Particle physics model of curvaton inflation in a stable universe

We investigate a particle physics model for cosmic inflation based on the following assumptions: (i) there are at least two complex scalar fields; (ii) the scalar potential is bounded from below and remains perturbative up to the Planck scale; (iii) we assume slow-roll inflation with maximally correlated adiabatic and entropy fluctuations 50–60 e-folds before the end of inflation. The energy scale of the inflation is set automatically by the model. Assuming also at least one massive right handed neutrino, we explore the allowed parameter space of the scalar potential as a function of the Yukawa coupling of this neutrino.


I. INTRODUCTION
The precise mechanism of cosmic inflation is one of the most pressing questions in our understanding of the early universe. Today the original idea for inflation [1] is not favoured because it is unclear how to define a proper mechanism to explain the required reheating of the universe. A popular solution to this question of reheating is the slow-roll scenario [2,3] in which the ground state starts from an unstable position and rolls down very slowly to a local or global minimum. The inflation stops when the potential energy function becomes too steep, which leads to a fast roll. In principle, the slow roll can start from a large field value and proceed towards a minimum with a smaller field value, or from a small (essentially vanishing) field value to a larger minimum. These two cases are referred to as large-and small-field slow roll [4]. A problem related to large-field slow-roll is the initial value problem, namely one has to explain why the ground state starts from a value much larger than the typical energy scale of inflation. Chaotic inflation [5] was devised to handle this problem, but then one has to assume very large -again larger than the scale of inflation -fluctuations.
The origin of inflation is still an open question in cosmology [6,7].
It is known that scalar fields can mimic the equation of state required for the exponential expansion of the early universe [2,3]. As the Higgs boson was discovered [8,9], we know that at least one doublet scalar field exists in nature. Hence, it may appear natural to assume that the Brout-Englert-Higgs (BEH) field is the inflaton (see for example Ref. [10]), but such * zoltanpeli92@gmail.com † nandori.istvan@science.unideb.hu ‡ Zoltan.Trocsanyi@cern.ch; http://pppheno.elte.hu/ a scenario was criticised, see for instance Ref. [11]. Many types of scalar potentials have already been discussed in the literature as viable scenarios for cosmic inflation [11]. There are three major categories of scalar inflaton potentials with minimal kinetic terms: (i) the large field, (ii) the small field and (iii) the hybrid models. In the third case one introduces more than one field, with one of those being the inflaton and the other field switches off the exponential expansion. In this sense it is not a real multifield model. The case of hybrid models is excluded by experimental observations because those predict a scalar tilt n s larger than one in contradiction with the observed structure of the thermal fluctuations of the cosmic microwave background radiation (CMBR) resulting in n s = 0.9677 ± 0.0060 [12,13]. The tensor and scalar power spectra of the CMBR suggest a small value for the tensor-to-scalar ratio r, consistent with zero, which emerges automatically in real multifield models with curvaton scenario [14][15][16][17].
In this letter we consider a simple multifield particle physics model of cosmic inflation with a curvaton scenario. We show that in a fairly constrained region of the parameter space, the model can provide a natural switch on and off mechanism of inflation.

II. PARTICLE PHYSICS MODEL
The particle content of the model coincides with that in the standard model of particle interactions, supplemented with one complex scalar field. We also allow for one (or more) Dirac-, or Majorana-type right-handed neutrinos. In this letter for the sake of definiteness we consider the case of Dirac neutrinos.
In addition to the usual SU (2)-doublet scalar field we assume the existence of a complex scalar χ that transforms as a singlet under the standard model gauge transformations. The potential energy of these scalar fields is assumed as where |φ| 2 = |φ + | 2 +|φ 0 | 2 and C =   2λ φ λ λ 2λ χ   is the coupling matrix. This potential energy function contains a coupling term λ|φ| 2 |χ| 2 of the scalar fields in addition to the usual quartic terms. The value of the additive constant V 0 is irrelevant for particle dynamics, but as we shall see, it is relevant for the inflationary model, hence we allow a non-vanishing value for it.
In order that this potential energy be bounded from below, we have to require the positivity of the self-couplings, λ φ , λ χ > 0, and also the coupling matrix be positive definite, Our model for cosmic inflation works only if λ < 0 [18]. If these conditions are satisfied, we find the minimum of the potential energy at field values φ = v and χ = w where the vacuum Using the VEVs, we can express the quadratic couplings as For λ < 0, the constraint (3) are satisfied simultaneously, which can be fulfilled [19] if at most one of the quadratic couplings is smaller than zero [20].
After spontaneous symmetry breaking and choosing unitary gauge, the scalar kinetic term leads to a mass matrix of the two real scalars [21]. We can diagonalize this matrix by an orthogonal rotation and find for the masses M h/H of the mass eigenstates: where M h ≤ M H by convention. At this point either h or H can correspond to the observed scalar boson. As M h must be positive, the condition has to be fulfilled. If both VEVs are greater than zero, as needed for two non-vanishing scalar masses, then this condition coincides with the positivity constraint (3).
We studied the ultraviolet behaviour of the scalar couplings of this model in Ref. [19] where we constrained the parameter space by requiring that (i) the scalar potential remains bounded from below and (ii) the couplings remain perturbative up to the Planck scale m P .

III. COSMOLOGICAL INFLATION
We now explore the cosmic inflation of the two-field model with potential energy defined in Eq. (2). We consider slow-roll inflation when the potential energy has a large, almost flat area for small field values and a global minimum at large values of the VEVs. Such a potential energy allows for slow roll of the fields from small values towards the global minimum, resulting in cosmic inflation. The required form of the potential energy function appears naturally at some high energy scale, for certain values of the scalar couplings at the mass of the t-quark m t . As Eq. (4) shows, the VEVs are inversely proportional to √ det C .
(note that η φφ + η χχ = η σσ + η ss ), while  the end of inflation. This corresponds to an even smaller , which reduces the tensor-to-scalar ratio, r = 16 (sin ∆) 2 to essentially zero. Such a small r is not excluded by cosmological measurements. The smallness of r however is in conflict with the traditional cosmological normalization V 0 r 1.6 · 10 16 GeV 4 .
This conflict may be resolved by assuming that the adiabatic and entropy fluctuations were maximally correlated at 50 − 60 e-folds before the end of inflation, implying cos ∆ = 1, and hence predicting zero for the tensor-to-scalar ratio. Consequently, we have to find different conditions to set the scale of inflation µ inf and for the normalization of the potential energy.
As suggested above, we provide the first from the particle physics model by identifying µ inf with the location of the wedge in the running of det C . The case of ∆ = 0, i.e. maximally correlated fluctuations are referred to as the curvaton scenario. In this case, the various tilts coincide. Neglecting , we have: Considering η ss as a function of V 0 (see Eq. (9) with V 0 in V in the denominator), we normalize it to produce the scalar tilt in agreement with the most recent data, n s 0.966, Having fixed the value of V 0 , we propose the following inflationary scenario. The scalar potential energy is given by Eq. scanning, we need to search only for those points where w(m t ) -given by [19] , with λ SM (m t ) = 1 2 m H (m t ) 2 /v(m t ) 2 0.126 -and the masses of the scalars given by Eq. (7) remain positive. We find that the parameter space is constrained to a shell on the surface of the region allowed by the condi-tions of stability and perturbativity of V . The width of the shell is affected by the allowed depth of the minimum of det C : the smaller det C , the thinner the shell. Furthermore, we have also found that the minimum value of the location of the wedge µ inf is around 10 14 GeV, depending slightly on c ν (m t ).
In Fig. 3 we present the results of such scan of the parameter space. These plots show different planar projections of the three dimensional parameter space, spanned by λ φ (m t ), λ χ (m t ) and λ(m t ). The shape and size of the supported regions is affected by the choice of c ν (m t ), as seen in the titles of the figures. We find that the parameter space of the scalar couplings is not empty, but constrained strongly if we assume that cosmic inflation took place as described above. This assumption constrains the smallest value of w to around 265 GeV.

V. CONCLUSIONS
In this letter we proposed a particle physics model of cosmic inflation. It requires at least two scalar fields. We found that in a small region of the parameter space of the scalar couplings, the determinant of the scalar quartic coupling matrix becomes very small at a scale around 10 16 GeV. As a result the global minimum of the scalar potential increases significantly, allowing for an accelerated expansion of the universe by a slow-roll model at this scale, called the scale of inflation. We assume the curvaton scenario of inflation, i.e. maximally correlated adiabatic and entropy fluctuations at 50 − 60 e-folds before the end of inflation, which implies vanishing tensor-to-scalar ratio. To set the normalization of the potential at vanishing field values, we required that the model reproduces the measured value of the scalar tilt. The inflation stops when the parameter that measures the acceleration of the fields starts to increase quickly. After this the global minimum of the potential decreases preventing the appearance of another period of inflation.