# Scalar fields with derivative coupling to curvature in the Palatini and the
metric formulation

Hamed Bouzari Nezhad  and Syksy Räsänen

###### Abstract

We study models where a scalar field has derivative and non-derivative
couplings to the Ricci tensor and the co-Ricci tensor with a view to
inflation. We consider both the metric formulation and the Palatini
formulation. In the Palatini case, the couplings to the Ricci tensor and the
Ricci scalar give the same result regardless of whether the connection is
unconstrained or the non-metricity or the torsion is assumed to vanish. When
the co-Ricci tensor is included, the unconstrained case and the zero torsion
case are physically different. We reduce all the actions to the Einstein frame
with minimally coupled matter, and find the leading order differences between
the metric case and the Palatini cases.

HIP-2023-11/TH

##  1 Introduction

Inflation is the most successful scenario for the early universe [1, 2, 3, 4,
5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15], and its predictions agree well with
observations [16]. The simplest candidate for driving inflation is a scalar
field. The field may be non-minimally coupled to curvature, as such couplings
are generated by loop corrections [17]. Direct coupling to the Ricci scalar is
the key feature of Higgs inflation [18, 19, 20, 21]. Derivatives of the field
can also couple to curvature [22, 23, 24, 25, 26, 27, 28, 29, 30]. In the
Higgs case, inflationary models with such couplings are called New Higgs
Inflation [31, 32, 33, 34, 35, 36, 37, 38, 39]. When both derivative and non-
derivative non-minimal couplings are present, the theories are sometimes
called hybrid models [40, 41, 42, 43, 44, 45, 46].

Generic actions with derivative couplings to the curvature, like generic
actions with higher order curvature terms, lead to higher than second order
equations of motion, which involve extra degrees of freedom that suffer from
the Ostrogradsky instability [47]. The most general scalar-tensor theories
with second order equations of motion, called Horndeski theories, are
explicitly known [48, 49, 50]. They are, however, not the most general stable
scalar-tensor theories, because it is possible that the theory is degenerate
and some degrees of freedom are not physical. On the gravity side, the
simplest example is $f(R)$ theory [47]. Degenerate higher order scalar-tensor
theories (DHOST) have been explicitly catalogued up to terms cubic in the
second derivatives of the field [49, 50]. The only such theories that are
phenomenologically viable (with propagating gravitational waves and a
Newtonian limit), at least at linear order in perturbation theory, are those
that are related to Horndeski theories by an invertible disformal
transformation [51] (see also [52, 53]). Beyond DHOST are U-degenerate scalar-
tensor theories, which are degenerate only in the unitary gauge, where the
gradient of the scalar field has to be timelike [54, 55, 56, 57, 58, 59]. They
have also been explicitly catalogued up to third order in second derivatives,
and the procedure for determining whether a theory with arbitrary powers of
second derivatives is DHOST or U-degenerate or neither is known.

These results are for the metric formulation of gravity. In other formulations
that are equivalent for the Einstein–Hilbert action with minimally coupled
matter but physically distinct for more complicated actions, the stability
properties of non-minimally coupled scalar fields have not been completely
categorised. (For Horndeski theories in teleparallel and symmetric
teleparallel gravity, see [60, 61].) One of the most common alternatives to
the metric formulation is the Palatini formulation, where the connection is an
independent variable [62, 63].111Some works have taken the metric formulation
Horndeski action and simply replaced the Levi–Civita connection with a
connection treated as an independent variable. In general, the resulting
theories do not have second order equations of motion and are not stable [64,
65, 66, 67]. Higgs inflation, where the field couples directly to the Ricci
scalar has been much studied in the Palatini formulation, and the predictions
are different than in the metric case [68, 69, 70, 71, 72, 73, 74, 75, 76, 77,
78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90]. Inflation in the case
when derivatives of the field couple directly to the curvature has also been
studied [91, 92, 93, 94]; in [95], such a theory was used for quintessence
(see also [96, 97, 98, 99]). Unlike in the case when only the field couples
directly to the curvature, in the derivative coupling case the results of the
metric and the Palatini formulation are close to each other. We extend
previous work by including the co-Ricci tensor in the cases when the
connection is taken to be metric-compatible or torsion-free a priori. When
parts of the connection are constrained in the action, the theory is in
general different from the unconstrained case. For example, the theory with an
Einstein–Hilbert term plus a term quadratic term in the antisymmetric part of
the Ricci tensor is stable in the zero torsion case, but unstable in the
unconstrained case [100, 101].

In section 2 we give the geometrical background for the Palatini formulation
and present the action. We shift to the Einstein frame with minimally coupled
matter by making a disformal transformation followed by solving the remaining
pieces of the connection from the equation of motion and inserting them back
into the action. We calculate the leading order differences between the
Palatini cases when the connection is unconstrained, when non-metricity or
torsion is put to zero, and the metric case. In section 3 we summarise our
findings and outline open questions. Some technical details are relegated to
appendices A and B.

##  2 Non-minimal coupling to kinetic terms

###  2.1 Curvature, non-metricity, and torsion

In the Palatini formulation the metric $g_{\alpha\beta}$ and the connection
$\Gamma^{\gamma}_{\alpha\beta}$ are independent variables. The connection,
defined with the covariant derivative as
$\nabla_{\beta}A^{\alpha}=\partial_{\beta}A^{\alpha}+\Gamma^{\alpha}_{\beta\gamma}A^{\gamma}$,
can be decomposed as

$\displaystyle\Gamma^{\gamma}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\mathring{\Gamma}^{\gamma}_{\alpha\beta}+L^{\gamma}{}_{\alpha\beta}=\mathring{\Gamma}^{\gamma}_{\alpha\beta}+J^{\gamma}{}_{\alpha\beta}+K^{\gamma}{}_{\alpha\beta}\
,$ (2.1)

where $\mathring{\Gamma}_{\alpha\beta}^{\gamma}$ is the Levi–Civita connection
of the metric $g_{\alpha\beta}$. We denote quantities defined with the
Levi–Civita connection by $\mathring{}$. In the second equality we have
decomposed the distortion tensor $L^{\gamma}{}_{\alpha\beta}$ into the
disformation tensor $J_{\alpha\beta\gamma}$ and the contortion tensor
$K_{\alpha\beta\gamma}$, defined as

$\displaystyle J_{\alpha\beta\gamma}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\left(Q_{\alpha\beta\gamma}-Q_{\gamma\alpha\beta}-Q_{\beta\alpha\gamma}\right)\
,\qquad
K_{\alpha\beta\gamma}\equiv\frac{1}{2}(T_{\alpha\beta\gamma}+T_{\gamma\alpha\beta}+T_{\beta\alpha\gamma})\
,$ (2.2)

where $Q_{\alpha\beta\gamma}$ and $T_{\alpha\beta\gamma}$ are the non-
metricity and the torsion, respectively, defined as

$\displaystyle\qquad
Q_{\gamma\alpha\beta}\equiv\nabla_{\gamma}g_{\alpha\beta}\ ,\qquad
T^{\gamma}{}_{\alpha\beta}$ $\displaystyle\equiv$ $\displaystyle
2\Gamma^{\gamma}_{[\alpha\beta]}\ .$ (2.3)

We have $Q_{\gamma\alpha\beta}=Q_{\gamma(\alpha\beta)}$,
$J_{\alpha\beta\gamma}=J_{\alpha(\beta\gamma)}$, and
$K^{\gamma}{}_{\alpha}{}^{\beta}=K^{[\gamma}{}_{\alpha}{}^{\beta]}$.

The Riemann tensor can be decomposed into the Levi–Civita and the distortion
contributions as

$\displaystyle
R^{\alpha}{}_{\beta\gamma\delta}=\mathring{R}^{\alpha}{}_{\beta\gamma\delta}+2\mathring{\nabla}_{[\gamma}L^{\alpha}{}_{\delta]\beta}+2L^{\alpha}{}_{[\gamma|\mu|}L^{\mu}{}_{\delta]\beta}\
.$ (2.4)

There are three independent first contractions of the Riemann tensor, called
Ricci-type tensors,

$\displaystyle R_{\alpha\beta}\equiv R^{\gamma}{}_{\alpha\gamma\beta}\
,\quad\hat{R}_{\alpha\beta}\equiv
g_{\alpha\epsilon}g^{\gamma\delta}R^{\epsilon}{}_{\gamma\delta\beta}\
,\quad\tilde{R}_{\alpha\beta}\equiv R^{\gamma}{}_{\gamma\alpha\beta}\ .$ (2.5)

The first is the Ricci tensor, the second is the co-Ricci tensor, and the
third is the homothetic curvature tensor. There is only one independent Ricci
scalar, $R=-\hat{R}$, $\tilde{R}=0$. Instead of the co-Ricci tensor, it can be
convenient to use the average of the co-Ricci tensor and the Ricci tensor.
Using the definition (2.5) and the decompositions (2.1), (2.2), and (2.4), we
see that the average vanishes when $Q_{\alpha\beta\gamma}=0$,

$\accentset{\Delta}{R}_{\alpha\beta}\equiv\frac{1}{2}(\hat{R}_{\alpha\beta}+R_{\alpha\beta})=g^{\mu\nu}\nabla_{[\beta}Q_{\mu]\nu\alpha}-\frac{1}{2}T^{\mu\nu}{}_{\beta}Q_{\mu\nu\alpha}\
.$ (2.6)

The Einstein tensor is

$\displaystyle G_{\alpha\beta}$ $\displaystyle\equiv$
$\displaystyle-\frac{1}{4}\epsilon_{\alpha\gamma}{}^{\mu_{1}\nu_{1}}\epsilon_{\beta}{}^{\gamma\mu_{2}\nu_{2}}R_{\mu_{2}\nu_{2}\mu_{1}\nu_{1}}=\frac{1}{2}(R_{\alpha\beta}-\hat{R}_{\alpha\beta}-g_{\alpha\beta}R)\
,$ (2.7)

where $\epsilon_{\alpha\beta\gamma\delta}$ is the Levi–Civita tensor.

###  2.2 The action

We consider a scalar field $\varphi$ whose kinetic term
$X_{\alpha\beta}\equiv\partial_{\alpha}\varphi\partial_{\beta}\varphi$ couples
linearly to the first traces of the Riemann tensor, while $\varphi$ can appear
non-linearly. (General non-linear couplings have been studied in [101].) The
homothetic curvature tensor does not appear because it is antisymmetric, so in
the Palatini case, we have couplings to $R_{\alpha\beta}$,
$\hat{R}_{\alpha\beta}$ and $R$, and the action is

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\bigg{[}\frac{1}{2}F(\varphi)g^{\alpha\beta}R_{\alpha\beta}-\frac{1}{2}K(\varphi)g^{\alpha\beta}X_{\alpha\beta}+\frac{1}{2}\alpha_{1}(\varphi)g^{\alpha\beta}g^{\gamma\delta}R_{\alpha\beta}X_{\gamma\delta}$
(2.8)
$\displaystyle+\frac{1}{2}\alpha_{2}(\varphi)g^{\alpha\gamma}g^{\beta\delta}R_{\alpha\beta}X_{\gamma\delta}+\frac{1}{2}\alpha_{3}(\varphi)g^{\beta\gamma}g^{\delta\mu}R^{\alpha}{}_{\beta\gamma\delta}X_{\alpha\mu}-V(\varphi)+\mathcal{L}_{\textrm{m}}(\Psi,\varphi,g^{\alpha\beta})\bigg{]}$
$\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2}(F+\alpha_{1}X)R-\frac{1}{2}KX+\frac{1}{2}(\alpha_{2}R^{\alpha\beta}+\alpha_{3}\hat{R}^{\alpha\beta})X_{\alpha\beta}-V+\mathcal{L}_{\textrm{m}}\right]\
,$

where $g=\det g_{\alpha\beta}$, $X\equiv g^{\alpha\beta}X_{\alpha\beta}$, and
$\mathcal{L}_{\textrm{m}}(\Psi,\varphi,g_{\alpha\beta})$ is a matter
action222Fermion kinetic terms involve the connection. We neglect them; it is
always possible to assume that they couple only to the Levi–Civita connection,
and thus do not contribute to the distortion., with $\Psi$ denoting any matter
degrees of freedom other than $\varphi$.

In the metric case $\hat{R}_{\alpha\beta}=-R_{\alpha\beta}$, so we can put
$\alpha_{3}=0$. Then when $\alpha_{1}=-\frac{1}{2}\alpha_{2}$, the action is
of the Horndeski form, and there are no extra degrees of freedom, otherwise
there is an extra ghost [48]. If also $\alpha_{2}>0$, the scalar degree of
freedom corresponding to $\varphi$ is healthy, otherwise it is a ghost [31].

In the Palatini case, the theory is different depending on which, if any,
constraints are imposed on the connection. The case without constraints has
been studied in [96, 97]. Solving the connection equation obtained by varying
(2.8) with respect to $\Gamma^{\gamma}_{\alpha\beta}$ and inserting the
solution into the action gives a metric theory with a modified scalar sector.
For an action including (2.8) but more general, it was shown in [97] that the
theory is at least U-degenerate (and can be DHOST or Horndeski). The reason is
that it is symmetric under the projective transformation
$\Gamma^{\gamma}_{\alpha\beta}\to\Gamma^{\gamma}_{\alpha\beta}+\delta^{\gamma}{}_{\beta}V_{\alpha}$,
where $V_{\alpha}$ is an arbitrary vector field. When the gradient of the
scalar field is timelike, the ghost is subsumed in the unphysical projective
mode.333In [55] it is argued that U-degenerate theories could be healthy.
However, it is not clear how the theory behaves when spatial gradients are
larger than the time derivatives [56], for example during reheating or close
to the vacuum at late times. In general, projective symmetry does not
guarantee the absence of ghosts, and whether ghosts appear can depend on the
background [101]. The results of [97] show that for the action (2.8), the
theory is in the DHOST class. (For the case when $X_{\alpha\beta}$ couples
only to the Ricci scalar and the Einstein tensor (2.7), i.e.
$\alpha_{3}=-\alpha_{2}$, this was shown already in [96].)

We will consider the case when either non-metricity or torsion is set to zero.

###  2.3 Disformal transformation

We could solve the connection separately in the cases with zero non-metricity
or zero torsion and insert the solution back into the action. However, it is
easier to first get rid of all non-minimal couplings except those to
$\accentset{\Delta}{R}_{\alpha\beta}$ with a disformal transformation. This
will also establish that the result is the same in the case when the
connection is unconstrained and when non-metricity is put to zero, and that in
the zero torsion case the difference arises only from
$\accentset{\Delta}{R}_{\alpha\beta}$. It has been shown that observables such
as inflationary power spectra are invariant under disformal transformations at
least for Horndeski theories [102, 103, 104, 105, 106] (see also [107]).

We will perform an invertible disformal transformation in the action (2.8)
such that only a coupling to $\accentset{\Delta}{R}_{\alpha\beta}$ remains
[108, 109, 102, 103, 104, 105, 106, 110, 111, 107, 101]:

$\displaystyle g_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\gamma_{1}(\varphi,\tilde{X})\tilde{g}_{\alpha\beta}+\gamma_{2}(\varphi,\tilde{X})X_{\alpha\beta}\
,$ (2.9)

where $\tilde{X}\equiv\tilde{g}^{\alpha\beta}X_{\alpha\beta}$. The inverse
transformation is

$\displaystyle\tilde{g}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\tilde{\gamma}_{1}(\varphi,X)g_{\alpha\beta}+\tilde{\gamma}_{2}(\varphi,X)X_{\alpha\beta}\
.$ (2.10)

The original and tilded transformation functions are related to each other as
$\tilde{\gamma}_{1}=1/\gamma_{1}$,
$\tilde{\gamma}_{2}=-\gamma_{2}/\gamma_{1}$. The inverse metric is

$g^{\alpha\beta}=\frac{1}{\gamma_{1}}\tilde{g}^{\alpha\beta}-\frac{\gamma_{2}}{\gamma_{1}(\gamma_{1}+\gamma_{2}\tilde{X})}\tilde{g}^{\alpha\mu}\tilde{g}^{\beta\nu}X_{\mu\nu}\
,$ (2.11)

and $\tilde{g}^{\alpha\beta}$ is given by the same expression with the
replacements $\gamma_{i}\to\tilde{\gamma}_{i}$, $\tilde{X}\to X$,
$\tilde{g}^{\alpha\beta}\to g^{\alpha\beta}$. These equations give us the
relation between $X$ and $\tilde{X}$

$X=\frac{\tilde{X}}{\gamma_{1}+\gamma_{2}\tilde{X}}\ .$ (2.12)

As the original and tilded variables are in a symmetric position, $X$ as a
function of $\tilde{X}$ is, again, given by the same equation with the
original and tilded quantities switched. The determinants of the metrics are
related by

$g=\tilde{g}\gamma_{1}^{3}(\gamma_{1}+\gamma_{2}\tilde{X})\ .$ (2.13)

Under the disformal transformation (2.9), the curvature coupling terms in the
action (2.8) transform as follows

$\displaystyle\sqrt{-g}g^{\alpha\beta}R_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\sqrt{-\tilde{g}}\gamma_{1}(1+\gamma\tilde{X})^{1/2}\left(\tilde{g}^{\alpha\beta}R_{\alpha\beta}-\frac{\gamma}{1+\gamma\tilde{X}}\tilde{g}^{\alpha\gamma}\tilde{g}^{\beta\delta}R_{\alpha\beta}X_{\gamma\delta}\right)$
$\displaystyle\sqrt{-g}g^{\alpha\gamma}g^{\beta\delta}R_{\alpha\beta}X_{\gamma\delta}$
$\displaystyle=$
$\displaystyle\sqrt{-\tilde{g}}(1+\gamma\tilde{X})^{-3/2}\tilde{g}^{\alpha\gamma}\tilde{g}^{\beta\delta}R_{\alpha\beta}X_{\gamma\delta}$
$\displaystyle\sqrt{-g}g^{\beta\gamma}g^{\delta\mu}R^{\alpha}{}_{\beta\gamma\delta}X_{\alpha\mu}$
$\displaystyle=$
$\displaystyle\sqrt{-\tilde{g}}(1+\gamma\tilde{X})^{-1/2}\tilde{g}^{\beta\gamma}\tilde{g}^{\delta\mu}R^{\alpha}{}_{\beta\gamma\delta}X_{\alpha\mu}\
,$ (2.14)

where $\gamma\equiv\gamma_{2}/\gamma_{1}$.

Applying the disformal transformation (2.9) to the action (2.8), using the
above results, writing the co-Ricci tensor $\hat{R}_{\alpha\beta}$ in terms of
$\accentset{\Delta}{R}_{\alpha\beta}$ defined in (2.6), and dropping the
tildes on $g_{\alpha\beta}$ and $X$, we get

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\Bigg{\\{}\frac{1}{2}(1+\gamma
X)^{1/2}\left(\gamma_{1}F+\frac{\alpha_{1}X}{1+\gamma
X}\right)R+\frac{\alpha_{3}}{(1+\gamma
X)^{1/2}}\accentset{\Delta}{R}^{\alpha\beta}X_{\alpha\beta}$ (2.15)
$\displaystyle+\frac{1}{2}\frac{1}{(1+\gamma
X)^{1/2}}\left[-F\gamma_{2}+\frac{\alpha_{2}-\alpha_{3}-(\alpha_{1}+\alpha_{3})\gamma
X}{1+\gamma X}\right]R^{\alpha\beta}X_{\alpha\beta}$
$\displaystyle-\frac{\gamma_{1}}{2(1+\gamma
X)^{1/2}}KX-\gamma_{1}^{2}(1+\gamma X)^{1/2}V$
$\displaystyle+\gamma_{1}^{2}(1+\gamma
X)^{1/2}\mathcal{L}_{\textrm{m}}\left[\Psi,\varphi,\frac{1}{\gamma_{1}}g^{\alpha\beta}-\frac{\gamma}{\gamma_{1}(1+\gamma
X)}g^{\alpha\mu}g^{\beta\nu}X_{\mu\nu}\right]\Bigg{\\}}\ .$

The non-minimal couplings to $R$ and $R_{\alpha\beta}$ are eliminated by
choosing

$\displaystyle(1+\gamma X)^{1/2}\left(\gamma_{1}F+\frac{\alpha_{1}X}{1+\gamma
X}\right)=1$ $\displaystyle
F\gamma_{2}+\frac{\alpha_{2}-\alpha_{3}-(\alpha_{1}+\alpha_{3})\gamma
X}{1+\gamma X}=0\ .$ (2.16)

From (2.3) we can solve for $\gamma_{1}$ and $\gamma_{2}$ in closed form. The
solutions are not very illuminating, so we do not write them down. For
$\alpha_{3}=0$ they simplify; the case $\alpha_{1}=-\frac{1}{2}\alpha_{2}$,
$\alpha_{3}=0$ is given in [93].

The disformal transformation is invertible and the original and transformed
metric describe the same physics when $\gamma_{1}>0$, $\gamma_{2}\geq 0$,
$\gamma_{1}+\tilde{X}\gamma_{2}>0$,
$\tilde{\gamma}_{1}-X\partial_{X}\tilde{\gamma}_{1}-X^{2}\partial_{X}\tilde{\gamma}_{2}\neq
0$. These conditions set a limit on the values $X$ can take. This is a
limitation of the disformal transformation. Large spatial gradients such as
may occur during preheating may mean that the coefficient of the Ricci tensor
is not positive, so that even in the case $\alpha_{3}=0$, the theory cannot be
mapped to a minimally coupled Einstein frame with a disformal transformation.
For study of slow-roll inflation in the super-Hubble regime, this is not a
problem.

Inserting $\gamma_{1}$ and $\gamma_{2}$ back into the action (2.15), we get
(dropping the matter Lagrangian)

$\displaystyle
S=\int\mathrm{d}^{4}x\sqrt{-g}\left(\frac{1}{2}R+\mathcal{G}_{1}\accentset{\Delta}{R}^{\alpha\beta}X_{\alpha\beta}-\frac{1}{2}\mathcal{G}_{2}KX-\mathcal{G}_{3}V\right)\
,$ (2.17)

where we have defined

$\displaystyle\mathcal{G}_{1}\equiv\frac{\alpha_{3}}{(1+\gamma X)^{1/2}}\ ,\
\mathcal{G}_{2}\equiv\frac{\gamma_{1}}{(1+\gamma X)^{1/2}}\ ,\
\mathcal{G}_{3}\equiv\gamma_{1}^{2}(1+\gamma X)^{1/2}\ .$ (2.18)

If $\alpha_{3}=0$, then $\mathcal{G}_{1}=0$. In this case the Ricci scalar is
the only term that contains the connection, so the connection equation of
motion gives the Levi–Civita connection. (In the case when there are no a
priori constraints on the connection, it is determined only up to a projective
transformation.) Inserting it back into the action we obtain a metric theory
with a minimally coupled scalar field. The physics related to the distortion
has been shifted to the modifications of the scalar field kinetic term and
potential (and the matter Lagrangian). In the Einstein frame all matter
couples to the scalar field and its kinetic term. We have not assumed anything
about the connection, showing that if the co-Ricci tensor does not appear in
the action, the physics is the same whether we keep the connection
unconstrained or put non-metricity or torsion to zero. This is also the case
in Palatini $f(R)$ theory, which can be reduced to the Einstein–Hilbert plus
minimally coupled matter form via field transformations [112, 113, 114, 115].
If the non-metricity is put to zero a priori, (2.6) shows that
$\hat{R}_{\alpha\beta}=-R_{\alpha\beta}$, so the co-Ricci tensor is not
independent, and there is no $\alpha_{3}$ coupling.

In any case, if $\alpha_{3}=0$, the action (2.17) reduces to

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2}g^{\alpha\beta}R_{\alpha\beta}-\frac{1}{2}\frac{\gamma_{1}}{(1+\gamma
X)^{1/2}}KX-\gamma_{1}^{2}(1+\gamma X)^{1/2}V\right]$ (2.19)
$\displaystyle\simeq$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\Bigg{\\{}\frac{1}{2}\mathring{R}-\frac{KX}{2F}\left[1-\left(\alpha_{1}+\alpha_{2}\right)X\right]$
$\displaystyle-\frac{V}{F^{2}}\left[1-\left(2\alpha_{1}+\frac{1}{2}\alpha_{2}\right)X+\left(\alpha_{1}^{2}+2\alpha_{1}\alpha_{2}+\frac{5}{8}\alpha_{2}^{2}\right)X^{2}\right]\Bigg{\\}}\
,$

where in the second equality we have expanded to second order in
$X_{\alpha\beta}$. For $\alpha_{1}=-\frac{1}{2}\alpha_{2}$, $\alpha_{3}=0$ the
result agrees with [93].

The action (2.19) is manifestly in the Horndeski class. We noted in section
2.2 that based on the results of [97], the action is of the DHOST form.
However, as written in the introduction, the only viable DHOST theories (at
least to cubic order in second derivatives) seem to be those that are related
to Horndeski theories by an invertible disformal transformation. There is no
physical difference between Horndeski and DHOST theories as regards physical
degrees of freedom and stability.

###  2.4 Zero torsion case with $\alpha_{3}\neq 0$

When $\alpha_{3}\neq 0$ and the non-metricity is non-zero, we have to solve
the connection equation of motion and insert the solution back into the
action. Let us consider the case with zero torsion. (The case with no
constraints was considered in [97].) Varying the action (2.17) with respect to
the distortion tensor (taking into account that it is symmetric in the last
two indices) gives the equation of motion

$\displaystyle 0$ $\displaystyle=$ $\displaystyle
g_{\beta\gamma}L^{\delta}{}_{\delta\alpha}-L_{(\beta\gamma)\alpha}-L_{(\beta|\alpha|\gamma)}+g_{\alpha(\beta}L_{\gamma)}{}^{\delta}{}_{\delta}$
(2.20)
$\displaystyle+\mathcal{G}_{1}\big{(}L^{\delta}{}_{\delta\alpha}X_{\beta\gamma}-g_{\beta\gamma}L^{\delta\epsilon}{}_{\alpha}X_{\delta\epsilon}-X_{\alpha}Y_{\beta\gamma}+g_{\beta\gamma}\mathring{\nabla}_{\delta}X_{\alpha}{}^{\delta}+g_{\alpha(\beta}\mathring{\nabla}^{\delta}X_{\gamma)\delta}+L_{(\beta}{}^{\delta}{}_{\gamma)}X_{\alpha\delta}$
$\displaystyle-
L_{(\beta|\alpha|}{}^{\delta}X_{\gamma)\delta}-L_{(\beta}{}^{\delta}{}_{|\alpha|}X_{\gamma)\delta}-L_{(\beta}{}^{\delta}{}_{|\delta}X_{\alpha|\gamma)}+L^{\delta}{}_{(\beta|\alpha|}X_{\gamma)\delta}-\frac{3}{2}\mathring{\nabla}_{\alpha}X_{\beta\gamma}+g_{\alpha(\beta}L_{\gamma)}{}^{\delta\epsilon}X_{\delta\epsilon}\big{)}$
$\displaystyle+\mathcal{G}_{1}^{\prime}\big{(}Xg_{\beta\gamma}X_{\alpha}+Xg_{\alpha(\beta}X_{\gamma)}-2X_{\alpha}X_{\beta\gamma}\big{)}$
$\displaystyle+\partial_{X}\mathcal{G}_{1}\big{(}g_{\beta\gamma}X_{\alpha\delta}\mathring{\nabla}^{\delta}X+g_{\alpha(\beta}X_{\gamma)}{}^{\delta}\mathring{\nabla}_{\delta}X-X_{\beta\gamma}\mathring{\nabla}_{\alpha}X-X_{\alpha(\beta}\mathring{\nabla}_{\gamma)}X\big{)}\
,$

where $X_{\alpha}\equiv\partial_{\alpha}\varphi$,
$Y_{\alpha\beta}\equiv\mathring{\nabla}_{\alpha}\mathring{\nabla}_{\beta}\varphi$,
and prime denotes partial derivative with respect to $\varphi$. The general
solution has the form

$\displaystyle L_{\alpha\beta\gamma}$ $\displaystyle=$ $\displaystyle
l_{1}g_{\beta\gamma}X_{\alpha}+l_{2}g_{\alpha(\beta}X_{\gamma)}+l_{3}X_{\alpha}Y_{\beta\gamma}+l_{4}\mathring{\nabla}_{\alpha}X_{\beta\gamma}+l_{5}g_{\beta\gamma}\mathring{\nabla}_{\alpha}X+l_{6}g_{\beta\gamma}\mathring{\nabla}_{\delta}X_{\alpha}{}^{\delta}$
(2.21)
$\displaystyle+l_{7}g_{\alpha(\beta}\mathring{\nabla}^{\delta}X_{\gamma)\delta}+l_{8}g_{\alpha(\beta}\mathring{\nabla}_{\gamma)}X+l_{9}X_{\alpha}X_{\beta\gamma}+l_{10}X_{\beta\gamma}\mathring{\nabla}_{\alpha}X+l_{11}X_{\beta\gamma}\mathring{\nabla}_{\delta}X_{\alpha}{}^{\delta}$
$\displaystyle+l_{12}g_{\beta\gamma}X_{\alpha\delta}\mathring{\nabla}^{\delta}X+l_{13}X_{\alpha(\beta}\mathring{\nabla}_{\gamma)}X+l_{14}g_{\alpha(\beta}X_{\gamma)}{}^{\delta}\mathring{\nabla}_{\delta}X+l_{15}X_{\alpha\delta}X_{\beta\gamma}\mathring{\nabla}^{\delta}X\
,$

Inserting (2.21) into (2.20), we solve for the coefficients
$l_{i}(\varphi,X)$. The result is rather lengthy and is given in appendix A.
Inserting $l_{i}$ into the action (2.17), we get the minimally coupled action

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!S$
$\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\bigg{(}\frac{1}{2}\mathring{R}-\frac{1}{2}\mathcal{G}_{2}KX-\mathcal{G}_{3}V+\mathcal{B}_{1}+\mathcal{B}_{2}Y+\mathcal{B}_{3}X^{\alpha\beta}Y_{\alpha\beta}$
(2.22)
$\displaystyle+\mathcal{A}_{1}Y_{\alpha\beta}Y^{\alpha\beta}+\mathcal{A}_{2}Y^{2}+\mathcal{A}_{3}X^{\alpha\beta}Y_{\alpha\beta}Y+\mathcal{A}_{4}X^{\alpha\beta}Y_{\alpha}{}^{\gamma}Y_{\beta\gamma}+\mathcal{A}_{5}X^{\alpha\beta}X^{\gamma\delta}Y_{\alpha\beta}Y_{\gamma\delta}\bigg{)}\,,$

where the coefficients $\mathcal{B}_{i}(\varphi,X)$ and
$\mathcal{A}_{i}(\varphi,X)$ are again relegated to appendix A. If
$\alpha_{3}=0$, then $\mathcal{B}_{i}=\mathcal{A}_{i}=0$, and (2.22) reduces
to (2.19). The terms on the second line are non-Horndeski, but the functions
$\mathcal{A}_{i}$ satisfy the conditions for the theory to be DHOST [49, 50].
In order to obtain a minimally coupled action, it was important to consider
coupling to $\accentset{\Delta}{R}_{\alpha\beta}$, which vanishes for the
Levi–Civita connection, rather than $\hat{R}_{\alpha\beta}$.

To second order in $X_{\alpha\beta}$, the action (2.22) reads

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\bigg{\\{}\frac{1}{2}\mathring{R}-\frac{KX}{2F}[1-(\alpha_{1}+\alpha_{2}-\alpha_{3})X]$
(2.23)
$\displaystyle-\frac{V}{F^{2}}\Big{[}1-\big{(}2\alpha_{1}+\frac{1}{2}[\alpha_{2}-\alpha_{3}]\big{)}X+\big{(}\alpha_{1}^{2}+\frac{5}{8}\alpha_{2}^{2}+2\alpha_{1}[\alpha_{2}-\alpha_{3}]-\frac{3}{4}\alpha_{2}\alpha_{3}+\frac{1}{8}\alpha_{3}^{2}\big{)}X^{2}\Big{]}$
$\displaystyle+\frac{5}{8}\alpha_{3}^{2}XY_{\alpha\beta}Y^{\alpha\beta}-\frac{13}{24}\alpha_{3}^{2}XY^{2}-\frac{11}{12}\alpha_{3}^{2}X^{\alpha\beta}Y_{\alpha\beta}Y+\frac{5}{6}\alpha_{3}^{2}X^{\alpha\beta}Y_{\alpha}{}^{\gamma}Y_{\beta\gamma}\bigg{\\}}\
.$

In [96, 97] where the connection was unconstrained, the couplings to
$\mathring{R}_{\alpha\beta}$ were instead eliminated by writing them in terms
of the commutator of the Levi–Civita covariant derivative, without
transforming to the Einstein frame. This leads to a different form of the
action; it is well known that a Horndeski or a DHOST theory can take quite
different-looking forms. Transforming the action in [96, 97] (keeping only the
same original terms that we have) to the Einstein frame gives, to second order
in $X_{\alpha\beta}$,

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\bigg{\\{}\frac{1}{2}\mathring{R}-\frac{KX}{2F}\big{[}1-(\alpha_{1}+\alpha_{2}-\alpha_{3})X\big{]}$
(2.24)
$\displaystyle-\frac{V}{F^{2}}\Big{[}1-\big{(}2\alpha_{1}+\frac{1}{2}[\alpha_{2}-\alpha_{3}]\big{)}X+\big{(}\alpha_{1}^{2}+\frac{5}{8}\alpha_{2}^{2}+2\alpha_{1}[\alpha_{2}-\alpha_{3}]-\frac{3}{4}\alpha_{2}\alpha_{3}+\frac{1}{8}\alpha_{3}^{2}\big{)}X^{2}\Big{]}$
$\displaystyle+\frac{1}{2}\alpha_{3}^{2}XY_{\alpha\beta}Y^{\alpha\beta}-\frac{1}{2}\alpha_{3}^{2}XY^{2}-\alpha_{3}^{2}X^{\alpha\beta}Y_{\alpha\beta}Y+\alpha_{3}^{2}X^{\alpha\beta}Y_{\alpha}{}^{\gamma}Y_{\beta\gamma}\bigg{\\}}\
.$

Comparing (2.23) and (2.24) shows that the theories agree for $\alpha_{3}=0$,
as then it makes no difference whether or not the torsion is constrained to be
zero, as shown in section 2.3. For $\alpha_{3}\neq 0$ the theory with an
unconstrained connection and the theory with zero torsion are physically
inequivalent.

In [101] it was shown that an action that depends on $\hat{R}_{\alpha\beta}$
has a ghost around Minkowski space in the zero torsion case, and that in the
unconstrained case there is a ghost around some FLRW backgrounds. This is not
in contradiction with our result and the results of [96, 97] that these cases
are stable. In [101] it was assumed that the Legendre transformation to the
Einstein frame is non-degenerate, which means that all degrees of freedom in
$\hat{R}_{\alpha\beta}$ are included in the Einstein frame action. In our case
with a scalar field, there are no vector or tensor modes. In [101] the FLRW
ghost was in the tensor sector.

###  2.5 Metric case

Let us compare the Palatini case result (2.22) to the metric case. We are
interested in the leading order differences in slow-roll inflation to see how
the cases could be distinguished observationally. We again start with the
action (2.8), now assuming that the connection is Levi–Civita. The action is
of the Horndeski form when the kinetic term couples only to the Einstein
tensor $G_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R$, not to
the Ricci tensor and the Ricci scalar separately, otherwise it has a ghost. So
we set $\alpha_{1}=-\frac{1}{2}\alpha_{2}$, $\alpha_{3}=0$. We again shift to
the Einstein frame with the disformal transformation (2.9). Now the
calculation is more involved, because the Riemann tensor depends on the metric
and its first and second derivative, unlike in the Palatini case. Hence, it is
not invariant under the disformal transformation, which now introduces second
order derivatives of $\varphi$. The transformation rules of the connection,
the Riemann tensor, the Ricci tensor and the Ricci scalar are somewhat
lengthy, and are given in appendix B. Inserting the result of the disformal
transformation into the action (2.8), expanding to second order in
$X_{\alpha\beta}$ and choosing the disformal functions $\gamma_{1}$ and
$\gamma_{2}$ so that the non-minimal couplings vanish, we get the Einstein
frame action

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\Bigg{\\{}\frac{1}{2}\mathring{R}-\frac{KX}{2F}\left(1-\frac{\alpha_{2}X}{2}\right)-\frac{3}{4}\frac{F^{\prime}{}^{2}}{F^{2}}X-\frac{V}{F^{2}}\left(1+\frac{\alpha_{2}X}{2}-\frac{\alpha_{2}^{2}X^{2}}{8}\right)$
(2.25)
$\displaystyle+\frac{3}{2}\frac{(\alpha_{2}F)^{\prime}}{2F}\frac{F^{\prime}}{F}X^{2}-\frac{(\alpha_{2}F)^{\prime}}{2F}XY+\frac{(\alpha_{2}F)^{\prime}}{2F}X^{\alpha\beta}Y_{\alpha\beta}$
$\displaystyle+\frac{1}{2}\alpha_{2}^{2}X^{\alpha\beta}Y_{\alpha\beta}Y-\frac{1}{2}\alpha_{2}^{2}X^{\alpha\beta}Y_{\alpha}{}^{\gamma}Y_{\beta\gamma}\Bigg{\\}}\
.$

To first order in $X_{\alpha\beta}$ and $Y_{\alpha\beta}$, we recover the
result of the original New Higgs Inflation paper [31], apart from the term
involving $F^{\prime}$. (The hybrid case with both $F^{\prime}\neq 0$ and
$\alpha_{2}\neq 0$ has been studied in [46].) Apart from the $F^{\prime}$
term, this leading order result agrees with the Palatini action (2.19) when
$\alpha_{1}=-\frac{1}{2}\alpha_{2}$, $\alpha_{3}=0$, as observed in [39]. This
is easy to understand: the distortion is sourced by $F^{\prime}$ and
$X_{\alpha\beta}$, and only appears in the action via the total derivative and
quadratic terms in the Riemann tensor (2.4) and the coupling to the kinetic
terms. So if $F^{\prime}=0$, the distortion only enters at second order in
$X_{\alpha\beta}$. The second order terms on the first line of (2.25) also
agree with the Palatini result, which is less obvious. It is only the non-
Horndeski terms that are different.

However, in the Palatini case we can obtain the same action to first (but not
second) order in $X_{\alpha\beta}$ and $Y_{\alpha\beta}$ by coupling to just
$R$, i.e. with $\alpha_{2}=\alpha_{3}=0$. In the metric case such a coupling
would lead to a ghost. Also, in the metric case the derivatives of $F$ and
$\alpha_{2}$ enter, unlike in the Palatini case. The terms involving
$Y_{\alpha\beta}$ are also different: in the Palatini case they appear only if
$\alpha_{3}\neq 0$. If the dynamics are dominated by the derivative coupling,
the differences are small in slow-roll, but if the non-derivative coupling is
important, the theories can have quite different predictions, as comparison of
[46] and [93] shows.

##  3 Conclusions

We have considered a theory where a scalar field kinetic term couples linearly
to the Ricci tensor and the co-Ricci tensor, which appear linearly in the
action, while the field itself can have non-linear non-minimal couplings. We
look at both the Palatini formulation and the metric formulation. Extending
previous Palatini work, we consider the case when either the non-metricity or
the torsion is taken to vanish a priori. To establish the stability properties
of the different cases and compare them side-by-side, we use a disformal
transformation, followed by solving for the connection and inserting the
solution back into the action. In this way we reduce the different cases to
metric gravity with the Einstein–Hilbert action minimally coupled to matter.

We find that all the Palatini cases we consider are ghost-free: they are
either in the Horndeski or DHOST class. If there is no coupling to the co-
Ricci tensor, the Palatini result is independent of the assumptions about the
connection. Otherwise, the case with unconstrained connection and the case
with zero torsion are physically different. (If non-metricity is zero, the co-
Ricci tensor vanishes.) We expand the actions up to second order in the scalar
field kinetic term and compare the differences.

At leading order, the metric case and all the Palatini cases all agree with
each other. However, in the Palatini case a much wider range of couplings is
stable, for example it is possible to simply couple the Ricci scalar to the
trace of the kinetic term, simplifying the model. At second order, the
Horndeski terms agree in the Palatini and metric cases, but the beyond
Horndeski terms are different. The detailed form of the terms beyond the
leading order might appear contrived if written in the metric formulation to
begin with, but in the original Palatini formulation they are simple. The
Palatini formulation can be seen as a selection principle to determine which
complicated derivative couplings should appear in a metric formulation action.

Higgs inflation driven by derivative couplings in the metric formulation does
not have a unitarity problem, unlike the metric formulation of the original
Higgs inflation scenario with a non-derivative coupling to the Ricci scalar
alone [33, 35, 36, 39]. The theory is however sensitive to loop corrections
[39]. It would be interesting to see whether these features change in the
derivative-driven or hybrid Palatini case. In the case with a non-derivative
non-minimal coupling to the Einstein tensor alone, the unitarity problem is
ameliorated in the Palatini formulation [69, 79, 82, 80, 83, 84, 85, 86].

It is an interesting question how to characterise the stability properties of
theories in the Palatini formulation without reducing the theory a metric
equivalent or calculating propagators. In [97] projective symmetry was used to
show that a theory is U-degenerate, as the ghosts appear only in the
unphysical projective mode. Projective invariance does not guarantee the
absence of ghosts in general [101], only in particular cases [97, 100, 99]. It
would be interesting to understand better theories whose structure is tuned to
the projective symmetry so that it makes them stable, and in particular
whether projective symmetry (which has only a vectorial gauge mode) can
prevent terms that would lead to tensor ghosts.

## Acknowledgments

We thank Katsuki Aoki and Keigo Shimada for helpful correspondence. HBN
acknowledges the Mathematica xAct package [116] used in the calculations.

##  Appendix A Details of the solution for the connection in the zero torsion
case

We give here details of the connection calculation in section 2.4 in the case
when $\alpha_{3}\neq 0$ and the torsion is zero. The general solution of the
connection equation of motion (2.20) in terms of the coefficients (2.21) is

$\displaystyle
l_{1}=\frac{\mathcal{G}_{1}\mathcal{G}_{1}^{\prime}X^{2}}{1+\mathcal{G}_{1}X}$
$\displaystyle
l_{2}=-\frac{2\mathcal{G}_{1}^{2}\mathcal{G}_{1}^{\prime}X^{3}}{1-\mathcal{G}_{1}^{2}X^{2}}$
$\displaystyle l_{3}=\frac{2\mathcal{G}_{1}}{2-\mathcal{G}_{1}X}$
$\displaystyle
l_{4}=\frac{6\mathcal{G}_{1}(1-\mathcal{G}_{1}X)}{(2-\mathcal{G}_{1}X)^{2}}$
$\displaystyle
l_{5}=-\frac{12\partial_{X}\mathcal{G}_{1}X+9\mathcal{G}_{1}^{3}X^{2}+5\mathcal{G}_{1}^{4}X^{3}-2\mathcal{G}_{1}^{2}X(19-6\partial_{X}\mathcal{G}_{1}X^{2})+6\mathcal{G}_{1}(2-5\partial_{X}\mathcal{G}_{1}X^{2})}{3(2-\mathcal{G}_{1}X)(1+\mathcal{G}_{1}X)(6-5\mathcal{G}_{1}X)}$
$\displaystyle
l_{6}=\frac{2\mathcal{G}_{1}(1-\mathcal{G}_{1}X)}{6-3\mathcal{G}_{1}X}$
$\displaystyle
l_{7}=\frac{4\mathcal{G}_{1}\big{[}1-\mathcal{G}_{1}X(1-\mathcal{G}_{1}X)\big{]}}{3(2-\mathcal{G}_{1}X)^{2}}$
$\displaystyle
l_{8}=-\frac{2}{3(2-\mathcal{G}_{1}X)^{2}(1+\mathcal{G}_{1}X)(6-5\mathcal{G}_{1}X)}\big{[}12\partial_{X}\mathcal{G}_{1}X+6\mathcal{G}_{1}^{4}X^{3}-5\mathcal{G}_{1}^{5}X^{4}+3\mathcal{G}_{1}^{3}\partial_{X}\mathcal{G}_{1}X^{4}$
$\displaystyle+6\mathcal{G}_{1}(2-\partial_{X}\mathcal{G}_{1}X^{2})-\mathcal{G}_{1}^{2}(8X+6\partial_{X}\mathcal{G}_{1}X^{3})\big{]}$
$\displaystyle
l_{9}=\frac{2(\mathcal{G}_{1}^{\prime}+\mathcal{G}_{1}^{2}\mathcal{G}_{1}^{\prime}X^{2})}{1-\mathcal{G}_{1}^{2}X^{2}}$
$\displaystyle
l_{10}=\frac{125\mathcal{G}_{1}^{2}+84\partial_{X}\mathcal{G}_{1}}{264-220\mathcal{G}_{1}X}+\frac{4\mathcal{G}_{1}^{2}}{(2-\mathcal{G}_{1}X)^{2}}-\frac{25\mathcal{G}_{1}^{2}+4\partial_{X}\mathcal{G}_{1}}{12(2-\mathcal{G}_{1}X)}-\frac{28(\mathcal{G}_{1}^{2}-\partial_{X}\mathcal{G}_{1})}{33(1+\mathcal{G}_{1}X)}$
$\displaystyle
l_{11}=\frac{2\mathcal{G}_{1}^{2}(1-2\mathcal{G}_{1}X)}{3(2-\mathcal{G}_{1}X)^{2}}$
$\displaystyle
l_{12}=\partial_{X}\mathcal{G}_{1}+\frac{\mathcal{G}_{1}^{2}}{12-6\mathcal{G}_{1}X}+\frac{5(\mathcal{G}_{1}^{2}-\partial_{X}\mathcal{G}_{1})}{11(1+\mathcal{G}_{1}X)}-\frac{125\mathcal{G}_{1}^{2}+84\partial_{X}\mathcal{G}_{1}}{66(6-5\mathcal{G}_{1}X)}$
$\displaystyle
l_{13}=\frac{2}{(2-\mathcal{G}_{1}X)^{2}(1+\mathcal{G}_{1}X)^{2}(6-5\mathcal{G}_{1}X)}\big{\\{}\mathcal{G}_{1}^{3}X\big{[}15+\mathcal{G}_{1}X(19-20\mathcal{G}_{1}X)\big{]}$
$\displaystyle+\partial_{X}\mathcal{G}_{1}(2-\mathcal{G}_{1}X)(6+10\mathcal{G}_{1}X-5\mathcal{G}_{1}^{2}X^{2}-\mathcal{G}_{1}^{3}X^{3})\big{\\}}$
$\displaystyle
l_{14}=2\partial_{X}\mathcal{G}_{1}+\frac{125\mathcal{G}_{1}^{2}+84\partial_{X}\mathcal{G}_{1}}{792-660\mathcal{G}_{1}X}+\frac{5\mathcal{G}_{1}^{2}+4\partial_{X}\mathcal{G}_{1}}{24-12\mathcal{G}_{1}X}-\frac{\mathcal{G}_{1}^{2}}{(2-\mathcal{G}_{1}X)^{2}}-\frac{\mathcal{G}_{1}^{2}+\partial_{X}\mathcal{G}_{1}}{1-\mathcal{G}_{1}X}$
$\displaystyle+\frac{31(\mathcal{G}_{1}^{2}-\partial_{X}\mathcal{G}_{1})}{33(1+\mathcal{G}_{1}X)}$
$\displaystyle
l_{15}=-\frac{2\mathcal{G}_{1}}{3(2-\mathcal{G}_{1}X)^{2}(1-\mathcal{G}_{1}X)(1+\mathcal{G}_{1}X)^{2}(6-5\mathcal{G}_{1}X)}\big{[}12\partial_{X}\mathcal{G}_{1}-126\mathcal{G}_{1}\partial_{X}\mathcal{G}_{1}X+10\mathcal{G}_{1}^{6}X^{4}$
$\displaystyle-12\mathcal{G}_{1}^{4}X^{2}(3+5\partial_{X}\mathcal{G}_{1}X^{2})-\mathcal{G}_{1}^{5}X^{3}(7-15\partial_{X}\mathcal{G}_{1}X^{2})+\mathcal{G}_{1}^{3}X(73+33\partial_{X}\mathcal{G}_{1}X^{2})$
$\displaystyle-2\mathcal{G}_{1}^{2}(26-57\partial_{X}\mathcal{G}_{1}X^{2})\big{]}\
.$ (A.1)

The coefficients of the final Einstein frame action (2.22) are

$\displaystyle\mathcal{B}_{1}=\frac{3\mathcal{G}_{1}^{2}\mathcal{G}_{1}^{\prime
2}X^{5}}{4-4\mathcal{G}_{1}^{2}X^{2}}$
$\displaystyle\mathcal{B}_{2}=-\mathcal{G}_{1}\mathcal{G}_{1}^{\prime}X^{2}$
$\displaystyle\mathcal{B}_{3}=\frac{\mathcal{G}_{1}\mathcal{G}_{1}^{\prime}X\big{[}1+\mathcal{G}_{1}X^{2}(2\mathcal{G}_{1}+3\partial_{X}\mathcal{G}_{1}X)\big{]}}{1-\mathcal{G}_{1}^{2}X^{2}}$
$\displaystyle\mathcal{A}_{1}=\frac{\mathcal{G}_{1}^{2}X(5-4\mathcal{G}_{1}X)}{2(2-\mathcal{G}_{1}X)^{2}}$
$\displaystyle\mathcal{A}_{2}=-\frac{\mathcal{G}_{1}^{2}X(13-12\mathcal{G}_{1}X+\mathcal{G}_{1}^{2}X^{2})}{6(2-\mathcal{G}_{1}X)^{2}}$
$\displaystyle\mathcal{A}_{3}=-\frac{1}{3}\mathcal{G}_{1}\Big{[}6\partial_{X}\mathcal{G}_{1}X+\mathcal{G}_{1}\frac{11-12\mathcal{G}_{1}X+4\mathcal{G}_{1}^{2}X^{2}}{(2-\mathcal{G}_{1}X)^{2}}\Big{]}$
$\displaystyle\mathcal{A}_{4}=\frac{4}{(2-\mathcal{G}_{1}X)^{2}(1+\mathcal{G}_{1}X)^{2}(6-5\mathcal{G}_{1}X)}\Big{\\{}2(\partial_{X}\mathcal{G}_{1})^{2}X^{2}+6\mathcal{G}_{1}^{5}X^{3}-5\mathcal{G}_{1}^{6}X^{4}$
$\displaystyle+5\mathcal{G}_{1}^{4}\partial_{X}\mathcal{G}_{1}X^{4}-\mathcal{G}_{1}^{3}\partial_{X}\mathcal{G}_{1}X^{3}(16-\partial_{X}\mathcal{G}_{1}X^{2})+\mathcal{G}_{1}\partial_{X}\mathcal{G}_{1}X(14+3\partial_{X}\mathcal{G}_{1}X^{2})$
$\displaystyle+\mathcal{G}_{1}^{2}\big{[}5+\partial_{X}\mathcal{G}_{1}X^{2}(5-4\partial_{X}\mathcal{G}_{1}X^{2})\big{]}\Big{\\}}$
$\displaystyle\mathcal{A}_{5}=-\frac{1}{3(2-\mathcal{G}_{1}X)^{2}(1-\mathcal{G}_{1}X)(1+\mathcal{G}_{1}X)^{2}(6-5\mathcal{G}_{1}X)}\Big{\\{}24(\partial_{X}\mathcal{G}_{1})^{2}X+20\mathcal{G}_{1}^{8}X^{5}$
$\displaystyle+12\mathcal{G}_{1}\partial_{X}\mathcal{G}_{1}(2+\partial_{X}\mathcal{G}_{1}X^{2})+12\mathcal{G}_{1}^{2}\partial_{X}\mathcal{G}_{1}X(1-25\partial_{X}\mathcal{G}_{1}X^{2})-\mathcal{G}_{1}^{7}(64X^{4}-60\partial_{X}\mathcal{G}_{1}X^{6})$
$\displaystyle+\mathcal{G}_{1}^{3}\big{[}2-48\partial_{X}\mathcal{G}_{1}X^{2}(9-5\partial_{X}\mathcal{G}_{1}X^{2})\big{]}+\mathcal{G}_{1}^{6}X^{3}\big{[}23-9\partial_{X}\mathcal{G}_{1}X^{2}(28-5\partial_{X}\mathcal{G}_{1}X^{2})\big{]}$
$\displaystyle-2\mathcal{G}_{1}^{4}X\big{[}62-3\partial_{X}\mathcal{G}_{1}X^{2}(61+25\partial_{X}\mathcal{G}_{1}X^{2})\big{]}$
$\displaystyle+\mathcal{G}_{1}^{5}X^{2}[125+3\partial_{X}\mathcal{G}_{1}X^{2}(62-63\partial_{X}\mathcal{G}_{1}X^{2})]\Big{\\}}\
.$ (A.2)

##  Appendix B Disformal transformation in the metric formulation

Under the disformal transformation (2.11), the Levi–Civita connection
transforms as

$\displaystyle\mathring{\Gamma}^{\gamma}{}_{\alpha\beta}$
$\displaystyle\rightarrow$
$\displaystyle\mathring{\Gamma}^{\gamma}{}_{\alpha\beta}+\frac{-\gamma_{1}^{\prime}g_{\alpha\beta}+2\gamma_{2}Y_{\alpha\beta}}{2(\gamma_{1}+\gamma_{2}X)}X^{\gamma}-\frac{1}{2\gamma_{1}}(\partial_{X}\gamma_{1}g_{\alpha\beta}+\partial_{X}\gamma_{2}X_{\alpha\beta})\mathring{\nabla}^{\gamma}X+\frac{\gamma_{1}^{\prime}}{2\gamma_{1}}\delta_{(\alpha}{}^{\gamma}X_{\beta)}$
(B.1)
$\displaystyle+\frac{\partial_{X}\gamma_{1}}{2\gamma_{1}}\delta_{(\alpha}{}^{\gamma}\mathring{\nabla}_{\beta)}X+\frac{1}{2\gamma_{1}(\gamma_{1}+\gamma_{2}X)}\Big{\\{}\partial_{X}\gamma_{1}\gamma_{2}g_{\alpha\beta}X^{\gamma}{}_{\delta}\mathring{\nabla}^{\delta}X$
$\displaystyle+\big{[}(-2\gamma_{1}^{\prime}\gamma_{2}+\gamma_{1}\gamma_{2}^{\prime})X^{\gamma}+\gamma_{2}\partial_{X}\gamma_{2}X^{\gamma}{}_{\delta}\mathring{\nabla}^{\delta}X\big{]}X_{\alpha\beta}$
$\displaystyle+2(-\partial_{X}\gamma_{1}\gamma_{2}+\gamma_{1}\partial_{X}\gamma_{2})X_{(\alpha}{}^{\gamma}\mathring{\nabla}_{\beta)}X\Big{\\}}\
.$

For the Riemann tensor we get

$\displaystyle\mathring{R}^{\alpha}{}_{\beta\gamma\delta}$
$\displaystyle\rightarrow$
$\displaystyle\mathring{R}^{\alpha}{}_{\beta\gamma\delta}+\frac{\partial_{X}\gamma_{1}g_{\beta[\gamma}\mathring{\nabla}^{\alpha}\mathring{\nabla}_{\delta]}X-\partial_{X}\gamma_{1}\delta_{[\gamma}{}^{\alpha}\mathring{\nabla}_{|\beta|}\mathring{\nabla}_{\delta]}X+\partial_{X}\gamma_{2}X_{\beta[\gamma}\mathring{\nabla}^{\alpha}\mathring{\nabla}_{\delta]}X}{\gamma_{1}}$
(B.2)
$\displaystyle+\frac{\gamma_{1}^{\prime}g_{\beta[\gamma}Y_{\delta]}{}^{\alpha}-\gamma_{1}^{\prime}\delta_{[\gamma}{}^{\alpha}Y_{|\beta|\delta]}+2\gamma_{2}(X^{\alpha}\mathring{\nabla}_{[\gamma}Y_{|\beta|\delta]}-Y_{\beta[\gamma}Y_{\delta]}{}^{\alpha})}{\gamma_{1}+\gamma_{2}X}$
$\displaystyle+\frac{1}{2\gamma_{1}(\gamma_{1}+\gamma_{2}X)^{2}}\Big{\\{}\big{[}-2\gamma_{1}^{2}\gamma^{\prime\prime}_{1}+2\gamma_{1}^{\prime
2}\gamma_{2}X+\gamma_{1}(3\gamma_{1}^{\prime
2}-2\gamma^{\prime\prime}_{1}\gamma_{2}X+\gamma_{1}^{\prime}\gamma_{2}^{\prime}X)\big{]}g_{\beta[\gamma}X_{\delta]}{}^{\alpha}$
$\displaystyle+\big{[}-2\gamma_{1}^{2}\partial_{X}\gamma_{1}^{\prime}+2\partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}X+\gamma_{1}(3\partial_{X}\gamma_{1}\gamma_{1}^{\prime}+\gamma_{1}^{\prime}\gamma_{2}-2\partial_{X}\gamma_{1}^{\prime}\gamma_{2}X+\gamma_{1}^{\prime}\partial_{X}\gamma_{2}X)\big{]}$
$\displaystyle
g_{\beta[\gamma}X^{\alpha}\mathring{\nabla}_{\delta]}X\Big{\\}}+\frac{1}{2\gamma_{1}^{2}}\big{[}(\partial_{X}\gamma_{1})^{2}\mathring{\nabla}_{\epsilon}X\mathring{\nabla}^{\epsilon}Xg_{\beta[\gamma}\delta_{\delta]}{}^{\alpha}-\partial_{X}\gamma_{1}\partial_{X}\gamma_{2}\mathring{\nabla}_{\epsilon}X\mathring{\nabla}^{\epsilon}X\delta_{[\gamma}{}^{\alpha}X_{|\beta|\delta]}$
$\displaystyle-(3(\partial_{X}\gamma_{1})^{2}-2\gamma_{1}\partial_{X}^{2}\gamma_{1})(g_{\beta[\gamma}\mathring{\nabla}_{\delta]}X\mathring{\nabla}^{\alpha}X-\delta_{[\gamma}{}^{\alpha}\mathring{\nabla}_{\delta]}X\mathring{\nabla}_{\beta}X)\big{]}$
$\displaystyle+\frac{1}{2\gamma_{1}(\gamma_{1}+\gamma_{2}X)}\big{[}\gamma_{1}^{\prime
2}Xg_{\beta[\gamma}\delta_{\delta]}{}^{\alpha}+2\partial_{X}\gamma_{1}\gamma_{1}^{\prime}X^{\epsilon}\mathring{\nabla}_{\epsilon}Xg_{\beta[\gamma}\delta_{\delta]}{}^{\alpha}$
$\displaystyle-2\partial_{X}\gamma_{1}\gamma_{2}X^{\epsilon}\mathring{\nabla}_{\epsilon}Xg_{\beta[\gamma}Y_{\delta]}{}^{\alpha}+2\partial_{X}\gamma_{1}\gamma_{2}X^{\epsilon}\mathring{\nabla}_{\epsilon}X\delta_{[\gamma}{}^{\alpha}Y_{|\beta|\delta]}+(4\gamma_{1}^{\prime}\gamma_{2}-2\gamma_{1}\gamma_{2}^{\prime})$
$\displaystyle
X_{\beta[\gamma}Y_{\delta]}{}^{\alpha}-2\gamma_{2}\partial_{X}\gamma_{2}X^{\epsilon}\mathring{\nabla}_{\epsilon}XX_{\beta[\gamma}Y_{\delta]}{}^{\alpha}+(2\partial_{X}\gamma_{1}\gamma_{2}-2\gamma_{1}\partial_{X}\gamma_{2})(X_{[\gamma}{}^{\alpha}\mathring{\nabla}_{|\beta|}\mathring{\nabla}_{\delta]}X$
$\displaystyle-
X_{[\gamma}Y_{|\beta|\delta]}\mathring{\nabla}^{\alpha}X+X_{[\gamma}Y_{\delta]}{}^{\alpha}\mathring{\nabla}_{\beta}X-X_{\beta}Y_{[\gamma}{}^{\alpha}\mathring{\nabla}_{\delta]}X)-2\partial_{X}\gamma_{1}\gamma_{2}g_{\beta[\gamma}X^{\alpha\epsilon}\mathring{\nabla}_{|\epsilon|}\mathring{\nabla}_{\delta]}X$
$\displaystyle-2\gamma_{2}\partial_{X}\gamma_{2}X_{\beta[\gamma}X^{\alpha\epsilon}\mathring{\nabla}_{|\epsilon|}\mathring{\nabla}_{\delta]}X\big{]}+\frac{1}{(\gamma_{1}+\gamma_{2}X)^{2}}\Big{\\{}(\gamma_{1}\gamma_{2}^{\prime}-\gamma_{1}^{\prime}\gamma_{2})X_{[\gamma}{}^{\alpha}Y_{|\beta|\delta]}+\big{[}\gamma_{2}(\partial_{X}\gamma_{1}$
$\displaystyle+\gamma_{2})-\gamma_{1}\partial_{X}\gamma_{2}\big{]}X^{\alpha}Y_{\beta[\gamma}\mathring{\nabla}_{\delta]}X\Big{\\}}-\frac{1}{2\gamma_{1}^{2}(\gamma_{1}+\gamma_{2}X)}\bigg{\\{}(\partial_{X}\gamma_{1})^{2}\gamma_{2}X_{\epsilon\zeta}\mathring{\nabla}^{\epsilon}X\mathring{\nabla}^{\zeta}Xg_{\beta[\gamma}\delta_{\delta]}{}^{\alpha}$
$\displaystyle+\partial_{X}\gamma_{1}(\partial_{X}\gamma_{1}\gamma_{2}-\gamma_{1}\partial_{X}\gamma_{2})\mathring{\nabla}_{\epsilon}X\mathring{\nabla}^{\epsilon}Xg_{\beta[\gamma}X_{\delta]}{}^{\alpha}+\big{[}2\gamma_{1}^{2}\gamma^{\prime\prime}_{1}-\gamma_{1}^{\prime
2}\gamma_{2}X-\gamma_{1}$ $\displaystyle(3\gamma_{1}^{\prime
2}-2\gamma^{\prime\prime}_{1}\gamma_{2}X+\gamma_{1}^{\prime}\gamma_{2}^{\prime}X)\big{]}\delta_{[\gamma}{}^{\alpha}X_{|\beta|\delta]}+(2\partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}+\gamma_{1}\gamma_{1}^{\prime}\partial_{X}\gamma_{2}-\gamma_{1}\partial_{X}\gamma_{1}\gamma_{2}^{\prime})$
$\displaystyle
X^{\epsilon}\mathring{\nabla}_{\epsilon}X\delta_{[\gamma}{}^{\alpha}X_{|\beta|\delta]}-\partial_{X}\gamma_{1}\gamma_{2}\partial_{X}\gamma_{2}X_{\epsilon\zeta}\mathring{\nabla}^{\epsilon}X\mathring{\nabla}^{\zeta}X\delta_{[\gamma}{}^{\alpha}X_{|\beta|\delta]}-\big{[}2\gamma_{1}^{2}\partial_{X}\gamma_{1}^{\prime}$
$\displaystyle-2\partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}X-\gamma_{1}(3\partial_{X}\gamma_{1}\gamma_{1}^{\prime}-2\partial_{X}\gamma_{1}^{\prime}\gamma_{2}X+\gamma_{1}^{\prime}\partial_{X}\gamma_{2}X)\big{]}g_{\beta[\gamma}X_{\delta]}\mathring{\nabla}^{\alpha}X$
$\displaystyle+\big{[}2\gamma_{1}^{2}\partial_{X}\gamma_{1}^{\prime}-2\partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}X-\gamma_{1}(3\partial_{X}\gamma_{1}\gamma_{1}^{\prime}-2\partial_{X}\gamma_{1}^{\prime}\gamma_{2}X+\gamma_{1}^{\prime}\partial_{X}\gamma_{2}X)\big{]}(\delta_{[\gamma}{}^{\alpha}X_{\delta]}\mathring{\nabla}_{\beta}X$
$\displaystyle+\delta_{[\gamma}{}^{\alpha}X_{|\beta|}\mathring{\nabla}_{\delta]}X)-\Big{\\{}(\partial_{X}\gamma_{1})^{2}\gamma_{2}-2\partial_{X}\gamma_{1}\partial_{X}\gamma_{2}(2\gamma_{1}+\gamma_{2}X)+\gamma_{1}\big{[}2\gamma_{1}\partial_{X}^{2}\gamma_{2}-((\partial_{X}\gamma_{2})^{2}$
$\displaystyle-2\gamma_{2}\partial_{X}^{2}\gamma_{2})X\big{]}\Big{\\}}X_{\beta[\gamma}\mathring{\nabla}_{\delta]}X\mathring{\nabla}^{\alpha}X-\partial_{X}\gamma_{1}(\partial_{X}\gamma_{1}\gamma_{2}-\gamma_{1}\partial_{X}\gamma_{2})g_{\beta[\gamma}X_{\delta]}{}^{\epsilon}\mathring{\nabla}^{\alpha}X\mathring{\nabla}_{\epsilon}X$
$\displaystyle+\partial_{X}\gamma_{1}(\partial_{X}\gamma_{1}\gamma_{2}-\gamma_{1}\partial_{X}\gamma_{2})\delta_{[\gamma}{}^{\alpha}X_{\delta]}{}^{\epsilon}\mathring{\nabla}_{\beta}X\mathring{\nabla}_{\epsilon}X+\partial_{X}\gamma_{1}(\partial_{X}\gamma_{1}\gamma_{2}-\gamma_{1}\partial_{X}\gamma_{2})$
$\displaystyle\delta_{[\gamma}{}^{\alpha}X_{|\beta|}{}^{\epsilon}\mathring{\nabla}_{\delta]}X\mathring{\nabla}_{\epsilon}X\bigg{\\}}+\frac{1}{2\gamma_{1}^{2}(\gamma_{1}+\gamma_{2}X)^{2}}\bigg{\\{}\big{[}\gamma_{1}(3\partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}-2\gamma_{1}\partial_{X}\gamma_{1}^{\prime}\gamma_{2}+\gamma_{1}\gamma_{1}^{\prime}\partial_{X}\gamma_{2}$
$\displaystyle-\gamma_{1}\partial_{X}\gamma_{1}\gamma_{2}^{\prime})+\gamma_{2}(2\partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}-2\gamma_{1}\partial_{X}\gamma_{1}^{\prime}\gamma_{2}+\gamma_{1}\gamma_{1}^{\prime}\partial_{X}\gamma_{2})X\big{]}X^{\epsilon}\mathring{\nabla}_{\epsilon}Xg_{\beta[\gamma}X_{\delta]}{}^{\alpha}$
$\displaystyle+\gamma_{2}\big{[}\gamma_{1}^{2}(2\partial_{X}\gamma_{1}^{\prime}+\gamma_{2}^{\prime})-2\partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}X-\gamma_{1}(3\partial_{X}\gamma_{1}\gamma_{1}^{\prime}+2\gamma_{1}^{\prime}\gamma_{2}-2\partial_{X}\gamma_{1}^{\prime}\gamma_{2}X$
$\displaystyle+\gamma_{1}^{\prime}\partial_{X}\gamma_{2}X)\big{]}X_{\beta[\gamma}X^{\alpha}\mathring{\nabla}_{\delta]}X-\Big{\\{}2\gamma_{1}^{3}\partial_{X}^{2}\gamma_{2}+3(\partial_{X}\gamma_{1})^{2}\gamma_{2}^{2}X-\gamma_{1}^{2}(2\partial_{X}^{2}\gamma_{1}\gamma_{2}$
$\displaystyle+4\partial_{X}\gamma_{1}\partial_{X}\gamma_{2}+\gamma_{2}\partial_{X}\gamma_{2}+(\partial_{X}\gamma_{2})^{2}X-2\gamma_{2}\partial_{X}^{2}\gamma_{2}X)+\gamma_{1}\gamma_{2}\big{[}4(\partial_{X}\gamma_{1})^{2}$
$\displaystyle-2\partial_{X}^{2}\gamma_{1}\gamma_{2}X+\partial_{X}\gamma_{1}(\gamma_{2}-2\partial_{X}\gamma_{2}X)\big{]}\Big{\\}}X_{[\gamma}{}^{\alpha}\mathring{\nabla}_{\delta]}X\mathring{\nabla}_{\beta}X+\big{[}\gamma_{1}^{2}(2\partial_{X}^{2}\gamma_{1}\gamma_{2}$
$\displaystyle+\partial_{X}\gamma_{1}\partial_{X}\gamma_{2})-3(\partial_{X}\gamma_{1})^{2}\gamma_{2}^{2}X-\gamma_{1}\gamma_{2}(4(\partial_{X}\gamma_{1})^{2}+\partial_{X}\gamma_{1}\gamma_{2}-2\partial_{X}^{2}\gamma_{1}\gamma_{2}X)\big{]}$
$\displaystyle
g_{\beta[\gamma}X^{\alpha\epsilon}\mathring{\nabla}_{\delta]}X\mathring{\nabla}_{\epsilon}X-\gamma_{2}\big{[}2\gamma_{1}^{2}\partial_{X}^{2}\gamma_{2}-2\partial_{X}\gamma_{1}\gamma_{2}\partial_{X}\gamma_{2}X-\gamma_{1}(3\partial_{X}\gamma_{1}\partial_{X}\gamma_{2}$
$\displaystyle+\gamma_{2}\partial_{X}\gamma_{2}+(\partial_{X}\gamma_{2})^{2}X-2\gamma_{2}\partial_{X}^{2}\gamma_{2}X)\big{]}X_{\beta[\gamma}X^{\alpha\epsilon}\mathring{\nabla}_{\delta]}X\mathring{\nabla}_{\epsilon}X\bigg{\\}}\
.$

The Ricci tensor transforms as

$\displaystyle\mathring{R}_{\alpha\beta}$ $\displaystyle\rightarrow$
$\displaystyle\mathring{R}_{\alpha\beta}+\frac{-2\gamma_{1}\gamma_{1}^{\prime}-3\gamma_{1}^{\prime}\gamma_{2}X+\gamma_{1}\gamma_{2}^{\prime}X}{2(\gamma_{1}+\gamma_{2}X)^{2}}Y_{\alpha\beta}+\frac{-\gamma_{1}^{\prime}g_{\alpha\beta}Y+2\gamma_{2}(Y_{\alpha\beta}Y+X^{\gamma}\mathring{\nabla}_{\gamma}Y_{\alpha\beta})}{2(\gamma_{1}+\gamma_{2}X)}$
(B.3)
$\displaystyle-\frac{(\partial_{X}\gamma_{1}g_{\alpha\beta}+\partial_{X}\gamma_{2}X_{\alpha\beta})\mathring{\nabla}_{\gamma}\mathring{\nabla}^{\gamma}X}{2\gamma_{1}}-\frac{1}{4\gamma_{1}(\gamma_{1}+\gamma_{2}X)^{2}}\bigg{\\{}X\big{[}2\gamma_{1}^{2}\gamma^{\prime\prime}_{1}+\gamma_{1}^{\prime
2}\gamma_{2}X$
$\displaystyle+\gamma_{1}(2\gamma^{\prime\prime}_{1}\gamma_{2}-\gamma_{1}^{\prime}\gamma_{2}^{\prime})X\big{]}g_{\alpha\beta}+\Big{\\{}4\gamma_{1}^{2}\partial_{X}\gamma_{1}^{\prime}+2\partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}X+\gamma_{1}\big{[}4\partial_{X}\gamma_{1}^{\prime}\gamma_{2}X$
$\displaystyle-\partial_{X}\gamma_{1}\gamma_{2}^{\prime}X-\gamma_{1}^{\prime}(\gamma_{2}+\partial_{X}\gamma_{2}X)\big{]}\Big{\\}}g_{\alpha\beta}X^{\gamma}\mathring{\nabla}_{\gamma}X-2\big{[}2\gamma_{1}^{2}\partial_{X}\gamma_{2}+\partial_{X}\gamma_{1}\gamma_{2}^{2}X$
$\displaystyle+\gamma_{1}\gamma_{2}(-\gamma_{2}+\partial_{X}\gamma_{2}X)\big{]}X^{\gamma}Y_{\alpha\beta}\mathring{\nabla}_{\gamma}X\bigg{\\}}-\frac{1}{4\gamma_{1}^{2}(\gamma_{1}+\gamma_{2}X)}\Big{\\{}\big{[}2\gamma_{1}^{2}\partial_{X}^{2}\gamma_{1}-(\partial_{X}\gamma_{1})^{2}\gamma_{2}X$
$\displaystyle+\gamma_{1}(2\partial_{X}^{2}\gamma_{1}\gamma_{2}+\partial_{X}\gamma_{1}\partial_{X}\gamma_{2})X\big{]}g_{\alpha\beta}\mathring{\nabla}_{\gamma}X\mathring{\nabla}^{\gamma}X+\big{[}2(\partial_{X}\gamma_{1})^{2}\gamma_{2}+\partial_{X}\gamma_{1}(-2\gamma_{1}\partial_{X}\gamma_{2}$
$\displaystyle+\gamma_{2}\partial_{X}\gamma_{2}X)+\gamma_{1}(2\gamma_{1}\partial_{X}^{2}\gamma_{2}-(\partial_{X}\gamma_{2})^{2}X+2\gamma_{2}\partial_{X}^{2}\gamma_{2}X)\big{]}X_{\alpha\beta}\mathring{\nabla}_{\gamma}X\mathring{\nabla}^{\gamma}X\Big{\\}}$
$\displaystyle+\frac{1}{4\gamma_{1}^{2}(\gamma_{1}+\gamma_{2}X)^{2}}\bigg{\\{}\big{[}-4\gamma_{1}^{3}\gamma^{\prime\prime}_{1}+3\gamma_{1}^{\prime
2}\gamma_{2}^{2}X^{2}+2\gamma_{1}^{2}(3\gamma_{1}^{\prime
2}-5\gamma^{\prime\prime}_{1}\gamma_{2}X+\gamma_{1}^{\prime}\gamma_{2}^{\prime}X)$
$\displaystyle+\gamma_{1}\gamma_{2}X(10\gamma_{1}^{\prime
2}-6\gamma^{\prime\prime}_{1}\gamma_{2}X+3\gamma_{1}^{\prime}\gamma_{2}^{\prime}X)\big{]}X_{\alpha\beta}+2\big{[}-\gamma_{1}^{3}(4\partial_{X}\gamma_{1}^{\prime}+\gamma_{2}^{\prime})$
$\displaystyle+4\partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}^{2}X^{2}+2\gamma_{1}\gamma_{2}X(5\partial_{X}\gamma_{1}\gamma_{1}^{\prime}-2\partial_{X}\gamma_{1}^{\prime}\gamma_{2}X+\gamma_{1}^{\prime}\partial_{X}\gamma_{2}X)+\gamma_{1}^{2}(6\partial_{X}\gamma_{1}\gamma_{1}^{\prime}$
$\displaystyle+\gamma_{1}^{\prime}\gamma_{2}-8\partial_{X}\gamma_{1}^{\prime}\gamma_{2}X+2\gamma_{1}^{\prime}\partial_{X}\gamma_{2}X)\big{]}X_{(\alpha}\mathring{\nabla}_{\beta)}X+\Big{\\{}3(\partial_{X}\gamma_{1})^{2}\gamma_{2}^{2}X^{2}+2\gamma_{1}\gamma_{2}X(4(\partial_{X}\gamma_{1})^{2}$
$\displaystyle-\partial_{X}^{2}\gamma_{1}\gamma_{2}X+\partial_{X}\gamma_{1}\partial_{X}\gamma_{2}X)-2\gamma_{1}^{3}(2\partial_{X}^{2}\gamma_{1}+\partial_{X}\gamma_{2}+\partial_{X}^{2}\gamma_{2}X)+\gamma_{1}^{2}\big{[}6(\partial_{X}\gamma_{1})^{2}+\gamma_{2}^{2}$
$\displaystyle+(\partial_{X}\gamma_{2})^{2}X^{2}+2\partial_{X}\gamma_{1}(\gamma_{2}+2\partial_{X}\gamma_{2}X)-2\gamma_{2}X(3\partial_{X}^{2}\gamma_{1}+\partial_{X}^{2}\gamma_{2}X)\big{]}\Big{\\}}\mathring{\nabla}_{\alpha}X\mathring{\nabla}_{\beta}X$
$\displaystyle+\big{[}-\gamma_{1}^{2}(4\partial_{X}\gamma_{1}^{\prime}\gamma_{2}+2\gamma_{1}^{\prime}\partial_{X}\gamma_{2}-2\partial_{X}\gamma_{1}\gamma_{2}^{\prime}+\gamma_{2}\gamma_{2}^{\prime})-2\partial_{X}\gamma_{1}\gamma_{1}^{\prime}\gamma_{2}^{2}X+\gamma_{1}\gamma_{2}(2\gamma_{1}^{\prime}\gamma_{2}$
$\displaystyle-4\partial_{X}\gamma_{1}^{\prime}\gamma_{2}X-\gamma_{1}^{\prime}\partial_{X}\gamma_{2}X+3\partial_{X}\gamma_{1}\gamma_{2}^{\prime}X)\big{]}X_{\alpha\beta}X^{\gamma}\mathring{\nabla}_{\gamma}X+2\big{[}2\gamma_{1}^{3}\partial_{X}^{2}\gamma_{2}+(\partial_{X}\gamma_{1})^{2}\gamma_{2}^{2}X$
$\displaystyle+\gamma_{1}\gamma_{2}(2(\partial_{X}\gamma_{1})^{2}+\partial_{X}\gamma_{1}\gamma_{2}-2\partial_{X}^{2}\gamma_{1}\gamma_{2}X)-\gamma_{1}^{2}(2\partial_{X}^{2}\gamma_{1}\gamma_{2}+2\partial_{X}\gamma_{1}\partial_{X}\gamma_{2}+\gamma_{2}\partial_{X}\gamma_{2}$
$\displaystyle+(\partial_{X}\gamma_{2})^{2}X-2\gamma_{2}\partial_{X}^{2}\gamma_{2}X)\big{]}X_{(\alpha|\gamma|}\mathring{\nabla}_{\beta)}X\mathring{\nabla}^{\gamma}X+\Big{\\{}\gamma_{1}\big{[}2\gamma_{1}\partial_{X}^{2}\gamma_{1}\gamma_{2}-\partial_{X}\gamma_{1}\gamma_{2}(2\partial_{X}\gamma_{1}+\gamma_{2})$
$\displaystyle+2\gamma_{1}\partial_{X}\gamma_{1}\partial_{X}\gamma_{2}\big{]}+\gamma_{2}(-(\partial_{X}\gamma_{1})^{2}\gamma_{2}+2\gamma_{1}\partial_{X}^{2}\gamma_{1}\gamma_{2}+\gamma_{1}\partial_{X}\gamma_{1}\partial_{X}\gamma_{2})X\Big{\\}}g_{\alpha\beta}X_{\gamma\delta}\mathring{\nabla}^{\gamma}X\mathring{\nabla}^{\delta}X$
$\displaystyle+\gamma_{2}\Big{\\{}2\gamma_{1}^{2}\partial_{X}^{2}\gamma_{2}+\partial_{X}\gamma_{1}\gamma_{2}\partial_{X}\gamma_{2}X-\gamma_{1}\big{[}(\partial_{X}\gamma_{2})^{2}X+\gamma_{2}(\partial_{X}\gamma_{2}-2\partial_{X}^{2}\gamma_{2}X)\big{]}\Big{\\}}$
$\displaystyle
X_{\alpha\beta}X_{\gamma\delta}\mathring{\nabla}^{\gamma}X\mathring{\nabla}^{\delta}X\bigg{\\}}-\frac{1}{2\gamma_{1}(\gamma_{1}+\gamma_{2}X)}\Big{\\{}(2\gamma_{1}^{\prime}\gamma_{2}-\gamma_{1}\gamma_{2}^{\prime})X_{\alpha\beta}Y+2(\partial_{X}\gamma_{1}\gamma_{2}-\gamma_{1}\partial_{X}\gamma_{2})$
$\displaystyle
X_{(\alpha}Y\mathring{\nabla}_{\beta)}X+\big{[}\partial_{X}\gamma_{1}\gamma_{2}X+\gamma_{1}(2\partial_{X}\gamma_{1}+\gamma_{2}+\partial_{X}\gamma_{2}X)\big{]}\mathring{\nabla}_{\alpha}\mathring{\nabla}_{\beta}X$
$\displaystyle-\partial_{X}\gamma_{1}\gamma_{2}g_{\alpha\beta}X^{\gamma}Y\mathring{\nabla}_{\gamma}X-\gamma_{2}\partial_{X}\gamma_{2}X_{\alpha\beta}X^{\gamma}Y\mathring{\nabla}_{\gamma}X+2(-\partial_{X}\gamma_{1}\gamma_{2}+\gamma_{1}\partial_{X}\gamma_{2})X_{(\alpha}Y_{\beta)\gamma}\mathring{\nabla}^{\gamma}X$
$\displaystyle+2(\partial_{X}\gamma_{1}\gamma_{2}-\gamma_{1}\partial_{X}\gamma_{2})X_{(\alpha|\gamma|}\mathring{\nabla}^{\gamma}\mathring{\nabla}_{\beta)}X-\partial_{X}\gamma_{1}\gamma_{2}g_{\alpha\beta}X_{\gamma\delta}\mathring{\nabla}^{\delta}\mathring{\nabla}^{\gamma}X$
$\displaystyle-\gamma_{2}\partial_{X}\gamma_{2}X_{\alpha\beta}X_{\gamma\delta}\mathring{\nabla}^{\delta}\mathring{\nabla}^{\gamma}X\Big{\\}}\
.$

Finally, the Ricci scalar transforms as

$\displaystyle\mathring{R}$ $\displaystyle\rightarrow$
$\displaystyle\frac{\mathring{R}}{\gamma_{1}}+\frac{1}{2\gamma_{1}(\gamma_{1}+\gamma_{2}X)^{2}}\Big{\\{}3X\big{[}\gamma_{1}^{\prime
2}+\gamma_{1}^{\prime}\gamma_{2}^{\prime}X-2\partial^{2}_{\varphi}\gamma_{1}(\gamma_{1}+\gamma_{2}X)\big{]}$
(B.4)
$\displaystyle-2(3\gamma_{1}\gamma_{1}^{\prime}+4\gamma_{1}^{\prime}\gamma_{2}X-\gamma_{1}\gamma_{2}^{\prime}X)Y+\big{[}6\partial_{X}\gamma_{1}\gamma_{1}^{\prime}+4\gamma_{1}^{\prime}\gamma_{2}-\gamma_{1}(12\partial_{X}\gamma_{1}^{\prime}+\gamma_{2}^{\prime})$
$\displaystyle-12\partial_{X}\gamma_{1}^{\prime}\gamma_{2}X+3\gamma_{1}^{\prime}\partial_{X}\gamma_{2}X+3\partial_{X}\gamma_{1}\gamma_{2}^{\prime}X\big{]}X^{\alpha}\mathring{\nabla}_{\alpha}X\Big{\\}}$
$\displaystyle+\frac{1}{2\gamma_{1}^{2}(\gamma_{1}+\gamma_{2}X)^{2}}\Big{\\{}\big{[}2\gamma_{1}^{2}\partial_{X}\gamma_{2}+\partial_{X}\gamma_{1}\gamma_{2}^{2}X-\gamma_{1}\gamma_{2}(\gamma_{2}-\partial_{X}\gamma_{2}X)\big{]}X^{\alpha}Y\mathring{\nabla}_{\alpha}X$
$\displaystyle+\big{[}2\gamma_{1}^{2}\partial_{X}\gamma_{2}+3\partial_{X}\gamma_{1}\gamma_{2}^{2}X+\gamma_{1}\gamma_{2}(2\partial_{X}\gamma_{1}-\gamma_{2}+\partial_{X}\gamma_{2}X)\big{]}\mathring{\nabla}^{\alpha}X\mathring{\nabla}_{\beta}X_{\alpha}{}^{\beta}\Big{\\}}$
$\displaystyle-\frac{\gamma_{2}(\mathring{R}^{\alpha\beta}X_{\alpha\beta}-\mathring{\nabla}_{\beta}\mathring{\nabla}_{\alpha}X^{\alpha\beta})}{\gamma_{1}(\gamma_{1}+\gamma_{2}X)}$
$\displaystyle-\frac{1}{4\gamma_{1}^{3}(\gamma_{1}+\gamma_{2}X)^{2}}\mathring{\nabla}^{\alpha}X\Big{\\{}\Big{[}-6(\partial_{X}\gamma_{1})^{2}\gamma_{2}^{2}X^{2}+\gamma_{1}\gamma_{2}X(-10(\partial_{X}\gamma_{1})^{2}+3\partial_{X}\gamma_{1}\gamma_{2}$
$\displaystyle+8\partial^{2}_{X}\gamma_{1}\gamma_{2}X+2\partial_{X}\gamma_{1}\partial_{X}\gamma_{2}X)+2\gamma_{1}^{3}(6\partial^{2}_{X}\gamma_{1}+3\partial_{X}\gamma_{2}+2\partial^{2}_{X}\gamma_{2}X)+\gamma_{1}^{2}\big{[}-6(\partial_{X}\gamma_{1})^{2}$
$\displaystyle-3\gamma_{2}^{2}-2(\partial_{X}\gamma_{2})^{2}X^{2}-2\partial_{X}\gamma_{1}(\gamma_{2}+\partial_{X}\gamma_{2}X)+\gamma_{2}X(20\partial^{2}_{X}\gamma_{1}+\partial_{X}\gamma_{2}+4\partial^{2}_{X}\gamma_{2}X)\big{]}\Big{]}\mathring{\nabla}_{\alpha}X$
$\displaystyle+2\big{[}-2\gamma_{1}^{3}\partial^{2}_{X}\gamma_{2}+3(\partial_{X}\gamma_{1})^{2}\gamma_{2}^{2}X+\gamma_{1}\gamma_{2}(5(\partial_{X}\gamma_{1})^{2}+2\partial_{X}\gamma_{1}\gamma_{2}-4\partial^{2}_{X}\gamma_{1}\gamma_{2}X$
$\displaystyle-\partial_{X}\gamma_{1}\partial_{X}\gamma_{2}X)+\gamma_{1}^{2}(-4\partial^{2}_{X}\gamma_{1}\gamma_{2}-2\partial_{X}\gamma_{1}\partial_{X}\gamma_{2}+\gamma_{2}\partial_{X}\gamma_{2}+(\partial_{X}\gamma_{2})^{2}X-2\gamma_{2}\partial^{2}_{X}\gamma_{2}X)\big{]}$
$\displaystyle
X_{\alpha\beta}\mathring{\nabla}^{\beta}X\Big{\\}}-\frac{1}{\gamma_{1}^{2}(\gamma_{1}+\gamma_{2}X)}\Big{\\{}\big{[}2\partial_{X}\gamma_{1}\gamma_{2}X+\gamma_{1}(3\partial_{X}\gamma_{1}+\gamma_{2}+\partial_{X}\gamma_{2}X)\big{]}\mathring{\nabla}_{\alpha}\mathring{\nabla}^{\alpha}X$
$\displaystyle-(2\partial_{X}\gamma_{1}\gamma_{2}+\gamma_{1}\partial_{X}\gamma_{2})X_{\alpha\beta}\mathring{\nabla}^{\beta}\mathring{\nabla}^{\alpha}X\Big{\\}}\
.$

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