-mating the parameters entering gravity’s effec-tive action. An actual discovery of primordialtensor modes can therefore be used to deter-mine these parameters at the inflationary scale,which may turn out to be close to the grandunification energy scale.To lowest order, the effective action for grav-ity can be parametrized asS =∫d4x√−g−M2p2 R + h0R2 + h1R3++c0C2 + e0E + ...],where Mp is the Planck mass. Beyond anexpansion in the Ricci scalar R, we formallyincluded the Weyl conformal tensor C2 andthe Euler four dimensional topological termE. However we can drop E since it is a totalderivative. Furthermore when gravity is quan-tized around the Friedmann Lemaitre Robert-son Walker metric the Weyl terms are sub–leading since the geometry is conformally flat[11]. We are left with an f (R) form of theEFT. Higher powers of R, C2 and E are nat-urally suppressed by the Planck mass scale.If inflation occurs at energy scales much be-low the Planck scale the EFT is accurate. Wemust, however, take into account also marginaldeformations including, for example, logarith-mic corrections to the action above. Because ofthe similarity between the EFT description ofgravity and the chiral Lagrangian for QuantumChromo Dynamics we expect the quantum-induced logarithmic corrections to play a fun-damental role for a coherent understanding oflow energy gravitational dynamics at the in-flationary scale. This is exactly what happensin hadronic processes involving pions at lowenergies.We encode these ideas in a simple f (R) formof the gravitational action formulated in theJordan frame:SJ =∫d4x√−g−M2p2 R + hM4αp R2(1−α) .(1)We assume that α is a real parameterwith 2|α| < 1 and h is a dimension-less parameter. We linearize the actionvia SJ =∫d4x√−g[f (y) + f ′(y)(R − y)]withf (R) = −M2p2R + hM4αp R2(1−α). The equation ofmotion for y implies R = y provided f ′′(y) doesnot vanish. Introducing the conformal modeψ = − f ′(y) with V(ψ) = −y(ψ)ψ− f (y(ψ)) we ar-rive, after having introduced the mass dimen-arXiv:1404.3558v3  [hep-ph]  10 Feb 20152sion one real scalar field φ via 2ψ −M2P = ξφ2at:SJ =∫d4x√−g−M2p + ξφ22 R − V(φ) , (2)with V(φ) = λφ4(φMp)4γ, α = γ/(1 + 2γ) andh1+2γ =(ξ41+2γ1+γ)2(1+γ) 1λ(1+2γ) . This providesthe explicit relation between (1) and the ef-fective quantum-corrected non-minimally cou-pled scalar field theory used in [12]. Here wecan simply set the non-minimal coupling valueof ξ associated to the φ2R term to unity. Thisis valid since this action is equivalent to (1),where only the two independent parameters hand α appear. In this way h depends only on γand λ. We will retain the explicit dependenceon ξ to ease the comparison with the resultsobtained in [12].An important difference with respect to thenon-minimally coupled theory of [12] is thepresence, already in the Jordan frame, of a ki-netic term for φ. In the f (R) framework thekinetic term for φ is absent in the Jordan framebut it is, however, generated via the followingconformal transformation of the metric:gµν → Ω(φ)2gµν, Ω(φ)2 = 1 +ξφ2M2p. (3)This transformation allows us to rewrite boththeories in terms of a propagating scalar fieldminimally coupled to ordinary Einstein grav-ity. This is the Einstein frame. Here the theoryreadsSE =∫d4x√−g−M2p2 R + 12 gµν∂µχ∂νχ −U(χ) , U(χ) ≡ (Ω−4V) (φ (χ)) . (4)The canonically normalized scalar field χ is re-lated to φ via12(dχdφ)2=M2p(σM2p + (σ + 3ξ) ξφ2)(M2p + ξφ2)2 . (5)The map from the Jordan frame of f (R) gravityto the Einstein frame with a canonically nor-malized field is obtained for σ = 0 and agreeswith other derivations [13]. The case σ = 1corresponds to a φ with an initial kinetic termin the Jordan frame. Because of this σ depen-dence it is, in principle, possible to disentanglethe two original models. To obtain an explicitrelation between χ and φ we assume that in-flation occurs at large values of the the scalarfield, i.e. φ Mp√ξyieldingχ ' κMp log √ξφMp , κ = √2σξ + 6 . (6)For large values of the non-minimal couplingξ it is not possible to differentiate between thetwo values of σ. The Einstein frame potentialtakes the formU (χ) =λM4pξ2(1 + exp[−2χκMp])−2︸                         ︷︷                         ︸φ4-Inflationξ−2γ exp[4γχκMp]︸             ︷︷             ︸Corrections from γ. (7)The underbraced ’φ4-Inflation’-term refers tothe potential one would obtain by settingγ = 0,that is, non-minimally coupled φ4-Inflation.This limit corresponds to the Starobinskymodel [1].The Starobinsky potential is recovered forα = 0. For 0 < α < 0.5 the potential growsexponentially and, as shown in [12], it leads toa successful inflationary model with nonzeroprimordial tensor modes. On the other hand3FIG. 1: Constraints on the parameter α bydirect comparison with the combined fit ofBICEP2 and Planck data provided in [5].forαnegative the potential is unable to produceenough e-folds. In Fig. 1 generic modificationsof the Starobinsky model are confronted withBICEP2 and PLANCK data. We observe thatcosmology may constrain the deformation pa-rameter α, and as we will show shortly, α holdsinformation regarding the generic particle con-tent embedded in this gravity model of infla-tion.Different extensions of the original Starobin-sky model including, for example, new degreesof freedom, have been investigated [14]. We be-lieve that our approach is a minimal one which,as we shall see, also leads to relevant testablephenomenological implications.We now argue that these marginal deforma-tions, needed from a purely phenomenologi-cal standpoint, arise naturally within a field-theoretical approach to quantum gravity. Togain insight we start by expanding (1) in pow-ers of α and writeSJ '∫d4x√−g−M2p2 R + hR21 − 2α log  RM2p + O(α2) .(8)The logarithmic term is reminiscent of whatone would obtain via trace-log evaluations ofquantum corrections. There are several pos-sible sources for these corrections. They mayarise, for example, by integrating out matterfields, or they can arise directly from gravityloops. To sum-up the entire series of logarith-mic corrections, and hence recover the R2(1−α),we expect that a renormalization group im-proved computation is needed. This stronglysuggests that one can determine α if a morefundamental theory were at our disposal. Inthe absence of a full theory of quantum grav-ity we start here by comparing different pre-dictions for the coefficient of the logarithmicterm in (8) stemming out from: i) Integratingout minimally coupled non-interacting NS realscalar fields [15] (only non–conformal invari-ant matter contributes); ii) gravity correctionsvia the effective field theory (EFT) approach[8–10]; iii) gravity corrections within higherderivative gravity (HDG) [16]. The action (8)naturally arises after a direct computation ofthe quantum corrections stemming from the Trlog ∆ operator. This is deduced via heat kernelmethods [17], where ∆ is the laplacian aris-ing as the Hessian of the minimally coupledscalar action, the Einstein-Hilbert action in theeffective field theory case or the HDG action.The R2 log R/µ2 term emerges upon evaluatingthe operator on S4. For dimensional reasons,further corrections can be parametrized by anh(R/µ2) R2 term, where h is now a function ofR/µ2, with µ the renormalization scale. Theexplicit computations via heat kernel methodsshows [17] that leading order quantum fluctua-tions indeed induce a logarithmic form for h asin (8). This fact alone immediately shows thelink between the exponent α and the coefficientof the beta function of the coupling of the R2term, as a scale derivative with respect to themass scale in (8) shows. But we can give a bet-ter argument noticing that, because h dependson the ratio R/µ2, we have 2R∂Rh = −µ∂µh andone can determine the R dependence once thebeta function, with respect to µ, of h is known.Non–local R2 log(−/µ2) quantum correctionscan also be derived in a similar way [8]. Cos-mological applications of the non-local termshave recently been studied in [18]; other kindsof non-local actions, not generated by quan-tum loops, have also been considered in [19]4and [20]. Here we are interested in the cos-mological role of non-analytical terms whichare present in the effective quantum gravityaction. Our analysis favors the idea that non-analytic terms are more significant for inflationthan non-local ones. To the lowest order thebeta function is µ∂µh = C(4π)2 with C a constantdepending on the source of quantum correc-tions considered. After an RG improvement,the equation for h readsR ∂Rh = −C2(4π)2h . (9)The improvement is related to the appearanceof a factor h(R/µ2) on the right hand-side of theequation above. If one sets h(R/µ2) = 1, on theright-hand side, we only obtain the first loga-rithmic correction of (8). Using (9) we constructthe log–resummed solutionh(R) = h(R0)( RR0)− C2(4π)2. (10)Here R0 = µ20 is a given renormalization scale.We therefore have α = C4(4π)2 and the constanth in (1) is h(R0). If C > 0 this would naturallylead to a positive α. An explicit evaluation ofC gives [15, 16]:C =NS72minimally coupled scalars,C =14EFT gravity, (11)C =536HDG.Remarkably we deduce a positive exponent re-gardless of the underlying theory used to de-termine the associated quantum corrections tothe gravitation action. Massive particles (weconsider scalars of mass m for simplicity) leadto the beta function µ∂µh = C(4π)2 (1 + m2/µ2)−1[16]. When the renormalisation scale is taken tobe Planck’s mass the effect of the mass term isnegligible. Smaller renormalisation scales gen-erally tend to reduce the value of C and thus ofα, but in particular they don’t affect it’s sign.From (11) we deduce that quantum gravita-tional contributions can account, at most, fora 3% increase in r as compared to the originalStarobinsky model. Therefore any larger valueof r can only be generated by adding matter cor-rections. This, in turn, can be used to constrainparticle physics models minimally coupled tof (R) gravity. Furthermore, as it is evident formFig 1, for small r the spectral index (ns) de-pends sensitively on the particular value of α.We can therefore provide the following generalconstraint at the one sigma confidence level onthe number of scalar fields minimally coupledto f (R) gravityNS ≤ 85 . (12)The corresponding r values cannot exceed0.007. To exemplify the power of our resultswe now compare (12) with popular modelsof grand unification (GUT) such as minimalSU(5) that features 34 scalars and (non)minimalSO(10) featuring (297) 109 scalars. It is clearthat only models with a low content of scalarsare preferred by current experiments.Values of r around and above 0.2 can beachieved at two sigma confidence level onlyby allowing for the presence of thousandsof scalars. This corresponds to the upperpart of Fig. 1. Here one might hope to usenon-minimal models of supersymmetric GUTswhich would otherwise be physically excludedwithin the paradigm investigated here.We have pointed out that non-analytic termsare presents in the effective quantum actionand can have a role in cosmology. We have alsochecked that higher order terms of the type R3cannot lead to sizable nonzero tensor modes.Therefore we arrive at the general conclusion,that if inflation is driven by an f (R) theory ofgravity, a natural form for this function is themarginally deformed Starobinsky action pro-vided in (1) with a positive α whose size isrelated to the microscopic theory dictating thetrace-log quantum corrections. This form canbe tested by current and future experimentalresults and constitutes a natural generalizationof the original Starobinsky action.Work supported by the Danish National Re-search Foundation DNRF:90 grant.∗ codello@cp3-origins.net† joergensen@cp3-origins.net‡ sannino@cp3-origins.net§ svendsen@cp3-origins.net[1] A. A. Starobinsky, JETP Lett. 30, 682 (1979)[Pisma Zh. Eksp. Teor. Fiz. 30, 719 (1979)]; idem;Phys. Lett. B 91, 99 (1980).[2] V. F. Mukhanov and G. V. Chibisov, JETP Lett.33, 532 (1981) [Pisma Zh. Eksp. Teor. Fiz. 33, 549(1981)].[3] A. H. Guth, Phys. Rev. D 23, 347 (1981).[4] A. D. Linde, Phys. Lett. B 108, 389 (1982).[5] P. A. R. Ade et al. [BICEP2 Collaboration],arXiv:1403.3985 [astro-ph.CO].5[6] R. Flauger, J. C. Hill and D. N. Spergel,arXiv:1405.7351 [astro-ph.CO].[7] L. M. Krauss and F. Wilczek, Phys. Rev. D 89(2014) 047501 [arXiv:1309.5343 [hep-th]].[8] J. F. Donoghue, Phys. Rev. D 50 (1994) 3874 [gr-qc/9405057].[9] C. P. Burgess, arXiv:0912.4295 [gr-qc].[10] J. F. Donoghue, AIP Conf. Proc. 1483 (2012) 73[arXiv:1209.3511 [gr-qc]].[11] M. Iihoshi, S. V. Ketov and A. Morishita, Prog.Theor. Phys. 118 (2007) 475 [hep-th/0702139].[12] J. Joergensen, F. Sannino and O. Svendsen,arXiv:1403.3289 [hep-ph]. To appear in Phys.Rev. D.[13] T. P. Sotiriou and V. Faraoni, Rev. Mod. Phys.82 (2010) 451 [arXiv:0805.1726 [gr-qc]].[14] S. Ferrara, A. Kehagias and A. Riotto,arXiv:1403.5531 [hep-th].[15] A. Codello, New J. Phys. 14 (2012) 015009[arXiv:1108.1908 [gr-qc]].[16] A. Codello, R. Percacci and C. Rahmede, An-nals Phys. 324 (2009) 414 [arXiv:0805.2909 [hep-th]].[17] I. G. Avramidi, Lect. Notes Phys. M 64 (2000) 1.[18] J. F. Donoghue and B. K. El-Menoufi, Phys. Rev.D 89 (2014) 104062 [arXiv:1402.3252 [gr-qc]].[19] S. Deser and R. P. Woodard, Phys. Rev. Lett. 99(2007) 111301 [arXiv:0706.2151 [astro-ph]].[20] M. Maggiore and M. Mancarella, Phys. Rev. D90 (2014) 023005 [arXiv:1402.0448 [hep-th]].

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Marginally Deformed Starobinsky Gravity

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