Inflaton mass in the nuMSM inflation

We analyse the reheating in the modification of \nuMSM (Standard Model with three right handed neutrinos with masses below the electroweak scale) where the sterile neutrino providing the Dark Matter is generated in decays of the additional inflaton field. We deduce that due to rather inefficient transfer of energy from the inflaton to the Standard Model sector reheating tends to happen at very low temperature, thus providing strict bounds on the coupling between the inflaton and the Higgs particles. This in turn translates to the bound on the inflaton mass, which appears to be very light 0.1 GeV<~ m_I<~ 10 GeV, or slightly heavier then two Higgs masses 300 GeV<~ m_I<~ 1000 GeV.


Introduction
In [1,2] it was shown that within the Standard Model (SM) complimented with three right-handed neutrinos N I with the masses smaller than the electroweak scale one can simultaneously explain both the dark matter and the baryon asymmetry of the universe [1,2,3,4,5,6,7,8,9,10,11,12]. This model dubbed as νMSM represents a particular realization of the seesaw extension of the SM and is fully consistent with the current experimental data from the light neutrino sector. However, generation of the proper Dark Matter abundance of the sterile neutrino is not simple during the thermal evolution of the Universe, and requires some amount of fine-tuning [13,14]. Being very weakly coupled, sterile neutrinos do not reach thermal equilibrium, so an interesting possibility is to generate them before the beginning of the standard thermal history. In [6] such mech-Email addresses: alexey.anisimov@desy.de (Alexey Anisimov), Fedor.Bezrukov@mpi-hd.mpg.de (Fedor L. Bezrukov). anism was proposed, where the νMSM model was extended by adding the inflaton field, which generates all the masses in the model and decays into the SM particles and sterile neutrinos after inflation, where Φ and χ are the Higgs and the inflaton fields correspondingly and is the νMSM Lagrangian with all the dimensional parameters being put to zero. The potential V (Φ, χ) is 1 1 In order to avoid the domain wall problem a cubic term µχ 3 can be introduced. It will be further assumed that µ α 3 /λ v EW . In that case such term has no influence on the dynamics of the model during the reheating stage, and the relation (4) for the values of the parameters considered in the Letter is not altered significantly either.
where V 0 = m 4 χ 4β was introduced in order to cancel the vacuum energy. Expanding (3) around its vacuum expectation value one has the relation between the inflaton 2 mass m I and the Higgs mass m H : If α > β/2 the inflaton mass is smaller then the Higgs mass and, therefore, the decay of the inflaton into the Higgs can only occur in a thermal bath. In what follows we will first concentrate on this particular case. Parameter β is fixed by the COBE normalization of the amplitude of scalar perturbations [15], β ≃ 1.3 × 10 −13 . Pure quartic potential inflation is currently disfavored by the WMAP5 data [16] because of the too large predicted value of the tensor to scalar amplitudes ratio. However, if one allows non-minimal coupling of the inflaton to gravity [17] one can bring this potential in agreement with the data. This, in turn, will influence the bounds on the inflaton mass. We will discuss this in the end of the Letter. The upper constraint on the value of α comes from the requirement that radiative corrections do not spoil the flatness of the inflaton potential and is given by α ≤ 3 × 10 −7 . This corresponds to the lower bound on the inflaton mass One should note that larger values of α (leading to smaller inflaton masses) may also be possible, but then the analysis of the loop corrections to the effective potential of the inflaton becomes important.
The lower bound on α comes from the requirement to have successful baryogenesis in νMSM [2]. To allow for efficient sphaleron conversion of the lepton asymmetry to baryon asymmetry requires the reheating temperature to be larger then roughly 150 GeV [18]. In [6] it was advocated that the resulting lower bound is α > β ∼ 10 −13 . Below we will argue that the lower bound is quite a bit stronger which leads to a narrow window for the inflaton mass. 2 Notations I and H will be used throughout the Letter to represent the diagonalized excitations above the vacuum expectation value for (3). I is the one mostly mixed with inflaton χ, and H mostly mixed with the SM Higgs Φ.

Reheating bounds
Reheating after inflation proceeds through a regime of the parametric resonance. The dynamics of the models with potentials similar to (3) in the parametric resonance regime was studied via analytic methods in, e.g. [19,20]. The analysis of the late stages of preheating was made possible with the lattice simulations package LatticeEasy [21,22,23,24]. In particular, the preheating in the model with the potential which contains only first two terms in (3) have been studied in [22].
At large values of the inflaton field χ the behavior is that of the pure quartic inflation. The expectation value of the Higgs field Φ is set along the flat direction: |Φ| 2 = α λ χ 2 . After the end of inflationary slow roll regime the inflaton field starts to oscillate. In the very beginning all the energy is stored in the zero (or homogeneous) mode of the inflaton χ 0 , and all other modes are absent. The oscillations of χ 0 initially excites the nonzero modes of both the Higgs and the inflaton. One can compare the contribution of the zero mode of the inflaton to the effective masses of the Higgs and the inflaton: If α > β the corresponding contribution to the effective mass of the Higgs is larger. Therefore at early stages of the evolution the energy transfer into the Higgs particles is the dominating process. This is in accord with [19,20], and can be inferred from the early time behavior of the number densities shown in Fig. 1. One could then expect that the whole energy of the inflaton field will be transferred exponentially fast to the Higgs particles. 3 Since the Higgs decay to the SM fields and their consequent thermalization are fairly fast compared to the Hubble rate one could then estimate the resulting reheating temperature as in [6] T R ∼ m Pl which for λ ∼ 0.1, the number of the SM d.o.f. g * ∼ 10 2 and α > β leads to the values of T R which greatly exceed the freeze-out temperature of the sphaleron processes.  Higgs selfcoupling is taken as λ = 10 −2 . Time is given in program units, see [23]. Preheating ends earlier for Higgs field (tpr 100) than for inflaton (tpr 500). For α = 10 −9 one has the border case when the average momenta of the fields are less then the lattice ultraviolet cutoff.
The lattice results, however, show that such exponential energy transfer into the Higgs particles for a broad range of parameters terminates before any significant part of the inflaton zero mode energy is depleted. The reason for that is the large Higgs boson self-coupling λ ∼ 0.1 which makes the re-scattering processes become important quite early. Unless the Higgs-inflaton coupling α is fairly large the re-scatterings terminate the resonance when only a negligible part of the energy in the inflaton zero mode is depleted. 4 On Fig. 2 one can see how the amount of the transferred energy depends on the value of the Higgs self coupling λ which we allowed to vary to small values just to demonstrate the importance of the re-scattering processes.
On Fig. 3 one can see the dependence of the total energy transferred into the Higgs field as a function of the inflaton-Higgs coupling α. One can draw the conclusion that parametric resonance effects only become important at α ∼ 10 −7 , which is too large a value. Thus, the reheating process proceeds by means of the simple decay of the inflaton (generated abundantly by parametric resonance) into the Higgs particle. This process will be analysed analytically in the next subsection, where we will advocate that this perturbative inflaton decay really reheats the Universe at lower values of the parameter α.

Light inflaton case (m I < 2m H )
While the parametric resonance regime for the Higgs is terminated quite early, the fluctuations of the inflaton field continue to grow exponentially. Since the amount  of the energy transferred into the Higgs field is practically negligible the dynamics of the inflaton field is very close to that of the pure quartic inflaton model which was analyzed numerically in [21,22]. In brief, the inflaton zero mode keeps driving the exponential grows of the nonzero modes until roughly half of its energy is transferred into the inflaton particles. After that the rescattering processes become important, slowly moving the inflaton particle distribution to thermal equilibrium. At some moment the scattering process 2I → 2H becomes important and the Higgs particle (together with all other SM particles) is generated and the standard thermal history of the Universe takes over. The easiest way to estimate the equilibration temperature of this process is to compare the mean free path nσ 2I→2H ∼ n α 2 For the thermal distribution of the inflaton particles this leads to the estimate However, the distribution of the inflaton excitations may be, generally, rather far from thermal equilibrium [21,22]. Evolution of the occupation numbers of the inflaton modes was found to be self similar in [21,22] n(k, τ ) = τ −q n 0 (kτ −p ); , where τ is the conformal time, k is the comoving momentum, and p = 1/5 for three particle interactions and 1/7 for four particle interactions, q ∼ 4p. The only relevant for us property of the function n 0 (kτ −p ) is that the average momentum in (9) at the beginning of reheating after inflation is β 1/4 m Pl . Thus, the average momentum at later times is smaller than expected from the total energy density, p avg /T ∼ (m Pl /T ) p β (1+p)/4 , where T ∼ ρ 1/4 is now not a real temperature, but rather a parameter defining the energy density 6 (cf. equilibration time description in [21,22]). This enhances the 2I → 2H cross section together with the I number density, increasing the estimate (8) by a factor (T /p avg ) 3 . This leads to the increase of the equilibration temperature by a factor 10 5 for four particle interaction, p = 1/7, and by a factor 10 2 for three particle interaction, p = 1/5. Exact calculation of the equilibration temperature requires extensive numerical study, but, in any case, the expression (8) should be considered as the lower bound, while 10 5 T R is the upper (most conservative) bound.
Requiring that T R > 150 GeV we can obtain the lower bound on α for the thermal estimate (8) and for the most conservative estimate of non-thermal distribution of the inflaton. 7 While the bound (10) roughly coincides with the one at which the energy transfer to the Higgs field becomes effective enough to significantly deplete the zero mode of the inflaton (see Fig. 3) while the value given by (11) is about two orders of magnitude smaller. We can, therefore, conclude that the upper bound on the inflaton mass is given by where the range corresponds to the thermal or the most conservative non-thermal estimates.

Heavy inflaton case (m I > 2m H )
In this case the inflaton mass allows for the direct decay of the inflaton into two Higgs particles. The corresponding decay rate is given by Comparing this rate with the Hubble parameter and requiring again for the reheating temperature T R > 150 GeV we get GeV .

(14)
Of course, in the case α β/8 the generation of the cosmological perturbations is different from the case of pure quartic inflation. The Higgs field becomes relatively light and the parameter space of the model is modified. In particular, isocurvature fluctuations which one would generically expect in the two-field model have to be somehow suppressed. This will put the restriction on the allowed values (α, β). The analysis of this parameter space is very involved. One can expect, for example, that the parameter β can differ from its value in the case of pure quartic inflation. That is one of the reasons why the parametric dependence on β is kept in (14). 8

WMAP constraints and non-minimal coupling
Finally let us discuss the constraints on the model from the WMAP data [16]. As was already mentioned in the inflationary regime the model is indistinguishable from the pure quartic potential inflation. One should then confront the fact that the amplitude of the tensor perturbations is too large. One possible resolution of this problem is to assume that the inflaton χ has nonminimal coupling to gravity [17]. We will repeat here the estimates following closely [25,26,27]. We will take the following action as an example where m ≃ m Pl . Even if the coupling ξ is zero at a tree level one can expect that it will be generated via radiative corrections. As it will be discussed below even for small values of ξ the coupling β will deviate from the one, obtained from the COBE normalization in the absence of the non-minimal coupling β| ξ=0 ∼ 1.3 × 10 −13 . The bound on the tensor-to-scalar ratio comes from the perturbations generated at N ≃ 62 e-foldings (see, e.g. [15]) before the end of inflation. In that regime the Higgs part of the model is not important and can be dropped to simplify the discussion. The inflaton part of (15) as it appears in Jordan frame by means of the conformal transformation can be rewritten as (hat denotes transformed quantities) and the new fieldχ is defined as The new potential is given by We assume that ξχ 2 e /m 2 Pl 1, where χ e is the value of the inflaton field at the end of inflation, so the contribution to the effective Plank mass vanishes after the inflationary period. In that case the dynamics of the model with the action (15) after inflation is not different from that of the νMSM model with the potential (3). This suggestion corresponds to ξ < 0.1, see (21). Following [17,25] one can find that the first slow-roll parameter ǫ is given by Slow-roll ends when ǫ = 1. From that one can find that The number of e-foldings from the moment when the inflaton field has the value χ N till the end of inflation is given by Since ξ ≪ 1 one can find that with a good accuracy χ N ≈ 2 2(N +1) 1+6ξ m Pl . The tensor-to-scalar ratio is given by [16] r ≡ 16ǫ = 128m 4 Pl χ 2 N (m 2 Pl + ξχ 2 N (1 + 6ξ)) ≈ 16(1 + 6ξ) (N + 1)(8ξ(N + 1) + 1) .
One can see [17] that roughly in the interval ξ = 0.001 ÷ 0.1 this ratio satisfies the WMAP constraints. The value of the inflaton self-coupling as a function of ξ can be found from the COBE normalization U (χ N )/ǫ(χ N ) = (0.027m Pl ) 4 . The corresponding behavior is shown in Fig. 4. This introduces slight growth of β with ξ, and thus increases all bounds simultaneously, which is demonstrated in Fig. 5.

Conclusions
In Fig. 5 we combined the bounds on the inflaton mass we have found so far. We can conclude, therefore, that the mass of the inflaton in the νMSM inflation [6] should be roughly in the range 0.1 GeV m I 10 GeV (24) in the case when it is light and in the range 300 GeV m I 1000 GeV (25) in the case when the inflaton-Higgs coupling is very small. These bounds could be evaded in models with arbitrary scalar field potentials, but the fact of the strong lower bound from reheating on the coupling between the inflaton and the Higgs should remain rather universal.
Values of ξ larger then 0.1 (and, therefore larger lower and upper bounds on the inflaton mass) are also allowed as well. However, since the dynamics of the model at preheating may be strongly modified from the one we have studied in this Letter it is hard for us to make any statements in that case, and we leave this for future analysis.