Eternal inflation and collapse theories

The eternal inflation problem continues to be considered one of standard's cosmology most serious shortcomings. This arises when one considers the effects of “quantum fluctuations” on the zero mode of inflaton field during a Hubble time in the inflationary epoch. In the slow-roll regime it is quite clear that such quantum fluctuations could dwarf the classical rolling down of the inflaton, and with overwhelming probability this prevents inflation from ever ending. When one recognizes that quantum fluctuations can not be taken as synonymous of stochastic fluctuations, but rather intrinsic levels of indefiniteness in the quantities in question, one concludes that the eternal inflation problem simply does not exist. However, the same argument would serve to invalidate the account for the generation of the primordial seeds of cosmic structure as has been amply discussed elsewhere [1,2,3]. In order to do address that issue, one must explain the breaking of homogeneity and isotropy of the situation prevailing during the early inflationary epoch (at both the quantum and classical levels of the description). For that one needs to rely on some additional element, beyond those present in the traditional treatments. The so called spontaneous collapse theories offer a viable candidate for that element, namely the stochastic and spontaneous state reduction characteristic of those proposals possesses the basic features to break those symmetries. In fact, a version of the CSL theory adapted to the cosmological context has been shown to offer a satisfactory account for the origin the seeds of cosmic structure with an adequate power spectrum [2], and will serve as the basis of our analysis. However, once such stochastic collapse is introduced into the theoretical framework the eternal inflation problem has the potential reappear. In this manuscript we explore those issues in detail and discuss an avenue that seems to allow for a satisfactory account for the generation of the primordial inhomogeneities and anisotropies while freeing the theory from the eternal inflation problem.


Introduction
The eternal inflation problem is taken to be such a serious problem for the inflationary cosmology paradigm, that even some the most enthusiastic early advocates of inflation have, nowadays, become some of the theory's strongest detractors [4,5,6] 1 .According to [7] the quantum treatment of the inflaton gives rise to the emergence of the eternal inflation problem due to "quantum fluctuations" that must be taken into account on top of the classical slow roll down of the inflaton field due to the combined effect of the inflationary potential and the effective friction associated with the rapid cosmic expansion.The argument is essentially the following: consider the classical change in the inflaton field, δ Class ϕ, as it "rolls down the potential" during a characteristic period of cosmic time δt, and compare it to the magnitude of the quantum fluctuations that can be expected during that period δ Quant ϕ.If the former is larger than the later, then the inflaton will evolve towards the potential's minimum, and inflation would eventually end (after a brief reheating period).
However if the latter is larger than the former, and considering the stochastic nature of the latter, one can expect that while in certain regions the overall change would bring the inflaton closer towards the potential minimum, in other regions the net change would be in a direction of an increasing value of the inflaton potential.The crucial point, however, is that regions characterized by a higher value of inflaton potential will undergo greater level of expansion (as a result of being dominated by an larger effective cosmological constant) compared to regions with a smaller potential value.Thus, over time, the universe would end up dominated by regions with larger values of the inflaton potential where the stochastic fluctuation increased the potential.That is a situation where it would be overwhelmingly likely for any given region of the universe to be one where the potential has always increased.Thus, except perhaps for regions of measure that tend to zero, we are faced with a runaway regime of ever increasing inflaton potential and never-ending accelerated expansion (not to be confused with much slower accelerated expansion that our universe is currently undergoing).The detailed analysis that have been done concerning this issue conclude that, quite generally [8], the result is that δ Quant ϕ >> δ Class ϕ, indicating the prevalence of eternal inflation among the inflationary models that have been studied [4,9,10,11,12] 2 .
1 These considerations, however, are subject to debate connected to the absence of a natural measure in General Relativity (GR) that can be used to estimate probabilities within the framework of GR [55]. 2 Eternal inflation has been atributed mainly to two factors.One of them [54] is the extended duration the inflaton can persist in a given region when the net displacement of the inflaton takes it away from the zero of the potential.Another factor [10] is that generically, when the field "rolls backward" the value of potential increases, and so does the expansion rate, and as time passes, regions with a higher expansion rate However, a close examination of the arguments above, unveils that several delicate issues are at play, and that several assumptions have been implicitly employed, assumptions that are not only far from inescapable, but often an problematic and not really appropriate.
Let start by noting that what drives the accelerated expansion during inflaton it is the zero mode of the inflaton field, the only mode that is supposed to be highly excited and sharply peaked, to the point that it is treated in classical terms in most works on inflation. 3The other modes of the field are supposed to be in the vacuum state ( more precisely a suitable adiabatic vacuum state), often taken as the Bunch Davies vacuum state, (in part as a result the extremely rapid cosmic expansion).The potential is then, up to small corrections, essentially determined by the value of the zero mode of the inflaton.On the other hand, as the zero mode is, by definition, invariant under spatial translations, any variation in the zero mode can not result in differences in what takes place in different locations, so the argument that the fluctuations would lead to increases or decreases in the value of the potential in different regions can not possibly refer to anything associated with the zero mode.
Next, let us point out that the identification of quantum fluctuations with stochastic fluctuations is not something that can be justified in the context of standard versions of quantum theory.The ground state of the hydrogen atom does not reflect -according to the textbook versions of quantum mechanics-an electron moving stochastically within the region where the value of wave function is large.That type of picture might arise in some modified versions of the theory but it is certainly not what the Copenhaguen or von Neumann interpretations say.In fact, standard versions of quantum theory contain absolutely no stochastic elements except for those that occur in a association with the reduction postulate.That is, stochasticity makes its only appearance in the theory in connection with the measurement process, for otherwise the evolution is provided by the Schrödinger equation which is completely deterministic regarding the wave function.Moreover as the theory explicitly states that the wave function provides a complete description of the physical situation, there is nothing else to which to associate a stocasticity.It is thus clear there is nothing that can possibly be said to behave stochastically unless one makes use of the come to predominate.
3 See [13] for a an example where the zero mode is given a full quantum treatment.
reduction postulate, which of course brings up the notion of observers or devices, which as stressed, elsewhere are hared to make sense of the cosmological context (See for instance [14]).
This last point in fact takes straight us into the delicate questions surrounding the so called measurement problem in quantum mechanics [15,16], and its extensions into the even more complex realm of quantum field theory [17] in curved space-time [13].It is not the intention of the present work to enter deeply into this discussion, as that has been the object of extensive discussions, and in fact various other works have focused on the specific form the problem takes in the contexts of inflationary cosmology [1,18].Lets just note that as discussed already in [19], the cosmology setting is one where the conceptual problems that besiege quantum theory are exacerbated by the fact that one is dealing with a situation where no measuring apparatuses or observers can be call upon to justify the use of the usual reduction postulate.When considering, in particular, the inflationary account of the emergence of the seeds of cosmic structure, one faces the further obstacle that, the evolution of such structure, leading the formation of galaxies, stars, black holes, etc, is a prerequisite for the emergence of observers and measuring devices.When attempting to face such difficulties, and more generally the measurement problem in quantum theory, one can be helped by the work [15] which helps to classify the viable logical paths open to deal with the measurement problem, as: i) hidden variable theories 4 , ii) spontaneous reduction theories [20,21,22,23], or iii) many worlds type approaches, which includes the bold attempt in [24,25,26], which however seems to suffer from very serious problems [27].The difficulties regarding of inflationary account of the emergence of the primordial inhomogeneities have been the subject of explorations based on all those approaches: The works [28,18] are examples of i), [2,29,30] are representative of ii), while an implicit acceptance of iii) seems to underlay most of the works on the subject that do not explicitly acknowledge the issue.There have been various works focused on the problem (such as [31,32], while trying to avoid taking a definite posture regarding options i) ii) and iii), have been found to suffer from serious shortcomings [33,34]).
The fact is that, having argued that in standard versions of quantum theory there is nothing stochastic unless a measurement is involved, and having noted that in the cosmological situation at hand there is no justification for calling on the reduction postulate, we recognize that something must be added into the picture in order to break the homogeneity and isotropy reflected both in the aspects treated classically, (often just the background) and in vacuum quantum state characterizing the conditions of the degrees of freedom that are given a quantum treatment.In other words, within the inflationary context all modes of the quantum fields involved other than the zero mode of the inflaton are supposed to be in the adiabatic vacuum and such state is completely homogeneous and isotropic, despite the quantum uncertainties for various quantities 5 , so whatever element we add to the theoretical picture, it should be able to account for the breakdown of such symmetries during the subsequent evolution.
Here we will focus on a treatments based on spontaneous collapse, or dynamical reduction theories which are based on the idea of unifying the unitary and deterministic evolution provided by the Schrödinger equation and the indeterministic changes normally associated measurement situations by adding to the former suitable stochastic terms.That modified evolution is taken to be universally valid, requiring no a priori distinction between situations involving measurements and those that do not.When applied to the inflationary cosmology setting it offers a novel account to the breaking the homogeneity and isotropy of the adiabatic vacuum.In that context the evolution of the quantum state is no longer fully deterministic despite the absence of measurements and observers, and in fact, stochastic elements now do appear in the dynamics [1,35], which can help to account for the emergence of the primordial seeds of cosmic structure [1], however, at the same time, those stochastic element in the dynamics resurrect the specter of eternal inflation.
In this work, we will consider a treatment based on the Continuous Spontaneous Localization theory [35], and adapted in a particular manner [2] to the inflationary cosmology setting, and which, offers a successful treatment of the problem including the possibility to account for the an adequate primordial power spectrum of density fluctuations 6 .
5 At this point we might note that just as the adiabatic vacuum state of such quantum fields is characterized by typical uncertainties, so is the ordinary Minkowski vacuum and that such uncertainties in no way affect the complete invariance of the Minkowski vacuum under the full Poincare group. 6We note that the predictions for the power spectrum of tensor modes in this approach differs substantially from the standard accounts [36,37].
The study carried out in [2] considered two scenarios for the choice of the collapse operator 7 , the field itself and its momentum conjugate.The conclusions indicate that in the first case, in order to generate an essentially scale invariant primordial spectrum, the collapse rate λ must, as is also suggested on dimensional grounds, depend on the wavelength of the collapsing mode according to λ = λ(k) = k λ, where k is the mode's co-moving wave number and λ is an universal parameter of order ∼ 10 −5 M pc −1 (which as noted in [2] exhibits favorable comparisons with the range of values typically associated with the collapse rate in the non-relativistic many-particle quantum mechanical framework for which the theory was initially developed to address).
For the second case, and again as suggested by dimensional considerations, it turns out that in order to generate the essentially scale invariant primordial spectrum, the collapse rate must depend on the wavelength of the collapsing mode according to where k is the mode's co-moving wave number and λ is again universal parameter.In fact a related study addressing the issue of eternal inflation in the context of spontaneous collapse theories have been carried out in [38] focusing explicitly in this second scenario.
I the present work we will consider the first scenario, which we find more natural given the lack of pathological behavior of the collapse rate in the vicinity on the zero mode ( i.e. as k → 0).
The conclusions for the case we examine, strongly suggests that the zero mode should not be affected by the spontaneous collapse dynamics, and thus would not suffer from stochastic fluctuations therefore fully evading the eternal inflation problem.Unfortunately things are not so simple, and one must explore the effect of the modes other than the zero mode but having sufficiently long wave length to behave "effectively" as the zero mode.
The detail analysis of that question is one of the main objectives of the present work.

Inflation and cosmic structure
The majority of contemporary cosmological models incorporate an early inflationary epoch characterized by an accelerated expansion, as an integral part.One of its most attractive aspects is it claim to offer a natural explanation for the emergence of the primordial seeds 7 The operator that accompanies the new stochastic terms that the theory adds to the Schrödinger evolution.
of cosmic structure as a result of quantum fluctuations.In this section we present a brief description of the standard accounts of that.
The model is described by the action, with the matter sector characterized by a scalar field ϕ, named the inflaton.This work is focuses on the "chaotic inflation model" associated with the quadratic potential, V (ϕ) = 1 2 m 2 ϕ 2 as a representative example 8 .The background it is given by the flat Friedman-Lemaître-Robertson-Walker (FLRW) space-time, whose line element is expressed in conformal time, η, is: The Einstein field equations are with the energy momentum tensor for the scalar field given by, The relevant Einstein equations are: and the Klein-Gordon equation is, The setting in which the process of interest takes place is taken to be that of the so call "slow-roll regime", corresponding to the well known slow-roll condition, where H(η) = a(η)H I and H I = 8πGV 3 .Combining (7) with (6) one obtains; Recall that to a good approximation we can take a(η) = −1 H I η , and set the inflationary period to correspond to −τ < η < η f (where inflation takes place, begins and ends) and η f < 0.
The above expression for the scale factor taken H I as a constant, is only an approximation.
Alternatively we can take H I to have a small time dependence due to the slow but nonvanishing changes in V as inflation proceeds.As is often done, we take the expression for a(η) to be a(η) = −1 , the factor is written in terms of slow roll parameter and In the standard treatment employed to account for the emergence of the first cosmic seeds structure one considers perturbations for both, the inflaton field ϕ = ϕ 0 + δϕ(η, ⃗ x), and takes the metric which (using a specific gauge and ignoring vectorial and tensorial perturbations) to have the form, It is customary to define a new variable linking ψ(η, ⃗ x) with δϕ(η, ⃗ x), Then, one proceeds to quantize this variable constructing the Fock-Hilbert space and to write the corresponding field in terms of creation and anihilation operators The modes are chosen in such a way that, at η → −∞ behave as the positive frequency standard modes in Minkowski space-time.This corresponds to the selection of a particular vacuum state |0⟩, via the condition â⃗ k |0⟩ = 0, namely the "Bunch Davies" state in de-Sitter space-time and to the so called "adiabatic vacuum" in the case of our slow rolling inflationary model.
One is interested in the characterization primordial seeds of structure as given by the relative density fluctuation during inflationary regime δ(η, ⃗ x) , where ρ(η) is the spatial average of the universe's density, ρ(η, ⃗ x), δρ(η, ⃗ x) ≡ ρ(η, ⃗ x) − ρ(η).The power spectrum characterizing those primordial inhomogeneities is taken to be characterized statistically by the correlation: where the quantity of interest is P δ (η, k) known as the power spectrum of density fluctuations.
The analysis proceeds by taken the latter as represented the two point function: Thus from the last expression the power spectrum, is extracted through: The result is essentially P (k) = Ck −3 (up to a small correction associated with the slow roll parameter ϵ which we will ignore henceforth).In other words one argues that the "quantum fluctuations" seed the primordial inhomogeneites and anisotropies that eventually evolve into all cosmic structure present in our universe.This result is considered as one of the biggest successes of the inflationary model.
At this point it is worthwhile emphasizing that the delicate issue of the meaning of "quantum fluctuations" in this context.Before doing so, it is important to consider the differences between the three following notions: Quantum fluctuations: This type of fluctuations usually describe indeterminacies of some operator when the system in question is in a given quantum state.An example is provided by a particle that does not have a definite momentum or position, such as, say, a harmonic oscillator in its ground state.
One must note that in the previous treatment the last three notions are taken as synonymous.In particular many cosmologists treat "quantum fluctuations", quite naturally as if they spacetime or statistical fluctuations.It is well known in quantum theory that quantum uncertainties play a significant role.In particular they characterize the dispersion of outcomes observed when measuring an observable on identical systems.According to the standard theory when a measurement is performed, the state of the system transitions from the state that in general does not have a well-defined value of the observable in question to a new state that possesses a well-defined value (or at least a better-defined value, if the measurement is not a perfectly accurate one).
However as noted in above (and amply discussed elsewhere) quantum fluctuations describe only indeterminacies and lack statistical behaviour.Thus, the analysis presented above, without some additional input is simply unjustified as an account of the emergence of structure.In particular it contains no element accounting for the beak down of the complete homogeneity and isotropy of the situation, because vacuum state possesses those symmetries and there is nothing in the account that would justify arguing that those are broken.Inspired in this problem, in reference [2,40] (following various previous works based on simplistic versions incorporating some kind of spontaneous state reduction [1]) consider a modified approach with an specific proposal: incorporate continuous spontaneous localization, a modified quantum theory, designed to solve the measurement problem in ordinary quantum mechanics, and adapt it to the context of semi-classical gravity.The following section describes the relevant results of [2], the work on which the subsequent analysis of the eternal inflation problem will be based.
3 Seeds of cosmic structure and collapse theories

Considerations about the treatment to be adopted
Here we will offer a brief review of the work in [2] presents an approach to account for the generation of primordial anisotropies and inhomogeneities during the infationary period that is explicit and clear regarding the mechanism by which the symmetries of the vacuum state are broken.Before discussing that it its worthwhile to touch on another delicate aspect in the standard treatment in order to clarify the steps taken in this work.Specifically, we address the challenge of working at the interface between quantum theory and gravitation, which is further complicated by the absence of a fully workable theory of quantum gravity (although certainly substantial efforts have been made in number of approaches to the subject).
The point is that while it is common to think that the difficult questions involving quantum gravity refer exclusively the Plank regime, problematic issues emerge in a much broader set of circumstances.One aspect of the problem can already be seen to occur in simple situations such as that discussed in [41] where an actual experiment involving a gravitating body that is arranged to go into macroscopic superposition of two different positions, and employed arguable to illustrate the invariability of the semi-classical General Relativity (GR).More specifically the claim is [42] that 1) If there are no quantum collapses, then semi-classical GR conflicts with the results of their experiment.2) If there are quantum collapses, or reductions of the wave function, then semi-classical GR equations are internally inconsistent.The last point refers to the fact that the left side of Einstein's equation, G µν = 8π ⟨ Tµν ⟩, is divergence free as a result of Bianchi's identities while that the expectation value of the energy momentum tensor, will not be divergence free if a collapse takes place.However, as we noted we do not consider that we have a really solid alternative and thus we are inclined to use the semi-classical GR and consider it, not as a complete and consistent theory on its own , but as an approximated description with limited domain of applicability.On the other hand we must acknowledge that other approaches have been used to incorporate the collapse theories in considering inflationary cosmology [56] [57]9 .
For truly fundamental characterization, we think a full theory of quantum gravity would be required.That is we take the view that a theory with spontaneous quantum collapses not only offers a viable resolution of measurement problem but that it can provide an approximate effective description of situations involving gravitation quantum matter, despite the fact that during the collapse events the equations can not be valid and require a delicate handling.
Therefore, we will consider semi-classical gravity as an analogous to say the Navier Stokes equations in hydrodynamics which although clearly not fundamental equations failing to reflect in full the atomic molecular nature of fluids, often provides a rather good characterization of their effective behavior.Such descriptions will often fail when the behavior of fundamental degrees of freedom of the fluid become sufficiently excited for the approximations involved in the hydrodynamic description cease to be valid (say at the onset of turbulence).Similarly we assume that semi-classical GR equations are valid before and after a spontaneous collapse, but not at the time in which it is occurring.A formalism describing the treatment of that kind of situation have been presented in [13] and further developed in other works [2,43] .The essence of that formalism relies on precise descriptions for the situations just before and just after a collapse, which are subject "gluing procedure" inspired on the treatment of [44].The precise characterization of the situations before and after a collapse are to be given in terms of Semi-classical Self-consistent Configuration (SSC), defined as follows: The set It is said to represents a SSC iff φ(x), π(x) correspond to suitable field and momentum conjugate operators acting on the Hilbert space H for a quantum field theory in the spacetime with metric g µν (x), and where |ξ⟩ is a state in the Hilbert space that satisfies : for each point x in the space-time.
Building an SSC is not trivial it is necessary give an ansatz, one must propose the appropriate metric of space-time, build quantum theory, find an appropriate state, |ξ⟩ ∈ H, that is compatible with the selected configuration of space time.
Using the SSC approach as shown in [13] one can explicitly describe the the transition from an early homogeneous and isotropic stage of the universe one that is inhomogeneous and anisotropic.The additional element is provided by the spontaneous and stochastic quantum collapse of the wave function of the matter fields.
The proposal, by incorporating the CSL theories, is to be able to explain this transition considering normal unit evolution, characteristic of standard quantum field theory but supplemented by quantum collapse, which has been suggested can somehow predict the gravitational degrees of freedom.
In this work they consider that any state can be understood as a particular SSC, which is why the transition from one state to another, mediated by collapse, can be read as Inspired by this work [13], which explains in a general manner how one evolves from one hypersurface to another through a collapse, the following scheme [2] is employed to account for the formation of primordial structure.
It is worth mentioning that recent substantial progress haas been achieved in putting those ides in firmer mathematical ground [45,46,43].

A revised account of the emergence the seeds of cosmic structure
The work [2] presents a consistent treatment to describe the emergence of the seeds of cosmic structure using an approach that incorporates semi-classical gravity and the CSL version of spontaneous collapse theories.Most of this section is a recount of that work with small adaptations required for our purposes.The staring point is the usual description in the inflation regime with its basic dynamics given by the action: leading to the field equations with T ab , Prior to conducting the analysis proceeds by separating the metric and scalar field into a spatially homogeneous-isotropic background and the perturbation part describing the possible departures from the exactly symmetric situation, where again one is working in the Newtonian gauge and the vector and tonsorial perturbations have been ignored.The scalar field is, (in following with the semi-classical spirit in which the space-time metric is given a classical treatment and the matter fields a full quantum one) treated in principle in quantum terms at the level of both background and perturbations.We however separate field into the space independent, or zero mode ϕ 0 , and the rest : where the δϕ(⃗ x, η) represents what is normally considered in terms of the inflaton field perturbation which is the only part that is usually given a quantum treatment.Expanding the action up to the second order in the scalar field "field perturbations" we obtain after making a convenient change of variable y ≡ aϕ, From this we obtain the corresponding Hamiltonian, where we have suppressed the implicit dependence on η.In fact it is convenient to work in the Shröedinger picture and focus on the behavior of the relevant expectation values of Fourier decomposition of field and momentum operators.In principle we should separate each of these into symmetric and anti-symmetric parts in order to work with bonna-fide hermitian operators, but as shown in [2], the results obtained in a simplistic analysis that avoids that step are the same as those found using more explicitly rigorous one.Then we write the field and conjugate momentum operators as: The commutation relations are then, As noted, in order to give a quantum treatment and work with hermitian operators the description is made in terms of symmetric and anti-symmetric components of the field and momentum finding a simple collection of independent modified-harmonic oscillators.We focus our attention on one specific Fourier mode k (and treat all in a similar manner).The operators that will turn out to be convenient are introduced: where d 3 k represents an "infinitesimal" volume in the space of the k's around ⃗ k (i.e. correspond to (2π/L) 3 if we "put the universe in a box" of size L and impose periodic boundary conditions).Then, ( 28) and ( 29) satisfy the standard discrete commutation rules [ Xk , Pk ] = iδ k,k ′ .The hamiltonian for the mode k is: In the treatment done above, the zero mode of the field is not included, because we will see it, in the approach we use it will naturally be unaffected by the modified for which a more detailed analysis will be given in the following sections.
And noted we will focus the analysis on a single mode k so in the reminder of the paper the sub-index k will be dropped so, for instance, Xk and Pk will be denoted by X and P respectively.

Quantum collapse free scenario
Before considering what happens once the CSL modifications are incorporated, one consider the behaviour of the relevant expectation values when only the standard unitary evolution is considered (i.e the CSL modifications are turned off).The Hamiltonian of the system is H k , but as we are working in the Schrödinger picture we have: in the following, we will be denote ⟨A⟩ = ⟨ψ, η| Â |ψ, η⟩ where the wave function for each mode, is that corresponding to the Bunch-Davies vacuum, which is just the harmonic oscillator ground state, at the start of inflation (η = −τ ) one has: Note that we have used lower case (x, p) to denote eigenvalues and corresponding eigenvectors associated with the operators X a P .
For the field operator X and the field momentum operator, P , the equations of motion are: and their solutions: For the equations of the second degree, the following change of variables is made: and it follows that, whose solutions are the boundary conditions, i.e, η = −τ , are used to evaluate the coefficients C 1 , C 2 , C 3 : when * denotes the complex conjugate.If one replaces the last in expression (37) one obtains, the last result, will result useful in the following section.

Quantum collapse scenario
Now, as in [2] we consider an adaptation to the context of the continuous spontaneous localization theory, CSL, which takes as the collapse operator Â with a collapse rate λ.The dynamic of this theory it is given by two equations, the first is a modified of Shrödinger equation, whose solution is with T the time-ordering operator, w(t) a random classical function of time and the second equation is its the probability of realization of a stochastic function in the band specified by the central values w(t i ) and widths dw(t i ) 10 , and is given by It is useful to possess a concise formulation for the density matrix that accurately represents the collective evolution within the ensemble.The evolution equation for the density matrix is given by the Lindblad equation: Consequently, the ensemble expectation value of the operator ⟨ Ô⟩ = T r Ôρ(t) satisfies the following expression: As mentioned before, the modification introduced by CSL dynamics tends to collapse state vector toward eigenstates of Â and it gives an ensemble of different evolutions of the state vector, each characterized by a particular realization of the stochastic function w(t).
In [2] CSL is to be applied to the modes described by the focus operators P , X, and the Hamiltonian, According with the ec.(66) of the [2] one obtains the following expression: The resulting evolution from the initial state has the form ⟨p|ψ, η⟩ = e [−A(η)p 2 +B(η)p+C(η)] , and is used in [2] with the aim is to find the spectrum, for that reason ⟨P ⟩ 2 is sought.They 10 For more details see the exposition in [20].
compute ⟨P 2 ⟩ and in turn the coefficients A, B and C. Finally, using the probabilistic rule and the initial conditions it is found that the ensemble average is : From this result and using expression (46), the equations ( 36) are examined in two cases: when the field acts (namely X) as collapse operator and when the conjugate momentum of the field (namely X) plays that role .In this way, two results are found for the functional form of k that would generate the required scale invariant power spectrum: for the first case λ = λk and λ = λ/k for the second case.Afterwards, using these results, the authors investigate the overall scale power spectrum and find the typical parameters that reproduce the observations from the CMB.

Auxiliary calculations
Since in this work we will focus in the case in which the quantum filed itself acts as collapse operator , it is useful to compute the Fourier transformation of the expression for the wave function (of the generic mode under consideration).
⟨x|ψ, η⟩ =N dp ⟨x|p⟩ ⟨p|ψ, η⟩ After rather direct calculation one finds which has the same functional form as the original expression ⟨p|ψ, η⟩, so one can write and identify the corresponding coefficients: Thus, one can use the calculations in [2] to obtain the analogous to expression for and in this case, correspond to: that in terms of A it turns into: For convenience we will use the first expression, i.e., in A ′ terms.Using the modified Schrödinger equation it is possible obtain the coefficients A ′ (η), B ′ (η), C ′ (η) and then calculate the expectation values of (36), keep in mind the previous change of variables (35), such expectation values could be obtained from (46).For the case in which Â = X it is obtain we get the coefficient A: being β = √ k 2 − 2iλ, and C is determined by the initial condition With the above mentioned it is possible to write the expectations values under CSL theory.
One complication we face is that while in the original work [2] interest centered on the behavior of the relevant quantities only at the end of inflation , for our purposes we need to the behavior relevant quantities throughout the inflationary regime.The second order equations (36) are unchanged except for the R, and the general solution is therefore the sum of previous homogeneous equations added an inhomogeneous solution part: The corresponding system is In the same way that we proceed in the collapse-free case we use the initial condition to find the coefficients C1 , C2 and C3 : now substituting the above expressions one finds the following spectrum: for the case λ = 0 the expression vanishes.The requirement for this spectrum to be scale invariant is that the dominant term goes as with λ ≈ 10 −5 M pc −1 .

Useful estimations
We proceed to review the magnitude estimates that will justify the approximations that will be made in this work.We start by saying that temperature fluctuations go like Considering the dominant term, we have where I represents a modes sum k, I = 10 2 10 −3 dk/k ≈ 11.5, the integration limits represent the observed range for k, 10 As discussed in [1], the effect on the reheating period is given by: covered by regions where there was more expansion.Thus, the inflationary potential grows and inflation never ends.
As we argued at the start of this work, there are flaws in this story that, however reappear now in a fully justified fashion as a result of the incorporation of a truly stochastic aspect to the dynamics of the quantum field, which we did in order to account for the emergence of cosmic seeds.That is, the problem re-emerges as a result of the inclusion of spontaneous collapse theories.

A strategy to address the problem
The inclusion into the inflationary account of the behaviour of the early universe of a spontaneous collapse theory brings in both, an explanation for the breaking of homogeneity and isotropy of the vacuum state of the inflaton field, a justification for the stochasticity which otherwise seems unfounded, and the potential to resurrect the problem of eternal inflation.However it also brings into the dynamics new elements which might be adjusted so as to deal with the last problem.We will thus consider a particular treatment of that kind together with the lessons learned regarding the empirical adequacy of the approach regarding the predictions for the power spectrum of scalar perturbations reflected both in the CMB and the studies of BAO [47,48], and study possible ways to make all that compatible with having no eternal inflation.
As we mentioned before in order to specify the version of the adaptation of that kind of theories to the situation involving relativistic quantum fields and gravitation, one must among other specify the quantity that one assumes plays the of the collapse operator 12 .In the work [2] two proposals are studied : one in which the collapse operator role is taken by momentum conjugate to the field, and a second in which it is the quantum filed itself that plays that role.In the analysis it is found that in order to generate a primordial spectrum of stochastic density fluctuations that is essentially scale invariant (as required for basic adjustment to empirical data) the collapse rate must depend on the wave number k of the mode.In fact what is required in each case is λ ∼ λ/k and λ ∼ λk, respectively.
The problem of eternal inflation using collapse theories and semi-classical gravity has been address before in [38], considering the case momentum as the collapse operator of the theory.Here we will focus on the case where the quantum filed filed plays the role of collapse operator.We find that proposal more attractive because taking λ = λk the collapse rate for zero mode vanishes, thus eliminating from the onset the collapses that could generate the eternal inflation problem.However one further consideration one notes that modes that have wavelengths that are too large to be associated with causal effects during inflation, would have to be considered as representing "effectively zero modes".That is, those would be the modes with wavelengths that are larger than the particle horizon.In most treatments of the issue of eternal inflation, one focuses on what takes place withing individual Hubble volumes , and thus, the magnitude of the quantum fluctuation that is compared with the classical displacement is estimated by focusing on the modes with wavelength that is larger than the Hubble radius (leading the "quantum fluctuation" as ⟨ϕ 2 ⟩ 1/2 ∼ H 2π (see [62] for an explicit calculation of a similar form).We do not consider that this would be the appropriate criterion to be employ in our analysis, as we have emphasized that the quantity driving inflation is the zero mode of the inflaton field, which is by definition homogeneous.Therefore any type of fluctuation (or indeterminacy) pertaining to it, would represent something affecting the whole universe, and not just one Hubble volume within it.However on further consideration one notes that modes that have wavelengths that are too large to be associated with causal effects during inflation, would have to be considered as representing "effectively zero modes".That is, it is modes with wavelengths that are larger than the particle horizon, (which is what determines causal contact), that must be considered as "effectively zero modes".The Hubble radius is a good estimate of regions in which a cosmic fluid could be expected to be in thermal equilibrium, but not necessarily as characterizing the regions over which quantum entanglement and correlations exists.The particle horizon is what determines causal contact, thus determines which modes with must be considered as "effectively zero modes".We thus analyze their effect and ascertain under what conditions they lead the eternal inflation problem and how is it possible to evade it. 13.
The size of particle horizon in the context of interest is: where a(t) = C exp H I t.Thus: The modes that could give rise to the eternal inflation problem are then those for which : This implies that, taking as initial condition, a(τ ) = 1 H I τ , we have The next step is estimate the magnitude of the cumulative stochastic displacement of the expected value of the field associated with the collapse of these modes, (79) Note that we have: ⟨X⟩ 2 λ=0 = 0, which is as expected.Namely if there is no collapse the quantum expectation value of the harmonic oscillator continues to vanishes at all times if it (and that of its momentum conjugate) vanished initially.Now we expand ⟨X⟩ 2 in Taylor series, up to the first order in λ , and to the eight order in k and obtain Using that δϕ = a −1 y and the expression for the Fourier mode of this field ỹ⃗ k = X( ⃗ k) (recalling we have employed the variable X⃗ k = √ d 3 k X( ⃗ k) where d 3 k represents an "infinitesimal " volume in the space of the k's around ⃗ k).We can now proceed to make the desired estimate.
However before doing so, and anticipating that the direct extrapolation of the simple for of the k dependence of λ encountered in the previous studies might not be adequate for the task at hand, we consider a slight generalization In fact as argued in previous works works one might expect that the current versions of the collapse theory are just effective versions of a more fundamental theory, it is likely that what we take as the collapse parameter λ might be an effective characterization of a quantity that depends on various other quantities including space-time curvature.This suggests that the collapse rate could exhibit variation depending on the specific context at hand.In fact it was already noted in [2] in order to avoid the divergences of the stochastic fluctuations of field itself something like, going beyond the simple dependence already mentioned that seems to be required.Thus we consider of form of collapse rate that, adjust to what is needed to account for the essentially flat power spectrum consistent with CMB and BAO data with some added flexibility at very low k so as deal with the issue of eternal inflation.Since we are considering that X acts as the collapse operator we need λ ∼ λk, so we take a slightly more general form: where b has the same units of k.Note that for α = 0 we recover λ = λk.The last expression also reduces to λ = λk in the limit k ≫ b, so we will recover an empirically adequate power spectrum provided k ≫ b for all k corresponding to the modes relevant for the CMB and BAO data.On the other hand we expect that for the modes that could play a role in eternal inflation b ≫ k so that for those we have, We assume that the displacements in the "discrete " variables X⃗ k are statistically independent (since the stochastic processes of CSL dynamics are assumed independent for each mode), so: Replacing in last expression the result (80) we find : Evaluating the integral we find: Using (84) we obtain, where we used a(η) = −1/H I η.We define the quantum displacement ∆Q = ⟨δϕ⟩ 2 an compared with the classical displacement over a Hubble time, arising from slow-roll regime (7), ∆C ≡ d dη ϕ 0 η Hub which can be expressed as: where η Hub = ηe has been used14 .Thus the condition for avoiding eternal inflation is: and equivalently : Using (85) in the last expression and rearranging terms, the condition takes the form: (4π)

Conclusions
In this work, we have analyzed the aspects that could contribute to the re-emergence of the problem of eternal inflation in the context in which a spontaneous collapse theory has been added to the standard inflationary account of the emergence of the primordial seeds of cosmic structure .We began by recognizing that within the framework of the standard theory, this problem is no present in the usually stated form once one recognizes that the zero mode of the field is independent of position.Furthermore, once one notes that quantum fluctuations can not justifiably be identified with statistical fluctuations, one again concludes that the problem does not exist, however that will also invalidate the standard account for the origin of cosmic.Based on a study that incorporates a new element accounting for the breaking of homogeneity and isotropy [2], we center our study around this aspect.In other words, if one seeks a consistent treatment to generate the primordial structure, the price to be paid is the reappearance of the problem of eternal inflation.Once again, focusing on one of the cases studied in [2], we note that the collapse rate is proportional to the wave number of mode of the field, and therefore, there would be no collapse for the zero mode of the field, seemingly resolving the problem outright.However, there are other modes to consider, those whose wavelength are close enough to to the zero mode.We identified the modes that have a physical wavelength greater than the particle horizon as those that could have an effect on the eternal inflation problem.We examined the effect of these modes utilizing the proposed collapse theory while considering a slightly modified rate that takes into account both the observable modes and those that could give rise to eternal inflation.At first glance, modifying this rate may appear unjustified.However, as previously mentioned, previous studies [49,52] provide clues that the collapse rate could be an effective parameter emergent from a more fundamental theory and effectively dependent on curvature, suggesting that it could vary from context to context where the collapse theory is applied.We find a condition on the parameters characterizing the the modified collapse rate, for which the eternal inflation problem does not arise.It is worthwhile emphasizing that although the present analysis concerns a very specific version of a spontaneous collapse theory and a rather add hoc parametrization of the collapse rate, the central point is that the search for a satisfactory account for emergence of the primordial seeds of cosmic structure through the inclusion of that type of modification of the standard versions of quantum theory, which in turn, are motivated by the quest for a consistent treatment of the measurement problem, have the potential to offer unexpected additional advantages.In this case it seemed that the implicit stochasticity in the dynamics of theory would lead back to the problem of eternal inflation, but as a result of the flexibility provided by the unknown exact form of the collapse theory, adjustments can be made to remove the problem.The situation is reminiscent of similar situations found in the works [52,49,53,43].
Space-time fluctuations: This kind of fluctuations describe a unique and wide object on which a locally defined quantity changes from one point to another, an example, the temperature on each part of fluid at a definite time.Statistical fluctuations: Those are associated to an ensemble, each element of which having definite value of the quantity of interest.This fluctuations describes the variations of such value among the various elements of the ensemble.

2 ϕ 2 0 M 4 p≈
expression and using that H I = (8πm 2 ϕ 2 0 /2M 2 p ) 1/2 and the estimates: M p ≈ 1019 GeV , m = ϕ 0 ≈ 10 15 GeV , so m 10 −16 , λτ ≈ 10 3 , τ η ≈ 10 30 we obtain the condition: , as anticipated the simple case where α = 0 does not avoid the eternal inflation problem.In other words, the above is the condition that avoids the eternal inflation problem.We must then look for α and b such that b < k for k in the range, 10 −3 M pc −1 ≤ k ≤ 10 2 M pc −1 [2].There values of (α, b) with b ≤ 10 −3 M pc −1 .Here we present a plot showing an interesting region in which the eternal inflation problem does not arise:

Figure 1 :
Figure 1: In the last plot, the units are in M pc −1 , the yellow region represents where there is NO eternal inflation problem.The α scale was chosen arbitrarily while b was selected to clearly comply with CMB and BAO bounds.