Warm Little Inflaton Becomes Cold Dark Matter

PHYSICAL REVIEW LETTERS 122, 161301 (2019)

† João G. Rosa* and Luís B. Ventura Departamento de Física da Universidade de Aveiro and CIDMA, Campus de Santiago, 3810-183 Aveiro, Portugal (Received 22 November 2018; published 24 April 2019)

We present a model where the inflaton can naturally account for all the dark matter in the Universe within the warm inflation paradigm. In particular, we show that the symmetries and particle content of the warm little inflaton scenario (i) avoid large thermal and radiative corrections to the scalar potential, (ii) allow for sufficiently strong dissipative effects to sustain a radiation bath during inflation that becomes dominant at the end of the slow-roll regime, and (iii) enable a stable inflaton remnant in the postinflationary epochs. The latter behaves as dark radiation during nucleosynthesis, leading to a non-negligible contribution to the effective number of relativistic degrees of freedom, and becomes the dominant cold dark matter component in the Universe shortly before matter-radiation equality for inflaton masses in the 10−4–10−1 eV range. Cold dark matter isocurvature perturbations, anticorrelated with the main adiabatic component, provide a smoking gun for this scenario that can be tested in the near future.

DOI: 10.1103/PhysRevLett.122.161301

Inflation [1] has inevitably become a part of the modern can smoothly take over as the dominant component at the cosmological paradigm. Observations are consistent with end of inflation, with no need for a reheating period its predictions of a flat universe with a nearly scale- [11,12]. Since dissipation induces small thermal fluctua- invariant, Gaussian, and adiabatic spectrum of primordial tions of the inflaton field about the background value ϕ, the density perturbations [2]. However, the exact nature of the primordial spectrum of curvature perturbations can differ inflaton field and its scalar potential remain open questions significantly in the cold and warm inflation scenarios that may hopefully be addressed with future measurements [13–20], the latter, e.g., predicting a suppression of the of B modes in the cosmic microwave background (CMB) tensor-to-scalar ratio in chaotic models [11,13,19]. Warm polarization, as well as possibly non-Gaussian features and inflation thus provides a unique observational window into isocurvature perturbations. inflationary particle physics. The present ambiguity is no less apparent when one Warm inflation models were, however, hindered by regards two consistent, yet distinct, descriptions of this several technical difficulties [21,22]. Interactions between 2 2 2 period of accelerated expansion: cold and warm inflation the inflaton and other fields, e.g., g ϕ χ or gϕψψ¯ , typically [3,4]. The former is described by the overdamped motion of give the latter a large mass. On the one hand, to keep the one (or more) field(s), coupled solely with gravity, whose fields light, and also avoid the associated large thermal nearly flat potential dominates the energy content of the corrections to the inflaton potential, one must consider small Universe, acting as an effective cosmological constant for a coupling constants that make dissipative effects too feeble to finite period. Inflaton quantum fluctuations then act as sustain a radiation bath for ∼50–60 e folds of inflation. On seeds for structure formation and CMB anisotropies. the other hand, heavy fields acting as mediators between the If, however, interactions between the inflaton, ϕ, and inflaton and the light degrees of freedom in the radiation bath other fields are non-negligible during inflation, dissipative can yield sufficiently strong dissipative effects [23–26],but effects transfer the inflaton’s energy into light degrees of at the expense of considering a large number of such freedom (DOF), sustaining a subdominant radiation bath mediators that may only be present in certain string con- that significantly changes the dynamics of inflation [5].In structions [27] or extradimensional models [28]. particular, the additional dissipation makes the inflaton These shortcomings were recently overcome in the warm field roll more slowly than in the cold case [6]. When little inflaton (WLI) scenario, which provides the only dissipation becomes sufficiently strong, the radiation bath consistent model of warm inflation with only two light fields coupled to the inflaton [29]. In this Letter, we show, for the first time, that the underlying symmetries and particle content of this model necessarily lead to a stable Published by the American Physical Society under the terms of inflaton remnant that survives until the present day, the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to naturally accounting for the cold dark matter component the author(s) and the published article’s title, journal citation, in our Universe. In this sense, warm inflation implies and DOI. Funded by SCOAP3. inflaton–dark matter unification.

0031-9007=19=122(16)=161301(6) 161301-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 122, 161301 (2019)

This is hard to implement within the cold inflation a chiral rotation. Hence, the only effects of the inflaton paradigm because the inflaton must decay efficiently at field’s interactions with the fermions are related to its the end of inflation to ensure a successful “reheating” of the dynamical nature; i.e., they correspond to nonlocal con- Universe, but cannot do so completely to provide a tributions to the effective action, in particular dissipative sufficiently long-lived dark relic [30]. This can be achieved effects. by introducing additional symmetries or cosmological The interchange symmetry corresponds to a Z2 reflec- phase(s) which, however, do not affect the inflationary tion for the inflaton field, ϕ ↔ −ϕ, that protects it from dynamics and have, in general, no direct observational decaying into any other fields besides the fermions ψ 1;2.As imprint [31–38]. discussed below, a significant dissipation of the inflaton’s In this Letter, we propose a radically different unification energy implies gM ≲ T ≲ M, which is only satisfied during scenario, where the inflaton decays during, but not after, inflation, thus ensuring the stability of the inflaton in the inflation. We show that this is possible due to the very same postinflationary Universe. symmetries that protect the scalar potential against thermal The fermion fields may decay into light scalar and and quantum corrections, independently of its form, while fermion fields, σ and ψ σ, respectively, with appropriate allowing for sufficiently strong adiabatic dissipation effects U(1) charges, through Yukawa interactions: that become exponentially suppressed once the inflaton X exits the slow-roll regime and radiation smoothly takes −Lψσ ¼ −hσ ðψ¯ iLψ σR þ ψ¯ σLψ iRÞ: ð3Þ over. These symmetries then ensure that the inflaton i¼1;2 behaves as a stable cold relic while oscillating about the minimum of its potential at late times, regardless of the full The dissipation coefficient resulting from the inflaton- form of the scalar potential. fermion interactions can be computed using standard ϕ The WLI model includes two complex scalar fields, 1;2, thermal field theory tools in the adiabatic regime with equal charge q under a U(1) gauge symmetry that is [5,25,39], where the fermions are kept close to thermal spontaneously broken by their identical vacuum expect- pffiffiffi equilibrium through decays and inverse decays, Γψ ≳ H ≫ ation values, hϕ1i¼hϕ2i ≡ M= 2. This yields masses of jϕ_ =ϕj [40]. Its dominant contribution corresponds to on- order M for the radial scalar components and the U(1) shell fermion production [26], being given by: gauge field that then decouple from the dynamics for ≲ temperatures T M. The remaining physical light degree 2 Z 3 g d p Γψ of freedom is the relative phase between ϕ1 and ϕ2, which ϒ ¼ 4 ðω Þ½1 − ðω Þ ð Þ 3 2 nF p nF p ; 4 we identify as the inflaton field, ϕ: T ð2πÞ mψ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M iϕ=M M −iϕ=M ¼ 2 2 þ 2 2 8 ϕ1 ¼ pffiffiffi e ; ϕ2 ¼ pffiffiffi e : ð1Þ where mψ g M h T = is the thermally corrected 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 fermion mass, ωp ¼ jpj þ mψ , Γψ is the fermion decay ψ 1 ψ 2 The model also includes a pair of fermions, and , width (see Ref. [29]) and nF is the Fermi-Dirac distribution. whose left-handed (right-handed) components have U(1) The momentum integral can be computed analytically for charge q (0), and we impose an interchange symmetry mψ =T ≪ 1 and mψ =T ≫ 1, yielding: ϕ1 ↔ ϕ2, ψ 1 ↔ ψ 2, such that the allowed Yukawa interactions are given by: 2 2 2g h 1 mψ ϒ ≈ T − ln ;mψ =T ≪ 1; 1 1 ð2πÞ3 5 T pffiffiffi ¯ pffiffiffi ¯ rffiffiffirffiffiffiffiffiffiffi −Lϕψ ¼ gϕ1ψ 1Lψ 1R þ gϕ2ψ 2Lψ 2R þ H:c: 2 2 2 2 2g h π mψ ϒ ≈ −ðmψ =TÞ ≫ 1 ð Þ 3 T e ;mψ =T : 5 iγ5ϕ=M −iγ5ϕ=M ð2πÞ 2 ¼ gMψ¯ 1e ψ 1 þ gMψ¯ 2e ψ 2; ð2Þ T where we have also imposed the sequestering of the “1” and In the high temperature regime relevant during inflation, the “2” sectors, which may be achieved, e.g., by considering dissipation coefficient is proportional to the temperature as additional global symmetries for each sector or by physi- in the original WLI model, although the proportionality cally separating them along an extra compact dimension. constant is smaller in this case. Inflationary observables can This is in contrast with the original WLI proposal [29], thus be computed as in Refs. [29,41,42], to which we refer providing the additional appealing feature that the fermion the interested reader. After inflation, dissipative effects are masses, m1 ¼ m2 ¼ gM, and therefore the associated exponentially suppressed as the ψ 1;2 fermions become radiative and thermal corrections to the effective potential nonrelativistic, halting the energy transfer between the are independent of the inflaton field. Note that if ϕ were inflaton condensate ϕ and the light DOF. constant throughout the whole space-time, one could The evolution of the background homogeneous inflaton remove it completely from the fermion Lagrangian through field and entropy density is then dictated by the following:

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FIG. 1. Cosmological evolution of the inflaton (solid blue line) and radiation (dashed red line) energy densities for a representative choice of parameters. The vertical dashed black lines mark the inflaton-radiation equality times at the end of inflation and of the radiation era. The inset plot shows that the inflaton behaves as a subdominant dark radiation component during nucleosynthesis FIG. 2. Evolution of ϕ=T (brown), T=H (red), Γψ =H (orange), ∼ 1 2 (BBN) (T MeV). The last scattering surface (CMB) Q (green), ϵ ≡ −H=H_ , Q ≡ ϒ=3H (purple), and T=M (red, ∼ 0 3 H (T . eV) and the present day are also highlighted. bottom plot) during inflation. In this example, h ¼ 1.8, g ¼ 0.2, 14 M ¼ 7.2 × 10 GeV, with horizon-crossing conditions ϕ ¼ 21 ¼ 0 0072 ϒϕ_ 2 MP, Q . . This yields 58 e folds of inflation, a ̈ _ scalar spectral index n ¼ 0.965 and a tensor-to-scalar ratio ϕ þð3H þ ϒÞϕ þ V;ϕ ¼ 0; s_ þ 3Hs ¼ ; ð6Þ s T r¼4.3×10−3 for nearly thermal inflaton fluctuations (see Refs. [29,41,42]). 2 ¼ðρ þ ρ Þ ð3 2Þ alongside the Friedmann equation H ϕ r = MP , _ 2 2 3 where ρϕ ¼ðϕ =2ÞþVðϕÞ, s¼ð2π =45ÞgT ¼4ρ =ð3TÞ, r The ratio ϕ=T remains constant throughout the dark and g is the number of relativistic DOF. Since the inflaton radiation phase (up to changes in g), and we find numeri- is a gauge singlet, the scalar potential is an arbitrary even cally that it is an Oð1–10Þ factor below its value at inflaton- function of the field, so we consider the simplest renorma- 4 2 2 4 radiation equality, ϕ=T ∼ 10 . This suppression is the result lizable potential, VðϕÞ¼m ϕ =2 þ λϕ , with the quartic ϕ of the faster decay of the inflaton amplitude compared to (quadratic) term dominating at large (small) field values. the temperature at the onset of inflaton oscillations. This A representative example of the cosmological dynamics assumes that the inflaton field does not decay into or scatter is illustrated in Figs. 1 and 2. The inflaton potential energy significantly off the particles in the thermal bath after ∼60 e dominates the energy balance for folds in the slow- inflation. The latter could, in particular, thermalize the roll regime, while sustaining a nearly constant radiation inflaton particles that would later decouple from the abundance through dissipative effects. These become thermal bath as a standard WIMP. However, since the Q ≳ 1 strong ( ) at the end of inflation, allowing for radiation temperature falls below the mass of the fermions just after to take over as the dominant component at a temperature inflation, inflaton interactions with the remaining light ∼ 1013–1014 TR GeV. Almost simultaneously, the temper- fields are significantly suppressed. ψ ature drops below the mass of the 1;2 fermions and they Consider first the four-body decay of inflaton, gradually decouple from the dynamics, therefore making ϕ → ψ¯ σψ σσσ, mediated by the heavy ψ 1;2 fermions in the inflaton effectively stable as we discuss below. At this 4 the darkp radiationffiffiffiffiffiffiffi phase where the effective inflaton mass is stage, the ratio T=H ∼ 10 becomes large enough to excite mϕ ≈ 12λϕ [39,45], for which we obtain: all the standard model DOF [43], resulting in a temperature ;eff drop by a factor ∼2, further suppressing dissipative 4 5 2 3 2 effects [44]. Γϕ λ = ϕ ϕ ≈ 10−19 h =T In the absence of significant dissipation, the inflaton H g2 10−15 M 104 P condensate starts oscillating about the origin in the quartic −4 4 −1 10 M ϕ potential, with amplitude ϕ ∝ a , hence behaving as dark × P sin2 ; ð7Þ M M radiation. Thispffiffiffiffiffi phase lasts until the amplitude drops below ϕ ¼ 2λ DM mϕ= , after which the quadratic term dominates and the inflaton remnant behaves as cold (pressureless) which vanishes at the origin due to the underlying Z2 dark matter until the present day. symmetry, and is in fact negligible throughout the whole

161301-3 PHYSICAL REVIEW LETTERS 122, 161301 (2019) radiation era in the parametric range relevant for a suc- The produced inflaton particlesp areffiffiffi relativistic, since they cessful inflationary dynamics, as in the example of Fig. 2. have typical momenta jpj ∼ ωϕ ∼ λϕ and mass mϕ ≪ ωϕ Low-momentum ϕ particles in the inflaton condensate once the condensate evaporates. They are also decoupled (jpϕj ≲ H) can also scatter off light particles in the thermal from the radiation bath since they only interact with the bath, e.g., ϕσ → ϕσψ¯ σψ σ and similar processes that could latter through the heavy ψ 1;2 fermions, so they will simply lead to the condensate’s evaporation and thermalization redshift as dark radiation until they become nonrelativistic, Δ ∼ ðω Þ ∼ ðϕ ϕ Þ of the inflaton particles. This process is again mediated by i.e., for Ne log ϕ=mϕ log = DM . Accordingly, the heavy ψ 1;2 fermions, with center of mass energies they behave as dark radiation for essentially the same ≈ 2 s mϕ;effT, yielding the following [39]: amount of time as the oscillating inflaton condensate had it not evaporated. This implies that the cosmological evolu- 2 3 tion of the produced inflaton particles is indistinguishable Γϕσ λ ϕ T ≈ 10−9h4g from that of the oscillating background field, and for H 10−15 M gM P simplicity we will henceforth consider the latter picture. −4 3 10 MP 2 ϕ The ratio nϕ=s remains constant in the cold dark matter × cos ; ð8Þ ϕ ϕ M M regime, for < DM, being given by rffiffiffi 3 nϕ ρϕ=mϕ λ 45 1 ϕ which is also negligible in the relevant parametric range. ¼ ¼ ; ð Þ ð2π2 45Þ ð Þ 3 2 2π2 9 Both the four-body decay and scattering processes s = g TDM TDM g T discussed previously are suppressed by the mass of the −1=3 ψ 1;2 fermions, which is larger than the temperature in the where we used that ðϕ=TÞg remains constant through- radiation era (T

161301-4 PHYSICAL REVIEW LETTERS 122, 161301 (2019) 12Q ϕ0 The scenario described in this Letter shows that inflaton– S ¼ 2 − R; ð12Þ c 3 þ 5Q ϕ dark matter unification is a natural feature in warm inflation. The underlying symmetries of the model allow where R ¼ δϕ=ϕ0 is the gauge-invariant (adiabatic) cur- the inflaton to decay significantly only during inflation, vature perturbation, and all quantities are evaluated when remaining essentially stable throughout the remainder of the relevant CMB scales become superhorizon during the cosmic history. The basic ingredients of the model can inflation. Since ϕ0 < 0, isocurvature modes are anticorre- nevertheless accommodate more general scenarios, with lated with the main adiabatic component for Q < 3, the different forms of the scalar potential and further inter- regime where the WLI model with a quartic potential is actions that could allow for further (partial) inflaton decay technically and observationally consistent [29,41,42]. after inflation or even its thermalization with the cosmic Their contribution to the primordial perturbation spectrum heat bath. Our model may also, in principle, be embedded β ¼ 2 ð 2 þ 1Þ can be parametrized by the ratio Iso Bc= Bc with within different standard model extensions, which could Bc ¼ Sc=R. For the example in Figs. 1 and 2, potentially yield novel observational signatures of inflaton– β ¼ 3 15 10−4 Iso . × , and, in the consistent parameter range, dark matter unification. we find β ∈ ½3; 4 × 10−4, below the Planck bounds on Iso L. B. V is supported by FCT Grant No. PD/BD/140917/ fully anticorrelated isocurvature perturbations, β < Iso 2019. J. G. R. is supported by the FCT Investigator 2 × 10−3 [2].SinceQ and ϕ are functions of the number Grant No. IF/01597/2015. This Letter was partially of e folds, the isocurvature spectral index, nI ≡ 1 þ Δ2 supported by the H2020-MSCA-RISE-2015 Grant d log I =dNe, differs from the adiabatic one. For in- ≈ 0 999 No. StronGrHEP-690904 and by the Center for Research stance, in our working example we find nI . and and Development in Mathematics and Applications Project n ≈ 0 965 s . . Future searches of cold dark matter isocurvature No. UID/MAT/04106/2019. We would like to thank Mar modes with the CMB-S4 mission [52] will therefore be Bastero-Gil, Arjun Berera and Rudnei Ramos for useful crucial to test our scenario. discussions. The second is the prediction of extra relativistic degrees of freedom during nucleosynthesis (BBN), whose main contribution is given by the oscillating condensate, Δ ¼ 4 4ρ ρ ρ Neff . ϕ= γ [53], with γ denoting the photon energy *[email protected] density, which is fixed by the value of ϕ=T at the onset of † [email protected] inflaton oscillations and the thermal evolution of the [1] A. A. Starobinsky, Phys. Lett. 91B, 99 (1980); K. Sato, standard model DOF. For the case represented above, 195 Δ ≈ 0 13 Δ Δ 0 20 Mon. Not. R. Astron. Soc. , 467 (1981); A. H. Guth, Neff . . BBN bounds on Neff, Neff < . [54] Phys. Rev. D 23, 347 (1981); A. Albrecht and P. J. are already strong enough to restrict the parameter space Steinhardt, Phys. Rev. Lett. 48, 1220 (1982); A. D. Linde, consistent with ∼50–60 e folds of inflation to g ∼ 0.1–0.2, Phys. Lett. 108B, 389 (1982). 14 15 −3 −2 h ∼ 1–2, M ∼ 10 –10 GeV, and Q ∼ 10 –10 . [2] Y. Akrami et al. (Planck Collaboration), arXiv:1807.06211. This contributionvanishes before recombination, once the [3] A. Berera and L.-Z. Fang, Phys. Rev. Lett. 74, 1912 (1995). condensate ϕ starts behaving as cold dark matter. The small [4] A. Berera, Phys. Rev. Lett. 75, 3218 (1995). inflationary thermal fluctuations about ϕ remain, however, [5] A. Berera, I. G. Moss, and R. O. Ramos, Rep. Prog. Phys. 72, 026901 (2009). relativistic until after recombination, since mϕ ≲ 0.1 eV. [6] Provided that dissipative effects are sufficiently strong, Although these consequently give only a subdominant warm inflation may satisfy the criteria of the string swamp- contribution to the present cold dark matter abundance, land conjecture [7–10]. they change the number of relativistic DOF at both BBN [7] G. Obied, H. Ooguri, L. Spodyneiko, and C. Vafa, arXiv: Δ th ¼ð4 7Þð43 4 Þ4=3 ≈ 0 027 and recombination by Neff = = g . , 1806.08362. where g ≈ 106.75 at the end of inflation [51], which may be [8] S. Das, arXiv:1809.03962. probed with the next generation CMB experiments and [9] S. Das, Phys. Rev. D 99, 063514 (2019). large-scale structure surveys [53]. [10] M. Motaharfar, V. Kamali, and R. O. Ramos, Phys. Rev. D Finally, we note that warm inflation itself has distinctive 99, 063513 (2019). 24 observational features, namely a modified consistency [11] M. Bastero-Gil and A. Berera, Int. J. Mod. Phys. A , 2207 (2009). relation between the tensor-to-scalar ratio and the tensor 55 spectral index [19,55] and non-Gaussianity with a charac- [12] A. Berera, Phys. Rev. D , 3346 (1997). [13] A. Berera, Nucl. Phys. B585, 666 (2000). teristic bispectrum [56]. All these observables thus provide, [14] L. M. H. Hall, I. G. Moss, and A. Berera, Phys. Rev. D 69, in conjunction, a very distinctive and testable scenario for 083525 (2004). – inflaton dark matter unification, where one can probe the [15] I. G. Moss and C. Xiong, J. Cosmol. Astropart. Phys. 04 inflationary nature of dark matter even if it interacts too (2007) 007. feebly with known particles to be found in direct searches [16] C. Graham and I. G. Moss, J. Cosmol. Astropart. Phys. 07 or collider experiments. (2009) 013.

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[17] R. O. Ramos and L. A. da Silva, J. Cosmol. Astropart. Phys. [39] See Supplemental Material at http://link.aps.org/ 03 (2013) 032. supplemental/10.1103/PhysRevLett.122.161301 for the [18] S. Bartrum, A. Berera, and J. G. Rosa, J. Cosmol. Astropart. derivation of the dissipation coefficient and the determi- Phys. 06 (2013) 025. nation of the decay and scattering rates of the inflaton. [19] S. Bartrum, M. Bastero-Gil, A. Berera, R. Cerezo, R. O. [40] Our results are derived under the technical assumptions Ramos, and J. G. Rosa, Phys. Lett. B 732, 116 (2014). of adiabaticity, near thermal equilibrium and flat space (at [20] M. Benetti and R. O. Ramos, Phys. Rev. D 95, 023517 scales ∼T−1), which restrict the parameter space of the (2017). model, although without any fine-tuning. We note, however, [21] A. Berera, M. Gleiser, and R. O. Ramos, Phys. Rev. D 58, that outside this parametric regime, nonequilibrium dissi- 123508 (1998). pative effects will still be present and need not be sup- [22] J. Yokoyama and A. D. Linde, Phys. Rev. D 60, 083509 pressed, so that one is not necessarily in a cold inflation (1999). regime. [23] A. Berera and R. O. Ramos, Phys. Lett. B 567, 294 (2003). [41] M. Bastero-Gil, S. Bhattacharya, K. Dutta, and M. R. [24] I. G. Moss and C. Xiong, arXiv:hep-ph/0603266. Gangopadhyay, J. Cosmol. Astropart. Phys. 02 (2018) 054. [25] M. Bastero-Gil, A. Berera, and R. O. Ramos, J. Cosmol. [42] M. Bastero-Gil, A. Berera, R. Hernandez-Jimenez, and J. G. Astropart. Phys. 09 (2011) 033. Rosa, Phys. Rev. D 98, 083502 (2018). [26] M. Bastero-Gil, A. Berera, R. O. Ramos, and J. G. Rosa, [43] E. W. Kolb and M. S. Turner, The Early Universe (Avalon J. Cosmol. Astropart. Phys. 01 (2013) 016. Publishing, New York, 1994). [27] M. Bastero-Gil, A. Berera, and J. G. Rosa, Phys. Rev. D 84, [44] This depends, of course, on how the WLI model is 103503 (2011). embedded within an extension of the standard model. [28] T. Matsuda, Phys. Rev. D 87, 026001 (2013). [45] H. M. Asatrian, A. Hovhannisyan, and A. Yeghiazaryan, [29] M. Bastero-Gil, A. Berera, R. O. Ramos, and J. G. Rosa, Phys. Rev. D 86, 114023 (2012). Phys. Rev. Lett. 117, 151301 (2016). [46] T. Appelquist and J. Carazzone, Phys. Rev. D 11, 2856 [30] A. R. Liddle, C. Pahud, and L. A. Urena-Lopez, Phys. (1975). Rev. D 77, 121301(R) (2008). [47] M. Bastero-Gil, A. Berera, and B. M. Jackson, J. Cosmol. [31] V. H. Cardenas, Phys. Rev. D 75, 083512 (2007);G. Astropart. Phys. 07 (2011) 010. Panotopoulos, Phys. Rev. D 75, 127301 (2007); K. Mukaida [48] K. Kainulainen, S. Nurmi, T. Tenkanen, K. Tuominen, and K. Nakayama, J. Cosmol. Astropart. Phys. 08 (2014) and V. Vaskonen, J. Cosmol. Astropart. Phys. 06 (2016) 062. 022. [32] A. R. Liddle, C. Pahud, and L. A. Urena-Lopez, Phys. [49] K. Ichikawa, T. Suyama, T. Takahashi, and M. Yamaguchi, Rev. D 77, 121301(R) (2008). Phys. Rev. D 78, 063545 (2008). [33] N. Bose and A. S. Majumdar, Phys. Rev. D 80, 103508 [50] C. Cosme, J. G. Rosa, and O. Bertolami, J. High Energy (2009); J. De-Santiago and J. L. Cervantes-Cota, Phys. Phys. 05 (2018) 129. Rev. D 83, 063502 (2011). [51] N. Aghanim et al. (Planck Collaboration), arXiv: [34] N. Okada and Q. Shafi, Phys. Rev. D 84, 043533 (2011); 1807.06209. A. de la Macorra, Astropart. Phys. 35, 478 (2012). [52] K. Abazajian et al. (CMB-S4 Collaboration), arXiv: [35] R. N. Lerner and J. McDonald, Phys. Rev. D 80, 123507 1610.02743. (2009); V. V. Khoze, J. High Energy Phys. 11 (2013) 215; T. [53] D. Baumann, D. Green, and B. Wallisch, J. Cosmol. Tenkanen, J. High Energy Phys. 09 (2016) 049. Astropart. Phys. 08 (2018) 029. [36] R. Daido, F. Takahashi, and W. Yin, J. Cosmol. Astropart. [54] R. H. Cyburt, B. D. Fields, K. A. Olive, and T.-H. Yeh, Rev. Phys. 05 (2017) 044. Mod. Phys. 88, 015004 (2016). [37] D. Hooper, G. Krnjaic, A. J. Long, and S. D. Mcdermott, [55] Y. F. Cai, J. B. Dent, and D. A. Easson, Phys. Rev. D 83, Phys. Rev. Lett. 122, 091802 (2019). 101301(R) (2011). [38] M. Bastero-Gil, R. Cerezo, and J. G. Rosa, Phys. Rev. D 93, [56] M. Bastero-Gil, A. Berera, I. G. Moss, and R. O. Ramos, 103531 (2016). J. Cosmol. Astropart. Phys. 12 (2014) 008.

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