Starobinsky-type Inflation in Dynamical Supergravity Breaking Scenarios
KKCL-PH-TH/2013- LCTS/2013-26
Starobinsky-type Inflation in Dynamical Supergravity Breaking Scenarios
Jean Alexandre a , Nick Houston a and Nick E. Mavromatos a,b a Theoretical Particle Physics and Cosmology Group, Physics Department,King’s College London, Strand, London WC2R 2LS. b Also currently at: Theory Division, Physics Department, CERN, CH-1211 Geneva 23, Switzerland.
In the context of dynamical breaking of local supersymmetry (supergravity), including the Deser-Zumino super-Higgs effect, for the simple but quite representative cases of N = 1, D = 4 super-gravity, we discuss the emergence of Starobinsky-type inflation, due to quantum corrections in theeffective action arising from integrating out gravitino fields in their massive phase. This type ofinflation may occur after a first-stage small-field inflation that characterises models near the ori-gin of the one-loop effective potential, and it may occur at the non-trivial minima of the latter.Phenomenologically realistic scenarios, compatible with the Planck data, may be expected for theconformal supergravity variants of the basic model. This short article serves as an addendum to our previ-ous publication [1], where we discussed dynamical break-ing of supergravity (SUGRA) theories via gravitino con-densation. In particular, we shall demonstrate the com-patibility of this scenario with Starobinsky-like [2] infla-tionary scenarios, which in our case can characterise themassive gravitino phase. As we shall argue, this is a sec-ond inflationary phase, that may succeed a first inflationwhich occurs in the flat region of the one-loop effectivepotential for the gravitino condensate field [3].Starobinsky inflation is a model for obtaining a de Sit-ter (inflationary) cosmological solution to gravitationalequations arising from a (four space-time-dimensional)action that includes higher curvature terms, specificallyof the type in which the quadratic curvature correctionsconsist only of scalar curvature terms [2] S = 12 κ (cid:90) d x √− g (cid:0) R + β R (cid:1) , β = 8 π M , (1)where κ = 8 πG , and G = 1 /m P is Newton’s (gravita-tional) constant in four space-time dimensions, with m P the Planck mass, and M is a constant of mass dimensionone, characteristic of the model.The important feature of this model is that inflation-ary dynamics are driven by the purely gravitational sec-tor, through the R terms, and the scale of inflation islinked to M . From a microscopic point of view, the scalarcurvature-squared terms in (1) are viewed as the resultof quantum fluctuations (at one-loop level) of conformal(massless or high energy) matter fields of various spins,which have been integrated out in the relevant path in-tegral in a curved background space-time [4]. The quan-tum mechanics of this model, by means of tunneling ofthe Universe from a state of “nothing” to the inflationaryphase of ref. [2] has been discussed in detail in [5]. Theabove considerations necessitate truncation to one-loopquantum order and to curvature-square (four-derivative)terms, which implies that there must be a region of va- lidity for curvature invariants such that O (cid:0) R /m p (cid:1) (cid:28) H I ≤ . × − m P = O (10 ) GeV(the reader should recall that R ∝ H I in the inflationaryphase).Although the inflation in this model is not drivenby fundamental rolling scalar fields, nevertheless themodel (1) (and for that matter, any other model wherethe Einstein-Hilbert space-time Lagrangian density is re-placed by an arbitrary function f ( R ) of the scalar cur-vature) is conformally equivalent to that of an ordinaryEinstein-gravity coupled to a scalar field with a potentialthat drives inflation [8]. To see this, one firstly linearisesthe R terms in (1) by means of an auxiliary (Lagrange-multiplier) field ˜ α ( x ), before rescaling the metric by aconformal transformation and redefining the scalar field(so that the final theory acquires canonically-normalisedEinstein and scalar-field terms): g µν → g Eµν = (1 + 2 β ˜ α ( x )) g µν , (2)˜ α ( x ) → ϕ ( x ) ≡ (cid:114)
32 ln (1 + 2 β ˜ α ( x )) . (3)These steps may be understood schematically via (cid:90) d x √− g (cid:0) R + β R (cid:1) (4) (cid:44) → (cid:90) d x √− g (cid:16) (1 + 2 β ˜ α ( x )) R − β ˜ α ( x ) (cid:17) (cid:44) → (cid:90) d x (cid:112) − g E (cid:0) R E + g E µ ν ∂ µ ϕ ∂ ν ϕ − V (cid:0) ϕ )) , where the arrows have the meaning that the correspond-ing actions appear in the appropriate path integrals, withthe potential V ( ϕ ) given by: V ( ϕ ) = (cid:16) − e − √ ϕ (cid:17) β = 3 M (cid:16) − e − √ ϕ (cid:17) π . (5) a r X i v : . [ g r- q c ] D ec The potential is plotted in fig. 1. We observe that it is V ( ' ) ' FIG. 1. The effective potential (5) of the collective scalar field ϕ that describes the one-loop quantum fluctuations of matterfields, leading to the higher-order scalar curvature correctionsin the Starobinski model for inflation (1). The potential issufficiently flat to ensure slow-roll conditions for inflation aresatisfied, in agreement with the Planck data, for appropriatevalues of the scale 1 /β ∝ M (which sets the overall scale ofinflation in the model). sufficiently flat for large values of ϕ (compared to thePlanck scale) to produce phenomenologically acceptableinflation, with the scalar field ϕ playing the role of theinflaton. In fact the Starobinsky model fits excellentlythe Planck data on inflation [7].Quantum-gravity corrections in the original Starobin-sky model (1) have been considered recently in [11] fromthe point of view of an exact renormalisation-group (RG)analysis [12]. It was shown that the non-perturbative beta-functions for the ‘running’ of Newton’s ‘constant’G and the dimensionless R coupling β − in (1) implyan asymptotically safe Ultraviolet (UV) fixed point forthe former (that is, G( k → ∞ ) → constant, for some 4-momentum cutoff scale k )), in the spirit of Weinberg [13],and an attractive asymptotically-free ( β − ( k → ∞ ) → R coupling, required for agreement with inflationary ob-servables [7], is naturally ensured by the presence of theasymptotically free UV fixed point.The agreement of the model of [2] with the Planckdata triggered an enormous interest in the current litera-ture in revisiting the model from various points of view,such as its connection with no-scale supergravity [9] and(super)conformal versions of supergravity and related ar-eas [10]. In the latter works however the Starobinskiscalar field is fundamental, arising from the appropriatescalar component of some chiral superfield that appearsin the superpotentials of the model. Although of greatvalue, illuminating a strong connection between super-gravity models and inflationary physics, and especiallyfor explaining the low-scale of inflation compared to thePlanck scale, these works contradict the original spirit of the Starobinsky model (1) where, as mentioned previ-ously, the higher curvature corrections are viewed as aris-ing from quantum fluctuations of matter fields in a curvedspace-time background such that inflation is driven bythe pure gravity sector in the absence of fundamentalscalars.In a recent publication [3], we have considered an al-ternative inflationary scenario, in which, in the spirit ofthe original Starobinsky model, the inflaton field was nota fundamental scalar but arose as a result of condensa-tion (in the scalar s -wave channel) of the gravitino fieldin simple supergravity (SUGRA) models with sponta-neous breaking of global supersymmetry (SUSY) via thesuper-Higgs effect [1], at a (mass) scale √ f . Dynamicalbreaking of SUGRA, in the sense of the generation of amass for the gravitino field ψ µ , whilst the gravitons re-main massless, occurs in the model as a result of the four-gravitino interactions characterising the SUGRA action,arising from the torsionful contributions of the spin con-nection, characteristic of local supersymmetric theories.The one-loop effective potential for the scalar gravitinocondensate field σ c ∝ (cid:104) ψ µ ψ µ (cid:105) has a double-well shape asa function of σ c which is symmetric about the origin, asdictated by the fact that the sign of a fermion mass doesnot have physical significance. Dynamical generation ofthe gravitino mass occurs at the non-trivial minima cor-responding to σ c (cid:54) = 0. The potential of the σ c field isalso flat near the origin, and this has been identified in[3] with the inflationary phase.In [1] the one-loop effective potential was derived byfirst formulating the theory on a curved de Sitter back-ground [14], with cosmological constant (one-loop in-duced) Λ >
0, and integrating out spin-2 (graviton)and spin 3/2 (gravitino) quantum fluctuations in a givenclass of gauges ( physical ), before considering the flat limitΛ → physical gauge , has demonstratedthat the dynamically broken phase is then stable (in thesense of the effective action not being characterised byimaginary parts) provided σ c ≤ f . (6)This result demonstrated the importance of the existenceof global SUSY breaking scale for the stability of thephase where dynamical generation of gravitino massesoccurs, which was not considered in the previous litera-ture [15] .The self-consistency of the Λ → Although performed in different gauges to our own, the resultof those references that imaginary parts prevent gravitino massgeneration would also be valid in the case we consider here, wereit not for the super-Higgs effect and the condition (6). Such aconclusion could however not be reached in [15], as the role ofthe super-Higgs effect, and the ‘eating’ of the Goldstino associ-ated with the global SUSY breaking by the gravitino, was notincluded. the non-trivial minima, which is a limiting case consis-tent with the supersymmetry breaking. This restricts thescale of the f and σ c in such a way that both scales mustbe of order of Planck if the simplest four dimensional N = 1 SUGRA model is considered. On the other hand,if one considers superconformal versions of SUGRA, e.g .those in ref. [16], then phenomenologically realistic scalesfor f and σ c of order of the Grand Unification Scale, canappear, for appropriate values of the expectation value ofthe conformal factor, implying inflationary scenarios inperfect agreement with the Planck data [3, 7], on equalfooting to the original Starobinski model. The inflation-ary period in this scenario is obtained by a simple em-bedding of the one-loop effective potential for the grav-itino condensate field in a standard Einstein-backgroundgravity, where higher curvature corrections are ignored,whilst the end of the inflationary period coincides withthe flat space-time limit that characterises the dynami-cal breaking of SUGRA at the non-trivial minima of theone-loop effective potential.In this note we would like to consider an extension ofthe analysis of [1], where the de Sitter parameter Λ is per-turbatively small compared to m P , but not zero, so thattruncation of the series to order Λ suffices. This is inthe spirit of the original Starobinsky model [2], with therole of matter fulfilled by the now-massive gravitino field.Specifically, we are interested in the behaviour of the ef-fective potential near the non-trivial minimum, where σ c is a non-zero constant. In our analysis, unlike Starobin-sky’s original work, we will keep the contributions from both graviton (spin-two) and gravitino quantum fluctua-tions. Notice that our one-loop analysis does not allowus to make any comment on asymptotic safety of the so-lution as in [11], as this would require detailed analysisbased on exact RG which we do not perform here.We firstly note that the one-loop effective potential,obtained by integrating out gravitons and (massive) grav-itino fields in the scalar channel (after appropriate eu-clideanisation), may be expressed as a power series inΛ: Γ (cid:39) S cl − π Λ (cid:0) α F + α B + (cid:0) α F + α B (cid:1) Λ+ (cid:0) α F + α B (cid:1) Λ + . . . (cid:1) , (7)where S cl denotes the classical action with tree-level cos-mological constant Λ (to be contrasted with the one-loop cosmological constant Λ): − κ (cid:90) d x √ g (cid:16) (cid:98) R − (cid:17) , Λ = κ (cid:0) σ − f (cid:1) , (8)with (cid:98) R denoting the fixed S background we expandaround ( (cid:98) R = 4Λ, Volume = 24 π / Λ ), and the α ’s in-dicate the bosonic (graviton) and fermionic (gravitino)quantum corrections at each order in Λ.The leading order term in Λ is then the effective action found in [1] in the limit Λ → Λ → (cid:39) − π Λ (cid:18) − Λ κ + α F + α B (cid:19) ≡ π Λ Λ κ , (9)and the remaining quantum corrections then, propor-tional to Λ and Λ may be identified respectively withEinstein-Hilbert R -type and Starobinsky R -type termsin an effective action (10) of the formΓ (cid:39) − κ (cid:90) d x √ g (cid:16)(cid:16) (cid:98) R − (cid:17) + α (cid:98) R + α (cid:98) R (cid:17) , (10)where we have combined terms of order Λ into curvaturescalar square terms. For general backgrounds such termswould correspond to invariants of the form (cid:98) R µνρσ (cid:98) R µνρσ , (cid:98) R µν (cid:98) R µν and (cid:98) R , which for a de Sitter background allcombine to yield (cid:98) R terms. The coefficients α and α absorb the non-polynomial (logarithmic) in Λ contribu-tions, so that we may then identify (10) with (7) via α = κ (cid:0) α F + α B (cid:1) , α = κ (cid:0) α F + α B (cid:1) . (11)To identify the conditions for phenomenologically ac-ceptable Starobinsky inflation around the non-trivialminima of the broken SUGRA phase of our model, weimpose first the cancelation of the “classical” Einstein-Hilbert space term (cid:98) R by the “cosmological constant”term Λ , i.e. that (cid:98) R = 4 Λ = 2 Λ . This conditionshould be understood as a necessary one characterisingour background in order to produce phenomenologically-acceptable Starobinsky inflation in the broken SUGRAphase following the first inflationary stage, as discussedin [3]. This may naturally be understood as a general-isation of the relation (cid:98) R = 2Λ = 0, imposed in [1] asa self-consistency condition for the dynamical generationof a gravitino mass.The effective Newton’s constant in (10) is then κ = κ /α , and from this, we can express the effectiveStarobinsky scale (1) in terms of κ eff as β eff ≡ α /α .This condition thus makes a direct link between the ac-tion (7) with a Starobinsky type action (1). Comparingwith (1), we may then determine the Starobinsky infla-tionary scale in this case as M = (cid:114) π α α . (12)We may then determine the coefficients α and α inorder to evaluate the scale 1 /β of the effective Starobin-sky potential given in fig. 1 in this case, and thus thescale of the second inflationary phase. To this end, weuse the results of [1], derived via an asymptotic expan-sion as explained in the appendix therein, to obtain thefollowing forms for the coefficients α F = ( − √ ) ˜ κ σ c log (cid:16) Λ µ (cid:17) + ( √ − ) ˜ κ σ c + ( √ − ) ˜ κ σ c log (cid:16) ˜ κ σ c µ (cid:17) ,α F = ( √ − ) log (cid:16) ˜ κ σ c µ (cid:17) + ( −√ ) log (cid:16) Λ µ (cid:17) + √ − , (13)and α B = √ − Λ log (cid:16) Λ3 µ (cid:17) + ( √ − ) Λ log (cid:16) − µ (cid:17) − ( √ − ) (1+log(2))198570 Λ ,α B = − ( √ − ) log (cid:16) Λ3 µ (cid:17) + ( √ − ) log (cid:16) − µ (cid:17) + −√ , (14)where ˜ κ = e −(cid:104) Φ (cid:105) κ is the conformally-rescaled gravita-tional constant in the model of [16] and (cid:104) Φ (cid:105) (cid:54) = 0 is thev.e.v. of the conformal (‘dilaton’) factor, assumed to bestabilised by means of an appropriate potential. In thecase of standard N = 1 SUGRA, (cid:104) Φ (cid:105) = 0. We note atthis stage that the spin-two parts, arising from integrat-ing out graviton quantum fluctuations, are not dominantin the conformal case [1], provided ˜ κ/κ ≥ O (10 ), whichleads [3] to the agreement of the first inflationary phaseof the model with the Planck data [7]. However, if thefirst phase is succeeded by a Starobinsky phase, it is thelatter only that needs to be checked against the data.We search numerically for points in the parameterspace such that; the effective equations ∂ Γ ∂ Λ = 0 , ∂ Γ ∂σ = 0 , (15)are satisfied, Λ is small and positive (0 < Λ < − M ,to ensure the validity of our expansion in Λ) and 10 − < M /M Pl < − , to match with known phenomenologyof Starobinsky inflation [7].For ˜ κ = κ (i.e. for non-conformal supergravity), wewere unable to find any solutions satisfying these con-straints. This of course may not be surprising, giventhe previously demonstrated non-phenomenological suit-ability of this simple model [1]. If we consider ˜ κ >> κ however, we find that we are able to satisfy the aboveconstraints for a range of values. We present this viathe two representative cases below, indicated in figs. 2, 3, where we have the gravitino mass [1] m / = (cid:114)
112 ˜ κσ c , (16) √ f is the scale of global supersymmetry breaking, andwe have set the normalisation scale via κµ = √ π .Every point in the graphs of the figures is selected (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) f (cid:144) M Pl (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) m (cid:144) (cid:144) M Pl FIG. 2. Results for ˜ κ = 10 κ . (cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230)(cid:230) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) f (cid:144) M Pl (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) m (cid:144) (cid:144) M Pl FIG. 3. Results for ˜ κ = 10 κ . to make the Starobinsky scale of order M ∼ − M Pl ,hence we able to achieve phenomenologically acceptableStarobinsky inflation in the massive gravitino phase, con-sistent with the Planck-satellite data [7].Exit from the inflationary phase is a complicated issuewhich we shall not discuss here, aside from the observa-tion that it can be achieved by coherent oscillations ofthe gravitino condensate field around its minima, or tun-nelling processes `a la Vilenkin [5]. We hope to addressthese issues in detail in a future work. ACKNOWLEDGEMENTS
The work of N.H. is supported by a KCL GTA stu-dentship, while that of N.E.M. is supported in part bythe London Centre for Terauniverse Studies (LCTS), us-ing funding from the European Research Council via theAdvanced Investigator Grant 267352 and by STFC (UK)under the research grant ST/J002798/1. [1] J. Alexandre, N. Houston and N. E. Mavromatos, Phys.Rev. D, in press, arXiv:1310.4122 [hep-th].[2] A. A. Starobinsky, Phys. Lett. B , 99 (1980).[3] J. Ellis and N. E. Mavromatos, Phys. Rev. D , 085029(2013) [arXiv:1308.1906 [hep-th]].[4] P. C. W. Davies, S. A. Fulling, S. M. Christensen andT. S. Bunch, Annals Phys. , 108 (1977); T. S. Bunchand P. C. W. Davies, Proc. Roy. Soc. Lond. A ,117 (1978); N. Birrell and P. Davies, Quantum Fields inCurved Space , (Cambridge Monogr.Math.Phys., 1982).[5] A. Vilenkin, Phys. Rev. D , 2511 (1985).[6] J. Martin, C. Ringeval and V. Vennin, arXiv:1303.3787[astro-ph.CO].[7] For Planck constraints on inflationary models,see P. A. R. Ade et al. [Planck Collaboration],arXiv:1303.5082 [astro-ph.CO]; for a general surveyof Planck results including inflation, see P. A. R. Ade et al. [Planck Collaboration], arXiv:1303.5062 [astro-ph.CO].[8] K. S. Stelle, Gen. Rel. Grav. , 353 (1978); B. Whitt,Phys. Lett. B , 176 (1984).[9] J. Ellis, D. V. Nanopoulos and K. A. Olive, Phys.Rev. Lett. , 111301 (2013) [arXiv:1305.1247 [hep-th]];JCAP , 009 (2013) [arXiv:1307.3537 [hep-th]]. [10] K. Nakayama, F. Takahashi and T. T. Yanagida,arXiv:1303.7315 [hep-ph]; arXiv:1305.5099 [hep-ph];R. Kallosh and A. Linde, JCAP , 028 (2013)[arXiv:1306.3214 [hep-th]]; W. Buchmuller, V. Domckeand K. Kamada, arXiv:1306.3471 [hep-th]; M. A. G. Gar-cia and K. A. Olive, arXiv:1306.6119 [hep-ph];F. Farakos, A. Kehagias and A. Riotto, Nucl. Phys. B , 187 (2013) [arXiv:1307.1137 [hep-th]]. D. Roest,M. Scalisi and I. Zavala, arXiv:1307.4343 [hep-th];S. Ferrara, R. Kallosh, A. Linde and M. Porrati,arXiv:1307.7696 [hep-th].[11] E. J. Copeland, C. Rahmede and I. D. Saltas,arXiv:1311.0881 [gr-qc].[12] For reviews see: D. F. Litim, arXiv:0810.3675 [hep-th];M. Reuter and F. Saueressig, New J. Phys. , 055022(2012) [arXiv:1202.2274 [hep-th]].[13] S. Weinberg, (1979), in General Relativity: An Einsteincentenary survey (ed. S. W. Hawking and W. Israel,1979), 790- 831.[14] E. S. Fradkin and A. A. Tseytlin, Nucl. Phys. B (1984) 509.[15] I. L. Buchbinder and S. D. Odintsov, Class. Quant. Grav. (1989) 1955; see also S. D. Odintsov, Phys. Lett. B ,7 (1988).[16] S. Ferrara, R. Kallosh, A. Linde, A. Marrani andA. Van Proeyen, Phys. Rev. D82