The Trans-Planckian Censorship Conjecture in different frameworks of viable inflation

We review the recently proposed Trans-Planckian Censorship Conjecture that stems from the Trans-Planckian problem of cosmological perturbations. We analyze the implications and constraints that TCC introduces in different frameworks of viable inflation. We revisit the case of slow-roll scalar field inflation and we investigate the cases of slow-roll $f(R)$ and $f(R,\phi)$-gravity. Finally, we consider the conjecture in the context of constant-roll scalar field inflation.


Introduction
The inflationary paradigm, according to which the universe underwent a brief period of earlytime rapid expansion, was initially introduced several years ago by Starobinsky [1,2], Guth [3] and Sato [4] and later by Linde [5], Albrech and Steinhardt [6]; in the last decades several theories have been suggested in order to describe inflation (see Ref. [7] for a nice review). Despite the fact that the arena of inflationary models is quite large, the huge amount of observational data [8,9] can be used to discriminate between them the viable ones. In particular, inflation provides a causal mechanism to generate the primordial inhomogeneities across the matter distribution in our universe, which evolve and persist in the universe today and which are object of cosmological observations. Fluctuations in both matter and gravitational waves are believed to have a quantum mechanical origin in terms of vacuum perturbations which originate inside the Hubble radius (or horizon) at the beginning of inflation. During inflation, when they cross the Hubble horizon, they become classical and later re-enter the horizon [1,10]. The study of these perturbations can be carried out by making use of field-theory computations without invoking any trans-Planckian physics. However, inflationary cosmology generally suffers from the so called 'trans-Planckian problem', which appears if the macroscopic fluctuations which cross the Hubble horizon trace back to trans-Planckian wavelengths at very early times. Since these fluctuations would contribute to the power spectrum, their computation involves low energy physics into regions where this physics is not applicable, which is clearly not desirable [11,12,13,14,15], unless one admits the possibility that trans-Planckian effects manifest themselves in the form of ultra-high energy particles at any point in time [16,17]. Recently, Bedroya and Vafa have proposed an alternative viewpoint that avoids the Trans-Planckian problem [18]. Their work is motivated by string theory, and is connected to the broader Swampland scenario, which encodes the low-energy effective field theories of gravity that are not compatible with (super)string theory [19,20]. In this respect different Swampland conditions have been formulated during the years, such as the de Sitter Conjecture [21] and the Distance Conjecture [22], limiting the number of theories that admit an ultraviolet completion (or that belong to the "string landscape"). Some examples of constraints emerging from the Swampland criteria can be found in Refs. [23,24].
The Trans-Planckian Censorship Conjecture (TCC) proposed by Bedroya and Vafa in the seminal paper [18] states that "a field theory consistent with a quantum theory of gravity does not lead to a cosmological expansion where any perturbation with length scale greater than the Hubble radius traces back to trans-Planckian scales at an earlier time". In other words, the TCC forbids Planck-scale perturbations to ever cross the Hubble horizon and enter the power spectrum. This statement can be formulated in the following mathematical form (in reduced Planck units), where a(t) and H ≡ H(t) =ȧ(t)/a(t) are the scale factor of the universe and the Hubble parameter, respectively, at a generic time t, the dot being the derivative with respect to time, and a(t0) is the scale factor of the universe at the early-time t0, when quantum fluctuations take place. The Planck length l P l is related to the reduced Planck Mass M P l as l P l = 1/M P l . As a consequence, when the TCC holds true, the length scales which exit the Hubble horizon preserve a wavelength bigger than the Planck length back into the past and the trans-Planckian quantum fluctuations remain quantum.
As an immediate consequence of the TCC we have that H(t) < M P l in an expanding universe. Moreover, if the expansion is decelerated only, the TCC is never violated due to the fact thatȧ Therefore, a possible violation of the TCC takes place if there is an accelerating expansion somewhere along the way.
As it is well known, the expansion of our universe today is accelerating and the so called 'dark energy' epoch is well described by the Cosmological Constant which implies a constant Hubble parameter (de Sitter space-time). Thus, if we assume the validity of the TCC we must accept that the de Sitter expansion cannot continue for an infinite amount of time and that there should be an upper bound for the lifetime of the universe (in Ref. [18] it is estimated as ∼ 2.4 trillion years). However, if the implications of the TCC on late-time universe are purely speculative, it is clear that they are extremely strong for inflation, which describes the early-time cosmic acceleration. In Ref. [25] it has been found that by assuming the TCC and in order to obtain a successful inflationary scenario for structure formation of galaxies, the energy scale of inflation has to be lower than 109 GeV. Moreover, for slow-roll inflationary scalar field models, a negligible amplitude of primordial gravitational waves is predicted with a severe fine-tuning of initial conditions.
In this paper we would like to generalize these studies. We will investigate the impact of the TCC on different models and different frameworks of viable inflation. By 'viable' we mean in agreement with cosmological observations, which constrain the values of the power spectrum of primordial fluctuations, the spectral index of scalar perturbations and the tensorto-scalar power spectra ratio. Our aim is to analyze if and under which conditions inflation free of trans-Planckian problem can be realized by starting from the TCC. In Sec. 2 we will revisit the consequences of the TCC in the classical picture of slow-roll scalar field inflation. In Secs. 3 and 4 we will consider the cases of f (R)-gravity and f (R, φ)-gravity, respectively. In Sec. 5 we will study the constant-roll scalar field inflationary scenario. Conclusions and final remarks are given in Sec. 6.
In our convention, the speed of light and the reduced Planck constant are c = = 1.

Scalar field slow-roll inflation
Let us start by considering a scalar field theory whose action is given by, where g is the determinant of the metric tensor and V (φ) is the potential of the scalar field φ. The metric of a flat Friedmann-Robertson-Walker (FRW) space-time is given by, Thus, the first Friedmann equation results to be, with the associated field conservation law, The dynamic of slow-roll inflation is described by the slow-roll parameters, which should be small during inflation. Thus, the scale-invariant power spectrum of primordial fluctuations when their wavelength amplitudes are equal to the horizon size reads, where k is the wavenumber of perturbation. Let us introduce the e-folds parameter, where te is the time when inflation ends. In order to solve the problem of the initial conditions of our Friedmann universe the perturbations must cross the horizon at N ∼ 55 − 65 before the inflation ends. Thus, N = 55 − 65 is the minimum expansion rate required for viable inflation. According with the inhomogeneities observed in our universe, P ∼ 10 −9 , while the Planck data [8] constrain the spectral index of scalar perturbations ns and the tensor-to-scalar power spectra ratio r as ns = 0.9649 ± 0.0042 at 68% CL and r < 0.06 at 95% CL. These quantities are given by (in first order approximation), and must be evaluated at N = 55 − 65. Now we will see how the TCC (1) introduces an upper bound for the Hubble parameter. We make use of the effective Equation of State (EoS) parameter, The slow-roll approximation ǫ1 , |ǫ2| ≪ 1 in Eqs. (5)-(6) leads to, such that where we used the fact that d/dt = −Hd/dN and made explicit the dependence of ω eff on N . As a consequence we obtain with N = 55 − 65. From (8) together with (1) we get, where we assumed P ∼ 10 −9 and where ǫ1 must be evaluated at N = 55 − 65. Here, N is the total e-folds from the beginning of inflation and in order to satisfy the TCC we minimized the Hubble horizon 1/H(t) in Eq. (1). Note that in any case the Hubble parameter should be almost a constant all through the inflation. Moreover, in order to check whether we meet the TCC condition, by taking into account that in slow-roll inflation the ǫ1 slow-roll parameter decreases with the e-folds number, we will take N as the e-folds when perturbations cross the horizon and we will pose N = 60. Thus, we arrive to the following inequality, which is our starting point to analyze viable scalar field inflation in terms of the effective EoS parameter. We will use a reconstructive approach following Refs. [26,27].
Inflation corresponds to a (quasi) de Sitter space-time, when the effective EoS parameter can be taken close but not equal to the value of minus one. Since we need an exit from inflation we also must require ω eff > −1 (quintessence inflation), due to the fact that, if ω eff passes through the value of minus one, the corresponding (exact) de Sitter space-time becomes a final attractor of the system and inflation never ends. Furthermore, we need ω eff to approach −1/3 in order eventually end acceleration. A reasonable ansatz for the EoS parameter in terms of the e-folds number is given by (see Ref. [26]), where α and β are positive numbers. For large values of N we have ω eff ≃ −1, while acceleration ends when N → 0. As a consequence, the spectral index and the tensor-to-scalar spectra ratio (14) are derived as, Since we are considering N = 60, the spectral index ns satisfies the Planck constraint only if α = 1 or α = 2, but in the first case β should be β ≃ 1/3 and the tensor-to-scalar ratio is ruled out by observations. Scalar field equation with effective EoS parameter in the form of (17) with α = 1 corresponds to power-law potentials [27] and the choice β = 1/3 leads to a quadratic potential, whose viability fell down due to the incompatibility with the observed tensor-to-scalar spectra ratio. Thus, we will focus on the case α = 2, namely in order to have which are in general in agreement with the Planck data. The TCC condition (16) reads, and we find the following upper bound on the parameter β, This result brings to r < 10 −44 , confirming the severe fine-tuning of initial conditions found in Ref. [25]. However, we can explicitly reconstruct a viable model which is compatible with the TCC predicting a strong suppression of the amplitude of primordial gravitational waves. As a matter of fact, the EoS parameter (19) corresponds to an exponential potential, as we can easily verify. By using the prime index to denote the derivative with respect to the e-folds number, in slow-roll approximation (12) we derive, such that by using (19) we obtain a differential equation for H whose solution reads Here, ρ0 is an integration constant whose physical meaning is the effective energy density of the universe at the beginning of inflation, when N is quite large. Now, by equaling 3H 2 M 2 P l to V (φ) we obtain, in slow-roll approximation, and together with Eq. (11) we geṫ where we assumedφ > 0 during inflation, when V (φ) is smaller and close to the initial (effective) energy density ρ0. Now, by making use of the second equation in (12) we are able to reconstruct the full form of the scalar field potential as where c1 is a positive dimensional integration constant and we are taking φ < 0.
In terms of the cosmological time the explicit solutions H ≡ H(t) and φ ≡ φ(t) of (12) are given by, where te is approximately the time when inflation ends and βM P l /(c1 √ ρ0) ≪ te such that |φ|/M P l ≫ 1 at the beginning of inflation. The e-folds N ≡ N (t) is given by, and the total amount of inflation results to be Finally, the slow-roll parameters (7) in terms of the cosmological time read As we observed above, when N = 60, the TCC condition is satisfied for β < 3 × 10 −42 . It means that the Hubble parameter is a constant during almost all the early-time acceleration and only at the very end of inflation goes to zero. As a last remark we note that the model with c1 = 2 and β = 1/2 corresponds to a scalar field inflation of the Starobinsky model in the Einstein frame [27], which clearly violates the TCC condition. In the next sections, we will investigate the TCC in inflationary modified gravity theories frameworks.

The case of f (R)-gravity
A different approach to inflation is given by the modified theories of gravity, where the gravitational Lagrangian is described as a general function of some curvature invariants. Generally speaking, one expects that at the early time some corrections to Hilbert-Einstein action arise, maybe related to quantum effects at high curvature [28,29,30]. In this Section we would like to analyze the simplest class of such models, namely f (R)-gravity, where the Lagrangian depends on the Ricci scalar only [31,32,33,34,35,36].
Let us consider the gravitational action, where f (R) is a function of the Ricci scalar R. The first Friedmann-like equation is given by, In the framework of f (R)-gravity slow-roll inflation is described by the slow-roll parameters [37,38,39], whose magnitude is assumed to be small during inflation 1 . We note that at the first order approximation the ǫ3 slow-roll parameter coincides with the opposite value of the ǫ1 slow-roll parameter and in the following expressions for the power spectrum and the spectral index we will pose ǫ3 ≃ −ǫ1 ≃Ḣ/H 2 . However, the tensor-to-scalar spectra ratio must be evaluated at the second leading order of ǫ1 + ǫ3 ≃ (Ḣ/H 2 )ǫ4, which implicitly defines ǫ3.
The power spectrum of cosmological perturbations results to be [31] while the spectral index and the tensor-to-scalar power spectra scalar ratio read (in the first and second order approximations), As well as in the previous case, it is convenient to introduce an effective equation of state parameter as in Eq. (24). In terms of the e-folds we get, with N = 55 − 65. Thus, the TCC condition holds true if Since we are interested in the sufficient condition to meet the TCC condition, we posed again the total e-folds from the beginning of inflation equal to the e-folds when perturbations cross the horizon, namely N = 60, and we considered the implicit form of F as a function of N .
As in §2, we can assume the ansatz (17) for the effective EoS parameter ω eff (N ). As a consequence we obtain In this case the Planck constraint on ns implies α = 1 [27,34], which leads to This choice corresponds to the Hubble parameter, which follows from Eq. (24). Here, ρe is an integration constant representing the effective energy density of the universe toward the end of inflation (at N ≃ 1). Now we can infer the implicit form of F (N ) from Eq. (34) which reads, with F ≡ F (N ). A simple analytic solution can be found for β = 1/3, namely By taking into account that we easily derive The f (R)-model can be finally fully reconstructed as where we posed c0 = 2 in order to recover the Hilbert-Einstein term of General Relativity (GR). This model is nothing else but the Starobinski model [2], which clearly violates the TCC condition (39). This fact is not surprising, since the Starobinsky model in the Einsteinframe leads to the scalar model with potential (28) and β = 1/2, c1 = 2. The conformal transformation between the two frames is given by, Now it is easy to verify that the power spectra of the two models coincide after the identification ρe = 3ρ0/2 (note that the inflation scales in the two frames do not coincide). One may be interested to see if Eq. (44) admits some solutions for small values of β, when the Hubble parameter reads Thus, an implicit solution of Eq. (44) to the leading-order term of β which allows to recover the Hilbert-Einstein contribution of GR is given by, In this case the TCC condition (39) is satisfied under the condition which brings to r < 2 × 10 −44 .
Due to the constraint on β the Hubble parameter remains a constant during inflation and starts to decrease at the very end of it. Moreover, as in the case of scalar field slow-roll inflation, in this class of viable f (R)-gravity models compatible with the TCC we have a strong suppression of the amplitude of the primordial gravitational waves and a severe finetuning of initial conditions. Despite the fact that our analysis is not exhaustive of the wide variety of f (R)-models for inflation, we can draw some conclusions. Since viable f (R)-gravity reduces to Einstein gravity at small curvature, it is clear that the TCC condition given in Eq. (39) introduces an upper bound on the effective EoS parameter as (1 + ω eff ) 10 −23 , such that the tensor-to-scalar spectra ratio in (37) results to be at most r ∼ 10 −44 . This result is independent of the ansatz on ω eff and is in agreement with (52)- (53). Thus, the suppression of the amplitude of the primordial gravitational waves seems to be a general feature of viable slow-roll f (R)-gravity compatible with TCC and since we have assumed as a minimal requirement that the total amount of inflation coincides with the e-folds at the perturbation horizon crossing, we get the fine-tuning problem of initial conditions. 4 The case of f (R, φ)-gravity: two specific examples of slow-roll inflation An important class of inflationary models is given by scalar-tensor theories, where the gravitational interaction is mediated by both a scalar and a tensor field [40,41]. In what follows, in the attempt to investigate the TCC in this framework, we will consider two specific examples of f (R, φ)-slow-roll inflation firstly presented in Ref. [42], where a scalar field is coupled with the Ricci scalar.
The general action of F (R, φ)-gravity is in the form, but in what follows we will assume V (φ) = 0.
In terms of the e-folds number, the slow-roll parameters describing slow-roll f (R, φ)inflation are given by [37], where F = ∂f /∂R , f ≡ f (R, φ), and The first Friedmann-like equation in slow-roll approximation leads to while the conservation law related to the field reads, The power spectrum of cosmological perturbations is given by, with As a check, we observe that when f (R, φ) = R and therefore Qs = φ ′2 , in slow-roll approximation we recover Eq. (8). On the other hand, if f (R, φ) = f (R), φ ′ = 0 and therefore Qs = 3M 2 P l F ′2 /(16πF ), we recover Eq. (36). The TCC introduces the following constraint, with N = 60, as per usual. Finally, we recall the expressions for the spectral index and the tensor-to-scalar power spectra ratio, with N = 55 − 65. As a check, we note that when f (R, φ) = R such that ǫ3 = ǫ4 = 0, by taking into account that, in slow-roll approximation, 2ǫ2 = −ǫ1 − H ′′ /H ′ , we correctly find the results of slow-roll scalar field inflation (10) ǫ4) and, by taking into account that in slow-roll approximation ǫ1 ≃ −ǫ3 and ǫ4 ≃ −3ǫ1 − ǫ ′ 1 /ǫ1, we recover the results of pure f (R)-gravity (37).
In Ref. [42] inflation is realized thanks to a sort of switching on the cosmological constant in two different models. The first model reads, where b is a positive parameter and λ is a positive cosmological constant (on curvature scale of inflation). Inflation starts with φ negative and very small (|φ| ≪ 0), such that we obtain a quasi-de Sitter expansion with Thus, from Eq. (58) with R ≃ 12H 2 we obtain, (65) The field grows during inflation which ends at N → 0. A direct evaluation of the slow roll parameters shows that Thus, a viable scenario with takes place. Moreover, since Qs ≃ φ ′2 , we observe that the TCC condition (61) holds true if and the amplitude of primordial gravitational waves is again strongly suppressed. By taking into account that R ∼ λ condition (68) also guarantees that the Hubble parameter is almost a constant until the very end of inflation, due to the fact that f turns out to behave as f ≃ R − 2λ, unless N is very close to zero.
The second model under consideration reads, with n, b positive parameters and λ a cosmological constant. Once again, inflation is supported by a quasi-de Sitter solution with φ negative and very small such that The field behaves as, and the early-time acceleration ends when N → 0. The slow-roll parameters are derived as As a consequence, and the spectral index ns is in agreement with the Planck observations only for large values of n. In this case the TCC is satisfied under the requirement confirming the suppression of the amplitude of primordial gravitational waves as the price to pay for the validity of the TCC condition.
Up to now we have considered models of slow-roll inflation. The slow-roll approximation is valid if all the slow-roll parameters are small during inflation. However, in order to obtain a constant Hubble parameter (or a flat potential, in the classical scenario of scalar field inflation) it is enough to require that ǫ1 ≪ 1, while the other horizon flow parameters can also be not so small, but constant. In the next Section, we will study the consequences of the TCC in the case of the so called 'constant-roll' inflation scenario.

Constant-roll scalar field inflation
Constant-roll inflation has some important and interesting properties. For example, it can generate large local non-gaussianities (which are negligible in the case of slow-roll inflation) and the curvature perturbations may grow on super-horizon scales [43,44,45,46]. In Ref. [47] constant-roll scalar field inflation has been studied and exact solutions for the inflaton potential have been found (see also Refs. [48,49,50,51] for constant-roll inflation in modified gravity). We recall these results in the context of the TCC.
For α = −3 we obtain the slow-roll approximation. We will investigate two models, for which all the possible values of α = −3 are covered. The first model reads, where 0 < M is a generic mass constant. We assume −∞ < φ < 0 and 0 <φ during inflation. If −3 < α < 0 the potential has a minimum (i.e. an attractor point) for φ → 0 − , while if 0 < α the field reaches a maximum of the potential when φ → 0 − . The second model is described by the following field potential, Here, we are assuming ∞ > φ > 0 ,φ > 0 and the potential has a maximum at φ = 0 + . The exact solutions of the field equations (5)- (6) in the case of (76) are When t → 0 + the Hubble parameter is not a constant (but we can still have an acceleration). Nevertheless, since the Hubble parameter approaches a constant when t → +∞ and the scale factor grows exponentially, we eventually have inflation. Additionally, in this case a transition phase at the end of inflation has to be assumed (see Refs. [52,53,54]). The exact solutions of (5)- (6) in the case of (77) are with −∞ < t < 0, such that the Hubble parameter is almost a constant when t → −∞ and we have inflation, while goes to zero when t → 0 − . We will denote with t0 , te the time when inflation starts and ends, respectively. The e-folds number (9) reads, For the model (76) we obtain, For the model (77) we derive, where we have evaluated a(t) = a(te)(cosh[(3 + α)M t]) 1/(3+α) and we have posed cosh[(3 + α)M te] = 1 (namely te = 0). We can now estimate the ǫ1 slow-roll parameter (7) in the two models as where we have introduced the total e-folds number N through the relation (9). Thus, for −3 < α, the bound of the ǫ1 slow-roll parameter at the time t = t0 (namely, N = N ) is given by ǫ1 = (3 + α)/2 and the parameter decreases with time through a quintessence region. A remark is in order. When ǫ1 > 1 the acceleration does not take place. However, it is clear that if (3 + α) ∼ O(1) the acceleration phase with ǫ1 ≪ 1 (namely, H almost a constant) is immediately reached. For this reason we still indicate with N the total amount of inflation. On the other hand, for α < −3, the ǫ1 parameter is negligible at the beginning of inflation and increases with the time. The power spectrum of scalar perturbations is given by [47], As a check, note that when α = −3 we recover the result of slow-roll inflation. Furthermore, the spectral index ns is related to α as such that in order to obtain ns = 0.96 with α > −3 (model (76)) we must fix α = 0.02, while with α < −3 (model (77)) we must require α = −3.02, namely we are near the slow-roll approximation region. Finally, the tensor-to-scalar power spectra ratio is still related to ǫ1 as where we remember that ǫ1 must be evaluated at the time when perturbations cross the horizon, at N = 55 − 65. In the case of α = 0.02 in order to satisfy the Planck data with r 0.06 we should require 61.1 N if ǫ1 is evaluated at N = 60, namely the total time of inflation must exceed by at least one the number of e-folds of the perturbation horizon crossing. Moreover, for α = −3.02 the tensor-to-scalar ratio is in agreement with the Planck data, leading to r ≃ 0.03 if ǫ1 is evaluated at N = 60. Now, the TCC condition leads to where ǫ1 is given by (83)-(84) with N = 60. Let us have a look for the viable model with α = −3.02. The sufficient condition for the validity of the TCC can be found as in the slow-roll inflation scenario and is realized when N assumes the minimal value, namely N = 60, when it is clearly violated. We remark that in this case we are near the slow-roll approximation region. In Ref. [47] it is argued that one can obtain r ≃ 3 × 10 −3 (like in Starobinsky inflation) by setting φ ∼ M P l at the beginning of inflation. Here the result can be derived directly from (84) and (87) by assuming N ≃ 60. Thus, the model is affected by Trans-Planckian problem.
The situation is different for constant-roll inflation with α = 0.02. In this case the ǫ1parameter decreases with the e-folds and the sufficient condition for the validity of the TCC is realized when N is much larger than its minimal value for viable inflation, namely N ≫ 61.1. Specifically, we find that for the TCC condition (88) is satisfied. The results show that in the constant-roll inflationary scenario it is is possible to deal with viable inflation in agreement with the TCC provided that inflation starts much before the time when perturbations cross the Hubble horizon. Also in this case the tensor-to-scalar ratio r, is extremely small and the model predicts a strong suppression of the amplitude of gravitational waves. However, at the beginning of inflation ǫ1 = 1.51 and is large enough to avoid a fine-tuning problem of the initial conditions, thanks to its peculiar behaviour.

Conclusions
In this paper we revisited the Trans-Planckian Censorship Conjecture in different models of viable inflation. As already observed in Ref. [25] and, more recently, in Ref. [55], the TCC tightly constrains slow-roll scalar field inflation. Here, we first extended the result to different frameworks of slow-roll inflation. For scalar field theory and f (R)-gravity, we used a general approach which permits to reconstruct the models that lead to the power spectrum of scalar perturbations, spectral index and tensor-to-scalar spectra ratio in agreement with Planck data and where the TCC holds true. For f (R, φ)-inflation, we proposed the study of two viable models. In this cases, we found that although under certain conditions we can obtain viable inflation free of the trans-Planckian problem, we get a severe fine-tuning of initial conditions. Moreover, these models predict a strong suppression of the amplitude of primordial gravitational waves as a direct consequence of the TCC. In the second part of the paper, moving away from slow-roll scenario, two examples of constant-roll scalar field inflation have been analyzed. Here we found that in principle it is possible to deal with viable inflation avoiding both trans-Planckian problem and fine-tuning problem of initial conditions, by asking that inflation starts much before the time when perturbations cross the Hubble horizon. We should stress that the result is related to a peculiar mechanism of inflation where the ǫ1 slow-roll parameter decreases with cosmological time and the de Sitter expansion is an attractor of the system, such that it is known that a transition phase at the end of inflation must be introduced. We may conclude that, since in order to avoid the presence of fluctuations which trace back to quantities beyond the Planck scale in the classical power spectrum the TCC imposes severe constraints on the majority of the inflationary models, different mechanisms (as in slow-roll inflation) or different approaches (like the cosmological bounce) are the natural implications of the conjecture itself.
A last remark is in order. As briefly considered in the introduction, it should be again emphasized that in the TCC, as well as in the consideration of the trans-Planckian problem, the investigation of generation of metric and fields is restricted only to the scenario where metric fluctuations become large and quasi-classical, which may be thought of as a deficiency of the conjecture. Specifically, the important case of particle creation when metric and field fluctuations remain quantum but show themselves in the form of ultra-high energy particles is neglected. In this instance, as shown in Ref. [17], any deviation of the quantum state of trans-Planckian modes from the adiabatic vacuum one would result in the appearance of super-high energy particles in any expanding universe and at any time, including the present time.