Challenges to Inflation in the Post-Planck Era

Space-based missions studying the cosmic microwave background (CMB) have progressively refined the parameter space in conventional models of inflation shortly (∼10−37 s) after the Big Bang. While most inflationary scenarios proposed thus far in the context of general relativity have since been ruled out, the basic idea of inflation may still be tenable, albeit with several unresolved conundrums, such as conflicting initial conditions and inconsistencies with the measured CMB power spectrum. In the new slow-roll inflationary picture, inflation arising in plateau-like potentials requires an initiation beyond the Planck time. This delay may be consistent with the cutoff, kmin , measured recently in the primordial power spectrum. However, the actual value of kmin would imply an initiation time too far beyond the Big Bang for inflation to solve the horizon problem. In this paper, we also describe several other undesirable consequences of this delay, including an absence of well-motivated initial conditions and a significant difficulty in providing a viable mechanism for properly quantizing the primordial fluctuations. Nevertheless, many of these inconsistencies may still be avoided if one introduces nonconventional modifications to inflation, such as a brief departure from slow-roll dynamics, possibly due to a dramatic change in the inflationary potential, inflation driven by multiple fields, or a nonminimal coupling to gravity. In addition, some of these difficulties could be mitigated via the use of alternative cosmologies based, e.g., on loop quantum gravity, which replaces the initial Big Bang singularity with finite conditions at a bounce-like beginning.


INTRODUCTION
Inflationary cosmology was introduced several decades ago (Starobinski ǐ 1979;Kazanas 1980;Guth 1981;Linde 1982).The rapid expansion it would have created in the early Universe might resolve several inconsistencies with the standard model, including a solution to the cosmic microwave background (CMB) temperature horizon problem, a dilution of the number density of magnetic monopoles to undetectable levels today, and an explanation for why the Universe is spatially flat.It may also have produced the primordial spectrum of fluctuations responsible for the eventual formation of large-scale structure (Mukhanov et al. 1992).But we still do not know inflation's underlying field and potential, Corresponding author: Fulvio Melia fmelia@email.arizona.eduand have very little guidance concerning how and when the inflationary phase may have started.Several other difficulties still cause great concern regarding its ultimate viability, e.g., given that inflation appears to conflict with several general relativistic constraints (Melia 2020a) and, most seriously, violates at least one of the fundamental energy conditions in Einstein's theory (Melia 2023).
Our attention in this paper will be focused on the conventional picture of inflation, i.e., the paradigm developed in the context of general relativity, based in part on the Friedmann-Lemaître-Robertson-Walker metric (FLRW) defined in Equation (5) below.We shall examine how the various constraints now available to us, principally the size of the homogeneous Universe today (Planck Collaboration et al. 2020), the inferred very rapid expansion starting right at the big bang singularity, inflation's energy scale measured by Planck, and the apparent cutoff in the primordial fluctuation spectrum measured in the CMB (Melia & López-Corredoira LIU & MELIA 2018;Melia et al. 2021;Sanchis-Lozano et al. 2022), cannot all be merged together into a self-consistent picture.We must emphasize at the outset, however, that alternative early universe scenarios have been proposed in recent years, in part to circumvent at least some of the problems we are highlighting in this paper.Though a period of inflation is still required in these models, the conditions leading up to the initiation of the exponential expansion can be drastically different from those in the conventional picture.
For example, a cosmology based on loop quantum gravity (Bojowald 2008;Ashtekar et al. 2003;Ashtekar & Singh 2011)-a modified version of general relativity that incorporates the Heisenberg uncertainty principle from quantum mechanics on scales approaching the Planck domain (see Eqs. 6 and 7)-replaces the initial singularity with a bounce (Ashtekar et al. 2006;Martín-Benito et al. 2009) and a small Hubble constant with a corresponding gravitational radius (Melia 2018) much larger than all of the other length scales.As we shall discuss more extensively in § 5, these changes can greatly mitigate some of the more serious defects in the standard inflationary paradigm (Taveras 2008;Bhardwaj et al. 2019;Ashtekar et al. 2020;Navascués & Mena Marugán 2021).Our analysis will not be directed towards such models.
But insofar as the conventional inflationary picture is concerned, prior to the Planck-2013 data release (Planck Collaboration et al. 2014), the basic paradigm was the so-called classic "chaotic" inflation (Linde 1983), in which a minimally-coupled scalar (inflaton) field, φ, emerged at the Planck time, t Pl , with an energy density set by the Planck scale (see Eq. 7 below).As the precision of the CMB measurements continued to improve, however, particularly with the most recent Planck analysis (Planck Collaboration et al. 2020), this simple picture has not continued to fare well.
The principal reason for this is that, after the Planck-2013 release, the simplest inflationary scenarios have been disfavoured in comparison with a particular class of models with a plateau-like potential, V (φ) (Planck Collaboration et al. 2014;Ijjas et al. 2013Ijjas et al. , 2014)).But in such cases, V (φ) has a minimum at the end of inflation, and must have changed slowly with φ at the beginning, which precludes any possibility of the Universe's energy density matching the Planck value at t Pl , unless the de Sitter expansion was initiated at t init ≫ t Pl .This undesirable behavior-which motivates the 'new' chaotic inflationary picture-is unavoidable when the various observables are fit to the data, especially the tensor to scalar ratio, R, and the spectral index, n s , of the scalar fluctuation spectrum or, more accurately, the dimensionless power spectrum where Throughout this paper, k is the comoving wavenumber of each mode, such that in terms of its proper wavelength, λ k , and the expansion factor, a(t), in the FLRW metric.
As we shall examine in this paper, however, the delayed initiation of the de Sitter phase creates several major problems with the conventional picture of inflation in the context of GR, the most significant of which is that its initial conditions (Guth 1981;Linde 1982) are then left unspecified (Ijjas et al. 2013(Ijjas et al. , 2014)).That is, why was the inflaton field homogeneous over distances spanning causally-disconnected regions?Slow-roll inflation needs φ to have been homogeneous when the slow-roll phase started at t init .This is required by the condition that the kinetic and spatial-derivative terms of φ be negligible compared to V (φ).
Furthermore, the primordial fluctuations in φ would themselves have been seeded at t init .But how then were they quantized?Such a delayed starting time would not have allowed all the modes to be seeded well below the gravitational horizon, R h ≡ c/H (i.e., the Hubble radius), to satisfy the Bunch-Davies vacuum conditions.These hurdles are in addition to the more obvious challenge of addressing the horizon problem when inflation is delayed too far beyond the big bang.For these reasons, the new chaotic inflation is far from established as a viable paradigm of the early Universe (see also Melia 2022 for a more detailed description of the current problems faced by the standard model).
In this paper, we shall first affirm the growing consensus that the most recent data unavoidably require t init ≫ t Pl in the context of slow-roll, plateau-potential models ( § 2).Then we examine how and why an inflationary expansion beginning at t init ≫ t Pl necessitates a sharp cutoff k min in the primordial power spectrum ( § 3), possibly explaining the large-angle anomaly seen in the CMB anisotropies.But we then also explain why the delay creates the negative features described above ( § 4).In our discussion, we shall include a brief survey of how some of these difficulties may be circumvented in alternative theories, such as loop quantum cosmology, and we end with our closing thoughts in § 6.

BACKGROUND
We adopt the FLRW metric for the cosmic spacetime, written in terms of the comoving coordinates (ct, r, θ, φ), and the spatial curvature constant, K (to properly distinguish it from the more commonly used symbol k denoting the mode wavenumber): dr 2 1 − Kr 2 + r 2 dθ 2 + sin 2 θ dφ 2 . (5) The observations suggest that the Universe is probably spatially flat, so we shall simply set K = 0 throughout this paper (see also Melia 2022).The universal expansion factor, a(t), evolves according to the imposed equation-of-state in the stress-energy tensor and, given that we adopt spatial flatness throughout this paper, we normalize it to 1 at t = t 0 , where t 0 is the current age of the Universe.
Our classical description of the Universe breaks down prior to the Planck time, t Pl ≡ l Pl /c, corresponding to the Planck spatial scale Numerically, we have l Pl ≈ 5.7 × 10 −33 cm and t Pl ≈ 1.9 × 10 −43 s.The Planck energy density is therefore simply given as As noted earlier, a significant challenge now faced by the 'new' chaotic inflationary paradigm in the context of GR is that the latest Planck measurements preclude the actual density in the early Universe from matching Equation (7) with a slow-roll inflaton potential.To make the discussion quantitative, we shall focus on the Higgs-like potential and demonstrate how it fails to satisfy the observational constraints.
This potential has the form: from which we can determine the so-called slow-roll parameters, and where V ,φ denotes the first derivative of V with respect to φ, and V ,φφ is its second derivative.In these expressions, M Pl is the reduced Planck mass, defined as M Pl = m Pl / √ 8π.Under the slow-roll approximation, we obtain: Adopting the Planck measurement, n s = 0.966, and the upper limit of the tensor to scalar ratio, R = 0.036 (Ade et al. 2021), we find that µ ≈ 18.7 M Pl and φ 0.002 ≈ 5.4 M Pl , denoting the value of φ measured at k = 0.002 Mpc −1 .The fact that (φ 0.002 /µ) 2 ≪ 1 confirms our inference that V would have been very nearly constant at the early stage of inflation.As long as V (φ) dominated the energy density of the Universe at that point, we conclude that H must also have been approximately constant during the early stages of inflation.
The dynamical evolution of φ, however, must allow its energy density ρ(φ) (∝ H 2 ) to self-consistently match ρ(t Pl ) at t Pl .With slow-roll (Linde 1983), But from the definition of the amplitude, A s , of the primordial power spectrum, we also have where k * is a pivot scale, usually taken to be 0.05 Mpc −1 .Thus, inserting the Planck measured value of the amplitude, From the Friedmann equation (with K = 0), we estimate a density Numerically, it appears that ρ(a init ) ≈ 1.7 × 10 84 kg m −3 .But from Equation ( 7), we have instead ρ(a Pl ) ≈ 5.2 × 10 96 kg m −3 , representing a significant difference of ∼ 10 12 in the density between these two times.The dynamical evolution of φ, which largely depends on V (φ), must allow sufficient time to pass (a init ≫ a Pl ) for ρ(φ) to drop well below its Planck value.

TRUNCATION OF THE PRIMORDIAL POWER SPECTRUM
On its own, the fact that t init ≫ t Pl already creates a significant challenge in ensuring self-consistency between V (φ) and the physical conditions in the early Universe.But a recent analysis of the CMB anisotropies measured by Planck provides a more severe constraint on the possible value of a init itself.This work demonstrates in a model-independent way how the angular correlation function of the temperature distribution unavoidably points to a distinct cutoff, k min , in the primordial power spectrum (Melia & López-Corredoira 2018;Melia et al. 2021;Sanchis-Lozano et al. 2022).
To be clear, this inference that the measured angular correlation function of the CMB is best explained with a clean cutoff, k min , is specifically associated with the conventional inflationary paradigm.Several other workers have probed the possibility that the missing angular correlation on large scales (and the missing power at small multipole moments) may instead be due to dynamical effects.Again using loop quantum cosmology as an evident example, only modes with the largest observable wavelengths in such models would have felt the quantum curvature effects near the bounce (Agullo & Morris 2015;Castelló Gomar et al. 2017;Ashtekar & Gupt 2017), undergoing a pre-inflationary phase of kinetic dominance that eventually evolved into the standard slow-roll profile.This sequence of steps can effectively suppress their power relative to the smaller k modes, thereby alleviating the CMB anomalies (Martín de Blas & Olmedo 2016; Ashtekar et al. 2020).
At first sight, this evidence of a cutoff, k min , is actually quite exciting in the context of a delayed initiation of inflation in standard GR.In the analysis of the CMB anisotropies (Hinshaw et al. 1996;Bennett et al. 2003;Planck Collaboration et al. 2020), the tacit assumption is usually made that quantum fluctuations in the inflaton field started exiting the Hubble horizon at t < t Pl , effectively implying no lower cutoff in the primordial power spectrum P s (k) since k min would then have been much smaller than any modes observable in the CMB today (see Melia 2020a and references cited therein).But the more recent analysis of the Planck data, focusing on the value of k min itself, has revealed that this simple picture is not consistent with the absence of large-angular correlations in the CMB temperature anisotropies (Liu & Melia 2020).
Three independent missions (Hinshaw et al. 1996;Bennett et al. 2003;Planck Collaboration et al. 2020) have now confirmed the lack of angular correlation in the CMB temperature profile at angles 60 • .The most likely explanation for this anomaly appears to be the presence of a hard cutoff, in the primordial power spectrum.As we shall see shortly ( § 4), this cutoff implies a specific time at which modes started exiting the horizon, and this is clearly well beyond the Planck time.
The interesting aspect of this finding is that-contrary to the prediction of classic chaotic inflation-a cutoff is unavoidable when a init ≫ a Pl .The issue is then whether the 'measured' value of k min consistently yields the same initiation time t init as that implied by the argument we have summarized in § 2.
Here, we shall demonstrate quantitatively why a delayed initiation for inflation must produce a truncated primordial power spectrum in the context of slow-roll models with a plateau-like potential, since these are the most commonly used scenarios.The almost scale free nature of the primordial power-spectrum implies that, during inflation, the Hubble radius R h = c/H (and thus the Hubble parameter H) changes very slowly.
The physical reason for this is not difficult to understand.Quantum fluctuations in φ oscillate and decay as long as their wavelength λ k ≡ 2πa(t)/k is smaller than 2πR h .Once λ k > 2πR h , however, they stop oscillating and their amplitude tends toward a constant value (Bardeen et al. 1983).Only their wavelength continues to change (in proportion to a) as the Universe expands.Horizon crossing corresponds to that moment at which λ k = 2πR h , and is thus an essential ingredient in the generation of observable density fluctuations in inflationary cosmology.A perfectly scale-free primordial spectrum would correspond to modes crossing R h with a constant H value.The fact that n s is not exactly one (but very close to it) implies that H is not exactly constant, but can vary only slightly during the time when the spectrum is created.
In this picture, the larger a mode is, the smaller is its wavenumber k, and therefore the earlier is its exit across the horizon.For simplicity, we consider the simple preinflationary scaling which leads to and A comparison of Equation ( 19) with (7) shows that ρ(t Pl ) matches the Planck energy density exactly when β = 0 and is very close to it as long as |β| ≪ 1.There is thus an additional motivation for considering this type of scaling in the preinflationary Universe, since it provides a 'natural' way for the measured energy of inflation to merge smoothly toward the Planck scale.The mismatch in densities would be difficult to address otherwise.In any case, Equation ( 17) is quite general because it could also represent a radiation-dominated universe (RD), with H ∝ a −2 , a kinetic-dominated universe (KD) (Contaldi et al. 2003), with H ∝ a −3 (Liu & Melia 2020), and even a string-dominated universe (SD), with H ∝ a −1 (Spergel & Pen 1997).Interestingly, the latter option would yield the simplest, most compelling solution for the energy density mismatch between t Pl and t init .
The growth of different modes and the Hubble radius then follow the trajectories illustrated in Figure 1.The yellow dots represent typical Horizon crossing times.As we can see , for all the cases, horizon crossing can happen during inflation (point 3), but for β ≤ 0, it cannot happen before the inflation.Indeed, there actually exists a largest possible mode (λ max ) whose crossing point corresponds to the beginning of inflation (at a init ).Horizon crossing prior to inflation can only occur for β > 0.Even here, however, we would still not see a primordial power spectrum without k min due to the requirements imposed by the quantization of the seed fluctuations (see below).In this regard, notice that all of the three examples we pointed to, RD, KD and SD, have β ≤ 0. It is thus clear that all the simple and well motivated models one can consider in the delayed initiation picture lead to the existence of a hard cutoff k min in the primordial power spectrum.

CHALLENGES DUE TO THE DELAYED INITIATION TIME
The cutoff k min corresponds to a specific time t init (and the corresponding a init ) at which the larget mode crossed the Hubble horizon.To estimate it, we consider the null condition in Equation ( 5), for a photon propagating radially between times t 1 and t 2 : Using the Hubble parameter, we may write for the comoving distance traveled by a photon from decoupling to us, and for the corresponding quantity from the Plank time to decoupling.
We can estimate r dec numerically using the Friedmann equation, where Ω m , Ω r and Ω Λ are the matter, radiation and cosmological-constant energy densities, respectively, scaled to the critical density today, ρ c ≡ 3c 2 H 2 0 /8πG.The result is r dec ≈ 13, 804 Mpc (Liu & Melia 2020), assuming the Planck optimized parameters in the standard model (Planck Collaboration et al. 2020).
The comoving distance r pre−cmb may be decomposed into the contribution before inflation (r pre−inf ), during inflation (r inf ) and after inflation (all the way up to decoupling).It is not difficult to see, however, that the dominant contribution is made prior to the end of the inflation.Furthermore, if we restrict our attention to slow-roll potentials, we may approximate r inf by setting a(t) equal to a simple exponential, the result of which is where, as usual, the subscript 'init' refers to the time at which inflation began.
To solve the horizon problem, we need Thus, Equations ( 25) and ( 26), with the measured upper limit on H init , give us a init 7.5 × 10 −57 .But a init must also correspond to the initial crossing time of the modes (Liu & Melia 2020), and the value of t init corresponding to k min is not necessarily constrained in the same way as the argument based on H init .Looking again at Equation (25), we now see that when k min = 0, the starting point of inflation can be found using the crossing condition of the largest mode (see Eq. 4): Let us now compare the value a init 7.5 × 10 −57 required to solve the horizon problem with that obtained using H init = H 0.002 and k min .Substituting these quantities into Equation ( 27), we estimate that a init = 9.7 × 10 −56 , too large by over an order of magnitude.We can see this more concretely by estimating the corresponding comoving distances.From Equations ( 25), ( 16) and ( 27), we find that r inf ≈ 3, 181 Mpc, which is much smaller than 2r dec ≈ 27, 608 Mpc.Thus, in order to solve the horizon problem while still maintaining consistency with the presence of the measured k min , one must rely on the pre-inflationary phase (i.e., r pre−inf ) to make up the difference.
Any self-consistent picture for establishing the initial conditions for inflation must therefore also be able to help inflation overcome the horizon problem.Of course, this is not the only challenge, because the existence of k min , which implies that t init ≫ t Pl , creates several other potential issues.At the very least, the pre-inflationary phase must account for the homogeneity of the inflaton field at the beginning of the slow-roll phase.And the framework for allowing this to happen must also provide a mechanism for properly quantizing the seed fluctuations in φ.
It doesn't take long to realize that a delayed initiation time for inflation merely pushes several problems that inflation was supposed to solve farther back into the pre-inflationary phase.The delayed initation time leaves the initial condition of inflation unspecified.Slow-roll models require the Universe to be dominated by a homogeneous inflaton field at the beginning of inflation, because they require the kinetic and spatial-derivative terms of φ to be negligible compared to V (φ).In other words, the homogeneous region at t init must be large enough to expand into the CMB sphere we see today.
As a result of this, the comoving distance a photon travelled before inflation, r preinf , should be longer than 2r dec ≈ 27, 608 Mpc.We can see if this is feasible using the simple pre-inflationary scaling we introduced in § 3. Combining Equations ( 17), ( 20) and ( 21), we have For β < 0 (which, as we have seen, is the better motivated scenario), the fact that a init ≫ a Pl implies that And combining this with Equation ( 27), with the measured value of k min , we find that |β| needs to be smaller than 0.13 to ensure that r preinf > 2r dec .This rules out many models, including the kinetic-dominated and radiation-dominated expansions.Furthermore, if inflation is delayed to a time when the inflaton field was already homogeneous, then the horizon problem would already have been solved during the preinflationary phase.This would raise the bigger question of why we would then even need inflation.An equally big challenge would be to understand how the seed fluctuations could have been quantized in this scenario.As we shall discuss in § 5, the principal difficulty with the normalization of these modes is due to the time dependence of their frequency, which arises from the changing spacetime curvature as they evolve.Selecting an otherwise seemingly random time to canonically quantize them is therefore poorly motivated, except in some alternative cosmologies that predict a bounce instead of an initial singularity.One may thus find reasons to justify a normalization at that initial time.
In the standard inflationary paradigm based on GR, canonical quantization can only occur in Minkowski space, where the frequency of the modes is time-independent.Of course, the cosmic spacetime never satisfies this condition if ä = 0, but with classical chaotic inflation, one could bypass this problem by assuming that the fluctuations first emerged far enough into the conformal past, well below the Planck scale (i.e., the 'Schwarzschild' radius at the Planck time) where the spacetime curvature could be neglected in estimating the mode amplitude.Known as the 'Bunch-Davies' vacuum (Bunch & Davies 1978), this remote patch of space has allowed classical chaotic inflation to provide a means of identifying the quantum modes in φ.
But this could be done in the classic "chaotic" picture because one could always find an early enough time, t ≪ t Pl , when λ k was much smaller than the Hubble radius, R h , of the Universe (corresponding to its gravitational horizon; Melia 2018.)With the new chaotic inflationary picture, however, the single field slow-roll phase beginning at t init ≫ t Pl no longer allows for this possibility.The larger modes, in particular, would never have transitioned through a Bunch-Davies vacuum, preventing us from establishing their canonical quantization condition.; wavelength of the mode (λmax) corresponding to kmin (straight, dashed), crossing the horizon at points 1 and 2; wavelength of another mode leaving the horizon earlier than λmax, at point 3.The early de-Sitter expansion (red Region I) ends at atrans, where the kinetic dominated (KD) phase (grey Region II) begins.
The KD expansion ends at ainit, where slow-roll inflation (blue Region III) begins.

DISCUSSION
All of our analysis thus far has been based on classic chaotic inflation.Not surprisingly, however, at least some of these inconsistencies may be mitigated with the introduction of unconventional modifications to the basic inflationary picture, with the effect of producing additional degrees of freedom.For example, a helpful change would be the assumption of a brief departure from slow-roll dynamics near the beginning of inflation.This could be due to various factors, including an ad hoc dramatic shift in the inflationary potential (Hunt & Sarkar 2007;Qureshi et al. 2017;Ragavendra et al. 2022), an early phase of inflation driven by multiple fields (Braglia et al. 2020), or a non-minimal coupling to gravity (Tiwari et al. 2023).
In reality, these types of modification do not completely solve the problems faced by inflation in the conventional context of GR either.For instance, in the inflationary picture, the strong energy condition (SEC) from general relativity is violated during the slow-roll phase (Melia 2023).Adding a brief non-slow-roll period at the beginning cannot eliminate this difficulty.
But to gauge how effective these changes may be in resolving the various issues we have raised in this paper, it helps to consider a simplified scenario that embraces many or all of the features required to make inflation work consistently with the data.To summarize, the minimal set of requirements include the following: (i) The inflaton field must be present sufficiently early (i.e., well below the Planck time) for its fluctuations to evolve through the Bunch-Davies vacuum for canonical quantization; (ii) the inflation energy, characterized by H(a init ) at the initiation of the 'new' chaotic slow-roll expansion, must be matched to the Planck energy, characterized by H(a Pl ), via the appropriate evolution of V (φ) prior to a init and the expansion history it generates; (iii) the entire inflationary phase, beginning with φ in the Bunch-Davies vacuum, must produce a sufficient number of e-folds to solve the horizon problem; and (iv) the fluctuation spectrum must display a clear cutoff at k min , as discussed above.
The schematic diagram in Figure 2 shows the evolution of proper distance as a function of the expansion factor a consistent with all four of these minimal conditions.To produce this kind of expansion history, the inflaton potential, V (φ), must be highly dynamic at a a init , consistent with the various proposals referenced earlier in this section.For example, the energy density must drop sharply between a Pl and a init , corresponding to roughly 12 orders of magnitude in H, producing the steep gradient in R h , from c/H pl to c/H init , during this time.In other words, the slow-roll phase cannot start until a init , which itself must correspond to the time at which the largest mode wavelength crosses the horizon (at point 1 in this figure).
As we have seen, however, the observed value of k min precludes the slow-roll expansion from solving the horizon problem on its own.And a simple KD phase (Region II) prior to that phase is not sufficient either.Thus, V (φ) must produce an earlier de Sitter (or quasi-de Sitter) expansion (Region 1) prior to a trans .This will solve the horizon problem, effectively splitting the impact of inflation in producing the primordial spectrum (at a > a init ) from its role in resolving the horizon anomaly (at a < a trans ).
Thus, it appears that such proposed modifications to the conventional chaotic inflationary scenario may satisfy at least three of the observational requirements.Unfortunately, it does not seem likely that all four can be upheld simultaneously.As one can see in Figure 2, any modification to V (φ) consistent with the first three conditions will fail to also satisfy the fourth.Though the value of a init is at least partially selected to comply with the observed cutoff k min , there would clearly be modes with larger wavelengths (λ k > λ max ) exiting the horizon prior to a init , e.g., at point 3 in the first de Sitter expansion.And these will have even larger amplitudes than they would have had if the slow-roll expansion had started prior to point 3.In other words, it is challenging to justify an earlier phase of de Sitter expansion, beginning within the Planck regime, given that it would produce a fluctuation spectrum inconsistent with the CMB data.
But were we to follow such a modification, we have for the KD phase (Region II) (Liu & Melia 2020): where M is a constant, and so using the fact that H remains roughly constant at its Planck value in Region I.But the first Friedmann equation yields and therefore using a init ≈ 9.7 × 10 −56 gives a trans ≈ 8.1 × 10 −58 .The Hubble constant must drop by about 12 orders of magnitude during this brief phase lasting an equivalent ∼ 5 e-folds.By assuming the expansion is de Sitter right from the Planck scale, this scenario mimics the earliest moments of classic chaotic inflation, setting the initial conditions in the same fashion.For example, since the scale factor a(t) can be arbitrarily small during this first de Sitter phase, the horizon problem can always be eliminated.All the modes will have been seeded prior to the Planck time (as in the traditional inflationary concept), so the Bunch-Davies vacuum condition can (in principle) be satisfied, allowing the fluctuations to be properly normalized.Nevertheless, the long-standing trans-Planckian anomaly is still present (Martin & Brandenberger 2001).
It would seem that unconventional modifications such as these may address some of the challenges discussed in this paper.But several long-standing problems would still remain, including inflation's violation of the SEC and the trans-Planckian anomaly (see also Melia 2020b, and references cited therein).And even so, it appears unlikely that all the minimal observational requirements can be satisfied simultaneously.
A much more drastic modification to the basic inflationary paradigm would be the introduction of alternative models, notably loop quantum cosmology (Bojowald 2008;Ashtekar et al. 2003;Ashtekar & Singh 2011;Martín-Benito et al. 2009;Taveras 2008;Bhardwaj et al. 2019;Navascués & Mena Marugán 2021).In these models, the big bang singularity is replaced with a bounce, where all the physical quantitites remain finite.In particular, the Hubble constant is actually zero at the very beginning, implying an arbitrarily large gravitational horizon-at least effectively-so that subsequent inflation does not seem necessary to solve a horizon problem in this restricted sense.The principle purpose of inflation is to produce the "correct" fluctuation spectrum, via the conventional mechanism of horizon crossing and freezing, which would require slow-roll conditions over most of the (presumably) de Sitter expansion while the inflaton field is dominant.
As we have noted earlier, the expansion in these models proceeds via a pre-inflationary regime of kinetic domi-nance that eventually morphs into the more traditional slowroll inflation.And the main effect of the loop quantum bounce on the primordial perturbations translates into a large infrared suppression in their spectrum, covering an even broader range of scales than one would see with a kineticallydominated pre-inflationary phase in the conventional picture.
The form and properties of this power spectrum, however, depend critically on the specific choice of the vacuum state.In traditional inflation, one merely assumes that the fluctuations were seeded far enough into the distant past that their wavelengths were much smaller than the spacetime curvature radius at that time.Born in what is effectively Minkowski space, these fluctuations could therefore be quantized canonically with a time-independent frequency.That is, one could invoke the existence of a Bunch-Davies vacuum for the conventional picture, though our analysis in this paper now suggests that even this approach is flawed for the new chaotic inflation.
But when the mode frequencies are time dependent, as they are in loop quantum cosmology, the Bunch-Davies vacuum is not an option.There is an ambiguity about the time at which the modes should be canonically quantized.Strides made in recent years (Ashtekar et al. 2020;Navascués & Mena Marugán 2021) have pointed to a possible resolution of this problem, with an indication that the fluctuations should be seeded at the time of the bounce itself.
Such alternative cosmologies may resolve some of the challenges imposed on the inflationary paradigm by the current observations.We point out, however, that much work would still need to be done to bring these models into full compliance with the data.A glaring example is provided by the state-of-the-art angular correlation function predicted by the version of loop quantum cosmology proposed by Ashtekar et al. (2020), which clearly improves upon the poor fit of the CMB data provided by the standard picture, but yet clearly also does very poorly compared to a simple, hard cutoff, k min , in the primordial spectrum.One can easily see this by inspection and a comparison of the C(θ) curves in Figure 3 of Ashtekar et al. (2020) and Figure 2 of Melia & López-Corredoira (2018).

CONCLUSION
The emergence of a cutoff k min in the primordial power spectrum, implying a delayed initiation time for inflation, affirms the view that the conventional quasi de Sitter expansion could not have started until well after the Planck time in order to avoid a conflict with the required initial energy density in the cosmic fluid.In this paper, we have demonstrated that this new picture faces several serious challenges.The first arises from the conflict in solving the horizon problem while simultaneously fitting the measured k min value.Second, the initial conditions of inflation are now unspecified, which may be solved during a pre-inflationary phase, but then raising the bigger question of why we would even need inflation to solve the horizon problem in the first place.Finally, this new inflationary scenario fails to provide a mechanism for properly quantizing the seed fluctuations.
Of course, an important caveat in this discussion is that we have focused largely on slow-roll models, with only a limited range of possible pre-inflationary expansions.Perhaps a more inventive approach can circumvent at least some of these difficulties, as we have illustrated with the simple toy model in § 5.In the end, however, the preservation of inflation may require an entirely new cosmological framework.Or inflation may turn out to be unnecessary after all (Melia 2013(Melia , 2020a)), and we shall learn that the challenges it now faces are an indication that it simply never happened.
We are grateful to the anonymous referees for their helpful comments that have led to several significant improvements in the presentation of this material.

Figure 1 .
Figure1.Proper length as a function of the universal expansion factor a(t), for the Hubble radius Rh ≡ c/H (solid, black when β = 0; red curves when β = 0), mode wavelength (λ k ) and maximum mode wavelength (λmax) (straight dashed lines).Slow-roll inflation (blue shaded region) initiates at ainit ≡ a(tinit), with tinit ≫ tPl.The linear pre-inflationary phase is indicated by the grey shaded region.The yellow circles illustrate several horizon crossings by various modes.

Figure 2 .
Figure2.Proper length as a function of a(t): Hubble radius Rh ≡ c/H (solid, black); wavelength of the mode (λmax) corresponding to kmin (straight, dashed), crossing the horizon at points 1 and 2; wavelength of another mode leaving the horizon earlier than λmax, at point 3.The early de-Sitter expansion (red Region I) ends at atrans, where the kinetic dominated (KD) phase (grey Region II) begins.The KD expansion ends at ainit, where slow-roll inflation (blue Region III) begins.