Curing inflationary degeneracies using reheating predictions and relic gravitational waves
aa r X i v : . [ g r- q c ] J a n Prepared for submission to JCAP
Curing inflationary degeneracies usingreheating predictions and relicgravitational waves
Swagat S. Mishra, a Varun Sahni a and Alexei A. Starobinsky b a Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune411 007, India b L. D, Landau Institute for Theoretical Physics RAS, 119334 Moscow, RussiaE-mail: [email protected], [email protected], [email protected]
Abstract.
It is well known that the inflationary scenario often displays different sets ofdegeneracies in its predictions for CMB observables. These degeneracies usually arise eitherbecause multiple inflationary models predict similar values for the scalar spectral index n S and the tensor-to-scalar ratio r , or because within the same model, the values of { n S , r } are insensitive to some of the model parameters, making it difficult for CMB observationsalone to constitute a unique probe of inflationary cosmology. We demonstrate that by takinginto account constraints on the post-inflationary reheating parameters such as the durationof reheating N re , its temperature T re and especially its equation of state (EOS), w re , it ispossible to break this degeneracy in certain classes of inflationary models where identicalvalues of { n S , r } can correspond to different reheating w re . In particular, we show howreheating constraints can break inflationary degeneracies in the T-model and the E-model α -attractors. Non-canonical inflation is also studied. The relic gravitational wave (GW)spectrum provides us with another tool to break inflationary degeneracies. This is becausethe GW spectrum is sensitive to the post-inflationary EOS of the universe. Indeed a stiff EOSduring reheating ( w re > /
3) gives rise to a blue tilt in the spectral index n GW = d log Ω GW d log k > w re < /
3) results in a red tilt. Relic GWs therefore provide us withvaluable information about the post-inflationary epoch, and their spectrum can be used tocure inflationary degeneracies in { n S , r } . Keywords:
Inflation, gravitational waves, reheating, early universe ontents α -attractors 32.2 Non-canonical inflation 6 α -attractor 173.5 Reheating constraints on the E-model α -attractor 193.6 Reheating constraints on non-canonical inflation 19 There has been a remarkable progress in our understanding of the early universe over thepast three decades fostered by new theoretical insights and reinforced by a plethora of pre-cision cosmological missions, ranging from Cosmic Microwave Background (CMB) to largescale structure (LSS) observations. As a result, the inflationary paradigm [1–6] has emergedas a key scenario for describing the early universe and for setting initial conditions for thehot Big Bang phase of expansion. One of the key predictions of the inflationary scenariois the quantum-mechanical production of primordial tensor fluctuations which give rise toa stochastic background of relic gravitational waves [7]. The reason for this lies in thefact that unlike other massless fields such as photons and massless neutrinos which cou-ple conformally to gravity and whose production is therefore suppressed in the conformallyflat Friedmann-Lemaitre-Robertson-Walker (FLRW) universe, gravitational waves in GeneralRelativity (GR) couple minimally to gravity [8] that results in their non-adiabatic productionin an expanding isotropic universe if the Ricci scalar R is non-zero . While several distinctpredictions of the single field slow-roll scenario of inflation have received spectacular observa-tional confirmation, both from CMB as well as LSS observations, the detection of primordialtensor fluctuations, both in the form of CMB B-mode polarization on large angular scales as More complicated situation may occur in modifed gravity. In particular, in f ( R ) gravity small oscillationsof the gravitational scalar degree of freedom with a non-zero R do not create gravitons; in quantum language,decay of scalarons into pairs of gravitons is suppressed [9]. – 1 –ell as a spectrum of relic gravitational waves (GWs), remains one of the major challengesconfronting observational cosmology in the coming decade.It is well known that GWs provide us with important information about the nature of theinflaton field and its potential. Of equal importance is the fact that their spectrum, Ω GW ( k ),and spectral index n GW = d log Ω g d log k , can serve as a key probe to physical processes occurringafter inflation. As originally shown in [10] the spectrum of relic gravitational radiation isexceedingly sensitive to the post-inflationary equation of state (EOS), w . In fact the GWspectrum has distinctly different properties for stiff/soft equations of state. For a stiff EOS, w > /
3, the GW spectrum shows a blue tilt: n GW >
0, that increases the GW amplitudeon small scales. Softer equations of state, w < /
3, on the other hand, lead to a red tilt,whereas the radiation EOS, w = 1 /
3, results in a flat spectrum with n GW ≃ w i . Of these, the most recentones are: the radiation dominated stage with w r ≃ /
3, the matter dominated stage with w m ≃ w de < − /
3. However, after theend of inflation and before the commencement of the radiation dominated stage, the universewent through the epoch of reheating during which the energy contained in the inflaton fieldwas transferred to the other matter/radiation degrees of freedom present in the universe.The nature of the reheating epoch, including its duration N re and EOS w re , dependscrucially upon how the inflaton couples with (and hence releases its energy into) other matterfields in the universe. If this process is slow then reheating takes place perturbatively andthe inflaton scalar field oscillates for a very long time, gradually releasing its energy intomatter/radiation. In this case, the EOS during the oscillatory regime is determined primarilyby the shape of the inflaton potential near its minimum value, about which the inflatonoscillates. Perturbative reheating in GR is expected to occur if the inflaton φ decays primarilyinto fermions (which soon decay into the standard model fields), its decay into bosons beingstrongly suppressed in the absence of the trilinear φχ interaction [11] .On the other hand, if the inflaton decays into bosons, χ , through a coupling g φ χ with g ≫ − , then oscillations of φ can lead to a parametric resonance during which quanta ofthe field χ are produced in copious amounts. This stage is usually referred to as preheating[11, 14–17]. The backreaction of χ on φ ends the resonance and the subsequent decay ofexcitations of the φ and χ fields into standard model (SM) fields gives rise to reheating andthe subsequent thermalization of the universe at a temperature T re . The duration of thepre-radiative epoch, which includes the end of inflation, the parametric resonance, the decayof the inflaton into bosons ( φ → χ χ ) and fermions ( φ → ψ ψ ) and thermalization can bequite long, and it is convenient to encode its physics by means of an effective EOS parameter w re . Since w re influences the spectrum of relic gravitational waves, observations of the GWspectrum can shed light on the complex, non-linear and out of equilibrium physics whichoperates during the reheating epoch.In addition to primordial tensor fluctuations which result in the GW background, a keyprediction of inflationary cosmology is the generation of primordial scalar fluctuations whichlater grow to form the LSS of the universe. Scalar and tensor perturbations generated duringinflation create an imprint in the cosmic microwave background (CMB) which can be used The situation is changing dramatically in the case of strong non-minimal coupling of bosons to gravity [12,13]. – 2 –o deduce the scalar spectral index n S and the tensor to scalar ratio r – two importantobservables which can be used to rule out competing inflationary models [6, 18]. Howeverit is well known that inflationary models often display degeneracies, with two (or more)models predicting essentially the same values of { n S , r } . This leads to so called ‘cosmologicalattractors’ or ‘universality classes’ of the inflationary scenario [19–21]. This degeneracy makesit difficult for CMB observations alone to constitute a unique probe of inflationary cosmology[22, 23].In this paper we show that reheating predictions including the reheating duration N re ,temperature T re and particularly the EOS w re , can help break degeneracies in inflationaryscenario’s in which identical values of { n S , r } can correspond to different w re ; also see [24–27].Since the gravitational wave spectrum Ω GW ( k ) is sensitive to the value of w re , observationsof Ω GW ( k ) by space-based GW observatories can shed valuable light both on the dynamicsof reheating as well as on the parameters of the inflationary potential.Our paper is organised as follows: section 2 demonstrates the existence of inflationarydegeneracies in the T-model and E-model α -attractors and in non-canonical inflation. Insection 3 we provide an introduction to reheating and discuss implications of reheating pre-dictions on inflationary degeneracies in the aforementioned models. The spectrum of relicGWs for the three inflationary models is determined in section 4 and a summary of ourresults is presented in section 5. It is well known that the inflationary scenario displays different sets of degeneracies in itspredictions for the CMB observables. These usually arise either because multiple inflationarymodels predict similar values for the scalar spectral index n S and the tensor-to-scalar ratio r ,or because within the same model, the values of { n S , r } are insensitive to some of the modelparameters. We focus on three separate inflationary models and show that the degeneracies in { n S , r } which they display can easily be broken by incorporating information obtained fromreheating predictions as well as using the associated relic gravitational wave background.The models discussed in this paper are: the T-model and E-model α -attractors and thenon-canonical m φ model. α -attractors The T-model α -attractor is associated with the potential V ( φ ) = V tanh p ( λφ/m p ) ; p = 1 , , .... (2.1)Large absolute values of | λφ | ≫ m p lead to a plateau-like, asymptotically flat potentialwith V ( φ ) ≃ V . On the other hand small values, | λφ | ≪ m p describe the minimum ofthe potential V ( φ ) ≃ V (cid:18) λφm p (cid:19) p (2.2)around which the scalar field oscillates after inflation. As originally shown in [28], ascalar field oscillating around the minimum of such a potential has the mean EOS h w φ i = p − p + 1 . (2.3) For the particular case p = 1, this was earlier derived in [29]. – 3 – − − − λ . . . . . . n S CMB bound
T-Model p = 1 p = 2 p = 3 − − − λ − − − − − r CMB bound r ≤ . T-Model p = 1 p = 2 p = 3 Figure 1 : This figure demonstrates the degeneracies of the T-model α -attractor (2.1). Inthe left panel, the scalar spectral index n S is plotted as a function of λ for three differentvalues of the parameter p in the potential (2.1). The right panel shows the tensor-to-scalarratio r as a function of λ for the same set of values for p . All the curves correspond to thenumber of e -folds N k = 60. Note that when λ > .
1, the scalar spectral index approaches aconstant value (e.g n S ≃ .
967 for N k = 60), whereas r decreases as λ − . The shaded regionrefers to the CMB 1 σ limits on n S and r as determined by Planck 2018 [22].It is interesting that when λ > .
1, the scalar ( n S ) and tensor ( n T ) spectral indices andthe tensor-to-scalar ratio ( r ) for the T-model (2.1) acquire the form n S − ≃ − N k , (2.4) n T ≃ − N k λ , (2.5) r ≃ N k λ , (2.6)which does not depend upon the value of p in (2.1). This interesting degeneracy of theT-model is illustrated in figure 1.2. The generalized Starobinsky potential or the E-model α -attractor isdescribed by V ( φ ) = V (cid:20) − exp (cid:18) − λ φm p (cid:19)(cid:21) p ; p = 1 , , .... (2.7)This potential reduces to the Einstein frame representation of Starobinsky’s R + R – 4 – − − − λ . . . . . . n S CMB bound
E-Model p = 1 p = 2 p = 3 − − − λ − − − − − r E-Model
CMB bound r ≤ . p = 1 p = 2 p = 3 Figure 2 : This figure illustrates the degeneracies of the E-model α -attractor (2.7). In theleft panel, the scalar spectral index n S is plotted as a function of λ for three different values ofthe parameter p in the potential (2.7) while, in the right panel, the value of tensor-to-scalar r is plotted as a function of λ for the same set of values for p . All the curves correspond to thenumber of e -folds N k = 60. Note that when λ > .
5, the scalar spectral index approaches aconstant value (e.g n S ≃ .
967 for N k = 60), whereas r decreases as λ − . The shaded regionrefers to the CMB 1 σ limits on n S and r as determined by Planck 2018 [22].model of inflation when λ = p / p = 1. For λ ≫ . n S − ≃ − N k (2.8) n T ≃ − N k λ , (2.9) r ≃ N k λ . (2.10)Note that these expressions do not depend upon the value of p in (2.7) as shown infigure 2.One therefore finds that both the T-model (2.1) as well as the E-model (2.7) displaydegeneracies, since the same values of { n S , r } can correspond to different values of p .Fortunately in both (2.1) and (2.7) this degeneracy can be broken once inflation endsand the scalar field begins to oscillate. As we shall see later, N k – the number of e -foldsbetween the Hubble exit of the CMB pivot scale and the end of inflation, is sensitiveto the post-inflationary reheating regime. Indeed in both models, close to its minimumvalue, the potential has the form V ∝ φ p , for which the mean EOS, w osc = h w φ i ,is described by (2.3). In the perturbative reheating regime, parameters such as thereheating duration N re and temperature T re are very sensitive to h w φ i = ( p − / ( p + 1)and hence to the value of p . Similarly the primordial GW background is also sensitiveto the value of p . Therefore a degeneracy in { n S , r } is easily broken if constraintson the CMB observables { n S , r } are determined by taking into account the reheating– 5 –OS. This can be further supplemented by the observations of the GW spectral densityparameter Ω g ( k ). Non-canonical scalars have the Lagrangian density [30] L ( X, φ ) = X (cid:18) XM (cid:19) α − − V ( φ ) , X = 12 ˙ φ , (2.11)where M has dimensions of mass while α is dimensionless. When α = 1 the Lagrangian(2.11) reduces to the usual canonical scalar field Lagrangian L ( X, φ ) = X − V ( φ ).The energy density and pressure have the form ρ φ = (2 α − X (cid:18) XM (cid:19) α − + V ( φ ) ,p φ = X (cid:18) XM (cid:19) α − − V ( φ ) , X ≡
12 ˙ φ , (2.12)which reduces to the canonical expression ρ φ = X + V , p φ = X − V when α = 1.One should note that the equation of motion¨ φ + 3 H ˙ φ α − (cid:18) V ′ ( φ ) α (2 α − (cid:19) (cid:18) M ˙ φ (cid:19) α − = 0 , (2.13)is singular at ˙ φ → φ remains finite in thislimit. This can be done by modifying the Lagrangian (2.11) to [31, 32] L R ( X, φ ) = (cid:18) X β (cid:19) β (cid:18) XM (cid:19) α − ! − V ( φ ) , (2.14)where β is a dimensionless parameter. In the limit when β ≫
1, equation (2.13) can beapproximated as¨ φ + 3 H ˙ φ α − (cid:18) V ′ ( φ ) ǫ + α (2 α −
1) (
X/M ) α − (cid:19) = 0 , X = 12 ˙ φ , (2.15)where ǫ ≡ (1 + β ) − is an infinitesimally small correction factor when β >> V ( φ ) = V φ n the average EOS duringscalar field oscillations is h w φ i = n − αn (2 α −
1) + 2 α . (2.16)For α = 1 the above expression reduces to the canonical result (2.3).Specializing to the quadratic potential with n = 2 one gets h w φ i = − (cid:18) α − α − (cid:19) (2.17)which informs us that the mean EOS during oscillations is negative and lies in the interval − / < D w NC φ E < α >
1. This should be contrasted to the EOS of oscillating canonical– 6 – . . . . . . . . α . . . . . . . . . n S CMB boundNon-canonical Inflation
Quadratic V ( φ ) ∝ φ . . . . . . . . α . . . . . . . . . r CMB bound r ≤ . Non-canonical Inflation
Quadratic V ( φ ) ∝ φ Figure 3 : The scalar spectral index n S (left panel) and the tensor-to-scalar ratio r (rightpanel) are determined for the potential V ( φ ) = m φ and shown as functions of the non-canonical parameter α defined in (2.11). Note that increasing α leads to a decrease in r buthas no effect on n S which is insensitive to this parameter; see (2.18).scalars (2.3) which lies in the interval 0 ≤ h w φ i <
1. One should note that the two ranges − / < D w NC φ E < ≤ h w φ i < h w φ i , can easily distinguish between canonical andnon-canonical models of inflation.Turning next to the fluctuation parameters { n S , r } one finds for V ( φ ) = m φ theexpression [31] n S = 1 − (cid:18) N k + 1 (cid:19) . (2.18)Surprisingly n S does not depend upon the value of α and coincides with the result for canon-ical scalars. Note that for N k ≫ n S − ≃ − N k (2.19)obtained earlier for the T-model and the E-model potentials.The value of r is given by r = (cid:18) √ α − (cid:19) (cid:18) N k + 1 (cid:19) (2.20)which reduces to the standard result r = (cid:18) N k + 1 (cid:19) (2.21)for canonical scalars ( α = 1). From (2.20) we find that increasing the value of the non-canonical parameter α reduces the scalar-to-tensor ratio r ; see figure 3. Comparing (2.20)with (2.6) & (2.10) we find that α plays the same role as the parameter λ in the potentials(2.1), (2.7). There therefore appears to be a close similarity, indeed a degeneracy, between– 7 –alues of { n S , r } in the non-canonical m φ model and in the T-model and Starobinskymodel, respectively. Interestingly, for a given value of N k all three models have the samevalue of n S . The value of r can also be identical in the three models by an appropriate choiceof λ and α . (Note that n S does not depend upon the free parameters λ and α present in the α -attractor models and non-canonical models, respectively.)The degeneracy in { n S , r } in the three inflationary models is easily broken by notingthat the equation of state during post-inflationary oscillations of the scalar field is markedlydifferent in canonical (2.3) and non-canonical (2.16) models. This implies that the spectrumof the relic gravitational wave background will differ in canonical and non-canonical models.To summarize, the relic GW background carries an important imprint of the post-inflationaryuniverse which can be used to break degeneracies in the inflationary parameters { n S , r } probed by the CMB. A key feature of inflationary cosmology is that it allows the universe to reheat by transferringthe energy localized in the inflaton to the matter/radiative degrees of freedom present in theuniverse.In potentials possessing a minimum, reheating can occur in two distinct ways: (i) per-turbatively (slowly), (ii) rapidly – via a parametric resonance. Which of these two ways isrealized depends upon the nature of the coupling between the inflaton and bosons/fermions.Below we provide a brief summary of perturbative and non-perturbative reheating in thecontext which is relevant for this paper. The reader is referred to [11, 14–17] for more detailson the subject of reheating. The perturbative theory of reheating was originally developed in the context of the newinflationary scenario in [33]. Phenomenologically it amounts to adding a friction term Γ ˙ φ to the classical equation of motion of the scalar field oscillating around the minimum of itspotential [33, 34] ¨ φ + 3 H ˙ φ + Γ ˙ φ + V ′ ( φ ) = 0 . (3.1)The perturbative theory of reheating works well if either (a) the inflaton decays only intofermions ψ through a hψ ¯ ψφ coupling with h ≪ m φ /m p , or (b) the coupling of the inflatonto bosons, χ , described by g φ χ is weak with g ≪ × − , making particle productionvia parametric resonance ineffective [11].As noted in the previous section, close to their minimum value, the potentials discussedin this paper have the general form V ( φ ) ∝ φ p . Equation (3.1) suggests that the amplitudeof scalar field oscillations around this minimum decreases as φ max ∝ a − (cid:16) p +1 (cid:17) exp (cid:18) − Γ t p (cid:19) (3.2)which reduces to the standard result [11] φ max ∝ a − / e − Γ t (3.3) The term ‘reheating’ is a misnomer carried over from early models of inflation in which inflation beganfrom a thermalized initial state. In later models, such as chaotic inflation and those discussed in this paper,the universe commences inflating from a non-radiation state and heats up only once, after inflation ends [5]. – 8 –or the chaotic potential V ∝ m φ .Reheating in this perturbative scenario is complete when the (decreasing) expansionrate becomes equal to the decay rate so that H ≃ Γ. Following thermalization, the reheatingtemperature is given by T r ≃ . p Γ m p which is independent of the duration of inflation andthe properties of V ( φ ). The fact that the coupling between matter and the inflaton can alter,via radiative corrections, the shape of V ( φ ), places strong constraints on the total decay rate:Γ < − m p . This in turn implies that the reheating temperature in perturbative modelscan be relatively small T r < GeV [11, 17].It therefore follows that the post-inflationary oscillatory stage in models with perturba-tive (slow) reheating can be quite long. It is important to note that during most of this stage(while H ≫ Γ) the EOS of the oscillating scalar field is given by (2.3) namely h w φ i = p − p +1 .As pointed out in [10] and discussed in detail in section 4, the spectrum of relic gravita-tional waves created during inflation is very sensitive to the post-inflationary EOS, w re , andhence to the value of the inflationary parameter p . Observations of the GW spectrum cantherefore help in breaking the degeneracy between inflationary models which was pointed outin section 2. For inflationary models in which the main source of reheating is through the decay of theinflaton into bosons, the universe thermalizes and reheats through a sequence of successivestages.1. The first stage, sometimes called preheating , sees the commencement of a parametricresonance brought about by coherent oscillations of the inflaton φ around the minimumof its potential. The resonance can be either narrow or broad depending upon (a) thevalue of coupling constant g in the interaction g φ χ between the inflaton and thebosonic field χ , (b) the scalar field amplitude Φ, (c) its effective mass, m φ = V ′′ . Ifthe resonance is broad ( g Φ /m φ > ∼
1) then coherent oscillations of φ give rise to anexponentially large number of quanta of the field χ in a discrete set of wave bands. (Theexistence of self-interaction, such as the presence of a λφ term in the Lagrangian, canalso result in the creation of quanta of the φ field during oscillations.)2. The second stage witnesses the backreaction of χ on φ (via scattering). This effect canbe quite significant and can lead to the termination of the resonance.3. During the third stage, which can be quite prolonged, quanta of the φ and χ fields trans-fer their energy into other matter fields including radiation. The interaction betweenthe quanta of different fields leads to their thermalization and results in the universeacquiring a reheating temperature T rh .Thus the end of the third stage sees the commencement of the radiation dominated stage ofexpansion during which the EOS in the universe is p ≃ ǫ/
3. The dynamics of the three stagesof reheating is quite complex and usually requires a numerical treatment [35]. It is howeverquite instructive if one characterizes the pre-radiation stages (1) - (3) by an effective EOSwhich, following [24–26], can be assumed to be a constant lying in the interval − / < w ≤ w , affectsthe spectrum of relic gravity waves produced during inflation. Future space-based GW ex-periments might therefore shed light on this important parameter, and through it on thephysics of the reheating epoch. – 9 –efore moving forward, let us designate the variables and parameters that are essentialto describe the reheating kinematics by listing them down systematically at one place, forthe convenience of the reader. We repeat the specifications in the text wherever necessary. • a k , H inf k , φ k : Scale factor, Hubble parameter and the inflaton field value, respectively,during the Hubble exit of a mode k , usually taken to be the CMB pivot scale k ≡ k ∗ =0 .
05 Mpc − . • a e , H e , ρ e , φ e : Scale factor, Hubble parameter, energy density and inflaton field value,respectively, at the end of inflation. • a re , H re , ρ re , T re , g re , g s re : Scale factor, Hubble parameter, energy density, temperature,effective number of relativistic degrees of freedom in energy and entropy, respectively,at the end of reheating. • a p , H p : Scale factor and Hubble parameter, respectively, at the Hubble re-entry epochof the pivot scale. • a BBN , H BBN , T BBN : Scale factor, Hubble parameter and temperature, respectively, atthe beginning of Big Bang Nucleosynthesis. • a eq , H eq , T eq , g eq , g s eq : Scale factor, Hubble parameter, temperature, effective numberof relativistic degrees of freedom in energy and entropy, respectively, at the epoch ofmatter-radiation equality. • a , H , T , g , g s : Scale factor, Hubble parameter, temperature, effective number ofrelativistic degrees of freedom in energy and entropy, respectively, at the present epoch. • N inf k = log ( a e /a k ) : Number of e -folds between the Hubble exit of scale k and the endof inflation. • N re = log ( a re /a e ) : Duration of reheating. • N RD = log ( a eq /a re ) : Duration of the radiation dominated epoch. • w re : Effective equation of state of the universe during the epoch of reheating. The main focus of this section will be on perturbative reheating. The epoch of reheating isusually characterized by a set of three parameters { w re , N re , T re } , namely the effective equa-tion of state (EOS) during reheating w re , the duration of reheating N re and the temperatureat the end of reheating T re , when the universe transits to a thermalized radiation dominatedhot Big Bang phase (see [24–26]). The duration of reheating can be defined by the numberof e -folds between the end of inflation a e and the end of reheating (commencement of theradiation dominated epoch) a re , given by N re = log ( a re /a e ). While N re and T re are inter-esting physical quantities in their own right describing the epoch of reheating, they are alsopotentially important for correctly interpreting the bounds on CMB observables such as thescalar spectral index n S and tensor-to-scalar ratio r , as we discuss next.Following the evolution of the comoving Hubble radius from the epoch of Hubble exit,at a k , of scale k , until its late-time re-entry at a p , one gets (see appendix A)log ka H = − N inf k − N re − N RD − log (1 + z eq ) + log H inf k H , (3.4)– 10 – − −
20 0 20 40 60 80 log e ( a ) − C o m o v i n g H ubb l e R a d i u s a H ∆ N ≃ S c a l e s o b s e r v a b l e b y C M B Pivot Scale λ ∗ Exit Re-entry
Slow-Roll Inflation M a tt e r - R a d i a t i o n E qu a li t y P R ≃ × − a k a e a re a p a eq R e h e a t i n g E nd s w r e N re N RD N inf k E nd o f I n fl a t i o n Figure 4 : This figure schematically illustrates the evolution of the comoving Hubble radius( aH ) − with scale factor. During inflation ( aH ) − decreases which causes physical scales toexit the Hubble radius. After inflation ends ( aH ) − increases, and physical scales begin tore-enter the Hubble radius. The CMB pivot scale, as used by the Planck mission, is set at k ∗ = 0 .
05 Mpc − . It enters the Hubble radius during the radiation dominated epoch when a p ∼ × − a . Note that the duration of reheating N re , and hence the duration of theradiation dominated epoch N RD , changes for different values of the reheating equation ofstate w re . Note that ( aH ) − ∝ a during the radiation dominated regime and ( aH ) − ∝ a − during inflation.where H inf k is the Hubble parameter at the time of the Hubble exit of the scale k , N inf k =log ( a e /a k ) is the number of e -folds between the Hubble exit (of scale k ) and the end ofinflation, N RD is the duration of the radiation dominated epoch and z eq is the redshift atthe epoch of matter-radiation equality. In general, k may correspond to any observableCMB scale in the range k ∈ [0 . , .
5] Mpc − . However, in order to derive constraintson the inflationary observables { n S , r } , we define k to be the CMB pivot scale, namely k ≡ k ∗ = 0 .
05 Mpc − , which makes its Hubble re-entry during the radiation dominatedepoch at a p ∼ × − a ; see figure 4.Our main goal is to characterize the epoch of reheating between the end of inflation a e and the commencement of the radiation dominated epoch a re . Assuming the effectiveequation of state w re during reheating to be a constant, allows one to match the density atthe beginning of the radiation dominated epoch to the density at the end of inflation by ρ re = ρ e (cid:18) a e a re (cid:19) w re ) , (3.5)– 11 –hich yields the following expression for the duration of reheating N re ≡ log (cid:18) a re a e (cid:19) = 13(1 + w re ) log (cid:18) ρ e ρ re (cid:19) . (3.6)Expressing ρ re in terms of the reheating temperature T re , one gets N re = 13(1 + w re ) log ρ eπ g re T re ! , (3.7)where g re ≡ g ( T re ) is the effective number of relativistic degrees of freedom at the end ofreheating. Applying entropy conservation to express T re in terms of a re , one finds (appendixA) T re = (cid:18) g s eq g s re (cid:19) (cid:18) a eq a re (cid:19) T eq , (3.8)where g s eq and g s re are the effective number of relativistic degrees of freedom in the entropy atthe epoch of matter-radiation equality and at the end of reheating respectively, while T eq isthe temperature at the matter-radiation equality. Incorporating (3.8) into (3.7), we obtain N re = 43(1 + w re )
14 log (cid:18) π g re (cid:19) + 13 log (cid:18) g s re g s eq (cid:19) + log ρ e T eq − N RD . (3.9)Substituting N RD from (3.4) into (3.9), we arrive at an important expression for the durationof reheating, namely N re = − w re )
14 log (cid:18) π g re (cid:19) + 13 log (cid:18) g s re g s (cid:19) + log ρ e H inf k + log (cid:18) ka T (cid:19) + N inf k + N re . (3.10)Note that if w re = 1 /
3, then the term N re cancels from both sides of (3.10), yielding thefollowing expression for N inf k N inf k = − log (cid:18) ka T (cid:19) + log ρ e H inf k + 14 log (cid:18) π g re (cid:19) + 13 log (cid:18) g s re g s (cid:19) . (3.11)This arises because the end of reheating, and hence the beginning of the radiation dominatedepoch, cannot be strictly defined within this framework if w re = 1 /
3. However for w re = 1 / N re from equation (3.10) N re = − − w re N inf k + log ρ e H inf k + log (cid:18) ka T (cid:19) + 14 log (cid:18) π g re (cid:19) + 13 log (cid:18) g s re g s (cid:19) . (3.12)Accordingly the expression for the reheating temperature T re in terms of the duration ofreheating N re and effective reheating EOS, w re , follows from (3.7) to be T re = (cid:18) ρ e π g re (cid:19) e − (1+ w re ) N re . (3.13)– 12 –aving expressed the duration of reheating N re and the reheating temperature T re in terms ofthe effective reheating EOS, w re , in (3.12) and (3.13) respectively, we now discuss how thesetwo quantities can be used to obtain tighter constraints on the CMB observables { n S , r } .Note that the expressions (3.12) and (3.13) are valid only for w re = 1 /
3. We return to thecase w re = 1 / N inf k ,following equation (3.11), is given in equation (3.28).In the context of single field slow-roll inflation with potential V ( φ ), the ‘potential slow-roll parameters’ are defined by ǫ V = m p (cid:18) V ′ V (cid:19) , (3.14) η V = m p (cid:18) V ′′ V (cid:19) , (3.15)and the slow-roll limit corresponds to ǫ V , η V ≪
1. The value of the inflaton field at the endof inflation φ e can be determined from the condition ǫ V ( φ e ) = m p (cid:18) V ′ V (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) φ e ≃ , (3.16)and the corresponding inflaton density at the end of inflation is given by (appendix B) ρ e ≡ ρ φ (cid:12)(cid:12)(cid:12)(cid:12) φ e = 12 ˙ φ + V ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) φ e ≃ V ( φ e ) ≡ V e . Substituting ρ e = V e in (3.13) results in the following expression for the reheating temper-ature T re = (cid:18) π g re (cid:19) V e e − (1+ w re ) N re . (3.17)Assuming k to be the CMB pivot scale, k ≡ k ∗ = a k H inf k = a p H p = 0 .
05 Mpc − in (3.12)and inserting the values of T , g s , g re and g s re , one arrives at the following formula whichexpresses the duration of reheating N re as a function of the reheating EOS, w re , on the onehand, and parameters of the inflationary potential V e , H inf k , on the other (see appendix A) N re = 41 − w re . − N inf k − log V e H inf k , w re = 1 / V ( φ ) = V f (cid:18) φm p (cid:19) , (3.19)the number of inflationary e -folds N inf k is given by N inf k = 1 m p Z φ k φ e d ˜ φ q ǫ V ( ˜ φ ) , (3.20)– 13 –here φ k is the value of the inflaton field at the Hubble exit of the scale k (which we take tobe the CMB pivot scale k ≡ k ∗ = 0 .
05 Mpc − ). Note that N inf k does not depend upon thevalue of V which is fixed by CMB normalization to be [22] A S ≡ π (cid:18) V m p (cid:19) f ( φ k ) ǫ V ( φ k ) = 2 . × − . (3.21)The expressions for the scalar spectral index n S and tensor-to-scalar ratio r , in the slow-rolllimit, are given by (appendix B) n S = 1 + 2 η V ( φ k ) − ǫ V ( φ k ) , (3.22) r = 16 ǫ V ( φ k ) . (3.23)Note that the CMB observables { n S , r } depend upon the value of inflaton field φ k at theHubble exit of the pivot scale k . On the other hand equation (3.20) informs us that φ k depends upon the number of e -folds N inf k between the Hubble exit of scale k and the end ofinflation. It therefore follows that { n S , r } ultimately depend upon N inf k . This implies thatconstraints on n S and r , for a given inflationary potential, directly translate onto a constrainton N inf k .The CMB 1 σ constraints on the scalar spectral index n S and the tensor-to-scalar ratio r from the recent CMB observations are given by n S = 0 . ± . r ≤ .
06 (3.25)The constraint on n S is especially strong and effectively restricts the scalar spectral index tothe interval n S ∈ [0 . , . n S to within 0 .
1% precision as discussed in [24–26]. For a given inflationary model, theCMB constraint on n S effectively restricts the value N inf k as discussed above. For a given N inf k , equation (3.20) can be inverted numerically to obtain the value of φ k (since φ e isdetermined from (3.16)). A knowledge of φ k can then be used to obtain the value of H inf k given by (cid:16) H inf k (cid:17) = 13 m p (cid:18)
12 ˙ φ + V ( φ ) (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) φ = φ k ≃ m p V ( φ k ) = 13 m p V f ( φ k ) , (3.26)with V determined from the CMB normalization (3.21). The latter can also be used toobtain the value of V e ≡ V ( φ e ) = V f ( φ e ) by substituting the value of φ e from (3.16).In the perturbative reheating scenario the effective reheating EOS is obtained from theinflationary potential via w re = h w φ i . For inflationary models in which the inflaton oscillatesaround the minimum of a V ∝ φ p potential the effective reheating EOS is given by (2.3),namely w re ≡ h w φ i = p − p +1 . The tight constraint on n S from CMB observations translate intoconstraints on N inf k , H inf k and V e for a given inflationary potential as discussed above. Onecan then use (3.18) and (3.17) to obtain constraints on the duration of reheating N re and thereheating temperature T re , respectively. The constraint on the tensor-to-scalar ratio r has been obtained by the combined observations of Planck2018 and BICEP-II [23] It is also possible to invert (3.20) to obtain φ k ( N inf k ) by using Lambert functions as described in [36]. – 14 – og e ( a ) − C o m o v i n g H ubb l e R a d i u s a H Pivot Scale M a tt e r - R a d i a t i o n E qu a li t y a k a e a p a eq w (1)re < w (2)re < w ( ) r e w ( ) r e N (1)re N (2)re N inf k E nd o f I n fl a t i o n log e ( a ) − C o m o v i n g H ubb l e R a d i u s a H Pivot Scale M a tt e r - R a d i a t i o n E qu a li t y a k a e a p a eq w (1)re > w (2)re > N inf k E nd o f I n fl a t i o n w ( ) r e w ( ) r e N (1)re N (2)re Figure 5 : This figure schematically illustrates the evolution of the comoving Hubble radius( aH ) − with scale factor of the universe and explicitly depicts the dependence of the durationof reheating on the reheating equation of state for a particular inflationary model with agiven N inf k . The top panel shows that for shallow reheating EOS w re < /
3, the durationof reheating N re is longer for a higher value of w re , namely N (1) re < N (2) re for w (1) re < w (2) re .The bottom panel demonstrates that for a stiffer reheating EOS w re > /
3, the durationof reheating N re is shorter for a higher value of w re , namely N (1) re < N (2) re for w (1) re > w (2) re ,in accordance with equation (3.18). Note that ( aH ) − ∝ a during the radiation dominatedregime and ( aH ) − ∝ a − during inflation. For comparison see figure 4.Equations (3.18) and (3.17) capture some of the essential implications of reheating kine-matics on CMB observables and possess important physical significance. For example, it iseasy to see, from equation (3.18), that for a softer reheating EOS with w re < /
3, a higher– 15 –alue of N inf k corresponds to a shorter reheating period N re , for a given model of inflation.Exactly the opposite is true for a stiffer EOS with w re > /
3. In this case the RHS of (3.18)flips sign so that a larger value of N inf k implies a larger N re and hence a longer duration ofreheating. Similarly equation (3.17) implies that the longer is the duration of reheating N re ,the lower will be the reheating temperature T re . Moreover this result is independent of thevalue of w re simply because 1 + w re > w re > − / N inf k (which satisfies the CMB bound on n S ∈ [0 . , . N re in-creases with an increase in the effective EOS w re as long as w re < /
3. This is demonstratedin the left panel of figure 5 in which N (1) re < N (2) re for w (1) re < w (2) re < /
3. Similarly N re increases with a decrease in the effective EOS, w re , if w re > /
3. This is shown in the rightpanel of figure 5 where N (1) re < N (2) re for w (1) re > w (2) re > /
3. These arguments also indicatethat w = 1 / T re , one notes that conservative upper and lower bounds on thisquantity can be placed from the following considerations. It is well known that the CMBupper bound on the tensor-to-scalar ratio, namely r ≤ .
06, translates into an upper boundon the inflationary Hubble scale H inf k ≤ . × GeV, which in turn sets an upper bound onthe energy scale of inflation T inf ≤ . × GeV, as described in appendix B. Since reheatinghappens after the end of inflation, one gets T re ≤ . × GeV as an absolute upper boundon the reheating temperature. Similarly, in order to preserve the success of the hot Big Bangphase, reheating must terminate before the beginning of Big Bang Nucleosynthesis (BBN)yielding the absolute lower bound T re ≥ ≤ T re ≤ GeV . (3.27)In accordance with the above discussion, one can obtain interesting reheating consistentbounds on the CMB observables for a given inflationary potential by proceeding in thefollowing systematic way (also see [24–27])1. Given a slow-roll inflationary potential V ( φ ) with V ∝ φ p as the asymptote duringreheating, the effective reheating equation of state is taken to be w re ≡ h w φ i = p − p +1 .2. Given the strong CMB constraint on the scalar spectral index, namely n S ∈ [0 . , . φ k from equation (3.22) and hence on N inf k fromequation (3.20). It is important to stress that this bound on N inf k has been obtainedpurely from CMB constraint on n S , without taking into account the reheating con-straints on N re and T re yet, which we shall do in the next step.3. We can then use equation (3.18) to translate the bound on N inf k to a bound on N re (as well as on T re using equation (3.17)). Note that N re might turn out to be negativefor some range of allowed values for N inf k . Hence in the next step, we will discard thecorresponding range of N inf k that yields unphysical negative values of N re . This putsadditional tighter constraint on N inf k .4. Next, as discussed above, imposing the condition N re > T re ∈ [1 MeV , GeV], we obtain a tighter bound on N inf k and hence subsequentlyon { n S , r } .5. We tabulate the final allowed range of values for N inf k , n S , r , N re and T re .– 16 –he importance of this procedure lies in the fact that it allows us to obtain reheating con-sistent constraints on the CMB observables { n S , r } , which are tighter than those obtained in[22]. In the following subsection we apply this methodology to determine reheating consis-tent values of { n S , r } in the T- and E- model α -attractors and in the non-canonical quadraticpotential. This will help us to break the inflationary degeneracies in these models. Later insection 4, we will discuss the implications of the reheating constraints for the spectrum ofrelic gravitational waves background.Before moving on, one should clarify that the above strategy is only applicable to infla-tionary models with w re = 1 /
3, which implies p = 2 in the potential V ∝ φ p . For w re = 1 / k to be the CMB pivot scale, i.e k ≡ k ∗ = 0 .
05 Mpc − andinserting the values of T , g s , g re and g s re , as was previously done for the case w re = 1 /
3, oneobtains the following strong prediction for N inf k . N inf k = 61 . − log V e H inf k , (3.28)which in turn translates into predictions for { n S , r } . α -attractor After the end of inflation in the T-model, the scalar field oscillates around the minimum ofthe T-model potential (2.1) which acquires the form V ( | λφ | ≪ m p ) ≃ V (cid:18) λ φm p (cid:19) p ; p = 1 , , ...., with the mean EOS during oscillations given by (2.3). Taking the effective reheating EOSto be w re = h w φ i = p − p +1 , in the context of perturbative reheating, the CMB bound on n S places constraints on the reheating parameters N re and T re . Additionally demanding N re > T re ∈ [1 MeV , GeV] allows us to obtain reheating-consistent constraints on the CMBobservables { n S , r } , as discussed above. Our results are illustrated in figure 6 and tabulatedin table 1.The red curve in each diagram of figure 6 corresponds to p = 1 and hence w re = 0in (2.3), while the green curve corresponds to p = 3 and hence w re = 1 /
2. The blue dotin each diagram, which corresponds to the case p = 2 with w re = 1 /
3, yields definitivepredictions for { n S , r } which can be inferred from (3.28). Notice that N re and T re occupydifferent regions of space in the diagram for different values of p . Also note that the reheatingtemperature is higher for p = 1 ( T re ≥ GeV) than for p = 3 ( T re ≥ e -foldings N inf k , and hencethe CMB observables { n S , r } , into different ranges of parameter space for different valuesof p . This facilitates the breaking of degeneracies associated with the T-model α -attractorpotential (2.1), which was illustrated in figure 1. Note that the tabulated constraints onthe CMB observables have been obtained by taking into account the conservative reheatingconstraints N re > T re ∈ [1 MeV , GeV].– 17 – .
950 0 .
955 0 .
960 0 .
965 0 .
970 0 .
975 0 . n S N r e T-Model λ = 0 . p = 1 p = 3 (a) .
950 0 .
955 0 .
960 0 .
965 0 .
970 0 .
975 0 . n S N r e T-Model λ = 0 . p = 1 p = 3 (b) .
950 0 .
955 0 .
960 0 .
965 0 .
970 0 .
975 0 . n S − T r e / M e V T-Model λ = 0 . BBN Constraint p = 1 p = 3 (c) .
950 0 .
955 0 .
960 0 .
965 0 .
970 0 .
975 0 . n S − T r e / M e V T-Model λ = 0 . BBN Constraint p = 1 p = 3 (d) Figure 6 : Constraints on the reheating parameters N re ( top row ) and T re ( bottom row ),given by equations (3.18) and (3.17) respectively, have been illustrated for the T-model α -attractor (2.1) for two different value of the inflationary parameter λ , namely λ = 0 . leftpanel ) and λ = 0 . right panel ). The red curve in each diagram corresponds to p = 1and hence w re = 0 in (2.3), while the green curve corresponds to p = 3 and hence w re = 1 / p = 2 with w re = 1 /
3, yieldsdefinitive predictions for { n S , r } as can be inferred from (3.28). We notice that the reheatingtemperature is typically higher for p = 1, notably T re ≥ GeV. The horizontal band in allfigures corresponds to the Planck bound n S = 0 . ± . N re and T re lyingoutside of this bound are disfavoured by CMB observations [22]. We therefore conclude thatdifferent values of p in the inflationary T-model potential (2.1) result in different relationsfor N re ( n S ) and T re ( n S ). This effectively breaks the CMB degeneracy illustrated in figure 1.– 18 – Observables p = 1, w re = 0 p = 2, w re = 1 / p = 3, w re = 1 / N inf k [50.4931, 55.686] 55.72 [55.724, 60.383] n S [0.9607, 0.9643] 0.9643 [0.96435, 0.967] r [0.003867, 0.004683] 0.003899 [0.003326, 0.003896] N re [0, 20.9624] 0 [0, 37.5836] T re [3 . × , 2 . × ] GeV 2 . × GeV [10 − , 2 . × ] GeV0.8 N inf k [50.6026, 55.17] 55.1749 [55.1761, 59.8198] n S [0.9607, 0.9639] 0.9639 [0.9639, 0.966706] r [0.001009, 0.001198] 0.00101 [0.0008604, 0.00101] N re [0, 18.4401] 0 [0, 37.4696] T re [2 . × , 2 . × ] GeV 2 . × GeV [10 − , 2 . × ] GeV Table 1 : This table demonstrates that by taking into account the reheating constraints N re > T re ∈ [1 MeV , GeV], the number of inflationary e -foldings N inf k and hencealso the associated CMB observables { n S , r } , get segregated into different ranges ofparameter space for different values of p . This allows one to break the degeneraciesassociated with the T-model α -attractor potential (2.1) illustrated in figure 1. α -attractor After inflation ends in the E-model, the inflaton begins to oscillate around the minimum ofthe E-model potential (2.7) which takes the form V ( φ ) ≃ V (cid:18) λ φm p (cid:19) p ; p = 1 , , ...., Assuming the effective reheating EOS to be w re = h w φ i = p − p +1 , similar conclusions aredrawn for the E-model as were obtained earlier for the T-model. Our results, described infigure 7, and tabulated in table 2, indicate that the existing degeneracies in the E-model,illustrated in figure 2, are easily broken by taking into consideration the kinematics of re-heating. Next we apply the techniques developed in section 3.1 to inflation in the non-canonicalscenario discussed in section 2.2. The unusual oscillatory EOS in the quadratic potential(2.17) namely , − / < D w NC φ E <
0, does not permit one to break the degeneracies which existin this model, and which arise because the scalar spectral index n S does not depend upon thenon-canonical parameter α ; see eqn. (2.18) and figure 3. However reheating considerationsdo succeed in placing strong constraints on the reheating temperature in this model which isconfined to fairly high values T re ≥ GeV, as shown in figure 8.
Relic gravitational waves are a generic prediction of the inflationary scenario [7]. Thesetensor fluctuations, which are created quantum mechanically, get stretched to super-Hubble– 19 – .
950 0 .
955 0 .
960 0 .
965 0 .
970 0 .
975 0 . n S N r e E-Model λ = 1 p = 1 p = 3 (a) .
950 0 .
955 0 .
960 0 .
965 0 .
970 0 .
975 0 . n S N r e E-Model λ = 2 p = 1 p = 3 (b) .
950 0 .
955 0 .
960 0 .
965 0 .
970 0 .
975 0 . n S − T r e / M e V E-Model λ = 1 BBN Constraint p = 1 p = 3 (c) .
950 0 .
955 0 .
960 0 .
965 0 .
970 0 .
975 0 . n S − T r e / M e V E-Model λ = 2 BBN Constraint p = 1 p = 3 (d) Figure 7 : Constraints on the reheating parameters N re ( top row ) and T re ( bottom row ),given by equations (3.18) and (3.17) respectively, have been illustrated for the E-model α -attractor (2.7) for two different value of the inflationary parameter λ , namely λ = 0 . leftpanel ) and λ = 0 . right panel ). The red curve in each diagram corresponds to p = 1and hence w re = 0 in (2.3), while the green curve corresponds to p = 3 and hence w re = 1 / p = 2 with w re = 1 /
3, yieldsdefinitive predictions for { n S , r } as can be inferred from (3.28). We notice that the reheatingtemperature is typically higher for p = 1, notably T re ≥ GeV. The horizontal band in allfigures corresponds to the Planck bound n S = 0 . ± . N re and T re lyingoutside of this bound are disfavoured by CMB observations [22]. We therefore conclude thatdifferent values of p in the inflationary E-model potential (2.7) result in different relationsfor N re ( n S ) and T re ( n S ). This effectively breaks the CMB degeneracy illustrated in figure 2.– 20 – Observables p = 1, w re = 0 p = 2, w re = 1 / p = 3, w re = 1 / N inf k [49.2795, 55.4335] 55.6419 [55.4944, 60.1534] n S [0.9607, 0.965] 0.9646 [0.9644, 0.9672] r [0.002372, 0.002873] 0.002513 [0.00211, 0.002471] N re [0, 24.8427] 0 [0, 37.5851] T re [2 . × , 2 . × ] GeV 2 . × GeV [10 − , 2 . × ] GeV2 N inf k [50.2202, 54.9935] 55.0077 [55.0125, 59.6497] n S [0.9607, 0.9641] 0.964 [0.9639, 0.9667] r [0.000639, 0.000764] 0.000645 [0.000552, 0.000648] N re [0, 19.3462] 0 [0, 37.413] T re [1 . × , 2 × ] GeV 2 × GeV [10 − , 2 × ] GeV Table 2 : This table demonstrates that by taking into account the reheating constraints N re > T re ∈ [1 MeV , GeV], the number of inflationary e -foldings N inf k and hencealso the associated CMB observables { n S , r } , get segregated into different ranges ofparameter space for different values of p . This allows one to break the degeneraciesassociated with the E-model α -attractor potential (2.7) which were illustrated in figure 2. .
950 0 .
955 0 .
960 0 .
965 0 .
970 0 .
975 0 . n S N r e Non-canonical m φ α = 5 α = 10 .
950 0 .
955 0 .
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970 0 .
975 0 . n S − T r e / M e V BBN Constraint
Non-canonical m φ α = 5 α = 10 Figure 8 : Constraints on the duration of reheating N re and the reheating temperature T re are shown for the non-canonical quadratic potential discussed in section 2.2. One finds thatthe inflationary degeneracies shown in figure 3 are not quite cured in this case, due to the factthat D w NC φ E <
0. Note that the reheating temperature is quite high, namely T re ≥ GeV.The horizontal band in both figures corresponds to the Planck bound n S = 0 . ± . (cid:3) h ij = 0 where h ij = φ k ( τ ) e − i k . x e ij and e ij is the polarizationtensor with τ being the conformal time defined by τ = R dt/a ( t ). As a result, each of thetwo polarization states of the graviton h × , + ( k ) = φ k ( τ ) m p e − i k . x , satisfies the equation φ ′′ k + 2 a ′ a φ ′ k + k φ k = 0 (4.1)where the derivative is with respect to the conformal time τ . For near-exponential inflation, a = τ /τ ( | τ | < | τ | ).Eqn. (4.1) implies that the amplitude of a tensor Fourier mode freezes to a constant valuein the super-Hubble limit. The corresponding dimensionless amplitude of a tensor mode isrelated to the inflationary Hubble parameter H inf k at Hubble exit by P GW ( k ) ≡ h × , + ( k ) ≃ π (cid:18) H inf k m p (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) k = aH . (4.2)The inflationary tensor fluctuations are often defined in terms of a different normalizationof the Fourier mode tensor amplitude φ k , which yields the following tensor power spectrum(see appendix B and [6, 39, 41]) P T = 2 π (cid:18) H inf k m p (cid:19) = 4 × P GW ( k ) , (4.3)and the tensor-to-scalar ratio is defined, in terms of P T , to be r = P T P R . (4.4)The power spectrum P GW ( k ) can be written as P GW ( k ) = P GW ( k ∗ ) (cid:18) kk ∗ (cid:19) n T , (4.5)where the tensor power at the CMB pivot scale k ∗ = 0 .
05 Mpc − is given, in terms of thescalar power (see appendix B), by A GW ≡ P GW ( k ∗ ) = 14 r A S = r × . × − , (4.6)and the tensor tilt is found to be n GW ≡ n T = d log P GW ( k ) d log k = − r , (4.7)which satisfies the consistency relation.The quantum mechanically generated tensor modes discussed above, which becomesuper-Hubble during inflation, make their Hubble re-entry at late times when k = aH andbehave like stochastic GWs in the universe [38, 39]. The physical frequency of these stochasticGWs at the present epoch is given by – 22 – = 12 π (cid:18) ka (cid:19) = 12 π (cid:18) aa (cid:19) H , (4.8)where a , H correspond to the scale factor and Hubble parameter of the universe during theepoch when the corresponding tensor mode makes its Hubble re-entry. In this work, wefocus on the relic GWs that become sub-Hubble prior to the matter-radiation equality, sothat their characteristic frequency is large enough to enable them to be detected by the GWobservatories in the near future. Expressing H in terms of temperature, we get Hm p = (cid:18) ρ m p (cid:19) = π g T T m p ! = π (cid:16) g T (cid:17) (cid:18) Tm p (cid:19) . (4.9)Using entropy conservation (discussed in appendix A), we obtain aa = (cid:18) a eq a (cid:19) (cid:18) g s eq g sT (cid:19) / (cid:18) T eq T (cid:19) . (4.10)Substituting H from equation (4.9) and a/a from equation (4.10) in (4.8), we obtainthe following important expression for the present day frequency of GWs in terms of theirHubble re-entry temperature. f = 7 . × − Hz (cid:18) g s g sT (cid:19) (cid:16) g T (cid:17) (cid:18) T GeV (cid:19) . (4.11)In figure 9, we illustrate the present day frequency of relic GWs as a function of theirHubble re-entry temperature along with the sensitivity bands of future GWs observatoriesLISA and BBO [42, 43]. The values of f corresponding to relic GWs that became sub-Hubbleat a number of important cosmic epochs are tabulated in table 3. Epoch Temperature T GW Present day f (in Hz) Matter-radiation equality ∼ . × − CMB pivot scale re-entry ∼ . × − Big Bang Nucleosynthesis ∼ . × − Electro-weak symmetry breaking ∼
100 GeV 2 . × − Table 3 : Present day frequencies of relic GWs have been tabulated for four differenttemperature scales, associated with the Hubble re-entry of the respective primordial tensormodes. In order to probe the epoch of reheating using relic GWs, the physical frequencycorresponding to tensor modes which become sub-Hubble during reheating must satisfy f > f
BBN ≃ − Hz in order to obey the BBN bound.The present day spectral density of stochastic GWs, defined in terms of the criticaldensity at the present epoch ρ c by [38, 39]– 23 – − − − T /
MeV − − − − f / H z BBOLISA
Figure 9 : The characteristic present day frequency of stochastic relic GWs is plotted as afunction of the temperature of the universe at which the corresponding tensor modes becamesub-Hubble. This figure also illustrates the frequency bands of future GWs detectors such asLISA and BBO. Ω GW ( f ) ≡ ρ c dρ ( f ) d log f , (4.12)is given by the following set of equations [10, 39, 40]Ω (MD) GW ( f ) = 38 π P GW ( f ) Ω m (cid:18) ff h (cid:19) − , f h < f ≤ f eq (4.13)Ω (RD) GW ( f ) = 16 π P GW ( f ) Ω r , f eq < f ≤ f re (4.14)Ω (re) GW ( f ) = Ω (RD) GW (cid:18) ff re (cid:19) (cid:16) w − / w +1 / (cid:17) , f re < f ≤ f e (4.15)where f h , f eq , f re , f e refer to the present day frequency of relic GWs corresponding to tensormodes that became sub-Hubble at: the present epoch ( f h ), the epoch of matter-radiationequality ( f eq ), at the end of reheating (commencement of the radiation dominated epoch, f re ) and at the end of inflation ( f e ). While the superscripts ‘MD’, ’RD’, ‘re’ in Ω GW referto matter dominated epoch, radiation epoch and the epoch of reheating respectively. Notethat f re > f BBN ≃ − Hz in order to satisfy the BBN bound on T re . Regarding the EOS w = w re during the epoch of reheating, it is important to keep in mind the following points. • In the case of perturbative reheating, the value of w re ≡ (cid:10) w φ (cid:11) is given by (2.3) forcanonical scalars, namely (cid:10) w φ (cid:11) = p − p + 1 , p ≥ D w NC φ E = − (cid:18) α − α − (cid:19) , α ≥ . (4.17)Note that the range permitted for non-canonical scalars − / < D w NC φ E ≤ ≤ (cid:10) w φ (cid:11) < . (4.19)Consequently, the GW spectrum (4.15) can easily distinguish between canonical andnon-canonical models of inflation as illustrated in figure 10. • In the case of non-perturbative reheating, the physics of the reheating epoch can bequite complex. In this case w re is sometimes assumed to be a constant, for the sake ofsimplicity [24–26].From the equations developed earlier in this section and (4.14),(4.15), it follows that thespectral density of stochastic GWs corresponding to modes that became sub-Hubble prior tomatter-radiation equality is Radiation dominated epoch: Ω (RD) GW ( f ) = (cid:0) π (cid:1) r A S (cid:16) ff ∗ (cid:17) n T Ω r , f eq < f ≤ f re , (4.20) During reheating: Ω (re) GW ( f ) = Ω (RD) GW ( f ) (cid:16) ff re (cid:17) (cid:16) w − / w +1 / (cid:17) , f re < f ≤ f e , (4.21)where we have used P GW ( f ) = P GW ( f ∗ ) (cid:16) ff ∗ (cid:17) n T , with A GW ≡ P GW ( f ∗ ) = r A S fromequation (4.6). Note that f ∗ is the physical frequency (of GW) corresponding to the CMBpivot scale comoving wave number k ∗ .Equations (4.5), (4.13), (4.14), (4.15) allow us to define a local post-inflationary gravi-tational wave (tensor) spectral index as follows n PI GW = d log Ω GW ( k ) d log k = d log Ω GW ( f ) d log f (4.22)where n PI GW = n GW + 2 (cid:18) w − / w + 1 / (cid:19) (4.23)which implies n PI GW > n GW for w > / n PI GW = n GW for w = 1 / n PI GW < n GW for w < / w is the background EOS and is given by w = 0 during matter domination, w = 1 / (cid:10) w φ (cid:11) in (2.3) and (2.16) during oscillations of canonicaland non-canonical scalars respectively.Note that since n GW ≃ − ǫ H , CMB constraints on the tensor-to-scalar ratio r = 16 ǫ H ≤ .
06, imply | n GW | ≤ . n GW is a very small quantity that does not generate anappreciable change in Ω GW ( k ) for more than 30 orders of magnitude variation in k (and hencein f ). Therefore (4.23) effectively reduces to n PI GW ≃ (cid:18) w − / w + 1 / (cid:19) . (4.24)– 25 – − − f/ Hz − − − − − Ω G W ( f ) h Canonical Inflation w re = h w φ i = p − p +1 T re = 10 GeV r = 0 . p = 1 p = 2 p = − − − − f/ Hz − − − − − Ω G W ( f ) h Non-canonical Inflation w re = h w NC φ i = − (cid:16) α − α − (cid:17) T re = 10 GeV α = 10 α = 5 Figure 10 : This figure demonstrates that the spectra of relic GWs Ω GW ( f ) can easilydistinguish between canonical and non-canonical inflation. Top panel shows the spectrumof relic GWs in the canonical case for which the post-inflationary EOS is described by (4.16)with p = 1 , Bottompanel shows the same for non-canonical inflation for which the post-inflationary EOS isdescribed by (4.17) with α = 5 and 10, plotted in solid and dashed blue curves respectively.Thus the post-inflationary EOS has a direct bearing on the spectral index of relic gravitationalradiation with n PI GW ≥ w > / n PI GW ≃ w = 1 / n PI GW < ∼ w < / − − − − − − f/ Hz − − − − − − − − − Ω G W ( f ) h LISABBO LIGOBBN Constraint T r e = M e V p = p = p = p = 2 , w re = − − − − − − f/ Hz − − − − − − − − − Ω G W ( f ) h LISABBO LIGOBBN Constraint T re = 100 GeV p = p = p = p = 2 , w re = Figure 11 : This figure illustrates the potential implications of blue tilted relic GWs with p > w re > /
3) from the perspective of near future GW observatoriessuch as the advanced LIGO, LISA and BBO.
Top panel depicts the spectrum of bluetilted relic GWs corresponding to p = 3 , ,
5, plotted in solid, dashed and dotted greencurves respectively, for a fixed reheating temperature T re = 1 MeV and tensor-to-scalar ratio r ≃ . bottom panel shows the same but with a higher reheating temperature T re = 100 GeV. The dotted green curve in the top panel indicates that relic GWs with a lowenough reheating temperature T re ≤
100 MeV and EOS stiffer than w re = 2 / p >
5) would violate the BBN constraint Ω GW h ≤ − . Theories predicting such signalscan therefore be regarded as being unphysical.in the universe.Setting n GW = 0 for simplicity, one gets, for the different cosmological epochs, the result • Matter domination ( w = 0) ⇒ n PI GW ( k ) (cid:12)(cid:12)(cid:12)(cid:12) MD = − • Radiation domination ( w = 1 / ⇒ n PI GW ( k ) (cid:12)(cid:12)(cid:12)(cid:12) RD ≃ − − − − − − f / Hz − − − − − − − − − Ω G W ( f ) h LISA BBO LIGOBBN Constraint T r e = M e V p = , w r e = p = 2 , w re = T-Model λ = 0 . T r e = M e V T r e = G e V p = 1 , w re = 0 Figure 12 : The spectrum of relic gravitational waves is shown for the T-model α -attractorpotential (2.1) for λ = 0 .
4. The dotted black curve corresponds to p = 2 for which thepost-inflationary EOS is radiation-like, with w re = (cid:10) w φ (cid:11) = 1 /
3, see (2.3), and the GWspectrum is flat. The solid and dashed green curves correspond to p = 3 and reheatingtemperatures T re = 1 MeV and 100 GeV respectively, for which the post-inflationary EOS is w re = (cid:10) w φ (cid:11) = 1 / T re ≤ GeV. The solid and dashed purple curves correspond toreheating temperatures T re = 1 MeV and 100 GeV respectively, and to a matter-like postinflationary EOS w re = (cid:10) w φ (cid:11) = 0 which arises for p = 1. Note that in this case GWs have ared tilt and their amplitude is suppressed relative to p = 2 , • During the pre-radiation epoch the GW spectrum depends upon the EOS during reheating.In the context of perturbative reheating, which is relevant for this work, one finds n PI GW ( k ) (cid:12)(cid:12)(cid:12)(cid:12) OSC = 2 (cid:18) p − p − (cid:19) canonical oscillatory epoch n PI GW ( k ) (cid:12)(cid:12)(cid:12)(cid:12) NC OSC = 2 (2 − α ) noncanonical oscillatory epoch (4.27)where p refers to the exponent in the inflationary potentials (2.1) & (2.7). Canonical os-cillatory epoch refers to post-inflationary oscillations of a canonical scalar field with p ≥ non-canonical scalar field with α ≥ n GW − n PI GW ( k ) ≤ n GW − n PI GW ( k ) ≥ p >
2, are potentially important from theobservational prospective, we discuss their implications in light of the ongoing and near futureGW observatories in figure 11. The top panel shows the spectrum of blue tilted relic GWscorresponding to p = 3 , ,
5, plotted in solid, dashed and dotted green curves respectively,for fixed reheating temperature T re = 1 MeV and tensor-to-scalar ratio r ≃ . T re = 100 GeV. Fromfigure 11, we conclude that • Relic GWs can be observed by LISA in the case of low reheating temperature, T re ∼ −
100 MeV and stiff enough reheating EOS w re > .
5, corresponding to p > • However for w re > /
3, corresponding to p >
5, with a low reheating temperature T re <
100 MeV, the spectrum of relic GWs would violate the BBN constraint Ω GW h ≤ − (as indicated by the dotted green curve in the top panel of figure 11). Hence thecorresponding parameter space of { p, T re } is ruled out by the BBN constraint, eventhough it would have been possible to detect the signal by the advanced LIGO detectors.Similar conclusions were also drawn in [39]. • The blue-tilted relic GW spectrum can be detected by the BBO, for a range of reheatingtemperatures T re ≤ GeV.In marked contrast to perturbative reheating, in models with non-perturbative reheatingthe reheating/preheating epoch can be a complex affair with explosive (resonant) particleproduction, backreaction and non-equilibrium field theory all playing a significant role untilthermalization is finally reached. For simplicity this epoch is usually characterised (see [24–27]) by a constant effective EOS parameter, w re , so that the general formulae (4.20) – (4.23)also have bearing on this scenario.In figures 1 and 2 we showed that the T and E model α -attractors exhibited a degeneracysince, for λ > ∼ .
2, different values of p in the T and E model potentials (2.1) & (2.7) gave riseto identical values of the CMB parameters { n S , r } . In section 3.4 we demonstrated that thisdegeneracy was easily broken if one took into account the reheating predictions encoded inthe parameters N re , T re . We now show that the degeneracy in { n S , r } can also be broken bythe GW spectrum since the latter depends explicitly on the reheating EOS, which, in turn,depends upon the value of p through (4.21) and (4.24).This can be seen from figure 12 for the T-model and from figure 13 for the E-model. Inboth models one notices that for p = 2 ⇒ w re = (cid:10) w φ (cid:11) = 1 / p such as p = 3 ⇒ w re = (cid:10) w φ (cid:11) = 1 / p = 1 ⇒ w re = (cid:10) w φ (cid:11) = 0, is red-tilted with n PI GW ( k ) ≃ −
2, and is suppressed relative to the other two cases.Note that the gravity wave spectrum for the T-model in figure 12 is shown with thevalue of the inflationary parameter λ in (2.1) set at λ = 0 .
4. For the E-model in figure 13,on the other hand, we have chosen λ = 1. Our choice for λ is motivated by the followingconsiderations:1. These values of λ correspond to the degeneracy regions in which different values of p result in the same values of { n S , r } ; see figures 1 & 2.– 29 – − − − − − − f / Hz − − − − − − − − − Ω G W ( f ) h LISA BBO LIGOBBN Constraint T r e = M e V p = , w r e = p = 2 , w re = p = 1 , w re = 0 E-Model λ = 1 T r e = M e V T r e = G e V Figure 13 : The spectrum of relic gravitational waves is shown for the E-model α -attractorpotential (2.7) for λ = 1. The dotted black curve corresponds to p = 2 for which thepost-inflationary EOS is radiation-like, with w re = (cid:10) w φ (cid:11) = 1 /
3, see (2.3), and the GWspectrum is flat. The solid and dashed green curves correspond to p = 3 and reheatingtemperatures T re = 1 MeV and 100 GeV respectively, for which the post-inflationary EOS is w re = (cid:10) w φ (cid:11) = 1 / T re ≤ GeV. The solid and dashed purple curves correspond toreheating temperatures T re = 1 MeV and 100 GeV respectively, and to a matter-like postinflationary EOS w re = (cid:10) w φ (cid:11) = 0 which arises for p = 1. Note that in this case GWs have ared tilt and their amplitude is suppressed relative to p = 2 , λ correspondsto a smaller value of the tensor-to-scalar ratio r . Our choice of λ = 0 . λ = 1 (E-model) corresponds to r ∼ − which lies within the observable range ofupcoming CMB missions such as the CMB-S4 [49] and the Simons Observatory [50]. The inflationary paradigm often exhibits degeneracies, with two (or more) models predictingessentially the same values of { n S , r } , leading to the existence of ‘cosmological attractors’ or‘universality classes’ of inflation [19–21]. Such degeneracies render difficulties for the CMBobservations alone to constitute a unique probe of the inflationary dynamics. Such degenera-cies usually emerge either because multiple inflationary potentials make similar predictionsfor the scalar spectral index n S and the tensor-to-scalar ratio r , or because within the samemodel, the predicted values of { n S , r } are insensitive to some of the model parameters in– 30 –he potential. In this work, we have demonstrated the existence of inflationary degeneraciesin two classes of α -attractor inflationary models, namely the T-model and E-model [19, 20]discussed in section 2.1. Inflationary degeneracies have also been shown to exist in the non-canonical framework of inflation [31, 32] discussed in section 2.2.In the context of the α -attractors, we have shown that the scalar spectral index n S be-comes insensitive to the potential parameter λ (related to the curvature of the superconformalK¨ahler metric [20]) as well as the exponent p , for λ > O (0 . r ,decreases with an increase in λ , and becomes insensitive to the exponent p for λ > O (0 . α in (2.11) innon-canonical inflation, as demonstrated in figure 3.In section 3 we provided an introduction to the kinematics of reheating in terms of thereheating parameters { w re , N re , T re } , and spelled out our strategy (developed along the linesof [24–26]) for yielding tighter constraints on the CMB observables by taking into accountreheating constraints developed for the case of perturbative reheating. In sections 3.4 and3.5, we demonstrated that the inflationary degeneracies of the T-model and the E-model α -attractors can be easily broken by noting that the reheating EOS is very sensitive to theparameter p . In particular, we showed that imposing the liberal reheating constraints N re > T re ∈ [1 MeV , GeV] on the α -attractor potentials, the CMB predictions for { n S , r } got segregated into different regions of space, as illustrated in figure 6 and 7.However for the case of quadratic potential in the non-canonical framework, we foundthat reheating constraints are not able to break the degeneracy in { n S , r } appreciably, owingto the fact that the non-canonical EOS, obeying D w NC φ E <
0, does not include the criticalpoint of segregation w re = 1 / { n S , r } , the corresponding spectra of relic GWs couldhelp distinguish between different values of p in the T-model and the E-model α -attractors.In particular, as illustrated in figure 12 and 13, we found that the relic GW backgroundgenerated for a stiffer reheating equation of state could be detectable by the future spacebased GWs observatory BBO. We also concluded that for p = 3, the GW spectra do notpossess enough power to reach LISA sensitivity, in agreement with the conclusions drawnin [39]. We plan to investigate the existence of degeneracies in a wider class of inflationarymodel, including the universality class corresponding to the non-minimally coupled power-lawpotentials [51] in a future work.In a scenario in which inflation is followed by a long duration of reheating ( N re ≥ w re = h w φ i > / w re ≥ V ( φ ) ∝ φ p during reheating while possessingasymptotically flat wing/wings for larger field values. In the context of α -attractors, theregime of scalar field fragmentation corresponds to λ ≫ p = 1, theeffective equation of state remains w re = h w φ i = 0, independently of whether fragmentationoccurs or not [53]. For p = 2, fragmentation occurs for λ ≫ e -folds, theeffective EOS becomes w re ≃ / h w φ i . So our analysis for p = 1 ,
2– 31 –emains unaltered independently of the value of λ . Coming to the case of p = 3, fragmentationoccurs for λ ≫ e -folds, becomes w re ≃ / h w φ i = 1 /
2. However fragmentation is effective only for λ ≫
1, for whichthe tensor to scalar ratio is extremely small, namely r ≪ − , rendering it undetectableby planned CMB missions. We therefore conclude that, as long as λ is not too large, ouranalysis remains robust even for the case of p = 3.Our analysis in this paper was carried out within the framework of the perturbativetheory of reheating. In the case of non-perturbative reheating, the physics of the reheatingepoch can be quite complex. In this case, one usually assumes the effective w re to be aconstant for the sake of simplicity [24–26]. During the initial stage of preheating , particleproduction occurs in a rapid and explosive manner due to parametric resonance. However thebackreaction of created particles usually leads to the termination of the resonance after whichthe universe reheats via the slow perturbative decay of the inflaton. As long as the durationof preheating is short, ∆ N ≤
1, and the inflaton dominates the energy budget of the universe,our analysis may also be extended to the case of non-perturbative reheating by assuming thata fraction ‘q’ of the energy density ρ e remains in the inflaton after the termination of theresonance. Accordingly the modified form of equation (3.18) can be written as N re = 41 − w re " . − N inf k − log ( q V e ) H inf k ! . (5.1)This and related issues will be discussed in greater detail in a companion paper. S.S.M. thanks the Council of Scientific and Industrial Research (CSIR), India, for financialsupport as senior research fellow. Varun Sahni was partially supported by the J. C. BoseFellowship of Department of Science and Technology, Government of India. A.A.S. waspartially supported by the Russian Foundation for Basic Research grant No. 20-02-00411.
A Kinematics during reheating
From entropy conservation in the universe during the post reheating radiation dominatedhot Big Bang epoch we can relate the temperature T at any epoch to the scale factor a (andhence redshift z ) of the universe, through the known temperature T eq and scale factor a eq atthe matter-radiation equality in the following way a g sT T = a g s eq T , (A.1)where g sT and g s eq are the effective relativistic degrees of freedom in entropy. Hence T = (cid:18) g s eq g sT (cid:19) (cid:16) a eq a (cid:17) T eq . (A.2)From the causal diagram in figure 4, the epoch at which an observable CMB scale makes itsHubble re-entry is giving by k = a ( z ) H ( z ) , (A.3)– 32 –here the Hubble parameter is given in terms of the radiation density ρ by H = 13 m p ρ = 13 m p π T . (A.4)Incorporating (A.2) into the above equation and substituting the subsequent expressionof Hubble parameter in (A.3), we can obtain the value of a mode k which made its Hubblere-entry at an epoch z with a temperature T . For example using the known values of z eq and T eq , we obtain the value of the CMB scale that made its Hubble re-entry during the matterradiation equality to be k eq = 0 .
013 Mpc − . (A.5)It is important to know that the given value ‘ x ’ of a CMB scale k , including the pivot scale k ∗ , should strictly be expressed in the form of k = x a Mpc − . However, following thestandard convention in the literature, assuming a = 1 implicitly, we will continue expressing k = x Mpc − . Similarly the scale corresponding to the Hubble radius at the present epoch isgiven by k = 2 . × − Mpc − . (A.6)The epoch z p at which the CMB pivot scale k = k ∗ = 0 .
05 Mpc − makes its Hubblere-entry is obtained to be 1 + z p ≃ . × , (A.7)which implies that the CMB pivot scale, which satisfies k ∗ = 0 .
05 Mpc − > k eq , consequentlybecame sub-Hubble during the radiation dominated epoch prior to the matter-radiationequality.In order to obtain expressions for the duration of reheating N re = log ( a re /a e ) as wellas the temperature T re at the end of reheating, we begin by matching the comoving Hubbleradius at the Hubble exit of the CMB pivot scale during inflation (see figure 4) k = a k H inf k = 0 .
05 Mpc − , (A.8) ⇒ a k H inf k a H = ka H , ⇒ (cid:18) a k a e (cid:19) (cid:18) a e a re (cid:19) (cid:18) a re a eq (cid:19) (cid:18) a eq a (cid:19) (cid:18) H inf k H (cid:19) = (cid:18) ka H (cid:19) . Taking logarithm of the above expression, we obtainlog (cid:18) ka H (cid:19) = − N inf k − N re − N RD − log (1 + z eq ) + log (cid:18) H inf k H (cid:19) . (A.9)Assuming the effective reheating equation of state w re to be a constant, we obtain the fol-lowing expression by matching the density at the end of reheating to the density at the endof inflation. ρ re = ρ e (cid:18) a e a re (cid:19) w re ) , (A.10)from which we can obtain the expression for the duration of reheating to be– 33 – re ≡ log (cid:18) a re a e (cid:19) = 13(1 + w re ) log (cid:18) ρ e ρ re (cid:19) . (A.11)In the radiation dominated epoch, since ρ re = π g re T re , equation (A.11) becomes N re = 13(1 + w re ) log ρ eπ g re T re ! , (A.12)where g re ≡ g ( T re ) is the effective number of relativistic degrees of freedom in energy densityat the end of reheating. From the entropy conservation, using (A.2), we get T re = (cid:18) g s eq g s re (cid:19) (cid:18) a eq a re (cid:19) T eq . (A.13)Using (A.13) in (A.12), we get N re = 13(1 + w re ) log (cid:18) π g re (cid:19) ρ e T eq (cid:18) a re a eq (cid:19) (cid:18) g s re g s eq (cid:19) , (A.14)which becomes N re = 43(1 + w re )
14 log (cid:18) π g re (cid:19) + 13 log (cid:18) g s re g s eq (cid:19) + log ρ e T eq − N RD . (A.15)Substituting the expression for N RD from (A.9) in (A.15), we obtain N re = − w re )
14 log (cid:18) π g re (cid:19) + 13 log (cid:18) g s re g s eq (cid:19) + log ρ e H inf k + log (cid:18) ka T eq (cid:19) + N inf k + log (1 + z eq ) . (A.16)Using the relation T eq = (1 + z eq ) T , we get N re = − w re )
14 log (cid:18) π g re (cid:19) + 13 log (cid:18) g s re g s (cid:19) + log ρ e H inf k + log (cid:18) ka T (cid:19) + N inf k + N re . (A.17)Assuming w = 1 / N re on the right hand side of the aboveequation to the left hand side, we obtain the following expression N re = − − w re N inf k + log ρ e H inf k + log (cid:18) ka T (cid:19) + 14 log (cid:18) π g re (cid:19) + 13 log (cid:18) g s re g s (cid:19) . (A.18)For slow-roll inflation, using the fact that ρ e = V e and using k = k ∗ = 0 .
05 Mpc − , T = 2 . g re = g s re = 106 .
75 and g s = 3 .
94, we obtain the following master formula forthe duration of reheating in terms of w re , N inf k and the inflationary parameters H inf k , V e . N re = 41 − w re . − N inf k − log V e H inf k , w re = 1 / . (A.19)– 34 –onsequently, from equation (A.12), the expression for the reheating temperature T re be-comes T re = (cid:18) π g re (cid:19) V e e − (1+ w re ) N re . (A.20)Note that for T re <
100 GeV, the effective number of relativistic degrees of freedom g re , g s re are smaller than 106.75 and vary with temperature. At the beginning of BigBang Nucleosynthesis, T BBN ≃ g re = g s re ≃ .
75. For T re ∈ (1 MeV ,
100 GeV), the variation of g re and g s re with temperature can be incorporatedby using lattice QCD calculations. However the variation has a small effect on N re and hencewe ignore it in our analysis. B CMB constraints on inflation
Consider the case of a canonical scalar field minimally coupled to gravity with potential V ( φ ) = V f (cid:18) φm p (cid:19) . (B.1)The potential slow-roll parameters are given by ǫ V = m p (cid:18) f ′ f (cid:19) , (B.2) η V = m p (cid:18) f ′′ f (cid:19) . (B.3)In the slow-roll limit ǫ V , η V ≪
1, the scalar power spectrum is given by [6] P R ( k ) = A S (cid:18) kk ∗ (cid:19) n S − , (B.4)with the amplitude of scalar power spectrum at the CMB pivot scale k ≡ k ∗ = 0 .
05 Mpc − is given by[22] A S ≡ P R ( k ∗ ) = 124 π V m p f ( φ k ) ǫ V ( φ k ) (cid:12)(cid:12)(cid:12)(cid:12) k = k ∗ , (B.5)and the scalar spectral index (with negligible running) is given by n S = 1 + 2 η V ( φ ∗ ) − ǫ V ( φ ∗ ) , (B.6)where φ ∗ is the value of the inflaton field at the Hubble exit of CMB pivot scale k ∗ . Similarlythe tensor power spectrum, in the slow-roll limit, is given by P T ( k ) = A T (cid:18) kk ∗ (cid:19) n T , (B.7)with the amplitude of tensor power spectrum at the CMB pivot scale is given by A T ≡ P T ( k ∗ ) = 2 π (cid:18) H inf k m p (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) k = k ∗ ≃ π V m p f ( φ k ) (cid:12)(cid:12)(cid:12)(cid:12) k = k ∗ , (B.8)– 35 –nd the tensor spectral index (with negligible running) is given by n T = − ǫ V ( φ ∗ ) . (B.9)The tensor-to-scalar ratio r is defined by r ≡ A T A S = 16 ǫ V ( φ ∗ ) , (B.10)yielding the single field consistency relation r = − n T . (B.11)From the CMB observations of Planck 2018 [22], we have A S = 2 . × − , (B.12)while the 1 σ constraint on the scalar spectral index is given by n S = 0 . ± . . (B.13)Similarly constraint on the tensor-to-scalar ratio r , from the combined observations of Planck2018 [22] and BICEPII/Keck [23], is given by r ≤ . , (B.14)which translates to the fact that A T ≤ × − A S , putting an upper bound on the inflationaryHubble scale H inf k from equation (B.8) as well as the energy scale during inflation T inf , givenby H inf k ≤ . × − m p = 6 . × GeV , (B.15) T inf ≡ (cid:18) m p (cid:16) H inf k (cid:17) (cid:19) / ≤ . × GeV . (B.16)Similarly the CMB bound on r translates directly to an upper bound on the first slow-rollparameter ǫ V ≤ . , (B.17)rendering the tensor tilt from equation (B.9) to be negligibly small | n T | ≤ . . (B.18)Given the upper limit on ǫ V , using the CMB bound on n S from (B.13) in (B.6), we infer thatthe second slow-roll parameter is negative and obtain interesting upper and lower limits onits magnitude, given by 0 . ≤ | η V | ≤ . . (B.19)The EOS w φ of the inflaton field is given by w φ = ˙ φ − V ( φ ) ˙ φ + V ( φ ) = − ǫ V ( φ ) (B.20)– 36 –n interesting consequence of (B.17) is the fact that, around the pivot scale, the EOS duringinflation is constrained to be w φ ≤ − . , (B.21)implying that the expansion of the universe during inflation was near exponential (quasi-deSitter like). End of inflation is marked by w φ = − / φ = V ( φ ).Hence the energy density of the inflaton at the end of inflation is given by ρ e ≡ ρ ( φ e ) = 12 ˙ φ + V ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) φ = φ e = 12 V ( φ e ) + V ( φ e ) = 32 V ( φ e ) (B.22) References [1] A. A. Starobinsky, Phys. Lett. B , 99 (1980).[2] A. H. Guth, Phys. Rev. D , 347 (1981).[3] A. D. Linde, Phys. Lett. B , 389 (1982).[4] A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. , 1220 (1982).[5] A. D. Linde, Particle Physics and Inflationary Cosmology , Harwood, Chur, Switzerland (1990).[6] D. Baumann, TASI Lectures on Inflation, [arXiv:0907.5424].[7] A. A. Starobinsky, JETP Lett. , 682 (1979).[8] L. P. Grishchuk, Sov. Phys. JETP , 409 (1975).[9] A. A. Starobinsky, JETP Lett. , 438 (1981).[10] V. Sahni, Phys. Rev. D , 453 (1990).[11] L. Kofman, in Relativistic Astrophysics: A Conference in Honor of Igor Novikov’s 60thBirthday , Proceedings, Copen- hagen, Denmark, 1996, edited by B. Jones and D. MarcovicCambridge University Press, Cambridge, England [astro-ph/9605155][12] Y. Ema, R. Jinno, K. Mukaida and K. Nakayama, JCAP (2017) 045 [arXiv:1609.05209[hep-ph]].[13] M. He, R. Jinno, K. Kamada, S. C. Park, A. A. Starobinsky and J. Yokoyama, Phys. Lett. B , 36 (2019) [arXiv:1812.10099 [hep-ph]].[14] L. A. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. Lett. , 3195 (1994).[15] Y. Shtanov, J. Traschen and R. Brandenberger, Phys. Rev. D , 5438 (1995); also seeY. Shtanov, Ukr. Fiz. Zh. , 1425 (1993) (in Russian).[16] L. A. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. Lett. , 1011 (1996).[17] L. A. Kofman, A. D. Linde and A. A. Starobinsky, Phys. Rev. D , 3258 (1997).[18] A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large Scale Structure, CambridgeUniversity Press, 2000.[19] R. Kallosh and A. Linde, JCAP07 (2013) 002 [arXiv:1306.5220].[20] R. Kallosh, A. Linde and D. Roest, JHEP (2013) 198 [arXiv:1311.0472].[21] D. Roest, JCAP (2014) 007 [arXiv:1309.1285 [hep-th]].[22] Planck collaboration: P. A. R. Ade et al. , Planck 2018 results. X. Constraints on inflation A&A641, A10 (2020) [arXiv:1807.06211][23] P. A. R. Ade et al. [BICEP2 and Keck Array], Phys. Rev. Lett. (2018) 221301[arXiv:1810.05216 [astro-ph.CO]]. – 37 –
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