Reheating Constraints on Inflaton, Dark Matter: Swampland Conjecture

In this paper, we studied the phenomenological consequences of recently proposed swampland conjecture on the inflationary models though constraints on reheating. If dark matter is assumed to be produced during reheating, the conjecture will provide further constraints on the dark matter parameter space through its current relic abundance. As has been pointed out already and also analyzed in our present paper any successful inflationary scenario is in clear tension with the aforementioned conjecture in its current form. However considering the swampland parameters to be free and constrained by the inflationary observables, we studied in detail its consequence on the reheating and dark matter phenomenology. We point out the connection between swampland conjecture and the scalar spectral index $n_s$ by PLANCK within current $2 \sigma$ range, and associated constraints imposed on the reheating temperature and the dark matter annihilation cross-section.


I. INTRODUCTION
Effective field theory framework has been the subject of intensive investigation for its universal appeal to diverse problems in physics. From large scale to small scale where ever there exists a hierarchy of scales in the problem, it proves to be a unique and logical tool to understand the low scale properties by integrating out the high scale modes supplemented with a finite number of scaledependent free parameters. However, procedure suggests the existence of theory at a high energy scale, which is in general difficult to define. Therefore, the usual approach is to construct the low energy theory order by order in terms of low energy modes based on some underlying symmetry principle which is assumed to be the full theory property. A natural question then one can ask is whether all possible effective field theory so constructed can have its ultraviolet completion. This is a very difficult question to answer. String theory has been proved to be a fantastic playing field in this regard. This is the only theory, we know, which is at least an ultraviolet complete theory of gravity.
Recently motivated by this question and taking help of various string theory constructions, a number of attempts have been made to put some constraints on the effective theory which will have consistent VU completion. One such proposal is the swampland conjecture [1], which has recently gained interest in the literature. The conjecture says that a low energy effective theory of scalar field minimally coupled with gravity must satisfy the following universal bound on its form of the potential, where c is a dimensionless constant with the magnitude of order unity and M p is the Planck mass.
However, there exists a refined version of the aforementioned swampland conjecture stated in [1] which is expressed as, This is a weaker condition on the possible form of the potential. c is another universal constant of order unity, and min(∇ i ∇ j V ) is the minimum eigenvalue of the Hessian of ∇ i ∇ j V matrix in an orthogonal frame. A large number of studies have been performed over the years to understand more on the theoretical understanding of this conjecture [2]- [45]. However, it would be important to mention the interesting debates going on in the literature on the existence of di-Sitter vacuum in sting theory [46] [47]. From the phenomenological point of view, the implication of this conjecture has been widely studied in the context of cosmology [48][49] [50]. But the main problem to connect this conjecture with the reheating era is the thermalization portion. However in [51], author introduce warm inflation to solve this problem. Starting from inflation to dark energy, the scalar field is ubiquitous and therefore, the conjecture can naturally put constraints on the model building.
More interestingly, the hope is that the inflationary, dark energy observation may shed light on UV physics through this conjecture.
In this paper, we will consider inflationary models with a specific interest on the reheating dynamics. We ask the following question: How does the swampland conjecture put constraints in the reheating dynamics and the dark matter phenomenology?. In our analysis, we will consider (c, c ) as free parameters. Taking constraints on those parameters from the inflationary dynamics, we will further study the reheating phase.
In the subsequent section we first briefly review the basic equations describing the constraints on reheating and consequently on the dark matter parameters considering the CMB anisotropy and the current dark matter abundance. We take four different types of inflationary model potentials and describe how the swampland conjecture restricts the reheating and the dark matter parameter space through the inflationary observables.

II. REHEATING AND DARK MATTER: METHODOLOGY
Reheating is the phase which connects the inflation and big-bang through explosive particle production. This phase also can play important role in the dark matter phenomenology. Even though inflation is severely constrained by a large number of cosmological observations, the reheating phase is generally unconstrained. This phase is parametrized by two important parameters called reheating temperature T re and e-folding number N re . To go beyond we further incorporate a possible dark matter candidate indirectly originating from the decaying inflaton. During reheating inflaton decays to radiation and then it annihilates to dark matter such that the process during reheating gives us correct relic abundance. As emphasized before, given the observational constraints on the inflationary dynamics, our goal of this paper would be to constrain the reheating and dark matter parameter space through the inflationary parameters considering the swampland conjecture.
For simplicity, we first follow the usual reheating constraint analysis [52] where the reheating parameters are calculated assuming the instantaneous conversion of inflaton energy into radiation at the instant of reheating. The evolution during reheating is parametrized by an effective constant equation of state w re . Following the assumptions and considering a particular inflation model, one can easily compute the reheating temperature to be where g re is the effective number of relativistic degrees of freedom at the instant of reheating.
(T 0 = 2.725K, a 0 ) are the CMB temperature, and the cosmological scale factor at the present time.
The number of reheating e-folding number during reheating can be expressed as [53] From the above equation, we can clearly see the appearance of inflationary parameters which are constrained by the swampland conjecture. Therefore, indirect constraints can be imposed on the reheating parameter space. We will be considering some simple canonical scalar field models of inflation, for which Hubble constant H k and the inflationary e-folding number, N k are defined as Where, one clearly sees the non-trivial dependence of reheating parameters on the scalar spectral index n k s , and tensor to scalar ratio r k through power spectrum of inflaton fluctuation A s (n k s ). The aforementioned inflationary parameters, in turn depend on the slow roll parameters related to the inflaton potential V (φ), which can now be constrained by the swampland conjecture, Most importantly all the above quantities are defined at a particular cosmological scale k. For CMB, we consider the pivot scale of PLANCK k/a 0 = 0.002 Mpc −1 . The end of inflation set the initial condition for the reheating dynamics. Therefore, dynamics will be mostly controlled by where, φ end is the inflaton field value at the end of inflation follows from the equation In the discussion so far we have not considered explicit decay of inflaton. However to shed light on the dark matter phenomenology we consider perturbative reheating process where inflation decays to radiation and then radiation to dark matter [56]. For this we will have three parameters, the inflaton decay constant Γ φ , thermal average of dark matter annihilation cross section σv , and the dark matter mass M X . In the perturbative reheating process, the dynamics of the inflaton energy density (ρ φ ), the radiation energy density (ρ R ) and dark matter particle number density (n X ) are modeled by following homogeneous Boltzmann equations [57].
where, for numerical purpose, new dimensionless variables are defined as We also rescale the cosmological scale factor by A = a a end , where a end is the scale factor at the end of inflation. Prime represents the derivative with respect to A. The inflation mass is m φ .
We assume that each X has energy E X = ρ X n X √ M 2 + 9T 2 . The equilibrium number density of dark matter particle of mass M X can be expressed in terms of modified Bessel function of the second kind: The constants c 1 and c 2 are defined as, The initial conditions to solve the Boltzmann equations are set at the end of reheating to be, In this process, we will define reheating temperature (T re ) from the radiation temperature (T rad ), at the instant of maximum energy transfer from inflaton to radiation for H(t) = Γ φ .
Combining this equation with eq.3 we can establish one to one correspondence between T re and Γ φ supplemented with the following condition for reheating As we mentioned earlier that our another interest is to constraint dark matter phenomenology through the swampland conjecture. Therefore, while solving the Boltzmann equation we also need to consider the following condition on the dark matter abundance parametrized by Ω X , which is expressed in terms of radiation abundance Ω R (Ω R h 2 = 4.3 × 10 −5 ), as where T F is the temperature at a very late time when co-moving dark matter, as well as radiation density, became constant. The present value of dark matter abundance imposes a constraint on the dark matter parameter space (M X , σv ) by the CMB anisotropies through the scalar spectral index n k s considering swampland conjecture. Therefore, inflationary dynamics, and considering the CMB temperature anisotropy, we will be able to constrain reheating as well as dark matter parameter space though the swampland conjecture. With all the ingredient discussed so far, and considering the condition of the refined swampland conjecture Eq.2, we can figure out the allowed region of (c, c ) with respect to inflation parameters (n k s , r k ). For our discussions, we will be considering (c, c ), as free parameters. The region of n k s will be considered to be bounded by the PLANCK 2σ region in (n k s , r) plane. With this consideration, a particular value of c, in general provides the maximum allowed value of n k s and that in turn, imposes restriction not only on the maximum value of the reheating temperature (T max re ) but also on the dark matter parameter space allowed by current dark matter abundance. Similarly, the maximum allowed value of reheating temperature, which is associated with a particular value of the spectral index, should also impose constraints on c. We show the resulting constraints on c and c for the maximum value of the scalar spectral index (n max s ) and the minimum value of the scalar spectral index (n min s ). In the following discussions, we consider various inflation model and discuss important results of our analysis. In all cases, we consider two different effective equation of state parameter for reheating, ω re = 0 and ω re = 1 6 .
A. Chaotic inflation [54] For usual chaotic inflation model potential looks like where n = 2, 4, 6 . . . . If we consider the absolute value of the field, n = 3, 5, . . . can also be included.
We consider only n = 2 for our numerical purpose. As mentioned earlier, the first condition of the refined swampland conjecture, M p V /V , transforms into the following inequality, where, inflaton field value for a particular scale of interest k, can be written as After combining equations (19) and (20), we see that a particular value of c gives an upper bound on the value of scalar spectral index n k s . This constraint sets a maximum possible value of the reheating temperature (T max re ). Because of implicit relation, the above constraint also sets a minimum possible value of dark matter scattering cross-section σv for a given dark matter mass. Similarly from the second condition of the conjecture, we find the following further constraint on the inflaton field value, Combing the equations (19), (20) and (21), we are able to find constraints on c, c . For n = 2, combine equation can be written as By using the above constraint relation along with any one of the swampland conjecture say equation Importantly considering the PLANCK observation, if we increase the value of c towards c max , this range of dark matter annihilation cross-section will be further narrowed down.
B. Natural inflation [58] The inflationary potential in this model is given by where, Λ is the height of the potential setting the inflationary energy scale, and f is the width of the  50M p )). With these ingredients the constraints on c will be obtained from swampland conjecture Eq.2 and Eq.21 which transforms into the following inequalities for the axion inflaton, One important fact to notice is that, at the point of instantaneous reheating (N re ∼ 0), the The resulting constraints in (c, c ) space has been plotted in the last plot of the fig.2, where the upper limit on c has been derived from the PLANCK constraints. In this section, we will consider a class of models which unifies a large number of inflationary models parameterized by a parameter α, first proposed in [59]. Conformal property of this class of models leads to a universal prediction for the inflationary observables. In the canonical form, the so-called α-attractor potential is expressed as This model is known as E-model. The mass scale Λ can be fixed from the CMB power spectrum. or where φ can be written in terms of spectral index (n s ) as [60] φ = 3α 2 M p ln 1 + 4n + 16 n 2 + 24 α n (1 + n) (1 − n s ) 3 α (1 − n s ) .
As in the natural inflation model, here also we have c tmax corresponding to instantaneous reheating, which are independent of equation of state but dependent on α. For example, we have c tmax = (0.0196, 0.088) for α = (1, 100). Nonetheless as already observed for other inflationary models,  D. Supergravity inspired minimal plateau model [61] In this section we will consider a special class of supergravity inspired inflation potential, The shape of the potential depends on the mass scale φ * and all other parameters are same as in the previous inflationary model. One of the striking features of this model is that unlike axion and α-attractor models, it fits well within PLANCK data for all possible values of φ * from super to sub-Planckian value. Our initial motivation was to figure out if with increasing φ * the value of (c, c ) increases towards unity. However we did not find such solutions. None the less for Swampland conjecture has gained significant attention mainly because of its potential to validate or invalidate large number of low energy effective theories proposed in diverse physics problems.
It mainly deals with a scalar field and its possible nature of the potential which is conjectured to follow certain constraints. Two parameters (c, c ) are conjectured to be of the order unity such that the field theory under consideration can have consistent ultraviolet completion when minimally coupled with gravity. One of the best candidates scalar field is inflaton which has been proved to be successful in explaining large volume of cosmological observations. However, it turned out that more successful a inflation scenario is more incompatible with the Swampland conjecture it becomes. In the present paper instead of taking swampland parameters to be of order unity, we considered them free and analyze its impact on the other cosmological parameters with special emphasis on the reheating phase. Based on our analysis so far let us try to point out the main outcomes. We have considered four different types of inflaton potential and studied the consequence of swampland conjectures on those and constrain the parameter space. However, we must say that the conventional slow roll potential is very much constrained by the conjecture. For all the models under consideration what we found is that the possible values of c is always less than unity. The maximum possible value one could get c 10 −1 is for chaotic inflation which predicts higher value of tensor to scalar ratio. Moreover we studied other models, such as axion, α-attractor, supergravity inspired inflation, which are consistent with PLANCK data, and maximum possible value of c turns out to even smaller for those model. This is intimately connected with smaller prediction of tensor to scalar ratio r. However, this has already been observed before. Our focus in this paper was more on the impact of this inflationary constraints of (c, c ) on the reheating phase and dark matter phenomenology. In Fig.1, 2, 3, and 4 we have considered different cosmological parameters such as (n s , T re , σv ), and studied their interdependence and constraints from the swampland conjecture.
In the usual reheating constraint analysis, the reheating temperature varies widely within 2σ range of n s , and consequently so does the dark matter cross-section σv given a dark matter mass. Since swampland conjecture is an inequality, for the model under considerations it provides us an upper   Considering different class of inflationary models, one of our important observation is the existence of maximum possible reheating temperature T max re 10 15 GeV irrespective of the models. Importantly, however, the associated swampland parameter c = c tmax are depending upon the models and their parameters. In the same way even more interesting fact is that for a fixed value of c within (c max , c tmax ), there exists associated maximum value of reheat-