Defying eternal inflation in warm inflation with a negative running

It was pointed out previously [1] that a sufficiently negative running of the spectral index of curvature perturbations from (ordinary i.e. cold) inflation is able to prevent eternal inflation from ever occurring. Here, we reevaluate those original results, but in the context of warm inflation, in which a substantial radiation component (produced by the inflaton) exists throughout the inflationary period. We demonstrate that the same general requirements found in the context of ordinary (cold) inflation also hold true in warm inflation; indeed an even tinier amount of negative running is sufficient to prevent eternal inflation. This is particularly pertinent, as models featuring negative running are more generic in warm inflation scenarios. Finally, the condition for the existence of eternal inflation in cold inflation — that the curvature perturbation amplitude exceed unity on superhorizon scales — becomes more restrictive in the case of warm inflation. The curvature perturbations must be even larger, i.e. even farther out on the potential, away from the part of the potential where observables, e.g. in the Cosmic Microwave Background, are produced.


Introduction
Inflation [2][3][4][5][6][7][8][9][10], a brief period of superluminal expansion in the early universe, is currently the most-compelling framework to produce the large scale homogeneity, isotropy, and oldness of the Universe, as well as provide a mechanism for generating the density fluctuations that ultimately give rise to the observed large-scale structures.
A large class of inflaton models involves a single scalar field slowly rolling down a nearly flat potential, inducing a quasi-de Sitter phase.Once the field nears the minimum of its potential, the accelerated expansion ends and the inflaton energy converts to radiation, thereby reheating the Universe.Hereafter, we refer to this standard scenario as cold inflation (CI) to emphasize the absence of substantial interactions between the inflaton field and other degrees of freedom during the inflationary period.
A well-established alternative framework to this standard picture is warm inflation (WI) [11] (see also Refs.[12][13][14] for reviews), where instead the inflaton field is thermally coupled to a bath of radiation, continuously sourcing its production throughout the accelerated expansion.Via these dissipative effects, WI is able to smoothly route the Universe to a radiation dominated era, alleviating the need for a separate reheating phase.The presence of dissipation and a radiation bath not only alter the inflaton dynamics but also its perturbations, which now become predominantly thermal in origin, with quantum fluctuations being subdominant in the limit of a large dissipation rate between the two sectors.For a given inflaton potential, this leads to significantly different predictions, with respect to those obtained in CI, for many primordial observables, such as the scalar spectral index n s and the tensor-to-scalar ratio r [15][16][17][18].These divergent predictions provide a means to distinguish between a CI and WI scenario and also allow the WI framework to reconcile some of the simplest inflaton potentials, otherwise excluded by data in the CI context [19].
In addition, from an effective field theory point of view, WI constructions can naturally overcome some of the issues in the standard CI picture.Specifically, the decay of the inflaton into radiation during inflation acts as a source of (thermal) friction, which allows WI models to occur for inflaton masses above the Hubble scale and with a subplanckian field excursion.This unique ingredient of the WI framework provides a solution to the η-problem [18], while also making WI consistent with the recently proposed Swampland Conjectures in String Theory [20][21][22][23][24][25][26], generally in strong tension with the slow-roll CI scenario.
Another important and generic feature of inflationary models is eternal inflation (EI) [27][28][29][30][31], which happens when the stochastic fluctuations of the inflaton field dominate over its deterministic evolution.For the regions where the field is systematically deviated upwards, a self-reproduction regime (SRR) of causally disconnected Hubble-sized regions will take place and inflation never ends.In CI, the stochastic fluctuations are only quantum in origin, while in WI both quantum and thermal fluctuations may be responsible for the onset of EI.However, in WI one also needs to account for the effect of the dissipation on the deterministic field excursion and the scale of the inflaton potential.In Ref. [32], by analyzing some physically motivated WI constructions, it was demonstrated that the dissipative and thermal characteristics of WI offer a mechanism for mitigating the onset of EI, thereby suggesting that EI may be prevented altogether in a strongly dissipative regime. 1 .
In this work, we aim to investigate this statement more generally and determine the requirements for the onset of EI in a WI setting, in terms of a bound on the amplitude of the curvature power spectrum and its functional form.In fact, in the previous work of Ref. [1], it was shown, in the context of CI, that there is a connection between the value of the running of the scalar spectral index α s of the curvature perturbations, and the onset of an eternally inflating regime.Specifically, EI is prevented if α s is sufficiently negative.
In the following, our main objective is to extend the bound obtained in Ref. [1] to the WI paradigm and investigate whether such new bound on α s , together with the obtained values of α s in physically motivated WI constructions [19,34], prevents EI and is consistent with the latest observational constraints from Planck [35].We will show that in the WI framework an even tinier amount of negative running is sufficient to prevent EI, and provide examples of several physically motivated WI constructions that produce a running of the spectral index that is sufficiently negative to prevent EI from ever occurring, while being consistent with CMB observations.Note that this is a nontrivial issue in WI, as there are four competing effects: (1) in WI there is a reduced deterministic field excursion due to the friction caused by the dissipation of the inflaton field into the radiation.At the same time, (2) there is an enhancement of the stochastic fluctuations of the inflaton field due to the addition of classical thermal fluctuations.Both of these effects act in a way to facilitate the onset of EI.On the other hand, (3) the inflaton fluctuations are also dumped into the radiation bath at a rate proportional to the strength of the dissipation.In addition, (4) the height of the inflaton potential, which sets the scale of the stochastic fluctuations, is suppressed relative to the CI case, such as to counteract the large thermal/dissipation enhancement factor in the scalar power spectrum in WI and reproduce the observed density perturbations.As we are going to explain and show, these latter effects surpass the former ones, which together with a smaller negative prediction for the running, acts to prevent EI altogether in WI.
The paper is organized as follows.In Sec. 2, we briefly review the relationship between EI and the amplitude of the curvature perturbation spectrum in the CI scenario, and introduce the Starobinsky stochastic formalism that more easily generalizes to the WI case.The stochastic (diffusion) and deterministic (drift) terms are discussed along with the definition of the fluctuation dominated regime (FDR).The FDR condition is then derived and expressed in terms of the scalar power spectrum.In Sec. 3, we extend the discussion to the case of WI, where the diffusion and drift terms now incorporate both quantum and thermal effects.A generalized FDR condition is derived in terms of the WI scalar power spectrum.Our main results start from Sec. 4, where we present the power spectrum in terms of the spectral index and its running, and derive the upper bound on α s necessary in order to prevent EI.We show some examples in order to compare CI and WI results regarding this upper bound.We then also discuss the higher order correction terms, i.e., the running of the running of the spectral index, β s , and the successive higher order terms γ s , δ s , etc, and their impact in the α s upper limit.Our final considerations and conclusions are presented in Section 5.

Eternal Inflation Regime in Standard Cold Inflation
In this section we review the conditions required for producing an eternal inflation (EI) regime in the case of standard CI.We begin with a simple comparison between quantum fluctuations of the rolling field with its classical evolution.Then we reproduce the same criterion using the Starobinsky stochastic formalism [31] that more easily generalizes when we turn to the WI scenario.
In CI, eternal inflation occurs when the quantum fluctuations in the inflaton field dominate over the classical field evolution.The amplitude of quantum fluctuations in the inflaton field during inflation is For EI to take place, this quantum fluctuation amplitude must be larger than the classical field variation over approximately a Hubble time (∆t ∼ H −1 ), Therefore, the condition for EI can be written as: We note that the fraction (2.3) is identical to the amplitude of the curvature perturbation for modes crossing the horizon during inflation, (2.4) In other words, the condition for EI in CI is that the curvature perturbations amplitude must exceed unity [1,36,37], This makes sense since in the standard CI scenario the curvature perturbations are equal to the amplitude of the quantum fluctuations in the inflaton, in units of the field variation in a Hubble time.
Starobinsky formulation: In order to be able to generalize these results beyond the case of the standard CI models, we show a derivation of the results in the CI case by expressing the dynamics of the inflaton field in terms of the Starobinsky stochastic inflation program.This approach provides a simple way for describing the backreaction of the short-wavelength inflaton modes on the dynamics of the long-wavelength ones.The equation for the homogeneous inflaton field ϕ is written as a Langevin-like equation of the form: where are the drift and diffusion coefficients, respectively, while ζ q (t) is a Gaussian noise term that accounts for the stochastic (quantum) fluctuations of the inflaton field, with correlation function The presence of an eternally inflating patch requires that the inflationary dynamics goes through a fluctuations dominated regime (FDR).This is equivalent to say that EI will occur when the stochastic fluctuations of the inflaton field, δϕ S , dominate over its deterministic evolution, δϕ D , over approximately a Hubble time, and where the deterministic and stochastic field excursions over approximately a Hubble time are given, respectively, by ) This then leads to the condition for the presence of a FDR, (2.11) The above condition emphasizes that when the diffusion term dominates over the drift one, the time evolution of the inflaton field is strongly nondeterministic.In CI, the FDR condition reads2 which matches the amplitude of the scalar curvature perturbations, Eq.(2.4).Hence, again we find the result of Eq. (2.5) above, that the condition for EI in CI is that the curvature perturbations amplitude must exceed unity [1,36,37], P (k) > 1.
In the standard CI scenario presented above, the stochastic fluctuations and hence the curvature perturbations were determined solely by the quantum fluctuations of the inflaton field.This is, however, not the case in WI, where classical thermal fluctuations in the inflaton field act as an additional source of curvature perturbations, generally dominating over the quantum fluctuations.In this sense, we also expect that the FDR condition in the WI context will depart from its simple form as given by Eq. (2.5).We explore this statement explicitly in the next section.

Eternal Inflation Regime in Warm Inflation
We now turn to the condition for eternal inflation in the case of WI.We introduce the inflaton field equation of motion in WI expressed in terms of the Starobinsky stochastic program, placing particular emphasis to the effects of thermal dissipation on the inflaton dynamics.We then use this formalism to derive the generalized condition for EI that is valid when temperature and dissipation effects become relevant.
The equivalent Langevin-like equation of motion of the inflaton field ϕ in WI reads [32]: where, in contrast to CI, we included a non-negligible dissipation rate Υ ≡ Υ(ϕ, T ), which accounts for the energy transfer between the inflaton field and the radiation bath (at temperature T ) present during inflation. 3In addition, we also introduced two stochastic Gaussian noise terms, ξ q and ξ T , which account for the quantum and thermal fluctuations, respectively, with two-point correlation functions given by [14,43,44]: ) Here, n * denotes the possible inflaton statistical distribution due to the presence of the radiation bath, generally assumed to be the equilibrium Bose-Einstein distribution, i.e., n * = [exp(H/T ) − 1] −1 ; and Q is the dimensionless ratio measuring the effectiveness at which the inflaton converts into radiation, defined as: For Q ≫ 1, a strongly dissipative WI (SDWI) regime is achieved, while Q < 1 represents the weak dissipative WI (WDWI) regime.In all cases, WI requires T > H, which is roughly the criterion for which thermal fluctuations dominate over quantum fluctuations [11].We note that for Q ≪ 1 and T < H, the CI limit is recovered.
For the purpose of this work, we are interested in the evolution of the nearly homogeneous inflaton field inside a Hubble volume and over a Hubble time.Hence, we must integrate out the spatial Dirac-delta function from the correlation functions.To do so, one simply notes that δ 3 (x − x ′ ) corresponds to an inverse volume factor.The natural volume to be taken is the de Sitter volume of the horizon, V H ≡ 4π 3 (1/H) 3 , such that we can simply take δ 3 (x − x ′ ) → 1/V H . Additionally, to recover the standard CI result, in the denominator we normalize the scale factor a, such that a(t = 1/H) = 1.With these substitutions, Eq. (3.1) becomes: where ζ q and ζ T are now the quantum and dissipation noises with the correlation functions given by ⟨ζ i (t)ζ i (t ′ )⟩ = δ(t − t ′ ), for i = {q, T }.In Eq. (3.5), we also have that f w (ϕ) is the drift coefficient, while D vac and D (2) diss are the quantum and thermal contributions to the diffusion coefficient, respectively.Their expressions are given by D (2)  vac ≡ The deterministic evolution and the stochastic fluctuations of the inflaton field now become 4 ) As previously mentioned, in WI the addition of thermal effects has also a significant impact on the form of the primordial power spectrum which in WI is given by [14]: where the multiplicative factor G(Q) accounts for the direct coupling of the inflaton and radiation fluctuations due to a temperature-dependent dissipation rate, and can only be determined numerically by solving the full set of perturbation equations [44][45][46].For a dissipation rate Υ ∝ T c , if c = 0 we have G(Q) = 1, while G(Q) is larger (smaller) than one for c > 0 (c < 0).We can now use Eq.(3.12) and rewrite the FDR condition in WI, Eq. (3.11), in terms of the amplitude of the curvature perturbations, similarly to what we previously did for the CI case.In formulas, the condition for EI in WI is where
P FDR defines the maximum amplitude of the curvature perturbations in order to avoid the onset of a fluctuations dominated regime.In other words, the condition for EI in WI is that the curvature perturbations amplitude must exceed P FDR .Note that in the limit of Q = 0, i.e. no dissipation, Eq. (3.13) reduces to the standard CI result: P (k) > 1.For Q > 0, we have that P FDR > 1.More specifically, there is a positive correlation between the dissipation strength Q and P FDR .This becomes evident in the SDWI regime, i.e. for Q ≫ 1.In this regime, we can approximate the scalar dissipation function G(Q) with a power-law function of , where both the prefactor a G and the exponent b G are generically ∼ O(1) numbers and additionally b G > 0 (< 0) for a positive (negative) temperature power dependence of the dissipation rate. 5Plugging this back into Eq.(3.13), we obtain that As long as b G > −1.5 (which is true for all dissipation rates of physical interest [44]), there is a positive power-law dependence on Q, such that As a whole, the generalized condition for eternal inflation given by Eq. (3.13) clearly shows that in a WI setting, the onset of EI is suppressed relative to the standard CI case, and this suppression is even more facilitated in the limit of strong dissipation.In other words, in WI the condition for the existence of EI is more restrictive than in the CI case: it is not sufficient for the amplitude of the curvature perturbations to exceed unity on superhorizon scales, it must be even larger (> P FDR ), i.e. even farther out on the inflaton potential, away from the part of the potential where observables, such as the Cosmic Microwave Background, are produced.
This conclusion may seem at first counter-intuitive since in the context of WI we expect: (1) a reduced deterministic field excursion due to the friction caused by the dissipation of the inflaton field into the radiation and (2) an enhancement of the stochastic fluctuations of the inflaton field due to the addition of classical thermal fluctuations.Effects (1) and ( 2) both act to decrease δϕ D and increase δϕ S .Hence, both of these effects contribute to enhance the ratio δϕ S /δϕ D , thereby facilitating the onset of EI.This however ignores two additional effects common to all WI constructions which suppress the ratio δϕ S /δϕ D and generally dominate over the first two effects.Specifically, in WI: (3) the inflaton fluctuations are dumped into the radiation bath6 at a rate proportional to the dissipation strength Q and (4) the height of the inflaton potential, which sets the scale of the stochastic fluctuations, is suppressed relative to the CI case, such as to counteract the large thermal enhancement factor in the power spectrum and to reproduce the observed density perturbations [47].Clearly, both effects (3) and ( 4) decrease the resulting δϕ S , thereby suppressing the onset of EI as anticipated above 7 .
In the next section, we investigate this statement analytically.More specifically, we extend the work of Kinney and Freese in Ref. [1] and derive a generalized upper bound on the running of the scalar spectral index to prevent EI to occur when in the presence of thermal dissipation.

An Upper Bound on the Running
The scalar power spectrum for curvature perturbations can be written in terms of the scalar spectral index n s and of the running α s , as where k * is a pivot scale, which is typically the scale relevant for CMB observations, i.e., k * = 0.05h Mpc −1 , for which the amplitude of the scalar power spectrum is constrained to be P * ≃ 2.1 × 10 −9 [48].Additionally, Planck data also constrain the values of n s and α s to be (at 68% C.L.) [35] 8 , While the value of the running α s is still poorly constrained, these CMB measurements clearly indicate that the spectral index is red, i.e., n s − 1 < 0, with a 99.7% confidence limit of approximately 0.95 < n s < 0.98.For a constant red spectral index and no running, i.e., α s = 0, EI is inevitable since there is always a scale k FDR ≪ k * such that for all k < k FDR , the condition for EI, Eq. (3.13), is satisfied.Thus, as emphasized in Ref. [1], the simplest case of interest is that of a constant negative running, α s < 0.
Notice that a constant negative running means that the spectral index gets redder on small scales k ≫ k * , and bluer on large scales, k ≪ k * , such that for a sufficiently negative α s , the spectral index for k ≪ k * will eventually exceed unity, n s − 1 > 0. This is sufficient to prevent EI as long as the scalar curvature power spectrum P (k) is bounded below P FDR on large scales, for all k < k * .In this scenario, the scalar power spectrum will have an extremum at some wavenumber k max < k * , given by [1] ln For EI to be prevented, we must simply enforce that the maximum of the curvature power spectrum is less than P FDR , This is equivalent to an upper bound on the running α s of Notice that this upper bound is generically looser in WI compared to the standard CI scenario, by a factor ln P * / ln(P * /P FDR ).As expected, for Q = 0, i.e.P FDR = 1, this bound reduces to the CI result obtained in Ref. [1].Given the updated CMB bounds on the spectral index (1 − n s < 0.05) and the amplitude of the scalar perturbations (P * ≃ 2.1 × 10 −9 ), the upper bound on the running in the CI limit is The bound on the running, generalized in the WI context, is therefore where ln P * / ln(P * /P FDR ) ≤ 1 for P FDR ≥ 1.In short, for any non-zero value of the dissipation strength Q, 9 the upper bound on the running moves to smaller negative values than in the CI case, i.e. it gets easier to avoid EI.This difference is significant in the case of strong dissipation (Q ≫ 1), and particularly for b G ≥ 0, which corresponds to a dissipation rate with a non-negative temperature dependence.For example, if we set Q = 100 and b G = 6.52, which corresponds to the value found for a dissipation rate Υ ∝ T c , with c = 3 [49], we obtain which is roughly a factor of 3 smaller than what we found in the CI case.Overall, the above discussion emphasizes one of the main takeaways of our work: in WI a tinier amount of negative running than was required in the CI framework is sufficient to prevent EI.This is particularly pertinent, as models featuring a relatively large negative running are quite generic in a WI scenario.
As an example, the Minimal Warm Inflation (MWI) model [42] is a successful WI construction where the inflaton is treated as an axionic field with a Chern-Simons coupling to non-Abelian gauge fields.MWI is realized in the SDWI regime (Q ≫ 1) and for a dissipation rate cubic in temperature, i.e.Υ ∝ T 3 ; hence the upper bound on the running in Eq. (4.10) roughly applies.The running of the spectral index associated with this model was recently computed in [34], with the result which, according to the bound in Eq. (4.10), is sufficiently negative to prevent EI from ever occurring.Similar values of the running were also found in Ref. [34] for a variant of MWI which implements an inflaton potential of the runaway type [26] and leads also to α s ≃ −4 × 10 −3 , which again is safely below the bound in Eq. (4.10).A relatively large negative running of the spectral index is also achieved for WI realizations in the WDWI regime (Q ≪ 1), as previously shown in Ref. [19], where, for a wide range of inflaton potentials and forms of the dissipation rates, α s ≲ −10 −4 .A few examples of models that produce such large negative values for the running include a quartic and a hilltop potential, respectively for a dissipation rate cubic and linear in temperature.
It is important to stress that the values of the running quoted here are also consistent with the latest observational constraints from Planck, Eq. (4.3), at the 1σ level.This emphasizes another relevant takeaway of our work: within the context of WI, CMB data are consistent with a large arrays of inflationary models that avoid the onset of EI, thereby restoring a simpler picture of the Universe evolution.
Finally, it is important to emphasize that in the above analysis we assumed that the running of the spectral index is constant, i.e. that the running of the running β s and higher order terms (γ s , δ s , etc.) are all set to zero.As already pointed out in [1] this restriction is not necessary: in order to prevent the onset of EI α s does not need to be constant, it simply needs to be sufficiently negative.More precisely, if we take higher-order terms in the spectral index to be non-zero, i.e. we take the scalar power spectrum to be of the form the bound in Eq. (4.9) to prevent eternal inflation becomes: which is always satisfied as long as Eq.(4.9) holds and This hierarchy, typical of CI constructions, is also generally expected in the context of WI (see, e.g., Ref. [34]), as higher order-terms in the spectral index will generally be proportional to larger powers of the slow-roll parameters ϵ W and η W . 10

Conclusions
In this paper, we have studied the conditions that lead to the so-called eternal inflation (EI), with a particular focus to the warm inflation (WI) framework, where thermal dissipation effects significantly alter the inflaton dynamics relative to the standard cold inflationary (CI) picture.Specifically, we have extended the work of Ref. [1] and generalized the upper bound on the running of the scalar spectral index α s necessary to prevent EI, when also incorporating the effects of thermal dissipation.We showed that, similarly to what was found in the context of CI, a sufficiently negative running on super-horizon scales is sufficient to prevent EI from ever occurring.Assuming a constant running, we derived the upper bound for α s , α s < −1.3 × 10 −4 × ln P * ln(P * /P FDR ) , where P * is the amplitude of the scalar power spectrum (at the relevant scale k * for CMB observations) and P FDR , defined in Eq. (3.14), represents the upper limit on the scalar power spectrum beyond which a fluctuations dominated regime (FDR) takes over, thereby causing the onset of EI.In standard CI, P FDR = 1, while in the context of WI, P FDR > 1 and specifically has a positive power-law dependence on the strength of the dissipation Q ≡ Υ/(3H) (see for instance Eq. (3.15)).This means that in a WI setting, an even tinier amount of negative running than the one that was required in the CI framework is sufficient to prevent EI from ever occuring.
In addition, we provided some examples of physically motivated WI constructions which are consistent with the CMB constraints on all of the relevant primordial observables (such as the tensor-to-scalar ratio r, the spectral index n s , as well as the running α s ) and also satisfy the bound in Eq. (5.1) i.e., they do not have EI.These include the Minimal Warm Inflation (MWI) [42] model and its variants [26] in the strongly dissipative regime (Q ≫ 1) [34]; as well as a quartic or a hilltop potential, respectively for a dissipation rate cubic and linear in temperature, in the weakly dissipative regime (Q ≪ 1) [19].This emphasizes one of the main takeaways of this work, which is that WI provides a natural framework to prevent EI from occurring, while still remaining consistent with the stringer constraints from CMB observations.Finally, as already pointed out in [1], we note that in a more realistic scenario where the running of the running β s , and higher order terms (γ s , δ s , etc.) are non-zero, as long as the scalar power spectrum remains smaller than P FDR , it is still the case that EI does not occur.This can be achieved as long as Eq.(5.1) holds and |α s | > 2 3 |β s | > 1 4 |γ s |, etc.This is an hierarchy generally expected in the context of WI, as higher order-terms in the spectral index will be proportional to larger powers of the slow-roll parameters ϵ W and η W [34].