Corpuscular Consideration of Eternal Inflation

We review the paradigm of eternal inflation in the light of the recently proposed corpuscular picture of space-time. Comparing the strength of the average fluctuation of the field up its potential with that of quantum depletion, we show that the latter can be dominant. We then study the full respective distributions in order to show that the fraction of the space-time which has an increasing potential is always below the eternal-inflation threshold. We prove that for monomial potentials eternal inflaton is excluded. This is likely to hold for other models as well.


I. INTRODUCTION
Cosmological inflation [1,2] is one of the central building blocks of our current understanding of the Universe. One of its simplest realizations, which is still compatible with observations, is via a single scalar field, called inflaton. Today's structure in the Universe is seeded by the quantum fluctuations of this field and of the space-time, and is in remarkable agreement with measurements (c.f. [3]). Depending on the value of the inflaton, it might experience large quantum fluctuations, also and in particular, up its potential, therefore inducing ever expanding inflationary patches of the Universe. This is the idea behind eternal inflation [4,5] (for a more recent review see [6]).
These considerations are, however, only valid if the semi-classical description of space-time is a faithful approximation. If gravity, like all other fundamental interactions, possesses a quantum description, inevitably the question arises when such a corpuscular picture of spacetime starts to become relevant. Recent progress by Dvali and Gomez (c.f. [7][8][9]), elaborating precisely on this topic, suggests that in certain situations one necessarily needs to take the graviton nature of what is classically regarded as space-time geometry into account. In fact, some phenomena like Hawking radiation, the Bekenstein Entropy, or, the information paradox can only be fully understood in this quantum picture [7][8][9] (c.f. also [10][11][12][13][14][15][16][17] for recent progress).
The mentioned attempt leads to regard space-time, such as black holes, de Sitter spaces, etc., as gravitationally bound states in the form of weakly/marginally bound states, or Bose-Einstein condensates, of gravitons with a mean wave-length equal to the curvature radius of that space-time. Due to the weak binding, quantum fluctuations are responsible for emptying the ground state of the condensate. This depletion is an intrinsically quantum effect which is entirely missed in any (semi-)classical treatment. In inflationary spaces, it acts like a quantum clock which works against the semi-classical one and, as we will investigate below, in particular against the fluctuations of the scalar field up the potential.
Recently, in [9] it has been argued that the corpuscular picture is incompatible with a positive cosmological constant. Also, in Ref. [7] the authors argue that quantum depletion sets a limit on the total number of e-foldings. Here, we investigate those qualitative considerations in more detail by quantitatively comparing the strengths of the relevant effects and considering, via the full respective probability distributions, the fraction of the space-time which has an increasing potential.

II. COMPETING FLUCTUATIONS
We consider a Universe filled with inflaton and graviton condensates. The number of inflatons is N φ and the number of gravitons is N . Working in Planck units (c = = M Pl = 1), the number of inflatons can easily be defined as where n φ is the number density of inflatons. The number of gravitons is given by When considering eternal inflation in view of the corpuscular description of gravity, we find that two competing quantum effects are active: the quantum fluctuations of the inflaton field due to the uncertainty principle, and the quantum depletion of the inflaton and graviton condensate due to graviton-inflaton scattering.
The typical quantum fluctuation due to the uncertainty principle reads (c.f. [4]) while quantum depletion of the gravitons is, to leading order in 1/N , given by (c.f. [7]) Here N ls is the number of species that are lighter than the energy of the gravitons in the condensate, which then present a possible decay channel. If no such lighter species exist the second term in Eq. (II.4) will just be −1/ √ N and represents graviton-graviton scattering. In most of the realistic scenarios this term will be a small correction, since the number of inflatons will vastly outnumber that of the gravitons.
The quantum fluctuations in the inflaton medium is the only source that may increase the energy of the inflaton medium, pushing the inflaton upwards in the potential. For eternal inflation to be realised, this effect has to be larger than the effect of depletion in a large enough fraction of the space, so that combined with the continued inflation of this fraction of space increases the volume of the inflating part of space-time. Since for each Hubble time this increase in volume is e 3 , this fraction must be < e −3 ≈ 1/20 = 0.05.
To compare the two effects we look for simplicity at an inflaton in the monomial potential where n > 0, and we define the effective mass scale We shall work in the slow-roll regime, |φ 2 | |V| and |φ 2 | 3 Hφ ∼ |V ,φ |, and use Eq. (II.4), the Friedmann equation where we assume that n = 1.
For quantum fluctuations in Eq. (II.3) to move the inflaton up the potential, the fluctuations have to be larger than the average depletion (II.4) moving the inflaton down. Using Eq. (II.7b) for N φ in Eq. (II.4), we realise that the magnitude of depletion is: We then insert Eqs. (II.7c) and (II.7d) into Eq. (II.3) to find the contribution toṄ stemming from a typical quantum fluctuation: Comparing (II.8) and (II.9) we find that the condition for the latter being dominant reads In order for eternal inflation to proceed when no corpuscular effects are present, the standard deviation of the Gaussian-distributed quantum fluctuations must be large enough for 1/e 3 ≈ 1/20 of the fluctuations to exceed the classical roll down the potential. This gives the criterion: H 2 /|φ cl | 3.8 (see [6]). Inserting for the corpuscular variables and our potential (II.5), this translates to the demand: To have eternal inflation, this bound must be fulfilled while the quantum depletion effects are still smaller than the regular quantum fluctuations. Taking the condition given for the vacuum fluctuations to dominate over depletion (II.10) and demanding that it holds above the bound yielding eternal inflation classically (II.11), we find n 7 |n − 1| 2500 π 2 N . (II.12) Since N can never go below 1, naïvely, eternal inflation at least does not to happen for monomial potentials V(φ) = 1/n! λ n φ n with n 3.7 . (II.13) Theories with n higher than this bound are excluded observationally [3]. However, the bound (II.13) should only be regarded as the naïve lowest bound one could possibly get. Though it is true that N cannot be lower than one, long before this bound is reached one would need to include higher orders in 1/N , which we neglected in our treatment. The bound (II.13) does thus not tell us that eternal inflation can happen for n 3.7, only that it naïvely will not occur for n 3.7.
Even if eternal inflation could seemingly occur for the higher values of n, eternal inflation in the true sense would not occur, as depletion at some point invariably will take the solution so far away from the classical that calling it inflation does not make sense any more (c.f. [7,9]). In this sense eternal inflation is a priori forbidden in the corpuscular view of gravity.
Here, though, we discuss eternal inflation in a more quantitative way, aiming also to exclude even extremely lengthy inflation.
Irrespective of the value of n, the above calculation was just done by comparing the typical quantum fluctuation and depletion. In order to get a more refined exclusion of eternal inflation, we must consider the two distributions properly. The quantum fluctuations approximately follow a Gaussian distribution centered around zero. The depletion process is given by a Poisson distribution [12].
For safe bounds on eternal inflation, we can compare the two distributions at the lower bound for the potential, so at the value of N m eff where standard (non-corpuscular) eternal inflation would occur [c.f. Eq. (II.11)]. Inserting this into Eq. (II.9) we find that here the Gaussian distribution has a standard deviation σ (n/2π)(n/3.8) 2/(2+n) (λ n /3n!) 1/(2+n) , whereas Eq. (II.8) implies that the Poisson distribution has an expectation value of λ 3.8 3/(n|n − 1|) /n. To compare the two competing effects we must then convolute the two probability distribution functions to find the fractional convoluted area that gives an increase in inflaton energy. That is, for each possible value for depletion, we multiply its probability with the probability of all quantum fluctuations that are large enough to dominate over it. In practice this is done by integrating the Gaussian up to where its contribution is the negative of each point on the Poisson curve along the Poisson distribution: (II.14) V up is the fraction of the space-time that has an increasing potential. Since the space-time volume in the inflating parts of the Universe is multiplied by twenty, eternal inflation can only occur when V up 0.05. Fig. 1 shows that, at least for small enough n, eternal inflation is excluded. Note that the true value of V up will be lower in practice because the classical flow will also pull the inflaton down its potential. Below we will include this effect in the convolution integral.
When investigating the effect of the classical flow down the potential along with the two quantum effects, we need to consider the convoluted volume for decreasing values of N , from the classical onset of eternal inflation, and down towards Planckian values N ≈ 1. The three effects in this regime can be written as: The naïve absolute maximum value for λ n , which is remotely sensible to consider, is the value for which eternal inflation can begin only at Planck scales N 1 in Eq. (II.11), that is λ n, max 3 n! n n 3.8 n .
(II.16) Again, as approaching N 1 lies beyond our approximations, the mentioned bound on λ n should be taken as a rough illustration only. In practice λ n would be much smaller than this. The minimum value of λ n is harder to obtain, however, as λ n decreases, the width of the Gaussian function for the quantum fluctuations decreases, making the quantum depletion effects more dominant.
For a given value of n and the self-coupling λ n we can calculate the fraction of the Universe undergoing eternal inflation by evaluating the convolution integral for a given value of N , In Fig. 2 we depict reults of the general convolution integral (II.17) for various values of the self-coupling λ n (10 −12 , 10 −10 , 10 −8 ) as well as for two values of N -once for a tenth of the maximum allowed value as given by (II.18), and once for a twentieth of it. We observe that the inclusion of the classical drift cures the seemingly dangerous increase of V up for larger n, as visible in Fig. 1. We also note that V up increases with increasing λ n . However, even for the absolute maximum, given by Eq. (II.16), we only get V up < 10 −5 . Thus, V up is below the eternal-inflation threshold for any value of n, and hence for monomial inflation we have proven that eternal inflation is excluded, therefore substantiating a corresponding claim in [7].

III. DISCUSSION AND OUTLOOK
In this work we investigated the paradigm of eternal inflation in view of the corpuscular picture of space-time for single-field inflation models with monomial potentials. We have compared the strength of the average fluctuation of the field up its potential with that of quantum depletion, and showed that the latter is dominant at least for small n.
In order to make a more refined statement, we then studied the fraction of space-time which has an increasing potential both with and without the effects of the classical roll present. For the case where we only considered quantum fluctuations versus quantum depletion, we could already prove the non-existence of eternal inflation for the observationallyrelevant small-n potential.
Including the classical effects we could show that the fraction of space-time moving up the potential is always, i.e. for any n, way below the eternal-inflation threshold. Summarized, we have proven that eternal inflation does strictly not occur for all canonical single-field inflation models with monomial self-interactions. This is a quantitative substantiation of the claim made in [7,9] that corpuscular gravity prohibits eternal inflation.
We believe that these findings are rather generic. In the case of more general potentials, such as for instance hilltop inflation, we still need the quantum fluctuations to be comparable to the classical evolution in order to drive eternal inflation. In these situations we also expect the depletion effects to become large, and more importantly dominate with respect to the usual quantum fluctuations, more or less regardless of the detailed shape of the potential.
We should stress that, in any case, at some finite point in time the quantum-depletion effects will accumulate to an extent that invalidates any (semi-)classical treatment. Then, the mean-field description is completely different from classical General Relativity and it will be impossible to reliably say that eternal inflation occurs. These statements are completely generic for any corpuscular treatment of inflation, and, in a forthcoming publication, we will further elaborate on this (non-eternal) inflation for generic single-field inflation models, using the quantitative methods developed in this work.
As the de Sitter solution might be approximated by the extreme slow-roll version of inflation, the fact that inflationary theories in the near-Planckian range are strongly dominated by the depletion effect also strengthens the argument found in [9] that the corpuscular view of gravity may have bearing on the cosmological constant problem.