Updated Constraints on Large Field Hybrid Inflation
aa r X i v : . [ a s t r o - ph . C O ] S e p Updated Constraints on Large Field Hybrid Inflation
S´ebastien Clesse ∗ and J´er´emy Rekier † Namur Center of Complex Systems (naXys), Department of Mathematics,University of Namur, Rempart de la Vierge 8, 5000 Namur, Belgium (Dated: July 19, 2018)We revisit the status of hybrid inflation in the light of Planck and recent BICEP2 results, takingcare of possible transient violations of the slow-roll conditions as the field passes from the largefield to the vacuum dominated phase. The usual regime where observable scales exit the Hubbleradius in the vacuum dominated phase predicts a blue scalar spectrum, which is ruled out. Butwhereas assuming slow-roll one expects this regime to be generic, by solving the exact dynamics weidentify the parameter space for which the small field phase is naturally avoided due to slow-rollviolations at the end of the large field phase. When the number of e-folds generated at small field isnegligible, the model predictions are degenerated with those of a quadratic potential. There existsalso a transitory case for which the small field phase is sufficiently long to affect importantly theobservable predictions. Interestingly, in this case the spectral index and the tensor to scalar ratioagree respectively with the best fit of Planck and BICEP2. This results in a ∆ χ ≃ . χ ≃ . PACS numbers: 98.80.Cq
I. INTRODUCTION
The inflationary paradigm provides an explanationto the horizon, flatness and monopole problems of thestandard hot Big-Bang cosmological scenario, as wellas a mechanism to generate Gaussian and nearly scale-invariant density perturbations from quantum fluctua-tions of one (or more than one) scalar field(s) duringinflation. Besides theoretical motivations, strong obser-vational evidences are consistent with a primordial phaseof quasi exponentially accelerated expansion. The ampli-tude A s and the spectral index n s of the power spectrumof primordial density perturbations have been measuredwith accuracy by experiments probing the Cosmic Mi-crowave Background (CMB) temperature anisotropies,such as the Planck spacecraft [1, 2], the Atacama Cos-mology Telescope [3] and the South Pole Telescope [4],giving A s = 2 . +0 . − . × − and n s = 0 . ± . f locNL = 2 . ± . , with a tensor to scalar ra- ∗ Electronic address: [email protected] † Electronic address: [email protected] Note that galactic dust could contribute more importantly to thesignal than initially expected, and therefore future observationswill be required to affirm the discovery of primordial gravitationalwaves [7–9] tio r = 0 . +0 . − . that favors super-planckian excursionsof the inflaton field and points towards an energy scaleassociated to inflation close to the Grand-Unified energy.In more than twenty years, hundreds of inflationarymodels and regimes have been proposed (for a recentreview of single-field models, see [10]). Among themthe class of hybrid models is particularly interestingbecause they can be embedded in various high energyframeworks like supersymmetry [11–17] and supergrav-ity [18, 19], Grand-Unified-Theory (GUT) [20–23], extra-dimensions [24, 25] and string theory [26–31]. The com-mon characteristics of hybrid models is that the field po-tential owns a nearly flat valley along which inflationcan occur and that inflation ends with a spontaneoussymmetry breaking when the field potential develops atachyonic instability in the direction of an extra auxil-iary field. During the so-called final waterfall phase theclassical field trajectories evolve towards one of the globalminima of the potential, whereas in a realistic scenario aphase of tachyonic preheating is triggered [32–34] whenthe tachyonic mass becomes larger than the Hubble rate.Usually the waterfall phase is assumed to be nearlyinstantaneous (lasting less than about one e-fold of ex-pansion), but there exists also a generic mild waterfallregime lasting for more than 60 e-folds [35]. In this casethe observable perturbation modes exit the Hubble ra-dius during the waterfall, changing the observable predic-tions of the model [35–40], and topological defects thatare formed at the critical instability point of the poten-tial can be conveniently stretched outside the observableUniverse by the subsequent phase of inflation.In the original version of the hybrid model [41, 42], ob-servable perturbation modes exit the Hubble horizon dur-ing the false-vacuum dominated phase at small field val-ues, which is very efficient to generate many e-folds of ex-pansion. This is translated in the primordial scalar powerspectrum by a slightly blue tilt, ruled out by Planck atmore than 5 σ [1, 2]. In its most well-known supersym-metric realizations, the F-term and D-term hybrid mod-els [11, 12], the scalar spectral index takes values between0 . . n s . II. SINGLE-FIELD DYNAMICSA. Background
Assuming that the Universe was filled by a homoge-neous scalar field φ , the Friedman-Lemaitre and Klein-Gordon equations describe the expansion and scalar fielddynamics, H = 13 M " ˙ φ V ( φ ) , (1)¨ aa = 13 M h − ˙ φ + V ( φ ) i , (2)¨ φ + 3 H ˙ φ + ∂V∂φ = 0 , (3)where a is the scale factor, H the Hubble rate, M pl ≡ m pl / √ π the reduced Planck mass, V ( φ ) is the scalarfield potential and where a dot denotes the derivativewith respect to the cosmic time t . The slow-roll approxi-mation consists in neglecting the kinetic terms in Eqs. (1)and (2) as well as the second time derivatives of the fieldin Eq. (3). Since we are interested by transient slow-rollviolations the exact dynamics have been integrated nu-merically. It is compared to the slow-roll approximationin Sec. III. Using the number of e-fold N ≡ ln a/a i as thetime variable, those equations can be rewritten as H = V ( φ )3 − (cid:16) dφdN (cid:17) , (4)1 H dHdN = − (cid:18) dφdN (cid:19) , (5)13 − (cid:16) dφdN (cid:17) d φdN + dφdN = − d ln Vdφ . (6)In this form, the field dynamics does not depend on theHubble rate. It is then usual to introduce the Hubbleflow functions, also referred as slow-roll parameters, ǫ ≡ − ˙ HH = 12 (cid:18) dφdN (cid:19) ≃ M (cid:18) V ,φ V (cid:19) , (7) ǫ ≡ d ln ǫ d N ≃ M "(cid:18) V ,φ V (cid:19) − V ,φφ V , (8) ǫ i> ≡ d ln | ǫ i − | d N , (9)where the approximate expressions are obtained underthe slow-roll approximation, valid as long as ǫ and ǫ are much smaller than one. B. Linear Perturbations
Measuring the temperature anisotropies and the B-mode polarization of the CMB gives access to the sta-tistical properties of the primordial curvature perturba-tions ζ and tensor perturbations h . These properties areencoded in the n -point correlation functions. The two-point correlation function is the integral of the adimen-sional power spectrum P ( k ) over the logarithm of thewavenumbers. By solving the perturbed Einstein equa-tions at second order in terms of slow-roll parameters,and assuming the initial states to be the Bunch-Davisvacuum, analytical expressions for the scalar and tensorperturbation power spectra can be derived [55]. Expand-ing these spectra around a chosen pivot scale k ∗ (usually k ∗ = 0 .
05 Mpc − ), one gets for scalar perturbations P ζ,h ( k ) = P ζ ,h × (cid:20) a + a ln (cid:18) kk ∗ (cid:19)(cid:21) , (10)with P ζ = H ∗ π M ǫ ∗ , (11)and where the star subscript denotes quantities evaluatedat the time t ∗ when the scale k ∗ exits the Hubble horizon, k ∗ = a ( t ∗ ) H ( t ∗ ). To first order, the coefficients of theexpansion read a (s)0 = 1 − C + 1) ǫ ∗ − Cǫ ∗ + O ( ǫ ) (12) a (s)1 = − ǫ ∗ − ǫ ∗ + O ( ǫ ) (13) a (t)0 = 1 − C + 1) ǫ ∗ + O ( ǫ ) (14) a (t)1 = − ǫ ∗ + O ( ǫ ) , (15)with C ≡ γ E + ln 2 − γ E being the Euler constant. Inthis notation, A s can thus be identified to P a (s)0 .At leading order, the power spectrum of tensor pertur-bations is given by P h = 2 H ∗ π M . (16)which gives a tensor to scalar ratio r ≡ P h P ζ = 16 ǫ ∗ , (17)The scalar spectral index is defined as n s − ≡ d ln P ζ ( k )d ln k . (18) At first order in slow-roll parameters, this gives n s = 1 − ǫ ∗ − ǫ ∗ , (19)and therefore one can relate the shape of the scalar andtensor power spectra to the background dynamics at thetime when the pivot scale exits the Hubble horizon, i.e.about N ∗ ∼
60 e-folds before the end of of inflation. InSec. III, those relations are applied to the hybrid poten-tial to derive the model observable predictions.
C. Reheating
The physical size of the pivot mode as it crosses thehorizon can be written as k ∗ a ∗ = k ∗ a a a end a end a ∗ . (20) k ∗ /a is the physical size of the pivot scale now. The evo-lution after inflation is contained in a /a end . The wholerelation can be conveniently parametrised by k ∗ a ∗ = k ∗ a (cid:18) ρ end ρ γ (cid:19) / R − a end a ∗ . (21)The new parameter R rad is equal to 1 in the case of in-stantaneous reheating after inflation. Otherwise it is re-lated to the mean equation of state parameter ¯ w reh dur-ing the reheating era and to the reheating energy ρ reh through [56]ln R rad = 1 − w reh w reh ) ln (cid:18) ρ reh ρ end (cid:19) . (22)The reheating parameter plays an important role in fixing N ∗ , the number of e-fold realized between t ∗ and theend of inflation. It is then convenient to introduce thereheating parameter R ≡ R rad ρ / /M pl , so that one has k ∗ a ∗ = k ∗ a M pl ρ / γ ! ρ / R − e N ∗ , (23)which makes N ∗ invariant under a rescaling of the scalarfield potential. In Sec. IV the reheating R rad parame-ter has been included within the Markov-Chain-Monte-Carlo analysis in order to derive reheating consistant con-straints on the large field hybrid model. III. HYBRID MODELA. Field Potential
The original two-field hybrid potential reads [41, 42] V ( φ, ψ ) = Λ "(cid:18) − ψ M (cid:19) + φ µ + 2 φ ψ φ M . (24)It owns a nearly flat valley in the direction ψ = 0 alongwhich inflation occurs, with the effective single-field po-tential V ( φ ) = Λ (cid:18) φ µ (cid:19) . (25)When the inflaton reaches the critical value φ c , the po-tential develops a tachyonic instability forcing the fieldsto reach one of the global minima of the potential, at( φ, ψ ) = (0 , ± M ). In the following, we do not assume aspecific high-energy framework and consider the possibil-ity to have inflation at field values larger than the Planckscale.The inflationary valley can be reached from field val-ues exterior to it without any important fine-tuning, asshown in Refs [52, 57–60]. Along the valley, the dynam-ics can be decomposed in two phases: i) at large fieldvalues φ > µ , when the potential is of quadratic form,and ii) at small field values φ < µ when the potential isdominated by the false vacuum term. At the end of in-flation, the waterfall phase takes place and we assume itto be nearly instantaneous throughout the paper, exceptin Sec. III B 4 where the case of a mild waterfall at largefield values is discussed briefly.Finally the parameter Λ fixes the energy scale of in-flation. It only influences the e-fold time N ∗ and has noimpact on the background field dynamics. That makesa 4-dim parameter space, to which will be added stan-dard cosmological parameters and nuisance parametersin Sec. IV.Along the valley, the Hubble-flow parameters in theslow-roll approximation are given by ǫ SR1 = 2 M φ µ µ (cid:16) φ µ (cid:17) , (26a) ǫ SR2 = 4 M (cid:16) − φ µ (cid:17) µ (cid:16) φ µ (cid:17) . (26b)They are represented on Fig. 1 and Fig. 2 for differ-ent values of the µ parameter. In the vacuum domi-nated regime, inflation stops at the critical point φ c be-low which the potential develops the tachyonic instabil-ity. Since ǫ ≪ ǫ <
0, a blue spectrum of scalarperturbation is expected. In the large field regime, onegets ǫ ≃ M /φ and ǫ ≃ M /φ as for the mas-sive potential, and thus the spectral index is red. How-ever, assuming the slow-roll dynamics is valid, the vac-uum dominated phase is so efficient in terms of e-foldsgeneration ( N ≫
60 generically) that observable scalesnecesseraly exit the horizon during this phase. But inthe case where µ . . M pl , the slow-roll conditions arenot satisfied during the transition between the large fieldand the vacuum dominated phase, and it has been shownthat in such a case the kinetic energy acquired by thethe inflaton field prevents inflation from taking place in the small field regime [52]. Another possibility is thatthe critical point is located at large field values. In bothcases, Hubble exit of observable scales occurs in the largefield phase, which generates a red spectrum possibly inagreement with observations. Those regimes are studiedin details in the next section. B. Large Field Regimes
1. Chaotic-like: φ c < µ ≪ φ ∗ and µ < M pl The first considered regime is the one similar to chaoticinflation with a quadratic potential. Inflation terminatesat the end of the large field phase and is not triggeredback afterwards. It is important to emphasize that thispossibility exists only due to the effect of slow-roll viola-tions during and after the transitory phase between largefield and small field values: when the slow-roll is stronglyviolated at the transition close to φ = µ , trajectories gainsufficient velocities to prevent them to reach back theslow-roll attractor at small field values [52]. This effectis not trivial since the slow-roll dynamics is violated atsmall field values whereas the slow-roll conditions are ap-parently satisfied ( ǫ SR1 ≪ ǫ SR2 ≪
1) and one has tointegrate for the exact dynamics to put this in evidence.Assuming slow-roll at small field values, one would ob-tain that the large field phase lies outside the range ofobservable modes. But in reality, instead of stopping atthe critical instability point after an efficient small fieldphase, inflation stops when the first Hubble flow param-eter ǫ reaches unity at the end of the large field phase.Fig. 1 shows the value of ǫ as a function of φ , for tworepresentative values of µ ( µ = 0 . M pl and µ = 0 . M pl )both with and without using the slow-roll approxima-tion. When µ is smaller than some threshold value, ǫ does not decrease below one at small field and inflationis not triggered again, contrarily to what is expected inthe slow-roll approximation with ǫ decreasing down totiny values. Fig. 2 shows the evolution of ǫ in a simi-lar fashion. Here again we find an important differencebetween slow-roll value and exact values. The influenceof µ on this effect is shown on Fig. 3 with the numberof e-fold of expansion after the maximum of ǫ plottedagainst µ . Below µ = µ thr ∼ . M pl , only a reducednumber of e-folds are realized, and for µ . . M pl it ismarginal (lower than unity). In the latter case, the smallfield phase does not affect significantly the observablepredictions, which corresponds to the chaotic-like regimeof hybrid inflation.For the derivation of the observable predictions, theslow-roll approximation can be used up to the pointwhere ǫ = 1, corresponding to a final field value φ end ≃ √ M pl s − √ µ M ! . (27)In the limit µ ≪ M pl , one recovers the expected value fora quadratic potential φ end ≃ √ M pl . Note that the exactvalue can differ significantly from this slow-roll value, butthis only has an non-significant effect on the value of φ ∗ .By integrating the slow-roll equationd φ d N = − M d ln V d φ , (28)one obtains µ M (cid:20) ln φ end φ ∗ + 12 µ ( φ − φ ∗ ) (cid:21) = N ( φ ∗ ) − N end . (29)which can be inverted to get the field value φ ∗ at thetime of Hubble crossing of the pivot scale k ∗ . It is thenstraightforward to calculate the scalar power spectrumamplitude and spectral index as well as the tensor toscalar ratio with the use of Eqs. (19) and (17). In thelimit µ ≪ M pl , one finds φ ∗ ≃ M pl r N ∗ + 12 ≃ . M pl , (30) ǫ ∗ ≃ N ∗ + 1 ≃ . , ǫ ∗ ≃ ǫ ∗ ≃ . , (31) n s ≃ − N ∗ + 1 ≃ . , r ≃ N ∗ + 1 ≃ . . (32)Those values degenerated with the predictions for aquadratic potential are obtained assuming N ∗ = 60 andare in agreement with both Planck and BICEP2 data.The chaotic-like regime corresponds to the bottom leftpart of Figs. 5 and 6 where the scalar spectral index andthe tensor to scalar ratio are represented in the plane(log µ, log φ c ) using the exact background dynamics.
2. Transitory: φ c < µ ∼ φ ∗ and µ ∼ M pl The second considered regime is the transitory casewhere µ is close to the threshold value µ thr below whichslow-roll violations prevent the last 60 e-folds of inflationto occur in the small field phase. As shown on Fig. 3between O (1) and O (60) e-fold can be realized in thevacuum dominated phase (small field values). Neverthe-less, observable scales still exit the Hubble radius dur-ing the large field phase. It results that ǫ ∗ and ǫ ∗ takelarger values than in the previous chaotic-like regime, de-pending on the duration of the vacuum dominated phase.Therefore the scalar spectral index is lowered and canaccommodate the best fit of Planck at n s = 0 . r = 0 . σ confidence level, as-suming N ∗ = 60.On Figs. 5 and 6 the predictions for the spectral indexand the tensor to scalar ratio are displayed, using theexact background dynamics and assuming N ∗ = 60. The ǫ φ/µµ = 0 . M pl µ = 0 . M pl (slow-roll) µ = 0 . M pl µ = 0 . M pl (slow-roll) FIG. 1: Evolution of the first Hubble-flow parameter ǫ asa function of φ/µ , computed by solving the full dynamics orusing the slow-roll approximation. The full dynamics solutiondiffers greatly from the slow-roll solution at small values of µ . The kinetic energy acquired at the transition between thelarge field and the vacuum dominated phase prevents inflationto take place at small field values. -16-14-12-10-8-6-4-2020.0001 0.001 0.01 0.1 1 10 ǫ φ/µµ = 0 . M pl µ = 0 . M pl (slow-roll) µ = 0 . M pl µ = 0 . M pl (slow-roll) FIG. 2: Evolution of the second Hubble-flow parameter ǫ as a function of φ/µ , computed by solving the full dynamicsor using the slow-roll approximation. As in Fig. 1 the fulldynamics differs from the slow-roll approximation for smallvalues of µ , leading to different observable predictions. figure shows how the observable predictions change whenvarying the parameters µ and φ c that fixes the end ofinflation. For φ c ≪ M pl , the spectral index is close tothe best fit of Planck when µ ∼ − M pl , as well as in avery thin band at µ ≃ M pl . Increasing φ c up to φ c ∼ µ ,one gets that the best fit is obtained at 3 M pl < µ < M pl .This is expected since the increase of φ c tends to reducethe number of e-folds generated in the vacuum dominatedphase, larger values of µ thus being necessary to make N e nd − N ǫ m a x µ FIG. 3: Number of e-fold produced after reaching the max-imum of ǫ as a function of µ . Below the threshold value µ thr ∼ . M pl , only a few e-folds are realized at small fieldvalues in contradiction with slow-roll predictions. this phase more efficient.Finally, note that the transition between the transitoryregime and the usual small field regime where all the rele-vant e-folds are realized in the vacuum dominated phase(and predicting a blue scalar power spectrum excludedby observations) is found to be very abrupt.
3. Large critical field value: µ < φ c < φ ∗ In this third regime, the critical instability point φ c below which field trajectories are destabilized is locatedat in the large field phase, so that the conditions µ < φ c and φ end = φ c > √ M pl are satisfied.The slow-roll approximation is valid prior to the crit-ical point, and thus Eq. (29) can be used to derive thecorresponding observable predictions. It can be invertedto find φ ∗ in terms of the the principal branch of theLambert function W ( z ), φ ∗ = µ W (cid:18) φ µ e φ N ∗ µ (cid:19) . (33)In the limit µ ≪ φ c , one has φ ∗ ≃ φ + 4 N ∗ M (34)which gives n s ≃ − M φ + 4 N ∗ M (35)and r ≃ M φ + 4 N ∗ M (36) independent of the value of the parameter µ . Values of φ c larger than the planck mass therefore correspond tospectral index values closer to unity, reducing the agree-ment with observations. On Fig. 4 we have plotted φ ∗ , n s and r as a function of φ end for several values of theparameter µ and assuming N ∗ = 60. Note that thoseanalytical results are in agreement with numerical ones,displayed on Figs. 5 and 6 (right part of the plot).As shown in Fig. 5, values of φ c & M pl are excludedby Planck at more than 2 σ level. Therefore hybrid infla-tion in the regime of large critical field show importantdeviations compared to the observable predictions of thechaotic large field regime, and Planck data constrain thecritical value that must be at maximum of the order ofthe Planck mass.
4. Mild waterfall: φ ∗ < φ c and Mµ & M Finally, it is important to mention that a mild water-fall phase is possible after crossing the instability point.In this case the last 60 e-folds of inflation can be realizedduring the waterfall, as first shown in Ref. [35]. Thispossibility and the resulting modifications of the observ-able predictions have been studied in details in Refs. [35–38, 40] for sub-planckian field values and in Ref. [39] forsuper-planckian fields. Below we briefly comment on thelarge field case.A condition for the waterfall to be mild is that µM & M . In order to calculate the corresponding observablepredictions, one has to solve the multi-field dynamicsboth at the background and linear perturbation levels, orto use of the δN formalism. The latter option has beenimplemented numerically in Ref. [39] and regions wherethe spectral index is in agreement with observations havebeen found. In the generic case where ψ ∗ ≪ M , a largelevel of non-gaussianity can be produced with f NL ≈ M M , (37)which is now ruled out by Planck if the parameter M is lower than the Planck mass. It is nevertheless pos-sible to find parameters for which the spectral index isclose to the Planck best fit and producing a negligibleamount of non-Gaussianity. We did not explore furtherthis regime that requires the implementation of the multi-field dynamics and the δN formalism, and we leave for afuture work the complete statistical analysis of the super-planckian mild waterfall case. IV. MODEL CONSTRAINTS
In this section we derive updated constraints on thehybrid model parameter space by performing a Markov-Chain-Monte-Carlo Bayesian statistical analysis. Forthis purpose we use a modified version of the
COSMOMC Φ end (cid:144) M p Φ * (cid:144) M p Φ end (cid:144) M p n s Φ end (cid:144) M p r FIG. 4: φ ∗ (top) and corresponding n s (central) and r (bot-tom) plotted as a function of φ end = φ c using Eq. 33, for µ = 1 M pl (blue), µ = 5 M pl (red), µ = 10 M pl (yellow) and µ = 15 M pl (green), assuming N ∗ = 60. The horizontal dottedlines in the central and bottom panels represent respectivelythe 2 σ regions of Planck and BICEP2. The dotted line in thetop panel is obtained by using the approximation of Eq. 34. numerical package [61]. In Bayesian inference the poste-rior probability of model parameters λ i given some data D (assuming that the model is the true one) are given byBayes’ theorem p ( λ i | D ) = p ( D | λ i ) π ( λ i ) R d λ i p ( D | λ i ) π ( λ i ) , (38)where π ( λ i ) is the prior probability distribution for theparameter λ i , and where the denominator is a normalisa- tion factor called the Bayesian evidence. For the purposeof constraining model parameters the Bayesian evidencecan be ommitted. A. Priors
The choice of the prior can play a crucial role. In thecase of the hybrid model, there is no a priori informa-tion on how small compared to the Planck scale can bethe position of the critical instability point φ c , the false-vacuum Λ, and the slope along the valley described bythe parameter µ . The magnitude of the reheating pa-rameter R is also not a priori known. As a consequencewe have considered Jeffrey’s priors on these parameters,which is an uniform prior on a logarithmic scale. Notethat an alternative choice of parameter is the scalar fieldmass m = √ Λ /µ , which remains small compared to thePlanck scale. But it is straightforward to derive the pos-terior probability of m (assuming a Jeffrey’s prior) fromΛ and µ posteriors by using importance sampling.Looking at Eq. (11) one sees that Λ and µ both con-tribute to the scalar spectrum amplitude, which is tightlyconstrained by Planck. So there is a high level of degen-eracy between Λ and µ , and the sampling method willloose a lot of efficiency if it probes these two parameters.As noticed in Ref. [5] it is more convenient to replaceΛ by the scalar spectrum amplitude with a logarithmicflat prior. In our analysis, Λ is thus a derived parametertogether with the energy density at the end of inflation ρ end . One could also derive the reheating energy ρ reh as-suming a mean equation of state parameter ¯ w . However,in the case of hybrid inflation, reheating does not pro-ceed with coherent field oscillations but with a phase oftachyonic preheating. Deriving ¯ w therefore requires theuse of lattice simulations, which is beyond the scope ofthis paper.Regarding the ranges of the parameters, µ cannot takearbitrary large values because at some point quantumstochastic effects are expected to dominate over the clas-sical dynamics [62, 63]. But for µ & M pl inflation occursin the vacuum dominated phase and the scalar powerspectrum is blue, which is ruled out by observations. Asa consequence the contribution to the Bayesian evidenceof this part of the parameter space will be negligible.At the opposite, tiny values of µ give the same observa-tional predictions than a quadratic potential and thereis no need to extend the range to values much smallerthan the Planck mass. We chose to probe log µ (fromnow µ and φ c are given in reduced Planck mass units tolighten the notation) within the range ( − , . φ c is much larger than the Planck mass, thespectral index is too low for being acceptable, and thusthe posterior probability is expected to be negligible. Theconsidered range for log φ c is ( − , -1-0.5 0 0.5 1 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 l og m log f c FIG. 5: Contour plot of the spectral index n s in the plane (log φ c , log µ ), using the exact background dynamics and assuming N ∗ = 60. The red dashed contours represent the 2 σ confidence interval for Planck. -1-0.5 0 0.5 1 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 l og m log f c FIG. 6: Contour plot of the tensor to scalar ratio r in the plane (log φ c , log µ ), using the exact background dynamics andassuming N ∗ = 60. The red dashed contours represent the 2 σ confidence interval for BICEP2. A s ) −0.5 0 0.5 1 −8 −4 0 −40 −20 0 l og m l og f c A s ) l n R m −1 −0.5 0 0.5 1−40−30−20−10010 log f c −8 −4 0−40−30−20−10010 FIG. 7: Marginalized one-dimensional and two-dimensional posterior probabilities for the hybrid model parameters (in reducedPlanck mass units) in the large field regime, for Planck+BICEP2. The red contours are the 1 σ and 2 σ regions of confidence.The black contours are the 1 σ and 2 σ regions for Planck only. In the 1D plots, the black/red solid lines show the marginalizedposterior distributions of the parameters respectively for Planck and Planck+BICEP2. The dotted lines represent the meanlikelihoods. have checked that below this value the observable pre-dictions are independent of φ c .For the reheating parameter ln R , we have followedRef. [5] and consider the range (-46,15), fixed so thatit encompasses all the possible reheating histories witha reheating energy that cannot be lower than the BBN scale. For ln As and other cosmological parameters(Ω b h , Ω c h , τ , θ ) we have considered the default boundsin COSMOMC , as well as for the 14 nuisance parameters ofPlanck.0 log m n s −1 −0.5 0 0.5 10.90.9511.05 log f c n s −8 −4 00.90.9511.05 ln R n s −40 −20 00.90.9511.05log m r −1 −0.5 0 0.5 100.10.20.30.40.5 log f c r −8 −4 000.10.20.30.40.5 ln R r −40 −20 000.10.20.30.40.5log m r end −1 −0.5 0 0.5 1−20−15−10 log f c r end −8 −4 0−20−15−10 ln R r end −40 −20 0−20−15−10 FIG. 8: Two-dimensional marginalized posterior probabilities for the hybrid model parameters (in reduced Planck mass units)as well as the derived parameters n s , r and ρ end for Planck+BICEP2. The red contours are the 1 σ and 2 σ regions of confidence.The black contours are the 1 σ and 2 σ regions for Planck only. B. Sampling method
We have modified
COSMOMC so that the primordialpower spectra are calculated using our external codefor integrating the exact homogeneous dynamics andderive the spectral index and tensor to scalar ratio.We thus include in
COSMOMC the additional parameterslog φ c , log µ and ln R , as well as the derived parame-ters n s , r, ρ end and Λ.The parameter space to probe is therefore 22nd dimen-sional. Posterior probabilities of hybrid model parame-ters are marginalized over the 18 other cosmological andnuisance parameters. Because of the high-dimensionalityof the parameter space, the sampling method must be such that the Markov chains must converge rapidly tooptimize the computational time cost. Therefore our ex-ternal code is called only when at least one of the fourmodel parameters is changed. This is of importance whenthe fast-slow and fast dragging options are used instead ofa simple vanilla Metropolis-Hastings algorithm. This effi-cient sampling method has been proposed and describedin Ref. [61]. In a few words, two types of parameters (fastand slow) are considered, depending on how much it iscomputationally expansive to derived the likelihood whenthey are changed. In addition, a fast dragging method isimplemented, which decorrelate some parameters by ro-tating the sampling directions. All together, this makesan important reduction of the computational cost com-pared to a standard Metropolis-Hastings algorithm. Fi-1nally, note that the MCMC temperature has to be ad-justed in order to optimize the sampling rate of the fastand slow parameter spaces. C. Statistical analysis results
The Bayesian analysis has been conducted forPlanck+BAO+BICEP2 data, as well as for Planck+BAOonly. The one-dimensional and two-dimensionalmarginalized posterior probability density distributionsfor our model parameters log µ , log φ c , ln A s andln R are displayed on Fig. 7. Posterior probabilities forthe standard cosmological parameters are identical to thePlanck analysis of a Λ-CDM model with n s , r and ln A s as primordial spectra parameters. This is expected giventhat our code derives n s and r for each set of hybridmodel parameters. The marginalized probabilities for thederived parameters r , n s and ρ end are displayed on Fig. 8We find that marginalized probabilities in the plane(log µ , log φ c ) are consistent with what was expectedfrom Fig. 5 and Fig. 6. The likelihood is higher in the re-gion corresponding to the transitory regime, and the bestfit values given in Tab. I are located in this region. ForPlanck+BICEP2+BAO data, the best fit corresponds toa ∆ χ ≃ . µ = 0 . M pl , φ c = 6 . × − M pl , ln R = −
5. Note however thatthe likelihood is reasonably flat in a rather wide regionof the parameter space, which makes difficult to identifythe best fit value.The 1 σ and 2 σ intervals are reported in Tab. I. ForPlanck+BICEP2+BAO we find that log µ < .
72 at 2 σ level. Above this value, the hybrid model corresponds tothe usual picture of inflation in the vacuum dominatedphase with a blue spectrum. There is no lower bound on φ c . Interestingly when BICEP2 data are included, thereheating parameter is constrained to be ln R > −
34. Atthe same time, the energy density at the end of inflationhas a maximum likelihood at ρ end ∼ × GeV, close tothe GUT scale. The energy scale of inflation lies withinthe range 4 . × GeV < ρ / < . × GeV at 2 σ level.The chaotic regime remains within the 1 σ bound,whereas for the large critical field regime we find thatlog φ c < . σ confidence level. This bound is largerthan what is expected from Fig. 4 with N ∗ = 60, but itis obtained after marginalization over ln R , which allows N ∗ (and corresponding φ ∗ ) to take lower values.Finally we have displayed in Fig. 9 the spectralindex and the tensor to scalar ratio in the plane(log µ, log φ c ) for 3000 points of the Markov chains.This figure illustrates that in the large critical field regimethe spectral index is enhanced whereas the tensor toscalar ratio decreases. It show also a few points at largervalues of µ corresponding to the small field regime thatgenerates a spectral index larger than unity.Those results therefore confirm the analysis of the pre- vious section for a fixed value of N ∗ and give new con-straints on the model parameters. −8 −6 −4 −2 0 2−1012 log f c l og m n s f c l og m r0.1 0.2 0.3 0.4 0.5 FIG. 9: Distribution of 3000 points within the Markov chainsin the plane (log µ, log φ c ). In the upper panel, the colorscale represents the corresponding spectral index value, in thelower panel it represents the corresponding tensor to scalarratio. V. CONCLUSION
In the light of experimental results from Planck andBICEP2 experiments, we have re-analyzed the status ofthe original hybrid model for values of the field above thePlanck scale. Compared to previous analyses [5, 64], wehave included the effect of slow-roll violations betweenthe large field and the vacuum dominated phases. Usingthe exact background dynamics, we have identified threeregimes of interests, with different observable predictions.Then we have performed a Bayesian statistical analysis ofthe model parameter space and derived new constraintson the parameters.A first regime of interest is the
Large critical instabilitypoint regime ( φ c > µ and φ c & M pl ). Inflation ends at2 Parameter Best-fit Mean 1 σ range 2 σ rangelog µ -0.27 -0.18 [ *, 0.045] [*, *]log φ c -2.9 -2.7 [-4.6, 1.4 ] [*, 1.20 ]ln R -8.4 -8.0 [-17, *] [-32,*]log ρ end -10.5 -10 [-11, -9.5] [-11, -8.3] n s r µ -0.24 -0.21 [ *, 0.021] [*, 0.72]log φ c -3.2 -2.5 [-4.0, 1.5 ] [*, 1.5 ]ln R -5.00 -10 [-17, *] [-34, *]log ρ end -10.5 -10 [-11, -8.9] [-11, -8.4] n s r σ and 2 σ intervals forhybrid model parameters (in units of reduced Planck mass),for Planck+BAO (upper part) and Planck+BAO+BICEP2(lower part). An asterisk denotes bounds not better than theprior limits. super-Planckian field values with a nearly instantaneoustachyonic instability, where the potential in the directionof the inflationary valley is dominated by the quadraticterm. The observable predictions differ from the massivesingle field model, with a spectral index closer to unityand a lower tensor to scalar ratio. The regime remainsnevertheless within the 2 σ confidence level of Planck andPlanck+BICEP2 at the condition that φ c . M pl .In the second Chaotic-like regime ( φ c < µ ≪ µ th ≃ . M pl ), the slow-roll is violated at the end of the largefield phase and the field gains sufficient kinetic energy tooverpass the vacuum dominated phase without reachingback the slow-roll attractor. This non-trivial effect im-plies that the last 60 e-folds of inflation are realized inthe large field phase, where the potential is dominatedby the quadratic term. Therefore the observable pre-dictions cannot be distinguished from the massive singlefield model. The best statistical agreement with experimental datais found in the third Transitory regime ( φ c < µ . µ th ).In this case, due to transient slow-roll violations, severale-folds (typical between one and ten) are produced inthe vacuum dominated phase but observable modes stillleave the Hubble radius at large field values. Comparedto a massive single field model, the spectral index takeslower values and the tensor to scalar ratio is enhanced.The best-fit to Planck data is found to be µ ∼ . M pl .The statistical analysis predicts a ∆ χ ≃ . χ ≃ . transitory regime parameters predicting a spectral index close toPlanck best fit generate simultaneously a tensor to scalarratio close to the central value observed by BICEP2. Fi-nally, note that future experiments such as COrE [65] orPIXIE [66] will have the ability to distinguish betweenthe three regimes identified above. The transitory regimecould also lead to an observable level of CMB distor-tions [67]. Acknowledgements
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