# Implications of single field inflation in general cosmological scenarios on
the nature of dark energy given the swampland conjectures

Oem Trivedi<EMAIL_ADDRESS>School of Arts and Sciences, Ahmedabad
University,Ahmedabad 380009,India

###### Abstract

Swampland Conjectures have attracted quite some interest in the Cosmological
Community. They have been shown to have wide ranging implications , like
Constraints on Inflationary Models, Primordial Black Holes etc. to name a few.
A particularly revealing insight on dark energy also shows that one can have
the dark energy equation of state for a quintessence scenario to be
signficantly different than -1 after one takes into account the refined dS
conjecture. Another interesting issue with the swampland conjectures is that
they have been shown to be incompatible with single field inflationary models
in GR based cosmology. In our previous work we have, however, showed that
single field inflationary models are quite compatible with swampland
conjectures in their usual string theoretic form in a large class of modified
cosmological scenarios. Building on that work, we now show that in modified
cosmological scenarios where the early universe expansion was driven by single
field inflation , one can have the dark energy equation of state to be
significantly different from -1 even if we just take into account the original
dS conjecture, let alone the refined form of that. We thereby show that one
does not need to apply a step function approach towards inflation in order to
have an observable distinction between constant and non constant dark energy
models in the context of the swampland conjectures.

††preprint: APS/123-QED

##  I Introduction

Inflationary Cosmology has achieved a tremendous amount of success in
describing the Very Early Universe sato1981first ; baumann2009tasi ;
guth1981inflationary ; linde1983chaotic ; linde1995quantum .A huge amount of
inflationary models martin2014encyclopaedia ; wands2008multiple ;
berera1995warm ; ashtekar2010loop ; langlois2008perturbations ;
golovnev2008vector ; setare2013warm ; kanti2015gauss ; dvali1999brane ;
alexander2013chern  have been shown to be very consistent with the most recent
datasets on the early universe aghanim2018planck ; akrami2020planck1 ;
akrami2018planck ; aghanim2020planck ; martin2014best .The consistency of
Inflation with observational data for large amount of diversely motivated
models is a really eye catching property of Inflationary Cosmology. Besides
explaining various problems of the early universe in standard big bang
cosmology, inflation has also provided a way to understand the current dark
energy epoch of the universe. The idea of Quintessential Inflation has
garnered quite a lot of interest recently and in this, the Inflaton field
itself becomes the cause of the accelerated expansion of the universe today
peebles1999quintessential ; caldwell1998cosmological ;
tsujikawa2013quintessence ; zlatev1999quintessence ; bento2002generalized ;
dimopoulos2019warm ; benisty2020quintessential ; dutta2008hilltop ;
chiba2009slow . The idea of Quintessence, like Inflation, has been thoroughly
studied in Modified Cosmologies as well gardner2007braneworld ;
sami2004quintessential ; koivisto2007gauss ; perrotta1999extended ;
van2017gauss ; benisty2020quintessential1 .
In recent years, a quest for a unified theory of everything has garnered a lot
of attention in high energy physics. Probably the most well known of these
”theory of everything” candidates, is String Theory green1981supersymmetrical
; green1982supersymmetric ; green1982supersymmetric1 ; gubser1998gauge ;
seiberg1999string ; garfinkle1991charged ; susskind2003anthropic ;
polchinski1998string ; lust1989lectures . It is one of the most viable
approaches for a theory of everything for many (if not the most viable), hence
one would expect that String theory has some implications for Cosmology and
could also hence allow us to understand both the very early universe and it’s
subsequent evolution in more detail. So it’s no surprise that a substantial
amount of work has been done on studying the cosmological implications of
String theory mcallister2008string ; gasperini1994dilaton ; gasperini2003pre ;
gasperini2007elements ; tseytlin1992elements ; krori1990some ; heckman2019f ;
heckman2019pixelated . One curious feature of this theory is the extremely
high amount of vacua states that is posits, going around
$\mathcal{O}(10^{500})$ and this constitutes the so called ”String Landscape”.
But then, a fundamental question that arises is how can one distinguish
between low energy effective field theories which are consistent with String
theory and those which are not. In a bid to understand this question in more
detail, Vafa in vafa2005string  gave the term ”Swampland” to refer to the set
of all low energy EFT’s which are inconsistent with the String paradigm. As
String theory is (as a consequence of being a theory of everything) seen as a
viable quantum gravitational setup by string theorists, then this would go on
to mean that the theories in the swampland are also not viable with a self
consistent theory of Quantum gravity. Further, to know as to if a given low
energy EFT belongs in the swampland, a number of field theoretic criterion
known as the ”Swampland conjectures ” have been proposed in recent years
obied2018sitter ; ooguri2016non ; mcnamara2019cobordism ; bedroya2019trans ;
garg2019bounds .The prominent Swampland Conejctures which gathered immediate
interest in the context of Cosmology were :

$1$ : Swampland Distance Conjecture (SDC) : This conjecture limits the field
space of validity of any effective field theory ooguri2016non  . This sets a
maximum range traversable by the scalar fields in an EFT as

$\Delta\phi\leq d\sim\mathcal{O}(1)$ (1)

where we are working in the Planck Units $m_{p}=1$ where $m_{p}$ is the
reduced Planck’s constant, d is some constant of $\mathcal{O}(1)$ , and $\phi$
is the Scalar Field of the EFT.

$2$ Swampland De Sitter Conjecture (SDSC) : This Conjecture states that it is
not possible to create dS Vacua in String Theory obied2018sitter . The
conjecture is a result of the observation that it has been very hard to
generate dS Vacua in String Theory dasgupta2019sitter ; danielsson2018if (
While it has been shown that creating dS Vacua in String Theory is possible in
some schemes ,like the KKLT Construction kachru2003sitter ). The Conjecture
sets a lower bound on the gradient of Scalar Potentials in an EFT ,

$\frac{|V^{\prime}|}{V}\geq c\sim\mathcal{O}(1)$ (2)

where c is some constant of $\mathcal{O}(1)$ , and V is the scalar Field
Potential. Another ” refined ” form of the Swampland De Sitter Conjecutre
(RSDSC) places constraints on the Hessian of the Scalar Potential (a finding
which first appeared in garg2019bounds  and later in ooguri2019distance  ).
Expressed in $m_{p}=1$ units, it reads

$\frac{V^{\prime\prime}}{V}\leq-c^{\prime}\sim\mathcal{O}(1)$ (3)

where $c^{\prime}$ is again some constant of Order 1.

The conjecture has had quite fascinating implications for single field
inflation, particularly those regimes in a General Relativistic Cosmology. It
was shown in kinney2019zoo  that the swampland conjectures Eq. (1-3) are not
consistent with the cosmological data on single field inflation in a GR based
cosmology and for them to be viable with this form of Inflation, the string
theory- motivated definitions of the conjectures would have to change. A lot
of work has since then been done to understand the issues of the distance and
de sitter conjectures with single field Inflation trivedi2020swampland ;
geng2020potential ; scalisi2019swampland ; scalisi2020inflation ;
ashoorioon2019rescuing .Besides these conjectures, another recently proposed
swampland conjecture by the name of the ” Trans Planckian Censorship
Conjecture(TCC)” bedroya2019trans  , implies that single field Inflation in a
GR Based Cosmology would have to be severely fine tuned in order for it to not
lie in the swampland bedroya2020trans  . If Inflation is plagued with severe
fine tuning problems itself, which were the kind of issues in standard big
bang cosmology which prompted work on Inflation in the fist case, then it is
certainly a very dire situation for Inflationary Cosmology keeping in mind
these conjectures. A lot of work has been done in order to understand the
issues of single field GR based inflation and the TCC in more detail
brahma2020trans ; brandenberger2020strengthening ; dhuria2019trans ;
kamali2020relaxing ; bernardo2020trans ; mizuno2020universal ; li2020trans ;
lin2020trans ; schmitz2020trans ; berera2020role . A particularly interesting
observation was made in brahma2020trans1  , which showed that the TCC can be
derived from the Distance conjecture (1) considering that criterion to be
true. The swampland conjectures have also had some pretty interesting
implications on the paradigm of eternal inflation matsui2019eternal ;
dimopoulos2018steep ; kinney2019eternal ; brahma2019stochastic ;
wang2019eternal ; trivedi2021rejuvenating .

The revelations from these works seemingly show that Inflation and the
Swampland conjectures are in unavoidable loggerheads with each other. However,
the true scenario is that it is specifically Single Field Supercooled
Inflation in GR Based Cosmology, which has serious issues with the swampland
criterion. Indeed, it has been shown that Inflation, even for single field
cases, can be quite consistent with the swampland conjectures in other regimes
of Inflation. One can take the example of warm inflation berera1995warm ,
which is on amicable terms with these conjectures even for single field cases
in a GR based cosmology berera2019trans ; motaharfar2019warm ; das2019warm ;
das2020swampland ; das2019note ; das2020distance ; motaharfar1810warm ;
mohammadi2020warm . Single field inflation in other cosmological scenarios
which are motivated by vastly different reasons, like string theory motivated
Braneworld Models randall1999alternative ; randall1999large ;
gogberashvili2002hierarchy  or regimes like Gauss-Bonet Cosmology
li2007cosmology  which are motivated by geometric requirements, have been
aptly studied in the context of the swampland conjectures and have proved to
be on good terms with these criterion as well lin2019chaotic ; kamali2020warm
; odintsov2020swampland ; yi2019gauss ; adhikari2020power ;
blumenhagen2017swampland . Particularly in trivedi2020swampland  , it was
shown that single field (cold) Inflation in a large class of non-GR based
cosmologies can be quite easily consistent with the swampland conjectures. For
example, this work also tangentially showed that single field inflation in
loop quantum cosmology ashtekar2010loop ; ashtekar2006quantum ;
ashtekar2006quantum1  and Chern-Simons Cosmology gomez2011standard  can also
be on good terms with the swampland conjectures. Finally, it has also been
shown that perhaps the Inflationary scenarios which fits the swampland
conjectures the best are multi field models bravo2020tip ; aragam2020multi .

The implications of the swampland conjectures on the current dark energy epoch
of the universe are quite interesting as well. Particularly the dS conjecture
(2) is in stark opposition of a cosmological constant form of dark energy and
favours quintessence to be a better realization of dark energy
agrawal2018cosmological ; cicoli2020out ; ibe2019quintessence ;
storm2020swampland ; marsh2019swampland ; olguin2019runaway ;
heisenberg2018dark ; van2019dark ; emelin2019axion ; brahma2019dark ;
odintsov2019finite . It was however shown in akrami2019landscape , that all
the existing string theory based models of quintessence with an exponential
potential or combinations of exponential potentials are ruled out by current
observations, implying that the swampland de Sitter conjecture is in tension
with viable quintessence models, similarly to the single-field slow-roll
inflation in a GR based cosmological scenario. The findings of that paper were
then further confirmed in raveri2019swampland  with even more data.
Furthermore, a significant amount of interest has also gone towards
investigating non conventional quintessence models in the view of the
swampland conjectures, with both multi-field and modified cosmological models
having been studied in this regard akrami2020multi ; cicoli2020out ;
oikonomou2020rescaled . There has also been some very insightful work on
studying quintessence based dark energy models in the view of the Hubble
Tension Riess:2016jrr ; Alam:2016hwk ; mortsell2018does ; vagnozzi2020new ;
di2020interacting ; di2020nonminimal ; visinelli2019revisiting  as a litmus
test for the swampland conjectures in colgain2019testing ; banerjee2020hubble
. Hence, there are quite a few reasons to consider quintessential inflation as
a viable paradigm of dark energy and more so in the context of the swampland
conjectures. If it is indeed the case for the current epoch of the universe,
one might expect that the equation of state parameter of the universe can be
quite different from the usual -1 in the case of a dS universe(and so a
cosmological constant form of dark energy). It has, however, been shown to not
quite be the case. Considering the universe went through (cold) single filed
inflation in it’s very early epoch and taking into account the swampland
conjectures (1-3), it was shown in chiang2019does  that equation of state
parameter can only diverge quite negligibly from -1. The point to note here is
that this work focused on a GR based cosmology and the work done on single
field inflation based in a GR based cosmology kinney2019zoo . We have however
showed in a recent paper trivedi2020swampland  that swampland conjectures have
pretty different implications on single field inflation in a general class of
modified cosmologies. Hence, in this paper we will be building on that work
and discuss the implications that the Distance and the dS conjectures have on
quintessence models in modified cosmological scenarios, given that the early
universe expansion was driven by (cold) single field inflation. In the next
section, we will be shortly reviewing the important aspects of the work
discussed in trivedi2020swampland . In Section III, we will discuss the
implications of that on the equation of state parameter of the universe and
see how one can get a viable distinction between constant and non-constant
dark energy models through the equation of state parameter in the view of the
swampland conjectures, without even having to rely to a step-function approach
to inflation . We will then conclude this work in section IV with some general
remarks.

##  II Single field inflation in general cosmologies and the swampland
conjectures

The central points of conflict which have been raised between single field
Inflation and the primary swampland conjectures were explored deeply in
kinney2019zoo . The most serious issues which arose between these two
paradigms were based on the string theory-motivated definitions of the
swampland conjectures, being in grave conflict with the available
observational data on single field inflation. We will quickly go over these
issues now. One of the central quantities which one encounters while studying
single field inflation are the slow roll parameters. A particularly important
slow roll parameter is the $\epsilon$ parameter, which can be written in
$m_{p}=1$ units in a GR based cosmology as,

$\epsilon=\frac{1}{2}\left(\frac{V^{\prime}}{V}\right)^{2}$ (4)

During inflation one needs $\epsilon<<1$ baumann2009tasi , and hence, it’s
immediately evident that the smallness requirement of the $\epsilon$ parameter
is in direct conflict with the dS conjecture (2). Another issue concerns the
e-fold number during Inflation, which is given the GR paradigm roughly as

$N\approx\frac{V\Delta\phi}{V^{\prime}}$ (5)

The e-fold number is a good measure of telling how much Inflation occurs in
some certain model. For inflation to solve the problems of the standard big
bang cosmology, one needs at least around 50 to 60 e-folds of inflationary
expansion akrami2020planck1 . But it can again be immediately clear, that if
one seriously considers the distance and dS conjectures (1-2), then the e-fold
number would not even be greater than unity. This is an incredible assertion
and one which would raise questions over the existence of any kind of single
field inflationary expansion in the context of the swampland conjectures.
Further in kinney2019zoo , the following relation between the scalar spectral
index aghanim2018planck ; akrami2020planck1  and the c and $c^{\prime}$
parameters from the swampland conjectures (1-2) , was derived

$1-n_{s}=[3c^{2};2c^{\prime}]$ (6)

Using the data on Inflation akrami2020planck1 , one can hence deduce that
$c\leq\mathcal{O}(0.1)$ and $c^{\prime}\leq\mathcal{O}(0.01)$, which is
clearly very different then the $\mathcal{O}(1)$ estimates of these parameters
in their string theoretic form (2-3). The above equation (6), had it’s roots
in the following relations between the slow roll and swampland conjecture
parameters,

$\epsilon\geq\frac{c^{2}}{2}$ (7) $\eta\leq-c^{\prime}$ (8)

This clearly shows that the swampland conjectures in their string theoretic
form are not consistent with the data on single field inflation, in a general
relativistic cosmology. While there have been works which have dealt with the
swampland conjectures in some particular cosmologies lin2019chaotic ;
odintsov2020swampland ; yi2019gauss ; blumenhagen2017swampland , in
trivedi2020swampland  it was shown using a very general approach that the
swampland conjectures will not have any unavoidable conflicts with single
field inflation in a large class of non GR cosmological scenarios. The general
Friedmann equation we considered in trivedi2020swampland , is given by

$F(H)=\frac{\rho}{3}$ (9)

where we are working in $m_{p}=1$ units. Here F(H) is some function of the
Hubble Parameter and $\rho$ is the energy density of the universe. Friedmann
equations of this form have been used to study Inflation in detail in the
paradigm of the Exact Solution Approach del2012approach ; trivedi2020exact ,
which is a generalization of the Hamilton-Jacobi Approach to Inflation to
general cosmologies kinney1997hamilton . Using this method, we can revaluate
whether the issues which persist between single field inflation and the
swampland conjectures in a GR based cosmology still hold tight in large class
of cosmologies described by the general friedmann equation (9) (for more
details on how general our friedmann equation in consideration is, please
refer to del2012approach ; trivedi2020exact ). The work in
trivedi2020swampland  showed that the main issues of conflict between single
field inflation and swampland conjectures highlighted in kinney2019zoo  , only
necessarily hold good in a GR based cosmology and are not sources of
unavoidable loggerheads between these two paradigms in the more general
cosmologies described by the Friedmann equation above.
The $\epsilon$ parameter, for example, can be as small as is needed for a
period of inflationary expansion as in general the $\epsilon$ parameter can be
written as trivedi2020swampland ,

$\epsilon=\frac{FH^{\prime}k}{H^{3}}$ (10)

where $k=\frac{V^{\prime}}{V}$ and considering the dS conjecture (2) , $k\geq
c\sim\mathcal{O}(1)$. During inflation, one needs $\epsilon<<1$ and we can
immediately start to see that in general cosmological scenarios ($F(H)\neq
H^{2}$ ), one can easily have the $\epsilon$ parameter as small as needed
whilst still satisfying the dS conjecture requirement $k>\mathcal{O}(1)$.
There is no severe issue of an insufficient e-fold number as the e-fold number
in this paradigm is given by ,

$N\approx\frac{H^{2}}{F}\frac{\Delta\phi}{k}$ (11)

where again for non-GR cosmologies, the e-fold number can satisfy both the
distance and dS conjectures (1-2), and still be high enough for inflation to
solve the fine tuning problems of standard big bang cosmology. And finally,
the scalar spectral index is related to the c and $c^{\prime}$ parameters as,

$\left(\frac{(1-n_{s})H^{2}}{2F}\right)\bigg{[}\frac{1}{\left(3-H\frac{F_{{}_{\prime}HH}}{F_{{}_{\prime}H}}\right)\frac{H^{\prime}}{H}-\frac{H^{\prime\prime}}{H^{\prime}}}\bigg{]}=\frac{F_{{}_{\prime}H}H^{\prime}}{F}\geq
c$ (12)
$\frac{1}{F_{{}_{\prime}H}}\bigg{[}F_{{}_{\prime}HH}H^{\prime}+\eta\frac{F_{{}_{\prime}H}H^{2}}{H^{\prime}}\bigg{]}\leq-c^{\prime}$
(13)

Again, the above equations (12-13) show that the $n_{s}$ can have it’s
appropriate observational value aghanim2018planck ; akrami2020planck1  whilst
still considering the string theoretic $\mathcal{O}(1)$ estimates of the c and
$c^{\prime}$ parameters. For the full discussion on how single field inflation
and swampland conjectures can be in a peaceful harmony in general cosmological
scenarios in the scope of our considered Friedmann equation (9), please refer
to trivedi2020swampland .

##  III Swampland conjecture implications on quintessence dark energy equation
of state in general cosmological scenarios

As mentioned earlier, there has been quite a lot of work on Quintessence
models in the light of the swampland criterion. The swampland conjectures
support the idea that the current dark energy epoch of our universe is due to
rolling quintessence field rather than a positive cosmological constant
agrawal2018cosmological . It was also recently shown that considering the
swampland conjectures to be true, one would not observe the equation of state
parameter of the universe to be greatly different than it’s dS value of -1
chiang2019does . This revelation was made considering a GR based cosmology and
as we are talking about only a single quintessence field, this result was also
made on the consideration that the early universe expansion occured through
single field inflation.
We will now,however, like to show that the equation of state parameter in
general cosmological scenarios described by (9), can indeed have a
signficantly different value from it’s dS scenario even after considering the
swampland conjectures. By definition, the equation of state parameter is given
by

$w=\frac{p}{\rho}$ (14)

where p and $\rho$ are the pressure and energy densities of the quintessence
field, respectively, given by $p=\frac{\dot{\phi}^{2}}{2}-V(\phi)$ and
$\rho=\frac{\dot{\phi}^{2}}{2}+V(\phi)$ . Hence,

$1+w=\frac{2\dot{\phi}^{2}}{\dot{\phi}^{2}+2V}$ (15)

For a rolling quintessence field, we have tsujikawa2013quintessence

$3H\dot{\phi}\approx-V^{\prime}(\phi)$ (16)

which allows us to express (15) as

$1+w=\frac{2{V^{\prime}}^{2}}{{V^{\prime}}^{2}+18H^{2}V}$ (17)

Further, for a rolling field in a general cosmological scenario our Friedmann
equation (9) becomes

$F\approx\frac{V}{3}$ (18)

and this allows us to further rewrite (17) as

$1+w=\frac{2{V^{\prime}}^{2}F}{{FV^{\prime}}^{2}+6V^{2}H^{2}}$ (19)

Finally, considering the dS conjecture (2) we can write the above equation as

$1+w\geq\frac{2c^{2}F}{Fc^{2}+6H^{2}}$ (20)

One can now see in the above inequality that for general non-GR based
cosmologies ($F\neq H^{2}$ ), w can have quite a different value than it’s
usual dS case even when we consider the dS conjecture. As one considers
general cosmological scenarios, the parameter c can retain it’s
$\mathcal{O}(1)$ estimate and still be consistent with data on single field
inflation. This makes a pivotal difference, as similar analysis for the
quintessence field in a GR based cosmology (with $c\approx 0.02$ kinney2019zoo
) resulted in a quite negligible deviation from -1 for w chiang2019does .
However, the c parameter retaining it’s string theoretic value coupled with
the fact that one can work with a range of different F functions corresponding
to different cosmologies (again, we recommend del2012approach ;
trivedi2020exact  for an overview of the generality of the Friedmann equation
in consideration), which would then give one a range of model-specific free
parameters which could then be tuned to get quite a significant deviation from
the dS value of w. It’s important to also note that in the Hamilton-Jacobi
approach one starts with an ansatz for the Hubble Parameter kinney1997hamilton
(contrary to the usual approach to inflationary studies where one starts off
with a potential and gets the physically interesting results from that) , and
in the generalizations of the Hamilton-Jacobi to general cosmologies
del2012approach ; trivedi2020exact  one also takes some predefined F to
ascertain the cosmology they want to study inflation in. Hence, the free
parameters for the model specific cases generate from these ansatz that one
starts off with before making concrete calculations. This shows that in
general cosmological scenarios one can have an appropriately observable
distinction between constant and non constant dark energy models, whilst still
considering the swampland conjectures to hold true in their string theoretic
forms. This results in an exciting prospect for experiments which are aiming
to measure w to within a percent’s accuracy, like DES dark2005dark , DESI
flaugher2014dark  , PFS takada2014extragalactic  , Euclid laureijs2010euclid
, HSS miyazaki2012hyper  , LSST lsst2012large  and WFIRST spergel2015wide  .

In chiang2019does , a step function approach towards inflation was taken and
both the forms of the dS conjectures were applied in different stages of the
inflationary regime owing to the tensions of the dS conjectures individually
with single field inflation and the consequent negligible deviation that the w
parameter has for a quintessence scenario when these conjectures are
considered in a GR based cosmology. As the dS conjectures and single field
inflation do not have any unavoidable conflicts with each other in general
cosmological scenarios trivedi2020swampland , one does not need adopt the step
function approach to inflation in these cosmological scenarios and can have
quite a different value of the w parameter from it’s positive cosmological
constant case.

##  IV Concluding Remarks and Discussion

In this paper we have discussed the status of dark energy quintessence models
in general cosmological scenarios whilst considering the swampland
conjectures. We started by discussing the core issues of conflict between
single fied inflation and swampland conjectues in a GR based cosmology and
what these issues have implied for quintessence models. Then we briefly
discussed how there are no such unavoidable conflicts between single field
inflation and swampland conjectures in general cosmological scenarios, which
are in the scope of a particular friedmann equation (9) and where the inflaton
( and consequently the quintessence field ) follows a usual Klein-Gordon form
. We then built upon that, and showed that the equation of state parameter in
general cosmological scenarios can be significantly different from it’s
cosmological constant value of -1 and hence, one can make a clear distinction
between constant and non constant dark energy models in such cosmologies even
after considering the swampland conjectures. We then concluded by noting that
the absence of any serious tension between single field inflation and the
conjectures in general cosmological scenarios, we hence showed that one does
not need to apply any step function approach to single field inflation in
order to have an equation of state parameter significantly different from -1
in the context of the swampland conjectures.

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