Measuring inflaton couplings via dark radiation as $\Delta N_{\rm eff}$ in CMB

We study the production of a beyond the Standard Model (BSM) free-streaming relativistic particles which contribute to $N_{eff}$ and investigate how much the predictions for the inflationary analysis change. We consider inflaton decay as the source of this dark radiation (DR) and use the Cosmic Microwave Background (CMB) data from $\textit{Planck}$-2018 to constrain the scenarios and identify the parameter space involving couplings and masses of the inflaton that will be within the reach of next-generation CMB experiments like SPT-3G, CMB-S4, $\text{CMB-Bh$\overline{a}$rat}$, PICO, CMB-HD, etc. We find that if the BSM particle is produced only from the interaction with inflaton along with Standard Model (SM) relativistic particles, then its contribution to $N_{eff}$ is a monotonically increasing function of the branching fraction, $B_X$ of the inflaton to the BSM particle $X$; $\textit{Planck}$ bound on $N_{eff}$ rules out such $B_X \gtrsim 0.09$. Considering two different analyses of $\textit{Planck}$+BICEP data together with other cosmological observations, $N_{eff}$ is treated as a free parameter, which relaxes the constraints on scalar spectral index ($n_s$) and tensor-to-scalar ratio ($r$). The first analysis leads to predictions on the inflationary models like Hilltop inflation being consistent with the data. Second analysis rules out the possibility that BSM particle $X$ producing from the inflaton decay in Coleman-Weinberg Inflation or Starobinsky Inflation scenarios. To this end, we assume that SM Higgs is produced along with the BSM particle. We explore the possibilities that $X$ can be either a scalar or a fermion or a gauge boson and consider possible interactions with inflaton and find out the permissible range on the allowed parameter space Planck and those which will be within the reaches of future CMB observations.


I. INTRODUCTION
For resolving the horizon problem, the flatness problem and lay the seed for structure formation in the late universe cosmic inflation [1][2][3][4][5][6] is the leading paradigm.This period in the very early universe involves an accelerated expansion epoch during which vacuum quantum fluctuations of the gravitational and matter fields were amplified to large-scale cosmological perturbations [7][8][9][10][11][12], that later seeded the anisotropies as observed in Cosmic Microwave Background Radiation (CMBR) and lead to the formation of the Large Scale Structure (LSS) of our Universe.
What we observe in CMBR can be accounted for in a minimal setup, where inflation is driven typically by a single scalar field ϕ with canonical kinetic term, minimally coupled to gravity, and evolving in a flat potential V (ϕ) in the slow-roll regime.Since particle physics beyond the electroweak scale remains elusive and given that inflation can proceed at energy scales as large as 10 16 GeV, even within this class of models, hundreds of inflationary scenarios have been proposed to match with the latest sophisticated measurements of CMB [13][14][15][16].A systematic Bayesian analysis reveals that one-third of them can now be considered as ruled out [17][18][19], while the vast majority of the preferred scenarios are of the plateau type, i.e., they are such that the potential V (ϕ) is a monotonic function that asymptotes a constant value when ϕ goes to infinity.
Cosmic inflation must be succeeded by a period during which the inflaton's energy must be transferred to relativistic SM particles, resulting in the formation of a hot universe consistent with present observations.The extremely adiabatic reheating epoch is important since it produces all matter in the cosmos as well as the relativistic SM fluid that raises the temperature of the universe.During this epoch, the standard picture is that the inflaton field vibrates coherently around the minimum of its potential, producing SM particles as well as viable BSM particles via gravitational interactions or other couplings that maybe available in the theory.This BSM particle may contribute to the dark matter sector or stay relativistic, contributing to dark radiation (DR).If they contribute to DR, they may influence the expansion rate of the universe (and thus the measured value of Hubble parameter), CMB anisotropy [20], and the perturbation for large scale structure-formation in the universe [21].
The predictions of inflationary models, such as n s and r, are solely determined by the parameters of the models and are independent of the presence of DR in the universe.However, the presence of extra relativistic species in the standard cosmology does have an impact on the selection of inflationary models.In particular, single-field slow-roll models, which are known to produce high values of the scalar spectral index in standard cosmology may be more favorable in the presence of extra relativistic species.Upcoming CMB experiments e.g.spt-3g, CMB Stage IV (CMB-S4), CMB-Bharat, PICO, and CMB-HD, all of which are highly sensitive to the additional relativistic degrees of freedom, are expected to offer more precise information on the extra relativistic BSM particles present along with CMB photons.Gaining this information is significant for developing a more complete and accurate theory of the physics of the early universe, including inflationary epoch followed by reheating era, and has important impacts on understanding the underlying physics of the thermal history, Hubble expansion rate, and other cosmological phenomena of the later universe.
In this work, we make the assumption that the DR is created during the reheating epoch as a relativistic non-thermal BSM particle together with relativistic SM particles, and that the inflaton is the only source of this radiation.We then apply the bounds on DR from the CMB to the branching fraction for the production of extra relativistic BSM particle.
Next, we explore the possibilities of the extra DR particle being a fermion, boson, or gauge boson and consider the possible leading-order interactions.Utilizing the present bound from current CMB observations and prospective sensitivity reaches on DR from future CMB experiments, we proceed to identify the parameter space that involves the inflaton mass, as well as the couplings between the inflaton and both the visible sector and the DR particle.
The paper is organized as follows: we begin with a discussion of N eff and ∆N eff in Section II.In Section III, we consider n s − r predictions from four disparate inflationary models and review whether they can be ruled out as viable inflationary models by the bound from Planck 2015+Bicep2 data while N eff is allowed to vary from its standard value.In Section IV, we consider the production of a BSM particle during the reheating era, which adds an extra relativistic degree to CMB.We also explore the connection between branching fraction for the production of that BSM particle and ∆N eff .In Section V, we look at probable interactions between the inflaton and the BSM particle, as well as the permitted parameter space for the couplings related to the bound on N eff from CMB.In Section VI, we summarize the findings.
In this work, we use the natural unit with ℏ = c = k B = 1, such that the reduced Planck mass is M P ≃ 2.4 × 10 18 GeV.Furthermore, we also assume that the signature of the space-time metric is (+, −, −, −).

II. EFFECTIVE NUMBER OF RELATIVISTIC DEGREES OF FREEDOM
Since the temperature of the CMB photons is a well-known quantity, the current total energy density of the relativistic species of the universe, ρ rad,tot , can be expressed in terms of the energy density of SM photon, ρ γ , as [21,22,[31][32][33][34][35] ρ rad,tot = ρ γ 1 + 7 8 4 11 where N eff , also known as effective number of relativistic degrees of freedom [21], parameterizes the contribution from non-photon relativistic particles, such as SM cosmic neutrinos.
There are three left-handed light cosmic neutrinos, and they were in thermal equilibrium with the SM relativistic particles in the hot early universe.When the temperature of the universe drops to 800 keV, neutrinos decouple from the SM photons just before the electronpositron annihilation.If we assume neutrinos decouple instantly, their contribution to N eff is expected to be 3.However, if we consider noninstantaneous decoupling of cosmic neutrinos [36], QED correction [31], and three-neutrino flavor oscillations on the neutrino decoupling phase [37] (see also [38]), then, neutrinos get partially heated during electron-positron decoupling from photon, resulting in slightly higher temperature of neutrinos.This leads to [39] (see also [40]) N eff ,SM = 3.046 . (2) 1 Non-instantaneous decoupling of neutrinos from e − e + pairs during the early universe leads to an increase in the number of equilibrium neutrino species by ∆N ν = 0.035 [41][42][43][44].The effective number of neutrino species receives an additional contribution ∆N ν = 0.011 resulting from the deviation of the electronpositron-photon plasma from an ideal gas state.For further details, see Refs.[43,44].Consequently, within the framework of the SM, the effective neutrino species amounts to N eff ,SM = 3.046.
Any measured value higher than Eq.(2) suggests the possibility of the existence of any relativistic BSM particle 2 .The contribution of the BSM particle in N eff is expressed as [32,[47][48][49] ∆N Current bounds on and prospective future sensitivity reaches on ∆N eff from that will be within the reach of future CMB measurements are mentioned in Table I.In this work, we  [50,51].

III. INFLATION AND DARK RADIATION
In this section, we review how the non-zero value of ∆N eff influences the selection of single-field slow-roll inflationary models.If V (ϕ) is the potential energy of a single real scalar inflaton ϕ, minimally coupled to gravity, then the action with canonical kinetic energy is [61] S ⊃ 2 If the BSM particle is thermal, with a higher value of couplings with electron-positron than with neutrinos, it may result in N eff < N eff ,SM [45,46].
Here, g is the determinant of the spacetime metric tensor g µν , and R is the Ricci scalar.
For slow roll inflationary scenario driven by ϕ, the first two potential-slow-roll parameters are [62] ϵ where prime symbolizes derivative with respect to ϕ.The slow-roll inflationary period is continued till φ2 ≪ V (ϕ) and φ ≪ 3H φ [63][64][65], with over dot implying a derivative with respect to physical time t and H ≡ H(t) being the Hubble parameter, are maintained.
Here, ϕ * is the value of inflaton corresponding to both cosmological scale factor a * and the length-scale of CMB observation, and a end is the cosmological scale factor at the end of inflation.When the value of inflaton is ϕ * , the length scale corresponding to e-fold N CMB leaves the causal horizon during inflation [66].The largest length scale of the CMB that can be observed today formed at∼ 60 e-folds before the end of inflation [67].As N CMB ≳ 60 and N CMB ≳ 40 are required to solve the horizon and the flatness problems, we varied N CMB from 50 to 60 in this work.On the other hand, quantum fluctuations can be generated by the inflaton during the inflationary epoch.The statistical nature of these primordial fluctuations is expressed in terms of the power spectrum.The scalar and tensor power spectrums for 'k'-th Fourier mode are defined as where A s and A t are the amplitudes [68] of the respective power spectrums, k * is the pivot scale corresponding to CMB measurements (also corresponding to ϕ * as previously mentioned), n s and n t are scalar and tensor spectral indexes, respectively.α s and β s are the running and running of running of scalar spectral index.The relation among n s and potential-slow-roll parameters at leading order is On the other hand, tensor-to-scalar ratio is defined as where the last relationship is only applicable in the case of slow-roll inflationary scenario.Now, bounds on the (n s , r) plane at 1 − σ and 2 − σ CL from Planck 2015 [69], and combined Planck 2015+Bicep2-Keck Array2015 data (ΛCDM+r model) [70] are shown in Fig. 1 (shaded contours with green and purple color, with dashed-lines as periphery The curves with a triangle on them correspond to N CMB = 50, while the curves with a square on them correspond to N CMB = 60.The green-shaded regions indicate the bound on (n s , r) plane at 68% and 95% CL from the Planck2015 [69], and the purple-colored regions show bounds from the combined analysis of Planck2015+Bicep2-Keck Array2015 [70] for a standard value of ∆N eff (∆N eff = 0).The regions with brown shading show the shifting of the same bounds when ∆N eff is varied (using Planck2015+Bicep2-Keck Array2015+baryon acoustic oscillation data + direct measured value of present-day Hubble constant, with ΛCDM+r + N eff as parameters lead ∆N eff = 0.254) [73,74].
Predictions from Hilltop and Starobinsky inflation are within 1 − σ CL from Planck2015+Bicep2-Keck Array2015 bound.However, when ∆N eff is used as a free parameter, S-I survives at 2 − σ bound for higher number of N CMB while H-I can satisfy (n s , r) values within proportional to the square of Ricci curvature as a modification to Einstein-gravity.The action of S-I in Jordan frame is given by [1, 80] Here, superscript JF implies that the corresponding parameter is defined in Jordan frame and m SI is the mass scale.This action can be converted to Einstein frame by a conformal transformation of the metric [81]

SI
. And then the action in g µν metric space takes a form similar to Eq. ( 4).Then the potential takes the form where the inflaton ϕ and Λ 4 are defined as [17,82] To satisfy CMB data of scalar fluctuations, m SI /M P ∼ O(10 −5 ).Now, the scalar spectral index and tensor-to-scalar ratio for the potential of Eq. ( 12) are The slow-roll inflationary epoch ends when ϵ V ∼ 1 happens.Furthermore, this S-I model predicts a very small value of r, and thus it is within 1−σ contour of the Planck 2015+Bicep2 CMB bound (see the magenta colored region (line) in Fig. 1 for 50 ≤ N CMB ≤ 60.Since n s and r do not depend on any parameter, this model cannot be further adjusted for the n s − r contour when ∆N eff is treated as a variable.

B. Natural Inflation (N-I)
For slow-rolling of the inflaton, the flatness of the potential needs to be maintained against radiative correction arising from self-interaction of the inflaton or its interaction with other fields.When the axion or the axionic field plays the role of inflaton, it provides the requisite mechanisms to protect the flatness of the potential.It also offers a proper explanation from particle physics, for the small values of the self-coupling and, thus, dilutes the issue of fine-tuning.That's why this inflationary model is called Natural Inflation (abbreviated as N-I).Axion is a pseudo-Nambu-Goldstone boson that arises as a result of the spontaneous breaking of global symmetry followed by additional explicit symmetry breaking [78].The axion potential which arises due to the spontaneous breaking of global shift symmetry or axionic symmetry is given by [78] V where Λ N is the U (1) explicit symmetry-breaking energy scale, which determines the inflation scale, f a is the axion decay constant, and ϕ is the canonically normalized axion field.The first spontaneous U (1) symmetry breaking happens when T ≃ f a (T represents the temperature of the universe).f a also reduces the value of the self-coupling of the ϕ by 1/f a [78].On the other hand, the axionic symmetry protects the flatness of the potential of axionicinflaton [83], at least up to tree level [78].
The potential of Eq. ( 15) has a number of discrete maxima at ϕ = nπf a , with n = 1, 3, At the vicinity of the location of the maximum of the potential, |η V | ≈ 1/2f 2 a ≪ 1, and this sets the limit of f a [83].This inflationary model, like S-I, comes to an end when ϵ V ∼ 1. n s and r can now be calculated as The values of n s and r for 10M P ≲ f a ≲ 1585M P are shown in Fig. 1 as blue colored region.
This inflationary model has been ruled out at 1 − σ CL by Planck 2015+Bicep2.

C. Hilltop Inflation (H-I)
In this inflationary model (abbreviated as H-I) inflaton starts rolling near the maximum of the potential and this automatically makes ϵ V ∼ 0 at the onset of inflation.The potential is given by [84,85] where Λ H and v are parameters, and the ellipsis represents the other higher-order terms that make the potential bounded from below, and may be responsible for creating the minimum.
The maximum of the potential of Eq. ( 17) is at ϕ = 0.This inflationary model also ends with ϵ V ∼ 1. n s and r are given by The values of n s and r for 0.01M P ≲ v ≲ 100M P exist well inside the 2 − σ range and are shown as cyan-colored region in Fig. 1 5 .

D. Coleman-Weinberg Inflation (C-I)
This potential of Coleman-Weinberg Inflation (abbreviated as C-I) is actually the effective potential of quartic self-interacting scalar field up to 1-loop order, and it is given by [86-89] Here f = ⟨ϕ⟩ is the renormalization scale and determined by the beta function of the scalar-quartic-coupling with inflaton.Here, we do not go into detailed models of interaction of ϕ with other fields, and we take A as a free parameter, and fixed it by the normalization to the amplitude of the CMB anisotropies.The model predicts n s and r as In comparison to previous inflationary models, here we are assuming small-filed inflationary

IV. INFLATON DECAY DURING REHEATING
Our discussion in the preceding section is independent of the origin of ∆N eff .In this section, we assume that a BSM particle which contributes to ∆N eff , is produced from the inflaton during the reheating epoch 7 .As soon as the slow-roll phase ends, the equation of state parameter becomes w > −(1/3), and inflaton quickly descends to the minimum of the potential and initiates damped coherent oscillations of inflaton about this minimum and the reheating era begins.The energy density of this oscillating field is assumed to behave as a non-relativistic fluid with no pressure when averaged over a number of coherent oscillations.
Therefore, the averaged equation of state parameter during reheating wr = 0.During this epoch, the energy density of this oscillating inflaton reduces due to Hubble expansion.In addition to that, the energy density of inflaton also decreases owing to interaction with the relativistic SM Higgs and possibly with a relativistic BSM particle, X, which eventually contributes to the effective number of neutrinos [102].During the initial phase of this epoch, Here, Γ ϕ is the effective dissipation rate of inflaton, and the small value of Γ ϕ is due to the small value of the couplings with inflaton.However, the value of H continues to decrease, and soon it becomes H ∼ Γ ϕ .Approximately at this time, the energy density of oscillating inflaton and the energy density of relativistic species produced from the decay of inflaton become equal.At this moment, the temperature of the universe, T RH , is given by [103-105] where g ⋆,RH denotes the effective number of degrees of freedom at the conclusion of the reheating era.We also assume that inflaton decays instantly and completely during the last stage of the reheating era, and the universe thereafter becomes radiation dominated.If a RH is the cosmological scale factor at the end of reheating, then, the number of e-folds during reheating, N RH , is given by [106] where ρ end is the energy density at the end of inflation.To derive this, we have used Eq. ( 22), and the fact that the universe becomes radiation dominated at the end of reheating, and also assumed that equation of state does not vary during reheating epoch.If slow roll inflation ends with ϵ V (ϕ end ) = 1 (i.e. with φ2 | ϕ=ϕ end = V (ϕ end )), then Again, rearranging Eq. ( 23), we get Now, if Γ ϕ→XX and Γ ϕ→hh are the decay width of inflaton to X and SM Higgs particle h, then the branching fraction B X for the production of X particle is defined as Here, Γ ϕ = Γ ϕ→XX +Γ ϕ→hh .Now, the time evolution of the energy density of inflaton, ρ ϕ , energy density of relativistic SM particles, ρ rad , and energy density of relativistic X particle, ρ X , can be computed by solving the following set of differential equations Here, we assume that X is so feebly interacting with H or other SM particles that we can ignore the interaction term.Besides, we also assume that X-particles are not self-interacting.
Therefore, ρ X only decreases due to Hubble-expansion of the universe.Furthermore, the Friedmann equation gives This X particle fails to establish thermal equilibrium with the thermal SM relativistic fluid of the universe, and hence does not share the temperature of photons or neutrinos.This BSM article remains relativistic up to today, and thus contributes to the total relativistic energy density of the present universe Following Sec.II, we claim that N eff takes care of the contribution of X to ρ rad,tot , and in the absence of X particle, N eff = N eff ,SM .The contribution of X in N eff is expressed as [107] ∆N eff = 43 7 10.75 g ⋆,s,RH The subscript H ≪ Γ ϕ implies that the integration has been done till the reheating process is over.In Eq. ( 32), g ⋆,s,RH is the effective degrees of freedom contributing to the entropy density of the universe at the time when ρ ϕ decays completely.We are assuming that g ⋆,s,RH ≈ 106.75, and it remains the same throughout the reheating process.  .I. If the non-thermal X-particle is created only from inflaton decay, then it should be B X ≲ 0.09 (Planck bound).If B X ≳ 0.009, then it can be tested further by future CMB observations.Now, using the bounds on ∆N eff and prospective future reaches of ∆N eff from Table I, ∆N eff as a function of B X from Eq. (32) (and assuming that B X does not depend on cosmological scale factor) is shown in Fig. 2. Because a higher value of B X suggests a higher production of X particles, ∆N eff grows monotonically with B X .This is true when ∆N eff is solely contributed by X particle.On the other hand, the terms on the right-hand-side of Eqs. ( 28) and ( 29) are the production rate of radiation and X, respectively, and only B X regulates the difference between those two production rates.Since ∆N eff ∼ ρ X /ρ rad , the result shown in Fig. 2 is independent of whether Γ ϕ is constant or varies with cosmological scale factor and temperature (e.g.[108]).Additionally, this figure depicts that Planck and Planck +BAO bounds draw an upper limit on the possible values of B X .B X > 0.09 is already eliminated by Planck data.Furthermore, 0.09 ≳ B X ≳ 0.066 can be tested by spt-3g, and B X ≳ 0.021, B X ≳ 0.014, B X ≳ 0.013, and B X ≳ 0.009 may be tested from other future CMB experiments such as CMB-S4/PICO, CMB-Bharat, CORE , and CMB-HD.
We use the relation between ∆N eff and B X from Fig. 2 in Fig. 3 where allowed ranges of and 2016 dataset from Planck High Frequency Instrument (HFI) ("tau") [77].The best fit value of ∆N eff at 68% CL are 0.504 for "PlanckTT + lowTEB", 0.094 for "TTTEEE+tau", 0.464 for "PlanckTTTEEE + lowTEB+BKP".Regarding PlanckTT + lowTEB dataset, all of these four inflationary models are ruled out at more than 2 − σ.Fig. 3 also shows that four inflationary models will be within 2 − σ contour of 'TTTEEE+tau' data if B X ≳ 0.033 (for N-I), B X ≳ 0.061 (for H-I), B X ≳ 0.097 (for C-I), B X ≳ 0.11 (for S-I).Similarly, to be within 2 − σ contour of 'TTTEEE+lowTEB+BKP' it is required that B X ≳ 0.046 (for N-I), B X ≳ 0.07 (for H-I), B X ≳ 0.13 (for C-I), B X ≳ 0.15 (for S-I).By using this now along with the Fig. 2, it is possible to determine whether any inflationary model is compatible with the assumption that inflaton is the only source of X and this X contributes solely and entirely to N eff .For instance, S-I and C-I inflation are incompatible with the aforementioned assumption regarding the 'TTTEEE+tau' and 'TTTEEE+lowTEB+BKP' dataset.If X originates from a different source, these inflationary models can still remain inside 1 − σ contour of 'TTTEEE+tau' data.In contrary, H-I is compatible with the assumption that inflaton is the sole source of X, at least, up to 2 − σ CL interval.It is also worth noting that, the best fit value of n s varies in presence of N eff , just like the best-fit value of r.As a result, Fig. 3 cannot be utilized to make the final decision to validate any inflationary models.In Fig. 3, the N-I model, for example, predicts r value within 2 − σ bounds, but this model is already ruled out in Fig. 1 due to the predicted small value of n s .

V. INFLATON DECAYING TO DARK RADIATION
Following the preceding section, we postulate that during reheating, the inflaton decays to SM Higgs doublet H and BSM particle X.The Lagrangian density describing such a process can be expressed as [103,106,110,111] where the first term on the right side of Eq. ( 33) encodes the decay of ϕ to the SM Higgs particle h, and hence this term is accountable for the generation of thermal relativistic SM plasma of the universe.λ 12,H is dimensionless in this case, and σ m , the mass scale, is taken to be equal to m ϕ .Subsequently, the decay width to SM Higgs particle h is In Secs.III A to III D, we have discussed four inflationary scenarios.The minimum of the potential of N-I is located at f π, whereas for S-I, it is at ϕ = 0, and for C-I, it is at ϕ = f .
Then, the masses of inflaton for the aforementioned three inflationary models are Since, Λ, Λ N , f a , A, and f are determined from the best-fit value of n s , r, A s obtained from CMB data and the number of e-folds N CMB , m ϕ for the aforementioned three inflationary models are determined by the data from CMB observations.However, it should be noted that N-I and C-I are already in discordance with n s − r contour from BICEP data, even at 2 − σ CL (see Figs. 1 and 7).Furthermore, the form of the potential of Hilltop model, as described in Sec.III C, is expected to be bounded from below by other terms without affecting the inflationary predictions.Therefore, it becomes necessary to consider a specific theory to define the mass of the inflaton in this scenario (See Ref. [88], for further details).
To keep the discussion of this section more general, we consider a broader range of inflationary scenarios beyond the four inflationary scenarios discussed above.For example, if we consider quartic inflationary potential, we need to add a bare mass term to study reheating (for example, see.Ref. [115]).This is why, instead of fixing a specific value, we explore variations in the m ϕ during our analysis.
Furthermore, L ϕ→X in Eq. ( 33) is the interaction term of ϕ with X.Let us also assume that λ ϕX is the coupling of inflaton-BSM particle interaction.Now, to make the discussion as generic as feasible, we suppose that X can be a light fermion χ, a BSM scalar φ, or a U (1) gauge boson γ.Therefore, possible interaction Lagrangian with ϕ includes [106, 110] Here y χ , λ 12,φ , λ 13,φ , and g ϕγ are dimensionless couplings with σ ′ m , Λ m as mass scales.F µν is the field strength tensor of γ, and F µν is its dual.Additionally, L scattering in Eq. (33) represents the scattering terms involving inflaton with both SM and BSM particles, and can be written as follows where λ 22 is dimensionless couplings.We assume that L ϕϕ→XX -channel is not effective in contribution to N eff .For instance, for ϕϕ → φφ channel, we can write [110] L ϕϕ→XX = + λ 22,φ 4 ϕϕ φφ , (ϕϕ → φφ ; λ ϕX ≡ λ 22,φ ) .
In Appendix C, we show that the contribution of φ producing via Eq.( 41) in ∆N eff is negligible compared to φ produced via ϕ → φφ decay channel, due to the reaction rate in the former case being dependent on the instantaneous value of inflaton energy density.
In Appendix D, we explore the range of these couplings to investigate whether DR can achieve thermal equilibrium with SM Higgs via inflaton exchange processes.Now, the reaction rates for the interactions from Eqs. ( 36), ( 37) and ( 39), in leading order, and related branching fractions are as follows: and ϕ → γγ) .(45) In this work, we assume σ ′ m = ϵ 1 m ϕ and Λ m = ϵ 2 m ϕ where ϵ 1 , ϵ 2 are dimensionless and ⊂ R >0 .Consequently, B χ , B φ (from Eqs. ( 43) and ( 44)), and B γ are independent of m ϕ .
Then, in Fig. 4, we explore the allowable area on the 2-dimensional parameter-space of (λ ϕX , λ 12,H ) for the interactions from Eqs. ( 36)- (39), regarding the present bounds on ∆N eff and prospective future reaches of ∆N eff that will be within the scope of future exploration by upcoming CMB observations listed in Table I.Here, we set all the mass scales to σ ′ m = Λ m = m ϕ .Furthermore, in order to prevent nonperturbative particle production from becoming significant during reheating, we maintain the values of the couplings below O(10 −4 ) [110,116].We observe that for a given value of ∆N eff , a greater value of λ 12,H implies a higher value of the coupling of the inflaton with the BSM particle.This is to be expected because a higher value of λ 12,H implies more generation of SM relativistic particles.To maintain N eff constant, a greater value of the coupling of the dark sector to inflaton is required.The red solid line stands for ∆N eff = 0.28 implying that Planck +BAO rules out the region above this line.The dashed lines are prospective future reaches that will be within the range of exploration from upcoming CMB experiments with higher sensitivity.We do not present the region plot separately for ϕ → φφ since B φ for this process on (λ 12,φ , λ 12,H ) plane has similar form of B χ on (y χ , λ 12,H ) plane.However, following the discussion of Appendix D, we should consider a lower range of λ 12,φ to prevent the scalar DR particles from reaching thermal equilibrium with SM Higgs.To remain within Planck bound, y χ /λ 12,H ≲ 0.31.Contrarily, for ϕ → φφφ, λ 13,φ /λ 12,H ≲ 19.37 and for ϕ → γγ, g ϕγ /λ 12,H ≲ 0.23.The presence of 1/2 and 384π 2 in the denominator of the branching fraction in Eqs. ( 44) and ( 45) results in these differing upper bounds on the coupling ratios.
In Fig. 5, we consider current bound on ∆N eff -bounds from Planck and only one prospective future reach of ∆N eff that could be observed by CMB-HD, to demonstrate the shift in the allowable region on the respective (λ ϕX , λ 12,H ) space for the choice of respective mass scales either greater or smaller than m ϕ for ϕ → φφ and ϕ → γγ.Similar to Fig. 4, here also the area above the lines are excluded or to be identified by present or future CMB observations.If we choose σ ′ m > m ϕ and σ ′ m < m ϕ , the allowed area decreases and increases for ϕ → φφ and it is displayed on the left panel of Fig. 5. Contrarily, the right panel of this figure exhibits that increment or reduction of the permissible area for the choice of Λ m less or larger than m ϕ .This conclusion is in congruence with the expression of B φ and B γ from Eqs. ( 43) and (45).4: Colored lines delimit regions corresponding to the value of ∆N eff ≤ current bounds and prospective future reaches that will be within the scope of upcoming CMB experiments, mentioned in Table .I. The solid line is for Planck+BAO bound, and the region above this line is already ruled out.The dashed-lines are for prospective future reaches that will be within the scope of exploration from upcoming CMB observations and the parameter space above those dashed-lines could potentially be measured by upcoming CMB observations.Different colors have been used to represent different bounds from different CMB observations.Top-left panel displays such regions for ϕ → χχ on (y χ , λ 12,H ) plane.Similar regions can be obtained for (ϕ → φφ) on (λ 12,φ , λ 12,H ) plane for σ ′ m = m ϕ (see the similarity between the branching fraction of Eq. (42) and Eq. ( 43)).Top-right panel and bottom panel illustrate such regions for ϕ → γγ on (g ϕγ , λ 12,H ) plane and for ϕ → φφφ on (λ 13,φ , λ 12,H ) plane, respectively.
Lines with fixed values of the coupling of inflaton-BSM particle interaction (inclined lines) on (∆N eff , λ 12,H ) plane are shown in Fig. 6.This figure shows that for a certain value of λ ϕX , ∆N eff grows as the value of λ 12,H decreases.A lower value of λ 12,H suggests less production of SM relativistic particle, and thus the ration in Eq. ( 32) increases.correspond to the bound on (n s , r) plane when N eff used as a free parameter in Fig. 1.

CMB-Bharat
Neff   TT,TE,EE+lowE+lensing+BAO [52] ).We assumed that the decay of inflaton during reheating epoch was the sole source of the production of an additional relativistic, free-streaming, non-self-interacting BSM particle contributing solely to N eff in the form of DR and used the CMB data to constrain the branching fraction for its production.This gives us novel constraints (from Planck ) on the parameter space involving couplings and masses of the inflaton.Moreover we explore the detectable prospects that will be within the reach of next-generation CMB experiments.We summarize our main findings below:

Inflation
• Fig. 1 depicts that the best-fit values of n s and r, obtained from the numerical simulation of Planck 2015+Bicep2 CMB data, alter depending on whether N eff is fixed to its standard value or it is regarded as a variable, which thus also affects the selection of preferred inflationary models.While predictions regarding the values of n s and r from three inflationary models -N-I, H-I, S-I are within 2σ contour from Planck 2015-Bicep2 data with ∆N eff = 0, as depicted in Fig. 1, only predictions from H-I and S-I are within 2 − σ bound when ∆N eff is allowed to vary to 0.254.
• We studied the simplest scenario in which a relativistic non-thermal BSM particle, X, acts as the only source contributing to ∆N eff .If X, together with SM relativistic particles, is produced from the decay of inflaton ϕ, with branching fraction for the production of X be given by B X , as defined in Eq. ( 26), then the contribution of X to ∆N eff , as defined in Eq. ( 32), is solely determined by B X and independent of Γ ϕ .
Additionally, as shown in Fig. 2, ∆N eff is a monotonically increasing linear function of B X .This is because a greater value of B X implies a larger generation of X particles.
Furthermore, Planck +BAO bound on N eff has already eliminated the possibility of B X > 0.09, and B X within the range 0.09 ≳ B X ≳ 0.009 can be testified by future CMB observations (e.g.CMB-HD) with better sensitivity for N eff .
• Using the combination of (r, B X ) best-fit contour (transformed from (r, ∆N eff ) bestfit contour from CMB data), the predicted value of r from inflationary models, and CMB bounds on N eff , we showed in Fig. 3 that we can conclude whether X can be completely produced from the decay of inflaton or whether that inflationary model remains a preferred one if X is entirely produced from inflaton decay.When inflaton acts the source of X, then Fig. 3 illustrates that four inflationary models will be within 2σ contour of 'TTTEEE+tau' data if B X ≳ 0.033 (for N-I), B X ≳ 0.061 (for H-I), B X ≳ 0.097 (for C-I), 0.11 (for S-I).Similarly, to be within 2σ contour of 'TTTEEE+lowTEB+BKP' it is required that B X ≳ 0.046 (for N-I), B X ≳ 0.07 (for H-I), B X ≳ 0.13 (for C-I), B X ≳ 0.15 (for S-I).This conclusion, along with the Fig. 2, indicates that the assumption that inflaton as the sole source of X which is the only particle that is contributing completely to ∆N eff , is not compatible for S-I and C-I (regarding 'TTTEEE+tau' data).However, for large field C-I, it is possible for the inflaton to be the sole source of X which is the only particle that is contributing In future, we may be able to extend our analysis to scenarios in which the DR from inflaton decay may alleviate the H 0 tension caused by the presence of extra DR, and how this may have profound implications for inflation model selection, but this is beyond the scope of the current manuscript and will be pursued in a subsequent publication.Array2018 (yellow shaded region) and future CMB observation (SO) with black shaded region.

Appendix B
As mentioned previously, the potential for H-I (Eq.( 17)) is not bounded from below.
Introducing other term to the potential to stabilize the potential of Eq. ( 17) affects the n s , r predictions of H-I except for the scenario where v < M P , where additional term ∝ ϕ/v can be introduced in the potential .Nevertheless, for such scenario, predicted value of n s even for N CMB = 60 is less (n s = 0.95) compared to best-fit 1 − σ contour from CMB data [88,118,120].To resolve these issues, the regularized form of the H-I (although this form of hilltop potential lacks significant theoretical motivation in contrast to the form of Eq. ( 17)) has been suggested for v ≳ M P as [88] V and C-I inflationary model (Eq.( 19)) but for f ≳ M P together with 1 − σ and 2 − σ contour from Planck2015, and Planck2015+Bicep2-Keck Array2015 and bound from analysis where ∆N eff is treated as a variable from [73,74], as described in Fig. 1.Right panel: olive-green shaded region shows the allowed region of r for regularized Hilltop/ C-I (with f ≳ M P ) inflationary models.Additionally, the best fit contour on (r, B X ) plane in the background is also depicted from Ref. [109], as mentioned in Fig. 3.
M H is the mass (≈ 125 GeV) of SM Higgs particles.Therefore, thermal Higgs particles are relativistic at that time with energy ≈ T .Furthermore, DR particles are expected to possess considerably low masses to contribute to ∆N eff .We can also assume that DR particles produced from the decay of inflaton, are also relativistic during the epoch we are interested.
Since, DR particles are non-thermal, their temperature need not necessarily align with that of SM Higgs i.e.T .Nevertheless, we can approximate that energy of DR particles ∼ O(T ).
Therefore, |v rel | ∼ 1 and n DR ∼ T 3 [119].For 2-to-2 scattering process, the total cross-section in center-of-mass frame where ).From Fig. 10 we conclude that even for T RH ∼ 10 7 GeV, along with (λ 12,H ≲ 10 −9 , λ 12,,φ ≲ 10 −9 ) thermal equilibrium between scalar DR and SM Higgs particles (via inflaton exchange process) is not feasible.This justifies the range of (λ 12,H , λ 12,φ ) we have considered (for instance, see Fig. 5).Reheating scenarios where T RH ≳ O(10 7 ) GeV may lead to an excessive production of gravitino [123].Additionally, several proposed scenarios suggest the plausibility of T RH being ∼ 10 4 GeV or even lower (for example, see Refs.[124,125]).For higher values of T RH , we need to consider smaller values of (λ 12,H , λ 12,φ ) to ensure that scalar DR particles cannot reach thermal equilibrium with SM Higgs.From Fig. 11, we find that achieving thermal equilibrium between DR and SM Higgs particles is not possible for T RH ∼ 10 7 GeV, considering values of couplings (λ 12,H ≲ 10 −6 , y χ ≲ 10 −6 ), and m ϕ ∼ 10 9 GeV.However, higher values of T RH necessitate either smaller values of (λ 12,H , y χ ), or a larger value m ϕ , or both, to ensure that fermionic DR particles cannot reach in thermal equilibrium with SM Higgs via inflaton exchange process.
When DR particles are U (1) gauge boson, and produced for the interaction in Eq. ( 39), the square of Feynman amplitude of 2-to-2 scattering process between DR and SM Higgs particles via inflaton mediation, is given by where f γ is ∼ (momentum of DR particle) 4 .This can be explained as follows: If A µ is the vector potential of γ, and we write it in terms of annihilation (a λp (k)) and creation operator (a † λp (k)), then A µ contains a term a † λp (k) ϵ µ * λp e ikx , where the bold symbol denotes 3-vectors, ϵ µ λp is the polarization vector, and λ p in the subscript is polarization index.Hence, ∂ µ A ν contains a term k µ a † λp (k) ϵ ν * λp e ikx .When we define F µν F µν and then, calculate |M| 2 we get These two conditions lead to ϵ V (ϕ), |η V (ϕ)| ≪ 1.By the time either of these two parameters becomes ∼ 1 at ϕ = ϕ end , slow roll epoch ends.The duration of the inflationary epoch is parameterized by the number of e-folds, N CMB , as, scenario with ϕ * ≪ f, ϕ end ≪ f and it ends with |η V | ∼ 1. n s − r predictions for C-I for different values of f are shown in Fig. 1 as red-colored region 6 .This model is already ruled even out at 2 − σ level by Planck 2015 data.Along with the Planck 2015 and Planck 2015+Bicep2-Keck Array2015 combined bounds and theoretical predictions for n s −r for the four above-mentioned inflationary models, Fig. 1 also displays n s − r contour (brown-colored region on (n s , r) plane) from Ref. [73] at 1 − σ and 2 − σ CL where ∆N eff is regarded as an independent variable.To draw this contour, the dataset used are Planck 2015 + Bicep2 + Keck Array B-mode CMB data + baryon acoustic oscillation (BAO) data + direct measured present-day-value of Hubble parameter H 0 with ΛCDM+r + N eff model.They found the best-fit parameter value ∆N eff = 0.254 and n s = 0.9787.Fig. 1 also exhibits that H-I can satisfy the (n s , r) values, even in 1 − σ range for N CMB ∼ 60.Predictions from S-I, on the contrary, can fit in 2 − σ region for larger values of N CMB while remaining outside of 2 − σ boundary for N CMB ∼ 50. 6For predictions regarding f ≳ M P C-I inflationary model, see Appendix B

FIG. 2 :
FIG.2: ∆N eff against B X following Eq.(32).The parallel horizontal lines indicate present bounds on ∆N eff and prospective future reaches that will be within the scope of the sensitivity of the upcoming CMB observations mentioned in Table.I.If the non-thermal X-particle is created for the four inflationary models are shown as colored horizontal-stripes on (r, B X /∆N eff ) plane.On the left panel, the green and yellow colored region exhibits allowed range for r for N-I and H-I models.The cyan and brown colored region on the right panel of Fig.3illustrates the allowed range for r for the other two inflationary models -C-I and S-I, respectively.The 68% and 95% CL contours depicted on the background on this (r, B X /∆N eff ) plane are taken from Ref.[109].These 2-dimensional contours are from 8-parameter analysis, including r and N eff , of Planck and Bicep data, assuming ΛCDM and flat universe.The datasets used here are the combined Planck 2015 temperature power spectrum (2 < ℓ < 2500, ℓ is the multipole number, and indicates an angular scale on the sky of roughly π/ℓ[44]) with polarization power spectra for (2 < ℓ < 29) ("PlanckTT + lowTEB")[75], high multipoles Planck polarization data with CMB polarization B modes constraints provided by the 2014 common analysis of Planck 2015, Bicep2 and Keck Array ("PlanckTTTEEE + lowTEB+BKP")[76], light green color indicates the allowed region of r predicted by N-I model, while the yellow-shaded region corresponds to the prediction from H-I inflationary model.Right panel: same bound on (r, B X ) plane as the left panel, but with predictions of r from C-I (cyan-colored region) and from S-I (brown-colored region).All of these models have been ruled out by 'PlanckTT + lowTEB' data on (r, B X /∆N eff ) plane.Although, N-I and H-I remains within 1 − σ limit of both 'TTTEEE+tau' and 'PlanckTTTEEE + lowTEB+BKP' data, C-I and S-I are within 1 − σ contour of only 'PlanckTTTEEE + lowTEB+BKP' data.

FIG. 5 :
FIG. 5: Illustration of the alteration of the permissible area on (λ ϕX , λ 12,H ) plane when σ ′ m , Λ m ̸ = m ϕ .Solid lines represent σ ′ m = Λ m = m ϕ whereas dashed and dashed-dot lines indicate σ ′ m = Λ m = 10 m ϕ and σ ′ m = Λ m = 0.1 m ϕ , respectively.Left-panel is for ϕ → φφ and right panel is for ϕ → γγ.Red color lines are corresponded to ∆N eff = 0.28 (current bound from Planck+BAO) and green colored lines belong to ∆N eff = 0.027 (prospective future reach of ∆N eff of upcoming observation -CMB-HD).The area above the lines are excluded (within the future reach) by Planck (CMB-HD).

Fig. 6
also gives a comparative view how the range of λ ϕX varies for a given λ 12,H (10 −5 ≥ λ 12,H ≥ 10 −10 ) and for the same range of ∆N eff (0.29 > ∆N eff ≥ 0.027) for different possible interactions of X with ϕ.For example, the required ranges for λ ϕX for different interaction channels are -10 −11 ≲ y χ ≲ 10 −6 , 10 −9 ≲ λ 13,φ ≲ 10 −4 .Additionally, this figure illustrates the allowed range of the value of the coupling λ 12,H regarding the Planck bound on ∆N eff for a given value of λ ϕX .Moreover, for a given value of λ ϕX , this figure shows the value of λ 12,H that are in compliance with Planck bound on ∆N eff .For instance, the value λ 12,H should be < 3.26 × 10 −9 for y χ = 10 −9 .Furthermore, the dashed-magenta colored line correspond to ∆N eff = 0.254, the best fit value predicted by Ref.[73].The point where this line intersects with the inclined lines in that figure, gives the values of the coupling-pairs (λ 12,H , λ ϕX )

FIG. 6 :
FIG. 6: Inclined lines correspond to fixed values of the inflaton-BSM particle couplings on (∆N eff , λ 12,H ) plane.For this figure, we set all mass scales ∼ m ϕ .The CMB experimental reaches on ∆N eff from Table I are shown by discontinuous-horizontal lines.The dashed line in magenta color indicates ∆N eff = 0.254, the value predicted by Ref. [73].Top-left panel, top-right panel and bottom panel are for ϕ → χχ, ϕ → γγ, and for ϕ → φφφ, respectively.

FIG. 10 :
FIG. 10: Presentation of representative sets of values for the couplings (λ 12,H , λ 12,φ ) mentioned in Eqs.(34) and (37), such that scalar DR particles cannot achieve thermal equilibrium with the SM Higgs via inflaton exchange diagram at the tree level.The solid lines correspond to H from Eq. (D5) for two different values of T RH , while the dashed lines correspond to Γ SC from Eq. (D4), for different values of (λ 12,H , λ 12,φ ).

FIG. 11 :
FIG. 11: Presentation of representative sets of values for the couplings (λ 12,H , y χ ) mentioned in Eqs.(34) and (36), such that fermionic DR particles cannot achieve thermal equilibrium with the SM Higgs via inflaton exchange diagram at the tree level.The solid lines correspond to H from Eq. (D5) for two different values of T RH , while the dashed lines correspond to Γ SC from Eq. (D7), for different values of (λ 12,H , y χ ) and m ϕ .

FIG. 12 :∼ 1 m 4 ϕ(λ 12 ,
FIG. 12: Presentation of representative sets of values for the couplings (λ 12,H , g ϕγ ) mentioned in Eqs.(34) and (39), such that DR particles, which are U (1) gauge bosons, cannot achieve thermal equilibrium with the SM Higgs via inflaton exchange diagram at the tree level.The solid lines correspond to H from Eq. (D5) for two different values of T RH , while the dashed lines correspond to Γ SC from Eq. (D9), for different values of (λ 12,H , y χ ) and m ϕ .

TABLE I :
Bounds on N eff (or ∆N eff ) from present CMB observations and prospective future reaches of N eff (or ∆N eff ) that upcoming CMB experiments may be able to observe
Table.III and it shows that T RH is well above Big Bang Nucleosynthesis (BBN) temperature (∼ 1 MeV) for our chosen range of coupling values.

TABLE III :
Estimation of T RH and N RH for different interaction with ϕ and for four inflationary models with benchmark value shown in TableII.For estimation of N RH , we use m ϕ = 10 15 GeV.(∆N eff = 0.06 correspond to future reach from PICO/CMB-S4).
(E 1 + E 2 ) is the sum of the energies of the incoming particles, |p f | is the magnitude of the 3-momentum for either of the outgoing particles, and |p i | is the magnitude of 3momentum for either of the incoming particles, M stands for the Feynman amplitude with the over-line indicating an average taken over unmeasured spins or polarization states of the involved particles, and the integration is performed over solid angle Ω.Following the aforementioned discussion, we assume E 1 , E 2 ∼ T and |p f | ∼ |p i |.For the scenario, when DR particles are scalar, and interact with SM Higgs particles with inflaton as the mediator, Hubble parameter can be defined as (such that at T = T RH , ρ ϕ = SM Higgs via inflaton exchange process.The values of (λ 12,H , λ 12,φ ) for which Γ SC < H is maintained, is shown in Fig.10.The solid lines in Fig.10represent H calculated from Eq. (D5) for two different values of T RH : T RH ∼ 10 4 GeV and T RH ∼ 10 7 GeV, and the dashed lines correspond to Γ SC for different values of (λ 12,H , λ 12,φ