Bare mass effects on the reheating process after inflation

We consider the effects of a bare mass term for the inflaton, when the inflationary potential takes the form $V(\phi)= \lambda \phi^k$ about its minimum with $k \ge 4$. We concentrate on $k=4$, but discuss general cases as well. Further, we assume $\lambda \phi_{\rm end}^2 \gg m_\phi^2$, where $\phi_{\rm end}$ is the inflaton field value when the inflationary expansion ends. We show that the presence of a mass term (which may be present due to radiative corrections or supersymmetry breaking) can significantly alter the reheating process, as the equation of state of the inflaton condensate changes from $w_\phi=\frac{1}{3}$ to $w_\phi=0$ when $\lambda \phi^2$ drops below $m_\phi^2$. We show that for a mass $m_\phi \gtrsim T_{\rm RH}/250$, the mass term will dominate at reheating. We compute the effects on the reheating temperature for cases where reheating is due to inflaton decay (to fermions, scalars, or vectors) or to inflaton scattering (to scalars or vectors). For scattering to scalars and in the absence of a decay, we derive a strong upper limit to the inflaton bare mass $m_\phi<350~{\rm MeV} (T_{\rm RH}/10^{10}~{\rm GeV})^{3/5}$, as there is always a residual inflaton background which acts as cold dark matter. We also consider the effect of the bare mass term on the fragmentation of the inflaton condensate.


I. INTRODUCTION
The hypothesis of a violent inflationary phase during the first moments of the Universe makes it possible to address several cosmological issues, ranging from the flatness of the Universe to the horizon or entropy problem [1].However, a complete inflationary model requires above all a mechanism for a graceful exit.Indeed, the prolonged period of exponential expansion must end with a sufficiently efficient transfer of the oscillation modes of the inflaton condensate ϕ to a thermal bath [2,3], i.e. reheating, that ensures a temperature ≳ 2 MeV to allow for standard big bang nucleosynthesis.Moreover, the density fluctuation spectrum produced during inflation should agree with observations of the CMB anisotropy spectrum [4], which in turn constrains the parameters of the inflaton potential V (ϕ).
The process of transferring the energy stored in inflaton oscillations to Standard Model particles is not instantaneous [5][6][7][8].Rather, in many models, an oscillating inflaton condensate decays or scatters progressively producing a bath of relativistic particles.The efficiency of the reheating process depends on the rate of the energy transfer as well as on the shape of the inflaton potential, V (ϕ), about its minimum [9,10].Even if the exact shape of the potential at the end of inflation is unknown it can often be approximated about its minimum by a polynomial function of ϕ.
In many models of inflation, the inflaton potential can be approximated about its minimum by a quadratic term, V (ϕ) = 1  2 m 2 ϕ ϕ 2 .The Starobinsky model [11] is one example.In this case, only one Fourier mode of the inflaton oscillation contributes to the reheating process.The energy density in radiation, ρ R , grows rapidly at first, and redshifts as ρ R ∝ a − 3  2 where a is the cosmological scale factor, as decays continue to add to the radiation bath.Because ρ ϕ ∝ a −3 , eventually, the radiation bath comes to dominate the total energy density, at which time we can define a reheating temperature.This occurs when the cosmological scale factor, a RH satisfies ρ R (a RH ) = ρ ϕ (a RH ).This occurs (up to a numerical factor) when H(a RH ) ≃ Γ ϕ , or T RH ≃ Γ ϕ M P , where H is the Hubble parameter, Γ ϕ is the width of the inflaton condensate, and M P = 1/ √ 8πG N ≃ 2.4 × 10 18 GeV is the reduced Planck mass.
For a potential whose expansion about its minimum is V (ϕ) = λϕ k , with k ≥ 4, the exercise is more subtle, and requires a more involved analysis [9,10].The reheating process will in general depend on the spin of the final state particles in either inflaton decays or scatterings.In fact, in some cases reheating does not occur.For example, for k = 4, the evolution of ρ ϕ ∝ a −4 is the same as the evolution of ρ R ∝ a −4 for inflaton decays or scatterings to vector bosons [12], precluding the condition ρ ϕ (a RH ) = ρ R (a RH ) to occur.However, we cannot exclude the presence of a bare mass term 1  2 m2 ϕ ϕ 2 , which may be subdominant at the end of inflation, and during the early phases of the oscillations, but which becomes dominant when ϕ has redshifted down to a point a m defined by λϕ 4 (a m ) = 12 m 2 ϕ ϕ 2 (a m ).The presence of this term, even if it is small, would then modify the reheating mechanisms, making for example reheating by decays to vector bosons possible in the case k = 4.
Many models of inflation have potentials which, when expanded about their minimum, are described by a series of self interactions beyond their mass term.For example, the well studied Starobinsky potential [11], contains a full series of interaction terms.However, for ϕ < ϕ end , where ϕ end = ϕ(a end ) is the inflaton field value when the inflationary expansion ends (when ä = 0), terms which are higher order than the quadratic (mass term), become greatly suppressed and do not substantially affect the subsequent evolution of the inflaton condensate.In contrast, models such the so-called α-attractor T -models of inflation [13], described by a potential of the form, contain only even interaction terms starting with λM 4−k P ϕ k yielding a massless inflaton for k ≥ 4.
A bare mass term may be present at the tree level, may be produced as a result of supersymmetry breaking in a supersymmetric model, or may be produced radiatively.Though we will treat the mass as a free parameter, we note that there are 1) upper limits on the mass imposed by slow-roll parameters which determine the inflationary observables, n s and r; 2) in the absence of fine-tuning, there is a lower bound on the mass derived from loop corrections to the potential à la Coleman-Weinberg.Both of these limits will be discussed below.In any case, the presence of a mass term seems unavoidable, at least at higher order, justifying a detailed analysis of its effect on the reheating process.
More specifically, the reheating phase in the T -models with k ≥ 4, as an example, is altered when a mass term is added to the potential in Eq. 1.As a result, for k = 4, the evolution of the energy density transitions from a radiation-dominated Universe ( If the reheating process is sufficiently slow, the quadratic term can come to dominate the inflaton energy density and would result in higher reheating temperature than would have been achieved from the quartic term alone.The presence of a bare mass term generalizes previous results [9,10].Furthermore, it was recently shown in [14,15] that the effects of the fragmentation of the inflaton condensate through its self interaction λϕ k , k ≥ 4 could considerably affect the reheating process.It was noticed that in the case of reheating generated via fermion decay, fragmentation stopped the reheating process too early, leaving the Universe with a bath of massless (and thus stable) particles (inflatons).This would be in contradiction with CMB/BBN observations.However, if the quadratic term 1 2 m 2 ϕ ϕ 2 were to dominate before the end of the fragmentation of the inflaton (a m < a F where a F is the scale factor when fragmentation is complete), the latter would stop, allowing the condensate to continue the reheating process safely through its decay, Γ ϕ ∝ m ϕ .For example, for k = 4, the conformal self-resonance responsible for the exponential growth of a narrow range of relativistic ϕ momentum modes is shut down as they become nonrelativistic.The effective frequencies lose the oscillatory driving, and become incapable of fragmenting the inflaton condensate [14,15].This will be discussed in more detail below.
The paper is organized as follows: in Section II, we describe the effect of the transition from a ϕ 4 → ϕ 2 potential on the evolution of the inflaton condensate and its impact on the reheating temperature.In Section III, we derive the upper limit to the inflaton mass from CMB observables and the bare mass expected from radiative corrections which in the absence of fine-tuning represents a lower limit to the mass.Then in Section IV, we derive the relations between the inflaton coupling to matter and the reheating temperature in view of the transition to a matter dominated expansion.These results are generalized to k ̸ = 4 in Section V and the consequences on the fragmentation of the inflaton condensate are discussed in Section VI.Our summary is found in Section VII.
We begin by supposing that the dominant contribution in a series expansion of the inflaton potential about its minimum is the quartic term and that at the end of inflation, this dominates over a quadratic mass term, so that For a > a end , the evolution of the energy density of ϕ is governed by the Friedmann equation for ρ ϕ Where ρ ϕ = ⟨V (ϕ)⟩ = V (ϕ 0 ), the mean being taken over the oscillation of ϕ and ϕ 0 is the envelope of the oscillations.More precisely, with P(t) a quasiperiodic function encoding the (an)harmonicity of short-timescale oscillations in the potential.
For k = 4, Eq. ( 3) gives where ρ end is the value of the density of energy of the inflaton at the end of inflation, when ä = 0.This condition is equivalent to w = −1/3 or φ2 end = V (ϕ end ).Hence, where for the T -models with potential given in Eq. ( 1) we have [10], The parameter λ in Eq. ( 1) is determined from the normalization of the CMB anisotropies [4].The normalization of the potential for different values of k can be approximated by [10] where N * is the number of e-folds from horizon crossing to the end of inflation and A S * ≃ 2.1 × 10 −9 is the amplitude of the curvature power spectrum.For N * = 56 e-folds we find λ = 3.3 × 10 −12 , and ρ As ϕ 0 decreases, eventually the evolution of the condensate will be governed by the quadratic term.This occurs at a = a m when Using ϕ 4 0 (a) = (ρ end /λ)( a end a ) 4 for a end < a < a m gives In deriving (10), we note that the envelope function ϕ 0 is determined by the average energy density ⟨ρ ϕ ⟩ = V (ϕ 0 ).Thus unless reheating occurs rapidly, the quadratic term will dominate the reheating process even if the quartic dominates after when oscillations begin.This will have huge consequences on the reheating temperature, as well as on the physics of fragmentation as we will see.
Indeed, if reheating occurs at a = a RH > a m , the process is affected by the bare mass term.For a > a m , the equation of state changes from w = 1/3 (for k = 4) to w = 0 (for k = 2) and the solution for a ≪ a m to the Friedmann equation becomes Furthermore, Combining Eqs.(10) and (11) we obtain This form for ρ ϕ dominates the energy density until reheating defined by ρ ϕ (a RH ) = ρ R (a RH ).Here, ρ R is the energy density transferred to the thermal bath via the Boltzmann equation From the above, we can determine the reheating temperature for a given mass, m ϕ for which the bare mass affects the reheating process, and therefore modifies the calculation of T RH .The condition a m < a RH implies that ρ m > ρ ϕ (a RH ) and thus the condition for the quadratic part to dominate the reheating process is given by from Eq. (12).
Defining ρ RH = αT 4 RH with α = gRHπ 2 30 for g RH relativistic degrees of freedom at a RH , we obtain which means that if the energy transfer between the condensate and the thermal bath is slow and the reheating temperature T RH lower than the limit obtained in the equation ( 16), we must take into account the quadratic term to determine T RH when a RH > a m .

III. LIMITS ON THE INFLATON BARE MASS
As noted earlier, the CMB observables impose an upper limit to m ϕ and in the absence of any fine-tuning, couplings of the inflaton to Standard Model fields (necessary for reheating), provide a lower bound to m ϕ from radiative corrections to the potential.
Planck [4] has determined with relatively high precision, the value for the tilt of the CMB anisotropy spectrum, n s = 0.9649 ± 0.0042 (68% CL).In addition, the tensor-to-scalar ratio, r < 0.036 is constrained by BI-CEP/Keck observations [16,17].To translate these limits to an upper limit on m ϕ , we use the T -model in Eq. (1) as an example.
Recall that the conventional slow-roll parameters for a single-field inflationary model are given by where ϕ * corresponds to the horizon exit scale k * = 0.05 Mpc −1 used in the Planck analysis.The scalar tilt and tensor-to-scalar ratio can be expressed in terms of the slow roll parameters as In a more precise model determination of N * , and n s , there is some dependence on the reheating temperature and equation of state [18,19].The computation is based on the self-consistent solution of the relation between N * and its corresponding pivot scale k * , 1/4 43 11 where the present Hubble parameter and photon temperature are given by H 0 = 67.36km s −1 Mpc −1 [4] and T 0 = 2.7255 K [20].For the T -Models dominated by a quadratic term, agreement with Planck/BICEP/Keck data requires N * between roughly 42 -56 [21].
In the absence of a mass, m ϕ = 0, N * ≃ 56 with ϕ * = 6.96MP and (n s , r) = (0.964, 0.0034), independently of the efficiency of reheating [10,15].Therefore, to set limits on a possible mass term for k = 4, we set N * = 56.For non-zero masses both n s and r increase, but the limit on m ϕ is determined mainly from n s .Figs. 1 and 2 show the numerically computed CMB observables n s and r for a variety of bare masses and inflaton-matter couplings.As is customary, the Planck (k * = 0.05 Mpc −1 ) and WMAP (k * = 0.002 Mpc −1 ) pivot scales are chosen for n s and r, respectively.For m ϕ ̸ = 0, the effective equation-of-state parameter evolves as w = −1/3 → 1/3 → 0 → 1/3 from the end of inflation to the end of reheating.The top panel of Fig.  Planck+BK18 regions [16].
depicts the bare mass dependence of the scalar tilt, as a function of an inflaton-matter Yukawa coupling (see Eq. ( 22)).For m ϕ = 0, N * ≃ 56 for any y, leading to the purple horizontal line.For m ϕ = 0.025 √ λM P , the smallest non-zero mass in the Figure, the resulting curve presents two regimes.At y ≳ 10 −1 , n s is independent of y since reheating is completed before matter domination, a RH < a m .However, for y ≲ 10 −1 , reheating is completed by the dissipation of the quadratic, harmonic oscillations of ϕ.A dependence of n s on y is induced, since now the last term of ( 21) is relevant for the determination of N * .For smaller y reheating is delayed, resulting in a smaller N * and as a consequence n s .In the case of larger masses, the pure quartic regime is reduced, or outright lost, and the relation of n s and y is determined by the duration of reheating in the matter dominated era, and the modification of the slow roll dynamics due to the presence of the large bare mass.Analogous conclusions can be drawn from the bottom panel of Fig. 1.In this case the addition of the bare mass increases the value of the tensor-to-scalar ratio, both from the modified inflation dynamics, and from the dependence on y of the number of e-folds N * .Fig. 2 compares the corresponding (n s , r) curves against the Planck+BK18 constraints [16].Here the range of couplings spans reheating temperatures from T RH ∼ 2 × 10 14 GeV for y = 1, to T RH ∼ O (10) MeV for y = 10 −15 .We note that for the smallest bare masses high reheating temperatures are favored by the CMB data.On the other hand, for the largest masses considered, lower T RH are preferred.At the nominal N * = 56, corresponding to y ≈ 1 in the figure, we find that m ϕ < 0.2 √ λM P ≃ 8.8 × 10 11 GeV at 68% CL with (n s , r) = (0.971, 0.0050) and m ϕ < 0.25 √ λM P ≃ 1.1 × 10 12 GeV at 95% CL with (n s , r) = (0.975, 0.0061).Above these masses, the values of n s and r rise very quickly and agreement with data is lost.Applying this limit on m ϕ in Eq. ( 16) gives T RH ≲ 2.8 × 10 14 GeV.In other words, for larger reheating temperatures, the energy transfer is sufficiently efficient to avoid any interference of a possible quadratic interaction without violating the CMB data.Allowing for the full range in coupling y or equivalently T RH and expanding the range in N * , we see from Fig. 2, that the 68% CL upper limit is m ϕ < 0.33 GeV (for y ≥ 10 −15 and a 95% CL upper limit of m ϕ ≲ 0.38 GeV.For larger masses it becomes impossible to simultaneously satisfy the Planck constraints to 2σ and the BBN bound T RH ≳ MeV. In addition to an upper bound to m ϕ , we expect that radiative corrections to the potential will provide finite mass which unless fine-tuned away, will determine a lower bound on the inflaton mass.We expect that through the coupling of the inflaton to either fermions or scalars, would lead to a mass term proportional to ym f or µ (see Eqs. ( 22) and ( 35) for couplings to fermions and scalars respectively.While the former is probably no larger than the weak scale, the coupling to scalars could generate a significant contribution to m ϕ .Furthermore, in a supersymmetric theory we would also expect contributions to the scalar mass of order the supersymmetry breaking scale.However, as noted, any lower limit to the inflaton mass would be subject to the degree of fine-tuning by canceling a bare mass term with any 1-loop corrections.Therefore unlike the upper limit discussed above, we do not apply a firm lower limit its mass, but recognize that it should not be surprising to generate weak scale masses, even in theories with the potential given in Eq. ( 1) for k ≥ 4.

IV. CONSEQUENCES OF THE INFLATON COUPLING TO MATTER
Reheating to create a thermal bath of Standard Model particles requires some coupling of the inflaton to the Standard Model.The relation between this coupling and the reheating temperature is dependent not only on the shape of the inflaton potential about its minimum, but also on whether the reheating is produced by inflaton decay (in to either fermions, scalars or vectors) or scattering.As in [9,10] we will study the three possible cases: fermion decay, scalar decay and scalar scattering, adding the vectorial final states (decay and scattering) analyzed in [12].

A. Inflaton decay to fermions
Given a Yukawa-like coupling of the inflaton to fermions, the inflaton decay rate is Here, the effective Yukawa coupling y eff (k) ̸ = y is defined by averaging over an oscillation.In general for k ̸ = 2, the effective coupling must be calculated numerically [10,22,23].
The general expressions for the reheating temperature, defined by ρ ϕ (a RH ) = ρ R (a RH ) and αT 4 RH = ρ R (a RH ), are given in the Appendix.T RH depends strongly on the spin of the final state decay products, and for decays to fermions, Eq. ( 91) gives with l = 1/2 − 1/k and k < 7 or Notable in Eq. ( 25) is that T RH exhibits a different dependence on the coupling and mass of the inflaton.In particular, a m < a RH .We will see that for sufficiently low coupling, the quadratic term can dominate the reheating process leading to a higher reheating temperature.
When the limit in Eq. ( 16) is satisfied, reheating is sufficiently late to be determined by the quadratic term (k = 2 in Eq. ( 25)) and that can be translated into a limit on the coupling y eff , We show in Fig. 3 the value of the reheating temperature as function of y eff for different values of the inflaton bare mass m ϕ = 10 3 , 10 9 and 10 11 GeV, neglecting the effects of an effective final state mass (see below) and thus y eff = y.To obtain the figure, we solved numerically the complete set of Friedmann equations for ρ R and ρ ϕ , taking the full potential V (ϕ) = 1 2 m 2 ϕ ϕ 2 +λϕ 4 .We also show for comparison with dashed lines, the analytical value of T RH obtained in Eqs.(25).We clearly see the change of behavior T RH = f (y eff ) below the limiting value in Eq. ( 26) where the bare mass term controls the final reheating temperature.For y eff ≲ y m eff , T RH ∝ y eff , whereas for larger values of y eff , when the reheating is dominated by the quartic part of the potential, the reheating temperature ∝ y 2 eff and is independent of m ϕ .A background field value for ϕ, however, induces an effective mass for the fermion, f , m eff = yϕ, and the rates for producing the fermions are suppressed by R −1/2 where R ∝ m 2 eff /m 2 ϕ ∝ y 2 (ϕ 0 /M P ) 4−k /λ [10].The mass of the inflaton is defined by When R ≫ 1, there is a significant suppression in the decay rate and y eff ≪ y.Note that in the case of a quartic potential, m ϕ ∝ ϕ.As m eff ∝ ϕ also, R is constant, R ≃ 1.4y 2 /λ ≃ 4.2 × 10 11 y 2 .In other words, the effect of R results in a suppression of the reheating efficiency by a constant factor R − 1 2 ≃ 1.5 × 10 −6 /y throughout the reheating process.This suppression begins to be efficient (R ≳ 1) for y ≳ 1.5 × 10 −6 [10].On the other hand, for a quadratic potential, R = 4(ϕ 0 /m ϕ ) 2 y 2 decreases with time, redshifting as a − 3 2 .Which means that if there is no suppression during the quartic dominated era (a < a m ), there is no suppression in the quadratic era (a > a m ).
The kinematic suppression in the effective coupling y eff for R ≫ 1 can be parametrized as [10] where c k is a k-dependent constant 2 and ω is the oscillation frequency.For k = 4, c 4 ≃ 0.5 and ω ≃ 0.49m ϕ .This leads to We note that only when R ∼ 0.1, do we recover y eff = y.Note also that unless y eff is relatively small, y eff ≲ 2 × 10 −3 , the Lagrangian coupling, y, is non-perturbative [10,14], where this perturbativity limit on y eff assumes y ≲ √ 4π.
For k = 2, c 2 ≃ 0.38 and ω = m ϕ .At the end of reheating, Then, for R ≫ 1, we can write and using Eq. ( 25) for T RH in terms of y eff we have In this case, non-perturbativity sets in unless y eff ≲ 1.5(m ϕ /ϕ 0 ) 1 2 , assuming that y eff < y.Note that for k > 4 2 There is an additional dependence of y eff on the sum of the Fourier modes associated with the inflaton oscillations in the potential V (ϕ) ∼ ϕ k , for each value of k.However, this additional dependence is O(1), as shown in [10].
the limit becomes more severe as R is larger and increases in time.
Because of the suppression in the decay rate, the relation between T RH and the decay coupling y shown in Fig. 3 needs to be reassessed.Indeed, when y ≳ 1.5 × 10 −6 , R ≳ 1 and the suppression effect should be taken into account.The relation between T RH and y when the effects of kinematic suppression are included is shown in Fig. 4. At very low values of y, R ≪ 1 and the suppression effects can be ignored.In this case, the relation between T RH and y is unaffected.However, when y eff ≤ y the relation is altered.From Eq. ( 32), this occurs when y > 1.3 × 10 −6 (m ϕ /GeV) 1 3 , or when y > 1.3 × 10 −5 (1.3 × 10 −3 )(6 × 10 −3 ) when m ϕ = 10 3 (10 9 ) (10 11 ) GeV.These values are seen in Fig. 4 when the solid curves begin to deviate from the dashed curves.The dashed curves show the relation in Fig. 3 when suppression effects are ignored.The expression for y eff in Eq. ( 32) can be inserted in Eq. ( 25) to obtain the relation between T RH and y for when suppression effects are included and reheating is governed by the quadratic term,  We saw previously in Eq. ( 16) that the reheating temperature is determined by the quartic term only if T RH ≳ 250m ϕ .When the kinematic suppression effects are ignored (y = y eff ), this occurs when y does not satisfy Eq. (26).In this case, we can use Eq.(29) to determine the relation between T RH and y, and thus we expect that reheating is determined by the quartic term when y > 1.7 × 10 −6 m ϕ /GeV.This occurs at y = 1.7 × 10 −3 for m ϕ = 10 3 GeV as can be seen in Fig. 4. For the larger masses shown, we see that the transition would only occur in the non-perturbative regime (with y ≫ 1) and so for the two higher masses, the reheating temperature is always determined by the quadratic mass term.

B. Decay to scalars
Another possibility is that reheating occurs predominantly through inflaton decay to scalars, through the coupling where b is a real scalar field.As was the case for the fermion decay, there is also an effect from the effective mass of the scalar field, and we parameterize it by considering an effective coupling µ eff .We note that µ eff is now a dimensionful parameter and is enhanced (and not reduced) by R 1/2 [10].The associated decay rate is given by For k = 2 this effective coupling reduces to the Lagrangian coupling µ but is different for k > 2. It is important to note that in this case, as m ϕ decreases with time, the decay rate increases with time.
For decays to scalars, l = 1/k − 1/2, and using the appropriate expression found in the Appendix for γ ϕ , we have We show in Fig. 5 the evolution of T RH as function of µ for the same set of masses m ϕ = 10 3 , 10 9 and 10 11 GeV, in the simplified case with m eff = 0. We clearly recognize the dependence T RH ∝ µ for the smaller values of µ and T RH ∝ µ 2/3 for the larger values, when reheating is dominated by the quartic part of the potential.The value of µ for which reheating is dominated by the quadratic term obtained from Eq. (38) which is effectively what is observed in Fig. 5. From Eq. ( 39), we see that the reheating temperature for m ϕ = 10 3 GeV (red curve) is always due to the quartic term, as the transition from quadratic to quartic occurs at a low value of µ beyond the range shown.For the larger values of m ϕ , Eq. ( 39) indicates when when the slopes of T RH vs. µ begins to change.Solid lines are obtained by solving numerically the Boltzmann equations for energy densities, while dashed lines are given by the analytical approximations in Eqs.(38).Here we neglect the effective mass of produced bosons, R = 0.
Then, the effects of the kinematic enhancement will occur when This can be see seen in Fig. 6 for m ϕ = 10 9 (10 11 GeV as the point when the solid curves break away from the dashed curves at µ ≃ 2.1 × 10 6 (9.8 × 10 8 ) GeV respectively.At lower values of µ, the effects of the kinematic suppression can be ignored.For m ϕ = 10 3 GeV, this occurs at a value of µ below the range shown.Solid lines are obtained by solving numerically the Boltzmann equations for energy densities, while dashed lines are given by the analytical approximations in Eqs.(38).Here we consider the effective mass of produced bosons, R = 8µϕ0/ω 2 .
In the region when µ eff > µ and quadratic reheating dominates, we can insert Eq. ( 41) into Eq.( 38) to obtain At higher values of µ the transition to quartic reheating occurs and using Eq. ( 40) with the expression for R for k = 4, we find that µ eff ≃ 2.5 GeV µ GeV 15 14 (44) which when inserted in Eq. (38) gives (k = 4) .
C. Decay to Vectors Recently, we have considered the possibility of inflaton decays to vectors [12] motivated by inflationary models in the context of no-scale supergravity [25] (which easily lend construction of the T -models considered here [9]).Often in such models, the inflaton couplings to matter fermions and scalars are highly suppressed [26][27][28] and reheating is only possible if the gauge kinetic functions contain inflaton couplings.The inflaton to vector couplings can be parameterized by From these Lagrangian couplings, we can derive the inflaton decay rate where α 2 eff = (g 2 eff + g2 eff )/(64π).Note the dependence of the width on m 3 ϕ , which is very different from the decay into fermions (∝ m ϕ ) and to scalars (∝ 1/m ϕ ).Γ ϕ→AµAµ decreases much more rapidly than Γ ϕ→f f , rendering the reheating much less efficient, even impossible as long as the reheating is dominated by the quartic term.Indeed, for decay to vectors, l = 3/2 − 3/k, and using the appropriate expression for γ ϕ , we have from Eq. ( 91) This expression is valid so long as k+8−6kl > 0, which is the case for k = 2, but not for k ≥ 4. For k + 8 − 6kl < 0, the reheating temperature is given by Eq. (92) for k > 4. For k = 4, the radiation density in Eq. (88) scales as a −4 as does the inflaton energy density in Eq. ( 84) and we never achieve the condition that ρ ϕ (a RH ) = ρ R (a RH ) and reheating never occurs.Thus we have Thus for a k = 4 inflationary potential, reheating via the decays to vector bosons does not occur in the absence of a bare mass term.The bare mass term is then necessary to ensure a successful reheating.However, the bare mass term should ensure T RH ≳ 2 MeV, which means This value is the minimal bare mass necessary to have reheating through decay to vectors for k = 4.
Finally we note that there are no kinematic enhancement/suppression effects in this case.Since the inflaton is coupled to F 2 (as opposed to A 2 ), no mass term is generated.Then g eff = g (and geff = g) for k = 2, and for k = 4 only differs by a Fourrier coefficient in an expansion of V (ϕ) [12].

D. Scattering to scalars
We can also consider the case where the inflaton transfers its energy through the coupling where b is a real scalar field.The associated decay rate is given by [10] where we have introduced the effective coupling σ eff obtained, as for y eff and µ eff , after averaging over oscillations of the background inflaton condensate [10].This effective coupling is equal to the Lagrangian coupling σ for k = 2 but is different for k > 2 and as in the case of decays to fermions there is a kinematic suppression.
For scattering to scalars, l = 3/k − 1/2, and using the appropriate expression found in the Appendix for γ ϕ , we have from Eq. ( 91) valid when k ≥ 4, (53) For k = 2, 8 + k − 6kl < 0 and ρ R redshifts as a −4 which is faster than ρ ϕ ∝ a −3 .Thus, in this case, reheating is not possible if the quadratic term becomes dominant before reheating is complete.The reheating temperature can then be written as, As one can see, the possibility of reheating through scattering to scalars is opposite the case of decays to vectors.Reheating is not possible when the quadratic part of the potential dominates the reheating process.Naively, when we neglect the kinematic suppression effects in R, reheating is therefore only possible if the limit in Eq. ( 16) is violated, namely For smaller couplings, the quadratic term will dominate before reheating is complete, and as a result never completes.We note in the expression for T RH in Eq. ( 54), the maximum value for σ eff that can be used is determined from a RH > a end , which gives Furthermore for self couplings this large, we expect that non-perturbative effects become non-negligible [24].For larger values, we have a maximum reheating temperature of 2 × 10 15 GeV, which is basically determined from ρ end .
As previously noted, for inflaton scattering to scalars, there is a kinematic suppression when R > 1.In this case, for k = 4, R ≃ 2.8σ/λ is a constant and 4 using c ′′ 4 = 1.22.Then the reheating temperature in terms of σ becomes In Fig. 8, we compare the reheating temperature as a function of σ when kinematic effects are ignored to the case where they are included.From Eq. ( 57), these effects become important when σ > 3.1 × 10 −10 .The dashed lines correspond to the solution when kinematic effects are ignored.The abrupt increase in T RH occurs when Eq. ( 55) is satisfied (and σ eff = 4σ).In contrast, the solid lines include the kinematic suppression and reheating is possible when Eq. ( 55) is used with Eq. (57) or when σ ≳ 6.4 × 10 −6 m ϕ 10 9 GeV 2 3 . (59) This limit accounts for the abrupt rise in T RH for the solid lines in Fig. 8.At higher coupling, the reheating temperature follows Eq. ( 58) and scales as σ 3 2 as opposed to σ 2 when the suppression effects are ignored.In the latter case, we see the curves flatten at large coupling since a RH is approaching a end and the approximation used in (54) breaks down.These curves end when a RH = a end , indicated by the vertical gray dotted line.The solid curves would end when σ ≃ 0.002.
In the absence of a decay term for the inflaton, a bare mass term will eventually lead to a non-zero relic density of inflatons after annihilations freeze out.Indeed, even if σ eff is sufficiently large and respects the condition (55), the presence of a quadratic term may dominate the energy budget of the Universe.Thus we can derive a limit 4 We neglect the dependence of σ eff on the sum of the Fourier modes associated with the inflaton oscillations for each value of k [10].Solid lines are obtained by solving numerically the Boltzmann equations for energy densities including the effect of R, while dashed lines neglect the effect of effective masses.The vertical gray dotted line corresponds to the limit Eq.( 56), when neglecting the effect of R.
on a combination of the inflaton mass, T RH and the coupling σ.Saturating the limit leaves us with the inflaton as a cold dark matter candidate!5 Indeed, for σ eff sufficiently large to ensure reheating with k = 4, for a > a RH , the evolution of ρ ϕ is determined from the Boltzmann equation including dissipative effects [10] and for k = 4, l = 1 4 and γ ϕ as given in the Appendix, we find that ρ ϕ scales as Here, we used H = ρ R /3M 2 P .In the absence of a mass term, since ϕ ∝ a −2 and after reheating, R ∝ a −2 , the ratio Γ/H remains constant and the scaling in Eq. (61) remains true indefinitely and the density of inflatons becomes negligibly small.However, when m ϕ ̸ = 0, eventually the mass term dominates over the quartic term (at a = a m ) and we can determine a m , when where the inflaton density is given by as was previously found in Eq. (12).
For a > a m , Eq. ( 60) can be solved, now with k = 2 and l = 1.In the limit that a ≫ a m , the residual inflaton density is given by so long as (m ϕ /M P ) ≪ (2λ) .001, which is always true given the upper limits on m ϕ discussed in Section III.Thus the presence of a mass term in the case where reheating is determined by a quartic coupling of the inflaton to scalars (which requires k > 2), leads automatically to cold dark matter candidate.
Given the inflaton density in Eq. ( 64), it is straightforward to compute the relic density today and in effect set a limit on the inflaton bare mass.Today, where ξ = (43/427)(4/11) ≃ 0.036 and relative to the critical density we have GeV , using Ω ϕ h 2 < 0.12.This is a remarkably strong limit on a bare mass term for the inflaton if it remains stable.

E. Scattering to Vectors
If the gauge kinetic function is quadratic in the inflaton, then scattering rather decay to vectors occurs.In this case, the inflaton to vector couplings can be parameterized by From these Lagrangian couplings, we can derive the inflaton decay rate where β 2 = (κ 2 eff + κ2 eff )/(4π).For scattering to vectors, l = 3/2 − 1/k, and using the appropriate expression for γ ϕ , we have from Eq. ( 92) since 8 + k − 6kl < 0 for k ≥ 2. However, Eq. ( 70) is only valid for k > 4. For k = 2(4), ρ ϕ ∝ a −3 (a −4 ) while ρ R ∝ a −4 for all k and reheating is not possible for k < 6.
For these specific cases, we then have In this case, the presence of a bare mass will not change the lack of reheating through the scattering to vectors.
As a conclusion, whereas in the case of decays to fermions or bosons, the presence of a quadratic term only acts on the value of T RH , decreasing the reheating temperature in the former case, increasing it in the latter case, the quadratic term when dominant removes the possibility of reheating through scattering to scalars but reopens the possibility of reheating through decay to vectors, but does not allow reheating through the scattering to vectors.

V. GENERALIZED POTENTIALS
The inflationary potential may be dominated by higher order terms if k > 4. In this section, we generalize some of the arguments made above in the event that the inflationary potential is approximated by about its minimum.In this case, the general expression for the scale factor when the mass term dominates is given by with ρ end given by Eq. ( 6) and λ by Eq. ( 8).Then which clearly reduces to Eq. ( 12) for k = 4.A parallel derivation leading to Eq. ( 16) implies that for the mass term to dominate at reheating.In terms of the reheating temperature, this amounts to . (76) For comparison with Eq. ( 16), we have using λ = 5.7 × 10 −13 and 9.5 × 10 −14 for k = 6 and 8, respectively.

VI. CONSEQUENCE ON THE INFLATON FRAGMENTATION
Recently, the authors of [14] and [15] have shown that fragmentation can significantly alter the reheating process.Indeed, the fragmentation of the inflaton condensate results in the population of an inflaton-particle bath, whose very low mass, proportional to the density of the condensate which remains unfragmented, may not allow reheating temperatures above the BBN bounds for inflaton decays to fermions.This fragmentation is due to the presence of a self-scattering term of type λϕ k , with k ≥ 4. The inflaton condensate does not fragment in the absence of self-interactions allowing for reheating to occur as discussed above.
However, the study [14] was carried out in the context of a monomial potential of the type V (ϕ) = λϕ k .It is then easy to see that the presence of a bare mass term of the type 1  2 m 2 ϕ ϕ 2 can change the conclusions of this study, in particular if the quadratic term begins to dominate before the fragmentation halts.If we define a F as the value of the scale factor at the end of fragmentation, then a F /a end = 180, 4.5×10 4 , 6×10 6 and 7×10 8 , for k = 4, 6, 8 and 10 respectively [14].In order for a quadratic term to affect the fragmentation process, we must have a m ≲ a F and using Eq.( 10) it becomes easy to compute, for each value of k, the minimal value of m ϕ necessary to ensure that the quadratic term dominates the potential before the end of fragmentation.The problem of a leftover bath of massless inflatons can then be avoided by stopping the fragmentation process.
More precisely, when reheating begins, self interactions can source the growth of the inflaton fluctuations δϕ(t, x) = ϕ(t, x) − φ(t), where φ denotes the homogeneous condensate.At early times, this growth can be captured by the linear equation of motion where For m ϕ = 0, the oscillating nature of this resulting effective mass term drives the resonant growth of δϕ and the eventual fragmentation, δϕ ≫ φ [14,15,[35][36][37][38][39][40][41][42][43].However, if m ϕ dominates before fragmentation, V ′′ ∼ const., strongly suppressing the oscillatory driving force. 6  Fig. 9 shows the evolution of the total inflaton energy density ρ ϕ , compared to the energy density in its fluctuations ρ δϕ , as computed numerically for a T-model of inflation [13] with k = 4 and three choices of the bare mass (see [15] for details).The top panel depicts the zero bare mass scenario.In it, the rapid growth of inflaton fluctuations driven by parametric resonance can be appreciated.This growth only stops when ρ ϕ ≃ ρ δϕ (a/a end ≃ 180), corresponding to the near-complete fragmentation of the inflaton condensate in favor of free ϕ-particles. 7For the 6 For a purely quadratic inflaton potential the growth of fluctuations is still present, albeit not exponentially enhanced, due to the coupling of δϕ with the fluctuations of the metric [44][45][46]. 7The fragmentation of the inflaton condensate is not total even for m ϕ = 0.A small but nonvanishing homogeneous component φ remains, and its presence can induce the decay of the free inflaton quanta δϕ [14,15].
bottom two panels we take m ϕ > 0. In both cases, the quartic → quadratic transition time has been chosen to be posterior to the complete fragmentation of the inflaton, a m > a F .A naive estimate from Eq. ( 79) would indicate that the resonant growth of δϕ would not stop until that is, the field would be fully fragmented before matter domination.However, the full numerical solution of the equation of motion (78) shows that the growth of fluctuations is in reality suppressed from a ≲ a m /2, as both panels of Fig. 9 demonstrate.Therefore, reaching quadratic dominance is a sufficient condition to avert full fragmentation.Note that for smaller masses than those used in Fig. 9, fragmentation would nearly completely destroy the condensate and potentially disrupt the reheating process entirely.On the other hand, for larger masses, the fragmentation process would not be operative at all.
A qualitative depiction of this result for potentials with k ≥ 4 is shown in Fig. 10, where we plot the limit on the mass m ϕ above which the bare mass term dominates over λϕ k in the potential as a function of k.We see that for larger value of k, where the fragmentation is less efficient due to the increasing difficulty for the self scattering to occurs for higher modes, even a small bare mass term can be sufficient to stop the fragmentation process and ensure a successful reheating.

VII. CONCLUSION
Reheating in most models of inflation is accomplished through either inflaton decay or scattering to Standard Model particles.This typically after inflationary expansion ends and a period of inflaton oscillations begins.When the potential is dominated by a quadratic term about its minimum, decays are necessary, as scatterings will not in general lead to a radiation dominated universe.However, potentials dominated by higher order interactions, k > 2, have anharmonic oscillations and scattering may lead to reheating, though these models may be subject to additional constraints arising from the fragmentation of the inflaton condensate.In addition, the details of the reheating process and the final reheating temperature depend on the spin of the final state particles produced in the decay or scattering.
In models of inflation for which the potential can be expanded about its minimum as V (ϕ) ∝ ϕ k , typically the lowest power, k appearing in the expansion dominates the reheating process.For k > 2, it is quite possible, as we have argued that in addition to the inflationary potential, a bare mass term in the full scalar potential is also present.This may arise from radiative corrections or supersymmetry breaking.In section III, we derived upper limits to this mass from CMB observables.These limits are sufficiently weak so that the presence of the mass term may affect the reheating process.Indeed, quite generally, if Eqs. ( 16) for k = 4 or (76) more generally are satified, the final reheating temperature will be determined by the quadratic rather than a higher order term.
The qualitative effect of the mass term also depends on the reheating mechanism (decays or scattering) as well as the spin of the final states.For decays to fermions, the reheating temperature is increased by the presence of mass term, while for scalars, it is decreased.For decays to vectors, reheating does not occur for k = 4 in the absence of a mass term and its presence allows for the possibility of reheating in this case.In contrast, if the mass term becomes important before the end of re- heating for scattering to scalars, the reheating process is halted.Furthermore, when reheating is accomplished through scattering to scalars with k ≥ 4, the density of inflatons is quickly redshifts (as a −8 ) until the mass term comes to dominate.In this case, the residual inflaton matter density acts as cold dark matter and a strong limit on the inflaton mass has been derived in Eq. (67).Finally we have seen that for scattering to vectors, reheating with k = 4 is not possible (k ≥ 6 is required) and the mass term does not come to the rescue in this case.
Understanding the reheating process after inflation is of great importance as it is not only responsible for providing an early period of radiation domination necessary for big bang nucleosynthesis, but may be the source of dark matter.Thermal production of dark matter in equilibrium remains an important mechanism, however, it is well established that non-equilibrium process just as freeze-in [47] may also be the ultimate source of dark matter in the Universe.For these cases, a detailed understanding of reheating is essential and here we examined the role of a bare mass term for the inflaton in models where the inflationary dynamics are governed by higher order interactions.
which implies that the temperature would simply redshift as T ∝ a −1 .

15 TFIG. 3 .
FIG.3.Reheating temperature as a function of the Yukawa coupling y when a bare mass term is added to a quartic potential (k = 4).Solid lines are obtained by solving numerically the Boltzmann equations for energy densities, while dashed lines are given by the analytical approximations in Eqs.(25)Here we neglect the effective mass of the final state fermion, R = 0 and y eff = y.

14 Tm φ = 10 3 m φ = 10 9 FIG. 5 .
FIG. 5.Reheating temperature as function of the bosonic coupling µ, for different values of the inflaton bare mass m ϕ = 10 3 GeV (red), 10 9 GeV (green) and 10 11 GeV (blue).Solid lines are obtained by solving numerically the Boltzmann equations for energy densities, while dashed lines are given by the analytical approximations in Eqs.(38).Here we neglect the effective mass of produced bosons, R = 0.

16 T 12 m φ = 10 3 m φ = 10 9 FIG. 7 .
FIG. 7.Reheating temperature as function of the scattering coupling σ, for different values of the inflaton bare mass m ϕ = 10 3 GeV (red), 10 9 GeV (green) and 10 11 GeV (blue).Solid lines are obtained by solving numerically the Boltzmann equations for energy densities including the effect of R, while dashed lines neglect the effect of effective masses.The vertical gray dotted line corresponds to the limit Eq.(56), when neglecting the effect of R.

8 FIG. 8 .
FIG.8.Minimal reheating temperature from Eq.(76), below which the inflaton mass term drives the process, as a function of the bare mass m ϕ and for different values of k, k = 4 (solid line), k = 6 (dashed), k = 8 (dotted).In the different shaded regions, reheating occurs while the inflaton oscillates in a quadratic potential (k = 2), given by its bare mass m ϕ .Above the lines, for different k, reheating occurs while the inflaton oscillates in the potential V (ϕ) ∼ ϕ k .
FIG. 9. Energy density of the inflaton fluctuations ρ δϕ compared to the total energy density ρ ϕ , for three values of the bare mass, for k = 4.The vertical dashed line corresponds to the value of am/a end when m ϕ ̸ = 0.In both of these cases, although am > aF, the exponential growth of δϕ is stopped by the transition to matter-domination.

FIG. 10 .
FIG.10.Region in the parameter space where the fragmentation happens after the domination by the bare mass term 1 2 m 2 ϕ ϕ 2 over λϕ k , allowing for a quadratic reheating.The dashed line is obtained from Eq.(73), while the solid line is obtained from Eq.(79).
end = ρ end /3M 2 P .At later times when a ≫ a end and 8 + k − 6kl > 0, we can approximate ρ R as ρ a≫a end R