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JHEP06(2019)070
where M
Pl
= 1/
√
8πG is the reduced Planck mass, M is an energy scale, α is a dim en-
sionless constant. S
m
and S
r
are the matter and radiation action. In the above, one can
rescale the parameter α to replace M by M
Pl
. It is also straightforward to see that setting
α = 0 gets us back to the standard quintessence action.
When potential V (π) is linear, the scalar field action preserves the Galilean shift sym-
metry π → π + b
µ
x
µ
+ c, where b
µ
and c are constants. However, an exponential po-
tential, which is what we shall consider in this work, results in the breaking of this shift
symmetry. Note that our primary aim is to show how the swampland conjectures can eas-
ily be satisfied when including non-canonical higher-derivative terms in the action, going
beyond quintessence models. As it is, quantum corrections can lead to the appearance
of higher derivative terms when the non-renormalizable theorem is violated [76], wh en
the Galileon symmetry is not weakly-broken. Our choice of the exponential potential is
purely phenomenological in spirit and takes this specific form to facilitate comparison with
quintessence [
13] and some classes of Horndeski theories [56], when combined with the
swampland conjectures.
1
Apart from the potential term, one also needs to make a choice for the higher derivative
terms to keep in the action. While the quartic/quintic terms in the covariant Galileon
formalism has been made nonviable since the detection of multi-messenger gravitational
wave astronomy [
79, 80],
2
the cubic term is highly disfavored by Integrated Sachs-Wolfe
effect (ISW) measurements [
83]. However, generalizations of the cubic Galileon models have
been shown to be compatible with the ISW data [
84]. Keeping these in mind, we shall only
consider th e cubic Galileon term, (∇π)
2
π, in our action. We remind the reader that not
only has this term not been ruled out by gravitational wave detection, neither it been shown
to be in conflict with the ISW data in the presence of a potential term [
71], as is the case
here. Fin ally, and most importantly, we emphasize that this specific mo del has been chosen
only to provide a concrete example to illustrate that parts of the parameter space of general
Horndeski theories remain viable even after the imposition of the swampland conjectures.
Varying the action (2.1) with respect to (w.r.t.) the metric g
µν
gives the Einstein
equation
M
2
Pl
G
µν
= T
(m)µν
+ T
(r)µν
+ T
(π)µν
, (2.2)
where subscripts m, r and π represent m atter, radiation and the scalar field respectively
and
T
(π)µν
= π
;µ
π
;ν
−
1
2
g
µν
(∇π)
2
− g
µν
V (π) +
α
M
3
h
π
,µ
π
;ν
π + g
µν
π
;λ
π
;λρ
π
;ρ
−π
;ρ
(π
;µ
π
;νρ
+ π
;ν
π
;µρ
)
i
, (2.3)
and varying w.r.t. the scalar field π gives the equation of motion of the scalar field
π +
α
M
3
h
(π)
2
− π
;µν
π
;µν
− R
µν
π
;µ
π
;ν
i
− V
′
(π) = 0 , (2.4)
where
′
denotes the derivative w.r.t. π.
1
However, independently from these considerations, one needs to add a potential term so as to get
ghost-free late-time acceleration in models of cubic Galileon [
77, 78].
2
For earlier work, see [
81, 82].
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