Constant-roll inflation: confrontation with recent observational data
CConstant-roll inflation: confrontation with recent observational data
Hayato Motohashi
1, 2 and Alexei A. Starobinsky
2, 3 Instituto de F´ısica Corpuscular (IFIC), Universidad de Valencia-CSIC, E-46980, Valencia, Spain Research Center for the Early Universe (RESCEU),Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan L. D. Landau Institute for Theoretical Physics RAS, Moscow 119334, Russia
The previously proposed class of phenomenological inflationary models in which the assumptionof inflaton slow-roll is replaced by the more general, constant-roll condition is compared with themost recent cosmological observational data, mainly the Planck ones. Models in this two-parametricclass which remain viable appear to be close to the slow-roll ones, and their inflaton potentials areclose to (but still different from) that of the natural inflation model. Permitted regions for the twomodel parameters are presented.
I. INTRODUCTION
The dramatic increase in accuracy of cosmological observational data occurred during the last few years, amongwhich the most important for the inflationary scenario of the early Universe are the value of the slope n s of theFourier power spectrum of primordial scalar (adiabatic matter density) perturbations P ζ ( k ) and the upper value onthe tensor-to-scalar ratio r ( k ), see [1] for the most recent data, gives the possibility to go beyond some of the mainassumptions on which all existing viable inflationary models are based and then to use the data to see to what extentthey do permit it. In particular, in this paper we want to abandon the standard slow-roll approximation for inflatonmotion during inflation which guarantees both smallness of scalar and tensor perturbations generated during inflationand “graceful exit” from it. *1 Of course, this can be done in many ways. We shall study the specific, but naturaland elegant generalization of inflation driven by a slow-rolling scalar field in General Relativity (GR) – constant-rollinflation.This is a two-parametric class of phenomenological inflationary models in GR in which the condition of a constantrate of the inflaton motion, see Eq. (1) below, is satisfied exactly . Then, neglecting spatial curvature and othertypes of matter apart from the inflaton φ itself and using the Hamilton-Jacobi-like equation for H ( φ ) [3, 4] where H = d ln a ( t ) /dt is the Hubble function, a ( t ) is the scale factor of a FRLW universe, it is straightforward to derivethe scalar field potential V ( φ ) needed for the constant-roll condition to be satisfied.The attempt of such a generalization first proposed in [5], and the inflaton potential was constructed so that itsatisfied the constant-roll condition approximately. Later in [6], the generic potential satisfying the constant-rollcondition exactly was found which has a rather simple and elegant form, see Eq. (2) below for one of its three possibletypes which is most interesting from the observational point of view. Thus, it is possible to expect that it mayappear in some more fundamental theory. The potential is novel by itself, though for some of its types and valuesof its parameters, it can be close to that in models studied previously, in particular, power-law inflation and naturalinflation. In addition, it was found in [6] in which cases the constant-roll solution represents an attractor for otherinflationary solutions and when curvature perturbation are approximately constant on super-Hubble scales (that isnot guaranteed generically for non-slow evolution of background due to mixing with the other, ‘decaying’ mode ofthem).The aim of this paper is to move constant-roll inflation closer to reality. Thus, in Sec. II we numerically calculatethe power spectrum of scalar (matter density) perturbations generated during it by using the exact solution forbackground, compare it with the most recent observational data and find a viable region for the two model parameters.Final conclusions are presented in Sec. III. II. CONSTANT-ROLL INFLATION
The constant-roll condition has the form [5, 6] *2 ¨ φ = βH ˙ φ. (1) *1 Note that the slow-roll approximation is not specific for inflation only. In fact, it was first used before the development of the inflationaryscenario, in the context of its main rival – a bouncing universe, in [2] for the V = m φ / *2 Note that β here is related to the parameter α used in [6] through β = − (3 + α ). a r X i v : . [ a s t r o - ph . C O ] M a r So, the slow-roll approximation corresponds to | β | (cid:28)
1. Now we want to determine the range of β permitted byobservations without assuming it to be small in advance. We consider only one of the three possible types of constant-roll potentials found in [6], V ( φ ) = 3 M M (cid:20) − β (cid:26) − cos (cid:18)(cid:112) β φM Pl (cid:19)(cid:27)(cid:21) , (2)since it has been already shown there that two other types (with purely exponential and hyperbolic potentials) arealready excluded by the data. This potential differs from that assumed in the case of natural inflation [7] by anadditional negative cosmological constant. As discussed below, this requires to cut it at some critical value φ = φ toget an exit from inflation (similar to the case of power-law inflation). The corresponding exact solution for a ( t ) , H ( t )and φ ( t ) is φ = 2 (cid:114) β M Pl arctan( e βMt ) , (3) H = − M tanh ( βM t ) = M cos (cid:32)(cid:114) β φM Pl (cid:33) , (4) a ∝ cosh − /β ( βM t ) = sin /β (cid:32)(cid:114) β φM Pl (cid:33) . (5)We consider β (cid:38) φ c where V = 0 is given by φ c = M Pl √ β arccos (cid:18) −
63 + β (cid:19) , (6)which is φ c /M Pl = 14 . . β = 0 .
02 and 0 .
01, respectively. We need to cut the potential before
V < φ should satisfy φ < φ c for a given value of β . For φ (cid:28) φ c the model looks likethe quadratic hilltop inflation with cutoff, whereas for φ (cid:46) φ c the model looks like the natural inflation with theadditional negative cosmological constant Λ = M (3 + α ) as was pointed above.By using the exact solution, we can evaluate a field position φ i , which we set is 55 e-folds back from φ c . We obtain φ i /M Pl = 3 .
38 and 8 .
68 for β = 0 .
02 and 0 .
01, respectively. We can choose the cutoff φ by either φ (cid:28) φ c or φ (cid:46) φ c , but in both cases generation of perturbations of our interest takes place within 0 < φ < φ i . Therefore, wecan investigate both cases at the same time, and the region 0 < φ < φ i includes all possible choices of φ .The potential slow-roll parameters are given by (cid:15) ≡ (cid:18) V (cid:48) V (cid:19) = β (3 + β ) sin ( √ βφ/M Pl )[ − − α + α cos( √ βφ/M Pl )] , (7) η ≡ V (cid:48)(cid:48) V = 2 β (3 + β ) cos( √ βφ/M Pl ) − β − (3 + β ) cos( √ βφ/M Pl ) , (8) ξ ≡ V (cid:48) V (cid:48)(cid:48) V = − β (3 + β ) sin ( √ βφ/M Pl )[ − β − (3 + β ) cos( √ βφ/M Pl )] , (9)which are at most O (10 − ) for 0 . < β < .
025 and 0 < φ < φ i . Therefore, for this parameter region the slow-rollapproximation applies, and the spectral parameters are well approximated by the potential slow-roll parameters as n s − − (cid:15) + 2 η, (10) r = 16 (cid:15), (11) dn s d ln k = 16 (cid:15)η − (cid:15) − ξ, (12)where n s is the spectral index of the scalar power spectrum and r is the ratio between tensor and scalar powerspectrum.We exploit the consistency relation for n s and r with the observational constraint from the joint analysis of Planckand BICEP2/Keck Array, namely, Fig. 7 in [1], and obtain the constraint in Fig. 1 for model parameters ( β, φ i /M Pl ). Φ c N (cid:61) N (cid:61) r (cid:61) (cid:45) Β Φ i (cid:144) M P l FIG. 1. Observational constraint on model parameter space ( β, φ i /M Pl ) of constant-roll inflation. 68% and 95% confidenceregions (blue), φ c (green), field position for 50 e-folds and 60 e-folds back from φ c (yellow), and r = 10 − (purple). Blue curves represent 68% and 95% confidence regions, which are within 0 < φ < φ i . Actually, these regions are morethan 60 e-folds back from φ c . We also plotted a purple curve for r = 10 − ≈ (1 − n s ) for which the hope to reach itin a not so remote future is realistic, see e.g. [8]. III. CONCLUSION
We have investigated most recent observational constraints on parameters of generic constant-roll inflationarymodels, in which the standard assumption of inflaton slow-roll is replaced by the more general, constant-roll condition.The inflaton potential producing such exact background behaviour is given, in the case permitted by observations, bya sinusoidal curve minus a small constant which is close to but still different from that in the natural inflation model.The corresponding exact analytic solution for the background space-time metric and the inflaton field appears to bean attractor for inflationary dynamics. Using the most recent cosmological observational constraints on the spectralindex n s and tensor-to-scalar ratio r obtained in [1], we have found a viable region for the two parameters of themodel presented in Fig. 1. It appears that though we did not assume the parameter β to be small in advance, thedata tell us that it should lie in the range (0 . , .
02) approximately – which is small indeed. However, the exactbackground constant-roll solution is still useful since it provides us with the possibility to calculate small correctionsto the slow-roll result in any order of the perturbation theory in slow-roll parameters and to check the convergence ofthe latter one. As for expected values of the tensor-to-scalar ratio r , the allowed region for them ranges from valuesas large as those permitted by the present upper limits ( r (cid:46) .
07) up to very small ones r (cid:28) (1 − n s ) ≈ − inthe small-field case φ i (cid:28) M Pl . However, for a sufficiently large area of allowed model parameters corresponding tolarge-field inflation with φ i > M Pl , r exceeds 10 − which will be accessible in near-future observations. ACKNOWLEDGMENTS
We thank the Research Center for the Early Universe, where part of this work was completed. H.M. is supportedin part by MINECO Grant SEV-2014-0398, PROMETEO II/2014/050, Spanish Grant FPA2014-57816-P of theMINECO, and European Unions Horizon 2020 research and innovation programme under the Marie Sk(cid:32)lodowska-Curie grant agreements No. 690575 and 674896. A.S. acknowledges RESCEU for hospitality as a visiting professor.He was also partially supported by the grant RFBR 17-02-01008 and by the Scientific Programme P-7 (sub-programme7B) of the Presidium of the Russian Academy of Sciences. [1] P. A. R. Ade et al. (BICEP2, Keck Array), Phys. Rev. Lett. , 031302 (2016), arXiv:1510.09217 [astro-ph.CO].[2] A. A. Starobinsky, Sov.Astron.Lett. , 82 (1978).[3] A. G. Muslimov, Class. Quant. Grav. , 231 (1990).[4] D. S. Salopek and J. R. Bond, Phys. Rev. D42 , 3936 (1990).[5] J. Martin, H. Motohashi, and T. Suyama, Phys. Rev.
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