# Trans-Planckian Censorship and Inflation in Grand Unified Theories

Kai Schmitz<EMAIL_ADDRESS>Theoretical Physics Department, CERN, 1211
Geneva 23, Switzerland

###### Abstract

The recently proposed trans-Planckian censorship conjecture (TCC) amounts to
the claim that inflation models with an inflationary energy scale larger than
$\Lambda_{\rm inf}^{\rm max}\sim 10^{9}\,\textrm{GeV}$ belong to the
swampland, i.e., cannot be embedded into a consistent theory of quantum
gravity. In this paper, we point out that this constraint can be readily
satisfied in D-term hybrid inflation (DHI), which is a well-motivated
inflation scenario in the context of supersymmetric grand unification. In DHI,
the amplitude of the primordial scalar power spectrum originates from a
Fayet–Iliopoulos term of the order of the unification scale, $\sqrt{\xi}\sim
10^{16}\,\textrm{GeV}$. At the same time, the TCC results in an upper bound on
the corresponding gauge coupling constant of $g_{\rm max}\sim 10^{-14}$. We
are able to show that this constraint translates into an upper bound on the
gravitino mass of $m_{3/2}^{\rm max}\sim 10\,\textrm{MeV}$, which opens the
possibility that dark matter is accounted for by thermally produced
gravitinos, if the reheating temperature is close to $T_{\rm reh}\sim
100\,\textrm{TeV}$. Interestingly enough, a somewhat similar gravitino mass
range has recently been derived in a model that aims at explaining dark energy
in terms of axion quintessence and resolving the Hubble tension by means of
decaying gravitino dark matter.

††preprint: CERN-TH-2019-171

Introduction. A standard assumption in modern cosmology is that the initial
conditions of the Hot Big Bang were determined by cosmic inflation, a stage of
accelerated expansion in the early Universe Starobinsky:1980te ; Guth:1980zm ;
Sato:1980yn ; Linde:1981mu ; Albrecht:1982wi . Inflation was originally
conceived to explain the vast size of the observable Universe as well as its
high degree of homogeneity and isotropy on cosmological scales; but in
addition, inflation can also provide the seeds for the postinflationary
formation of structure on smaller scales (see Linde:2005ht ; Lyth:1998xn  for
reviews). One of its key properties is that the spatial extent of causally
connected Hubble patches remains roughly constant during inflation, while at
the same time, microscopic quantum fluctuations are stretched to macroscopic
size. This means that primordial (scalar and tensor) perturbation modes
continuously exit the Hubble horizon during inflation, whereupon they “freeze
out” and classicalize. During the subsequent decelerated expansion, these
classical perturbations then re-enter the Hubble horizon, which causes them to
“thaw” and evolve further. The primordial scalar perturbations thus lead to
the temperature anisotropies in the cosmic microwave background (CMB)
Aghanim:2018eyx ; Akrami:2018odb  and eventually to the large-scale structure
of the Universe, while the primordial tensor perturbations give rise to a (yet
unobserved) stochastic background of gravitational waves Guzzetti:2016mkm
(see also DEramo:2019tit ).

Physical length scales are exponentially stretched during inflation. This
characteristic feature is essential in explaining how it is possible that the
entire observable Universe was initially contained in a single causal patch.
At the same time, it implies that sub-Planckian length scales, $\ell<\ell_{\rm
Pl}=M_{\rm Pl}^{-1}$ (where $M_{\rm Pl}\simeq 2.435\times
10^{18}\,\textrm{GeV}$ denotes the reduced Planck mass) can be easily
stretched to super-horizon size if inflation lasts long enough. This is known
as the trans-Planckian problem of inflationary cosmology Martin:2000xs  (see
also Brandenberger:2000wr ; Niemeyer:2000eh ; Kempf:2001fa ; Easther:2001fi ;
Bozza:2003pr ; Brandenberger:2012aj ). Indeed, regarding general relativity
(GR) as the low-energy effective field theory (EFT) of some unknown theory of
quantum gravity, one expects perturbative calculations in GR to remain valid
only up to some cut-off scale $\Lambda_{\rm EFT}\sim M_{\rm Pl}$. For modes on
sub-Planckian length scales, the standard approach of calculating the
inflationary power spectrum therefore ventures beyond the validity range of
the EFT. In line with this observation, Bedroya and Vafa recently conjectured
that such a situation may never occur in a consistent theory of quantum
gravity Bedroya:2019snp . That is, according to the trans-Planckian censorship
conjecture (TCC), any inflation model that stretches modes on sub-Planckian
length scales to the extent that they eventually exit the Hubble horizon must
lie in the swampland Vafa:2005ui . The TCC is thus the newest member in the
family of quantum gravity swampland conjectures ArkaniHamed:2006dz ;
Ooguri:2006in ; Obied:2018sgi ; Agrawal:2018own ; Garg:2018reu ;
Ooguri:2018wrx  (see Palti:2019pca  for a review).

The TCC has important implications for inflationary cosmology Bedroya:2019tba
(see also Cai:2019hge ; Tenkanen:2019wsd ; Das:2019hto ; Mizuno:2019bxy ;
Brahma:2019unn ; Dhuria:2019oyf ; Torabian:2019zms ; Cai:2019igo ;
Kadota:2019dol ). In particular, it severely constrains the allowed range for
the total number of $e$-folds $N_{\rm inf}$ during inflation, which translates
into upper bounds on the inflationary Hubble rate $H_{\rm inf}$, energy scale
$\Lambda_{\rm inf}=V_{\rm inf}^{1/4}$ (where $V_{\rm inf}$ is the inflaton
potential), and tensor-to-scalar ratio $r$. The main purpose of this paper is
to demonstrate that these bounds can be readily satisfied in D-term hybrid
inflation (DHI) Binetruy:1996xj ; Halyo:1996pp , which is a well-motivated
inflation scenario in the context of supersymmetric grand unified theories
(GUTs). A remarkable outcome of our analysis is that the dimensionful input
parameter of DHI, namely, the Fayet–Iliopoulos (FI) term Fayet:1974jb  can
still be chosen to be of the order of GUT scale, $\sqrt{\xi}\sim
10^{16}\,\textrm{GeV}$, despite the severe TCC bound on the inflationary
energy scale. This hierarchy of scales can be achieved in a technically
natural way by choosing a tiny gauge coupling constant, $g\lesssim 10^{-14}$.
At the same time, the best-fit value for the scalar spectral index,
$n_{s}^{\rm obs}\simeq 0.9649$ Aghanim:2018eyx ; Akrami:2018odb , can be
reproduced for gravitino masses in the sub-GeV range. This points toward
gravitino dark matter (DM) and the mediation of supersymmetry (SUSY) breaking
via gauge mediation.

Trans-Planckian censorship conjecture. We begin by reviewing the TCC bounds on
inflation. According to the TCC, physical length scales that are initially
Planckian must not exit the Hubble radius $H^{-1}$ during inflation,

$\displaystyle e^{N_{\rm inf}}\,\ell_{\rm Pl}<H_{\rm end}^{-1}\,.$ (1)

$N_{\rm inf}=\ln\left(a_{\rm end}/a_{\rm ini}\right)$ is the total number of
$e$-folds during inflation (where $a_{\rm ini}$ and $a_{\rm end}$ denote the
Friedmann-Lemaître-Robertson-Walker scale factor at the beginning and at the
end of inflation, respectively), and $H_{\rm end}^{-1}$ is the Hubble radius
at the end of inflation. Eq. (1) results in an upper bound on the number of
$e$-folds, $N_{\rm inf}<N_{\rm inf}^{\rm max}$,

$\displaystyle N_{\rm inf}^{\rm max}=\ln\left(\frac{M_{\rm Pl}}{H_{\rm
end}}\right)\simeq 43.5+\ln\left(\frac{0.3\,\textrm{GeV}}{H_{\rm
end}}\right)\,,$ (2)

where we anticipated an exceptionally small $H_{\rm end}$.

Meanwhile, inflation is also required to solve the horizon and flatness
problems of the Hot Big Bang, which means that the largest observable scale
today, $\ell_{0}$, must have initially been contained in a single causal
volume. Let us denote the number of $e$-folds before the end of inflation when
the corresponding comoving scale crossed outside the horizon by
$N_{\ell_{0}}$. We are then able to write

$\displaystyle\ell_{0}=\frac{a_{0}}{a_{\rm reh}}\frac{a_{\rm reh}}{a_{\rm
end}}\,e^{N_{\ell_{0}}}H_{\ell_{0}}^{-1}\,,$ (3)

where $H_{\ell_{0}}^{-1}$ is the Hubble radius at the time of horizon
crossing, and $a_{\rm reh}$ and $a_{0}$ denote the scale factor at the end of
reheating and at the present time, respectively. The ratio $a_{\rm reh}/a_{\rm
end}$ depends on the energy density at the end of inflation, $\rho_{\rm
end}=3\,M_{\rm Pl}^{2}H_{\rm end}^{2}$, the energy density at the end of
reheating, $\rho_{\rm reh}=\pi^{2}/30\,g_{*,\rho}^{\rm reh}\,T_{\rm reh}^{4}$,
and the equation-of-state parameter $w=p/\rho$ during reheating,

$\displaystyle N_{\rm reh}=\ln\left(\frac{a_{\rm reh}}{a_{\rm
end}}\right)=\frac{1}{3\left(1+w\right)}\ln\left(\frac{\rho_{\rm
end}}{\rho_{\rm reh}}\right)\,.$ (4)

The ratio $a_{0}/a_{\rm reh}$, on the other hand, can be determined if one
assumes entropy to be conserved after reheating,

$\displaystyle N_{0}=\ln\left(\frac{a_{0}}{a_{\rm
reh}}\right)=\frac{1}{3}\ln\bigg{(}\frac{g_{*,s}^{\rm
reh}}{g_{*,s}^{0}}\bigg{)}+\ln\left(\frac{T_{\rm reh}}{T_{0}}\right)\,.$ (5)

Below, we will treat the Hubble rate $H_{\rm end}$ and the reheating
temperature $T_{\rm reh}$ as free parameters, while fixing all other
parameters in Eqs. (4) and (5) at specific values. We will restrict ourselves
to the standard case of matter-dominated expansion during reheating, $w=0$;
the CMB photon temperature at the present time is given by $T_{0}\simeq
2.725\,\textrm{K}\simeq 2.349\times 10^{-13}\,\textrm{GeV}$ Fixsen:2009ug ;
and for the effective numbers of relativistic degrees of freedom, we will use
the values in the minimal supersymmetric standard model (MSSM),
$g_{*,\rho}^{\rm reh}=g_{*,s}^{\rm reh}=915/4$ and $g_{*,s}^{0}=43/11$.

In any viable model of inflation, $N_{\ell_{0}}$ must be smaller than the
maximal number of $e$-folds $N_{\rm inf}^{\rm max}$. Combining all of the
expressions above, this requirement, $N_{\ell_{0}}<N_{\rm inf}^{\rm max}$, can
be used to derive an upper bound on the Hubble rate,

$\displaystyle H_{\rm inf}<H_{\rm inf}^{\rm max}=M_{\rm
Pl}\left[\bigg{(}\frac{g_{*,s}^{\rm
reh}}{g_{*,s}^{0}}\bigg{)}^{1/3}\left(\frac{90}{\pi^{2}g_{*,\rho}^{\rm
reh}}\right)^{\alpha/2}\left(\frac{M_{\rm Pl}}{T_{\rm
reh}}\right)^{2\alpha-1}\frac{1}{T_{0}\,\ell_{0}}\right]^{1/\left(2-\alpha\right)}\,,\qquad\alpha=\frac{2}{3\left(1+w\right)}\,,$
(6)

where $\alpha$ characterizes the time dependence of the scale factor during
reheating, $a\left(t\right)\propto t^{\alpha}$. Formally, the Hubble rate
$H_{\rm inf}$ in Eq. (6) is defined as $H_{\rm inf}=H_{\ell_{0}}^{\beta}H_{\rm
end}^{1-\beta}$, where $\beta=1/\left(2-\alpha\right)$. In this sense, the
bound in Eq. (6) is completely general.111In particular, it is more general
than the bounds in Bedroya:2019tba ; Tenkanen:2019wsd ; Mizuno:2019bxy , which
are based on additional assumptions, such as instantaneous reheating or a
quasi-constant Hubble rate during inflation. In practice, however, one can
work with $H_{\rm inf}\approx H_{\ell_{0}}\approx H_{\rm end}$, as any
inflation model consistent with the bound in Eq. (6) anyway results in a small
variation of the Hubble rate during inflation, $-\dot{H}/H^{2}\ll 1$.

In passing, we also mention that Eq. (6) can be generalized to account for the
possibility of entropy production after reheating. Suppose the comoving
entropy density changes by a factor $\gamma\geq 1$ between reheating and
today,

$\displaystyle\gamma=\frac{g_{*,s}^{0}T_{0}^{3}a_{0}^{3}}{g_{*,s}^{\rm
reh}T_{\rm reh}^{3}a_{\rm reh}^{3}}\,.$ (7)

Late-time entropy production can thus be described by rescaling $g_{*,s}^{\rm
reh}$ by one power of the factor $\gamma$ Ji:2019gfy , such that

$\displaystyle H_{\rm inf}^{\rm
max}\rightarrow\gamma^{1/3/\left(2-\alpha\right)}H_{\rm inf}^{\rm max}\,.$ (8)

This rescaling allows one to weaken the bound on $H_{\rm inf}$ by considering
a large amount of entropy production after reheating, $\gamma\gg 1$. In the
remainder of this paper, we will, however, ignore this option and focus on the
$\gamma=1$ case.

To evaluate the right-hand side of Eq. (6), one needs to specify the scale
$\ell_{0}$. The authors of Bedroya:2019tba ; Tenkanen:2019wsd ; Dhuria:2019oyf
identified $\ell_{0}$ with the present Hubble radius, $H_{0}^{-1}\simeq
3.0/h\,\textrm{Gpc}$ (where
$H_{0}=100\,h\,\textrm{km}/\textrm{s}/\textrm{Mpc}$ and $h_{\rm cmb}\simeq
0.6736$ according to the 2018 PLANCK data Aghanim:2018eyx ; Akrami:2018odb ).
Meanwhile, the authors of Mizuno:2019bxy ; Torabian:2019zms  worked with the
comoving scale that was of horizon size at the time of matter-dark-energy
equality, $a_{0}/k_{\Lambda}=2^{-1/2}\,\Omega_{\Lambda}^{-1/6}\,\Omega_{\rm
m}^{-1/3}H_{0}^{-1}\simeq 3.3/h\,\textrm{Gpc}$, which is very close to the
largest comoving scale that ever became horizon-sized during the decelerated
expansion, $a_{0}/k_{\rm
min}=2^{1/3}\,3^{-1/2}\,\Omega_{\Lambda}^{-1/6}\,\Omega_{\rm
m}^{-1/3}H_{0}^{-1}\simeq 3.4/h\,\textrm{Gpc}$. In our analysis, we will
consider yet another scale — the optical horizon $d_{\rm opt}\simeq
9.4/h\,\textrm{Gpc}$ of our Universe Mukhanov:2005sc ,

$\displaystyle\frac{d_{\rm opt}}{H_{0}^{-1}}=\int_{a_{\rm
dec}}^{a_{0}}\frac{da}{a_{0}}\left[\Omega_{\rm r}+\frac{a}{a_{0}}\,\Omega_{\rm
m}+\left(\frac{a}{a_{0}}\right)^{4}\Omega_{\Lambda}\right]^{-1/2}\,,$ (9)

where $a_{\rm dec}$ denotes the scale factor at CMB decoupling. Note that
$d_{\rm opt}$ is almost identical to our particle horizon, which follows from
Eq. (9) by taking the limit $a_{\rm dec}\rightarrow 0$.

Upon identifying $\ell_{0}$ with $d_{\rm opt}$, Eq. (6) results in

$\displaystyle H_{\rm inf}^{\rm max}\simeq
0.35\,\textrm{GeV}\left(\frac{h}{h_{\rm
cmb}}\right)^{3/4}\left(\frac{100\,\textrm{TeV}}{T_{\rm reh}}\right)^{1/4}\,,$
(10)

where we now fixed all other parameters as described above. This constraint
immediately translates into upper bounds on the inflationary energy scale,
$\Lambda_{\rm inf}<\Lambda_{\rm inf}^{\rm max}$,

$\displaystyle\Lambda_{\rm inf}^{\rm max}\simeq 1.2\times
10^{9}\,\textrm{GeV}\left(\frac{h}{h_{\rm
cmb}}\right)^{3/8}\left(\frac{100\,\textrm{TeV}}{T_{\rm reh}}\right)^{1/8}\,,$
(11)

and the tensor-to-scalar ratio, $r=A_{t}/A_{s}<r_{\rm max}$,

$\displaystyle r_{\rm max}\simeq 2.0\times 10^{-30}\left(\frac{h}{h_{\rm
cmb}}\right)^{3/2}\left(\frac{100\,\textrm{TeV}}{T_{\rm reh}}\right)^{1/2}\,.$
(12)

Here, we used the measured value of the amplitude of the scalar power
spectrum, $A_{s}^{\rm obs}\simeq 2.10\times 10^{-9}$ Aghanim:2018eyx ;
Akrami:2018odb , as well as two standard relations in the description of slow-
roll inflation, $V_{\rm inf}=3\,M_{\rm Pl}^{2}H_{\rm inf}^{2}$ and
$A_{t}=2\,(H_{\rm inf}/(\pi M_{\rm Pl}))^{2}$.

D-term hybrid inflation. The bounds in Eqs. (10), (11), and (12) represent
severe constraints on the dynamics of cosmic inflation. Nonetheless, it is
straightforward to satisfy these constraints in DHI, as we will now discuss.
First of all, we recall that supersymmetric hybrid inflation is an attractive
model of inflation on general grounds because it allows one to establish a
connection between cosmology (inflation) and particle physics (grand
unification). The crucial observation is that hybrid inflation ends via a
rapid second-order phase transition Linde:1991km ; Linde:1993cn . This so-
called waterfall transition can be identified as the spontaneous breaking of
(a subgroup of) a local GUT symmetry, such as, e.g., $U\left(1\right)_{B-L}$
Buchmuller:2010yy ; Buchmuller:2011mw ; Buchmuller:2012wn ; Buchmuller:2012bt
; Buchmuller:2013lra ; Schmitz:2012kaa ; Buchmuller:2013dja ; Domcke:2017xvu ;
Domcke:2017rzu .

In this paper, we will not specify an explicit GUT embedding of DHI but simply
focus on the minimal scenario based on a $U(1)$ FI term. Similarly, we will
also refrain from considering F-term hybrid inflation (FHI) Copeland:1994vg ;
Dvali:1994ms  as an alternative to DHI. This model comes with less parametric
freedom than DHI (unlike in DHI, the $U(1)$ gauge coupling constant $g$ plays
no role in FHI) and requires a more complicated treatment due to the fact that
it is a two-field model of inflation Buchmuller:2014epa  (see also
Rehman:2009nq ). A detailed discussion of both FHI and DHI can be found in
Schmitz:2018nhb . Let us now review some of the results on DHI derived in
Schmitz:2018nhb  that will allow us to satisfy the TCC bound in Eq. (6).

As in Schmitz:2018nhb , we will consider DHI in combination with a SUSY-
breaking Polonyi sector Polonyi:1977pj  (see also Buchmuller:2000zm ). The
superpotential of our model is therefore given as follows,

$\displaystyle W=\kappa\,S\,\Phi\bar{\Phi}+\mu_{X}^{2}\,X+W_{0}\,.$ (13)

Here, $S$ denotes the inflaton field, $\Phi$ and $\bar{\Phi}$ are two
oppositely charged waterfall fields, and $X$ is the Polonyi field. $\kappa$
represents the dimensionless inflaton Yukawa coupling, $\mu_{X}$ is the SUSY-
breaking scale in the Polonyi sector, and $W_{0}$ is an $R$-symmetry-breaking
constant that is fixed by the requirement that inflation must end in a
Minkowski vacuum with zero cosmological constant. We assume that the Polonyi
field is stabilized at $\left<X\right>=0$ in consequence of additional hidden-
sector interactions. $\mu_{X}$ and $W_{0}$ can thus be related to the
gravitino mass $m_{3/2}$ as follows,

$\displaystyle\mu_{X}^{2}=\sqrt{3}\,M_{\rm Pl}\,m_{3/2}\,,\quad
W_{0}=m_{3/2}\,M_{\rm Pl}^{2}\,.$ (14)

We will work with a canonical Kähler potential for all fields in Eq. (13),
supplemented by a Planck-suppressed coupling between the inflaton and the
Polonyi field,

$\displaystyle K\supset\frac{\chi}{M_{\rm
Pl}^{2}}\,S^{\dagger}S\,X^{\dagger}X\,.$ (15)

This operator is allowed by all symmetries and expected to be present in the
EFT at energies below $\Lambda_{\rm EFT}\sim M_{\rm Pl}$. Below, we will set
the $\mathcal{O}\left(1\right)$ Wilson coefficient $\chi$ to unity, for
simplicity (see also Schmitz:2018nhb ). Finally, we assume a nonvanishing FI
term in the D-term scalar potential,222Constant FI terms cannot be
consistently coupled to supergravity Komargodski:2009pc ; Dienes:2009td  (see
also Cribiori:2017laj ; Kuzenko:2018jlz ; Antoniadis:2018cpq ). We therefore
assume an effective field-dependent FI term, i.e., an FI term that is
generated in consequence of spontaneous symmetry breaking and whose magnitude
is controlled by the vacuum expectation values (VEVs) of a number of moduli,
$\xi=\xi\left(\left<\varphi_{i}\right>\right)$ (see Domcke:2014zqa  for an
explicit model).

$\displaystyle
V_{D}=\frac{g^{2}}{2}\left[\xi-\left(\left|\phi\right|^{2}-\left|\bar{\phi}\right|^{2}\right)\right]^{2}\,,$
(16)

which is responsible for the nonvanishing vacuum energy density that drives
inflation, $V_{D}^{0}=3\,M_{\rm Pl}^{2}H_{\rm inf}^{2}=1/2\,g^{2}\xi^{2}$.

Eqs. (13) to (16) fully specify our inflaton model. In particular, they enable
us to compute the scalar potential for the real inflaton field,
$s=\sqrt{2}\left|S\right|$. In this potential, inflation takes place at field
values larger than the critical field value $s_{\rm crit}=g/\kappa\,v$, which
marks the location of the tachyonic instability in the potential that triggers
the waterfall phase transition. Here,
$v=\sqrt{2}\left<\Phi\right>=\sqrt{2\xi}$ denotes the VEV of the symmetry-
breaking waterfall field after the phase transition. In the following, we will
consider inflaton Yukawa couplings $\kappa$ of $\mathcal{O}\left(0.1\cdots
1\right)$. As discussed in Schmitz:2018nhb , this implies that the CMB pivot
scale $\ell_{*}=20\,\textrm{Mpc}$ exits the Hubble horizon at a field value
far away from the instability, $s_{*}\gg s_{\rm crit}$. In addition, we will
assume a tiny $U(1)$ gauge coupling constant, $g\ll\kappa$, in order to
suppress the inflationary Hubble rate.333The opposite scenario, $\kappa\ll g$,
has been considered in Buchmuller:2014rfa ; Buchmuller:2014dda  (see also
Bryant:2016tzg ; Bryant:2016sjj ; Ishiwata:2018dxg ; Gunji:2019wtk ), which
results in a super-Planckian critical field value, $s_{\rm crit}>M_{\rm Pl}$.
In this case, inflation occurs during the waterfall transition and leads to an
$\mathcal{O}\left(0.1\right)$ tensor-to-scalar ratio. In this parameter
regime, we obtain the following scalar potential,

$\displaystyle V\simeq
V_{D}^{0}+\frac{1}{2}\,m_{s}^{2}\,s^{2}+V_{1\ell}^{0}\,\ln\left(\frac{s}{s_{\rm
crit}}\right)\,.$ (17)

The mass parameter $m_{s}$ stems from soft SUSY breaking,

$\displaystyle m_{s}^{2}=\left(1-3\,\chi\right)m_{3/2}^{2}\,,$ (18)

while the logarithmic term arises in consequence of one-loop radiative
corrections when integrating out the heavy waterfall fields $\Phi$ and
$\bar{\Phi}$ along the inflationary trajectory,

$\displaystyle V_{1\ell}^{0}=\frac{m_{D}^{4}}{8\pi^{2}}\,,\quad
m_{D}=g\sqrt{\xi}\,.$ (19)

Note that, in consequence of our choice of $\chi$, the inflaton mass turns out
to be tachyonic during inflation, $m_{s}^{2}<0$. If this was not the case,
i.e., for $\chi<1/3$, we would not be able to reproduce the best-fit value of
the spectral index.

Based on the potential in Eq. (17), one is now able to perform a standard
analysis of slow-roll inflation. We begin by writing down the slow-roll
equation of motion,

$\displaystyle\frac{ds^{2}}{dN}=2\,\Delta\left(s^{2}-s_{\rm
max}^{2}\right)\,,$ (20)

where the quantities $\Delta$ and $s_{\rm max}$ are defined as follows,

$\displaystyle\Delta=\frac{m_{s}^{2}M_{\rm Pl}^{2}}{V_{D}^{0}}\,,\quad s_{\rm
max}=\left|\frac{V_{1\ell}^{0}}{m_{s}^{2}}\right|^{1/2}\,.$ (21)

$s_{\rm max}$ marks the position of a hilltop (i.e., local maximum) in the
potential. Slow-roll inflation takes place at field values below $s_{\rm max}$
and above $s_{\rm fast}$, where the latter represents the field value at which
the slow-roll condition $\eta=M_{\rm Pl}^{2}\,V^{\prime\prime}/V<\eta_{\rm
max}=10^{-0.5}$ becomes violated,444For $\kappa\lesssim 10^{-3}$, the hilltop
moves in the direct vicinity of the tachyonic instability, $s_{\rm max}\simeq
s_{\rm crit}$, such that successful inflaton can only be realized in a very
small region in field space Schmitz:2018nhb . This problem is absent for
$\kappa\sim 0.1\cdots 1$, which leads to $s_{\rm max}\gg s_{\rm crit}$.

$\displaystyle s_{\rm fast}=\left|\frac{\Delta}{\eta_{\rm
max}+\Delta}\right|^{1/2}\,s_{\rm max}\,.$ (22)

Imposing the boundary condition $s=s_{\rm fast}$ at $N=0$, the slow-roll
equation in Eq. (20) has the following solution,

$\displaystyle s^{2}\left(N\right)=s_{\rm
max}^{2}\left[1+F\left(N\right)\right]\,,$ (23)

with $F$ as a function of the number of $e$-folds $N$ given as

$\displaystyle F\left(N\right)=-\left(1+\frac{\Delta}{\eta_{\rm
max}+\Delta}\right)e^{2N\Delta}\,.$ (24)

This solution for $s\left(N\right)$ allows one to work out the slow-roll
parameters $\varepsilon=1/2\,M_{\rm Pl}^{2}\left(V^{\prime}/V\right)^{2}$ and
$\eta=M_{\rm Pl}^{2}\,V^{\prime\prime}/V$,

$\displaystyle\varepsilon=\left(\frac{s_{\rm max}}{M_{\rm
Pl}}\right)^{2}\frac{\left(F\Delta\right)^{2}}{2\left(1+F\right)}\,,\quad\eta=\frac{\left(2+F\right)\Delta}{1+F}\,,$
(25)

which in turn can be used to compute the amplitude,

$\displaystyle A_{s}=\left.\frac{V}{24\pi^{2}\,\varepsilon\,M_{\rm
Pl}^{4}}\vphantom{\left(\frac{\sqrt{\xi}}{M_{\rm
Pl}}\right)^{4}}\right|_{N_{*}}=\left.\frac{1+F}{6\left|\Delta\right|F^{2}}\left(\frac{\sqrt{\xi}}{M_{\rm
Pl}}\right)^{4}\right|_{N_{*}}\,,$ (26)

and index of the primordial scalar power spectrum,

$\displaystyle
n_{s}\approx\left.1+2\eta\vphantom{\frac{2\left(2+F\right)\Delta}{1+F}}\right|_{N_{*}}=\left.1+\frac{2\left(2+F\right)\Delta}{1+F}\right|_{N_{*}}\,.$
(27)

Here, we neglected $\varepsilon\ll\eta$ in computing
$n_{s}=1+2\,\eta-6\,\varepsilon$. All $N$-dependent quantities in Eqs. (26)
and (27) need to be evaluated at $N_{*}$, i.e., the number of $e$-folds
between the time of CMB horizon exit and the end of inflation,

$\displaystyle N_{*}$ $\displaystyle=\ln\left(\ell_{*}H_{\rm
inf}\right)-N_{0}-N_{\rm reh}$ (28) $\displaystyle\simeq
36.8+\frac{1}{3}\ln\left(\frac{H_{\rm
inf}}{0.3\,\textrm{GeV}}\right)+\frac{1}{3}\ln\left(\frac{T_{\rm
reh}}{100\,\textrm{TeV}}\right)\,.$

Remarkably enough, the expressions in Eqs. (26) and (27) offer enough
parametric freedom to simultaneously satisfy the two constraints
$A_{s}=A_{s}^{\rm obs}\simeq 2.10\times 10^{-9}$ and $n_{s}=n_{s}^{\rm
obs}\simeq 0.9649$. For $\eta_{\rm max}=10^{-0.5}$ and $N_{*}=36.8$, the
condition $n_{s}=n_{s}^{\rm obs}$ results in a unique value for $\Delta$,

$\displaystyle\Delta=-\frac{2}{3}\left(\frac{m_{3/2}}{H_{\rm
inf}}\right)^{2}\simeq-3.0\times 10^{-3}\,.$ (29)

With $\Delta$ fixed at this value, $A_{s}$ becomes a function of the FI
parameter $\xi$ only. $A_{s}=A_{s}^{\rm obs}$ is then satisfied for

$\displaystyle\sqrt{\xi}\simeq 0.80\times 10^{16}\,\textrm{GeV}\,,$ (30)

or $v\simeq 1.1\times 10^{16}\,\textrm{GeV}$ in terms of the VEV of the
waterfall field $\Phi$. This essentially coincides with the value of the GUT
scale that one expects in SUSY GUT models, thus underlining the close
connection between DHI and supersymmetric grand unification. With $\Delta$
determined by $n_{s}^{\rm obs}$ and $\sqrt{\xi}$ by $A_{s}^{\rm obs}$, the
Hubble rate $H_{\rm inf}$ is now solely controlled by the gauge coupling
constant $g$,

$\displaystyle H_{\rm inf}$ $\displaystyle\simeq
0.11\,\textrm{GeV}\left(\frac{g}{10^{-14}}\right)\,,$ (31)

and similarly for the gravitino mass $m_{3/2}$,

$\displaystyle m_{3/2}$ $\displaystyle\simeq
7.2\,\textrm{MeV}\left(\frac{g}{10^{-14}}\right)\,.$ (32)

Eqs. (31) and (32) are the main technical results of this paper. We observe
that, by choosing a very small gauge coupling constant $g$, DHI always allows
one to satisfy the TCC bounds on inflation, while at the same time being in
accord with the best-fit values for $A_{s}^{\rm obs}$ and $n_{s}^{\rm obs}$.
Indeed, by comparing $H_{\rm inf}$ in Eq. (31) with the maximal value $H_{\rm
inf}^{\rm max}$ in Eq. (10), we obtain an upper bound on $g$,

$\displaystyle g_{\rm max}\simeq 3.3\times 10^{-14}\left(\frac{h}{h_{\rm
cmb}}\right)^{3/4}\left(\frac{100\,\textrm{TeV}}{T_{\rm reh}}\right)^{1/4}\,,$
(33)

which immediately translates into a maximal $m_{3/2}$ value,

$\displaystyle m_{3/2}^{\rm max}\simeq 23\,\textrm{MeV}\left(\frac{h}{h_{\rm
cmb}}\right)^{3/4}\left(\frac{100\,\textrm{TeV}}{T_{\rm reh}}\right)^{1/4}\,.$
(34)

Our results in this section serve as a proof of principle that a small gauge
coupling constant in a GUT-inspired scenario can lead to an inflationary
energy scale that is parametrically smaller than the GUT scale and thus
consistent with the TCC bound in Eq. (6). Of course, our model does not
address the question why $g$ should be so small, nor does it explain why the
supergravity and radiative corrections should conspire in such a way that the
scalar potential ends up being extremely flat. These questions are beyond the
scope of this work and need to be addressed in the UV completion of our model
(see, e.g., Halyo:1999fd ; Halyo:1999wn  for related earlier work in string
theory).

Discussion. In light of the results derived in the previous section, several
comments are in order.

Distance and de Sitter conjectures. First of all, we point out that DHI also
trivially satisfies the swampland distance conjecture Ooguri:2006in , which
claims that scalar fields can at most traverse an $\mathcal{O}\left(M_{\rm
Pl}\right)$ distance in field space. In our case, the inflaton field range is
always bounded from above by the local maximum at $s_{\rm max}$, by
construction,

$\displaystyle s_{\rm max}=\frac{g\,M_{\rm
Pl}}{2\pi\sqrt{\left|\Delta\right|}}\simeq 7.1\times
10^{4}\,\textrm{GeV}\left(\frac{g}{10^{-14}\,\textrm{GeV}}\right)\,,$ (35)

which is parametrically smaller than the Planck scale. The small gauge
coupling constant $g$ is therefore instrumental in satisfying both the TCC and
the distance conjecture. At the same time, there is a slight tension between
DHI and the refined de Sitter conjecture Garg:2018reu ; Ooguri:2018wrx , just
as in most other models of single-field slow-roll inflation. The refined de
Sitter conjecture imposes constraints on the gradient and curvature of the
potential, which can be expressed in terms of the slow-roll parameters,

$\displaystyle\varepsilon=\frac{M_{\rm
Pl}^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2}>\frac{c^{2}}{2}\quad\textrm{or}\quad\eta=M_{\rm
Pl}^{2}\,\frac{V^{\prime\prime}}{V}<-c^{\prime}\,,$ (36)

where $c$ and $c^{\prime}$ are two positive numbers of
$\mathcal{O}\left(1\right)$. In DHI, $\varepsilon$ is always severely
suppressed, $\varepsilon\ll\eta$; at the local maximum at $s_{\rm max}$, it
even vanishes. The relevant condition in our case is therefore the constraint
on $\eta$. However, as we require our model to reproduce the best-fit value
for $n_{s}$, we know that $\left|\eta\right|$ cannot be of
$\mathcal{O}\left(1\right)$ or even larger. Instead, we have
$\eta\simeq\left(n_{s}^{\rm obs}-1\right)/2\simeq-0.01755$ at the time of CMB
horizon exit. This means that the refined de Sitter conjecture can only be
satisfied if the coefficient $c^{\prime}$ should turn out to be suppressed by
a factor of $\mathcal{O}\left(10^{-2}\right)$ compared to the naive
expectation $c^{\prime}\sim 1$. This discrepancy requires more work; however,
it is important to note that it is not specific to DHI but rather concerns
most models of single-field slow-roll inflation.

Initial conditions. In the parameter regime considered in this paper, DHI
amounts to a particular realization of hilltop inflation. This indicates that
it should be possible to satisfy the TCC bound also in other inflation models
that equally feature a very flat local maximum in the potential. In our case,
the hilltop arises from the interplay of logarithmic radiative corrections and
a soft SUSY-breaking mass in supergravity; but it is well conceivable that a
potential similar to the one in Eq. (17) could also have a different origin at
high energies. Meanwhile, inflaton near a hilltop also turns the spotlight on
the initial conditions of inflation. In order to obtain viable inflation that
ends in a waterfall phase transition, the initial inflaton field value $s_{\rm
ini}$ must be located on the correct side of the hilltop, $s_{\rm ini}<s_{\rm
max}$. At face value, this suggests that $s_{\rm ini}$ must be tuned. On the
other hand, one may speculate that the inflaton field initially starts out on
the other side of the hilltop, $s>s_{\rm max}$, and then jumps over or tunnels
through the potential barrier. Such a mechanism may require modifications of
the potential at large field values and we leave a more detailed investigation
for future work. Alternatively, one may suppose that the GUT embedding of DHI
at high energies provides a dynamical mechanism that automatically positions
the inflaton field near the local maximum in the potential.

In addition to the initial inflaton field value $s_{\rm ini}$, the initial
inflaton velocity $\dot{s}_{\rm ini}$ also appears to be tuned. As pointed out
in Bedroya:2019tba , this follows from the fact that the TCC automatically
implies that $\dot{s}_{\rm ini}^{2}/V_{\rm
ini}^{\vphantom{2}}\sim\varepsilon\ll 1$, which means that the initial kinetic
energy of the inflaton field must be severely suppressed compared to its
initial potential energy. There is no straightforward reason why this should
be the case. Naively, one would rather expect the initial energy density to be
more or less equilibrated, $\dot{s}_{\rm ini}^{2}\sim V_{\rm
ini}^{\vphantom{2}}$. The small initial inflaton velocity at the beginning of
the inflationary trajectory thus represents another open question that needs
to be addressed in the GUT embedding of our model at high energies. As before,
we stress that this issue is not specific to DHI but rather a general
implication of the TCC that applies to essentially all models of small-field
inflation.

Observational signatures. There are several ways to test the scenario proposed
in this paper. For instance, one way to falsify it would consist in the
observation of a large tensor-to-scalar ratio in future CMB experiments. An
$r$ value as large as, say, $r\sim 0.001\cdots 0.1$ would contradict the TCC
bound in Eq. (12) and hence point to a different origin of the primordial
tensor spectrum than the usual vacuum fluctuations of the metric tensor. In
this case, one would have to consider inflaton models that provide additional
sources of primordial tensor perturbations, or abandon the paradigm of cosmic
inflation altogether.

Another important phenomenological aspect is that our model, as defined in
Eqs. (13) to (16), results in the production of cosmic strings during the
waterfall phase transition after inflation. The tension $\mu_{\rm cs}$ of
these strings is determined by the energy scale of spontaneous symmetry
breaking, i.e., the VEV of the waterfall field $\Phi$,

$\displaystyle G\mu_{\rm cs}\simeq 2.1\times
10^{-6}\left(\frac{v}{10^{16}\,\textrm{GeV}}\right)^{2}\,,$ (37)

where $G=\left(8\pi M_{\rm Pl}^{2}\right)^{-1}$ denotes Newton’s constant and
where we used that the strings produced after DHI satisfy the Bogomolny limit,
$\mu_{\rm cs}=\pi v^{2}$ Bogomolny:1975de . The string tension in Eq. (37)
exceeds the current CMB bound on $G\mu_{\rm cs}$ by roughly an order of
magnitude, $G\mu_{\rm cs}^{\rm cmb}\lesssim 10^{-7}$ Charnock:2016nzm ;
Lizarraga:2016onn . At first sight, this represents a problem. However, at
second sight, the production of cosmic strings after DHI could also turn into
virtue; namely, if they should be unstable, which would be the case if the
gauge group of our model was embedded in a semisimple GUT gauge group at
higher energies Vilenkin:1982hm  (see also Monin:2008mp ; Dror:2019syi ). This
is a reasonable assumption and could lead to a promising scenario where the
string network first emits an observable signal in gravitational waves
Auclair:2019wcv  and then decays during the radiation-dominated era, so as to
circumvent the CMB bound on $G\mu_{\rm cs}$. We will come back to such a
scenario in future work Buchmuller:2019gfy . As an alternative to decaying
cosmic strings, one may also extend our model by additional symmetry-breaking
fields that are marginally coupled to the waterfall fields of DHI and that
break the $U(1)$ symmetry already during inflation. In this case, cosmic
strings would be produced at early times and subsequently strongly diluted in
consequence of the exponential expansion during inflation (see, e.g.,
Domcke:2017xvu ; Domcke:2017rzu ).

Finally, we mention that our model has interesting implications for the
particle spectrum at low energies. The key observation is that the spontaneous
$U(1)$ breaking during the waterfall transition results in a massive vector
multiplet with a supersymmetric mass contribution

$\displaystyle
m_{V}=gv=100\,\textrm{GeV}\left(\frac{g}{10^{-14}}\right)\left(\frac{v}{10^{16}\,\textrm{GeV}}\right)\,.$
(38)

This multiplet consists of the real $U(1)$ Higgs boson, the $U(1)$ Higgsino,
the $U(1)$ gaugino, and the massive $U(1)$ vector boson. At the end of the
day, the masses of these particles are potentially vastly separated from each
other, depending on the details of SUSY breaking and its mediation to the
waterfall sector. It is still an interesting observation that the combination
of the GUT-scale VEV $v$ and the tiny gauge coupling constant $g$ readily
results in a vector mass in the vicinity of the electroweak scale. Therefore,
depending on the details of the coupling between the waterfall fields and the
standard model, some of the particles contained in the massive $U(1)$ vector
multiplet could be within the reach of collider experiments. More work in this
direction is certainly desirable.

Reheating temperature. The coupling between the inflaton sector and the
standard model also determines the reheating temperature after inflation. In
the following, we will not specify the exact nature of this coupling, which
requires further model building and is thus beyond the scope of this work.
Instead, we will merely assume that the duration of reheating is controlled by
the perturbative decay of the $U(1)$ Higgs field $\phi$ (see also
Buchmuller:2010yy ; Buchmuller:2011mw ; Buchmuller:2012wn ; Buchmuller:2012bt
; Buchmuller:2013lra ; Schmitz:2012kaa ; Buchmuller:2013dja ; Domcke:2017xvu ;
Domcke:2017rzu ). Then, if we parametrize the $\phi$ decay rate
$\Gamma_{\phi}$ in a model-independent way by an effective coupling constant
$\lambda_{\rm eff}$,

$\displaystyle\Gamma_{\phi}=\frac{\lambda_{\rm eff}^{2}}{8\pi}\,m_{\phi}\,,$
(39)

the reheating temperature can be estimated as follows,

$\displaystyle T_{\rm reh}$
$\displaystyle=\left(\frac{90}{\pi^{2}g_{*,\rho}}\right)^{1/4}\sqrt{\Gamma_{\phi}\,M_{\rm
Pl}}$ (40) $\displaystyle\simeq 140\,\textrm{TeV}\left(\frac{\lambda_{\rm
eff}}{10^{-4}}\right)\bigg{(}\frac{m_{\phi}}{100\,\textrm{GeV}}\bigg{)}^{1/2}\,.$

Here, we have chosen a scalar mass $m_{\phi}$ of the same order of magnitude
as the vector mass in Eq. (38) and an effective coupling constant
$\lambda_{\rm eff}$ that allows us to avoid the thermal overproduction of
gravitinos (see below).

At temperatures of $\mathcal{O}\left(100\right)\,\textrm{TeV}$, the baryon
asymmetry of the Universe can be produced via thermal leptogenesis
Fukugita:1986hr , provided there is a slight degeneracy in the heavy-neutrino
mass spectrum that resonantly enhances the $CP$ asymmetry in heavy-neutrino
decays Pilaftsis:1997dr ; Pilaftsis:1997jf ; Pilaftsis:2003gt ; Dev:2017wwc
(see also Brdar:2019iem ; Brivio:2019hrj  for recent work on resonant
leptogenesis). Alternatively, there is a variety of leptogenesis scenarios in
the literature that are based on minimal extensions of the simplest type-I
seesaw model and that accomplish to successfully generate the baryon asymmetry
at temperatures $T\lesssim 100\,\textrm{TeV}$ (see Hugle:2018qbw ;
Borah:2018rca ; Alanne:2018brf  for three recent examples based on scalar
extensions of the type-I seesaw model).

Weak-gravity conjecture. Our inflation model is also consistent with the weak-
gravity conjecture (WGC) ArkaniHamed:2006dz . The electric WGC requires the
presence of states with charge $q$ and mass $m<m_{\rm
WGC}=\sqrt{2}\,gq\,M_{\rm Pl}$ in the particle spectrum. This requirement is
readily satisfied by the particles in the massive vector multiplet, whose mass
is induced by the GUT scale rather than the Planck scale, $m_{V}/m_{\rm
WGC}\sim v/M_{\rm Pl}\ll 1$ [see Eq. (38)]. The magnetic WGC meanwhile states
that the effective field theory describing the dynamics of the inflaton field
is only valid up to a UV cutoff scale $\Lambda_{\rm WGC}\sim gM_{\rm Pl}$,
which is at most of $\mathcal{O}\left(100\right)\,\textrm{TeV}$ in our
scenario. The scale $\Lambda_{\rm WGC}$ needs to be compared to the inflaton-
dependent energy scales in the scalar potential: the inflaton field value,
which is bounded by $s_{\rm max}\lesssim 100\,\textrm{TeV}$ [see Eq. (35)],
and the supergravity and radiative terms, whose fourths roots do not become
larger than $\mathcal{O}\left(100\right)\,\textrm{GeV}$. All relevant
interactions of the inflaton field thus occur at energies below the UV cutoff
scale $\Lambda_{\rm WGC}$. By contrast, the energy density stored in the
vacuum, $V_{D}^{0}=1/2\,g^{2}\xi^{2}$, is assumed to originate from a separate
sector (see footnote 2). This GUT sector is responsible for GUT symmetry
breaking, the generation of the FI term $\xi$, and necessarily requires a
larger UV cutoff scale.555The inflaton field is obviously not responsible for
GUT symmetry breaking. It is thus not necessary to have a reliable description
of the inflaton dynamics all the way up to the GUT scale. In this paper, we
shall not specify the GUT embedding of our model. For our purposes, it
suffices to assume that the GUT sector contains a set of appropriately charged
moduli whose VEVs induce a $U(1)$ FI term in the inflaton sector. Given the
fact that the inflaton is a $U(1)$ singlet, it is reasonable to assume that
this kind of communication between the GUT and inflaton sectors has no further
unwanted consequences for the dynamics of inflation. A more detailed
investigation of possible GUT embeddings is left for future work.

Gravitino dark matter. A gravitino mass of
$\mathcal{O}\left(10\right)\,\textrm{MeV}$ [see Eq. (32)] in combination with
a reheating temperature of $\mathcal{O}\left(100\right)\,\textrm{TeV}$ [see
Eq. (40)] are the perfect basis for DM in the form of thermally produced
gravitinos. To leading order in the strong gauge coupling constant,
supersymmetric QCD results in the following present-day abundance of thermally
produced gravitinos Bolz:2000fu ; Pradler:2006qh ,

$\displaystyle h^{2}\Omega_{3/2}\simeq 0.39\left(\frac{T_{\rm
reh}}{100\,\textrm{TeV}}\right)\left(\frac{10\,\textrm{MeV}}{m_{3/2}}\right)\bigg{(}\frac{m_{\tilde{g}}}{3\,\textrm{TeV}}\bigg{)}^{2},$
(41)

where $m_{\tilde{g}}$ denotes the gluino mass at energies around the $Z$-boson
mass. Including additional processes in the MSSM coupled to gravity, the total
gravitino abundance can still be enhanced by an additional factor of
$\mathcal{O}\left(1\right)$ Rychkov:2007uq ; in the following, we will,
however, content ourselves with the simple estimate in Eq. (41). Based on this
estimate, we can now solve the condition that gravitinos should account for
the entire DM relic density, $h^{2}\Omega_{3/2}=h^{2}\Omega_{\rm dm}\simeq
0.12$ Aghanim:2018eyx ; Akrami:2018odb , for the gluino mass $m_{\tilde{g}}$,

$\displaystyle m_{\tilde{g}}\simeq
1.7\,\textrm{TeV}\bigg{(}\frac{m_{3/2}}{10\,\textrm{MeV}}\bigg{)}^{1/2}\left(\frac{100\,\textrm{TeV}}{T_{\rm
reh}}\right)^{1/2}\,.$ (42)

For $T_{\rm reh}=100\,\textrm{TeV}$, the upper bound on $m_{3/2}$ in Eq. (34)
thus results in an upper bound on the gluino mass of $m_{\tilde{g}}^{\rm
max}\simeq 2.5\,\textrm{TeV}$. This defines an interesting gluino mass range
and opens the exciting possibility to probe the scenario of gravitino DM in
future collider experiments.

In closing, we mention that a similar mass range for gravitino DM as in our
scenario has recently been identified in Choi:2019jck , however, for a
completely different reason. The starting point of Choi:2019jck  is the
extension of the MSSM by a supersymmetric axion multiplet that couples to the
electroweak gauge bosons. This electroweak axion receives an extremely flat
potential from $SU(2)_{L}$ instantons, which turns it into a quintessence
field that serves as an explanation of dark energy Nomura:2000yk ; Ibe:2018ffn
. The authors of Choi:2019jck  consider the decay of unstable gravitino dark
matter into the degrees of freedom contained in the electroweak axion
multiplet, $\tilde{G}\rightarrow a+\tilde{a}$, where $a$ denotes the axion
quintessence field and $\tilde{a}$ is the corresponding axino. They argue that
this decay represents a concrete realization of the decaying-DM mechanism
proposed in Vattis:2019efj  to resolve the so-called Hubble tension, i.e., to
reconcile the CMB measurement of the Hubble parameter $H_{0}$ by the PLANCK
satellite with the local $H_{0}$ value obtained via the cosmic distance ladder
and time delay measurements of strongly lensed quasars Wong:2019kwg . The main
idea behind this solution to the Hubble tension is to slowly transfer energy
from cold dark matter to massless radiation (i.e., axions in Choi:2019jck )
and a subdominant component of warm dark matter (i.e., axinos in Choi:2019jck
). The energy in radiation is then quickly redshifted away, which moves the
onset of the dark-energy-dominated era to earlier times. This in turn leads to
a slower decrease of the Hubble rate in the late Universe, which eventually
results in a larger $H_{0}$ value at the present time than what one would
expect in the standard $\Lambda$-cold-dark-matter ($\Lambda$CDM) picture.

A necessary condition for this mechanism to work is that the lifetime of the
decaying DM particle, $\tau$, falls in the range
$1.3\lesssim\log_{10}\left(\tau/\textrm{Gyr}\right)\lesssim 2.2$
Vattis:2019efj . The authors of Choi:2019jck  use this and other constraints
on the model to derive a viable mass range for decaying gravitino dark matter,
$m_{3/2}\sim 100\,\textrm{MeV}\cdots 1\,\textrm{GeV}$. This differs from our
gravitino mass range only by a factor $\mathcal{O}\left(10\cdots 100\right)$.
In our case, the gravitino is unfortunately too long-lived. Neglecting any
phase space factors, we can estimate the gravitino lifetime due to the decay
$\tilde{G}\rightarrow a+\tilde{a}$ as follows,

$\displaystyle\tau\sim\frac{192\pi M_{\rm Pl}^{2}}{m_{3/2}^{2}}\sim
6000\,\textrm{Gyr}\left(\frac{23\,\textrm{MeV}}{m_{3/2}}\right)^{3}\,,$ (43)

which exceeds the upper bound on $\tau$ derived in Vattis:2019efj  by about a
factor $40$. Nonetheless, it is an remarkable coincidence that the TCC bound
on DHI points to a mass range for gravitino DM that is close to the one
considered in Choi:2019jck . We believe that the possible connection between
the TCC, DHI, axion quintessence, and a resolution of the Hubble tension due
to decaying gravitino DM deserves more attention in future work. It would be
interesting to consider, e.g., scenarios with a relaxed upper bound on the
gravitino mass in consequence of nonstandard dynamics during reheating,
$-1/3<w<0$ Mizuno:2019bxy .

Conclusions. In this paper, we discussed the implications of the recently
proposed trans-Planckian censorship conjecture (TCC) for single-field slow-
roll inflation and showed how the resulting constraint on the inflationary
energy scale can be readily satisfied in a minimal D-term hybrid inflation
(DHI) model. Our main observation was that a tiny gauge coupling constant $g$
always allows us to parametrically suppress the energy scale of inflaton
compared to the GUT scale $\sqrt{\xi}\sim 10^{16}\,\textrm{GeV}$. In future
work, it would be interesting to construct an explicit GUT embedding of our
model that sheds more light on the question where such a small value of $g$
might come from.

We also discussed: the relation between our model and other quantum gravity
swampland conjectures; several issues related to the initial conditions of
inflation; the production of cosmic strings; the prospects to probe our
scenario in CMB observations, in gravitational-wave experiments, and at
colliders; the thermal generation of gravitino dark matter; and the
possibility to relate our model to scenarios that explain dark energy in terms
of axion quintessence and that resolve the Hubble tension by means of decaying
gravitino dark matter. In this discussion, we identified masses of
$\mathcal{O}\left(10\right)\,\textrm{MeV}$ as the relevant mass range for
gravitino dark matter and masses of
$\mathcal{O}\left(100\right)\,\textrm{GeV}$ as the mass range of the degrees
of freedom in the waterfall sector that are responsible for reheating (and
which should thus couple to the standard model). In summary, we conclude that
DHI based on a tiny gauge coupling constant results in an interesting
theoretical framework as well as in an exciting phenomenology. Both aspects of
our model await further exploration.

Acknowledgements. I wish to thank Wilfried Buchmüller, Valerie Domcke, and
Hitoshi Murayama for discussions on unstable cosmic strings and the related
signal in gravitational waves; and I wish to thank Tsutomu Yanagida for
comments on the relation between the model studied in this paper and the weak-
gravity conjecture. This project has received funding from the European
Union’s Horizon 2020 Research and Innovation Programme under grant agreement
number 796961, “AxiBAU”.

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