Reheating and Post-Inflationary Production of Dark Matter

PHYSICAL REVIEW D 101, 123507 (2020)

Reheating and post-inflationary production of dark matter

† ‡ Marcos A. G. Garcia ,1,* Kunio Kaneta,2, Yann Mambrini,3, and Keith A. Olive2,§ 1Instituto de Física Teórica (IFT) UAM-CSIC, Campus de Cantoblanco, 28049 Madrid, Spain 2William I. Fine Theoretical Physics Institute, School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA 3Universit´e Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay, France

(Received 27 April 2020; accepted 22 May 2020; published 8 June 2020)

We perform a systematic analysis of dark matter production during postinflationary reheating. Following the period of exponential expansion, the inflaton begins a period of damped oscillations as it decays. These oscillations and the evolution of temperature of the thermalized decay products depend on the shape of the inflaton potential VðΦÞ. We consider potentials of the form, Φk. Standard matter-dominated oscillations occur for k ¼ 2. In general, the production of dark matter may depend on either (or both) the maximum temperature after inflation, or the reheating temperature, where the latter is defined when the Universe becomes radiation dominated. We show that dark matter production is sensitive to the inflaton potential and depends heavily on the maximum temperature when k>2. We also consider the production of dark matter with masses larger than the reheating temperature.

DOI: 10.1103/PhysRevD.101.123507

I. INTRODUCTION the price of complexifying the model by introducing new physics above ≃3 TeV. In this sense, the “WIMP Since the first computation indicating the presence of a miracle” is not as miraculous as it was believed to be dark component in our Galaxy by Poincar´e in 1906 [1], there were observations of the Coma cluster by Zwicky [2] in the first place. Even if better motivated, the minimal in 1933 and the analysis of the Andromeda rotation curve supersymmetric standard model [18,19] has a large – by Babcock in 1935 [3], leading to the proposition of a region of its parameter space [20 23] in tension with – microscopic dark component by Steigman et al. in 1978 LHC results [24 27]. [4]. However, despite technological developments, and an In this context, it becomes important to look for alter- increase in the size of new generations of experiments on natives. The WIMP miracle is based on the hypothesis of a every continent, not a single dark matter (DM) particle has dark matter particle in thermal equilibrium with the been observed in direct detection experiments [5–7]. The Standard Model over a period of time in the early WIMP (weakly interacting massive particle) paradigm Universe. The dark matter relic density is then independent appears to be in tension with observations (see [8] for a of initial conditions and is determined by the freeze-out of recent review). Classic WIMP candidates such 100 GeV annihilations [28,29]. Relaxing this hypothesis opens up neutral particles with standard weak interactions have interesting cosmological scenarios and potentially new elastic cross sections which are over 6 orders of magnitude candidates. The popular Feebly Interacting Massive larger than current direct detection limits. Indirect detection Particle (FIMP) [30,31] paradigm is one of them. The has been equally unsuccessful. visible and dark sectors can be secluded because of the There are many “minimal” extensions of the Standard smallness of their couplings, even Planck suppressed as in – Model (SM) such as the Higgs portal [9–14] or Z0 portals the case of the gravitino [19,32 37]. Another possibility is [15–17] that can still evade experimental constraints, but at that the two sectors communicate only through the exchange of very massive fields, that may be more massive than the reheating temperature. This is the case in unified *[email protected] 0 † SO(10) scenarios [38,39] or anomaly-free Uð1Þ construc- [email protected] ‡ [email protected] tions [40]. It is also possible that both a tiny coupling and a §[email protected] heavy mediator seclude the visible and dark sectors, as in high-scale supergravity [41–45], massive spin-2 portal Published by the American Physical Society under the terms of [46], or moduli-portal dark matter [47] models. It is easy the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to to understand that mass-suppressed interactions (either a the author(s) and the published article’s title, journal citation, Planck-suppressed coupling or the exchange of a heavy and DOI. Funded by SCOAP3. mediator) generate production rates that are highly

2470-0010=2020=101(12)=123507(15) 123507-1 Published by the American Physical Society GARCIA, KANETA, MAMBRINI, and OLIVE PHYS. REV. D 101, 123507 (2020) dependent on the energy of the primordial plasma. It is the coupling of the inflaton to other fields. Clearly, some crucial, therefore, to treat the interactions in the early coupling to Standard Model fields is necessary to produce a Universe with great care, especially if one wants to take into thermal bath. The inflaton may also couple directly to a account noninstantaneous reheating [42,48–51] or thermal- dark sector, or dark matter may be produced out of the ization [52–56] after inflation. thermal bath. Depending on the coupling of the dark matter Typically, after the period of exponential expansion with the Standard Model, the dark matter may or may not has ended, the reheating process takes place in a matter- ever come into thermal equilibrium. The reheating process dominated background of inflaton oscillations. As the itself may be disassociated from the period of inflation. inflaton begins to decay, the decay products begin to That is, the part of the potential that drives inflation (the thermalize and the temperature of this dilute plasma climbs exponential expansion) may be distinct from the part of the – quickly to a maximum temperature, Tmax [42,48 50]. potential which leads to a slow reheating process in which −3 Subsequently, the temperature falls as T ∝ a 8, where a energy stored in scalar field oscillations is converted to the is the cosmological scale factor, until the Universe becomes thermal bath. dominated by the radiation products at Treh. If the dark In this paper, we will indeed separate the inflationary era matter production cross section scales as Tn, the dark matter from reheating. As an example of this type of model, we 6 density is determined by Treh for n< and is sensitive to consider T-attractor models [58] (described in more detail ≥ 6 Tmax for n . below). In these models, the inflationary part of the In the reheating scenario described above, it commonly potential is nearly flat as in the Starobinsky model [59]. assumed that the inflaton undergoes classic harmonic However, there is considerable freedom for the shape of the oscillations about a minimum produced by a quadratic potential about the minimum. If inflaton decay is suffi- potential. If, however, the oscillations are anharmonic, and ciently slow, the details of reheating and particle production result from a potential other than a quadratic potential, the depend on the potential which controls the oscillatory equation of state during reheating will differ from that of a behavior of the inflaton and the equation of state during matter-dominated background and will affect the evolution reheating. of the thermalization process [57]. We start with the energy density and pressure of a scalar In this paper, we consider, the effect of oscillations λ k field which can be extracted from the stress-energy tensor, produced by a potential of the form VðΦÞ¼ −4 jΦj . Mk Tμν, yielding the standard expressions These oscillations alter the equation of state during reheating and affect the evolution of temperature as the Universe 1 1 ≠ 2 ρ ¼ Φ_ 2 þ ðΦÞ; ¼ Φ_ 2 − ðΦÞ ð Þ expands. It is important to note that for k , the mass of the Φ 2 V PΦ 2 V ; 1 inflaton is not constant, and hence the change in the equation of state also affects the inflaton decay width, and as a where we have neglected contributions from spatial gra- consequence, the evolution of the temperature of the pri- dients. Conservation of Tμν leads to mordial plasma. We will show that the resulting dark matter 2 abundance has increased sensitivity to Tmax when k> . ρ_Φ þ 3HðρΦ þ PΦÞ¼0; ð2Þ It is also possible to produce dark matter with masses in excess of the reheating temperature (so long as its mass is ¼ a_ where H a is the Hubble parameter. Inserting Eq. (1) into less than Tmax). As the temperature decreases from ¼ Eq. (2), we obtain the equation of motion for the inflaton T Tmax, dark matter particles are produced until reheating is complete. However, if the dark matter mass is ̈ _ 0 ≃ Φ þ 3HΦ þ V ðΦÞ¼0; ð3Þ mDM >Treh, production ends at T mDM and the dark matter abundance is suppressed. 0ðΦÞ¼∂ ðΦÞ The paper is organized as follows. In Sec. II,we where V ΦV . generalize the reheating process in the case of an inflaton As noted above, we will assume a generic power-law potential VðΦÞ ∝ Φk, analyzing in detail its consequences form for the potential about the minimum in noninstantaneous reheating. In Sec. III, we apply our jΦjk results to the computation of dark matter production from VðΦÞ¼λ : ð4Þ thermal bath scattering and inflaton decay. We consider Mk−4 dark matter masses below and above the reheating temper- Here, M is some high energy mass scale, which we can ature. We present our conclusions in Sec. IV. 1 take, without loss of generality, to be the Planck scale, MP. II. THE REHEATING PROCESS This form of the potential can be thought of as the small field limit of T-attractor models [58] and can be derived in A. The context 1 18 The process of reheating is necessarily model dependent. We will use throughout our work MP ¼ 2.4 × 10 GeV for It will depend not only on the inflaton potential, but also on the reduced Planck mass.

123507-2 REHEATING AND POST-INFLATIONARY PRODUCTION OF … PHYS. REV. D 101, 123507 (2020) no-scale supergravity [60–62]. The full potential exhibits friction and its decay into light particles (radiation). The Starobinsky-like inflation [59] for values of Φ >MP. More evolution of the energy density of this radiation, ρR, and details are given in the Appendix. Note that we use the thus of the instantaneous temperature3 T, as a function of T-attractor model as a UV-derivable example, but our time (or the scale factor a) is determined by the solution of analysis does not depend at all on the specifics of the the following set of Boltzmann-Friedmann equations: example. The value of λ can be fixed from the normalization of CMB anisotropies. Upon exiting from the infla- ρ_R þ 4HρR ¼ ΓΦρΦ; ð9Þ tionary stage, the inflaton will begin oscillations about the Φ ¼ 0 2 ρΦ þ ρ ρΦ minimum at . H2 ¼ R ≃ ; ð Þ 3 2 3 2 10 During the period of inflaton oscillations, the equation of MP MP state parameter, w ¼ PΦ=ρΦ, also oscillates taking values between −1 when Φ is at its maximum to þ1 when Φ ¼ 0. in addition to Eq. (8). The approximate equality in (10) It is useful, therefore, to compute an averaged equation of applies to a universe dominated by the inflaton field ρ ≫ ρ state, given by hPΦi¼whρΦi (see [63] for more details). ( Φ R), as is true in the early stages of reheating. Multiplying Eq. (3) by Φ and taking the mean over one Although we will make use of this approximation in our oscillation, we obtain analytical computations, we do not impose it in our numerical analysis. The key aspect of our treatment of hΦΦ̈ iþhΦV0ðΦÞi ¼ 0; ⇒ hΦ_ 2i¼hΦV0ðΦÞi: ð5Þ reheating consists in the realization that, for k ≠ 2, the inflaton decay rate is not constant in time. One deduces from Eq. (1) Assuming an effective coupling of the inflaton to Standard Model fermions f of the form yΦff¯ , we can 4 k hΦki write hρΦi¼ þ 1 λ ; 2 Mk−4 P y2 k Γ ¼ ð Þ ð Þ k hΦ i Φ 8π mΦ t ; 11 hPΦi¼ − 1 λ ; ð6Þ 2 k−4 MP where the effective mass mΦðtÞ is a function of time (and so that thus of the temperature of the thermal bath). In the adiabatic approximation,5 it can be written as hPΦi k − 2 w ¼ ¼ : ð7Þ 2 2 hρϕi k þ 2 mΦ ≡∂ΦVðΦÞ ¼ λkðk − 1ÞΦk−2M4−k If we allow the possibility for the inflaton to decay with a P 2ð4− Þ Γ k 2 k−2 width Φ, we can then rewrite Eq. (2) as ¼ ð − 1Þ k λkρ k ð Þ k k MP Φ : 12 2 ρ_ þ 3 k ρ ¼ −Γ ρ ð Þ To arrive at the expression for the effective mass in terms of Φ þ 2 H Φ Φ Φ: 8 k ρΦ, we are using an “envelope” approximation for ρΦ. This approximation is defined in the following way: one may Note that while we use the average equation of state (7) in approximate the oscillating inflaton as ΦðtÞ ≃ Φ0ðtÞ · PðtÞ. the evolution of the energy density, it is sufficient (and The function P is periodic and encodes the (an)harmonicity simpler) to use the energy density given entirely from the of the short time scale oscillations in the potential, while the potential. That is using the amplitude or envelope of the envelope Φ0ðtÞ encodes the effect of redshift and decay, oscillations. and varies on longer time scales. The instantaneous value Before looking at the detailed production of dark matter of Φ0 satisfies the equation [64] in a universe dominated by the density of energy ρΦ, it will be useful to discuss the process of reheating in the case of a 3 ðΦÞ¼λjΦjk k−4 Throughout this paper, we assume that the decay products of generic potential V =MP . the inflaton thermalize instantaneously after they are produced. 4A more careful analysis reveals that the decay rate of Φ, B. The process of reheating obtained by averaging over one oscillation the damping rate of the energy density of the oscillating inflaton condensate, corrects After inflation ends, the inflaton undergoes a damped this expression by an Oð1Þ factor, weakly dependent on k [64,65]. (anharmonic) oscillation about its minimum, due to Hubble We omit it from our analysis for simplicity, though it is included in our numerical results. Our main conclusions are unaffected by this omission. 2The absolute value in (4) is necessary only for k ¼ 3 5We will not consider the violations of adiabaticity that occur preventing this case from being derived from the supergravity at the time scale of the oscillation of the inflaton, which is much models discussed in the Appendix. shorter than the duration of reheating.

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VðΦ0ðtÞÞ ¼ ρΦðtÞ: ð13Þ

Using the envelope is advantageous because one can then immediately ignore short time scales in the analysis, in particular for the effective mass. In order to study any (particle production) process during reheating, it is indispensable to know the value of the temperature of the radiation bath at any moment of time (or scale factor a). At early times, when the Universe is dominated by inflaton oscillations, we find the solution ρΦ ¼ ρΦðaÞ from Eq. (8) and subsequently implement it in Eq. (9) to determine the evolution ρR ¼ ρRðaÞ and therefore T ¼ TðaÞ. In the early stages of the reheating, the decay rate of the FIG. 1. Scale-factor dependence of the instantaneous temper- inflaton is much smaller than the expansion rate H ature during reheating for selected values of k with y ¼ 10−5. The (ΓΦ ≪ H). The right-hand side of Eq. (8) can then be scale factor is rescaled by its value at the end of inflation. The star ρ ¼ ρ neglected, and straightforward integration then gives signals inflaton-radiation equality, Φ R, corresponding to T ¼ T . reh − 6k ρ ð Þ¼ρ a kþ2 ð Þ Φ a end ; 14 1 30ρ ð Þ 4 aend ð Þ¼ R a ð Þ T a 2 ; 16 π g ρ where end and aend denote the energy density and scale factor at the end of inflation, respectively. While the latter where g denotes the effective number of relativistic (aend) is simply a reference point for the scale factor, the degrees of freedom. Note that for a ≫ a , ρ end value of end does enter into our physical results. It is −3k−3 defined as the energy density at the moment when the slow T ∝ a 2kþ4: ð17Þ roll parameter, ϵ ¼ 2M2 ðH0ðΦÞ2=HðΦÞ2Þ¼1 or when P −3 ¼ −1 3 ρ ¼ 3 ðΦ Þ For k ¼ 2, we recover the well-known T ∝ a 8 for the w = [63]. At that moment, end 2 V end and clearly depends on the potential. In the Appendix, we redshift of the temperature during dustlike reheating Φ [48–50]. compute end for the T-attractor model [58] as a function of k. Note that for larger k, the temperature has a steeper −3 For k ¼ 2, we recover the classical evolution of a dust- dependence on the scale factor, e.g., T ∝ a 4 for radiation- −3 ¼ 4 dominated universe (ρΦ ∝ a ), whereas for k ¼ 4 we are like reheating with k . Indeed, in this case, the energy −4 in the presence of a “radiationlike inflaton”-dominated density of Φ, ρΦ, redshifts as a (14), faster than for a −4 ρ ∝ −3 universe (ρΦ ∝ a ). This difference in behavior will have dustlike inflaton where Φ a . This is to be expected as dramatic consequences on the temperature evolution and for k ¼ 4, Φ is massless at the minimum and evolves as the production of dark matter. Substitution of the decay rate radiation. Subsequently, this radiation will be further (11) and the effective mass (12) into (9), together with the redshifted by expansion. The temperature in the bath is, þ solution for ρΦ (14), we obtain in a sense, doubly redshifted (production expansion) compared to a dustlike inflaton decay. Figure 1 exemplifies 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi y 1 4 k þ 2 this steeper redshift of the temperature during reheating for ρ ðaÞ¼ 3kðk − 1ÞλkMk ¼ 3 ¼ 2 R 8π P 14 − 2k k and 4, compared to k . As a consequence, for k ¼ 4, the Universe begins to be dominated by the radiation 4 14−2k k−1 a a kþ2 ρ k end − 1 ð Þ when its scale factor is 3 orders of magnitude larger than × end : 15 a aend it would be for a dustlike inflaton (k ¼ 2). The effect is anything but an anecdote, as it corresponds to more than Note that the dependence of ρ on a found here is very R 5 orders of magnitude of difference in Treh. This comes −3k=ðkþ2Þ different from that in [57], where ρR ∼ a compared from the fact that the decay width of the inflaton, ð6−6kÞ=ðkþ2Þ with ρR ∼ a in Eq. (15), though the two expres- proportional to mΦðtÞ, decreases with time for k ¼ 4. sions agree for k ¼ 2. This is presumably because the We will explain this phenomena in detail in a dedicated decay width was held fixed in [57], whereas for k>2,any section, below. We note that for k ¼ 4 and particularly width proportional to the inflaton mass will vary with its for low y, the reheating temperature may drop so low as evolution. to be problematic with baryogenesis, and perhaps nucleo- In thermal equilibrium, the temperature of the inflaton synthesis. We comment further on this possibility in the decay products will be simply given by Appendix.

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C. The maximum and reheating temperatures Note that this definition of the reheating temperature is Inflation leads to a cold, empty universe that is repopu- slightly different from the more common definition used in lated during reheating. In the instantaneous thermalization instantaneous reheating where the moment of reheating is Γ ¼ 3 approximation, the temperature of the radiation plasma defined to be 2 H, which is sensible for exponential 2 4 initially grows until it reaches a maximum temperature decay but loses meaning for k> .TheratioofTreh from Γ¼3 ð25k−6 32k−2Þðð7− Þ Tmax, after which it decreases to the temperature at the end 2H to the expression in Eq. (21) is = k = ð þ2ÞÞk ¼ 2 of reheating, Treh, and below (see Fig. 1). k .Fork ,thismeansthatpffiffiffiffiffiffiffiffi Treh from Eq. (21) is In order to calculate the maximum temperature, one must smaller by a factor of 3=5 relative to that used in first compute the scale factor amax for which the temper- instantaneous reheating. The definition used in this paper ature (16) or the energy density (15) is maximized. From is better suited for generic k. this procedure, we obtain that We show in Fig. 1 the points (marked by stars) where the Universe begins to be dominated by radiation (ρΦ ¼ ρ ). kþ2 R 2 þ 4 14−2 ¼ k k ð Þ Note that the steeper scale-factor dependence leads to a amax aend : 18 3k − 3 lower Treh for larger k for a given inflaton-matter coupling. To be more precise, this comes from the fact that reheating This in turn implies that is delayed for larger values of k, delay which implies lower values of ρΦ (and as a consequence ρR) at reheating. 3ðk−1Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 −1 4 15y 1 k 3k − 3 7−k The decay width, given in Eq. (11), is proportional to ¼ 3 ð − 1Þλk k ρ k ð Þ Tmax 3 k k MP end : 19 mΦðtÞ. While mΦðtÞ is constant for k ¼ 2, it decreases 16gπ 2k þ 4 with time for k>2. The smaller decay rate causes the delay in reheating and thus results in a lower temperature, To compute Tmax, we must specify the potential to ρ T . On the other hand, the maximum temperature is determine end which is discussed in more detail for the reh T-attractor model [58] in the Appendix. The value of λ is set only weakly dependent on k and smaller only by a factor ∼0 9 ¼ 4 ¼ 2 from the normalization of CMB anisotropies and also of . for k relative to the value at k , ≃ 2 3 1012 depends on k as further discussed in the Appendix. For Tmax . × GeV. This is because the k dependence ρ the case of the T-attractor model, the monomial in Eq. (4) is in end nearly cancels the explicit k dependence in Eq. (19) a good approximation and numerically we find very similar and the implicit dependence in λ. ¼ 2 In the following section, which contains our computation values for Tmax for all three cases, k , 3, and 4 (particularly when plotted on a log scale as in Fig. 1). of the dark matter abundance, the ratio Tmax=Treh will play To compute the reheating temperature, one must first a key role. This ratio is shown in Fig. 2 as a function of the define what signals the end of reheating. We consider that Yukawa coupling y for k ¼ 2, 3, 4. We show both our reheating ends when the radiation density begins to analytic solution for the ratio given in Eqs. (22)–(24) below 6 dominate over the inflaton density, i.e., when ρΦ ¼ ρ . (dashed) and full numerical result (solid). As one can see, R 7 From Eqs. (14) and (15), we find the following approxi- the numerical analysis is in perfect agreement with the 1−k ∝ 2 mation for the scale factor at equality: analytical solutions, i.e., Tmax=Treh y . For the selected values of k, the following simple 3k k=2 1=2 a kþ2 16πð7 − kÞ ρ expressions can be derived: reh ≃ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi end y−k; λ 4 sffiffiffi aend ðk þ 2Þ 3kðk − 1Þ MP 1 2 1 T 5 3 40 π ρ 8 ð20Þ max ¼ end ; ð Þ 233 λ 4 22 Treh k¼2 y MP from which we determine the reheating temperature, 1 3 1 T 8 1 8 π ρ 6 max ¼ end ; ð23Þ k 2k 9 4 15½3kðk − 1Þ2y λ k þ 2 k Treh ¼3 y 3 × 5 λM T4 ¼ M4 : ð21Þ k P reh 24k−1π2þk 7 − P g k 4 3 T 1 π ρ 16 max ¼ pffiffiffiffiffiffiffi end : ð24Þ ρ 3 λ 4 As one can see, the dependence on end has disappeared Treh k¼4 2y MP from Treh. For the T-attractor model described in the ¼ð2 3 4Þ ¼ 10−5 λ ¼ ρ ¼ 3 ðΦ Þ Appendix, for k ; ; and y , we find As noted earlier, the value of end 2 V end depends on ð2 0 9 0 3Þ 10−11 ¼ð4 1 109 1 0 107 ; . ; . × and Treh . × ; . × ; the potential and the type of inflationary model we 3.2 × 104Þ GeV, respectively. consider. As an example, in Fig. 2, we use the value of

6Note that under this definition, the production of entropy from 7Throughout our work, the numerical results are obtained by inflaton decay continues for some time after the end of reheating. solving the full Boltzmann-Friedmann system (8)–(10).

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8 whereas one obtains H ∝ T 3 in the case k ¼ 4. This observation further confirms that for radiation-like reheating, the temperature decreases faster than in the dustlike scenario: for a given value of H, we have a larger temperature for larger k. As a function of time, the Hubble parameter takes the simple form

k þ 2 H ≃ ; ð26Þ 3kt

for aend T , ð540 2 3 105 7 1 107Þ ¼ð2 3 4Þ DM reh ; . × ; . × for k ; ; using the val- and we consider this possibility as well. ρ λ ues of end= given in the Appendix. The ratio is larger by 5 orders of magnitude for k ¼ 4 than for k ¼ 2. We empha- A. DM from thermal bath scattering size that this is not an enhancement in Tmax, but rather a 2 reduction in the value of Treh for k> for a given value of The DM number density, which we will simply y, as we discussed previously when commenting on Fig. 1. denote by n, corresponds to the solution of the Boltzmann We have avoided extrapolating our results for the equation temperature ratio in Fig. 2 for y ≳ 10−5. For any value of the inflaton-SM coupling, the decay products f acquire dn time-dependent masses induced by the oscillating inflaton þ 3Hn ¼ RðtÞ; ð27Þ −5 dt background, mf ¼yΦ.Fory≳10 and/or k>4, one finds in general that m2=m2 ≳ 1 at some stage of reheating. The f Φ ð Þ perturbative decay of the inflaton can therefore become where R t denotes the production rate of DM (per unit kinematically suppressed, or dominated by nonperturba- volume per unit time). This rate contains the contribu- tive particle production. We leave the detailed study of tion from scatterings in the plasma as well the con- DM production and reheating beyond these bounds for tribution from the direct decay of the inflaton into DM. future work. Depending on the magnitude of R compared to Hn, dark matter may or may not ever come into thermal equilibrium. For small R, DM remains out of equilibrium as D. The Hubble parameter in the case of gravitino production in supersymmetric We conclude this section by finding an explicit models [19,33,34] and in many generic freeze-in models expression for the Hubble parameter as a function of the [30].Makinguseof(25) and (26), we can rewrite the temperature during reheating. This relation will aid our Boltzmann equation in terms of the instantaneous computation of the DM relic abundance in the following temperature as follows: section. Substitution of (14) and (16) into the Friedmann equation (10) gives pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dn 2k þ 4 n 2k þ 4 RðTÞ 2k þ ¼ : ð28Þ ρ ð Þ −1 Φ Treh T k dT 1 − k T 3 − 3k THðTÞ H ¼ pffiffiffi ð25Þ 3MP Treh −2kþ4 ≫ ¼ 2 −1 for areh >a aend. Note that for k the previous Equivalently, if we introduce the DM yield Y ≡ nT k , expression recovers the well-known result H ∝ T4, we can write

123507-6 REHEATING AND POST-INFLATIONARY PRODUCTION OF … PHYS. REV. D 101, 123507 (2020) pffiffiffiffiffi s 10 2 þ 4 2 5 þ3 m n ðT0Þ dY k −1 k s 2 DM ¼ − pffiffiffiffiffi k 1−k ð Þ ð Þ Ω h ¼ MPTrehT R T : 29 DM −2 dT π g k − 1 ρch 2 s π g ðT0Þm nγðT0Þn ðT Þ ¼ s DM reh We parametrize the production rate from out-of- 3 −2 2ζð3Þg ðT ÞT ρ h equilibrium scatterings in the following way8: s reh reh c sð Þ ¼ 5 9 106 −1 mDMn Treh ð Þ Tnþ6 . × GeV 3 ; 34 RsðTÞ¼ : ð30Þ Treh Λnþ2 where g ðT0Þ¼43=11 is the present number of effective Here the superscript s denotes production via scatterings in s relativistic degrees of freedom for the entropy density, the plasma, and the mass scale Λ is identified with the −3 nγðT0Þ ≃ 410.66 cm is the number density of CMB beyond the Standard Model scale of the microscopic model ρ −2 ≃ 1 0534 10−5 −3 under consideration. Note that this effective description is photons, and ch . × GeV cm is the criti- ð Þ¼ valid for the duration of reheating provided that Λ ≳ T . cal density of the Universe [70]. We take gs Treh max ð Þ¼ The suppression by the UV scale ensures that DM g Treh greh, and consider for definiteness the high- ¼ 427 4 annihilation can be neglected. Integration of (29) after temperature Standard Model value greh = . substitution of (30) leads to the following results: 10−2k B. Production from scattering when mDM > Treh (i) For n< k−1 , sffiffiffiffiffi Ωs In the above derivation of DM, we have implicitly 10 M 2k þ 4 Tnþ4 sð Þ¼ P reh ð Þ assumed that mDM Treh, we must cut off the integral at mDM. ¼ 10−2k ¼ ρ ρ (ii) For n k−1 , However, at T mDM, R < Φ, and the density of DM sffiffiffiffiffi matter will be further diluted by the subsequent decays of þ4 sð 10M 2kþ4 Tn T the inflaton. Therefore, we compute n mDM) and scale it to sð Þ¼ P reh max ð Þ ≤ ð10 − 2 Þ ð − 1Þ n Treh nþ2 ln : 32 Treh using Eq. (17).Forn k = k , we find the g π k−1 Λ T reh following: 10−2k 10−2k (i) For n< k−1 , (iii) For n> k−1 , sffiffiffiffiffi sffiffiffiffiffi 10 M 2k þ 4 10 M 2k þ 4 sð Þ¼ P sð Þ¼ P n Treh n Treh π − − 10 þ 2 g π n − nk þ 10 − 2k g kn n k 2kþ6 nþ4 2kþ6 nþ4 Treh k−1 m T k−1 T DM ð Þ reh max ð Þ × nþ2 : 35 × nþ2 : 33 mDM Λ Tmax Λ ¼ 10−2k Note that these results are a generalization of [45,50] (ii) For n k−1 , applicable to the monomial potential given in Eq. (4) after sffiffiffiffiffi inflation. For the typical potential with k ¼ 2, i.e., oscil- 10 M 2k þ 4 sð Þ¼ P n Treh lations of a massive inflaton, the density of dark matter is g π k − 1 mainly sensitive to the reheating temperature if n<6, nþ4 nþ4 whereas it is mainly sensitive to the maximum temperature Treh mDM Tmax ð Þ × nþ2 ln : 36 prior to the end of reheating if n>6. We see that for k ¼ 4, mDM Λ mDM ≥ 1 dark matter production is sensitive to Tmax for n . This means that we expect significant production of dark matter Note that for n>ð10 − 2kÞ=ðk − 1Þ, the result in Eq. (33) 9 in many models. For example, in models where the dark is unchanged. and visible sectors are connected by massive mediators as in SO(10) [38,39,66–69] or moduli-portal models [47], the C. Production from inflaton decay production and final density of dark matter will be sensitive DM can also be produced during reheating by the direct to the postinflationary scalar potential. decay of the inflaton. When the decay rate for both the The DM number density produced by scatterings in the dominant decay products of Φ and the DM particle is plasma given in Eqs. (31)–(33) can be converted to the DM contribution to the critical density using 9 When Treh

123507-7 GARCIA, KANETA, MAMBRINI, and OLIVE PHYS. REV. D 101, 123507 (2020) rffiffiffiffiffiffiffiffiffi 4−2k 2 2 proportional to mΦ, the production rate in the Boltzmann T k−1 16π π g 7 − k T L ¼ reh ð Þ equation (28) takes the form 2 90 þ 2 : 39 Treh y k mDMMP y2 dð Þ¼ ρ ð Þ R T 8π BR Φ T In this case, 2 2 4 4k pffiffiffiffiffiffiffiffiffi y π gT T k−1 4 ≃ reh ð Þ 10g T k−1 8π BR 30 ; 37 ndðT Þ¼ ðk þ 2ÞB y2M reh T2 : ð40Þ Treh reh R P reh 480 TL where B denotes the branching ratio of the decay of the R ¼ inflaton into DM and includes the multiplicity of DM The crossover from Eq. (38) to Eq. (40) occurs when TL particles in the final state. After a straightforward integra- Treh and is easily obtained from Eq. (39). We call the mass tion, we obtain the following expression for the DM at the crossover mL and 10 number density originating directly from inflaton decay : rffiffiffiffiffiffiffiffiffi 2 2 16π π g 7 − k T pffiffiffiffiffiffiffiffiffi m ¼ reh : ð41Þ 10g L 2 90 þ 2 ndðT Þ¼ ðk þ 2ÞB y2M T2 ; ð Þ y k MP reh 480 R P reh 38 for low m (the crossover mass is defined shortly). For DM D. The total dark matter relic abundance high mDM, we must cut off the integration at a temperature ¼ 2 at which mΦ mDM. Recall that for k> , mΦ evolves We next combine the dark matter densities produced by with Φ. We can use Eq. (12) to determine the temperature scattering and inflaton decay to obtain the following ð Þ¼ TL such that mΦ TL mDM and find expression for the total present-day relic abundance: B 427=4 1=2 y 2 1010 GeV m Ω h2 ≃ 0.1 × ðk þ 2Þ R DM DM 10−5 g 10−5 T 1 GeV 8 reh reh > 1 <> ;mDM Treh k−1 :> ;mDM >mL; TL 1016 GeV nþ2 T nþ1 427=4 3=2 m þ 1.4 × 103−6n reh DM Λ 1010 GeV g 1 GeV 8 reh > 2kþ4 10−2k > − þ10−2 ;n<−1 mDM n nk k k > 2 þ6 > k −n−4 > T k−1 > 2kþ4 reh 10−2k > n−nkþ10−2k ;nTreh; > mDM > <> T i 2kþ4 ln max ;n¼ 10−2k m Treh 42 > > T > 2kþ4 max ¼ 10−2k > −1 ln ;n−1 mDM >Treh; > k m k > DM > 2k−10þn > T k−1 > 2kþ4 max 10−2k : nk−nþ2k−10 ;n>k−1 ; Treh

∝ 2 where the first term corresponds to the production from Treh y , and therefore the decay contribution does not decays, while the second, Λ-dependent term corresponds to depend on the reheating temperature. It depends only on the freeze-in production through scattering. For the former square of the ratio of the inflaton-DM and inflaton-SM term, it is worth noting that for k ¼ 4, Eq. (21) implies that couplings, encoded in BR, and the DM mass. In the case of scatterings, we see clearly here the enhancement in (T =T ) for n>ð10 − 2kÞ=ðk − 1Þ. 10This result is modified if the decay rate of the inflaton to DM max reh In Fig. 3, we display the value of Λ [in Eq. (30)]asa has a different dependence on mΦ. The corresponding DM αþ1 function of the DM mass, m , needed to obtain Ωs h2 ¼ number density for ΓΦ→DM ∝ mΦ for generic α is left for DM DM future work. 0.1 in Eq. (34) for k ¼ 2, 3, 4. In Fig. 3(a), we have chosen

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(a) (b)

2 FIG. 3. Contours ΩDMh ¼ 0.1 showing the required value of Λ as a function of the DM mass. We assume an inflaton decay coupling y ¼ 10−5 and a production rate with n ¼ 0 (a) n ¼ 2 (b) and n ¼ 6 (c). For (d) we assume y ¼ 10−7 and n ¼ 2. In all cases, we have set BR ¼ 0.

n ¼ 0 which is characteristic of a production rate for occurs at high Λð>MPÞ and is off the scale of the plot. The Λ ∼ gravitinos in supersymmetric models when MP.In relative slope seen for mDM >Treh can also be understood ¼ 10−5 9−k this figure, we have chosen y . According to Fig. 1, from Eq. (42), noting that Ωs scales as mk−1 =Λ2, so that ∼ 1012 DM DM this corresponds to a value of Tmax GeV and for k ¼ 2, Λ must decrease steeply, whereas for k ¼ 4,it T ∼ 1010 GeV. For k ¼ 2, one gets the expected result reh remains an increasing function of mDM. All of the curves that the density of gravitinos accounts for the DM when fall off when m approaches T . ∼ 100 Λ ∼ DM max m3=2 GeV, for MP. In Fig. 3(b), we show the corresponding result when As discussed in the earlier sections, fixing the inflaton n ¼ 2. Models with n ¼ 2 could correspond to inter- decay coupling, y, fixes the maximum and final reheating actions mediated by the exchange of a massive particle Λ ¼ 3 temperature depending on the value of k. The relic density with mass >Tmax. In this case, k is the critical sð Þ – depends on Treh through n Treh as given in Eqs. (31) value, and the density becomes sensitive to Tmax as in (33).Butns also depends on Λ−ðnþ2Þ. In Fig. 3(a), for Eq. (32). For larger k, the density is given by Eq. (33) n ¼ 0, the density is given by Eq. (31) and we see from and exhibits a stronger dependence on Tmax. For this Ωs Λ2 Λ Eq. (42) that DM scales as mDM= which accounts for the reason, we see that a larger value of is required to Ωs 2 ¼ 0 1 ¼ 4 ¼ 3 slope in the figure. We also see that the required value of Λ obtain DMh . for k than for k . Overall, decreases with increasing k to compensate for the lower however, we see that lower values of Λ are required for reheat temperature when k>2. Suitable DM masses range n ¼ 2 compared with n ¼ 0. Λ ¼ 1014 ¼ 2 ¼ 2 from 0.1 to Tmax for GeV to MP. For n , we again see a change in slope for k ¼ Ωs 5 Λ2 In Fig. 3(a), we also see changes in the slopes of the lines when mDM Treh. In this case, DM scales as mDM= at ¼ ¼ 3 for all three values of k. These occur when mDM Treh as large DM masses. For k , the change in slope is more ¼ 2 Λ2 discussed earlier. For k and 3, the change in slope subtle as the dependence on mDM goes from mDM= to

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(a) (b)

2 FIG. 4. Contours ΩDMh ¼ 0.1 showing the required branching ratio as a function of the DM mass, assuming an inflaton decay coupling y ¼ 10−5 (a) and y ¼ 10−7 (b). Here, we ignore production due to scattering.

ð Þ Λ2 ¼ 4 mDM log Treh=mDM = , and for k , there is no change and since Treh drops significantly with increasing k, BR as the density is primarily sensitive to Tmax. must be smaller for larger k, as seen in the figure. For a Ω ∝ In Fig. 3(c), we show the corresponding result when given value of k, DM BRmDM, accounting for the slope n ¼ 6. Rates with n ¼ 6 could correspond to the produc- in the lines shown in the figure. tion of gravitinos in high-scale supersymmetry [41– This behavior is strictly true in the whole kinematically ¼ 2 43,45,72] in which two gravitational vertices are required, allowed range only for k , or for a relatively small mDM or in models with spin-2 mediators [46]. In the case of high- for k>2. As discussed earlier, this is because for k>2, 2 Λ ∼ m3 2M scale supersymmetry, we expect that = P. Thus, when mDM >mL, decays to DM cannot continue all the way ∼ 1 Λ ∼ 10−5 for m3=2 EeV, we should find MP which is down to Treh. As a consequence, the slope of the required what is seen in Fig. 3(c). In this case, we are sensitive to branching ratio BR vs mDM changes. From Eqs. (39) and ¼ 2 Λ ð1− Þ ð4−2 Þ Tmax even for k and the value of needed is now T ∝ m k = k Ωd h2 ∝ ¼ 4 ¼ 2 ¼ 3 (40), we see that L DM and hence DM greater for k than both k and k . Note that there 4=ð1−kÞ ð8−2kÞ=ð4−2kÞ ¼ 2 B ndm ∝ B T m ∝ B m .Fork ¼ 3, is again a subtle change in slope for k when mDM R DM R L DM R DM ¼ 2 Ωd 2 ∝ ¼ 4 Ωd 2 ∝ becomes larger than Treh as was the case for n this gives DMh BR=mDM and for k , DMh BR, and k ¼ 3. independent of mDM, thus explaining the slopes seen in We can also use Eq. (42) to compare predictions for the the figure. relic density for a given scattering rate (ignoring decays) for In Fig. 4(b), we show the dependence of BDM on mDM for ¼ 10−7 ∼ different values of k. Consider first the case with n ¼ 2. y . Note that there is some dependence on y in Treh ¼ 2 ¼ 4 ¼ 2 k=2 Ω 2 ∝ 2−k=2 When comparing k and k for n , the large drop y as seen in Eq. (21). Therefore, DMh BDMy . 3 ¼ 2 ¼ 4 ¼ 2 in Treh when going from k to k is not compensated For k , we see that the required branching ratio is larger ð Þ4=3 ¼ 4 by the enhancement in Tmax=Treh and thus for a given compared with that in Fig. 4. However, for k , the value of Λ, we require larger masses for k ¼ 4, as seen in dependence on y drops out, and the required branching Fig. 3(b). In contrast when looking at the n ¼ 6 case, the ratio is unchanged. In this case, the crossover mass, mL,is 7 ¼ 4 drop in Treh is more than compensated for by the enhance- lower, and for k , it is below 1 GeV. ð Þ16=3 ment in Tmax=Treh and in this case for a given value of Λ, we require smaller masses for k ¼ 4, as seen in Fig. 3(c). IV. CONCLUSION AND DISCUSSION To see the dependence of these results on y, we show in Fig. 3(d) analogous results with y ¼ 10−7 for n ¼ 2. Since While inflation was designed to resolve a host of the reheating temperature is lower (for lower y), the cosmological problems, such as flatness and isotropy, it necessary value of Λ is also lower. Once again we see a also seeds the fluctuations necessary for structure in the ¼2 ¼ change in slope for k at mDM Treh, the logarithmic Universe. Also needed to form structure is the existence of change for k ¼ 3, and no change in slope for k ¼ 4,asin dark matter. If dark matter interactions with Standard Fig. 3(b). Model particles are so weak that they never attain thermal In Fig. 4(a), we show the necessary branching ratio to equilibrium with the Standard Model bath, their existence Ω 2 ¼ 0 1 ¼ 10−5 obtain DMh . from decay, assuming y , using may also be a result of an early inflationary period. More Eqs. (38) and (40) and ignoring any possible contribution precisely, the origin of dark matter may reside in process of Ω 2 ∝ −1 from scattering. Since DMh Treh [see Eq. (42) above], reheating after inflation.

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A classic example of dark matter born out of reheating is matter with masses in excess of the reheating temperature. the gravitino. With Planck-suppressed couplings, the abun- For completeness, we have also considered the effect of the dance of gravitinos is proportional to the reheating temper- equation of state on the dark matter abundance originating ature after inflation [19], though there may also be a directly from inflaton decays. nonthermal component from decays of either the inflaton, In this paper, we have focused on the effects of the or the next lightest supersymmetric particle. Dark matter equation of state during reheating. We have limited our production during reheating may be the dominant produc- discussion to inflaton decays to fermions, neglecting thermal tion mechanism for a class of candidates known as effects, and assumed that the decay width of the inflaton is FIMPs [30]. simply proportional to the inflaton mass. Both assumptions While certain quantitative aspects of dark matter pro- affect our quantitative results, and these will treated more duction can be ascertained from the instantaneous reheating generally in future work. We have also neglected the delay of approximation, the dark matter abundance may be grossly the onset of thermal equilibrium and the self-interaction of underestimated if the dark matter production rate has a the inflaton, which we will also consider in future work. The strong temperature dependence as in the case of the scenario outlined in this paper also does not exhaust all DM gravitino in high-scale supersymmetric models [41] or production channels. DM can also be produced by the decay dark matter interactions mediated by spin-2 particles of the heavy particles that may be produced at the early stages [46]. Instantaneous reheating refers to the approximation of reheating, or (in)directly by nonadiabatic particle pro- that all inflatons decay in an instant (usually defined to be duction [74–78]. We also leave the study of these scenarios in the time when the inflaton decay rate, ΓΦ ≃ H). At the same a generic reheating stage for future work. moment, the Universe becomes radiation dominated with a temperature determined by the energy density stored in the ACKNOWLEDGMENTS inflaton at the time of decay. However, inflaton decay is never instantaneous [42,48– The authors want to thank especially Christophe 50]. If the inflaton decay products rapidly thermalize, a Kulikowski for very insightful discussions. This work was supported by the France-US PICS MicroDark. The radiation bath is formed even though ρR ≪ ρΦ. Depending on the coupling of dark matter to the Standard Model, dark work of Marcos Garcia was supported by the Spanish matter production may begin soon after the first decays Agencia Estatal de Investigación through Grants occur. Indeed, the Universe will first heat up to a temper- No. FPA2015-65929-P (MINECO/FEDER, UE) and ≫ No. PGC2018095161-B-I00, IFT Centro de Excelencia ature Tmax Treh, and the maximum temperature may ultimately determine the dark matter abundance. Severo Ochoa SEV-2016-0597, and Red Consolider Inflation occurs in the part of field space where the scalar MultiDark FPA2017-90566-REDC. Marcos Garcia and potential is relatively flat. The exit from the inflationary Kunio Kaneta acknowledge support by Institut Pascal at phase occurs as the inflaton settles to its minimum and Universit´e Paris-Saclay during the Paris-Saclay Particle begins the reheating process. Often it is assumed that the Symposium, with the support of the P2I and SPU research departments and the P2IO Laboratory of Excellence (pro- potential during reheating can be approximated by a “ ’ ” quadratic potential. In this paper, we studied the reheating gram Investissements d avenir ANR-11-IDEX-0003-01 process in the case of a generic inflaton potential which can Paris-Saclay and ANR-10-LABX-0038), as well as the jΦjk IPhT. This project has received funding/support from the be expressed as VðΦÞ¼λ k−4 about its minimum. For MP European Unions Horizon 2020 research and innovation k ≠ 2, the Universe does not expand as if it were dominated programme under the Marie Skodowska-Curie grant agree- by matter, rather it is subject to an equation of state given by ments Elusives ITN No. 674896 and InvisiblesPlus RISE w ¼ðk − 2Þ=ðk þ 2Þ [57,73]. Here, we have shown that the No. 690575. The work of Kunio Kaneta and Keith A. Olive presence of an effective mass mΦðtÞ affects strongly the was supported in part by the DOE Grant No. DE- evolution of the temperature, especially near the end of SC0011842 at the University of Minnesota. the inflation where the reheating temperature Treh is highly dependent on k and can be significantly smaller than in the APPENDIX: INFLATIONARY MODELING vanilla quadratic case k ¼ 2. We have parametrized the dark matter production rate as 1. T-attractors and supergravity R ∝ Tnþ6. In the case, of k ¼ 2, for n<6, the dark matter The most recent measurements of the tilt of the primor- abundance is primarily determined by the reheating temper- dial scalar power spectrum ns and the null detection of ature. For n ¼ 6, the abundance is enhanced by primordial tensor modes by the Planck Collaboration ð Þ 6 log Tmax=Treh and for n> , it is primarily determined [79,80] appear to favor plateaulike potentials, characterized 2 by Tmax [50]. This picture changes, however, when k> . by relatively low energy densities. In this light, a lot of The critical value of n decreases with increasing k, and for interest has been focused on a class of models, which ¼ 4 k , the dark matter abundance is sensitive to Tmax for include the Starobinsky model [59] and converge in their n ≥ 1. In these cases, it is also possible to produce dark predictions to the “attractor point” [58,81–87],

123507-11 GARCIA, KANETA, MAMBRINI, and OLIVE PHYS. REV. D 101, 123507 (2020) 2 12 pffiffiffi Φ k ≃ 1 − ≃ ð Þ ns ;r2 : A1 VðΦÞ¼λ 6 tanh pffiffiffi : ðA9Þ N N 6

Here r denotes the tensor-to-scalar ratio, and N is the Alternatively, choosing number of e-folds between the horizon crossing of the pivot scale k and the end of inflation. pffiffiffi pffiffiffi k 2T − 1 2 Many of these models can be constructed [88] from no- ¼ λϕð2 Þ 6 ð Þ W T 2 þ 1 A10 scale supergravity [60–62] defined by a Kähler potential of T the form yields the same potential given in Eq. (A9) and both jϕj2 provide Planck-compatible completions for our potential ¼ −3 þ ¯ − ð Þ K ln T T 3 ; A2 (4) at large field values [58]. In all of the above expressions, we have worked in units of MP. In the remainder of this 11 where T is a volume modulus and ϕ is a matter like field. Appendix, we will restore powers of MP. Depending on the form of the superpotential, either T or ϕ can play the role of the inflaton [82,89]. For example, the 2. Normalization of the potential Starobinsky model is derived from a simple Wess-Zumino- ρ In order to determine end, one must find the inflaton like superpotential [88], field value at the end of inflation, defined where ä¼ 0 or Φ_ 2 ¼ ðΦÞ ϕ2 ϕ3 equivalently end V [63]. An approximate solution W ¼ M − pffiffiffi ; ðA3Þ for this condition yields using (A9) 2 3 3 rffiffiffi 3 1 pffiffiffiffiffiffiffiffiffiffiffiffiffi where Φ is related to the canonically normalized inflaton Φ ≃ þ k ð þ 2 þ 3Þ ð Þ end MP ln k k : A11 through 8 2 3 pffiffiffi Φ For k ¼ð2; 3; 4Þ, this yields Φ ¼ð0.78; 1.19; 1.50ÞM , ϕ ¼ 3 tanh pffiffiffi ; ðA4Þ end P 6 respectively, which can be compared with the Starobinsky Φ ¼ 0 62 ρ ¼ result, end . MP [63]. Recall in addition that end 3 ðΦ Þ ρ λ 4 ¼ð0 86 2 0 4 8Þ ¼ð2 3 4Þ yielding the scalar potential 2 V end so that end= MP . ; . ; . for k ; ; . pffiffiffi On the inflationary plateau, a series expansion of the pffiffiffi 2 pffiffi tanhðΦ= 6Þ 3 2 − 2Φ 2 inflationary potential allows us to relate the number of V ¼ 3M pffiffiffi ¼ M 1 −e 3 ; 1 þ tanhðΦ= 6Þ 4 e-folds with the potential and its derivative. Namely, with [58] ðA5Þ pffiffi pffiffi VðΦÞ − 2 Φ 2 −2 2 Φ when hTi¼1=2. Here the inflaton mass, M is fixed in a ¼ 1 − 2ke 3MP þ Oðk e 3MP Þ; ðA12Þ λ 4 6k=2 similar manner as is λ from the CMB normalization as MP discussed below. Alternatively, if hϕi¼0 is fixed, the superpotential [90] the number of e-folds in the slow-roll approximation can be computed as pffiffiffi 1 W ¼ 3Mϕ T − ðA6Þ Z rffiffiffi 2 1 Φ ðΦÞ 3 ≃ Φ V ≃ V ð Þ N 2 d 0 0 : A13 Φ ðΦÞ 2 MP end V MPV yields the same Starobinsky potential (A5) when pffiffi Substitution into the slow-roll expression for the amplitude 1 2Φ ¼ 3 ð Þ of the curvature power spectrum T 2 e : A7 3 A similar class of models sharing the attractor points V A ≃ ðA14Þ S 12π2 6 ð 0 Þ2 in (A1) can be derived from a superpotential of the form MP V kþ1 kþ3 pffiffiffi ϕ2 ϕ2 kþ1 finally gives W ¼ 24 λ − : ðA8Þ k þ 2 3ðk þ 6Þ 11We note that for k ¼ 3, this formulation does not lead to a The resulting scalar potential is then stable minimum at Φ ¼ 0.

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inflation and the duration of reheating, parametrized by the equation-of-state parameter. Concretely [92,93], 2 1=4 1 3 1 π 43 = T0 k N ¼ ln pffiffiffi − ln 3 30 11 H0 a0H0 1 2 1 − 3 ρ þ V þ wint reh 4 ln 4 ρ 12ð1 þ Þ ln ρ MP end wint end 1 − ð Þ 12 ln greh; A16 where the present temperature and Hubble parameter, as determined from CMB observations, are given by T0 ¼ −1 −1 2.7255 K and H0 ¼ 67.36 km s Mpc [91,94]. The FIG. 5. Number of e-folds from the exit of the Planck pivot present scale factor is normalized to a0 ¼ 1. The e-fold −1 scale k ¼ 0.05 Mpc to the end of inflation, as a function of k average of the equation of state parameter during reheating and y. For definiteness, we have used the Standard Model value w ¼ 427 4 is denoted by int, and for our purposes we will approxi- greh = . In the shaded region, our simple perturbative 1 mate it by w given by (7). The energy density at the end of analysis breaks down (right), or Treh < MeV (left). ρ reheating is denoted by reh. The value of the potential at ≃ 6k=2λ 4 λ horizon crossing is approximated as V MP, with 18π2 λ ≃ AS ð Þ given by (A15). k=2 2 : A15 6 N The numerical solution of Eq. (A16) for the perturbative decay of the inflaton into fermions f is shown in Fig. 5 for −1 10 ¼ 2 At the Planck pivot scale, k ¼ 0.05 Mpc ,lnð10 ASÞ¼ k , 3, 4. The excluded region in gray corresponds either 3.044 [79,91]. The number of e-folds must in general be to the kinematic suppression regime (to the right), in which 2 2 computed numerically, as it is determined by the duration the effective masses satisfy the relation mf >mΦ at some of reheating, which in turn is determined by the energy point during reheating, and therefore where our analysis density at the end of reheating, dependent on N. does not apply, or to reheating temperatures lower than ∼1 MeV, incompatible with big bang nucleosynthesis (to 3. The number of e-folds the left) [95] (see also [96]). Note that for all the values of y Finally, we provide numerical values for the number shown, N > 46. Therefore, ns and r here lie within the of e-folds between the exit of the horizon of the Planck 95% CL region of the Planck+BICEP2/Keck (PBK) con- pivot scale k and the end of inflation for T-attractor straints [80]. Moreover, for N ≳ 50, the model is com- inflation. patible with PBK at the 68% CL. Finally, we note that the Assuming no entropy production between the end of curves become less steep for larger k.Atk ¼ 4, N ≃ 55.9 reheating and the reentry to the horizon of the scale k in independently of the decay rate, consistent with the fact that the late-time radiation or matter-dominated Universe, the radiation domination (i.e., w ¼ 1=3) is reached immedi- number of e-folds will depend on the energy scale of ately after the end of inflation.

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