aainstitutetext: Departamento de Física, Universidad de Guanajuato,
Loma del Bosque No. 103 Col. Lomas del Campestre C.P 37150 Leon, Guanajuato,
Mexico.bbinstitutetext: Departamento de Ingeniería Mecánica, Universidad de
Guanajuato,
Carretera Salamanca-Valle de Santiago Km 3.5+1.8 Comunidad de Palo Blanco,
Salamanca, Mexico

# Inflationary implications of the Covariant Entropy Bound and the Swampland
de Sitter Conjectures

Dibya Chakraborty b Cesar Damian a Alberto González Bernal a and Oscar Loaiza-
Brito<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a proposal to relate the de Sitter Conjecture (dSC) to the
Covariant Entropy Bound (CEB). By assuming an early phase of accelerated
expansion where the CEB is satisfied, we take into account a contribution from
extra-dimensions to the four-dimensional entropy which restricts the values of
the usual slow-roll parameters. We show in this context that the dSC
inequalities follow from the CEB $-$including their mutual exclusion$-$ in
both single and multi-field inflationary scenarios. We also observe that the
order one constants, $c$ and $c^{\prime}$ in the conjecture are given in terms
of physical quantities such as the change in entropy over time, the Hubble
constant and the dynamics of the effective scalar fields. Finally, we give a
simple example to illustrate a possible contribution to the four-dimensional
entropy from a flux string scenario.

###### Keywords:

Fluxes, entropy, swampland, multifield inflation.

##  1 Introduction

The Swampland program has reached great advances in the last few years in its
pursuit to characterize those features distinguishing effective theories that
can be consistently completed into a quantum gravity theory (Landscape) from
those which do not (Swampland)Vafa:2005ui ; Ooguri:2006in ; Palti:2019pca ;
vanBeest:2021lhn . The boundary between the Landscape and Swampland is usually
defined by a series of bounds on quantities of the proper effective theory
which in turn are expressed in terms of Planck mass $M_{p}$. In the limit
where gravity decouples, i.e., $M_{p}\rightarrow\infty$, the Swampland
constraints vanish. A question to be answered is whether these boundaries can
be traced back to some essential microscopic physics in the Quantum Gravity
Theory, or accordingly, to some fundamental constraints in the 10-dimensional
String Theory formulation.

Many different bounds have been proposed as Swampland Conjectures, although it
is expected that all of them are related in some way through more fundamental
principles (see review Palti:2019pca  and lectures vanBeest:2021lhn  for
references). Some of them are based on solid grounds as the Distance
Conjecture and the Weak Gravity Conjecture, while others, as the de Sitter
Conjectures, were initially motivated by some empirical evidence based on
string models Danielsson:2018ztv ; Obied:2018sgi , and appropriately refined
into its final form in order to be compatible with some well known effective
scenarios such as the Higgs potential Denef:2018etk  among others. The refined
version of the conjecture was motivated by stability of de Sitter extrema
Andriot:2018wzk , by cosmological arguments through the slow-roll parameter
$\eta$ Garg:2018reu  as well as by entropy arguments on the dS space
Ooguri:2018wrx . Many logic and solid arguments have been followed in the
literature Andriot:2018ept ; Conlon:2018eyr ; Dasgupta:2018rtp ;
Andriot:2018mav ; Geng:2019zsx ; Atli:2020dni ; Dasgupta:2019vjn ;
Blumenhagen:2019kqm ; Damian:2019bkb ; Andriot:2020vlg  so far with the
purpose to give stronger ground basis to this conjecture. A matter of
particular interest to us is the series of arguments supporting the de Sitter
Conjecture (dSC) based on the laws of thermodynamics Seo:2019mfk ;
Gong:2020mbn ; Blumenhagen:2020doa ; Kobakhidze:2019ppv . Relating the dSC to
thermodynamic principles, such as a positive temperature phase and the
concavity of the entropy functional, respectively have also been considered in
the literature. Furthermore, as stochastic effects become relevant, it is
found by thermodynamic arguments that slow-roll conditions either violate the
second law or the swampland dSC Wang:2019eym ; Brahma:2019iyy .

In its original form, the de Sitter Conjecture states that

$\begin{split}\frac{|\nabla V|}{V}\geq
c\,\qquad\text{or}\qquad\frac{\text{min}\nabla_{i}\partial_{j}V}{V}\leq-c^{\prime}\,,\end{split}$
(1)

with $c$ and $c^{\prime}$ positive constants of order one in Planck units and
$V$ the effective scalar field potential. Besides the well known implication
of discarding the existence of dS minima, dSC also excludes the presence of
inflation driven by a single field Agrawal:2018own . However, as shown in
Achucarro:2018vey , dSC seems to be compatible with a multi-field inflationary
scenario.

Recently, cosmological implications of the dSC have been proven to be a matter
of great interest Heisenberg:2018yae ; Heisenberg:2018rdu ; Akrami:2018ylq ;
Wang:2018duq ; Chiang:2018lqx ; Matsui:2018bsy ; Schimmrigk:2018gch ;
Agrawal:2018rcg ; Damian:2018tlf ; Trivedi:2020xlh ; Trivedi:2020wxf ;
Das:2020xmh ; Brandenberger:2020oav . Roughly speaking we could say that there
are actually two scenarios: the first one concerns the study of the so called
late-acceleration of the Universe, to which the dSC seems to choose a
Dynamical Dark Energy mechanism over a cosmological constant related to a dS
minimum. A prototypical example of such a model is Quintessence Cicoli:2018kdo
; Blumenhagen:2018hsh ; Marsh:2018kub ; Olguin-Tejo:2018pfq ; Ibe:2018ffn ;
Brahma:2019kch ; Cicoli:2020noz ; the second one focuses on the study of the
early inflationary stage, prior to reheating, where the simplest model
involves a field slowly rolling down an almost flat scalar potential. As
mentioned above, the dSC seems to discard inflation driven by a single scalar
field.

It is then interesting for our purposes to review an (heuristic) argument
supporting the dSC in both scenarios above mentioned. For an accelerated
expansion, the dSC is expected by taking into account how entropy increases
over distance in a de Sitter space Palti:2019pca . Essentially it invokes the
association between an entropic system with the temperature given to the dS
Apparent Horizon. The number of microstates of the corresponding Hilbert space
contributes to the entropy which increases with the field distance. Since it
is possible to estimate that the entropy goes like the inverse of the
cosmological constant, one can conclude that the latter cannot be a constant
after all. Since it is conjectured that the mass of the field exponentially
decreases with field distance, one concludes that $V^{\prime}\sim cV$.

In contrast to late acceleration, during the early inflationary stage of the
Universe, there is almost no contribution to entropy, a reason being that
during an accelerated expansion, the comoving entropy density remains constant
Bousso:2002ju . This is true if one considers the 4-dimensional Universe a
closed system, but opens up the possibility to an inflow of entropy from extra
dimensions in a string theory scenario.

In this paper we are interested in studying the implications of the CEB
Bousso:2002ju ; bousso1999covariant ; Bousso:2015eda ; Bousso:2014sda
$-$under the assumption of an entropy inflow from extra dimensions$-$ into a
four-dimensional expanding Universe.

A direct outcome of relating the CEB to the dSC is the construction of
explicit expressions for the involved constants $c$ and $c^{\prime}$ in the
dSC innequaities (1) in terms of physical quantities, among which time
variations of entropy play a leading role. We also obtain that the exclusion
between the two inequalities in the dSC is a natural and direct consequence of
the CEB under the assumptions stated above. Therefore, the validity of the dSC
relies on the actual value of the entropy change for which it would be
necessary to compute the entropy contribution from a specific string
compactification scenario. A simple example concerning a toroidal flux
compactifcation is presented in this context.

However, it is important to emphazise that we are not generating an expanding
(inflationary or not) four-dimensional Universe through fluxes. The source of
an expanding Universe from a string theory point of view is beyond the scope
of this paper. Our interest focuses on the compatibility of an inflationary
Universe with the CEB111The source of entropy in extra dimensions may rely on
some basic and fundamental quantities in string theory such as the presence of
NS-NS and R-R fluxes. They play a crucial role in many phenomenological
applications such as generating the correct hierarchies of scales in four-
dimensional effective field theories by warping the geometry and particularly
in inflationary model building Grana:2005jc ; Blumenhagen:2017cxt ;
Blumenhagen:2015xpa ; Blumenhagen:2015kja ; Blumenhagen:2014nba  among others.
The Tapole constraints also lead to important implications on effective
scenarios CaboBizet:2019sku ; Betzler:2019kon ; Sati:2019tqq ;
CaboBizet:2020cse ; Plauschinn:2020ram ; Bena:2020xrh ..

The paper is organized as follows. In section 2 we briefly review some basics
regarding the CEB and establish a relation between the change of entropy over
time and the slow-roll parameter $\epsilon_{H}$ (Eq. 6). In section 3 we use
this relation to study its implications on single field and multi-field
inflationary scenarios. Our main result is that we are able to reproduce the
inequalities of the dSC as well as the mutual exclusion between them in terms
of the time variation of entropy sourced from physics in extra dimensions as
well as the explicit form of the constants $c$ and $c^{\prime}$. A simple
example concerning a toroidal flux compactification is presented to show how
fluxes contribute to the four-dimensional entropy. Finally we discuss our
conclusions and some final comments.

##  2 Covariant Entropy Bound and Inflation

The Covariant Entropy Bound (CEB) is a well tested conjecture in many
gravitational scenarios including cosmology bousso1999covariant ;
Bousso:2002ju ; Bousso:2014sda ; Bousso:2015eda . Basically, it states that
for a codimension 2 surface $B$, there exist at least two light-sheets $L$,
defined as the codimension 1 null hypersurfaces orthogonal to $B$ spanned by
light-type trajectories terminating at a singularity or a caustic, with a
negative expansion rate $\theta$. These light-sheets can be future or past-
directed. The general statement is that the entropy associated to the light-
sheet $S(L[B])$, at a given time, is bounded by the area $A$ of the surface
$B$ as222 In terms of universal constants, $S\leq\frac{c^{3}\kappa_{B}}{4\hbar
G}A$. In this paper we take all constants -including $M_{\text{Pl}}$ to be
equal to 1.

$S(L[B])\leq\frac{A(B)}{4}.$ (2)

In a flat 4-dimensional expanding Universe described by the Friedmann-
Robertson-Walker (FRW) metric

$\begin{split}ds^{2}_{4}=-dt^{2}+a^{2}(t)(dr^{2}+r^{2}d\Omega^{2}),\end{split}$
(3)

for a given surface $B$ with area $A$ at a time $t$, there are 4 different
null-hypersurfaces $L$, which can be past(future)-directed and are labeled by
their in(out)-going direction with expansion rates denoted by
$\theta_{(\pm,\pm)}=\pm\dot{a}\pm(1/r)$. In this scenario, it is possible to
define the Apparent Horizon (AH) as the surface $B$ with at least two null
expanding rates, specifically with $\theta_{+-}=\theta_{-+}=0$. As shown in
Bousso:2002ju , this is given by defining the radius of the AH at a time $t$
as $r_{AH}=(aH)^{-1}$, with $H$ the Hubble constant defined as $H=\dot{a}/a$,
where $\dot{a}$ denotes the time derivative with respect to the proper time
$t$. As the Universe expands, the comoving Hubble radius $r_{AH}$ shrinks and
the AH comoving area given by

$A_{AH}=4\pi a^{2}r_{AH}^{2}=\frac{4\pi^{2}}{H^{2}}$ (4)

increases, as expected by a negative $\dot{H}$. Notice that the corresponding
past directed light-sheet has an expansion rate $\theta_{--}=-2\dot{a}$ which
is negative for $\dot{a}>0$.

For any area, bigger or smaller than $A_{AH}$, it is conjectured that the CEB
will be always fulfilled. In particular, since we are considering an expanding
Universe, smaller areas are past-directed and the corresponding light-sheets
are represented by the expansion rate $\theta_{-,\pm}$. Hence, for
$t_{1}<t_{2}$, the area $A(t_{2})$ is connected to a smaller area $A(t_{1})$
by a light-sheet with expansion rate $\theta_{-,\pm}$ which is referred to as
an anti-trapped surface Bousso:2002ju . The CEB can be strengthened by
considering the entropy associated to the light-sheet between the surfaces
$A(t_{2})$ and $A(t_{1})$, as shown in Bousso:2014sda , with the entropy
change bounded as $\Delta S\leq(A(t_{2})-A(t_{1}))/4$, and where $\Delta S$ is
thought to be the von Neumann entropy related to the difference between matter
associated to the light-sheets and the entropy of vacuum.

We are interested in applying this bound for the AH area in an expanding flat
FRW Universe. One can wonder about the entropy production between some time
interval during inflation. By assuming an adiabatic expansion, the comoving
entropy density $s=dS/dV$ is constant, indicating that there is no entropy
variation. This can be understood as a consequence of considering the Universe
as a closed system in which the production of matter has not yet occurred
(reheating) and therefore there is no variation of entropy over time in
contrast to the AH area. However, we can think that in the framework of a
quantum theory of gravity there should be extra contributions to the entropy
allowing a condition of the form $\dot{S}\equiv\frac{dS}{dt}\neq 0$. This can
actually happen in the presence of extra dimensions as in string theory.

Following Bousso:2014sda , let us consider the AH surface at different times,
$t_{2}$ and $t_{1}$ with $t_{2}>t_{1}$. We then have the entropy variation
satisfies the bound

$\begin{split}\Delta S\leq\frac{A_{AH}(t_{2})-A_{AH}(t_{1})}{4},\end{split}$
(5)

where $\Delta S$ is the difference between the entropy at time $t_{2}$ and
$t_{1}$. Therefore, an infinitesimal variation of time helps us to rewrite
Eq.(5) in the form333 Although the CEB was proved to be consistent only for
scenarios in which the Null Energy Condition (NEC) holds, it was shown in
Bousso:2014sda  that Buosso bound can also be satisfied independently of the
NEC.

$\gamma\equiv\frac{H\dot{S}}{2\pi}\leq\epsilon_{H},$ (6)

with $\epsilon_{H}=-\dot{H}/H^{2}$ is the usual slow-roll inflationary
parameter.

Notice that this holds even for non-constant $H$. For the specific case in
which we have an exponential expansion, $\epsilon_{H}$ must be much smaller
than 1, and in consequence $\gamma$ must also be very small. Slow-roll
inflation is then ruled out if $\gamma\sim{\cal O}(1)$ or for $\gamma\gg 1$.

In the following we shall assume that a variation of entropy in time comes
from extra dimensions within a string theory based scenario. In this context
we expect that entropy must be a function of the Hubble parameter $H$, time
and other string parameters which we shall ignore to our present purpose.

It is the goal of this work to study the implications of a non-zero gamma
(Eq.(6)) on the formulation of the dSC.

##  3 Inflation and Effective Field Theory

In this section, we study the implications of assuming a non-zero flux entropy
$\gamma$ in an effective field theory involving scalar fields incorporating
both single and multi-field scenarios.

###  3.1 Single field inflation

From the inequality (6) and the usual Friedmann-Robertson-Walker (FRW)
equations of motion, we obtain

$\begin{split}\gamma\leq\epsilon_{H}=\frac{1}{2}\left(\frac{A+\frac{\partial_{\phi}V}{V}}{1+B}\right)^{2}\\\
\end{split}$ (7)

where $A=\frac{\ddot{\phi}}{V}$ and $B=\frac{\dot{\phi}^{2}}{2V}$. Then one
can immediately write

$\begin{split}\left|\frac{\partial_{\phi}V}{V}\right|\geq|1+B|\sqrt{2\gamma}-|A|\equiv
c.\end{split}$ (8)

The value of $c$ clearly depends on different scenarios, which we now proceed
to describe:

  1. 1.

$\gamma\ll 1$: this means that $\gamma\leq\epsilon_{H}\ll 1$. This implies
that the factor $(1+B)$ must be much greater than $A$ for
$\partial_{\phi}V/V\ll 1$. This allows a very small value for $c$ and the
usual identification $\epsilon_{H}\approx\epsilon_{V}$. It follows that an
accelerated expansion is a possibility. This case also involves the presence
of slow-roll inflation by taking $A\sim B\sim 0$. However, it also allows
values of $c$ to be of order 1 or bigger independently of $\gamma$ (for
adequate values of $A$ and $B$).

  2. 2.

$\gamma\geq 1$: if $\gamma$ is of order one or bigger, $\epsilon_{H}$ cannot
be sufficientely small to allow an inflationary stage of the Universe. The
value of $c$ can also be smaller, bigger or of order 1. Under the usual slow-
roll conditions, $c=\sqrt{2\gamma}$ and its value is determined absolutely by
the entropy variation during expansion. If $\gamma$ is of order 1 or bigger,
inflation is not allowed and the first inequality in the dSC is obtained as

$\begin{split}|\partial_{\phi}V|\geq\sqrt{2\gamma}~{}|V|.\end{split}$ (9)

Again the outcome is that an accelerated expansion and inflation are not
allowed.

We shall come back to these points once we consider the eigenvalues of the
Hessian matrix which will allow us to observe if we can reproduce the mutually
exclusive dSC (1).

Let us now be emphatic on two important aspects: a compatible inflationary
scenario requires at least $\gamma<\epsilon_{H}<1$; in a closed 4-dimensional
Universe $\gamma$ is almost zero since entropy is almost constant during
expansion Bousso:2002ju , and we have

$\begin{split}\epsilon_{H}\approx\epsilon_{V}\equiv\frac{1}{2}\frac{|\partial
V|^{2}}{V^{2}}\geq 0,\end{split}$ (10)

allowing $\epsilon_{V}<1$ as well. Inflation can be translated into conditions
on the effective scalar potential for an almost null $\gamma$.

###  3.2 Multi-field inflation

Let us generalize the above results to an $N$-field scenario. As shown in
Pinol:2020kvw , it is more convenient to switch to the kinematic $-$Frenet-
Serret$-$ frame to compute the equations of motion of an effective theory in
the presence of a scalar potential $V=V(\phi_{a})$ with $a=1,\dots N$. See
Appendix A for more details.

The equations of motion can be written in terms of the unitary tangential
vector $\hat{e}_{1}$, the normal $\hat{e}_{2}$ and the $(N-2)$ binormal
$\hat{e}_{i}$, as

$(\ddot{\phi}+3H\dot{\phi}+V_{1})\hat{e}_{1}+(-\dot{\phi}\Omega_{1}+V_{2})\hat{e}_{2}+V_{i}\hat{e}_{i}=0,$
(11)

with $i=3,\dots,N$ and $\dot{\phi}$ is the norm of the vector field
$\vec{\dot{\phi}}=(\dot{\phi}^{1},\dots,\dot{\phi}^{n})$. Since the Frenet-
Serret frame $\\{\hat{e}_{1},\hat{e}_{2},\hat{e}_{i}\\}$ is an orthonormal
basis, we find that

$\displaystyle\ddot{\phi}+3H\dot{\phi}+V_{1}$ $\displaystyle=$ $\displaystyle
0,$ (12) $\displaystyle-\dot{\phi}\Omega_{1}+V_{2}$ $\displaystyle=$
$\displaystyle 0,$ (13) $\displaystyle V_{i}$ $\displaystyle=$ $\displaystyle
0.$ (14)

Notice all projections of $\vec{\nabla}V$ along normal vectors $\hat{e}_{i}$,
denoted by $V_{i}$, vanish for $i=3,\dots,n$, implying that we can focus only
on the field trajectory on the osculating plane.

Now, together with the Einstein-Friedmann equation

$\begin{split}3H^{2}=\frac{\dot{\phi}^{2}}{2}+V,\end{split}$ (15)

and by the usual definitions of the slow-roll parameters,

$\begin{split}\epsilon_{V}\equiv\frac{1}{2}\left(\frac{\vec{\nabla}V}{V}\right)^{2}=\tilde{\epsilon}_{V}+\hat{\epsilon}_{V},\end{split}$
(16)

with

$\tilde{\epsilon}_{V}=\frac{1}{2}\left(\frac{V_{1}}{V}\right)^{2},\quad\hat{\epsilon}_{V}=\frac{1}{2}\left(\frac{V_{2}}{V}\right)^{2},$
(17)

we can construct a relation between $\epsilon_{H}$ and $\hat{\epsilon}_{V}$ by

$\epsilon_{H}^{2}-(6+\alpha)\epsilon_{H}+9=0,$ (18)

where

$\begin{split}\alpha=\frac{\Omega_{1}^{2}}{H^{2}\hat{\epsilon}_{V}}=\frac{2V^{2}}{H^{2}\dot{\phi}^{2}},\end{split}$
(19)

with $\hat{\epsilon}_{V}\neq 0$. Notice that $\alpha\rightarrow 0$ implies
$\epsilon_{H}\rightarrow 3$ for which an accelerated expansion $-$and
therefore inflation$-$ is ruled out. An almost zero $\alpha$ means two things:
one is that the kinetic energy is much bigger than $V$; the second possibility
involves $\hat{\epsilon}_{V}$ growing faster than the ratio $\Omega_{1}/H$.

Let us revisit the inequality (6). By using $\epsilon_{H}$ as a function of
$\hat{\epsilon}_{V}$ from (18) we obtain:

$\displaystyle\gamma$ $\displaystyle\leq$
$\displaystyle\left(3+\frac{\alpha}{2}\right)\pm\sqrt{\left(3+\frac{\alpha}{2}\right)^{2}-9},$
(20)

which provides us different implications for the following two possible
scenarios:

  1. 1.

$\gamma\geq 3+\frac{\alpha}{2}$: since $\alpha\geq 0$, $\epsilon_{H}>1$ and an
accelerated expansion is discarded. It follows from Eq.(20) that

$\hat{\epsilon}_{V}\leq\frac{\Omega_{1}^{2}}{H^{2}}\frac{\gamma}{(\gamma-3)^{2}}.$
(21)

We observe that $\gamma$ can take values from 3 to $\infty$. However, in this
case it is not possible to fix a lower bound for $\epsilon_{V}$.

  2. 2.

$\gamma\leq 3+\frac{\alpha}{2}$: from Eq.(20) we have

$\hat{\epsilon}_{V}\geq\frac{\Omega_{1}^{2}}{H^{2}}\frac{\gamma}{(\gamma-3)^{2}}.$
(22)

Then a lower bound for $\epsilon_{V}$ follows in the form of

$\displaystyle\frac{1}{2}\frac{|\nabla V|^{2}}{|V|^{2}}$ $\displaystyle=$
$\displaystyle\widetilde{\epsilon}_{V}+\hat{\epsilon}_{V},$ (23)
$\displaystyle\geq$
$\displaystyle\widetilde{\epsilon}_{V}+\frac{\Omega_{1}^{2}}{H^{2}}\frac{\gamma}{(\gamma-3)^{2}}.$

From $\epsilon_{H}=-\dot{H}/H^{2}$ and from the fact that
$\dot{V}=V_{1}\dot{\phi}$, we obtain

$\widetilde{\epsilon}_{V}=\frac{1}{2}\left[A+\sqrt{2\epsilon_{H}}(1+B)\right]^{2},$
(24)

with $A$ and $B$ defined in Eq.(LABEL:eH). This result together with

$\epsilon_{H}=\frac{\dot{\phi}^{2}}{2H^{2}}=\frac{V^{2}_{2}}{2H^{2}\Omega_{1}^{2}},$
(25)

implies that

$\displaystyle\frac{|\nabla V|^{2}}{|V|^{2}}$ $\displaystyle\geq$
$\displaystyle\left(A+\frac{V_{2}}{H\Omega_{1}}(1+B)\right)^{2}+\frac{\Omega_{1}^{2}}{H^{2}}\frac{2\gamma}{(\gamma-3)^{2}},$
(26)

from which the parameter $c$ of the first inequality of (1) reads

$c^{2}=\left(A+\frac{V_{2}}{H\Omega_{1}}(1+B)\right)^{2}+\frac{\Omega_{1}^{2}}{H^{2}}\frac{2\gamma}{(\gamma-3)^{2}}.$
(27)

In this case, $\gamma$ can be smaller than 1 and inflation is in principle not
discarded, indicating the fact that the dSC is fulfilled if $c$ is of order 1.

  3. 3.

Notice that, in the slow-roll limit, we have
$\epsilon_{H}\sim\widetilde{\epsilon}_{V}$, which implies that $A\sim B\sim
0$, inflation seems to be allowed for $\gamma\sim\epsilon_{H}$ since

$\frac{|\nabla V|^{2}}{|V|^{2}}\geq
2\gamma\left(1+\frac{\Omega_{1}^{2}}{H^{2}}\frac{1}{(\gamma-3)^{2}}\right).$
(28)

For $\gamma\ll 1$ it is straightforward to see that the above expression
reduces to

$\epsilon_{V}\sim\epsilon_{H}\left(1+\frac{\Omega_{1}^{2}}{9H^{2}}\right).$
(29)

Therefore, in a multi-field scenario, inflation is allowed with $\epsilon_{V}$
of order 1, as long as $\gamma$ is very small. This is precisely the statement
by Achúcarro and Palma in Achucarro:2018vey . A stringy example in which it is
possible to fulfill the Swampland Constraint in a multi-field scenario by
considering fat inflatons was studied in Chakraborty:2019dfh .

Observe that expression (27) allows us to determine the physical origin of the
constant $c$ and its direct relation, under our initial anzatz, to a quantum
theory of gravity from a 4d effective field theory point of view.

###  3.3 Refined Swampland de Sitter Conjecture

Now, let us try to reproduce the second inequality in the dSC (1) and
determine whether both of those inequalities are mutually exclusive or not.
First of all, let us assume that $\gamma$ is defined in a way such that we can
write

$\begin{split}\dot{\gamma}\leq\mathcal{O}(1)\dot{\epsilon}_{H}.\end{split}$
(30)

This is a reasonable assumption since $\gamma$ must be smaller than
$\epsilon_{H}$ during a finite time interval in which an exponential
accelerated expansion occurs. Since we are assuming $\gamma/\epsilon_{H}\leq
1$, we can restrict to a time interval during which
$\dot{\gamma}/\gamma\leq\dot{\epsilon}_{H}/\epsilon_{H}$ and then

$\begin{split}D_{t}\log\gamma\leq D_{t}\log\epsilon_{H}\,.\end{split}$ (31)

The left hand side of Eq.(31) can be written in terms of
$\eta_{H}=-\ddot{H}/(2H\dot{H})$ and $\epsilon_{H}$, as

$\begin{split}-\frac{1}{2H}D_{t}\log\gamma\geq\eta_{H}-\epsilon_{H}.\end{split}$
(32)

On the other hand, by taking the derivative of the Einstein-Friedmann
equations of motion,

$\begin{split}\eta_{H}+\epsilon_{H}=\frac{D_{t}V_{1}}{3H^{2}\dot{\phi}}+\omega^{\prime},\end{split}$
(33)

for $\omega^{\prime}=\dddot{\phi}/3\dot{\phi}H^{2}$ (this expression is
consistent with the well known result, $\eta_{H}+\epsilon_{H}\approx\eta_{V}$
for $\eta_{H}=-\frac{\ddot{\phi}}{H\dot{\phi}}$ and
$\eta_{V}=\frac{D_{t}V_{1}}{3H^{2}\dot{\phi}}$). Thus, inequality (32) can be
written as

$\begin{split}\frac{D_{t}V_{1}}{3H^{2}\dot{\phi}}\leq-\frac{1}{2H}D_{t}\log\gamma+2\epsilon_{H}-\omega^{\prime}.\end{split}$
(34)

Now, since

$\begin{split}\frac{D_{t}V_{1}}{3H^{2}\dot{\phi}}=\frac{Q_{1}-\Omega_{1}^{2}}{3H^{2}}\end{split}$
(35)

where $Q_{1}$ is a quadratic form given by

$\begin{split}Q_{1}=V_{ab}e^{a}_{1}e^{b}_{1},\end{split}$ (36)

with $e^{a}_{1}=\dot{\phi}^{a}/\dot{\phi}$ (see Appendix A) it follows that

$\begin{split}Q_{1}\leq-\frac{3H}{2}D_{t}\log{\gamma}+\Omega_{1}^{2}+3H^{2}(2\epsilon_{H}-\omega^{\prime}).\end{split}$
(37)

Under the condition of orthonormality $\sum_{a}|e^{a}_{1}|^{2}=1$, it is
straightforward to show that the quadratic form $Q_{1}$ has a minimum
(maximum) value given by the smallest (largest) eigenvalue $\lambda_{min}$
($\lambda_{max}$) of the Hessian matrix $V_{ab}$ in the basis
$\\{e^{a}_{i}\\}$. Hence we obtain

$\begin{split}\lambda_{min}\leq-\frac{3H}{2}D_{t}\log{\gamma}+\Omega_{1}^{2}+3H^{2}(2\epsilon_{H}-\omega^{\prime}).\\\
\end{split}$ (38)

Let us now stress the point that inflation claims that
$\gamma\leq\epsilon_{H}\ll 1$ with $A\sim B\sim 0$ and in turn
$\epsilon_{H}\sim\widetilde{\epsilon}_{V}$. Under these circumstances we have
the following options:

  1. 1.

If $\Omega_{1}^{2}<H^{2}$ then from (28)

$\begin{split}\frac{|\nabla V|^{2}}{|V|^{2}}\geq 2\gamma=c,\end{split}$ (39)

where $c$ is not of order 1. However, since $\gamma\ll 1$ we also have
$\frac{1}{H}D_{t}\log\gamma\gg 1$ for $H$ not very small. Furthermore, using
the inequality $HD_{t}\log\gamma>\Omega_{1}^{2}$ and Eq. (LABEL:lambda) we can
write

$\begin{split}\frac{\lambda_{min}}{V}\leq-\frac{1}{2H}D_{t}\log\gamma\equiv-c^{\prime}.\end{split}$
(40)

Observe that, while the first inequality of the dSC is not fulfilled, there is
no obstruction for the second one to be satisfied. This also includes the case
with $\Omega_{1}=0$, corresponding to a single scalar field scenario.

  2. 2.

If $\Omega_{1}^{2}>H^{2}$ it follows from (28) that $c$ can be of order 1. On
the other hand, in this limit there is always a value for $H$ such that
$D_{t}\log\gamma\gg H$. Then it is possible that $\Omega_{1}^{2}$ and
$HD_{t}\log\gamma$ are of the same order, implying that the difference is not
necessarily negative. Therefore, while the first inequality in (1) can be
fulfilled by an appropriate value of $\Omega_{1}$, the second one is
compromised.

Notice that, under the slow-roll conditions and with $\gamma\ll 1$, our two
inequalities (26) and (LABEL:lambda) are mutually exclusive as proposed in the
original dSC while inflation is still allowed. This depends clearly on the
value of $\gamma$ which according to our anzatz, is different from zero due to
quantum gravity effects, also in accordance with the Swampland conjectures.

Notice that for $\gamma\sim cV^{n-2/4}$ for some $n$, the bound (39) limits
the presence of eternal inflation Blumenhagen:2020doa .

###  3.4 Lyth’s bound and the Swampland Distance Conjecture

It has been found that, for a canonical single-field slow-roll model of
inflation, generally the overall field displacement $\Delta\varphi$
experienced by the inflaton during the quasi-de Sitter phase must satisfy a
lower bound, which is known as the Lyth bound

$\begin{split}\Delta\phi\simeq\sqrt{2{\epsilon}_{H}}\Delta N,\end{split}$ (41)

where $\Delta N$ is the effective number of efolds during inflation needed to
be at least 60. So, we can rewrite (41) as,

$\begin{split}\Delta\phi\simeq\sqrt{2}\cdot
60\sqrt{{\epsilon}_{H}}=\kappa\sqrt{{\epsilon}_{H}},\end{split}$ (42)

where $\kappa$ is a number of $\mathcal{O}(10)$. Hence, during inflation field
displacement is expected to be of order 1 in Planck units, although larger
orders have been considered in the literature and known as trans-Planckian
distances. However, recently it has been conjectured (see Palti:2019pca ;
vanBeest:2021lhn  and references therein) that field displacements have an
upper bound, probably connecting to other Swampland Conjectures, as the
Distance Conjecture.

Now, in the case of multifield inflation, the field excursion $\Delta\phi$ has
an extra contribution coming from non-trivial angular motion $\Omega_{1}$ and
entropy mass $M$ Achucarro:2018vey ,

$\begin{split}\Delta\phi\gtrsim\kappa\sqrt{\frac{{\epsilon}_{H}}{\beta}}\end{split}$
(43)

where $\beta$ is a function depending on $\Omega_{1}$ and $M$. Now following
Achucarro:2018vey , if the fluctuating mode crosses the horizon in the linear
regime, i.e., if the condition $(1-c_{s}^{2})H<c_{s}M$ is satisfied, then
$\beta\simeq c_{s}$, where $c_{s}$ is the propagating speed of adiabatic
perturbation also known as the speed of sound, given by

$\begin{split}c_{s}=\left(1+\frac{4\Omega_{1}^{2}}{M^{2}}\right)^{-1/2}\end{split}$
(44)

where $M^{2}$ is given by,

$\displaystyle M^{2}=Q_{2}+{\epsilon}_{H}H^{2}\mathbb{R}-\Omega_{1}^{2},$ (45)

with the quadratic form $Q_{2}$ given by

$\begin{split}Q_{2}=&e_{2}^{a}e_{2}^{b}V_{ab},\end{split}$ (46)

where $e^{a}_{2}$ are the normal unitary directions in the Frenet-Serret
frame, and $\mathbb{R}=\mathbb{R}_{abcd}e_{1}^{a}e_{2}^{b}e_{1}^{c}e_{2}^{d}$.
Notice that, the maximum and minimum values of $Q_{2}$ are given by the
eigenvalues of the Hessian matrix $V_{ab}$ denoted $\lambda_{\text{min}}$ or
$\lambda_{\text{max}}$, with $\mathbb{R}_{abcd}$ and $\mathbb{R}$ the Riemann
tensor and Ricci scalar of the scalar field manifold.

Hence the Lyth bound (43) can be written as,

$\begin{split}\Delta\phi\geq\kappa\sqrt{\epsilon_{H}}\left(1+\frac{4\Omega_{1}^{2}}{M^{2}}\right)^{1/4}=\kappa\sqrt{\epsilon_{H}}\left(1+\frac{4\Omega_{1}^{2}}{\lambda-\Omega_{1}^{2}+\epsilon_{H}H^{2}\mathbb{R}}\right)^{1/4},\end{split}$
(47)

where $\lambda\in[\lambda_{min},\lambda_{max}]$.

Let us also comment on different scenarios, by considering the CEB (6),

  1. 1.

If $c_{s}\simeq 1$, it looks like the bound can be made even stronger than in
(41). This is because $c_{s}=1$ implies that either $\Omega_{1}=0$ $-$which
holds for the case of a single field$-$ or $M^{2}\gg\Omega_{1}^{2}$. For
$\epsilon_{H}\ll 1$ the last case implies that $\lambda$ cannot be negative,
indicating that the second inequality of the dSC does not hold and that
$\lambda\gg\Omega_{1}^{2}$. Also for inflation to take place, we need
$\gamma\ll 1$ which makes the bound for $\Delta\phi$ stronger compared to the
bound fixed by $\epsilon_{H}$.

  2. 2.

If $c_{s}<1$, $\Omega_{1}$ cannot be zero. This case then applies to multi-
field scenarios. Notice that the bound on $\Delta\phi$ is now weaker than the
bound for single field models (41) if $\lambda<\Omega_{1}^{2}$. As shown in
Scalisi:2018eaz , if $\Delta\phi\geq 10$ in Planck units, there is still room
to satisfy the dSC. So now we have to see under which conditions the ratio
$\sqrt{\gamma/c_{s}}$ remains unaltered, allowing the dSC to be satisfied. If
$\lambda-\Omega_{1}^{2}>0$ the bound can become stronger.

An extra interesting implication of the Lyth bound is the posibility to
construct an upper bound on the field displacement $\Delta\phi$ under the
assumptions of the previous section. Since $\epsilon_{H}\geq\gamma$,
expression (47) can we rewritten for $\lambda_{min}$ as

$\lambda_{min}\geq\frac{\kappa^{4}\gamma^{2}(\epsilon_{H}H^{2}\mathbb{R}+3\Omega_{1}^{2})-\Delta\phi^{4}(\epsilon_{H}H^{2}\mathbb{R}-\Omega_{1}^{2})}{(\Delta\phi^{4}-\kappa^{4}\gamma^{2})},$
(48)

where we have assumed $\Delta\phi>\kappa\sqrt{\gamma}$. By comparing
innequalities (48) and (LABEL:lambda), we obtain

$\begin{split}\Delta\phi\leq\kappa\sqrt{\epsilon_{H}}\left(1-\frac{4\Omega_{1}^{2}}{\frac{3H}{2}D_{t}\log\gamma-3H^{2}(2\epsilon_{H}-\omega^{\prime}+\frac{1}{3}\epsilon_{H}\mathbb{R})}\right)^{1/4}.\end{split}$
(49)

Let us consider the case in which $\gamma\leq\epsilon_{H}\ll 1$ with
$\omega^{\prime}\ll 1$. By assuming a non-zero but very small $\gamma$ (such
that $D_{t}\log\gamma$ is much larger than $\epsilon_{H}$) it follows that it
is possible to have an upper bound for the field displacement $\Delta\phi$.
The bounds for $\Delta\phi$ now read

$\begin{split}\kappa\sqrt{\epsilon_{H}}\left(1+\frac{4\Omega_{1}^{2}}{\lambda_{min}-\Omega_{1}^{2}}\right)^{1/4}\leq\Delta\phi\leq\kappa\sqrt{\epsilon_{H}}\left(1-\frac{8\Omega_{1}^{2}}{3HD_{t}\log\gamma}\right)^{1/4}.\end{split}$
(50)

This result is consistent only if
$\lambda_{min}\leq-(3H/2)D_{t}\log\gamma+\Omega_{1}^{2}$ as implied by
(LABEL:lambda). It then follows that there must be a bound for field
displacement, in agreement with the Swampland Distance Conjecture. In
particular we observe that trans-Planckian displacements for the field require
large values of $D_{t}\log\gamma$.

In summary, by assuming a non-zero contribution to the entropy (from some
fundamental quantum quantities connected to string theory), we see that it is
possible to obtain the dSC with explicit expressions for the constants $c$ and
$c^{\prime}$ depending on the entropy $\gamma$, the Hubble constant $H$, the
curvature radius $\Omega_{1}$ and the usual slow-roll inflationary parameters.
Importantly enough, we can also reproduce the mutually exclusive inequalities
in the dSC.

We now proceed to consider a simple toy example in which the entropy
contribution from a flux string scenario is computed.

###  3.5 Entropy from a flux compactification on $T^{6}/{Z}_{2}$

Let us study a simple flux compactification on an isotropic six-dimensional
torus threaded by fluxes and in the presence of an orientifold $O3$-plane. We
are not considering the presence of $D3$-branes for which we are taking RR
potential $C_{4}=0$. We will follow the notation of Kachru:2002he  in this
subsection.

Consider a factorizable torus $T^{6}=T^{2}\times T^{2}\times T^{2}$ with
coordinates as in Fig. 1,

$y^{1}$$y^{2}$$y^{2}$$y^{3}$$y^{4}$$\times$$y^{5}$$y^{6}$$\times$ Figure 1:
Coordinates of the compact space.

and let us take the ten-dimensional metric given by

$\begin{split}ds_{10}^{2}=ds_{4}^{2}+\sum_{i}(dy^{i})^{2}\,,\end{split}$ (51)

where $ds_{4}^{2}$ is the homogeneous and isotropic FRW metric (3). Since we
are interested in computing an amplitude transition of a given flux
configuration of $N$ units of RR flux $F_{3}$ and $M$ units of NS-NS flux
$H_{3}$ which satisfy the supergravity equations of motion and the stringy
constraints, we shall consider that during transition the fluxes are time
dependent. Let us start by considering a specific example in which the closed
string potentials $C_{2}$ and $B_{2}$ depending on time, are given by

$C_{2}=f(t)Ny^{6}dy^{2}\wedge dy^{4},\quad\text{and}\quad
B_{2}=f(t)My^{5}dy^{1}\wedge dy^{3},$ (52)

because the function $f(t)$ depends on time. The corresponding action in the
string frame

$\mathcal{A}_{\text{flux}}=-\frac{1}{2}\int
d^{10}x\sqrt{-g_{10}}~{}\left(F_{3}^{2}+e^{-2\phi}H_{3}^{2}\right),$ (53)

reduces to

$\mathcal{A}=\frac{1}{2}\int
d^{10}x~{}a^{3}(t)\left[N(\dot{f}(y^{6})^{2}-f^{2})+Me^{-2\phi}(\dot{f}(y^{5})^{2}-f^{2})\right]$
(54)

where the 3-form fluxes are defined as

$\displaystyle F_{3}$ $\displaystyle=$ $\displaystyle
N(\dot{f}y^{6}dt+fdy^{6})\wedge dy^{2}\wedge dy^{4},$ $\displaystyle H_{3}$
$\displaystyle=$ $\displaystyle M(\dot{f}y^{5}dt+fdy^{5})\wedge dy^{1}\wedge
dy^{3},$ (55)

Since we are considering the isotropic case, we take all the internal
coordinates $y_{i}$ on equal footing $(y)$, and the above integral can be
written as

$\mathcal{A}_{\text{flux}}=\frac{1}{2}\int
d^{10}x~{}a^{3}(t)(N+e^{-2\phi}M)\left(\dot{f}(t)y^{2}-f(t)^{2}\right),\\\ $
(56)

which mimics the behavior of a harmonic oscillator in a compact space. Thus
the partition function related to the above action can be found explicitly.
The first step is to integrate over the internal coordinates from 0 to $2\pi
R$ and external space coordinate $r$ from $0$ to $r_{\ast}$, for some finite
$r_{\ast}$ from which one obtains

$\mathcal{A}_{\text{flux}}=-\int_{0}^{t_{\ast}}dt~{}f(t)\hat{D}f(t),$ (57)

where

$\displaystyle\hat{D}$ $\displaystyle=$
$\displaystyle\alpha(t)\left(\frac{d^{2}}{dt^{2}}+3H\frac{d}{dt}+\omega_{0}^{2}\right),$
(58)

with $\omega_{0}^{2}=3/(2\pi R)^{2}$ and

$\alpha(t)=\frac{(2\pi)^{9}R^{8}r_{\ast}^{3}}{9}a^{3}(t)(N+e^{-2\phi}M),$ (59)

with the boundary condition $f(0)=f(t_{\ast})$, which states that at time
$t=0$ there is a given distribution of fluxes with $N$ units of RR flux and
$M$ units of NS-NS flux. Between time 0 and $t_{\ast}$ this distribution of
fluxes changes such that at time $t_{\ast}$, the original distribution of
fluxes is recovered. It is important to notice that $\alpha(t)$ is finite due
to the tadpole cancellation condition and the fact that the dilaton has been
fixed by the fluxes.

Therefore the partition function can be given by

$\mathcal{Z}_{(N,M)}(0,t_{\ast})=\int\mathcal{D}[f(t)]e^{-\frac{i}{2}\int_{0}^{t_{\ast}}dt~{}\alpha(t)f(t)\hat{D}f(t)},$
(60)

which can be used to compute the transition amplitude between an initial flux
configuration $(N,M)$ and the final one specified by the same flux
configuration. After performing the integral of (60) (and replacing $t_{\ast}$
by $t$), and by taking the case in which
$\Omega^{2}=\frac{9H^{2}}{4}-\omega_{0}^{2}>0$, one obtains

$\mathcal{Z}_{(N,M)}(0,t)=\left(\frac{-i\alpha(t)\Omega}{2\pi\text{sinh}(\Omega
t)}\right)^{1/2}.$ (61)

Notice that, our model behaves like a harmonic oscillator with a time-
dependent mass $\alpha(t)$.

Now, after using standard methods in Euclidean time, it is possible to compute
the entropy $S(t)$ from $\mathcal{Z}_{(N,M)}$ which reads

$S(t)=\frac{1}{2}\log\left[-\frac{i}{2\pi}\alpha(t)\Omega\text{csch}(\Omega
t)\right]+\frac{1}{2}\left(\dot{\Omega}t+\Omega\right)t~{}\text{coth}(\Omega
t)-\frac{t}{2}\left(\frac{\dot{\alpha}}{\alpha}+\frac{\dot{\Omega}}{\Omega}\right)$
(62)

where

$\frac{\dot{\alpha}}{\alpha}=3H,\qquad\text{and}\qquad\frac{\dot{\Omega}}{\Omega}=\frac{9}{4}\frac{H\dot{H}}{\Omega^{2}}=-\left(\frac{3H}{2\Omega}\right)^{2}H\epsilon_{H},$
(63)

while time variation of $S$ is given by

$\displaystyle\dot{S}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\left[\Lambda
H\epsilon_{H}t-1\right]^{2}\Omega^{2}t~{}\text{csch}^{2}(\Omega t)$ (64)
$\displaystyle+\Lambda\epsilon_{H}H\left[\frac{H}{2}\left((\epsilon_{H}+2\eta_{H})-\Lambda\epsilon_{H}\right)t-1\right]\Omega
t~{}\text{coth}(\Omega t)$
$\displaystyle+\frac{1}{2}\left[3\epsilon_{H}+\Lambda\epsilon_{H}(2\Lambda\epsilon_{H}-\epsilon_{H}-2\eta_{H})\right]H^{2}t,$

where $\Lambda=\left(\frac{3H}{2\Omega}\right)^{2}$. In the following we are
going to consider appropriate limits in order to see if the obtained value for
$\gamma$ is consistent or not, meaning a positive value for $\dot{S}$ for some
time interval.

####  3.5.1 Swampland implications

Remember we are interested in studying inflation which requires the
conditions, $\epsilon_{H},\eta_{N}\ll 1$ and check if the model is compatible
with CEB by obtaining $\gamma\ll 1$. For the case we are considering, the only
limit that makes sense is when one takes a very small value of $\Omega$ which
implies that444Notice that $\Omega$ cannot be arbitrarily large. $\Lambda$ is
very large and that the volume of the internal manifold produces a lower bound
given by

$\text{Vol}(T^{6})\gtrsim\frac{1}{27\pi^{6}H^{6}},$ (65)

implying that the internal volume is approximately given by the Hubble radius
during an accelerated expansion. Now, we are taking a special limit in which

$\Lambda\sim\frac{1}{\epsilon_{H}},$ (66)

for which $\gamma$ is given by

$\gamma\approx-\frac{H}{4\pi}(Ht-1)^{2}\Omega^{2}t~{}\text{csch}^{2}(\Omega
t)-\frac{H^{2}}{2\pi}(1+\frac{Ht}{2})\Omega t~{}\text{coth}(\Omega
t)+\frac{H^{3}}{2\pi}t.$ (67)

As depicted in Fig. 2, $\gamma$ is positive and smaller than 1 for very large
values of $\Lambda$, making our whole model consistent. This means, as we have
witnessed in section 3, that we can reproduce the dSC and the mutual
exclusiveness among the corresponding inequalities.

a)  b)  $t$$t$

$\gamma$

$\gamma$

Figure 2:  By taking a small Hubble constant, $H_{inf}<2.5\times 10^{-5}$ in
units of Planck mass Akrami:2018odb  and small values for the internal volume
we plot $\gamma$ as a function of time for $\Omega\ll H$ or equivalently as in
(66) for $\Lambda\gg 1$ implying small $\Omega$. a)
$\Omega^{2}=\left(\frac{3H}{2}\right)^{2}-\omega_{0}^{2}>0$ for which $\gamma$
depends on hyperbolic trigonometric functions and b) $\Omega^{2}<0$ for which
$\gamma$ depends on trigonometric functions. In both cases $\Omega$ large
leads to negative values for $\gamma$.

Since we are dealing with an effective single field inflationary scenario,
from (39) the first inequality in the dSC reads

$\left(\frac{V^{\prime}}{V}\right)^{2}\geq-\frac{H}{2\pi}(Ht-1)^{2}\Omega^{2}t~{}\text{csch}^{2}(\Omega
t)-\frac{H^{2}}{\pi}(1+\frac{Ht}{2})\Omega t~{}\text{coth}(\Omega
t)+\frac{H^{3}}{\pi}t,$ (68)

which for small $H$ can be written as

$\left(\frac{V^{\prime}}{V}\right)^{2}\geq aH\sim aV^{1/2},$ (69)

where $a$ is derived from the $\gamma$ expression. Therefore, for this case,
since $\gamma<1$, eternal inflation seems to be allowed and a quantum breaking
time is actually happening for $t\sim 1/H^{3}$ Dvali:2018jhn . Notice also
that while the first inequality in the dSC is not fulfilled, the second one is
and in consequence inflation cannot last for sufficient amount of time.

For the second case in which $\Omega^{2}<0$ the volume of the internal
manifold is bounded as

$\text{Vol}(T^{6})\lesssim\frac{1}{27\pi^{6}H^{6}}.$ (70)

We get a similar expression for $\gamma$ in the limit of small $\Omega$ with
the hyperbolic trigonometric functions replaced by trigonometric ones. In this
case, there is also a time interval at which $\gamma$ is positive, but not
restricted to be small (see Fig. 2 b). Therefore, there are zones with
$\gamma$ of order 1 and for later times, $\gamma>1$. While $\gamma<1$
inflation is allowed, but when we reach times at which $\gamma$ is of order 1
or higher, inflation stops. Then, we can see that entropy produced from fluxes
is a natural mechanism to stop an accelerated expansion of the Universe.

##  4 Conclusions and Final comments

The de Sitter Conjectures (dSC) have stringent implications on four-
dimensional effective field theories derived from a theory of quantum gravity
such as string theory. For instance, if the conjectures are true, single field
inflationary models are ruled out. However, the situation is comparatively
optimistic for the multi-field cases. Given the implications of these
conjectures on the understanding of our Universe, it is desirable to relate
them to other well-tested conjectures like the Covariant Entropy Bound,
involving some fundamental microscopic features of the Universe. This will
test those conjectures from a more fundamental view point and put them in a
stronger ground.

By using the Covariant Entropy Bound on an early phase of a four-dimensional
accelerated universe, we present a bound on the slow-roll parameter
$\epsilon_{H}$ in terms of the time derivative of the entropy, given by
$\gamma<\epsilon_{H}$. In a thermodynamically closed four-dimensional Universe
on which $\gamma\approx 0$, there is no obstruction for inflation to occur.
However, by extending this effective model to include extra dimensions
(indicating that we have drastically modified the theory to be compatible with
string theory), it is possible to consider an extra entropy source. In this
case, $\gamma$ does not necessarily vanish, opening up the possibility to
either compromising inflation or establishing some constraints in the slow-
roll parameters such as $\epsilon_{V}=\nabla V/2V$.

In fact, for the single-field models we find that
$\epsilon_{V}<\sqrt{2\gamma}$, ruling out inflation for $\gamma$ of order 1 or
higher in concordance with the dSC. A simple toroidal flux compactification
shows that fluxes constribute to the four-dimensional entropy while fulfilling
the dSC.

For the multi-field scenario, we reproduce (in the Frenet-Serret frame) both
inequalities in the de Sitter Conjecture (1) as well as the fact that they are
mutually exclusive. For $\gamma\approx 0$, we reproduce the result first
presented by Achúcarro and Palma Achucarro:2018vey . For $\gamma\neq 0$, we
provide an expression for $c$ and $c^{\prime}$ in terms of physical quantities
in the effective field theory such as $H$, $\gamma$ and the field dynamics,
whose values determine the order of the constants. Besides, in agreement with
the distance conjecture, the length displacement of a scalar field is bounded
from above, where the limiting bound is described in terms of entropy changes.
In this sense we observe that all these swampland criteria are compatible with
the CEB.

Acknowledgments

We thank Bruno Valeixo Bento for his observations and suggestions on our
manuscript and to Nana Cabo, Yessenia Olguin and Ivonne Zavala for very useful
discussions. The work of D.C and A. G B was partially supported by a CONACyT
graduate scholarship. C. D and O. L-B are partially supported by CONACyT
project CB-2015-258982.

##  Appendix A Frenet-Serret frame

In order to have a more general description we shall consider $N$ fields. For
that, let us define a map

$\vec{\phi}:\mathbf{I}\rightarrow\mathbf{R}^{n},$ (71)

such that

$\vec{\phi}(t)=(\phi^{1}(t),\dots,\phi^{n}(t))$ (72)

where the entries are given in terms of a time independent basis
$\hat{u}_{a}$, i.e.,

$\vec{\phi}(t)=\phi^{a}(t)\hat{u}_{a},$ (73)

with

$||\vec{\phi}(t)||^{2}=\phi^{a}\phi^{b}h_{ab}=\phi^{2},$ (74)

where $h_{ab}$ is the metric of the moduli space we are assuming to be
explicitly independent of time.

As time runs, the trajectory in the field space has a tangential vector
defined as

$\hat{e}_{1}(t)=\frac{\dot{\vec{\phi}}}{\dot{\phi}}=\frac{\dot{\phi}^{a}}{\dot{\phi}}\hat{u}_{a}=T^{a}\hat{u}_{a},$
(75)

with $\dot{\vec{\phi}}(t)=\dot{\phi}^{a}(t)\hat{u}_{a}$ and
$||\dot{\vec{\phi}}(t)||^{2}=\dot{\phi}^{a}\dot{\phi}^{b}h_{ab}=\dot{\phi}^{2}$.
The normal unitary vector can be constructed as follows. Consider

$\ddot{\vec{\phi}}(t)=\ddot{\phi}^{a}(t)\hat{u}_{a},$ (76)

and the normal vector $\vec{e}_{2}(t)$ given by

$\vec{e}_{2}(t)=\ddot{\vec{\phi}}(t)-\langle\ddot{\vec{\phi}}(t),\hat{e}_{1}\rangle\hat{e}_{1}(t).$
(77)

After some calculations, the normalized normal vector is given by

$\hat{e}_{2}(t)=N^{a}\hat{u}_{a},$ (78)

with

$N^{a}=\frac{\dot{\phi}^{2}\ddot{\phi}^{a}-\dot{\phi}^{a}\dot{\phi}^{b}\ddot{\phi}_{b}}{\dot{\phi}(\dot{\phi}^{2}\ddot{\phi}^{2}-(\dot{\phi}^{b}\ddot{\phi}_{b})^{2})^{1/2}}.$
(79)

Now, in terms of an orthonormal basis at each point of the curve
$\vec{\phi}(t)$, i.e. the FS frame, we have that

$D_{t}\begin{bmatrix}\hat{e}_{1}\\\ \hat{e}_{2}\\\ \hat{e}_{3}\\\ \vdots\\\
\hat{e}_{n-1}\\\ \hat{e}_{n}\end{bmatrix}=\begin{bmatrix}0&-\Omega_{1}&0&&&\\\
\Omega_{1}&0&-\Omega_{2}&&&\\\ 0&\Omega_{2}&0&-\Omega_{3}&&&\\\ &&&\ddots&&\\\
&&&\Omega_{n-2}&0&-\Omega_{n-1}\\\
&&&0&\Omega_{n-1}&0\end{bmatrix}\begin{bmatrix}\hat{e}_{1}\\\ \hat{e}_{2}\\\
\hat{e}_{3}\\\ \vdots\\\ \hat{e}_{n-1}\\\ \hat{e}_{n}\end{bmatrix}$ (80)

and in consequence $D_{t}\hat{e}_{1}=-\Omega_{1}\hat{e}_{2}$. From the
corresponding equations of motion

$D_{t}\dot{\phi}^{a}+3H\dot{\phi}^{a}+V^{a}=0,$ (81)

we can deduce the expression for $\Omega_{1}$, where

$\displaystyle D_{t}\dot{\phi}^{a}$ $\displaystyle=$
$\displaystyle\ddot{\phi}^{a}+\Gamma^{a}_{bc}\dot{\phi}^{b}\dot{\phi}^{c},$
(82) $\displaystyle\ V^{a}$ $\displaystyle=$
$\displaystyle\vec{\nabla}V\cdot\hat{u}_{a}.$ (83)

and $\hat{u}_{a}$ is a time independent basis for the $N$-dimensional field
space.

The $N$ equations can be written in a vector notation as

$(\ddot{\phi}+3H\dot{\phi})\hat{e}_{1}+\dot{\phi}(D_{t}\hat{e}_{1})+\vec{\nabla}V=0,$
(84)

where $\hat{e}_{1}$ is the unitary tangent vector to the field trajectory.
Since $D_{t}\hat{e}_{1}=-\Omega_{1}\hat{e}_{2}$, where $\hat{e}_{2}$ is the
unitary normal vector to the trajectory, we have

$(\ddot{\phi}+3H\dot{\phi})\hat{e}_{1}-\dot{\phi}\Omega_{1}\hat{e}_{2}+\vec{\nabla}V=0,$
(85)

Now we want to express components of $\vec{\nabla}V$ in terms of the FS basis,
this is

$\vec{\nabla}V\cdot\hat{e}_{j}(t)=V_{a}e^{a}_{j}=V_{j},$ (86)

where we have used $a$ as an index for the basis $\hat{u}_{a}$ and $j$ for the
index in the FS system.

## References

  * (1) C. Vafa, _The String landscape and the swampland_ , hep-th/0509212.
  * (2) H. Ooguri and C. Vafa, _On the Geometry of the String Landscape and the Swampland_ , _Nucl. Phys. B_ 766 (2007) 21 [hep-th/0605264].
  * (3) E. Palti, _The Swampland: Introduction and Review_ , _Fortsch. Phys._ 67 (2019) 1900037 [1903.06239].
  * (4) M. van Beest, J. Calderón-Infante, D. Mirfendereski and I. Valenzuela, _Lectures on the Swampland Program in String Compactifications_ , 2102.01111.
  * (5) U. H. Danielsson and T. Van Riet, _What if string theory has no de Sitter vacua?_ , _Int. J. Mod. Phys. D_ 27 (2018) 1830007 [1804.01120].
  * (6) G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa, _De Sitter Space and the Swampland_ , 1806.08362.
  * (7) F. Denef, A. Hebecker and T. Wrase, _de Sitter swampland conjecture and the Higgs potential_ , _Phys. Rev. D_ 98 (2018) 086004 [1807.06581].
  * (8) D. Andriot, _On the de Sitter swampland criterion_ , _Phys. Lett. B_ 785 (2018) 570 [1806.10999].
  * (9) S. K. Garg and C. Krishnan, _Bounds on Slow Roll and the de Sitter Swampland_ , _JHEP_ 11 (2019) 075 [1807.05193].
  * (10) H. Ooguri, E. Palti, G. Shiu and C. Vafa, _Distance and de Sitter Conjectures on the Swampland_ , _Phys. Lett. B_ 788 (2019) 180 [1810.05506].
  * (11) D. Andriot, _New constraints on classical de Sitter: flirting with the swampland_ , _Fortsch. Phys._ 67 (2019) 1800103 [1807.09698].
  * (12) J. P. Conlon, _The de Sitter swampland conjecture and supersymmetric AdS vacua_ , _Int. J. Mod. Phys. A_ 33 (2018) 1850178 [1808.05040].
  * (13) K. Dasgupta, M. Emelin, E. McDonough and R. Tatar, _Quantum Corrections and the de Sitter Swampland Conjecture_ , _JHEP_ 01 (2019) 145 [1808.07498].
  * (14) D. Andriot and C. Roupec, _Further refining the de Sitter swampland conjecture_ , _Fortsch. Phys._ 67 (2019) 1800105 [1811.08889].
  * (15) H. Geng, _Distance Conjecture and De-Sitter Quantum Gravity_ , _Phys. Lett. B_ 803 (2020) 135327 [1910.03594].
  * (16) U. Atli and O. Guleryuz, _A Solution to the de Sitter Swampland Conjecture $versus$ Inflation Tension via Supergravity_, 2012.14920.
  * (17) K. Dasgupta, M. Emelin, M. M. Faruk and R. Tatar, _How a four-dimensional de Sitter solution remains outside the swampland_ , 1911.02604.
  * (18) R. Blumenhagen, M. Brinkmann and A. Makridou, _A Note on the dS Swampland Conjecture, Non-BPS Branes and K-Theory_ , _Fortsch. Phys._ 67 (2019) 1900068 [1906.06078].
  * (19) C. Damian and O. Loaiza-Brito, _Some remarks on the dS conjecture, fluxes and K-theory in IIB toroidal compactifications_ , 1906.08766.
  * (20) D. Andriot, P. Marconnet and T. Wrase, _Intricacies of classical de Sitter string backgrounds_ , _Phys. Lett. B_ 812 (2021) 136015 [2006.01848].
  * (21) M.-S. Seo, _Thermodynamic interpretation of the de Sitter swampland conjecture_ , _Phys. Lett. B_ 797 (2019) 134904 [1907.12142].
  * (22) J.-O. Gong and M.-S. Seo, _Instability of de Sitter space under thermal radiation in different vacua_ , 2011.01794.
  * (23) R. Blumenhagen, C. Kneissl and A. Makridou, _De Sitter Quantum Breaking, Swampland Conjectures and Thermal Strings_ , 2011.13956.
  * (24) A. Kobakhidze, _A brief remark on convexity of effective potentials and de Sitter Swampland conjectures_ , 1901.08137.
  * (25) Z. Wang, R. Brandenberger and L. Heisenberg, _Eternal Inflation, Entropy Bounds and the Swampland_ , 1907.08943.
  * (26) S. Brahma and S. Shandera, _Stochastic eternal inflation is in the swampland_ , 1904.10979.
  * (27) P. Agrawal, G. Obied, P. J. Steinhardt and C. Vafa, _On the Cosmological Implications of the String Swampland_ , _Phys. Lett. B_ 784 (2018) 271 [1806.09718].
  * (28) A. Achúcarro and G. A. Palma, _The string swampland constraints require multi-field inflation_ , _JCAP_ 02 (2019) 041 [1807.04390].
  * (29) L. Heisenberg, M. Bartelmann, R. Brandenberger and A. Refregier, _Dark Energy in the Swampland_ , _Phys. Rev. D_ 98 (2018) 123502 [1808.02877].
  * (30) L. Heisenberg, M. Bartelmann, R. Brandenberger and A. Refregier, _Dark Energy in the Swampland II_ , _Sci. China Phys. Mech. Astron._ 62 (2019) 990421 [1809.00154].
  * (31) Y. Akrami, R. Kallosh, A. Linde and V. Vardanyan, _The Landscape, the Swampland and the Era of Precision Cosmology_ , _Fortsch. Phys._ 67 (2019) 1800075 [1808.09440].
  * (32) D. Wang, _The multi-feature universe: Large parameter space cosmology and the swampland_ , _Phys. Dark Univ._ 28 (2020) 100545 [1809.04854].
  * (33) C.-I. Chiang, J. M. Leedom and H. Murayama, _What does inflation say about dark energy given the swampland conjectures?_ , _Phys. Rev. D_ 100 (2019) 043505 [1811.01987].
  * (34) H. Matsui and F. Takahashi, _Eternal Inflation and Swampland Conjectures_ , _Phys. Rev. D_ 99 (2019) 023533 [1807.11938].
  * (35) R. Schimmrigk, _The Swampland Spectrum Conjecture in Inflation_ , 1810.11699.
  * (36) P. Agrawal and G. Obied, _Dark Energy and the Refined de Sitter Conjecture_ , _JHEP_ 06 (2019) 103 [1811.00554].
  * (37) C. Damian and O. Loaiza-Brito, _Two-Field Axion Inflation and the Swampland Constraint in the Flux-Scaling Scenario_ , _Fortsch. Phys._ 67 (2019) 1800072 [1808.03397].
  * (38) O. Trivedi, _Implications of single field inflation in general cosmological scenarios on the nature of dark energy given the swampland conjectures_ , 2011.14316.
  * (39) O. Trivedi, _Swampland conjectures and single field inflation in modified cosmological scenarios_ , 2008.05474.
  * (40) S. Das and R. O. Ramos, _Runaway potentials in warm inflation satisfying the swampland conjectures_ , _Phys. Rev. D_ 102 (2020) 103522 [2007.15268].
  * (41) R. Brandenberger, V. Kamali and R. O. Ramos, _Strengthening the de Sitter swampland conjecture in warm inflation_ , _JHEP_ 08 (2020) 127 [2002.04925].
  * (42) M. Cicoli, S. De Alwis, A. Maharana, F. Muia and F. Quevedo, _De Sitter vs Quintessence in String Theory_ , _Fortsch. Phys._ 67 (2019) 1800079 [1808.08967].
  * (43) R. Blumenhagen, _Large Field Inflation/Quintessence and the Refined Swampland Distance Conjecture_ , _PoS_ CORFU2017 (2018) 175 [1804.10504].
  * (44) M. C. David Marsh, _The Swampland, Quintessence and the Vacuum Energy_ , _Phys. Lett. B_ 789 (2019) 639 [1809.00726].
  * (45) Y. Olguin-Trejo, S. L. Parameswaran, G. Tasinato and I. Zavala, _Runaway Quintessence, Out of the Swampland_ , _JCAP_ 01 (2019) 031 [1810.08634].
  * (46) M. Ibe, M. Yamazaki and T. T. Yanagida, _Quintessence Axion Revisited in Light of Swampland Conjectures_ , _Class. Quant. Grav._ 36 (2019) 235020 [1811.04664].
  * (47) S. Brahma and M. W. Hossain, _Dark energy beyond quintessence: Constraints from the swampland_ , _JHEP_ 06 (2019) 070 [1902.11014].
  * (48) M. Cicoli, G. Dibitetto and F. G. Pedro, _Out of the Swampland with Multifield Quintessence?_ , _JHEP_ 10 (2020) 035 [2007.11011].
  * (49) R. Bousso, _The Holographic principle_ , _Rev. Mod. Phys._ 74 (2002) 825 [hep-th/0203101].
  * (50) R. Bousso, _A covariant entropy conjecture_ , _Journal of High Energy Physics_ 1999 (1999) 004.
  * (51) R. Bousso and N. Engelhardt, _Generalized Second Law for Cosmology_ , _Phys. Rev. D_ 93 (2016) 024025 [1510.02099].
  * (52) R. Bousso, H. Casini, Z. Fisher and J. Maldacena, _Proof of a Quantum Bousso Bound_ , _Phys. Rev. D_ 90 (2014) 044002 [1404.5635].
  * (53) M. Grana, _Flux compactifications in string theory: A Comprehensive review_ , _Phys. Rept._ 423 (2006) 91 [hep-th/0509003].
  * (54) R. Blumenhagen, I. Valenzuela and F. Wolf, _The Swampland Conjecture and F-term Axion Monodromy Inflation_ , _JHEP_ 07 (2017) 145 [1703.05776].
  * (55) R. Blumenhagen, C. Damian, A. Font, D. Herschmann and R. Sun, _The Flux-Scaling Scenario: De Sitter Uplift and Axion Inflation_ , _Fortsch. Phys._ 64 (2016) 536 [1510.01522].
  * (56) R. Blumenhagen, A. Font, M. Fuchs, D. Herschmann, E. Plauschinn, Y. Sekiguchi et al., _A Flux-Scaling Scenario for High-Scale Moduli Stabilization in String Theory_ , _Nucl. Phys. B_ 897 (2015) 500 [1503.07634].
  * (57) R. Blumenhagen, D. Herschmann and E. Plauschinn, _The Challenge of Realizing F-term Axion Monodromy Inflation in String Theory_ , _JHEP_ 01 (2015) 007 [1409.7075].
  * (58) N. Cabo Bizet, C. Damian, O. Loaiza-Brito and D. M. Peña, _Leaving the Swampland: Non-geometric fluxes and the Distance Conjecture_ , _JHEP_ 09 (2019) 123 [1904.11091].
  * (59) P. Betzler and E. Plauschinn, _Type IIB flux vacua and tadpole cancellation_ , _Fortsch. Phys._ 67 (2019) 1900065 [1905.08823].
  * (60) H. Sati and U. Schreiber, _Equivariant Cohomotopy implies orientifold tadpole cancellation_ , _J. Geom. Phys._ 156 (2020) 103775 [1909.12277].
  * (61) N. Cabo Bizet, C. Damian, O. Loaiza-Brito, D. K. M. Peña and J. A. Montañez Barrera, _Testing Swampland Conjectures with Machine Learning_ , _Eur. Phys. J. C_ 80 (2020) 766 [2006.07290].
  * (62) E. Plauschinn, _Moduli stabilization with non-geometric fluxes – comments on tadpole contributions and de-Sitter vacua_ , 2011.08227.
  * (63) I. Bena, J. Blabäck, M. Graña and S. Lüst, _The Tadpole Problem_ , 2010.10519.
  * (64) L. Pinol, _Multifield inflation beyond $N_{\mathrm{field}}=2$: non-Gaussianities and single-field effective theory_, 2011.05930.
  * (65) D. Chakraborty, R. Chiovoloni, O. Loaiza-Brito, G. Niz and I. Zavala, _Fat inflatons, large turns and the $\eta$-problem_, _JCAP_ 01 (2020) 020 [1908.09797].
  * (66) M. Scalisi and I. Valenzuela, _Swampland distance conjecture, inflation and $\alpha$-attractors_, _JHEP_ 08 (2019) 160 [1812.07558].
  * (67) S. Kachru, M. B. Schulz and S. Trivedi, _Moduli stabilization from fluxes in a simple IIB orientifold_ , _JHEP_ 10 (2003) 007 [hep-th/0201028].
  * (68) Planck collaboration, Y. Akrami et al., _Planck 2018 results. X. Constraints on inflation_ , _Astron. Astrophys._ 641 (2020) A10 [1807.06211].
  * (69) G. Dvali, C. Gomez and S. Zell, _Quantum Breaking Bound on de Sitter and Swampland_ , _Fortsch. Phys._ 67 (2019) 1800094 [1810.11002].