Gravitational neutrino reheating

Despite having important cosmological implications, the reheating phase is believed to play a crucial role in cosmology and particle physics model building. Conventionally, the model of reheating with an arbitrary coupling of inflaton to massless fields naturally lacks precise prediction and hence difficult to verify through observation. In this paper, we propose a simple and natural reheating mechanism where the particle physics model, namely the Type-I seesaw is shown to play a major role in the entire reheating process where inflaton is coupled with all the fields only gravitationally. Besides successfully resolving the well-known neutrino mass and baryon asymmetry problems, this scenario offers successful reheating, predicts distinct primordial gravitational wave spectrum and non-vanishing lowest active neutrino mass. Our novel mechanism opens up a new avenue of integrating particle physics and cosmology in the context of reheating.


I. INTRODUCTION
If the dominant decay channel of the inflaton is mediated by gravitons, can the universe be populated by radiation?Such a question has gained significant interest [1][2][3][4][5], particularly in the context of the modelindependent reheating scenario.Recently proposed gravitational reheating (GRe) [1,6] is one such interesting example, where the inflaton equation of state (w ϕ ) and inflationary energy scale are the only controlling parameter.Due to Planck suppression the gravitational decay of inflaton is expected to be maximal near the end of the inflation due to the large inflaton amplitude.Consequently the produced radiation redshifts as ρ R ∼ a −4 from the very beginning of reheating while inflaton goes as ρ ϕ ∼ a −3(1+w ϕ ) .This essentially renders the inflaton equation of state (w ϕ ) to be stiff w ϕ > 0.65 to achieve reheating temperature above BBN temperature T BBN ∼ 4 MeV [7][8][9].Further, w ϕ > 0.65 condition predicts blue tilted Primordial Gravitational Waves (PGWs) spectrum, and that is severely constrained by total massless degrees of freedom at BBN [10][11][12][13][14].This fact turned the GRe scenario viable only for a kination-like equation of state w ϕ ∼ 1 [1].
It is in this parlance we invoke a particle physicsinspired Type-I seesaw extension of the standard model (SM) and investigate its impact on the aforesaid GRe scenario.Type-I seesaw [15][16][17][18][19][20] is a minimal extension in the SM neutrino sector, which is known to generate active neutrino mass [21][22][23] and baryon asymmetric universe [14].In this submission, we show a new attributes to this model that it can also successfully reheat our universe without invoking any new physics in the inflaton sector.We call it gravitational neutrino reheating 1 .Additionally, unlike the GRe scenario, gravitational neutrino reheating is shown to be consistent with the BBN bounds of PGWs and further predicts the non-vanishing mass of the lowest active neutrino.
The paper is organized as follows: In Sec.II, we briefly discuss our reheating framework and its connection with the Type-I seesaw model.In Sec.III and IV, we review the gravitational production of RHNs and purely gravitational reheating (GRe) scenarios, respectively.In Sec.V, we discuss in detail the gravitational neutrino reheating.In Sec.VI, we analyze the possible constraints of leptogenesis.Finally, conclude in Sec.VII

II. TYPE-I SEESAW AND REHEATING FRAMEWORK
Type-I seesaw consists of three right-handed SM singlet massive Majorana neutrinos ν i R (i = 1, 2, 3) with the following lepton number violating Lagrangian where H = iσ 2 H * , with H(L) is the SM Higgs (lepton) doublet.Except usual inflaton parameters, Yukawa coupling matrix y ij , and Majorana mass M = diag(M 1 , M 2 , M 3 ) are the only beyond the SM parameters.After the inflation, all the particles are produced from the inflaton condensate through gravitational interaction only.For the reheating dynamics to occur, there-fore, the relevant Boltzmann equations are, where ρ ϕ , ρ r are the energy density of the inflaton, radiation respectively.n i 's are the number density of the RHNs.R i ϕ 's are the gravitational production rate of the RHNs [2] .Γ T ϕ is the total inflaton decay width, Γ ϕϕ→hh is the gravitational decay width [26] of inflaton to SM Higgs, and Γ i = (y † y) ii M i /(8π)'s are the decay width of the RHNs.Out of all the parameters, the following three {β = (y † y) 33 , M 3 , w ϕ }, related to ν 3 R and inflaton will be shown to control the reheating dynamics.The remaining parameters will give rise to active neutrino mass and baryogenesis.

III. RHNs PRODUCTION FROM INFLATON
Before its decay, the evolution of RHN number density (n i ) is expected to be governed by the Boltzmann equation, where, R i ϕ the production rate for ν i R from inflaton scattering mediated by gravity [2].The factor Σ i ∼ 10 −2 , is nearly constant and γ is related with the oscillating inflaton condensate 2 at the minimum of its potential.For inflation, we consider the α-attractor E-model potential [34][35][36] , where Λ ∼ 8 × 10 15 GeV is the inflation mass-scale fixed by the CMB power spectrum .The parameters (α, n) control the shape of the potential.The inflaton equation 2 γ is the frequency of the oscillation in the unit of m ϕ , γ = .
of state w ϕ is directly related with n, via the relation w ϕ = (n − 1)/(n − 1) [37].For this model, the timedependent inflaton mass can be expressed as [37,38] m 2 ϕ ∼ (m end ϕ ) 2 (a/a end ) −6 w ϕ , m end ϕ is the inflaton mass defined at the end of the inflation, We consider α = 1 throughout unless otherwise stated.
During the initial period of reheating inflaton is typically the dominant energy component and neglecting the decay term in Eq.(2a), ρ ϕ evolves as, ρ ϕ (a) = 3M 2 p H 2 end (a/a end ) −3(1+w ϕ ) , H end being the Hubble constant at the end of the inflation.M p = 2.4 × 10 18 GeV is the reduced Planck mass.Solving Eq.3 one obtains, where the effective initial number density of RHNs at the end of inflation is In the above expression, all are written in GeV and assume w ϕ = 0.With this ingredient in hand, we now discuss various causes and conditions of different phases of reheating.

IV. GRAVITATIONAL REHEATING (GRe)
In this scenario, the radiation dynamics is assumed to be solely controlled by inflaton via gravitational scattering, and using Eq.2b, one obtains, For the exact expression see, Ref. [2].The radiation produced from the neutrino decay can be solved by combining Eq.5 and Eq.2b as Using this we, therefore, define a critical decay width, Particularly, if the coupling is very large satisfying β > β c ϕ , immediately after its production, neutrinos are expected to decay and cannot lead the reheating process.Therefore, GRe would prevail.However, we mainly discuss the alternative possibilities.

V. GRAVITATIONAL NEUTRINO REHEATING
As mentioned earlier Γ 3 ≪ Γ 1 , Γ 2 , and hence ν 3 R is a long-lived particle compared to the other two.Because of that ν 3 R can compete with inflaton and alter the reheating process.Hence we have three major players to deal with; inflaton and ν 3 R and radiation.The energy density of the ν 3 R before its decay evolves as, The immediate observation from Eq.( 10) is that for w ϕ > 0, since inflaton dilutes faster than the ν 3 R , it starts to dominate at a = a ν where ρ ϕ ∼ ρ 3 .
Once ν 3 R dominates over the inflaton, its subsequent decay into SM fields would populate the universe.Since it dominates at a ν , ν 3 R decay before this point should be subdominant.Hence, one indeed expects another critical β c ν value below which only ν 3 R can dominate over the inflaton, and it must be at the instant when Γ 3 = β 2 M 3 /(8π) ∼ H(a ν ).This condition immediately gives us, ν .Using these parameters, we can say from Eq.( 11) that our universe becomes neutrino-dominated when a ν /a end ≃ 4 × 10 7 , roughly after 17 e-folding number from the beginning of reheating.We, therefore, have two distinct possibilities for the reheating, A. Neutrino dominating case: β ≤ β c ν As stated earlier, for this condition ν 3 R dominates over the inflaton after a = a ν , and universe becomes matter dominated as ρ 3 ∝ a −3 (see Eq.10).Hence, reheating will end once ν 3 R decays completely into radiation at Γ 3 ∼ H.During this period, the Hubble parameter evolves as H(a) = H end (a ν /a end ) −3(1+w ϕ )/2 (a/a ν ) −3/2 for a ≥ a ν , Utilizing this in Eq.(2b), radiation evolves as The evolution of radiation energy density is independent of w ϕ as expected.However, For (a < a ν ) radiation is populated by the inflaton via gravitational decay.The final reheating temperature (T re ) is determined by ν 3 R decay, as indeed can be seen in the top plot of Fig. 1, and this is realized only for w ϕ ≥ 1/3.The reheating ends at a re where Γ 3 ∼ H(a re ) is satisfied which leads to, Where, we utilized the relation ρ r = ϵ T 4 , for radiation temperature T with ϵ = (π 2 /30)g ⋆r .The relativistic degrees of freedom in the radiation g ⋆r ≃ 100.For example if we assume T re = T BBN = 4 MeV, one gets  For this case, whereas the inflaton controls the background reheating dynamics, the process of heating the thermal bath is still dominated by RHNs and determines the final reheating temperature.After the neutrino heating starts dominating, the radiation from ν 3 R decay evolves as (solving Eq.2b This evolution will continue till the instant say at a = a νH ≤ a re , for which Γ 3 ∼ H satisfied. After this instant a νH (< a re ), any additional entropy injection into the thermal bath will be ceased, and the radiation component falls as a −4 .With this fall off, reheating completes at ρ ϕ (a re ) ∼ ρ r (a re ) (see the bottom plot of Fig. 1), and it is possible only when the inflaton EoS w ϕ > 1/3.Hence, T re can be expressed as, Reheating temperature depends nontrivially on the neutrino coupling, T re ∝ β −1/(3w ϕ −1) .For example, if we assume w ϕ = 1/2, the T re ∝ β −2 , and for w ϕ = 9/11, we got T re ∝ β −11/16 which we recover numerically as shown in Fig. 2 with solid lines.For this neutrino heating case, the maximum T re should be at β = β c ν , and it is the same as the neutrino dominating case where both meet in (β, T re )-plane (see Fig. 2).The minimum T re is set by the GRe temperature if w ϕ > 0.65 (and β = β c ϕ ), otherwise it would be T BBN .

C. Constraining inflation
Using same M 3 value, we have projected our results in n s − r plane for two different w ϕ = 9/11, 0.5 (see top Fig. 3) and shown the prediction of maximum α value from the observational 68% and 95% CL constraints from BICEP/Keck data.For w ϕ = 9/11, α = 1, we found the range of n s lying within 0.96491 ≤ n s ≤ 0.96825, which is well inside the 1 σ bound.That in turn fixes inflationary e-folding number (N I ) within 57 ≤ N I ≤ 64.Similar bounds can also be obtained for all allowed values of α.

D. PGW Constraints
One of the profound predictions of inflation is the existence of PGWs [39][40][41].It acts as a unique probe of the early universe.The amplitude and the evolution of the PGWs spectrum are sensitive to the energy scale of the inflation and the post-inflationary expansion.We are particularly interested in those modes between k re < k < k end , which re-enter the horizon during reheating.Depending on the post-inflationary expansion, the PGW spectrum traces out the features of those phases.The PGW spectral tilt generically assumes the form n GW = 6w−2 1+3w [10,11] which predicts a blue-tilted spectrum for an EoS w > 1/3 and red-tilted for w < 1/3.Neutrino dominating case: For this case, there is an intermediate ν 3 R dominated matter-like phase (w = 0).Corresponding to this phase there exists a particular scale k ν = a ν H(a ν ), that re-enters the horizon at the inflaton and neutrino equality point a ν .Where k end = e NI k * is the mode that enters the horizon at the beginning of reheating and is related to the inflationary efolding number N I calculated at the CMB pivot scale k ⋆ = 0.05 Mpc −1 .Since ν 3 R domination behaves as dust, following the expression for the spectral tilt mentioned before, the present-day PGWs spectrum corresponding to those modes lies in between k ν > k > k re will be, Where k re = a re H(a re ) is the mode that enters the horizon at the end of reheating, and its value naturally depends on T re as, Due to the red tilted spectrum with k −2 , as shown in the blue dashed line in the bottom plot of Fig. 3. On the other hand, modes within k end ≥ k ≥ k ν enter the horizon during the early inflaton-dominated phase, and the PGWs spectrum assumes the following form, PGW spectrum is always shown to be blue tilted as reheating happens for w ϕ ≥ 1/3.For w ϕ = 9/11, the Ω k GW h 2 behaves as k 16/19 (see solid blue line in bottom plot of Fig. 3).
Neutrino heating case: For this case, inflaton is the dominant component during the whole reheating period.All the modes that enter the horizon during reheating, the PGWs spectrum at the present time can be written as [11,42] Here, the PGW spectrum is also shown to be blue tilted (solid orange line).The PGWs spectrum would be maximum when k = k end , and it is bounded by the BBN bound of Ω k end GW h 2 ≤ 1.7 × 10 −6 [12][13][14] which is depicted by red straight line.Therefore, using the BBN bound of PGW strength one obtains a lower bound on the reheating temperature for w ϕ > 1/3 as, Setting the above temperature with the BBN energy scale ∼ 4 MeV, we can see that the BBN bound of PGWs are only important when w ϕ ≥ 0.60.Moreover, note that to satisfy this restriction on the reheating temperature, we can always estimate a β value (see, for instance, Eq.17) below which our analysis is true.

VI. LEPTOGENESIS AND CONSTRAINTS
Since RHNs are gravitationally produced from the inflaton, they will undergo CP-violating out-of-equilibrium decay and produce lepton asymmetry.By the well-known non-perturbative sphaleron processes [43,44], those lepton asymmetries are then converted into baryon asymmetry.For our analysis we considered the following mass hierarchy M 1 ≲ m end ϕ ≪ M 2 .The CP asymmetry parameter (ϵ ∆L ) generated from the decay of ν 1 R , using the seesaw mechanism [45][46][47][48] is expressed as , The effective CP-violating phase (δ eff ) in the neutrino mass matrix is assumed to be δ eff = 1 for our present analysis.
where s(T re ) = 2π 2 45 g ⋆s T 3 re = 4 ϵ 3 g⋆s g⋆r T 3 re is the entropy density at the end of reheating.Where g ⋆s counts the number of entropic degrees of freedom.The number density, n 1 (T re ) at the time of reheating from Eq.5, We will now use this expression in Eq. 24, to constrain our model parameters.Utilizing the expression of RHN number density for two different reheating scenarios, we arrived at the final exclusion plots in (w ϕ , β) plane (see Fig. 4).
The baryon asymmetry provides the strongest constraint on our reheating scenario.The green-shaded region in Fig. 4 corresponds to the neutrino dominating case, and the cyan-shaded region corresponds to the neutrino heating case.Particularly, the deep green and deep cyan shaded regions are where right baryon asymmetry is produced.On the other hand, the light green and light cyan-shaded regions represent the under-abundant baryon asymmetry.The magenta and red-shaded regions are ruled out from the BBN bound of the reheating temperature and PGWs, respectively.For Fig. 4, we particularly considered M 3 = 5 × 10 11 GeV.Combining two reheating cases, we found the overall allowed equation of state should be w ϕ ≳ 0.5, for which the right baryon asymmetry is generated, and the lightest active neutrino mass is predicted to be within (10 −10 ≳ m 1 ≳ 10 −19 ) eV.Although we have shown all the results for a particular ν 3 R mass, rather wider mass ranges of ν 3 R within 10 10 − 10 14 GeV is observed to be allowed.And corresponding mass range of ν 1 R for successful leptogenesis is within M 1 ≃ (8 − 350) × 10 11 GeV.

VII. CONCLUSIONS
In summary, we proposed a new reheating mechanism by invoking the Type-I seesaw neutrino model, which simultaneously resolves the SM neutrino mass and baryon asymmetry problems and also offers a successful reheating scenario.This reheating turned out to further constrain the inflation models within a very narrow range in the n s − r plane.We found two possible scenarios.If the reheating is controlled by neutrino domination, we found the reheating temperature T re ∝ β M 1/2 3 , and reaches its maximum at β = β c ν .For example, if w ϕ = 9/11, the maximum temperature turned out to be as high as 10 6  GeV, and β shoud be > 10 −17 with fixed by T BBN .For this case within k re < k < k ν , GW spectrum behaves as Ω k GW h 2 ∝ k −2 , and for k end < k < k ν , the spectrum behaves as Ω k GW h 2 ∝ k

N e u t r in o h e a t in g N e u t r in o d o m in a t inFIG. 4 .
FIG.4.β versus w ϕ parameter space for a given M3 = 5×1011  GeV.The deep cyan-shaded and green-shaded regions correspond to the regions where the observed baryon asymmetry is possible.The light cyan-shaded and green-shaded regions lead to the underabundance of YB.