LMU-ASC 36/11
ROM2F/2011/10

# Introducing the Slotheon: a slow Galileon scalar field in curved space-time

Cristiano Germani<EMAIL_ADDRESS>Arnold Sommerfeld Center, Ludwig-
Maximilians-University, Theresienstr. 37, 80333 Muenchen, Germany  Luca
Martucci<EMAIL_ADDRESS>I.N.F.N. Sezione di Roma “Tor Vergata” &
Dipartimento di Fisica, Università di Roma “TorVergata”, Via della Ricerca
Scienti ca, 00133 Roma, Italy  Parvin Moyassari
<EMAIL_ADDRESS>Excellence Cluster Universe, Boltzmannstr. 2,
85748 Garching, Germany  Arnold Sommerfeld Center, Ludwig-Maximilians-
University, Theresienstr. 37, 80333 Muenchen, Germany

###### Abstract

In this paper, we define covariant Galilean transformations in curved
spacetime and find all scalar field theories invariant under this symmetry.
The Slotheon is a Galilean invariant scalar field with a modified propagator
such that, whenever gravity is turned on and energy conditions are not
violated, it moves “slower” than in the canonical set-up. This property is
achieved by a non-minimal derivative coupling of the Slotheon to the Einstein
tensor. We prove that spherically symmetric black holes cannot have Slotheonic
hairs. We then notice that in small derivative regimes the theory has an
asymptotic local shift symmetry whenever the non-canonical coupling dominates
over the canonical one.

##  I Introduction

Undoubtedly, the search for theories with special symmetries is a key issue in
theoretical physics. Indeed, usually, such theories have the advantage of
being quantum mechanically under control.

One of the simplest possible symmetries is the shift invariance of a scalar
field $\pi$, i.e. the symmetry under the shift

$\displaystyle\pi\rightarrow\pi+c\ ,$ (1)

where $c$ is a constant. However, such a symmetry is not very interesting as
any theory involving only derivatives of $\pi$ would be invariant under (1).
The question is then whether such symmetry may be generalized to a more
complicated shift

$\displaystyle\pi\rightarrow\pi+f(x)\ ,$ (2)

where $f(x)$ is some specific function of space-time coordinates, depending on
some constants parametrizing the independent symmetries encoded in (2). The
class of Lagrangians invariant under (2) will be more and more constrained,
depending on the degree of arbitrariness of $f(x)$. The extreme case is the
one in which $f(x)$ is a completely arbitrary function and then (2) can be
regarded as a gauge symmetry.

The next to trivial shift symmetry is what is commonly called Galileon shift
galilean . This symmetry, formulated solely in flat (Minkowski) space, is an
on-shell symmetry. In other words, the equation of motion are invariant under
the Galileon shift

$\displaystyle\pi\rightarrow\pi+c+c_{\mu}x^{\mu}\ ,$ (3)

with $c$ and $c_{\mu}$ respectively a constant and a constant vector, whereas
the action shifts by a total derivative which gives a non-vanishing boundary
contribution. Mainly inspired by the decoupling limit of the Dvali-Gabadadze-
Porrati (DGP) model dgp , the Authors of galilean  showed that in flat space,
there exist only four forms of scalar field Lagrangians with second order
field equations and invariant under the Galileon symmetry. These theories,
turned out to admit a non-renormalization theorem. In other words, it is
proven that the mass parameters in the Galileon terms do not get renormalized
non ; non2 111Note however that any other operator can be generated by loops,
see for example nr ; referee ..

In a subsequent analysis, the Authors of covariant  showed that healthy
covariantization of the flat space Galileon invariant theories, would
generically break the flat space Galileon invariance. This is mainly due to
the fact that the constant form $c_{\mu}$, is not shear free, i.e.
$\nabla_{(\alpha}c_{\beta)}\neq 0$, where we defined
$v_{(\alpha\beta)}=\frac{1}{2}\left(v_{\alpha\beta}+v_{\beta\alpha}\right)$.
Indeed, by inserting the transformation (3) in the equation of motion for the
scalar field one always get terms proportional to the shear of
$c_{\alpha}$222Note that since we like to obtain a scalar equation, only shear
and not vorticity (the antisymmetric part) of the covariant derivative of
$c_{\alpha}$ enters in the shifted equations..

The question is then whether any symmetric scalar field theory under the shift
(2) can be constructed in a fixed curved space-time. Moreover, we will always
implicitly consider only theories with equations of motion (for both gravity
and scalar field) that are up to second order, although we will not explicitly
state it anymore.

In a manifold with covariantly constant Killing vectors, a Galileon symmetry
similar to (3), may indeed be realized. In this case however, as we shall
prove it, only the following Lagrangians can be constructed in contrast to the
flat space case. They are 333Note that in addition there are also “tadpole”
terms such as  ${\cal L}_{\rm
tp}=M_{tp}^{3}\pi\left(1+\frac{R}{\mu_{1}^{2}}+\frac{GB}{\mu_{2}^{4}}\right)\
,$ where $M_{tp}$ and $\mu_{1,2}$ are mass scales and $GB$ is the Gauss-Bonnet
term. We will not discuss them here as we focus on source free equation of
motion for $\pi$. Nevertheless, even considering them, they will not be
invariant under the approximate gauged shift symmetry (2).

$\displaystyle{\cal L}_{2}={\cal L}_{2}^{\text{m}}+{\cal
L}_{2}^{\text{nm}}\equiv-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\pi\partial_{\nu}\pi+\frac{1}{2M_{2}^{2}}G^{\mu\nu}\partial_{\mu}\pi\partial_{\nu}\pi\
,$ (4) $\displaystyle{\cal L}_{3}={\cal L}_{3}^{\text{m}}+{\cal
L}_{3}^{\text{nm}}\equiv\pm\frac{1}{2M_{3}^{3}}(\partial\pi)^{2}\square\pi\pm\frac{1}{2M_{5}^{5}}{}^{**}R^{\alpha\beta\mu\nu}\partial_{\alpha}\pi\partial_{\mu}\pi\nabla_{\beta}\nabla_{\nu}\pi\
.$ (5)

where $M_{i}$ are mass scales for the operators of dimension $i+4$.
$G_{\mu\nu}$ and ${}^{**}R^{\alpha\beta\mu\nu}$ are respectively the Einstein
and double dual Riemann tensors 444The double dual Riemann tensor is defined
as
${}^{**}R^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}\equiv-\frac{1}{4}\mathcal{E}^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}~{}\mathcal{E}^{\nu_{1}\nu_{2}\nu_{3}\nu_{4}}R_{\mu_{3}\mu_{4}\nu_{3}\nu_{4}}\
,$ where
$\mathcal{E}^{\mu_{1}\mu_{2}\mu_{3}\mu_{4}}=-\frac{1}{\sqrt{-g}}\delta_{1}^{[\mu_{1}}\delta_{2}^{\mu_{2}}\delta_{3}^{\mu_{3}}\delta_{4}^{\mu_{4}]}\
.$ . Note that, at this level, only the quadratic canonical Lagrangian may
have a definite sign in order to avoid ghost propagation around Minkowski.
Instead, the sign of ${\cal L}_{2}^{\rm nm}$ is chosen in such a way to avoid
ghosts when the weak energy condition $G^{tt}\geq 0$ is satisfied 555We would
like to stress that this condition is not enough to guarantee the absence of
ghost propagation whenever the metric is dynamical..

Next, one can remove the requirement of the existence of covariantly conserved
Killing vector and couple the theory (4) to a dynamical metric, by adding the
standard Einstein-Hilbert term and possibly also a potential for $\pi$. In
this case, we will show that for parity invariant Lagrangians
($\pi\rightarrow-\pi$), in the small derivatives regime of the scalar, an
approximate infinitesimal shift symmetry (2) emerges for the theory ${\cal
L}_{2}^{\text{nm}}$, if and only if an appropriate shift of the metric is also
considered. The theory ${\cal L}_{2}^{\text{nm}}$, in this regime, is the base
for the Gravitationally-Enhanced-Friction (GEF) models of inflation yuki ; new
; uv . Therefore, thanks to the additional symmetry, the GEF models are
endowed, during inflation (small scalar field derivatives) with a protection
against quantum corrections to the effective Lagrangian up to the Planck
scales, if the potential terms only softly break the gauged shift symmetry
(2). In fact, although other terms may be generated by loops that are
invariant under the symmetry (2), they will be generically suppressed by
either slow roll or higher powers of Planck mass. This is mainly due to the
fact that in this theory, the gravity strong coupling is still at the Planck
scale, as we shall show.

By adding the standard Einstein-Hilbert term to ${\cal L}_{2}$, and possibly a
non trivial potential for $\pi$, one gets a simple though rich gravitational
theory, with some nice peculiarities. In particular, in regimes in which the
analogue of the weak energy condition is valid, the field $\pi$ moves ‘slower’
than in the cousin canonical theory. For this reason, we dub $\pi$ as the
Slotheon and the Slotheonic nature of this theory is, in fact, at the origin
of the efficiency of the GEF models.

Furthermore, we show that the Slotheonic theory has only spherically symmetric
black hole solutions with no scalar hairs and we find indications that this
property should hold for any black hole solutions. This important result
combined with previous analysis in homogeneous and isotropic space-times yuki
, is a step forward to prove the stability of this theory.

##  II Shift and accidental symmetries

In this section we would like to find all the possible scalar field
Lagrangians (collectively denoted as ${\cal L}$), that do not contain purely
potential terms, according to some symmetry principle.

First of all, being $\pi$ a real scalar, it is natural to impose on ${\cal L}$
the shift symmetry (1). In other words, we assume that ${\cal L}$ depends only
on derivatives of $\pi$. Furthermore, in order to avoid problems with ghosts,
due to the so called Ostrogradski instability ghost  we also impose that the
equations of motion contain at most second order derivatives.

These requirements still leaves a large number of possible ${\cal L}$. In
order to further restricts the form of the action we require that, for certain
background metrics, the shift symmetry (1) enhances to accidental point-
dependent shift-symmetries of the form (2).

The prototype example of such an effect is provided by the Galileon symmetry
for flat space-time galilean , in which (1) enhances to (3). We would like to
preserve the Galileon symmetry for the case of flat space-time and generalized
it to more general, though still restricted, curved space-times.

As we shall discuss, the on-shell flat space-time symmetry (3) can be
naturally generalized in background metrics with a certain number of
covariantly constant vectors $\xi_{a}=\xi_{a}^{\mu}\partial_{\mu}$. Requiring
that the equations of motion derived from ${\cal L}$ preserve such accidental
symmetries will lead us to consider only the set ${\cal L}_{2,3}$.

###  II.1 Galileon symmetry in curved space-time

In flat spacetime, the transformations (3) can be written in a covariant way
by introducing the translation Killing vectors of Minkowski, i.e.
$\xi_{a}^{\mu}=\delta_{a}^{\mu}$ in Cartesian coordinates. These Killing
vectors are very special as they have the properties of being convariantly
constant, i.e. $\nabla_{\mu}\xi_{a}=0$.

Our definition of Galilean transformation in curved spacetime will just be the
straightforward generalization of the flat case. Let us introduce the one
forms $\xi^{a}=\xi^{a}_{\mu}\mbox{{\rm d}}x^{\mu}$ dual to the Killing
vectors: $\xi^{a}_{\mu}=g_{\mu\nu}\xi^{\nu}_{a}$. Hence, we will require that
our theories are invariant under

$\displaystyle\pi_{\mu}\rightarrow\pi_{\mu}+c_{a}\,\xi^{a}_{\mu}\ ,$ (6)

where $c_{a}$ are constants, latin indices are contracted with Euclidean
metric and we have used the notation

$\displaystyle\pi_{\mu_{1}\ldots\mu_{k}}\equiv\nabla_{\mu_{1}}\ldots\nabla_{\mu_{k}}\pi\
.$ (7)

Consistency of (6) requires the one forms $\xi^{a}$ to be closed:

$\displaystyle\mbox{{\rm d}}\xi^{a}=0\ .$ (8)

In other words, $\xi_{a}$ must be covariantly constant and we will loosely say
that the Killing vectors must be integrable 666Note, that one could find
different symmetries. For example, if one relaxes the constant shift
invariance (1) in curved space-time, one can find other theories invariant
under specific shifts $\pi\rightarrow\pi+c(x)$, where $c(x)$ is a function of
curvatures. A typical example is given in burrage ; goon1 . We thank Claudia
de Rham for pointing this out..

Space-times admitting integrable Killing vectors are of particular type
Stefani . A Killing vector $\xi^{\mu}$ can be covariantly constant only if
$\xi$ satisfies the algebraic condition

$\displaystyle R^{\mu}{}_{\nu\rho\sigma}\xi^{\nu}=0\ ,$ (9)

which can be obtained from the consistency condition
$[\nabla_{\rho},\nabla_{\sigma}]\xi^{\mu}=0$. In other words, the holonomy
group of space-time must be reduced to a subgroup of SO$(1,3)$. Explicitly, if
the vector is non-null, the space-time metric is of the form

$\displaystyle\mbox{{\rm d}}s^{2}=g_{ij}(x^{k})\mbox{{\rm d}}x^{i}\mbox{{\rm
d}}x^{j}+\kappa\,\mbox{{\rm d}}y^{2}\,,~{}~{}~{}i,j,k=1,2,3\ ,$ (10)

where $\kappa=+1,-1$ for spacelike or timelike $\xi^{\mu}$, respectively, or
for a null $\xi^{\mu}$

$\displaystyle\mbox{{\rm d}}s^{2}=g_{ij}(x^{k})\mbox{{\rm d}}x^{i}\mbox{{\rm
d}}x^{j}+\mbox{{\rm d}}z\mbox{{\rm d}}y\,,~{}~{}~{}i,j,k=1,2,3\ ,$ (11)

where $z$ is any coordinate in the $i$’s directions.

Given a set of integrable Killing vectors $\xi_{a}$ we can easily integrate
(6) into a curved Galileon transformation

$\displaystyle\pi(x)\rightarrow\pi(x)+c+c_{a}\int^{x}_{\gamma,x_{0}}\xi^{a},$
(12)

where we have chosen a certain reference point $x_{0}$ and a curve $\gamma$
connecting $x$ with $x_{0}$. Thanks to (8), this quantity is well defined.
Indeed, it does not change under continuous deformation of the curve $\gamma$.
Furthermore, the change of the reference point $x_{0}$ can be reabsorbed into
a shift of $c$.

The transformation (12) represents our proposal of curved Galileon symmetry.
Let us revisit the Minkowski case in this covariant language. In that case,
the integrable Killing vectors are the four generators of the translations.
Fixing $x^{\mu}$ to be the Minkowskian coordinates, the associated one-forms
take the form

$\displaystyle\xi^{a}\Big{|}_{\rm Mink_{4}}=\delta^{a}_{\mu}\mbox{{\rm
d}}x^{\mu}\ .$ (13)

By choosing $x_{0}$ as the origin $x^{\mu}=0$ it is immediate to see that (12)
reproduces (3), with $c_{\mu}\equiv c_{a}\delta^{a}_{\mu}$.

###  II.2 Galileon invariant theories in curved space-time

We would like now to find a Galilean invariant theory in curved spacetime
where the metric is non-dynamical. Later on we will drop this last
requirement. The Lagrangian defining the theory we look for is either 1) the
covariantized version of the Galileon Lagrangian in Minkowski or 2) made of
terms which vanish once restricted to flat space.

We start with theories that are not trivial once restricted to flat space,
i.e. the case 1). In flat space, Galilean invariant theories were classified
by galilean . To keep the equation of motion second order in curved spacetime,
the Authors in covariant  showed that the original flat space self couplings
of the scalar field derivatives must be supplemented by non-minimal couplings
to curvatures.

We can now check directly what theories among the one classified in covariant
are invariant under the Galilean symmetries in curved spacetime (6). The key
point to bare in mind is that the shift of the scalar field derivative is
covariantly constant and, the same shift, once contracted to curvatures
vanishes as in (9). Therefore, if the scalar field equation of motion contains
terms proportional to one derivative of $\pi$ without contraction to
curvatures, then, the theory is not Galilean invariant in curved spacetime. An
example of those terms is $R(\partial\pi)^{2}$.

Inspecting the set of four Lagrangians found in covariant , the only Galilean
invariant theories in curved spacetime which are not trivial in the flat limit
are ${\cal L}_{2}^{\rm m},{\cal L}_{3}^{\rm m}$.

We now consider theories that are solely non-minimally coupled, i.e. they
vanish in the flat limit. These theories, and in fact all possible scalar-
tensor theories with second order equation of motion, are classified by
Hordenski in hord . We can then easily check that the only theories invariant
under the Galilean symmetry in curved spacetime are ${\cal L}_{2}^{\rm
nm},{\cal L}_{3}^{\rm nm}$. This is again due to the fact that terms
proportional to one derivative of $\pi$ without contraction to curvatures are
not Galilean invariant in curved spacetime.

In conclusion, the only invariant theory under Galilean symmetry in curved
spacetime (6), parameterized by the masses $M_{i}$, is

$\displaystyle S=-\frac{1}{2}\int\mbox{{\rm d}}^{4}x\sqrt{-g}\left({\cal
L}_{2}+{\cal L}_{3}\right)\ .$ (14)

It is interesting to note that the theory (14), as in the flat case non ,
follows a non-renormalization theorem for the mass parameters $M_{i}$,
whenever gravity is non dynamical. This is due to the fact that in each cubic
vertex, the algebraic structure of derivatives is not different from the one
of the flat space, thanks to Bianchi identities. In other words, one can
easily check that cubic interactions only produce effective higher-derivatives
operators and therefore cannot renormalize $M_{i}$. Specifically, one sees,
following non  for the flat case, that vertexes of type

$\displaystyle\sqrt{-g}{}^{**}R^{\alpha\beta\mu\nu}\partial_{\alpha}\pi_{\rm
ext}\partial_{\mu}\pi_{\rm int}\nabla_{\beta\nu}\pi_{\rm
int}=-\sqrt{-g}{}^{**}R^{\alpha\beta\mu\nu}\nabla_{\alpha\nu}\pi_{\rm
ext}\partial_{\mu}\pi_{\rm int}\partial_{\beta}\pi_{\rm int}+{\rm boundaries}\
,$ (15)

where $\pi_{\rm ext}$ and $\pi_{\rm int}$ are respectively the external and
the internal legs of a diagram involving loops, cannot renormalize $M_{3}$ as
they are equivalent to higher-derivatives powers in $\pi_{\rm ext}$.

Obviously, as the quadratic Lagrangian has no any interactions in the case of
non-dynamical metric, $M_{2}$ is not renormalized as well.

This conclusion would change in the case in which gravity is dynamical.
Nevertheless, in this case, the parameters $M_{i}$ would only have runnings
suppressed by $M_{\rm P}$, as we shall discuss later on.

##  III Gauging the shift invariance in curved space-time

As we showed before, only few manifolds may support a Galileon symmetry. We
may however ask whether the Galileon invariance introduced earlier can be
recovered, even in some approximate sense, in space-times with no integrable
Killing vectors.

An obvious requirement is that, if such an approximate symmetry exists, it
should be also realized point-wise. Locally indeed we can always define
Riemann coordinates $(x_{R}^{\mu})$ around any point $P$ such that, for any
constant form $c_{\mu}$ in this coordinates

$\displaystyle\nabla_{\mu}c_{\nu}={\cal O}(x_{R})\ .$ (16)

In this case we can ask whether there exists any theory invariant under the
local Galileon symmetry

$\displaystyle\pi\rightarrow\pi+c+c_{\mu}x_{R}^{\mu}\ ,$ (17)

up to order ${\cal O}(x_{R})$. Note that, although the Christoffel symbols
vanish up to order ${\cal O}(x_{R})$, curvatures do not vanish. Let us then
consider the scalar field equations of motion of theories (4) and (5)

$\displaystyle E_{2}:$ $\displaystyle=$
$\displaystyle\left(g^{\mu\nu}-\frac{G^{\mu\nu}}{M^{2}_{2}}\right)\pi_{\mu\nu}\
,$ (18) $\displaystyle E_{3}:$ $\displaystyle=$
$\displaystyle\pm\left(\pm\frac{M_{3}^{3}}{M_{5}^{5}}{}^{**}R^{\mu_{1}\mu_{2}\nu_{1}\nu_{2}}+g^{\mu_{1}\nu_{1}}g^{\mu_{2}\nu_{2}}-g^{\mu_{1}\nu_{2}}g^{\mu_{2}\nu_{1}}\right)\left(\pi_{\mu_{1}\nu_{1}}\pi_{\mu_{2}\nu_{2}}+R^{\alpha}{}_{\mu_{1}\nu_{1}\mu_{2}}\pi_{\alpha}\pi_{\nu_{2}}\right)\
.$

We can easily see that only $E_{2}$ is invariant under the approximate shift
(17) up to distance $x_{R}\sim\ell$ where $\ell$ is the local curvature radius
of the spacetime.

At this level however, the approximate symmetry (17) cannot be extended far
away from the point $P$. It is then clear that gravity should participate to
the Galileon shift in order to extend this symmetry at distances such that the
Christoffel symbols cannot be neglected. We will then only focus on the theory
${\cal L}_{2}$. This theory can be singled out by requiring the action to be
invariant under the following additional discrete symmetry

$\displaystyle\pi\rightarrow-\pi\ ,$ (19)

which we will loosely call $\pi$-parity.

Let us then study the following action:

$\displaystyle S(g,\pi)=\int\mbox{{\rm d}}^{4}x\sqrt{-g}\left[\frac{M_{\rm
P}^{2}}{2}R(g)+{\cal L}_{2}\right]\ .$ (20)

Now we can make the metric $g_{\mu\nu}$ participate actively, enlarging the
possibility of identifying the relevant (approximate) symmetries of the form
(2).

Consider the following small derivative expansion regime

$\displaystyle\varepsilon\sim\frac{(\partial\pi)^{2}}{M_{2}^{2}M^{2}_{\rm
P}}\ll 1\ ,$ (21)

and note that fab

$\displaystyle\int d^{4}x\sqrt{-g}G^{\alpha\beta}\pi_{\alpha}\pi_{\beta}$
$\displaystyle=$ $\displaystyle M_{2}^{2}M^{2}_{\rm P}\int
d^{4}x\frac{\delta{\sqrt{-g}R}}{\delta g_{\alpha\beta}}\delta
g_{\alpha\beta}\Big{|}_{\delta
g_{\alpha\beta}=-\frac{\partial_{\alpha}\pi\partial_{\beta}\pi}{M_{2}^{2}M_{\rm
P}^{2}}}+{\rm boundaries}\ .$ (22)

We find that (20) can be found as an expansion of ${\cal O}(\varepsilon^{2})$
of the following action 777Our quadratic action (23) agrees with kurt  and
disagrees with claudia . This can be seen by noticing that the purely
derivative quadratic terms in $\pi$ of kurt  (Eq. (31) of the cited paper) is
nothing else than the Ricci scalar coupled to the kinetic term of $\pi$ plus
boundary terms. The disagreement is due to a missing factor upon passing from
the correct expansion Eq.(77) to the Lagrangian Eq.(79) of claudia .

$\displaystyle\hat{S}(h,\pi)=\frac{1}{2}\int\mbox{{\rm
d}}^{4}x\,\sqrt{-h}\left[M_{\rm
P}^{2}R(h)-h^{\mu\nu}\partial_{\mu}\pi\partial_{\nu}\pi\right]\ ,$ (23)

where

$\displaystyle h_{\mu\nu}\equiv
g_{\mu\nu}-\frac{\partial_{\mu}\pi\partial_{\nu}\pi}{M_{2}^{2}M_{\rm P}^{2}}\
,$ (24)

and $h^{\mu\nu}$ is the inverse of $h_{\mu\nu}$ which is also known as Finsler
metric. Explicitly, we have

$\displaystyle\hat{S}(h,\pi)=S(g,\pi)+{\cal O}\left(\varepsilon^{2}\right)\ .$
(25)

Notice that, in the regime (21) and to leading order in the perturbative
$\varepsilon$-expansion, the canonical action $\hat{S}(h,\pi)$ can be regarded
as the Einstein frame action of the theory (20). Clearly, this is not true if
(21) is violated, as the two theories are substantially different.

The metric $h_{\mu\nu}$ is exactly invariant if we consider the combined
transformation

$\displaystyle\pi\rightarrow\pi+f(x)\ ,\quad g_{\mu\nu}\rightarrow
g_{\mu\nu}+2\frac{\partial_{(\mu}f\partial_{\nu)}\pi}{M_{2}^{2}M_{\rm P}^{2}}\
,$ (26)

where we can assume that

$\displaystyle\frac{\partial f}{M_{2}M_{\rm P}}\sim{\cal
O}(\sqrt{\varepsilon})\ .$ (27)

In this way, the transformed $\pi$ continues to satisfy the small derivative
condition (21). This simple observation has an immediate consequence in a
regime in which the Lagrangian $G^{\mu\nu}\pi_{\mu}\pi_{\nu}/M_{2}^{2}$
dominates over the canonical kinetic term in (20), i.e.

$\displaystyle\frac{{\cal L}_{2}^{\rm nm}}{{\cal L}_{2}^{\rm m}}\gg 1\ .$ (28)

We call this the high friction regime for reasons which will become clear in
the following discussions.

If the system is in high friction regime the theory (20) can be recast as a
first order expansion of the Einstein-Hilbert action for the metric $h$

$\displaystyle S_{\rm EH}(h)=\frac{M_{\rm P}^{2}}{2}\int\mbox{{\rm
d}}^{4}x\sqrt{-h}\,R(h)\ .$ (29)

It is now easy to see that, in the small derivative high-friction regime
defined by (21) and (28), the Slotheon action $S(g,\pi)$ is invariant under
the transformation (26) up to terms of order ${\cal
O}\left(\varepsilon^{2}\right)$. Hence, we conclude that in this regime the
action (20) has an approximate symmetry (26) which ‘gauges’ the constant shift
symmetry $\pi\rightarrow\pi+c$ by mixing $\pi$ and metric degrees of freedom.
As it is clear from the above discussion, this gauge symmetry simply removes
the physical degrees of freedom encoded in $\pi$, which recombines with $g$
into the physical Einstein metric $h$, at least to first order in
$\varepsilon$.

It is interesting to compare this symmetry with the curved Galileon symmetry
discussed in section II. Consider a certain metric $g$ with a certain set of
Killing vectors $\xi_{a}$ and take $f(x)$ to have the form given by (12), i.e.

$\displaystyle f(x)=c+c_{a}\int^{x}_{x_{0},\gamma}\xi^{a}\ .$ (30)

By requiring (21), we must impose $c_{a}\xi^{a}_{\mu}/(M_{2}M_{\rm
P})\sim\sqrt{\varepsilon}$. Having fixed the metric, the equation of motion
for $\pi$ is clearly invariant under (30). However now, differently from the
curved Galileon symmetry in section II, the symmetry (26) acts also on the
metric $g_{\mu\nu}$, under which the equations should be approximately
invariant by construction, if (21) and (28) are satisfied. This effect can be
understood by observing that the transformation for $g_{\mu\nu}$ can be
regarded as an infinitesimal $\pi$-dependent ‘diffeomorphism’

$\displaystyle g_{\mu\nu}\rightarrow g_{\mu\nu}+\nabla_{(\mu}w_{\nu)}\,,\qquad
w_{\mu}=2\frac{c_{a}\xi^{a}_{\mu}\pi}{M_{2}^{2}M_{\rm P}^{2}}\ .$ (31)

We would like to end this section by commenting on ${\cal L}_{3}$. As
discussed in fab ,

$\displaystyle\int
d^{4}x\sqrt{-g}{}^{**}R^{\alpha\beta\mu\nu}\pi_{\alpha}\pi_{\mu}\pi_{\beta\nu}$
$\displaystyle=$ $\displaystyle-\frac{1}{4}M_{2}^{2}M_{\rm P}^{2}\int
d^{4}x\frac{\delta{\sqrt{-g}\pi GB}}{\delta g_{\alpha\beta}}\delta
g_{\alpha\beta}\Big{|}_{\delta
g_{\alpha\beta}=-\frac{\partial_{\alpha}\pi\partial_{\beta}\pi}{M_{2}^{2}M_{\rm
P}^{2}}}+{\rm boundaries}\ ,$ (32)

where $GB$ is the Gauss-Bonnet combination. Therefore, ${\cal L}_{3}$ might be
also rewritten in terms of a Finsler metric in the high friction regime.
However, the presence of the tadpole term $\pi GB$, would not be invariant
under the symmetry (26).

We then focus in the rest of the paper only on the following $\pi$-parity
invariant Lagrangian:

$\displaystyle S_{\rm sloth}=\frac{1}{2}\int\mbox{{\rm
d}}^{4}x\sqrt{-g}\left[M_{\rm
P}^{2}R-\left(g^{\alpha\beta}-\frac{G^{\alpha\beta}}{M^{2}}\right)\partial_{\alpha}\pi\partial_{\beta}\pi\right]\
,$ (33)

where we replaced $M_{2}\rightarrow M$ for notational simplicity.

###  III.1 On the strong coupling in high friction regime

Because of the non-trivial coupling of gravity with the Slotheon, the
identification of the tree-level strong coupling scale of the theory (33) is
in general strongly background dependent. In other words, in order to
calculate the perturbative cut-off scale of the theory (33) as an expansion of
fields in specific backgrounds, one should take care upon identifying the
correct propagating degrees of freedom which are generically a combination of
the Slotheon, the graviton and the background quantities.

In order to identify the physical perturbative degrees of freedom one should
rewrite the theory as an expansion around gaussian fixed points. Although the
obvious Minkowski cut-off of the theory (33) is $\Lambda_{\text{cut-
off}}=(M^{2}M_{\rm P})^{1/3}$, it has been shown in yuki ; new  that for a
slow rolling Slotheon in a homogeneous and isotropic background, (i.e. when
$\varepsilon\ll 1$) the strong coupling scale of the theory is enhanced to
$\Lambda_{\text{cut-off}}=M_{\rm P}+{\cal O}(\varepsilon)$. This result can be
readily generalized by using the previous arguments. Indeed, let us consider
an expansion around the $\varepsilon\ll 1$ solution, which we loosely call
small derivative regime.

If there exists a non-trivial background for the scalar field, one can always
reparameterize time in order to reabsorb the perturbative scalar degree of
freedom into the metric. Explicitly, let us consider the expansion of the
Slotheon around a background solution $\pi_{0}$. At first order (higher orders
are easily generalizable) we have

$\displaystyle\pi=\pi_{0}(t,\vec{x})+\delta\pi(t,\vec{x})\ ,$ (34)

where $\delta\pi$ is the perturbation.

We can now consider the first order coordinate transformation $t\rightarrow
t+\delta t$ to obtain, at first order

$\displaystyle\pi=\pi_{0}+\dot{\pi}_{0}\delta t+\delta\pi\ .$ (35)

Therefore, by choosing the gauge $\delta t=-\delta\pi/\dot{\pi}_{0}$ we obtain
the desired result of reabsorbing the scalar degree of freedom into the
metric. This gauge is called unitary gauge in cosmology and widely used to
calculate (quantum) correlation functions maldacena .

Using the unitary gauge, during the small derivative regime of the background
scalar field, the theory (33) is well approximated by (29) plus the canonical
kinetic term, i.e.

$\displaystyle S=\frac{1}{2}\int d^{4}x\sqrt{-h}\left[M_{\rm
P}^{2}R(h)-(\partial\pi)^{2}\right]\ .$ (36)

Therefore, the true (gaussian) degrees of freedom become a self interacting
$h_{\mu\nu}$ with cut-off scale $M_{\rm P}$ and the free scalar $\pi$.

We thus proved that the strong coupling scale of the Slotheonic theory in a
background in which $\varepsilon\ll 1$ is

$\displaystyle\Lambda_{\text{cut-off}}=M_{\rm P}+{\cal O}(\varepsilon)\ ,$
(37)

which matches direct computations in homogeneous and isotropic backgrounds
yuki ; new . The presence of a possible (renormalizable) potential term for
$\pi$ obviously does not alter this result.

In this sense then, the background $\varepsilon\ll 1$ is always in weak
coupling if $M$ and curvatures are below the Planck scale.

##  IV The Slotheon: a “slow” scalar field

Let us now investigate the properties of the theory (33). In particular, we
focus on the dynamics which governs the temporal evolution of the scalar field
$\pi$.

Let us take the ADM decomposition ADM ; wald  where the metric can be written
as

$\displaystyle\mbox{{\rm d}}s^{2}=-N^{2}\mbox{{\rm
d}}t^{2}+\gamma_{ij}(\mbox{{\rm d}}x^{i}-N^{i}\mbox{{\rm d}}t)(\mbox{{\rm
d}}x^{j}-N^{j}\mbox{{\rm d}}t)\ .$ (38)

In this parameterization of the metric the action related to ${\cal L}_{2}$
looks like

$\displaystyle S=\frac{1}{2}\int\mbox{{\rm
d}}^{4}x\,N\sqrt{\gamma}\left[\left(\frac{1}{N^{2}}+\frac{G^{tt}}{M^{2}}\right)\dot{\pi}^{2}+2\left(\frac{N^{i}}{N}+\frac{G^{ti}}{M^{2}}\right)\dot{\pi}\partial_{i}\pi-(\gamma^{ij}-\frac{G^{ij}}{M^{2}})\partial_{i}\pi\partial_{j}\pi\right]\
.$ (39)

The momentum conjugate to $\pi$ is therefore defined as

$\displaystyle\Pi=\frac{\delta
S}{\delta\dot{\pi}}=N\sqrt{\gamma}\left[\left(\frac{1}{N^{2}}+\frac{G^{tt}}{M^{2}}\right)\dot{\pi}+\left(\frac{N^{i}}{N}+\frac{G^{ti}}{M^{2}}\right)\partial_{i}\pi\right]\
.$ (40)

The contribution of the Hamiltonian density coming from the scalar field
kinetic term is then

$\displaystyle{\cal
K}_{\pi}=\frac{1}{2}\sqrt{\gamma}\,\alpha^{2}\frac{\dot{\pi}^{2}}{N^{2}}\ ,$
(41)

where we defined

$\displaystyle\alpha^{2}\equiv 1+N^{2}\frac{G^{tt}}{M^{2}}\ .$ (42)

We would now like to focus on the regimes in which $G^{tt}\geq 0$. This
condition can be regarded as the analogous of standard weak energy condition
in our non-canonical theory and immediately implies that

$\displaystyle\alpha^{2}\geq 1\ .$ (43)

Considering the same background geometry, we would like now to compare the
kinetic energies of a canonical scalar field and of the Slotheon. In order to
do that we must fix the time lapse to be the same for the two theories. The
simplest choice is to use the synchronous gauge $N=1$. In this case, for a
given kinetic energy per unit volume (${\cal K}_{\pi}$) we have

$\displaystyle\dot{\pi}^{2}\sim\frac{{\cal K}_{\pi}}{\alpha^{2}}\leq{\cal
K}_{\pi}\ .$ (44)

It is then clear that the time derivative of the Slotheon is smaller than the
corresponding one (with the same energy density) of a canonical scalar field
(where $\alpha=1$). In this sense the Slotheon is slower than a canonical
scalar field.

The same conclusion can be readily drawn also by adding a positive definite
potential. Although we imposed the $\pi$-parity invariance to select the
theory all our results in this section and in the following sections are also
valid for potentials breaking this invariance.

The Slotheonic theory is then

$\displaystyle\tilde{S}=\frac{1}{2}\int\mbox{{\rm
d}}^{4}x\sqrt{-g}\left[M_{\rm
P}^{2}R-\left(g^{\mu\nu}-\frac{G^{\mu\nu}}{M^{2}}\right)\pi_{\mu}\pi_{\nu}-2V(\pi)\right]\
,$ (45)

with $V(\pi)\geq 0$ and it is easy to see that this modification does not
modify the above arguments. Notice also that the slowing of the Slotheon is
due solely to gravitational interaction. This is profoundly different from
self-interacting theories which have similar properties only in specific
backgrounds (see for instance selfinteraction ).

A typical example of a Slotheonic theory in action can be seen on de Sitter or
almost-de Sitter (inflationary) backgrounds, where $G^{tt}=3\Lambda^{2}$ and
$\Lambda$ is roughly constant. In this case the scalar field kinetic energy of
a canonical scalar field is modified as

$\displaystyle\dot{\pi}^{2}\rightarrow(1+3\frac{\Lambda^{2}}{M^{2}})\dot{\pi}^{2}\
.$ (46)

Redefining the effective time of the scalar field as

$\displaystyle\mbox{{\rm d}}t_{\rm Slotheon}=\frac{\mbox{{\rm
d}}t}{\sqrt{1+3\frac{\Lambda^{2}}{M^{2}}}}\ ,$ (47)

we find

$\displaystyle\mbox{{\rm d}}t_{\rm Slotheon}\leq\mbox{{\rm d}}t\ .$ (48)

Therefore, one may interpret the proper clock of the Slotheon to be slower
than the clock of an observer tight to the Universe expansion. From a
different point of view, the slowness of the Slotheon in the previous example
was obtained by increasing the friction term acting on the scalar field. This
mechanism has been dubbed the Gravitational-Enhanced-Friction mechanism in
yuki  for inflationary scenarios and it is the base of New Higgs Inflation new
and UV-Protected inflation uv ; yuki .

##  V No-(Slotheonic) Hair Theorem

In this section we will prove that the only spherically symmetric black hole
solution of the Slotheonic theory (45) is the vacuum solution, i.e. the
Schwarzschild solution. This result is an important step to prove the
stability of the Slotheonic theory in general curved space-time (which we
postpone for future work). In fact, it is widely believed that ghost-like or
unstable scalar theories may support scalar hairs outside a black hole horizon
bekenstein95 .

Since we assumed that the potential does not violate energy conditions, we can
restrict our proof to the massless case. In fact, since a mass implies a
faster decay of the scalar field than the massless case, proving the
impossibility of massless scalar hairs will be enough. We will then restrict
our attention to the theory

$\displaystyle S=\frac{1}{2}\int\mbox{{\rm d}}^{4}x\sqrt{-g}\left[M_{\rm
P}^{2}\,R-\left(g^{\mu\nu}-\frac{G^{\mu\nu}}{M^{2}}\right)\pi_{\mu}\pi_{\nu}\right]\
.$ (49)

In order to prove that the only spherically symmetric solution is trivial for
the Slotheon we will closely follow bekenstein  with the help of the gravity
and scalar field equations obtained by varying the action (49) with respect to
$\pi$ and metric. The equations are respectively (see also shush )

$\displaystyle(g^{\mu\nu}-\frac{G^{\mu\nu}}{M^{2}})\pi_{\mu\nu}=0,$ (50)
$\displaystyle G_{\mu\nu}=M_{\rm P}^{-2}T_{\mu\nu}\ .$

Where

$\displaystyle
T_{\mu\nu}=\pi_{\mu}\pi_{\nu}-\frac{1}{2}g_{\mu\nu}(\partial\pi)^{2}+\frac{\Theta_{\mu\nu}}{M^{2}}\
,$ (51)

and

$\displaystyle\Theta_{\mu\nu}=\frac{1}{2}\pi_{\mu}\pi_{\nu}R-2\pi_{\alpha}\pi_{(\mu}R^{\alpha}_{\nu)}+\frac{1}{2}\pi_{\alpha}\pi^{\alpha}G_{\mu\nu}-\pi^{\alpha}\pi^{\beta}R_{\mu\alpha\nu\beta}-\pi_{\alpha\mu}\pi^{\alpha}_{\nu}+\pi_{\mu\nu}\pi_{\alpha}^{~{}\alpha}+\frac{1}{2}g_{\mu\nu}[\pi_{\alpha\beta}\pi^{\alpha\beta}-(\pi_{\alpha}^{~{}\alpha})^{2}+2\pi_{\alpha}\pi_{\beta}R^{\alpha\beta}]\
.$

###  V.1 Spherically symmetric case

Let us start by imposing spherical symmetry. In this case the metric will be

$\displaystyle\mbox{{\rm d}}s^{2}=-A(r)^{2}\mbox{{\rm
d}}t^{2}+B(r)^{2}\mbox{{\rm d}}r^{2}+r^{2}\mbox{{\rm d}}\Omega^{2}\ ,$ (52)

where $d\Omega^{2}=\mbox{{\rm d}}\theta^{2}+\sin\theta^{2}\mbox{{\rm
d}}\phi^{2}$.

The equation of motion for the scalar field reads

$\displaystyle\left(g^{\alpha\beta}-\frac{G^{\alpha\beta}}{M^{2}}\right)\nabla_{\alpha}\nabla_{\beta}\pi=0\
.$ (53)

Multiplying it by the scalar field $\pi$ and integrating in the closed region
$S$ of Fig.1, delimited by an horizon at $r_{H}$ and two time slices
$\Sigma_{\pm}$, we get

$\displaystyle\int_{S}\mbox{{\rm
d}}^{4}x\sqrt{-g}\pi\left(g^{\alpha\beta}-\frac{G^{\alpha\beta}}{M^{2}}\right)\nabla_{\alpha}\nabla_{\beta}\pi=0\
.$ (54)

Figure 1: Integration region.

Integrating by parts (54) we obtain

$\displaystyle\int_{S}\mbox{{\rm
d}}^{4}x\sqrt{-g}\left(g^{\alpha\beta}-\frac{G^{\alpha\beta}}{M^{2}}\right)\nabla_{\alpha}\pi\nabla_{\beta}\pi=\int_{H}\mbox{{\rm
d}}^{3}x\sqrt{-g}n_{\alpha}\left(g^{r\alpha}-\frac{G^{r\alpha}}{M^{2}}\right)\pi\pi^{\prime}\
,$ (55)

where the sum of the boundary integrals over $\Sigma_{\pm}$ vanish because of
staticity and the integral at infinity vanishes because the assumption of
asymptotic flatness. In (55) $H$ is the horizon surface, $n^{\alpha}$ is the
normal to the horizon and ${}^{\prime}=d/dr$. By definition an horizon is a
light-like surface, i.e. $n_{\alpha}n^{\alpha}=0$ and for a static metric
$n_{t}=0$ on the horizon. By using the Cauchy inequality

$\displaystyle 0\leq(n_{i}A^{i})^{2}\leq n_{i}n^{i}A_{j}A^{j}=0\ ,$ (56)

where the last equality is valid if and only if $A_{j}A^{j}<\infty$, we find
that the left hand side of (55) vanishes. Taking

$\displaystyle A^{j}=\left(g^{rj}-\frac{G^{rj}}{M^{2}}\right)\pi\pi^{\prime}\
,$ (57)

we see indeed that $A_{j}A^{j}$ cannot diverge for a smooth space-time and
non-divergent scalar field. We are then left with the integral equation

$\displaystyle\int_{S}\mbox{{\rm
d}}^{4}x\sqrt{-g}\left(g^{\alpha\beta}-\frac{G^{\alpha\beta}}{M^{2}}\right)\nabla_{\alpha}\pi\nabla_{\beta}\pi=\int_{S}\mbox{{\rm
d}}^{4}x\sqrt{-g}\left(g^{rr}-\frac{G^{rr}}{M^{2}}\right)\pi^{\prime 2}=0\ .$
(58)

We are now interested in finding the form of $G^{rr}$. The gravity equations
are (we fix here $M_{\rm P}$=1)

$\displaystyle G_{\alpha\beta}=T_{\alpha\beta}\ ,$ (59)

where $T_{\alpha\beta}$ is given in Eq. (51). With the metric (52) we find

$\displaystyle
G_{rr}=\frac{\frac{1}{2}-\frac{1}{M^{2}r^{2}}}{1+\frac{3}{2}\frac{\pi^{\prime
2}}{B^{2}M^{2}}}\pi^{\prime 2}\ .$ (60)

Plugging the previous result into the integral (58) we get

$\displaystyle\int_{S}\mbox{{\rm
d}}^{4}x\sqrt{-g}\frac{B^{2}+\frac{\pi^{\prime
2}}{M^{2}}(1+\frac{1}{r^{2}})}{B^{4}(1+\frac{3}{2}\frac{\pi^{\prime
2}}{B^{2}M^{2}})}\pi^{\prime 2}=0\ .$ (61)

Since the integrand is positive definite in (61), the only solution is
$\pi^{\prime}=0$, i.e. the only solution for a spherically symmetric black
hole is with no Slotheonic hairs. The black hole solution is then a solution
of the Einstein equation in vacuum that has as the unique solution the
Schwarzschild metric

$\displaystyle\mbox{{\rm d}}s^{2}=-(1-\frac{2m}{r})\mbox{{\rm
d}}t^{2}+(1-\frac{2m}{r})^{-1}\mbox{{\rm d}}r^{2}+r^{2}\mbox{{\rm
d}}\Omega^{2}\ ,$ (62)

where $m$ is the black hole mass.

###  V.2 No-hair theorem: a re-interpretation and a conjecture

We can now re-interpret the no-hair theorem proved previously in the theory
(33). If we consider the canonical theory (36) we can obviously use the
standard no-hair theorem. In that case there is no non-trivial solution for
the scalar field $\pi$ and the only spherically symmetric solution is the
Schwarzschild solution

$\displaystyle\mbox{{\rm d}}s^{2}=-(1-\frac{2m}{r})\mbox{{\rm
d}}t^{2}+(1-\frac{2m}{r})^{-1}\mbox{{\rm d}}r^{2}+r^{2}\mbox{{\rm
d}}\Omega^{2}\ .$ (63)

Of course, the theories (33) and (36) are equivalent only up to fist order in
the small derivative perturbative regime (21). Hence, the no-hair theorem for
the canonical theory (36) can only be used to easily conclude that black hole
solutions in (33) cannot have perturbative Slotheonic hairs. This provides a
non-trivial confirmation of our direct proof of the no-hair theorem in Sec.V
for spherically symmetric black holes and it automatically extends to non-
spherically symmetric black holes, within the perturbative regime. This
encourages us to conjecture that there are no black hole solutions with non-
perturbative Slotheonic hairs. We leave the investigation of this important
conjecture for future work.

##  VI Asymptotic local shift symmetry and inflation

Standard inflationary models enjoy an asymptotic shift symmetry of the scalar
creminelli

$\displaystyle\pi\rightarrow\pi+c\ ,$ (64)

due to the fact that, under such a shift, the Inflaton potential only shifts
at next to leading order in the slow roll expansion. This shift however, does
not protect the theory under new derivative couplings and it is expected to be
anyway broken by Quantum Gravity effects. In order to avoid these potential
problems one may then try to “gauge” the symmetry (64) to

$\displaystyle\pi\rightarrow\pi+f(x)\ .$ (65)

In a spatially flat Friedamn-Robertson-Walker (FRW) geometry

$\displaystyle\mbox{{\rm d}}s^{2}=-\mbox{{\rm d}}t^{2}+a(t)^{2}\mbox{{\rm
d}}\vec{x}\cdot\mbox{{\rm d}}\vec{x}\ ,$ (66)

the field and gravity evolution equations are new

$\displaystyle H^{2}=\frac{1}{3M_{\rm
P}^{2}}\left[\frac{\dot{\pi}^{2}}{2}(1+9\frac{H^{2}}{M^{2}})+V\right],$ (67)
$\displaystyle\partial_{t}\left[a^{3}\dot{\pi}(1+3\frac{H^{2}}{M^{2}})\right]=-a^{3}V^{\prime}\
,$ (68)

where $H=\frac{\dot{a}}{a}$ and $(\dot{})=\mbox{{\rm d}}/\mbox{{\rm d}}t$.

In GEF of yuki ; new ; uv , the Inflaton (a Slotheon) is non-minimally coupled
to gravity as in (33) so that slow roll may be naturally obtained. With this
coupling, even very steep potentials for the scalar field, $V(\pi)$, would
produce a successful inflationary scenario, thanks to a huge gravitational
friction acting on the Inflaton during inflation. Specifically, one can then
always choose the mass $M$ small enough such that, during inflation,
$H^{2}/M^{2}\gg 1$. Note, as explained before, that no strong coupling happens
here thanks to the canonical normalization of the field $\pi$ new ; yuki .
This regime is called the high friction regime yuki . In this regime, for any
given potential $V$, a quasi-de Sitter solution always exists for $M$ small
enough. This is the basis for the New Higgs Inflation new  and the UV-
protected Inflation uv . A quasi-de Sitter background implies that the slow
roll parameters are small, i.e.

$\displaystyle\epsilon\equiv-\frac{\dot{H}}{H^{2}}\ll 1\ ,\
\delta\equiv\Big{|}\frac{\ddot{\pi}}{H\dot{\pi}}\Big{|}\ll 1\ .$ (69)

We will firstly focus on the case in which the Inflaton potential has small
curvatures (chaotic type inflation), at least during inflation. We then ask
that the “canonical” slow roll conditions are satisfied

$\displaystyle\epsilon_{\rm can}\equiv\frac{V^{\prime 2}}{2V^{2}}M_{\rm
P}^{2}\ll 1\ ,\ \eta_{\rm can}\equiv\frac{V^{\prime\prime}}{V}M_{\rm P}^{2}\ll
1\ ,$ (70)

and assume a monomial potential for the Inflaton so that the above conditions
generically require $\pi\gg M_{\rm P}$.

In high friction limit ($H\gg M$), during slow roll, one finds that yuki

$\displaystyle\epsilon\simeq\frac{3}{2}\frac{\dot{\pi}^{2}}{M^{2}M_{\rm
P}^{2}}\simeq\epsilon_{\rm can}\frac{M^{2}}{3H^{2}}\ll 1\ ,\
V=V_{0}\left[1+{\cal O}(\sqrt{\epsilon})\frac{\delta\pi}{M_{\rm P}}\right]\ ,$
(71)

where $\delta\pi$ is a shift on the background value for $\pi$ and $\epsilon$
is the true slow roll parameter defined in (69). Thus, it is exactly in this
regime that the symmetry (26) is realized for the kinetic and gravitational
parts of the action as, in this regime, $\varepsilon\sim\epsilon\ll 1$. The
potential term also would break the symmetry (26) only at higher order in slow
roll. This can be easily seen from the action. There, the potential term would
shift as

$\displaystyle V\sqrt{-g}\rightarrow V\sqrt{-g}\left(1+{\cal
O}(\sqrt{\epsilon})\frac{f}{M_{\rm P}}\right)\ ,$ (72)

thanks to (71). In other words, the local shift symmetry (26) is only softly
broken by the potential if (70) are satisfied.

Let us now suppose that the potential generating inflation violates the
conditions (70). For monomial potential this would mean sub-Planckian field
values. The GEF mechanism would nevertheless work in order to fulfill (69) for
$M$ small enough. This can be easily seen from the first equation in (71),
which is always valid in high friction limit yuki . In this case the symmetry
(26) would in general be badly broken by the potential, unless the potential
does not introduce any self interactions. In other words, the symmetry (26)
may still be softly broken by a mass term, i.e. in the case in which
$V=V_{0}\pm\frac{1}{2}m^{2}\pi^{2}$, for any field value during inflation.
Because no self-interactions are introduced in the potential, one would indeed
expect that quantum corrections to the propagator would still be suppressed by
slow roll, i.e. they would still obey the asymptotic symmetry (26) during
inflation 888For a similar discussion see also galinf .. Thanks to that, the
UV-protected inflation of uv ; yuki , has an extra quantum protection in the
high friction limit: the local shift symmetry (26).

We then found that during inflation and in high friction regime, the
Slotheonic Lagrangian (45) enjoys an asymptotic gauge symmetry (26) protecting
chaotic type inflationary set-up and inflationary set-up with mass potentials
from quantum corrections to both the potential and the kinetic terms. Note
that extra-derivative couplings that could be added and are invariant under
the approximate symmetry (26), can only come from further expanding the action
(23). Therefore, extra-derivative couplings may only modify the equation to
higher order in slow roll, in this sense they are completely negligible and
the inflationary trajectory is stable. One may still wonder about couplings of
the Slotheon to matter fields. These couplings would generically produce a
(Coleman-Weinberg) logarithmically corrected potential for the effective
canonically normalized field $\psi=\frac{H}{M}\pi$. The strength of these
corrections depends upon the particular couplings chosen and may or may not be
important for the Inflationary evolution. We leave this important discussion
for a future work.

Finally, let us comment on possible renormalization of the mass parameter
suppressing the non-minimal coupling ${\cal L}_{2}^{\rm nm}$. Since, during
the high friction regime, the graviton is still canonically normalized with
the Planck scale (see Sec. III.1), we expect that the running of $M$ is
suppressed by the Planck scale and therefore negligible for an inflationary
trajectory where the total energy is far below $M_{\rm P}$. The study of the
exact running of $M$ is left for future work.

## Acknowledgements

We wish to thank Alex Vikman for careful reading and important comments on the
first draft of the paper. CG wishes to thank Paolo Creminelli and Guido
D’Amico, for important comments. CG and PM would also like to thank Yuki
Watanabe for useful discussions. PM wishes to thanks Claudia de Rham for
comments on the first draft. The Authors thanks Alex Kehagias for earlier
participation to the project and important discussions. Finally, the Authors
wish to thank the anonymous referee for pointing out an extra Galilean
invariant term. CG and PM are supported by Alexander Von Humboldt Foundation.
LM is partially supported by the ERC Advanced Grant n.226455 “Superfields”, by
the Italian MIUR-PRIN contract 20075ATT78 and by the NATO grant
PST.CLG.978785.

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