Connecting Cosmic Inflation to Particle Physics with LiteBIRD, CMB-S4, EUCLID and SKA

We show that next generation Cosmic Microwave Background experiments will be capable of the first ever measurement of the inflaton coupling to other particles, opening a new window to probe the connection between cosmic inflation and particle physics.This sensitivity is based on the impact that the reheating phase after cosmic inflation has on the redshifting of cosmic perturbations. For our analysis we introduce a simple analytic method to estimate the sensitivity of future CMB observations to the reheating temperature and the inflaton coupling. Applying our method to LiteBIRD and CMB-S4 we find that, within a given model of inflation, these missions have the potential to impose both an upper and a lower bound on the inflaton coupling. Further improvement can be achieved if CMB data is combined with optical and 21cm surveys. Our results demonstrate the potential of future observations to constrain microphysical parameters that can provide an important clue to understand how a given model of inflation may be embedded in a more fundamental theory of nature.


Introduction:
The current concordance model of cosmology, known as ΛCDM model, can explain almost all properties of the observable universe at an astonishing accuracy with only a handful of free parameters [1,2]. 1  Leaving aside the composition of the Dark Matter (DM), the model is firmly based on the Standard Model (SM) of particle physics and the theory of General Relativity (GR), implying that the most fundamental laws of nature that we know from earth [4] hold in the most distant regions of the observable universe. 2 However, to date it is unknown what mechanism set the initial conditions for the hot big bang, including the initial overall geometry of the observable Universe and the temperature T re of the primordial plasma at the onset of the radiation dominated epoch. 3The former -in particular overall homogeneity, isotropy, and spacial flatness reflected in the Cosmic Microwave Background (CMB)is amongst the most compelling mysteries of modern cosmology.Cosmic inflation [7][8][9] offers an elegant solution for these problems and can in addition explain the observed correlations amongst the small perturbations in the CMB.However, very little is known about the mechanism that may have driven the exponential growth of the scale factor a. A wide range of theoretical models of inflation exist (see e.g.[10] for a partial list), but the observational evidence is not conclusive enough to clearly single out one of them.Moreover, even less is known about the embedding of cosmic inflation into a more fundamental theory of nature and its connection to theories of particle physics beyond the SM.In the next decade the observational situation will change drastically.Upgrades at the South Pole Observatory [11] and the Simmons Observatory [12] aim at pushing the uncertainty in the scalar-to-tensor ratio r down to δr ∼ 3 × 10 −3 .In the 2030s JAXA's LiteBIRD satellite [13] and the ground-based CBM-S4 program [14] are expected to further reduce this to δr < 10 −3 for r = 0.

arXiv:2208.07609v3 [hep-ph] 20 Jul 2024
is inherently sensitive to the inflaton couplings to other fields, i.e., microphysical parameters that connect inflation to particle physics [62][63][64], as these interactions mediated the energy transfer.However, past studies have almost exclusively focused on T re , ignoring the possibility to constrain microphysical (particle physics) parameters.The fundamental limitations on the possibility of constraining the microphysical coupling constant g associated with the interaction through which the universe was primarily reheated were laid out in [64], where it was estimated that such a measurement may be within reach of nextgeneration instruments.However, neither there nor in any of the few phenomenological works addressing the relation to microphysical parameters [47,50,55] a systematic study was performed to quantify the feasibility of such a measurement with realistic instrumental sensitivities.In the present work we introduce a simple analytic method to quantify the sensitivity of observations to g for given instrumental sensitivities to the amplitude of the scalar perturbations in the CMB A s , the spectral index n s and the tensor-to-scalar ratio r. 6 We apply this method to show for the first time that upcoming observations will be capable of the first ever measurement of both g and T re , where we define a measurement as the ability to impose both an upper and a lower bound on the respective quantity.In this Letter we present the main results of our research, a more detailed analysis is presented in [65].Imprint of reheating in the CMB: The primary goal of this work is to quantify constraints on g and T re from current and future CMB data.We consider inflationary models that can effectively be described by a single field Φ and assume that the effective single field description holds throughout both, inflation and the reheating epoch. 7 Defining φ = ⟨Φ⟩ as the quantum statistical expectation value of Φ = φ+ϕ with fluctuations ϕ and φ end as its value at the end of inflation, the energy density at the end of inflation ρ end ≃ 4  3 V(φ end ) ≡ 4 3 V end and the spectrum of primordial perturbations are fixed by the effective potential V(φ). 8Assuming a standard cosmic history after reheating (and leaving aside foreground effects), the observable spectrum of CMB perturbations can be predicted from V(φ) once the expansion history during reheating is known.The latter requires knowledge of the duration of the reheating epoch N re in terms of expansion e-folds N (defined as the logarithm of the scale factor growth) and the average equation of state during reheating wre = 1 Nre Nre 0 w(N )dN .Since the total energy density of 6 Throughout we fix As = 10 −10 e 3.043 [1] and neglect the uncertainty σ As ; we checked in [65] that this does not affect the conclusions. 7See [64,66,67] and references therein for a discussion of effects that limit the validity of this assumption during reheating. 8Throughout we work at leading order in the slow roll parameters ϵ and η defined after (13); we confirmed in [65] that this approximation does not affect our conclusions.
the universe ρ is still dominated by the energy density ρ φ of φ during reheating, wre is determined by specification of V(φ), and N re is the only relevant quantity that is not fixed by the choice of V(φ), i.e., is sensitive to g.Within a given model of inflation one can obtain information about N re by comparing the observed CMB spectrum to the model's prediction.Using the general redshifting relation ρ ∝ exp(−3N (1 + w)), this can then be translated into a constraint on the energy density at the end of reheating ρ re = ρ end exp(−3N re (1 + wre )) [68], often expressed in terms of an effective reheating temperature defined as π 2 g * 30 T 4 re ≡ ρ re with g * the effective number of relativistic degrees of freedom, In order to further translate knowledge on N re into knowledge on microphysical parameters, we utilise the fact that reheating ends when H = Γ , where Γ is an effective dissipation rate for φ and H = ȧ/a is the Hubble rate.
Together with the Friedmann equation with M pl = 2.435 × 10 18 GeV the reduced Planck mass.The RHS of ( 1) and ( 2) only contain quantities that are either calculable for given V(φ) or can be obtained from CMB observations; we summarise the relations to CMB observables in the appendix.Meanwhile Γ on the LHS depends on microphysical parameters of the particle physics model in which V(φ) is realised.
Measuring the inflaton coupling in the CMB: We classify microphysical parameters in three categories.A model of inflation is defined by specifying the effective potential V(φ).Ignoring quantum corrections to the φ-trajectory, this is equivalent to fixing the set of coefficients {v i } of all operators in the action that can be constructed from Φ alone, e.g., by Taylor expanding the inflaton potential around its minimum as 4! φ 4 + . ... The set of inflaton couplings {g i } comprises coupling constants (or Wilson coefficients) associated with operators that are constructed from Φ and other fields.A complete particle physics model contains a much larger set of parameters than the combined set {v i } ∪ {g i }, including the masses of the particles produced during reheating as well as their interactions amongst each other and with all other fields.We refer to the set of all parameters in the action that are not contained in {v i } ∪ {g i } as {a i }.This set, e.g., contains the parameters of the SM.
Γ in (2) necessarily depends on the {g i } and {v i }.For instance, for reheating through elementary particle decays, one typically finds Γ = g 2 m ϕ /c, with g ∈ {g i } a coupling constant, m ϕ ∈ {v i } the inflaton mass, and c a numerical factor.However, in general feedback effects from produced particles on the ongoing reheating process introduce a dependence of Γ on a large sub-set of {a i }, making it impossible to determine g from the CMB in a model-independent way, i.e., without having to specify the details of the underlying particle physics model and the values of the parameters {a i }. 9 The conditions under which g can be constrained model-independently have been studied in detail in [64], a conservative estimate is with D the mass dimension of the interaction term under consideration, j the power at which Φ appears in that operator, and Λ a scale that can be identified with m ϕ for D ≤ 4 and represents a UV cutoff of the effective theory for D > 4. The conditions (3) and ( 4) ensure that the production of particles proceeds slow enough that redshifting spreads their momenta over a sufficiently broad phase space volume for the occupation numbers in each mode to remain low enough to avoid sizeable feedback effects, such as a parametric resonance.The condition (4) practically restricts the possibility to constrain g modelindependently to scenarios where the φ-oscillations occur in a mildly non-linear regime.
Application to specific models: In the following we apply the previous considerations to two models of inflation, namely radion gauge inflation (RGI) [78,79] and α-attractor T-models (α-T) [80][81][82][83], with the potentials The scale M can be expressed in terms of other parameters with the help of ( 14), and condition (4) implies n = 1.Within these families of models (16) implies the relations α = for the RGI and α-T models, respectively.This defines a line in the n s -r plane, the position along which is given by N re (and hence g), cf.Fig. 1.Condition (4) implies α > 2.4 in (5) and α > 1/4 in (6).For our analysis we pick α = 19 in ( 5) and α = 6 in (6).When conditions (3) and ( 4) are fulfilled we may parameterise Γ = g 2 m ϕ /c [64] with (g, c) = (g/m ϕ , 8π) for a scalar coupling gΦχ 2 [84], (g, c) = (y, 8π) for a Yukawa coupling yΦ ψψ [73], and (g, c) = (σm ϕ /Λ, 4π) for an axion-like coupling σ Λ ΦF µν F µν [85], where we neglected the produced particles' rest masses.We shall assume a Yukawa coupling y in the following, bounds on other interactions can be obtained by simple rescaling according to c [65].  3) and ( 4) are fulfilled in the green part, the gray parts are ruled out by the conditions Nre > 0 and Tre < T BBN .Ellipses indicate current constraints and future sensitivities to ns and r, N k is the number of efolds between the horizon crossing of a perturbation with wave number k and the end of inflation (12).

CMB constraints on the inflaton coupling:
With the above considerations and the relations given in the appendix, n s and r in a given model of inflation are simple functions of x ≡ log 10 g.Prior to any measurement of (A s , n s , r) it is known that N re > 0 and that there is a lower bound T re > T BBN to allow for success-ful big bang nucleosynthesis (BBN).When ( 3) and ( 4) are fulfilled one can use the standard estimate T re ≃ Γ M pl × (90/(π 2 g * )) with θ the Heaviside function, and γ a function that allows for a re-weighting of the prior P (x).The constant C 1 can be fixed from the requirement dxP (x) = 1.We now quantify the gain in knowledge about x that can be obtained from data D.This gain can be quantified by the posterior distribution P (x|D) = P (D|x)P (x)/P (D) with P (D) = dxP (D|x)P (x).Current constraints from the data D obtained by Planck and BICEP/Keck [2] can be approximated by the likelihood function with N (n s , r|n s , σ ns ; r, σ r ) a two-dimensional Gaussian10 and γ another weighting function.The constant C 2 is fixed by normalising P (D|x) to unity.We fix the fiducial values to ns = 0.967, r = 0.01 and estimate the errors σ ns = 0.005, σ r = 0.018 based on Fig. 5 in [2].The result is shown in Fig. 2. While it is known that present data already provides information about the reheating epoch [86], current CMB observations do not provide a significant information gain on x with respect to P (x).
The scalar-to-tensor ratio will be constrained with much higher accuracy in the future [87].To quantify the expected information gain on x we repeat the analysis for r = 0.02 with σ ns = 0.002 and σ r = 0.0012, which reflects the sensitivity anticipated by LiteBIRD [13] or CMB-S4 [14].11Fig. 2 shows that in both models future data can rule out previously allowed values of T re .In the α−T model the posterior peaks in a region where condition (3) is violated, implying that Γ depends on a potentially large number of model parameters {a i }, and it is impossible to translate a constraint on N re into a model-independent constraint on g.This is a result of the fact that the currently allowed region in Fig. 1 is very close to the N re = 0 line.One can nevertheless obtain constraints log 10 (T re /GeV) = 13.1 ± 1.4 and 0.00515 < M/M pl < 0.00526 (the latter from ( 14)).In the RGI model, on the other hand, the posterior peaks in a region where condition (3) is fulfilled, so that future CMB data will permit measuring g independently of the {a i }.For the fiducial parameters chosen here, the mean values and variances for the posteriors read log 10 y = −6.5 ± 2.2, log 10 (T re /GeV) = 8.4 ± 2.1 and M/M pl = 0.00529 ± 0.00007.Finally, we estimate the improvement that can be made with data from the EUCLID satellite [88] and Square Kilometre Array (SKA) [89] by using σ ns = 0.00085 [90].The resulting posteriors in Fig. 2  Conclusions: We introduced a simple analytic method to quantify the information gain on the inflaton coupling g and the reheating temperature T re from observational constraints on n s and r.When applying it to future CMB observations with LiteBIRD and CMB-S4 we showed for the first time that these missions will be capable of performing the first ever measurement of T re in both models considered here.For the chosen fiducial values this can directly be translated into a measurement of g in the RGI model, while in the α-T model such a translation would require a specification of further parameters {a i }.Adding information from optical and 21cm surveys can further reduce the error bar on g, and may help to constrain α and g simultaneously from data by including observational knowledge on quantities not considered here, such as non-Gaussianities or the running of n s .The inflaton coupling g did not only crucially shape the evolution of the observable universe through its impact on T re , but it is also a key parameter that connects models of inflation to theories of particle physics.Measuring this microphysical parameter, even with large error bars, will open up a new window to probe the connection between cosmology and fundamental physics.Hence, our findings add a qualitatively new dimension to the physics cases of future observatories.
Acknowledgements: We would like to thank Marcos Garcia, Jan Hamann, Jin U Kang, Eiichiro Komatsu, Isabel Oldengott, Christophe Ringeval, Evangelos Sfakianakis, and Yvonne Wong for helpful discussions.We also thank Björn Garbrecht, Andrea Giammanco, Jan Hamann and Fabio Maltoni for feedback on the revised version of this manuscript.L.M. acknowledges the State Scholarship Fund managed by the China Scholarship Council (CSC) and the Project funded by China Postdoctoral Science Foundation (2022M723677).MaD would like to thank the University of New South Wales for their hospitality under the Gordon Godfrey visitor program during part of the work on this project.
Appendix: Relation to observables: In this appendix we give the relations between the RHS of (2) and observables.A detailed derivation can be found in [47] and has been adapted to our notation in [50].N re can be obtained from with g s * ≈ g * , a 0 and T 0 = 2.725 K the scale factor and the temperature of the CMB at the present time, respectively, and The subscript notation H k , φ k etc. indicates the value of the quantities H, φ etc. at the moment when a pivot-scale k crosses the horizon.φ k can be expressed in terms of n s and r by solving the relations n s = 1 − 6ϵ k + 2η k , r = 16ϵ k (13) with the slow roll parameters ϵ = (∂ φ V/V) 2 M 2 pl /2 and η = M 2 pl ∂ 2 φ V/V.In the slow roll regime, we find with A s = 10 −10 e 3.043 [1].T re can be expressed in terms of the observables (n s , A s , r) by plugging (11) with ( 12) into (1), φ k is found by solving (13) for φ k , and V end , and φ end can be determined by solving ϵ = 1 for φ.From ( 13) we obtain from which we find by using the definitions of ϵ and η.Together with (14) this provides three equations that can be used to relate the effective potential and its derivatives to the observables (n s , A s , r).That is sufficient to express wre and N re in (11) in terms of observables, which is all that is needed to determine the RHS of (2).

Figure 1 :
Figure 1: The diagonal line represents values of ns and r predicted by the RGI model (upper panel) and α-T model (lower panel) for fixed α, with the inflaton coupling varying along the curve.The black discs indicate the predictions for specific values of a Yukawa coupling y, with (log 10 y, log 10 Tre/GeV) given in the legend.Conditions (3) and (4) are fulfilled in the green part, the gray parts are ruled out by the conditions Nre > 0 and Tre < T BBN .Ellipses indicate current constraints and future sensitivities to ns and r, N k is the number of efolds between the horizon crossing of a perturbation with wave number k and the end of inflation(12).