Probing the reheating phase through primordial magnetic field and CMB

Inflationary magnetogenesis has long been assumed to be the most promising mechanism for production of Large scale magnetic filed in our universe. However, it is generically shown that such models are plagued with either backraction or strong coupling problem within the standard framework. In this paper we have shown that reheating phase can play very crucial role in alleviating those problems. Assuming electrical conductivity to be in-effective during the entire period of reheating, the classic Faraday electromagnetic induction changes the dynamics of magnetic field drastically. Our detailed analysis reveals that this physical phenomena not only converts a large class of magnetogenesis model observationally viable without any theoretical problem, but also can uniquely fix the reheating equation of state given the specific values of scalar spectral index and the large scale magnetic field. Our analysis also opes up a new avenue towards constraining the inflationary and magnetogenesis model together via reheating.


I. INTRODUCTION
Reheating is one of the most important early phase of our universe. It essentially links the standard thermal universe with its pre-thermal phase namely the inflationary universe through a complicated non-linear process. Over the years major cosmological observations [1][2][3][4] have given us ample evidences in understanding the theoretical as well as observational aspects of both the thermal and the non-thermal inflationary universe cosmology to an unprecedented label. However, the intermediate reheating phase is still at its novice stage in terms of both theory and observation. From the cosmic microwave background (CMB) anisotropy [1], one can estimate the baryon content of the universe which agrees extremely well with the theoretical prediction of big-bang nucleosynthesis (BBN). Furthermore, with the successful standard big-bang model we have a very good understanding over a large time scale of the universe from the present (redshift z = 0) to BBN stage (z ∼ 10 9 ) at an energy scale ∼ O(1) MeV. The tiny fluctuation of CMB anisotropy can be successfully linked with the almost scale-invariant density fluctuation predicted by the inflation in the early Universe [6,8]. Therefore, precession CMB data has provided us significant insight into how our universe evolves during inflation.
However, in this paper, our goal is to understand the intermediate phase which joins the end of inflation and the BBN. This phase is largely ill-understood due to a lack of observational evidences. It is generically described by the coherently oscillating inflaton and its non-linear decay into the radiation field. In the Boltzmann description, the phase is parametrized by the reheating temperature (T re ) and the reheating equation of state (ω re ). Still now both the parameters remain unconstrained except the reheating temperature which is approximately bounded within 10 16 GeV > T re > T BBN ∼ 10 MeV.
However, if one takes into account non-perturbative reheating in the beginning, the upper bound on reheating temperature could be within 10 10 − 10 13 GeV [79]. Attempts to understand this phase in the literature can be broadly classified into two categories: By studying the background dynamics during reheating reveals useful information about the deep connection among the inflationary scalar spectral index (n s ) and the reheating temperature (T re ), and reheating equation of state (ω re ) [43,51,56,58].
Evolution of inflationary stochastic gravitational waves has been shown to encode valuable information when passing through this phase [7], [5] and reference therein.
In this paper we consider present-day Large Scale Magnetic Field (LSMF) combined with the CMB anisotropy to probe the reheating phase of the universe followed by the standard inflationary phase.
For LSMF we consider simple model of primordial magnetogeneis [25][26][27]51]. While probing the reheating phase through those observables, we observe how the magnetogensis models itself will be constrained by the observables as well. Inflationary magnetogenesis models have been studied quite extensively in the literature [23,51,[74][75][76][77]. Mechanisms known so far are to introduce the interaction Lagrangian which explicitly breaks the conformal invariance in the electromagnetic sector. However, this mechanism generically suffers from either strong coupling or backreaction problem [28] which will be elaborated as we go along. In this regard reheating phase has recently been shown to play a very important [25] role. As stated earlier the primary motivation of our present study will be to see in detail how both the problems can be successfully resolved by the reheating phase for various inflationary models, and simultaneously provide constraints on the reheating. Taking into account both the CMB anisotropic constraints on the inflationary power spectrum and the present value of the large scale magnetic field, our analysis reveals an important connection among the reheating parameters (T re , w re ), magnetogensis models and inflationary scalar spectral index (n s ).
The universe is observationally proved to be magnetized over a wide range of scales. Zeeman splitting, synchrotron emission, and Faraday rotation are some of the fundamental physical mechanisms by which the existence of a magnetic field can be probed. Various astrophysical and cosmological observations of those quantities tell us that our universe is magnetized over scales starting from our earth, the sun, stars, galaxies, galaxy clusters, and also the intergalactic medium (IGM) in voids. In the galaxies and galaxy clusters of few to hundred-kilo parsecs (kpc) scale, the magnetic fields have been observed to be of order a few micro Gauss [9,11,14]. Various observational evidence suggests that even the intergalactic medium (IGM) in voids can host a weak ∼ 10 −16 Gauss magnetic field, with the coherence length as large as Mpc scales [15]. At the astrophysical scale, time-evolving plasma helps the magnetic field to persist. The magnetic field at the cosmological scale can evolve and survive today due to the presence of the early plasma state of our universe before the structure formation.
The time-evolving plasma, therefore, proves to be an ideal environment to have a sustainable evolution and growth of the magnetic field. However, any physical processes responsible for the successful magnetogenesis inside the time-evolving plasma, it is the tiny seed initial magnetic field which plays a significant role. In the cosmological context, the most popular mechanism in this regard is the primordial inflationary magnetogenesis. Inflation provides us an outstanding mechanism for producing coherent magnetic fields for a wide range of scales. Mpc scale magnetic field can survive until today as a cosmological relic whose magnitude could be ∼ 10 −16 G. On the other hand at small scales, this tiny inflationary magnetic field can be the seed field which will be further enhanced to galactic scale µG order magnetic field by the well known Galactic dynamo mechanism [11,16,17].
We consider a standard scenario of inflationary magnetogenesis where electromagnetic field kinetic term is conformally coupled with a scalar field. Background inflation dynamics naturally produces a large-scale electromagnetic field which subsequently evolves through the reheating phase. Instead of going into the details of the magnetogenesis mechanism, we concentrate on dynamics during reheating considering various inflationary models. In our analysis we assume the negligible Schwinger effect on the magnetogenesis. The paper is arranged as follows: in section II we discuss the general analysis of the primordial magnetic fields from inflation and also the reheating dynamics used to constraint the parameters in the scenario of inflationary magnetogenesis. In section II we also discuss different inflationary models and their dynamics, which are used to constrain the parameters. Subsequently in section IV we finally show how our analysis constrain the reheating as well as magnetogenesis model considering few observationally viable inflationary scenario.

II. INFLATIONARY MAGNETOGENESIS: GENERAL DISCUSSION
During inflation the large scale magnetic field is generated out of quantum vacuum, and then subsequently evolves though various phases of our universe. Therefore, the evolving magnetic field must encode the valuable information about the reheating. Considering the present value of the large scale magnetic field we, therefore, can place constraints not only on the parameters of the reheating phase, but also on the magnetogensis model itself. Contrary to the convention, the important point of our present analysis is the assumption of conductivity being negligible until the end of reheating. The reason being the production of the radiation plasma occurs nearly at the end of reheating. From large number of studies [43,44,46,56] so far almost entire reheating phase has been shown to be primarily inflaton energy dominated. In the context of inflationary magnetogensis scenario, this particular assumption has recently been proposed to be important [25] during reheating. Conventionally after the end of inflation, the magnetic field on the super-horizon scales is assumed to be redshifted with the scale factor a as B 2 ∝ 1/a 4 provided inflaton energy density transfers into plasma and the universe become good conductor instantly right after the end of inflation. Hence, the electric field ceases to exist. However, in the reference, [25] it has been shown that if the conductivity remains small redshifts of magnetic energy density becomes slower, B 2 ∝ 1/a 6 H 2 , due to electromagnetic Faraday induction.
Here H is the Hubble parameter. This helps one to obtain the required value of the present day large scale magnetic field considering large class inflationary model which we describe below, A. Quantizing the Ratra model: Electromagnetic power spectrum The simplest magnetogenesis scenario which one can think of is the well known scalar-gauge field model with the following interaction Lagrangian I(φ) 2 F F , well known as Ratra model [27]. In this interacting Lagrangian, the conformal symmetry is explicitly broken by the scalar field coupling function I(φ) in the gauge field sector. During inflation, the model generally predicts strong primordial electric field than magnetic field, and that can backreact to invalidate the mechanism itself. First we discuss the model in detail and show how the reheating phase can come as our rescue of this backreaction problem. In the frame of a comoving observer having four-velocity u µ (u i = 0, u µ u µ = −1), the magnetic and electric fields are defined as Where, F µν = ∂ µ A ν − ∂ ν A µ is the electromagnetic field tensor and µνρσ is a totally antisymmetric tensor. The background is the well known FLRW metric with the time dependent scale factor a(τ ) expressed in conformal coordinate, As already described before, the gauge field action is taken to be, At this point let us point out that one can consider other inflationary magnetigeneis models with axionelectromagnetic field coupling [80][81][82], higher curvature coupling [83,84] and apply our methodology presented here to not only constraint the reheating phase but also make the models under consideration viable.
In the inflationary magnetogenesis scenario, the essential idea is to quantize the electromagnetic field in the classical inflationary background. Here I(τ ) 2 ≡ I(φ(τ )), therefore, is the time dependent coupling arising form some classical background scalar field. To maintain generality we do not specify any background dynamics of the scalar field. Through this coupling the electromagnetic field experiences the spatially flat expanding FLRW background. In order to quantize the field components A µ are expressed in terms of irreducible scalar and vector components as follows, In the standard canonical quantization procedure, one writes V i in terms of the annihilation (a k ) and the creation operator (a † k ) as here the (p) i (k) is the polarization vector corresponding to the two polarization direction p = 1, 2, which satisfy the following relations, All the dynamics of the field will be encoded into the mode function which satisfy the following equation of motion, Where the prime denotes derivative with respect to the proper time τ . Conventionally the electromagnetic power spectrum is expressed in terms of those mode functions as follows, However, it would look physically elegant, if we express the power spectrum in terms Bogliubov coefficient which has direct physical interpretation in terms of quantum particle production. In order to do that let us first write down the Hamiltonian in terms of time independent creation and annihilation operator as follows, This is clearly not diagonal. Therefore, in order to diagonalize, we employ the Bogoliubov transformation. In this transformation, new set of time dependent creation and annihilation operators ) are defined in term of old ones. And the new basis of the Hilbert space so constructed diagonalizes the above Hamiltonian, b (p) Where α The Bogoliubov coefficients follow the following normalization condition With all these ingredients one represents the power spectrum in terms of Bogoliubov coefficients, α k as follows, Considering the super-horizon limit, the above expressions for the electromagnetic power spectrum can be further simplified by extracting the amplitude and phase of those coefficients. By using the normalization condition Eq (13) one can write, Where, the phase factor is expressed as arg(α k . In the following discussion, we consider a specific model of the electromagnetic coupling function.

Magnetogenesis: Modelling the coupling function
In order to study further we consider the following widely considered power law form of the coupling function [27] where, a end is the scale factor at the end of inflation. At this stage let us point out that, in all the previous analysis in the literature, n has been considered to be integer. However, for our present discussion we keep the value of n arbitrary. As one of the important focuses of the analysis also is to constrain the magnetogenesis model itself namely the value of n in accord with the present day large scale magnetic field and CMB anisotropy. Further more after the end of inlation the value of the function is so chosen that the usual conformal electrodynamics is restored.
Our main interest is to understand the large scale magnetic field which is assumed to be produced during the initial stage of inflation. Therefore, through out our analysis, we assume the Hubble parameter to be constant. This also helps us to give clear picture in terms of analytic solutions.
The perfect di-Sitter background with Hubble parameter H inf , one obtains the solution for the mode function as, which leads to Bunch-Davis vacuum state at the sub-Horizon scale. H Here ν = −n ± 1 2 . Defining a new variable z = k/aH inf , the time dependent Bogoliubov coefficient are expressed as, This essentially suggests that any mode starting from the Bunch-Davis vacuum at the sub-horizon scale transforms into highly squeezed state, parametrized by |β (p) k | 2 1, after its horizon exit during inflation. Using this condition one gets the following simplified form of the electromagnetic power spectrum, For any arbitrary value of n, the final expression for the spectrum are as follows, In our final analysis, we will be considering this expression. To this end it is important to remind the reader, the widely studied case for n being positive integer for which the electromagnetic power spectra assumes the following simple forms in the superhorizon limit (k aH inf ) as [25].
With this expression, subsequent magnetic field evolution has been widely studied considering the magnetic energy density decreasing as |B| 2 ∼ 1/a 4 . The required value of large scale magnetic field of order 10 −16 G can be obtained only if inflation scale H inf assumes low value which has been proved to be difficult in conventional inflationary model in the effective theory framework. We also have observed this in our numerical analysis. Simultaneously we also observed how this problem can be alleviated by the electromagnetic induction during reheating. Furthermore, we will assume arbitrary value of n.

After inflation dynamics: reheating
In order to associate the observed current magnetic field with the magnetic field produced during inflation, it is essential to study the subsequent evolution. Most of the studies so far considered the fact that when I 2 becomes constant at the end of the inflation, the co-moving photon density |β (p) k | 2 is conserved. Consequently the magnetic power redshifts as P B (k) ∝ a −4 until today. Before we embark on our original analysis, for the sake of completeness let us briefly discuss this widely studied case. Thus at the end of the inflation, the phase parameter (θ end k ) and the photon density (|β end k | 2 ) is identified as Here we define z end ≡ k/a end H inf . For any arbitrary values of n, the magnetic power spectrum is For positive integer n, the expression above (27) boils down to at the super-horizon scale k aH inf . Considering nontrivial dynamics, the magnetic power spectrum at the present universe can be correlated with the magnetic power at the end of the inflation through the following standard equation Where a end /a 0 can be expressed in terms of inflationary e-folding number (N k ) as Here k /a 0 = 0.05M −1 pc is taken as pivot scale set by Planck observation. By combining equations (28), (29), and (30), the expression for the magnetic power considering integer values of n, follow the equation It has already been observed and also shown in Fig.1 the well known fact that magnetic strength of order 10 −15 − 10 −22 Gauss on 1 Mpc scales [51,54] can not be obtained by the above magnetic power spectrum for high-scale inflation model such as well studied Higgs-Starobinsky inflation. Therefore, number of models have been constructed just to elevate this problem without much success with regard to the theoretical issues which we discuss in the next section. Form here itself we will advocate the need for re-looking into the reheating effect more seriously than model building. In the recent paper [25], authors have showed that considering negligible electrical conductivity during reheating the Faraday's law of electromagnetic induction plays an interesting role in modifying the magnetic field evolution.
Therefore, we want to see how this induction effect can safely generate presently observed magnetic field for different high scale inflationary models. Interestingly incorporating the CMB anisotropy into the analysis will further reveal an intricate interconnection among various apparently disconnected cosmological parameters such as (n s , P B0 , T re , w re ). Our analysis, therefor, opens up an interesting new possibility of probing the reheating dynamics, which is otherwise difficult, and simultaneously constraining the inflationary as well as magnetogenesis model parameters through the evolution of the primordial magnetic field.
After the end of inflation the gauge kinetic function I(φ) is assumed to be unity, rendering the fact that the post inflationary evolution of the electromagnetic field is essentially standard Maxwellian.
Therefore, the conformal invariance is restored, and consequently the gauge field production from the quantum vacuum ceases to exist. The electromagnetic field produced during inflation will cross the horizon and turned into the classical one which will subsequently evolve during this phase. In the Fourier space, the mode function solution of the free Maxwell equation is, here τ end represents the end of inflationary era. For arbitrary value of n, the phase factor is calculated Where the elapsed conformal time is obtained as Since there is no further production of gauge field, the photon number density |β k | 2 becomes independent of time and follows the same equation (26) as before. However, as emphasized already, the Farady induction will come into play during this phase. In order to see this let us first express the magnetic power spectra at the end of reheating parametrized by the scale factor (a re ) as where θ re k is the phase parameter at the end of the reheating era, which is defined as, Notable term of the above expression is the second one which leads to the non-conventional dynamics of the magnetic field. Assuming the constant equation of state during reheating dynamics the special term boils down to the following simple from, The phase term contributes to the dynamics of the magnetic power spectrum in two different ways.
The conventional one will be associated with the redshift factor ∝ a −4 emerging from the first term in the right-hand side of the Eq. (33). Important one is associated with the redshift factor ∝ a −6 H −2 emerged out from k/aH term. As the expansion of the universe is decelerating after inflation, leading contribution to the evolution of the magnetic power would be controlled by the latter one Eq.(33).
However, the electric field energy dilutes following the conventional form For large scale (k aH), the above expression boils down to, for integer values of n. After the end of reheating, inflaton energy is converted into highly conducting plasma containing all the standard model particels. Due to large electrical conductivity primordial electric field decays to zero and comoving magnetic energy density freezes to a constant value until today. Therefore, final general expression of our interest is the present day magnetic field strength given as If we take integer value of n, at super horizon scale, the above expression will transforms into [25] where a 0 is the scale factor at the present time. In the next section we will briefly discuss about standard theoretical issues related to the magnetogenesis model.
In the context of simplest magnetogenesis model proposed in [27], the gauge kinetic function I(τ ) can essentially be interpreted as time dependent effective electromagnetic coupling. By considering field theoretic argument and experimental observations the reference [28] generically argued that the model either suffers from strong coupling problem or backreaction problem.
In our previous discussion, we have chosen the effective electromagnetic coupling function as a monomial function of the scale factor, where N k = ln(a/a end ) associated with a particular scale k is identified as the e-folding number during inflation. It is clear from the above expression that once the value of n is chosen to be negative, the gauge kinetic function increases during inflation. Now the behavior of I(τ ) is so chosen that it boils down to unity after inflation. Therefore, during inflation, its magnitude must be less than unity. This is where the origin of the strong coupling problem in the electromagnetic sector lies. Under the field redefinition A µ → √ IA µ , the effective electromagnetic coupling α ef f , defined through the fermion-gauge field interaction L int = eA µ J µ modified as e is the charge of the fermionic field contributing to the electric current. α is the standard electromagnetic coupling. Nonetheless, as one goes early in the inflationary phase, the effective electromagnetic coupling necessarily becomes very large, which turns the theory non-perturbative. Moreover, the perturbative computation of magnetogenesis, discussed in our previous section, will no longer be valid.
Keeping this problem aside, if we still do our magnetogenesis computation as before, due to large effective electromagnetic coupling α ef f , the production of electromagnetic energy density may be sup-pressed compared to the background energy density (3M 2 p H 2 Inf ). Therefore, backreaction problem will not come into play.
On the other hand, if one considers positive values of n ≥ 0, the whole reasoning expressed just now will be reversed or, more precisely, throughout the inflationary period, the gauge kinetic function I(τ ) will be way larger than unity. Based on our previous discussions, the perturbative magnetogenesis analysis appears to be still valid. However, considerable reduction of the effective electromagnetic coupling α ef f ∝ 1/I(τ ), specifically during the early inflationary stage of our interest, significantly enhances the production of electromagnetic energy. In such a scenario the quantum production of electromagnetic energy may take over the background energy density. This is precisely the backreaction problem, which jeopardies the fixed background magnetogenesis analysis instead.
To avoid the backreaction problem, the energy density of the gauge field must be smaller than the total background energy density ρ tot = 3M 2 p H 2 Inf . The parameter which quantifies the amount of background energy density during inflation against the gauge field energy density ρ A is the ratio, Where ζ is identified with another physical quantity associated with the amplitude of the curvature perturbation measured from CMB anisotropy. From the CMB analysis the amplitude of the curvature perturbation is measured to be, ζ = 4.58 × 10 −5 from Planck [1]. Total gauge field energy density is defined as In the above expression, for carrying out k-integral, we consider the limit k from k IR to k U V . Associated with our observable universe, the IR cut-off k IR corresponds to the highest mode that exits the horizon at the beginning of the inflation, and the UV cut-off k U V , the lowest scale set at the end of the inflation.
Afterwards scaling k during inflation as k ≈ aH end , one can find Based on the above expression and the inequality relation (44), the upper bound on the total EM energy density associated with a particular value of coupling parameter n max follow the following Moreover, considering integer values of n, at super horizon limit, the preceding expression turns out From the constraint Eqs. (47) and (48), we can set a limit on the coupling parameter as n max . Thus considering the coupling parameter within the aforementioned range, we are able to take care of both strong coupling and backreaction problems.
For example the scale-invariant electric power spectrum which corresponds to the power n = 2, the total gauge field energy density during inflation roughly of the order of Here, the ratio between the gauge field and the total energy density is always less than the amplitude of the curvature perturbation (ρ A /ρ tot ∼ 10 −10 ) for all allowed values of the spectral index. In addition to that, the model with power n = 3, which produce a scale-invariant magnetic power spectrum, the ratio ρ A /ρ tot turns out as which is much smaller than the amplitude of the curvature perturbation. So for both cases, the scaleinvariant electric and magnetic field power spectrum, the back reaction as well as strong coupling problem can be avoided.

B. Reheating dynamics: Connecting Reheating and Primordial magnetic field via CMB
By now it becomes clear that the primary importance of the reheating phase is to enhance the strength of the large scale magnetic field to the required order. We understood the fact that Faraday's law of electromagnetic induction plays a crucial role in this regard. Therefore, to obtain the correct order of the current magnetic field, understanding the reheating dynamics as well as the evolution of the magnetic field during this period will be of utmost importance. In order to do that, we will consider two possible reheating models and compare the result.
Case-I: For this we follow the effective one fluid description of reheating dynamics proposed in [43], where inflaton energy is assumed to converted into radiation instantaneously at the end of reheating. The dynamics is parametrized by an effective equation of state ω ef f , reheating temperature T re and duration N re (e-folding number during reheating era). In this reheating model, following the approximation as mentioned earlier, one can easily derive the expression for T re and N re in terms of some inflationary parameters as [46] T re = 43 11g s,re where the present CMB temperature T 0 = 2.725 K, the pivot scale k /a 0 = 0.05 M −1 pc . a 0 is the present cosmological scale factor. Here for simplicity we have taken both the values of the degrees of freedom for entropy at reheating (g s,re ), and the effective number of relativistic species upon thermalization (g re ) is same g s,re = g re ≈ 100. From the above expressions (51) and (52), we can clearly see that the inflationary parameters put constraints on the reheating parameters T re and N re . Considering simple canonical inflation potential V (φ), the inflation model-dependent input parameters, the inflationary e-folding number (N k ), and the inflationary Hubble constant (H k ) for a particular CMB scale k are known to be written as here the inflaton field φ end defined at the end of the inflation. Furthermore, the scalar spectral index n k s , and tensor to scalar ratio r k can be related with the above inflationary parameters (H k , N k ) through the following equations Here the slow-roll parameters expressed as The above expressions manage to write reheating parameters in terms of the scalar spectral index (n k s ) for a given CMB scale k. After identifying all the required parameters, we will be able to connect CMB and reheating through inflation.
Reheating parameters and primordial magnetic field : Now in the context of inflationary magnetogenesis, as we mentioned earlier, the electric field continues to exist after the post inflationary era until the universe becomes perfect conductor. This is the non-zero electric field during reheating whose dynamics will significantly change the dynamics of magnetic field and produce the strong magnetic field today.
For this present reheating model, the phase parameter at the end of the reheating θ re k in equation (36) can be defined as, where H re is the Hubble rate at the end point of reheating. Using evolution of effective density with the normalized scale factor A re = a re /a end = e Nre . For integer values of n, the earlier expression of the phase parameter turns out as Furthermore, utilizing equations (58) and (35), one can obtain the magnetic power spectrum during the reheating epoch as, After reheating, the conductivity of the universe becomes sufficiently large. In consequence of that, the electric field dies out very fast, and the magnetic field redshifts as P B ∝ a −4 till today. Since the comoving magnetic power spectrum is conserved after reheating, the present-day magnetic field obeys the following relation [25] where the ratio between the scale factor a 0 and a re , considering entropy conservation can be expressed as a 0 a re = 11g s,re 43 From the preceding expression, we can see that the magnetic field's present strength explicitly depends on the reheating parameters as well as some inflationary parameters. Therefore, this opens up the window for probing the early stage of the universe, particularly the reheating phase through the current observational amplitude of the magnetic field. Basically, for a specific value of the spectral index, any present value of the magnetic field P B 0 has a direct one to one correspondence with the effective equation of state of reheating ω ef f and reheating temperature T re . Therefore, important conclusion we arrived at that the effective equation of state is no longer a free parameter rather it can be fixed by the present value of P B 0 via CMB.
Case-II: For the previous case we did not take into account explicit decay of the inflaton field and additionally the effective equation of state was assumed to be constant. In this case we consider perturbative reheating scenario [56] constrained by CMB anisotropy. Therefore, as opposed to the previous case, the effective equation of state is time-dependent. However, average inflaton equation of state is taken to be constant. For example inflaton potential V (φ) ∝ φ p gives rise to the value of average equation of state ω φ = (p − 2)/(p + 2) considering virial theorem [55]. Through out our discussion we consider p = 2. Furthermore, for simplicity we assume inflaton decays into radiation only. The Boltzmann equation for the inflaton energy density (ρ φ ) and radiation energy density (ρ r ) are, Where the comoving densities in terms of the dimensionless variable are used, In the above equations to increase the stability of the numerical solution, we use the inflaton mass (m φ ) to define the dimensionless scale factor as A = a/a end = am φ . For solving the Boltzmann equation, the natural initial conditions will be set at the end of inflation as follows, .
For this mechanism the reheating temperature is identified from the radiation temperature T rad at the point of H(t re ) = Γ φ , when maximum inflaton energy density transfer into radiation.
T re = T end rad = 30 To establish one to one correspondence between T re and Γ φ , we combine the equations (65) and (51).
To fixed the values of decay width Γ φ in terms of spectral index (n s ), we use one further condition at the end of the reheating Connecting reheating and primordial magnetic field : In order to connect the reheating and primordial magnetic field through the CMB, it is necessary to understand the cosmological evolution of the electromagnetic field during the post inflationary epoch, especially during reheating, which modifies the present strength of the magnetic power spectrum. As we consider the perturbative decay of the inflaton field during reheating, the phase parameter (36) now explicitly depends on the evolution of the two energy components, ρ φ and ρ R with time Furthermore, for integer values of coupling parameters at the super horizon scale, one can find [25] θ re where the time-dependent density components will be followed from Eq.
In the preceding expression, we can clearly see the appearance of the reheating parameters, which are the function of inflationary observables. Therefore, we definitely will be able to put indirect bounds on the reheating parameters which in turn can constrain the inflation model. As we emphasized before, from the measurement of the CMB anisotropy, our goal of this paper would be to constraints the reheating dynamics through inflationary parameters considering the present strength of the magnetic field P 1/2 B 0 .

III. MAGNETIC POWER SPECTRUM IN THE PRESENT UNIVERSE IN TERMS OF RE-HEATING PARAMETERS: AN ANALYTIC STUDY
Before employing the numerical analysis, in this section we present approximate calculation for the present value of the magnetic power spectrum and estimate the reheating parameters (T re , N re ) in terms that. Following our previous work [79], the radiation temperature assumes the following form consider Case-II reheating scenario, where β = π 2 g re /30. Further, utilizing the above expression of the radiation temperature, we can calculate the approximate expression of the reheating temperature. Subsequently, at the point of A re , where the condition H(A re ) = Γ φ is satisfied, one can define the reheating temperature as Here the decay width Γ φ and the normalized scale factor at the end of reheating (A re ) can be approximated as, From the Eq.35 it is obvious that during reheating phase the electromagnetic power spectrum is crucially dependent upon the evolution of the phase θ k which is giving rise to induced magnetic field. Importantly after the reheating phase ends large scale magnetic field freezes inside the plasma.
Therefore, it is the hierarchy between the inflationary and reheating scale which set the current strength of the magnetic field today after the inflation. Hence naturally one can obtain interesting constraints on reheating evolution parameters such as (T re , N re , w ef f ) though the current value of the large scale magnetic field. As the inflaton energy density dominates large part of the duration of the reheating phase, we approximate the solution of the inflaton energy density as This helps us to further obtaining the approximate Hubble parameter as With all the above approximate expressions, one can arrive the following expression of the present magnetic power spectrum Where η = 5H inf 2Γ φ . Interestingly, our analytical expression of the magnetic power spectrum roughly matches with the numerical values.

IV. INFLATION MODELS AND NUMERICAL RESULTS
The main aim of our study is to see how the present observational limit on the magnitude of large scale magnetic field as well as the CMB anisotropy can be used to probe the reheating dynamics and put combined constraints on the reheating and inflationary parameter space. We consider different inflationary models ans study how the present limits on the magnetic field strength constrain the effective reheating equation of state ω ef f in terms of inflationary scalar spectral index n s . Further, the allowed range of the current magnetic field can be shown to impose an upper limit on the values of the spectral index n max s and effective equation of state ω max ef f , which in turn provides bound on the maximum possible reheating temperature T max re . Moreover, to connect reheating parameters such as (T re , ω ef f ) with the present magnetic field power spectrum P 1/2 B 0 , we take in to account two different power law form of the gauge kinetic function with power n = (2, 3). It is well known that inflationary magnetogenesis model with n = 2 produces scale invariant electric power spectrum and n = 3 gives rise to scale invariant magnetic power spectrum. Furthermore, we also analyze the the maximum possible value of n = n max that will be allowed for different inflationary model within 2σ range of n s from Planck [1]. In the following discussion, we consider different inflationary models and discuss the constraints on the reheating parameters.

A. Model independent results
In our present paper we have considered a specific magnetogenesis model with dilatonic gauge kinetic function, I ∝ φ −n . As we have described before there exists two special values of n = (2, 3) for which we have scale invariant electric field and magnetic field configuration respectively. Our analysis shows that irrespective of the inflation models, n = 2 gauge coupling function can not produce enough magnetic field strength within the observable limits 10 −9 G > P  3)). In the conventional study the magnetic energy density is assumed to be diluted adiabatically with the expansion of the universe as 1/a 4 starting from the end of inflation. This framework never gives rise to enough present day magnetic field strength for high reheating temperature, which is believed to be generic prediction of inflation in the effective field theory framework. Therefore, in the inflationary magnetogenesis scenario non-trivial magnetic field dynamics during reheating should be an important physical phenomena that has to be understood quite well. Furthermore, during this phase itself the inflaton decay should happen in such a manner that the electric field survives for Faraday's effect to play the role.
Most important point of our study is the constraints set on the effective reheating equation of state ω ef f . A particular present day value of P B 0 , the CMB anisotropic constraint and after reheating entropy conservation law automatically fixes the reheating equation of state uniquely. Most interestingly, this would place severe restriction on the inflaton potential which we will study in our later paper. As has already been discussed, most of our discussions will be centered around two different forms of the dilatonic gauge kinetic functions.
Scale-invariant electric power spectra : Even though we are considering scale invariant electric field, important to remember that it will survive only until the end of reheating. The generated large scale magnetic field will be constrained by this condition. In this case how the reheating and inflationary parameters and the large scale magnetic field are intertwined each other are clearly depicted in fig.(2). From the first panel of fig.(2), we can clearly predict a unique value of ω ef f associated with a specific choice of the present magnetic field once we fixed scalar spectral index n s . Additionally, for a given ω ef f , the reheating temperature is also determined uniquely. Therefore, taking into account CMB constraints the reheating phase can be uniquely probed by the evolution of the primordial magnetic field. In the Table-I For case-I reheating scenario, irrespective of models under consideration large scale magnetic field constrains the effective equation of state within 0.15 < ω ef f < 0.33, and consequently predicts low value of reheating temperature. For the minimum observational limit of the present-day magnetic field strength (P 1/2 B 0 = 10 −22 G), set the maximum allowed value of the reheating temperature around ∼ 1 TeV, which turns out to be inflationary model-independent. As has already been pointed out for pertrubative reheating (case-II), since the inflaton equation of state is taken to be zero, we found it hard to get the strong magnetic field within the observable limit.
Scale-invariant magnetic power spectra : Here we will discuss another special case n = 3 when magnetic power spectra itself is scale invariant. Unlike n = 2, inflationary magnetogenesis model with scale invariant magnetic field turned out to be viable for a wide range of reheating temperature.
The reheating temperature can be as high as ∼ 10 14 GeV without any backreaction problem which was the main concern of all the previously studied magnetogenesis scenario. Furthermore for case-I reheating scenario the effective reheating equation state now ranges from − 1 3 < ω ef f < 0.3 irrespective of models under consideration. Negative reheating equation of state during reheating generically indicates the unconventional matter field production from inflaton which has not been studied in the literature extensively. Hence for the time being if we exclude this possibility scale invariant magnetic field generation is very tightly constraints. As opposed to the scale invariant electric field model, for the present case the observable value of large scale magnetic field can be successfully generated within the perturbative reheating framework shown in the second panel of the Fig.(3). However, as can be observed, the the model is viable only within a very narrow range of magnetic field within 10 −10 G ∼ 10 −12 G. Similar to the previous case for any specific choice of the current magnetic field strength within the observational limit, there exits maximum allowed values of the spectral index n max s and associated ω max ef f and that naturally leads to the maximum permissible value of the reheating temperature which are provided in the tables II, IV, VI. Through out our discussion, for n = 3 case we consider three distinct values of P 1/2 B 0 = (10 −9 , 10 −12 , 10 −15 ) G. With this general discussion, in the following sections we consider various inflationary models and discuss about their quantitative predictions.
B. Natural inflation [49] In natural inflationary model, the potential is defined by here Λ is the height of the potential, and f is the width, also known as the axion decay width. To fit this model with Planck data, this model needs a super-Planckian value of axion decay constant. This is why we have taken f = 50M p for our numerical analysis purpose. Here the overall scale of the inflation Λ fixes by the CMB normalization.
In order to connect the primordial magnetic field with those observed in the present universe through reheating dynamics, we need to define inflationary e-folding number N k and tensor-to -scaler ratio r k .
In the framework of the natural inflation model, the inflationary parameters N k and r k in terms of n s are defined as In addition with that for the perturbative reheating model, the initial conditions to solve the Boltzmann equations for two density component are set at the end of the inflation to be where Here, the primordial scalar amplitude of the inflationary scalar fluctuation is A δφ ∼ 2.19 × 10 −9 from Planck [1]. As we have the connection relation (Eqs. 60,69) between the reheating parameters and the parameters of magnetogenesis model, in the following, we discuss their implications and various constraints for two distinct scenarios which we have discussed before. C. α−attractor model [50] In this section, we will consider the α−attractor model, which is a class of theoretical models that unifies a large number of the existing inflationary models proposed in [50]. Conformal transformation of a large class of non-canonical inflaton field lagrangian, a canonical exponential potential obtained    in the following form

Scale
The canonical property of this class of model predicts inflationary observable (n s , r), compatible with Planck observation [1,66]. Here the mass scale Λ of the E-model fixed from CMB normalization.
However, this class of models includes the Higgs-Starobinsky model [67,68] for n = 1, α = 1. For numerical study, we have taken Higgs-Starobinsky model. Before we jump into our analysis details, let us first determine the relationship between the inflationary parameters with potential parameters. The inflationary e-folding number, N k and tensor to scalar ratio, r k , can be express interms of n s and model parameter as In addition to that, the initial conditions to solve the Boltzmann equations for different energy components, considering the perturbative reheating model in the context of the present scenario, can be expressed as where Λ = M p 3π 2 rA s 2 Since we want to study all reasonable constraints on the reheating parameter space through inflationary magnetogenesis, we use equations (60), (69) with the details of the potential parameters, as we have done in the previous inflationary model. This section will briefly review the essential outcomes Scale-invariant electric power spectra : In our present analysis, for case-I reheating model, once  1 3 ), and set the upper limit of the reheating temperature associated with the lower limit of the magnetic field strength (P . In addition to that, the perturbative analysis of reheating is efficient to produce the observational strength of the magnetic field of the order (10 −10 ∼ 10 −11 ) GeV.

D. Minimal plateau model
In this section, we will introduce a special class of the inflationary model, the minimal plateau models proposed in [64]. The potential of this type of model is a non-polynomial modification to the simple power-law potential φ n and is given by where m and φ * are two mass scales. The parameter λ and the scale m are fixed from WMAP normalization [65]. Only even values of the index p are taken, as was the case for the chaotic inflation model. The new scale φ * can be shown to fix the scalar spectral index and the scalar-to tensor ratio within the observational limit from Planck [1,66]. For the numerical purpose, we choose φ * = 0.01M p with n = 2.
Before we go to the quantitative discussion to acquire one to one correspondence between the reheating parameters (ω ef f , T re ) with P 1/2 B 0 , let us point out the usual inflationary parameters for this class of minimal models we discussed. The inflationary parameters N k and r k can be written as, Moreover, similar to the other inflation model, the initial conditions to solve the differential equation in the case of the perturbative reheating model are set as where (87) We set Λ = 1 for p = 2.
Details constraints on the reheating parameter space can be read from Figures (2)   We discussed earlier in section-II A that any values of the coupling parameter n in the conventional inflationary magnetogenesis model are not permissible. To overcome either strong coupling or the backreaction problem coupling parameter must lie between 0 ≤ n ≤ n max , where n max can be determined from equation (47). In Figure.4, we show the numerical value of the maximum allowed coupling parameter as a function of the scalar spectral index for various inflationary models. From   Fig.4, we can clearly see that the permissible range of n max for various inflationary models nearby to 5 except for the minimal plateau model where n max is predicted to lie around 6. Among all magnetogenesis models, the inflationary magnetogenesis models are well-motivated to explain the origin of the large scale magnetic field in our universe. Allowing the exchange of the power between two fields as a simple consequence of Maxwell's law of induction, the primordial magnetic field does not decrease as radiation during the decelerating era of the universe, especially during reheating. Moreover, we have given that during inflation, the electric field is much stronger than the magnetic field, so the electric field continues to be the source of the magnetic field and can secure the magnetic field to slow redshift. Therefore the magnetic field power no longer scales as ∝ a −4 during the post inflationary era till the end of the reheating. After the end of the reheating electric field dies out fast due to the large electric conductivity and magnetic power decay as radiation until today.
Considering this effect during the reheating epoch, the present-day amplitude of the magnetic field arising from inflationary magnetogenesis can match with the observational limit of the intergalactic magnetic field on the Mpc scale.
So far, we have discussed mainly understanding the connection between reheating parameters and the parameters of the inflationary magnetogenesis in consideration of the two different reheating scenarios, the conventional reheating model and usual perturbative reheating. We show that the large-scale magnetic field gives non-trivial constrain on both the reheating parameters (ω ef f , T re ) and the inflationary parameter n s . This bound can provide stringent limits on reheating parameters with those already derived from the cosmic microwave background by assuming a high energy scale inflation model. We also derive the possible constraints on the coupling parameter by analyzing the gauge field's gravitational backreaction and the strong coupling problem in the electromagnetic sector. The maximum allowed coupling parameter is nearly 5 for the natural and Higgs-Starobinsky model. However, for the minimum plateau model n max lies around 6.
Interestingly, once we fixed the scalar spectral index, our analysis clearly predicts a unique value of the effective equation of state associated with the specific choice of the present-day magnetic field. Once we fix ω ef f , the reheating temperature can be determined uniquely. The corresponding results for different inflationary models are shown tables I,II and III. As reheating and inflationary parameters are connected, the present-day strength of the magnetic field also provides possible limits on the inflationary scalar spectral index. This extra bound from inflationary magnetogenesis essentially narrows down the possible value of n s within the 1σ range of n s = 0.9649 ± 0.0042 (68% CL, Planck TT,TE,EE+lowE+lensing) from Planck [1]. (shown in tables II, IV, and VI for different inflationary models). Furthermore, irrespective of the inflationary model under consideration for conventional reheating scenario (case-I) with scale-invariant electric power spectra model set the effective equation of state within a narrow range of (0.15 < ω ef f < 1