PHYSICAL REVIEW D 101, 103502 (2020)

Trans-Planckian censorship and inflationary cosmology

† ‡ Alek Bedroya ,1,* Robert Brandenberger ,2, Marilena Loverde,3, and Cumrun Vafa1,§ 1Jefferson Physical Laboratory, , Cambridge, Massachusetts 02138, USA 2Department of Physics, McGill University, Montr´eal, H3A 2T8, 3C.N. Yang Institute for Theoretical Physics, Department of Physics & Astronomy, Stony Brook University, Stony Brook, New York 11794, USA

(Received 31 December 2019; accepted 15 February 2020; published 1 May 2020)

We study the implications of the recently proposed Trans-Planckian censorship conjecture (TCC) for early Universe cosmology and in particular inflationary cosmology. The TCC leads to the conclusion that if we want inflationary cosmology to provide a successful scenario for cosmological structure formation, the energy scale of inflation has to be lower than 109 GeV. Demanding the correct amplitude of the cosmological perturbations then forces the generalized slow-roll parameter ϵ of the model to be very small (<10−31). This leads to the prediction of a negligible amplitude of primordial gravitational waves. For slow- roll inflation models, it also leads to severe fine-tuning of initial conditions.

DOI: 10.1103/PhysRevD.101.103502

I. INTRODUCTION that fluctuations in both matter [6] and gravitational waves [7] originate as quantum vacuum perturbations that Cosmological observations provide detailed information exit the Hubble radius during the early phase, are squeezed about our Universe on the largest observable scales. and classicalize, and then reenter the Hubble radius at late Cosmic microwave background (CMB) measurements times to produce the CMB anisotropies and matter density [1–3], for instance, demonstrate that fluctuations in the perturbations that we observe today. matter and energy persist on cosmological scales. There is In [8] (see also [9]) it was realized that if the inflationary no causal explanation for the origin of these fluctuations in phase lasts somewhat longer than the minimal period, then standard big bang cosmology. Scenarios of early Universe the length scales we observe today originate from modes cosmology such as the inflationary Universe [4] provide a that are smaller than the Planck length during inflation. causal mechanism to generate these fluctuations. A key This was called the trans-Planckian problem for cosmo- aspect of both inflationary cosmology and of other scenar- logical fluctuations (see also [10]). This problem was ios that provide an explanation for the origin of structure in viewed not so much as an issue with a particular model, the Universe (see e.g., [5] for a comparative review) is the but more as a question of how to treat trans-Planckian existence of a phase in the early Universe during which the −1ð Þ ð Þ modes in such a situation. It has been conjectured [11], Hubble horizon H t , where H t is the Hubble expan- however, that this trans-Planckian problem can never arise sion rate as a function of time t, shrinks in comoving in a consistent theory of quantum gravity and that all the coordinates. The Hubble horizon provides the limiting models which would lead to such issues are inconsistent length above which causal physics that is local in time and belong to the swampland. This is called the trans- cannot create fluctuations. In both inflationary cosmology Planckian censorship conjecture (TCC). and in the proposed alternatives, comoving length scales According to the TCC no length scales which exit the which are probed in current cosmological experiments were Hubble horizon could ever have had a wavelength smaller inside the Hubble horizon at early times. It is postulated than the Planck length. In standard big bang cosmology no modes ever exit the Hubble horizon, and the TCC has no implications (indeed the TCC is automatically satisfied for *[email protected] † all models with a w ≥−1=3). However, in all early [email protected] ‡ [email protected] Universe scenarios which can provide an explanation §[email protected] for the origin of structure, modes exit the Hubble horizon in an early phase. If ai is the value of the cosmological Published by the American Physical Society under the terms of scale factor at the beginning of the new early Universe the Creative Commons Attribution 4.0 International license. a Further distribution of this work must maintain attribution to phase, and f is the value at the time of the transition from the author(s) and the published article’s title, journal citation, the early phase to the phase of standard big bang and DOI. Funded by SCOAP3. expansion, the TCC reads

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a M whereas they freeze out and become squeezed when the f < pl ; ð1Þ ai Hf wavelength is larger than the Hubble radius. During a phase of accelerated expansion, the proper wavelengths of fluc- where Hf is the radius of the Hubble horizon at the final tuations initially smaller than the Hubble scale can be stretched to super-Hubble scales. This transition from sub- time tf and Mpl is the reduced Planck mass. In [11], the relationship between the TCC and other Hubble to super-Hubble is referred to as horizon crossing. swampland conjectures [12] which have recently attracted The TCC prohibits horizon crossing for modes with initial a lot of attention (see e.g., [13] for reviews) was discussed. wavelengths smaller than the Planck length. Here, we focus on the consequences of the TCC for We will be considering models of inflation in which a ϕ inflationary cosmology. canonically normalized scalar field with potential energy ðϕÞ It is clear from the form of (1) that the TCC will have V constitutes the matter field driving the accelerated strong implications for inflationary cosmology. In the case expansion of space. We will be using units in which the speed of sound, Boltzmann’s constant and ℏ are set to 1. of de Sitter expansion, af is exponentially larger than ai, and hence (1) strictly limits the time duration of any inflationary phase. The implications for some alternative II. IMPLICATIONS OF THE TCC FOR THE early Universe scenarios are weaker. For example, in string ENERGY SCALE OF INFLATION gas cosmology [14], the early phase is postulated to be quasistatic. Hence, the condition (1) is satisfied: no modes In this section we work in the approximation that the which were larger than the Hubble scale at the beginning of Hubble expansion rate during the period of inflation is the standard cosmology phase ever had a wavelength constant. In order for inflation to provide a solution to the smaller than the Planck length. The same is true for the structure formation problem of standard big bang cosmol- various bouncing scenarios (e.g., the matter bounce [15], ogy, the current comoving Hubble radius must originate and the pre-big-bang [16] and ekpyrotic [17] scenarios) inside the Hubble radius at the beginning of the period of [18], where the initial phase is one of contraction. This is inflation (see Fig. 1). This condition reads true as long as the energy scale at the bounce point is smaller than the Planck scale. In the following we will study the consequences of the TCC for inflationary cosmology. The outline of this paper is as follows: In the following section we discuss general constraints imposed by the TCC on the energy scale of inflation and the resulting conse- quences for the amplitude of gravitational waves. These conclusions do not depend on what drives inflation, only that it occurred. In Sec. III, we then specialize to slow-roll inflation models, and show that consistency with the TCC leads requires fine-tuning of the initial conditions. We will work in the context of homogeneous and isotropic cosmologies with four space-time dimensions. For simplicity, we assume spatially flatness so that the metric can be written in the form

ds2 ¼ dt2 − aðtÞ2dx2; ð2Þ where x are the spatial comoving coordinates and aðtÞ is FIG. 1. Space-time sketch of inflationary cosmology. The the scale factor [which can be normalized such that vertical axis is time, the horizontal axis represents physical distance. The inflationary period lasts from t to t . Shown are aðt0Þ¼1, where t0 is the present time]. The Hubble i R −1ð Þ λ ð Þ λ ð Þ expansion rate is the Hubble radius H t and two length scales 0 t and 1 t (fixed wavelength in comoving coordinates). For inflation to a_ provide a possible explanation for the observed fluctuations on HðtÞ ≡ ; ð3Þ large scales, the scale λ0ðtÞ corresponding to the current Hubble a horizon must originate inside of the Hubble radius at the beginning of inflation. This leads to the condition (4). The and its inverse is the Hubble radius. As is well known TCC, on the other hand, demands that the length scale λ1ðtÞ (see [19] for an in-depth review of the theory of cosmo- which equals the Hubble radius at the end of inflation was never logical fluctuations and [20] for an overview), quantum- trans-Planckian. In the sketch, both conditions are marginally mechanical fluctuations oscillate on sub-Hubble scales, satisfied.

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1=3 1 a T gðT Þ 1 Equation (7) for Nþ and the upper bound (8) on Nþ coming · eNþ · R · R R ≃ ; ð4Þ 1=3 from the TCC are compatible only provided that the H aend T0gðT0Þ H0 condition where H is the Hubble scale during inflation, aend and aR pffiffiffi 3=4 3 2 ð Þ1=2 ð Þ are the values of the scale factor at the end of inflation and V < Mpl T0Teq 9 when reheating is completed, and g indicates the number of spin degrees of freedom in the thermal bath. Here Nþ is is satisfied. Inserting the values of T0, Teq and Mpl we the number of e-foldings accrued during the inflation after obtain the CMB-scale modes exit the horizon, T0 is the temper- 1=4 6 108 ∼ 3 10−10 ð Þ ature of the CMB at the present time, and TR is the V < × GeV × Mpl: 10 corresponding temperature after reheating. Equation (4) can be summarized as follows. We start with a Hubble Note that this conclusion is independent of the assumption horizon scale which at the time of the inflation is 1=H,by that quantum fluctuations during inflation are the seeds for the end of inflation it is magnified by eNþ, by reheating it primordial structure formation. While we have used a has grown again by aR=aend, and between reheating and the potential V to describe the energy density during inflation, present day it grows by the ratio of the TR=T0 (corrected by our analysis holds for more general scenarios and Eq. (10) the number of degrees of freedom). To solve the horizon can be interpreted as a bound on the energy density during problem, this scale should then be larger than the Hubble the inflationary epoch. scale of the current Universe 1=H0. To obtain the order of We now add the assumption that quantum fluctuations of magnitude of the constraints, we consider rapid reheating the inflaton are responsible for the origin of structure. In and take the reheating temperature to be given by the this case, the power spectrum P of the curvature fluctuation potential energy at the end of inflation, and hence set R (see [19]) is given by ∼ aR aend. For simplicity we also assume that the ratio of   2 g’sis1. 1 ð Þ P ð Þ¼ H k ð Þ Under the assumption that the period of reheating lasts R k 2 ; 11 8π ϵ Mpl less than one Hubble time TR is given by the potential ≈ 1=4 energy V during inflation via TR V . Using the where k is the comoving wave number of the fluctuation Friedmann equation, the Hubble scale 1=H0 is given by mode and HðkÞ is the value of H at the time when the mode the current energy density ρ0 via k exits the Hubble radius. The parameter ϵ determines the deviation of the equation of state in the inflationary phase 1 pffiffiffi ¼ 3ρ−1=2 ð Þ compared to pure de Sitter: 0 Mpl: 5 H0   3 p ϵ ≡ þ 1 ; ð12Þ In turn, ρ0 can be reexpressed in terms of the temper- 2 ρ ature T0: where p and ρ are pressure and energy densities, respec- T 1 ρ ≈ 4 eq ð Þ tively. For inflation to provide the source of structure in the 0 T0 Ω ; 6 T0 m Universe, we need [21] where Ω is the fraction of energy density in matter and T −9 m eq PRðkÞ ∼ 10 : ð13Þ is the temperature at the time of equal matter and radiation, and we have used the fact that the matter energy density 4 Combining (10), (11), and (13) leads to an upper bound today is larger than the radiationp energyffiffiffi density T0 by the on ϵ ¼ 1=2 ð 3 Þ factor Teq=T0. Using H V = Mpl , the condition (4)   then becomes 1 HðkÞ 2 V ϵ ∼ 109 ∼ 109 < 10−31: ð Þ 8π2 24π2 4 14 Mpl Mpl V1=4 pffiffiffiffiffiffiffi V1=4 eNþ ≃ Ω ∼ : ð7Þ ð Þ1=2 m ð Þ1=2 T0Teq T0Teq Since the power spectrum of gravitational waves is given by In the approximation of constant value of H during   inflation, the TCC condition (1) can be written in the form HðkÞ 2 PhðkÞ ∼ ; ð15Þ Mpl Mpl eNþ < : ð8Þ H the tensor to scalar ratio r is given by

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r ¼ 16ϵ < 10−30; ð16Þ The slow-roll equation of motion is where the factor 16 comes from the different normalization 3Hϕ_ ¼ −V0: ð20Þ conventions for the scalar and tensor spectra. While the discussion above assumed that the inflaton dominated the The field range Δϕ which the inflaton field ϕ moves scalar perturbations it is important to note that the TCC during the period of inflation is given by constrains the absolute amplitude of the primordial gravi- tational waves. The bound on r therefore relies only on the jΔϕj ≃ jϕ_ Δtj; ð21Þ TCC bound on the energy in Eq. (10) and the observed P amplitude of R. Allowing scalar perturbations from where Δt is the time period of inflation. We will show that additional fields or a modified sound speed for the inflaton, Δϕ is very small compared to the Planck mass. In this case, for example, will not relax Eq. (16). it is self-consistent to assume that H and ϕ_ are constant. In From (16) we draw the conclusion that any detection of this case using the TCC primordial gravitational waves on cosmological scales   would provide evidence for a different origin of the Mpl primordial gravitational wave spectrum than any infla- Δt ¼ H−1N

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00   1 − 1 2 4 2 V ns ϕ 1− ð3M T0T Þ2 M ≃ ; ð26Þ CMB ns pl eq ð Þ pl 2 < 3 : 33 V ϕ 4 i V0 where n ¼ 1 þ 2η − 6ϵ is the tilt parameter and η ¼ s −9 2 00 Plugging ϕCMB from (27) in (33) with P ≃ 2 × 10 , MplV =V is the second slow-roll parameter. This fixes 4 n ≃ 0.96, T0 ≃ 3K, and T ∼ 10 K leads to V00 from observation. From Eqs. (11) and (19) we find s eq       1 03 2 2 V0 . ϕ V0 n − 1 6 6 10−12 i ð Þ ≃ Pð Þ s ð Þ 4 < . × · : 34 2 2 2 k ; 27 12π ϕ 2 Mpl Mpl Mpl CMB ϕ The above inequality is necessary for the potential to be where CMB is the value of the field when the modes on CMB scales exited the Hubble horizon. Assuming H consistent with the TCC, but it is not sufficient. This is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi because a potential is consistent with the TCC if the ≃ 3 2 remains almost constant H V0= Mpl during the inequality (1) is satisfied for every expansionary trajectory, slow-roll inflation, one can show not just one particular trajectory. 1=4 ¼ 10−10   For energy scale V0 Mpl the potential ϕ ð1 − 0 02ϕ2Þ ½ϕ ϕ ¼ 2 ≃ jηj ðϕ → ϕ Þ ð Þ V0 . defined over the field range i; f ln ϕ N 1 2 ; 28 ½9 7 10−16 2 4 10−15 1 . × Mpl; . × Mpl satisfies all the criteria (10), (30), and (34). These criteria were imposed by where Nðϕ1 → ϕ2Þ is the number of e-folds accrued as ϕ observation and consistency with (1) for the slow-roll ϕ ϕ ϕ ¼ ϕ ϕ ¼ ϕ goes from 1 to 2. If we plug 1 CMB and 2 f trajectory. By numerical analysis, we further verified the into the above identity and use Eq. (7), we find consistency of this potential with the inequality (1) for every expansionary trajectory. This is an example of a jηj jηj 4 2 simple potential that can explain the observation and be ϕf V0 Ωm ¼ jηjNþ ≃ ð Þ e jηj : 29 consistent with the TCC at the same time, however, due to ϕ ð Þ 2 CMB T0Teq its short field range, it suffers from the fine-tuning problem.

Plugging ϕ from (27) into (29) leads to CMB IV. CONCLUSIONS AND DISCUSSION

1þjηj jηj   2 4 2 0 505 We have studied the implications of the recently proposed V Ω V0 . ϕ ≃ 0 m ≃ 3.9 × 105 · ; the TCC for inflationary cosmology. Demanding that the f 2 jηj 1 1 ð Þ 2 122P2πjηj Mpl Mpl T0Teq TCC holds and that the largest scales that we currently probe ð30Þ in cosmology are sub-Hubble at the beginning of the inflationary phase (a necessary condition for the causal generation mechanism of fluctuations of inflationary cos- where in the last step we substituted jηj ≃ 0.02, P ≃ 2 10−9 ≃ 3 ∼ 104 Ω ≃ 0 3 mology to work) leads to an upper bound on the energy scale × , T0 K, Teq K, and m . . This fixes 109 ϕ of inflation which is of the order of GeV. Demanding the end of the field range f in terms of the energy scale V0. that the amplitude of the cosmological perturbations agrees ϕ The only free parameters left are V0 and i. with observations then leads to an upper bound on the Now we impost the TCC for the slow-roll trajectory to ϵ ϵ < 10−31 ϕ ϕ ¼ ϕ generalized slow-roll parameter of the order of . find a constraint in terms of i and V0. Plugging 1 i As a consequence, the tensor to scalar ratio is predicted to be and ϕ2 ¼ ϕ in (28) gives −30 CMB smaller than 10 . A detection of primordial gravitational

1−ns waves via B-mode polarization, pulsar timing measure- ϕ ≃ ϕ 2 N− ð Þ CMB ie ; 31 ments or direct detection, assuming the TCC, would then imply that the source of these gravitational waves is not due where N− is the number of e-folds accrued before the to quantum fluctuations during inflation. modes on CMB scales exit the horizon. The total number of The above conclusions are independent of any assump- ¼ þ e-foldings is Ntotal N− Nþ. From (7) we find tions on the possible single-field nature of inflation. If we then specialize the discussion to the case of single field V1=4 eN ≃ eN− : ð32Þ slow-roll inflation with a canonically normalized inflaton ð Þ1=2 Δϕ T0Teq field, we find that the range which the inflaton field traverses during the inflationary phase is of the order of ϵ1=2 On the other hand, from the TCC, we know that the total Mpl. This raises an initial condition problem for most of N number of e-folds is bounded by e

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We proposed an inverted parabola potential as a simple developed. R. B. thanks the Institute for Theoretical example that is consistent with the TCC and can explain the Physics of the ETH for hospitality. The research at observation at the same time. McGill is supported in part by funds from NSERC and from the Canada Research Chair program. M. L. is ACKNOWLEDGMENTS supported by DOE DE-SC0017848. The research of We thank the Simons Center for Geometry and C. V. is supported in part by the NSF Grant No. PHY- Physics for hospitality during the Summer Workshop on 1719924 and by a grant from the Simons Foundation String Theory during which the ideas for this project were (602883, CV).

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