HIP-2018-27/TH

# Higgs inflation in the Palatini formulation with kinetic terms for the
metric

Syksy Räsänen<EMAIL_ADDRESS>University of Helsinki, Department of
Physics and Helsinki Institute of Physics,
P.O. Box 64, FIN-00014 University of Helsinki, Finland
Birzeit University, Department of Physics
P.O. Box 14, Birzeit, West Bank, Palestine

###### Abstract

We consider scalar field inflation in the Palatini formulation of general
relativity. The covariant derivative of the metric is then non-zero. From the
effective theory point of view it should couple to other fields. We write down
the most general couplings between it and a scalar field that are quadratic in
derivatives. We consider both the case when the torsion is determined by the
field equations and the case when it is assumed to be zero a priori. We find
the metric derivative terms can significantly modify inflationary predictions.
We specialise to Higgs inflation and terms of only up to dimension 4.
Transforming to the Einstein frame, we show that by tuning the coefficients of
the new terms, we can generate various effective inflationary potentials,
including quadratic, hilltop-type, $\alpha$-attractor and inflection point.
Some of these can give inflation in agreement with observations, including
with a large tensor-to-scalar ratio, even if the non-minimal coupling is zero.

##  1 Introduction

Inflation is the most successful scenario for the primordial universe. It
alleviates the homogeneity and isotropy problem [1, 2, 3, 4, 5], explains
spatial flatness and has (in its simplest variants) predicted that primordial
perturbations are mostly adiabatic, close to scale-invariant, highly Gaussian,
predominantly scalar and statistically homogeneous and isotropic to a high
degree [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. This is in
excellent agreement with observations [20]. Higgs inflation [21] (for reviews,
see [22, 23, 24]) is an attractive model where inflation is driven by the only
fundamental scalar field in the Standard Model (SM) of particle physics. This
is not possible with the SM action alone, as the Higgs potential is not
sufficiently flat [25, 26, 27].111False vacuum inflation is a possibility, but
requires physics beyond the SM for graceful exit [27, 28, 29, 30, 31, 32, 33].
However, including a non-minimal coupling between the Higgs field and gravity
decelerates the field sufficiently. (In the Einstein frame the potential
becomes exponentially flat, in the Jordan frame there is extra friction.) The
predictions of the classical action agree well with observations [20]. Loop
corrections complicate the picture [34, 35, 36, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62,
63, 64], and can change the qualitative behaviour to produce near-inflection
point inflation [44, 51, 50, 52, 53, 56, 58, 57, 59, 65, 66, 67, 68], hilltop
inflation [56, 59, 60], hillclimbing inflation [61, 62] and generate higher
order curvature terms in the action that change the evolution [69, 70, 66, 71,
72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83]. The issue of unitarity also
remains unsettled [84, 38, 85, 86, 42, 87, 48, 88, 89, 90, 91, 92, 93, 77].

Another complication is the dependence of the predictions on the choice of the
gravitational degrees of freedom. (On different formulations of general
relativity and related theories, see [94, 95, 96, 97, 98, 99, 100, 101, 102,
103, 104, 105, 106, 107, 108, 109, 110, 111].) In the metric formulation of
general relativity, the only independent gravitational degree of freedom is
the metric, and all geometric properties of the manifold are derived from it.
In particular, the connection is the Levi–Civita connection that is uniquely
determined by the metric. However, the metric and the connection describe
different aspects of the manifold. The metric defines distances in spacetime
and dot products between vectors in the tangent space, whereas the connection
sets straight lines and gives the derivatives of tensor fields. In the
Palatini formulation, the metric and the connection are taken to be
independent degrees of freedom.222The Palatini formulation is somewhat of a
misnomer, as it was developed by Einstein [112, 113]. It is also called the
first order formulation, as only first derivatives appear in the gravitational
action, unlike in the metric formulation (also referred to as the Hilbert
formulation). Metric-affine is perhaps a more descriptive term, but it is
often used to refer to the case when the general connection is used in both
the gravitational and the matter action, with the name Palatini restricted to
the case when the general connection is reserved for the gravitational sector,
and the Levi–Civita connection is used in the matter sector [114, 115, 116].
In the metric formulation, the only tensors that can be constructed from the
gravitational degrees of freedom are the metric, the Levi–Civita tensor and
the Riemann tensor. In the Palatini formulation, applying the covariant
derivative to the metric gives a new tensor, the non-metricity tensor
$Q_{\gamma\alpha\beta}=\nabla_{\gamma}g_{\alpha\beta}$. If the connection is
not symmetric, its antisymmetric part, the torsion, is a third gravitational
tensor. A number of new gravitational terms can thus be added to the action.

In the Palatini case, the equations of motion do not determine the metric and
the connection uniquely for all actions [94, 117, 96, 97, 98, 118, 119]. This
happens for the Einstein–Hilbert action if the connection is not taken to be
symmetric. Rather than a drawback of the Palatini formulation, this can be
seen as a constraint on the action, showing that extra terms are needed to
break the projective invariance that is the reason for the system being
underdetermined. An alternative viewpoint is to use Lagrange multipliers to
impose the desired properties of the connection [120, 96, 98, 118, 119, 121,
106, 109, 107], or simply assume them a priori. However, there are actions,
for example the $R^{2}$ action, for which the equations of motion do not
determine the degrees of freedom even if the connection is symmetric.

For the Einstein–Hilbert action plus a matter action that is coupled only to
the metric, not to the connection, the equation of motion for a symmetric
connection gives the Levi–Civita connection, so the metric and the Palatini
formulation are physically equivalent. When the gravitational action is more
complicated [111, 117, 122, 123, 124, 125, 114, 115, 116, 126, 127, 128, 119,
118, 106, 110, 129, 130, 131, 107, 109, 108, 80, 81, 132, 133, 134, 135, 136]
or the matter action couples to the connection [137, 138, 139, 140, 86, 141,
59, 60, 63, 142, 143, 144], this is no longer true. In particular, this is the
case when a scalar field is coupled directly to the Ricci scalar [137, 138,
139, 140, 86, 141, 59, 60, 63, 143, 142, 145, 146]. In Higgs inflation, this
means that the same term that enables inflation breaks the equivalence between
the metric and the Palatini formulation [140, 86, 59, 60, 63, 68].

If we combine general relativity and the SM of particle physics, in the metric
case the only new dimension 4 term that could not have been written down with
either theory alone is the non-minimal coupling $\xi|H|^{2}R$, where $H$ is
the Higgs doublet. In the Palatini case, the non-metricity tensor
$Q_{\alpha\beta\gamma}$ can couple to SM fields, in particular to the gradient
of the Higgs field. Both the non-minimal coupling and the kinetic mixing
between the metric and the Higgs then act as sources for
$Q_{\alpha\beta\gamma}$.

Assuming that the connection enters only via the Riemann tensor and the
covariant derivative, we write down the most general classical action built
from the metric, the connection and a scalar field that involves only up to
two derivatives and no boundary terms, excepting the Holst term and Riemann
tensor squared terms. We will consider separately the cases when the
connection is assumed to be symmetric a priori, and when the antisymmetric
part is left free to be determined by the field equations. In section 2 we
present the action, derive the equations of motion and transform to the
Einstein frame. In section 3 we specialise to SM Higgs inflation and terms of
only up to dimension 4 and study the effect of the new terms on the effective
Higgs potential. In section 4 we discuss our results and open issues, and in
section 5 we summarise our findings.

##  2 Inflation with kinetic terms for the metric

###  2.1 Action and degrees of freedom

#### The degrees of freedom.

We will consider three independent degrees of freedom: scalar field $h$,
symmetric metric $g_{\alpha\beta}$ and connection
$\Gamma^{\gamma}_{\alpha\beta}$. We will comment on fermions and gauge fields
in section 4. We decompose the connection as

$\displaystyle\Gamma^{\gamma}_{\alpha\beta}=\mathring{\Gamma}^{\gamma}_{\alpha\beta}+L^{\gamma}_{\
\alpha\beta}\ ,$ (1)

where $\mathring{\Gamma}_{\alpha\beta}^{\gamma}$ is the Levi–Civita connection
defined by the metric $g_{\alpha\beta}$. As the difference of two connections,
$L^{\gamma}_{\ \alpha\beta}$ is a tensor, known as the deformation tensor. We
can decompose $L^{\gamma}_{\ \alpha\beta}$ into parts that depend on torsion
tensor and on the non-metricity tensor defined as, respectively,

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

Note that $Q_{\gamma\alpha\beta}=Q_{\gamma(\alpha\beta)}$ and
$\nabla_{\gamma}g^{\alpha\beta}=-Q_{\gamma}^{\ \alpha\beta}$. We also define
$Q^{\gamma}\equiv g_{\alpha\beta}Q^{\gamma\alpha\beta}$,
$\hat{Q}^{\beta}\equiv g_{\alpha\gamma}Q^{\alpha\beta\gamma}$ and
$T^{\beta}\equiv g_{\alpha\gamma}T^{\alpha\beta\gamma}$. Denoting the
covariant derivative defined by the Levi–Civita connection by
$\mathring{\nabla}$, we have for an arbitrary vector $A^{\alpha}$

$\displaystyle\nabla_{\beta}A^{\alpha}=\mathring{\nabla}_{\beta}A^{\alpha}+L^{\alpha}_{\
\beta\gamma}A^{\gamma}\ .$ (3)

As $\mathring{\nabla}_{\gamma}g_{\alpha\beta}=0$ by definition, we get

$\displaystyle Q_{\gamma\alpha\beta}$ $\displaystyle=$
$\displaystyle\nabla_{\gamma}g_{\alpha\beta}=-L_{\alpha\gamma\beta}-L_{\beta\gamma\alpha}\
.$ (4)

We can write $L_{\alpha\beta\gamma}$ as

$\displaystyle
L_{\alpha\beta\gamma}=\frac{1}{2}\left(Q_{\alpha\beta\gamma}-Q_{\gamma\alpha\beta}-Q_{\beta\alpha\gamma}\right)+K_{\alpha\beta\gamma}\
,$ (5)

where $K^{\gamma\ \ \beta}_{\ \ \alpha}=K^{[\gamma\ \ \beta]}_{\ \ \alpha}$ is
the contortion tensor, given in terms of the torsion as

$\displaystyle
K_{\alpha\beta\gamma}=\frac{1}{2}(T_{\alpha\beta\gamma}+T_{\beta\alpha\gamma}+T_{\gamma\alpha\beta})\
.$ (6)

We will discuss separately the case when no symmetry assumptions are made
about the connection and the case when the torsion is taken to be zero a
priori (i.e. $L^{\gamma}_{\ \alpha\beta}$ is symmetric).

#### The action.

We want write a bulk action from which equations of motion can derived without
having to add boundary terms to cancel terms arising in the variation. (In the
metric formulation the York–Gibbons–Hawking boundary term is needed to cancel
a boundary term involving the variation of the first derivative of the metric
[147, 148].) We will include $L_{\alpha\beta\gamma}$ only as part of the
connection, so it appears only in covariant derivatives and in the Riemann
tensor. We cannot get kinetic terms for $L_{\alpha\beta\gamma}$ from covariant
derivatives of the metric, because they would involve the second derivative of
$g_{\alpha\beta}$, leading to terms proportional to the variation of the first
derivative on the boundary, which does not vanish. Without kinetic terms, the
connection reduces to an auxiliary variable, so the Palatini formulation does
not introduce new degrees of freedom compared to the metric case, it only
changes the relation between existing degrees of freedom.333In the case when
the equation of motion of the connection has no derivatives, the Palatini
formulation can be viewed as the low-energy limit of a theory with a
connection field (more precisely, $L_{\alpha\beta\gamma}$) that has high mass
[99, 102, 103]. For related considerations of confinement of the spin
connection, see [149, 150]. We will not consider terms non-linear in the
Riemann tensor, nor the parity-violating Holst term. We will be mainly
interested in terms quadratic in the derivatives and only up to dimension 4,
in which case such Riemann tensor terms do not couple to the scalar field.
Also, unless the coefficients of the second order Riemann terms are large,
they are not important for inflation in the observed range of e-folds, because
the curvature is much smaller than the Planck scale,
$R/M_{{}_{\mathrm{Pl}}}^{2}=2.7\times 10^{-9}r/0.07$ [20], where $r<0.07$ is
the tensor-to-scalar ratio. The effect of Ricci scalar squared terms in non-
minimally coupled scalar field inflation in the Palatini formulation has been
studied in [80, 81].

The most general action built from scalar field $h$, metric $g_{\alpha\beta}$
and connection $\Gamma^{\gamma}_{\alpha\beta}$ containing only up to quadratic
derivative terms is then

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2}F(h)g^{\alpha\beta}R_{\alpha\beta}(\Gamma,\partial\Gamma)-\frac{1}{2}K(h)g^{\alpha\beta}\nabla_{\alpha}h\nabla_{\beta}h-V(h)\right.$
(8)
$\displaystyle+A_{1}(h)\nabla_{\alpha}h\nabla_{\beta}g^{\beta\alpha}+A_{2}(h)g^{\alpha\beta}g_{\gamma\delta}\nabla_{\alpha}h\nabla_{\beta}g^{\gamma\delta}$
$\displaystyle+B_{1}(h)g^{\alpha\beta}g_{\gamma\delta}g_{\epsilon\eta}\nabla_{\alpha}g^{\gamma\epsilon}\nabla_{\beta}g^{\delta\eta}+B_{2}(h)g_{\gamma\delta}\nabla_{\alpha}g^{\beta\gamma}\nabla_{\beta}g^{\alpha\delta}+B_{3}(h)g_{\alpha\beta}\nabla_{\gamma}g^{\gamma\alpha}\nabla_{\delta}g^{\delta\beta}$
$\displaystyle+B_{4}(h)g^{\alpha\beta}g_{\gamma\delta}g_{\epsilon\eta}\nabla_{\alpha}g^{\gamma\delta}\nabla_{\beta}g^{\epsilon\eta}+B_{5}(h)g_{\gamma\delta}\nabla_{\alpha}g^{\alpha\beta}\nabla_{\beta}g^{\gamma\delta}$
$\displaystyle\left.+C(h)\epsilon^{\alpha\beta}_{\ \ \
\gamma\delta}g_{\epsilon\eta}\nabla_{\alpha}g^{\gamma\epsilon}\nabla_{\beta}g^{\delta\eta}\right]$
$\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2}F(h)g^{\alpha\beta}R_{\alpha\beta}(\Gamma,\partial\Gamma)-\frac{1}{2}K(h)g^{\alpha\beta}\nabla_{\alpha}h\nabla_{\beta}h-V(h)\right.$
$\displaystyle-
A_{1}(h)\nabla_{\alpha}h\hat{Q}^{\alpha}-A_{2}(h)\nabla_{\alpha}hQ^{\alpha}$
$\displaystyle+B_{1}(h)Q_{\gamma\alpha\beta}Q^{\gamma\alpha\beta}+B_{2}(h)Q_{\gamma\alpha\beta}Q^{\beta\gamma\alpha}+B_{3}(h)\hat{Q}_{\alpha}\hat{Q}^{\alpha}+B_{4}(h)Q_{\alpha}Q^{\alpha}+B_{5}(h)Q_{\alpha}\hat{Q}^{\alpha}$
$\displaystyle\left.+C(h)\epsilon^{\alpha\beta\gamma\delta}g^{\epsilon\eta}Q_{\alpha\gamma\epsilon}Q_{\beta\delta\eta}\right]\
,$

where $g$ is the determinant of $g_{\alpha\beta}$,
$R_{\alpha\beta}=R^{\gamma}_{\ \,\alpha\gamma\beta}$ is the Ricci tensor444In
the Palatini case $R_{\alpha\beta}$ is not in general symmetric, but the
antisymmetric part does not contribute to the action or the equations of
motion. The quantity $R_{\alpha\beta}$ is not the only independent first
contraction of the Riemann tensor [128]. However, there is only one total
contraction, i.e. the Ricci scalar is unique. and
$\epsilon_{\alpha\beta\gamma\delta}$ is the Levi–Civita tensor. If we did not
exclude boundary terms, there would be more possible terms, including those
involving the Riemann tensor built from the Levi–Civita connection alone and
terms that mix it and the full Riemann tensor [125, 130]. In the action, the
covariant derivatives acting on $h$ are of course partial derivatives, and the
connection that they will be paired with in the equations of motion is
determined by the vanishing of boundary terms when varying the action. The
kinetic mixing terms on second line of (8) that couple derivatives of the
scalar field and of the metric have been discussed in [139, 151, 142, 144],
but do not seem to have been studied in the context of inflation before.555The
term kinetic mixing may be slightly inappropriate because in the equations of
motion the partial derivatives of the metric cancel with the Levi–Civita part
of the connection, and only terms algebraic in $Q_{\alpha\beta\gamma}$ remain.
They will source $Q_{\alpha\beta\gamma}$, which will in turn change the
evolution of the scalar field. The quadratic terms on the third and fourth
lines have been considered in different contexts in [139, 99, 152, 104, 102,
110, 106, 109, 107, 108, 144]. In (8) we have written the action in terms of
$Q_{\alpha\beta\gamma}$ to more clearly display the structure where we
effectively have two extra vector fields of mass dimension one (in addition to
the traceless part of $Q_{\alpha\beta\gamma}$, which we have not separated
out). However, note that we do not vary the action with respect to them, but
with respect to $g_{\alpha\beta}$ and $\Gamma^{\gamma}_{\alpha\beta}$. The
$Q_{\alpha\beta\gamma}$ terms contribute to both variations. If we were to
make the split (1) already in the action before applying the variational
principle, we would get boundary terms involving the variation of the first
derivative of the metric, which would not vanish, just as in the metric case.
With only scalar field matter, the Levi–Civita term on the last line does not
contribute to the equations of motion, so we drop it henceforth.

With the set of independent variables
$\\{h,g_{\alpha\beta},\Gamma^{\gamma}_{\alpha\beta}\\}$, the variation of the
action reads

$\displaystyle\delta S$ $\displaystyle=$ $\displaystyle\delta h\frac{\delta
S}{\delta h}+\delta g_{\alpha\beta}\frac{\delta S}{\delta
g_{\alpha\beta}}+\delta\Gamma^{\gamma}_{\alpha\beta}\frac{\delta
S}{\delta\Gamma^{\gamma}_{\alpha\beta}}$ (9) $\displaystyle=$
$\displaystyle\delta h\frac{\delta S}{\delta h}+\delta
g_{\alpha\beta}\frac{\delta S}{\delta
g_{\alpha\beta}}+\delta\Gamma^{\gamma}_{(\alpha\beta)}\frac{\delta
S}{\delta\Gamma^{\gamma}_{(\alpha\beta)}}+p\delta\Gamma^{\gamma}_{[\alpha\beta]}\frac{\delta
S}{\delta\Gamma^{\gamma}_{[\alpha\beta]}}\ .$

The first variation gives the scalar field equation of motion, the second
gives the generalisation of the Einstein equation, and the third determines
the connection in terms of the scalar field and the metric. On the second line
we have split the variation with respect to the connection into the symmetric
and the antisymmetric part. In the unconstrained case we have $p=1$; if the
connection is taken to be symmetric a priori, we have $p=0$.

When varying the fields, it is important to note that the contraction of the
covariant derivative with a vector does not reduce to a boundary term, because
the connection is not Levi–Civita. Instead, (3) gives

$\displaystyle\nabla_{\alpha}A^{\alpha}$ $\displaystyle=$
$\displaystyle\mathring{\nabla}_{\alpha}A^{\alpha}-\frac{1}{2}Q_{\alpha}A^{\alpha}-T_{\alpha}A^{\alpha}\
,$ (10)

where the Levi–Civita covariant derivative
$\mathring{\nabla}_{\alpha}A^{\alpha}$ gives a boundary term as usual.

Often in the Palatini formulation the connection is taken to be symmetric a
priori, i.e. the torsion is set to zero by hand. However, if the connection is
taken to be an independent degree of freedom, from the effective theory point
of view is no reason to assume that its antisymmetric part is zero. For the
Einstein–Hilbert action, the equations of motion only show that a linear
combination of $Q_{\alpha\beta\gamma}$ and $T_{\alpha\beta\gamma}$ vanishes, a
part is left undetermined, because of the the invariance of the action under
the projective transformation
$\Gamma^{\gamma}_{\alpha\beta}\rightarrow\Gamma^{\gamma}_{\alpha\beta}+\delta^{\gamma}_{\
\,\alpha}A_{\beta}$, where $A_{\beta}$ is an arbitrary vector [96]. If we
assume that either $Q_{\alpha\beta\gamma}$ or $T_{\alpha\beta\gamma}$
vanishes, then it follows that the other one is zero as well. This can be seen
as a motivation to extend the Einstein–Hilbert action so that the equations of
motion fully determine the evolution of the degrees of freedom, rather than a
reason to set some of the degrees of freedom to zero by hand. We will see that
the quadratic $Q_{\alpha\beta\gamma}$ terms we have introduced break the
projective invariance, and the equations motion fully determine the
connection.

###  2.2 Connection equation of motion

#### General case.

Variation of the action with respect to $\Gamma^{\gamma}_{\alpha\beta}$ gives,
making the split (1) in the equation of motion and taking into account
$\mathring{\nabla}_{\gamma}g_{\alpha\beta}=0$,

$\displaystyle
8B_{1}Q_{\alpha\beta\gamma}+8B_{2}Q_{(\beta\gamma)\alpha}+4g_{\alpha(\beta}(2B_{3}\hat{Q}_{\gamma)}+B_{5}Q_{\gamma)})+4g_{\beta\gamma}(2B_{4}Q_{\alpha}+B_{5}\hat{Q}_{\alpha})$
$\displaystyle-
FQ_{\gamma\alpha\beta}+Fg_{\alpha\gamma}\hat{Q}_{\beta}+Fg_{\alpha[\beta}Q_{\gamma]}+FT_{\alpha\beta\gamma}+2Fg_{\alpha[\beta}T_{\gamma]}$
$\displaystyle=4A_{1}g_{\alpha(\beta}\partial_{\gamma)}h+4A_{2}g_{\beta\gamma}\partial_{\alpha}h-2F^{\prime}g_{\alpha[\beta}\partial_{\gamma]}h\
,$ (11)

where prime denotes derivative with respect to $h$. This shows how non-
metricity and torsion are sourced by the derivative of the non-minimal
coupling $F^{\prime}$ and the kinetic mixings $A_{i}$.

The general solution of (2.2) has the form

$\displaystyle Q_{\gamma\alpha\beta}$ $\displaystyle=$ $\displaystyle
q_{1}(h)g_{\alpha\beta}\partial_{\gamma}h+2q_{2}(h)g_{\gamma(\alpha}\partial_{\beta)}h$
$\displaystyle T_{\alpha\beta\gamma}$ $\displaystyle=$ $\displaystyle
2t(h)g_{\alpha[\beta}\partial_{\gamma]}h\ .$ (12)

Inserting (2.2) into (2.2), we get

$\displaystyle(a-2Ft)g_{\alpha\beta}\partial_{\gamma}h+(b+c+2Ft)g_{\gamma(\alpha}\partial_{\beta)}h+p(b-c+2Ft)g_{\gamma[\alpha}\partial_{\beta]}h=0\
,$ (13)

where

$\displaystyle a$ $\displaystyle\equiv$
$\displaystyle-2A_{1}+F^{\prime}+(4B_{2}+4B_{3}+8B_{5}+F)q_{1}+(8B_{1}+4B_{2}+20B_{3}+4B_{5}+F)q_{2}$
$\displaystyle b$ $\displaystyle\equiv$
$\displaystyle-2A_{1}-F^{\prime}+(4B_{2}+4B_{3}+8B_{5}-F)q_{1}+(8B_{1}+4B_{2}+20B_{3}+4B_{5}+3F)q_{2}$
$\displaystyle c$ $\displaystyle\equiv$
$\displaystyle-4A_{2}+(8B_{1}+32B_{4}+4B_{5})q_{1}+(8B_{2}+16B_{4}+20B_{5}-F)q_{2}\
.$ (14)

#### Unconstrained case.

In the unconstrained case ($p=1$), (13) gives $a=2Ft,a+b=0,c=0$, leading to

$\displaystyle q_{1}$ $\displaystyle=$
$\displaystyle\frac{-A_{1}(8B_{2}+16B_{4}+20B_{5}-F)+4A_{2}(4B_{1}+2B_{2}+10B_{3}+2B_{5}+F)}{2M}$
$\displaystyle q_{2}$ $\displaystyle=$
$\displaystyle\frac{2A_{1}(2B_{1}+8B_{4}+B_{5})-4A_{2}(B_{2}+B_{3}+2B_{5})}{M}$
$\displaystyle t$ $\displaystyle=$
$\displaystyle\\{-A_{1}(8B_{1}+8B_{2}+48B_{4}+24B_{5}-F)+4A_{2}(4B_{1}+4B_{2}+12B_{3}+6B_{5}+F)$
$\displaystyle+2[16B_{1}^{2}-8B_{2}^{2}-36B_{5}^{2}+144B_{3}B_{4}+4B_{1}(2B_{2}+10B_{3}+16B_{4}+4B_{5})$
$\displaystyle+B_{2}(-8B_{3}+16B_{4}-32B_{5})+(4B_{1}+B_{2}+B_{3}+16B_{4}+4B_{5})F]F^{\prime}/F\\}/(4M)$
$\displaystyle M$ $\displaystyle\equiv$ $\displaystyle
16B_{1}^{2}-8B_{2}^{2}-36B_{5}^{2}+4B_{1}(2B_{2}+10B_{3}+16B_{4}+4B_{5})+144B_{3}B_{4}$
(15)
$\displaystyle+B_{2}(-8B_{3}+16B_{4}-32B_{5})+F(4B_{1}+B_{2}+B_{3}+16B_{4}+4B_{5})\
.$

If $M=0$, the equations do not in general have a solution. A particular case
of this is $B_{i}=0$. If we put all $Q_{\alpha\beta\gamma}$ terms to zero in
the action, $A_{i}=B_{i}=0$, we get $2t-q_{1}=F^{\prime}/F,q_{2}=0$, showing
how only a particular linear combination of the torsion and non-metricity is
determined. If we keep the quadratic terms but put the linear terms to zero,
$A_{i}=0,B_{i}\neq 0$, we get $q_{1}=q_{2}=0,t=F^{\prime}/(2F)$. In other
words, without the kinetic mixing terms, the non-minimal coupling generates
torsion, but not non-metricity. If we also remove the non-minimal coupling
(i.e. put $F^{\prime}=0$), the connection becomes Levi–Civita. So if were to
only include the terms quadratic in $Q_{\alpha\beta\gamma}$, they would
successfully break the projective invariance and reduce the connection to
Levi–Civita without otherwise affecting the equations of motion. However, when
also $A_{1},A_{2}$ and/or $F^{\prime}$ is non-zero, the functional form of the
quadratic terms is relevant, and they can have a significant effect on the
evolution.

#### Zero torsion.

If torsion is taken to be zero a priori ($t=0,p=0$), (13) gives $a=0,b+c=0$,
leading to

$\displaystyle q_{1}$ $\displaystyle=$
$\displaystyle[-2A_{1}(8B_{2}+16B_{4}+20B_{5}+F)+4A_{2}(8B_{1}+4B_{2}+20B_{3}+4B_{5}+F)$
$\displaystyle+(16B_{1}+16B_{2}+40B_{3}+16B_{4}+28B_{5}+3F)F^{\prime}]/N$
$\displaystyle q_{2}$ $\displaystyle=$ $\displaystyle
4[A_{1}(4B_{1}+16B_{4}+2B_{5}-F)-A_{2}(4B_{2}+4B_{3}+8B_{5}+F)$
$\displaystyle-(2B_{1}+2B_{2}+2B_{3}+8B_{4}+5B_{5})F^{\prime}]/N$
$\displaystyle N$ $\displaystyle\equiv$ $\displaystyle
64B_{1}^{2}-32B_{2}^{2}-144B_{5}^{2}+8B_{1}(4B_{2}+20B_{3}+32B_{4}+8B_{5})-4B_{2}(8B_{3}-16B_{4}+32B_{5})$
(16)
$\displaystyle+576B_{3}B_{4}-(8B_{1}+20B_{2}+44B_{3}-16B_{4}+32B_{5}+3F)F\ .$

If $N=0$, the equations do not in general have a solution. In contrast to the
case with torsion, $N\neq 0$ even if $B_{i}=0$. This corresponds to the fact
that the projective invariance has been broken by setting the torsion to zero
by hand. Here $F^{\prime}$ sources non-metricity, unlike in the case with
torsion, where it exclusively sources torsion and only $A_{i}$ act as sources
for $Q_{\alpha\beta\gamma}$. If we take $A_{i}=B_{i}=0$, we get
$q_{1}=-F^{\prime}/F,q_{2}=0$. This is the usual case of non-minimally coupled
Higgs inflation in the Palatini formulation, studied originally in [140].

#### Zero non-metricity.

If we impose $Q_{\alpha\beta\gamma}=0$ a priori instead of
$T_{\alpha\beta\gamma}=0$, the $Q_{\alpha\beta\gamma}$ terms in the action
obviously vanish, so we are left only with the non-minimal coupling $F$, and
get the solution $t=F^{\prime}/(2F)$. If $F^{\prime}=0$, the torsion is zero,
and we again recover the Levi–Civita connection, having set one of the degrees
of freedom to zero by hand. It is straightforward to show that the torsion
generated by $F^{\prime}\neq 0$ has exactly the same effect on the equations
of motion of the scalar field and the metric as the non-zero
$Q_{\alpha\beta\gamma}$ in the above case when the torsion is set to zero and
only the non-minimal coupling is relevant (i.e. $A_{i}=B_{i}=0$). This has to
be the case, because the non-minimal coupling can be eliminated via a
conformal transformation of the metric, which knows nothing about the
symmetries of the connection. So we see that the effect of the non-minimal
coupling in usual Higgs inflation in the Palatini formulation can be mapped
not only from the non-minimal coupling and non-metricity to the kinetic term
or the potential, but also between the non-metricity and the torsion. (Mapping
between non-metricity and torsion has also been studied in [133, 136, 144].)
Let us now look at field transformations in more detail.

###  2.3 Field transformations

#### General case.

The equations of motion written in terms of the original variables
$\\{h,g_{\alpha\beta},\Gamma^{\gamma}_{\alpha\beta}\\}$ are rather
complicated. Therefore, rather than solving them directly, as we did above for
the connection, we simplify the problem by choosing suitable coordinates in
field space. Without introducing higher order derivatives into the action, we
can make the following field transformations:

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!h$ $\displaystyle\rightarrow$
$\displaystyle\chi(h)$ (17)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!g_{\alpha\beta}$
$\displaystyle\rightarrow$ $\displaystyle\Omega(h)^{-1}g_{\alpha\beta}$ (18)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\Gamma^{\gamma}_{\alpha\beta}$
$\displaystyle\rightarrow$
$\displaystyle\Gamma^{\gamma}_{\alpha\beta}+\Sigma^{\gamma}_{\
\alpha\beta}=\Gamma^{\gamma}_{\alpha\beta}+g^{\gamma\delta}\left[\Sigma_{1}(h)g_{\alpha\beta}\partial_{\delta}h+2\Sigma_{2}(h)g_{\delta(\alpha}\partial_{\beta)}h+2\Sigma_{3}(h)g_{\delta[\alpha}\partial_{\beta]}h\right]\,,$
(19)

so we have 5 free functions $\chi(h)$, $\Omega(h)$ and $\Sigma_{i}(h)$.666The
term $\Sigma_{4}(h)\epsilon^{\gamma\delta}_{\ \ \
\alpha\beta}\partial_{\delta}h$ could also be added to the torsion, but in our
case where the torsion is generated by a scalar field its totally
antisymmetric part is zero, so this term will not be needed. The formulation
of non-minimally coupled scalar field theories that is covariant with respect
to transformations of the scalar field (17) and the metric (18) [153, 154,
155, 156, 157, 158, 159] can be extended to also cover transformations of the
connection (19) [142]. For more general mapping of gravitational degrees of
freedom into matter degrees of freedom in the Palatini formulation, see [132,
133, 134, 135].

The scalar field redefinition (17) simply gives

$\displaystyle\nabla_{\alpha}h$ $\displaystyle\rightarrow$
$\displaystyle\frac{\mathrm{d}h}{\mathrm{d}\chi}\nabla_{\alpha}\chi\ .$ (20)

Under the conformal transformation of the metric (18), we have

$\displaystyle g^{\alpha\beta}$ $\displaystyle\rightarrow$
$\displaystyle\Omega g^{\alpha\beta}$ $\displaystyle\sqrt{-g}$
$\displaystyle\rightarrow$ $\displaystyle\Omega^{-2}\sqrt{-g}$
$\displaystyle\nabla_{\gamma}g^{\alpha\beta}$ $\displaystyle\rightarrow$
$\displaystyle\Omega(\nabla_{\gamma}g^{\alpha\beta}+g^{\alpha\beta}\partial_{\gamma}\omega)\
,$ (21)

where we have denoted $\omega\equiv\ln\Omega$.

Under the translation of the connection (19), we have

$\displaystyle\nabla_{\gamma}g^{\alpha\beta}$ $\displaystyle\rightarrow$
$\displaystyle\nabla_{\gamma}g^{\alpha\beta}+2\Sigma^{(\alpha\ \ \beta)}_{\ \
\ \gamma}$ $\displaystyle g^{\alpha\beta}R_{\alpha\beta}$
$\displaystyle\rightarrow$ $\displaystyle
g^{\alpha\beta}R_{\alpha\beta}+g^{\alpha\beta}\nabla_{\gamma}\Sigma^{\gamma}_{\
\ \alpha\beta}-g^{\alpha\beta}\nabla_{\alpha}\Sigma^{\gamma}_{\ \
\gamma\beta}+\Sigma^{\alpha\beta}_{\ \ \ \beta}\Sigma^{\gamma}_{\
\gamma\alpha}-\Sigma_{\alpha\beta\gamma}\Sigma^{\gamma\alpha\beta}-T_{\alpha\beta\gamma}\Sigma^{\gamma\alpha\beta}$
(22) $\displaystyle=$ $\displaystyle
R-\left(\frac{1}{2}Q_{\alpha}+T_{\alpha}\right)(\Sigma^{\alpha\beta}_{\ \ \
\beta}-\Sigma_{\beta}^{\
\beta\alpha})-T_{\alpha\beta\gamma}\Sigma^{\gamma\alpha\beta}+Q_{\gamma\alpha\beta}\Sigma^{\gamma\alpha\beta}-\hat{Q}^{\alpha}\Sigma^{\beta}_{\
\beta\alpha}$ $\displaystyle+\Sigma^{\alpha\beta}_{\ \ \
\beta}\Sigma^{\gamma}_{\
\gamma\alpha}-\Sigma_{\alpha\beta\gamma}\Sigma^{\gamma\alpha\beta}+\mathring{\nabla}_{\alpha}(\Sigma^{\alpha\beta}_{\
\ \ \beta}-\Sigma_{\beta}^{\ \beta\alpha})\ .$

where we have applied (10). In the unconstrained case where torsion is
present, this transformation generates a term proportional to
$T^{\alpha}\partial_{\alpha}h$ in the action. As we have assumed that the
torsion enters only via the connection that in turn enters only via the
Riemann tensor and covariant derivatives, we have no such term in the action
to begin with. Demanding that such terms are not generated requires
$T_{\alpha}(\Sigma^{\alpha\beta}_{\ \ \ \beta}-\Sigma_{\beta}^{\
\beta\alpha})+T_{\alpha\beta\gamma}\Sigma^{\gamma\alpha\beta}=0$, which
according to (17) is equivalent to $\Sigma_{3}=\Sigma_{1}-\Sigma_{2}$. Then we
also have $\Sigma^{\alpha\beta}_{\ \ \ \beta}-\Sigma_{\beta}^{\
\beta\alpha}=0$, and the boundary term in (2.3) vanishes. (If we had a
$T^{\alpha}\partial_{\alpha}h$ term in the original action, we would fix
$\Sigma_{3}$ by demanding that the new term cancels it.) In the constrained
case where the torsion is put to zero a priori, we instead have
$\Sigma_{3}=0$.777In this case we are also left with a boundary term, which we
discard. We used the principle of not adding or subtracting boundary terms
when writing down the action. However, the field transformation is just a
calculational device. We would get the same result by solving the field
equations without making field redefinitions.

Redefining the functions of $h$ in the action to absorb these changes, for the
scalar field transformation $h\rightarrow\chi$ we get

$\displaystyle K$ $\displaystyle\rightarrow$
$\displaystyle\tilde{K}=\left(\frac{\mathrm{d}h}{\mathrm{d}\chi}\right)^{2}K$
$\displaystyle A_{i}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{A}_{i}=\frac{\mathrm{d}h}{\mathrm{d}\chi}A_{i}\ .$ (23)

For the metric transformation
$g_{\alpha\beta}\rightarrow\Omega^{-1}g_{\alpha\beta}$ applied to the action
(8) we get

$\displaystyle F$ $\displaystyle\rightarrow$
$\displaystyle\tilde{F}=\Omega^{-1}F$ $\displaystyle K$
$\displaystyle\rightarrow$
$\displaystyle\tilde{K}=\Omega^{-1}[K-2(A_{1}+4A_{2})\omega^{\prime}-2(4B_{1}+B_{2}+B_{3}+16B_{4}+4B_{5})\omega^{\prime}{}^{2}]$
$\displaystyle V$ $\displaystyle\rightarrow$
$\displaystyle\tilde{V}=\Omega^{-2}V$ $\displaystyle A_{1}$
$\displaystyle\rightarrow$
$\displaystyle\tilde{A}_{1}=\Omega^{-1}[A_{1}+(2B_{2}+2B_{3}+4B_{5})\omega^{\prime}]$
$\displaystyle A_{2}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{A}_{2}=\Omega^{-1}[A_{2}+(2B_{1}+8B_{4}+B_{5})\omega^{\prime}]$
$\displaystyle B_{i}$ $\displaystyle\rightarrow$
$\displaystyle\tilde{B}_{i}=\Omega^{-1}B_{i}\ .$ (24)

Finally, the connection transformation
$\Gamma^{\gamma}_{\alpha\beta}\rightarrow\Gamma^{\gamma}_{\alpha\beta}+\Sigma^{\gamma}_{\
\alpha\beta}$ gives, using the decomposition (19),

$\displaystyle K\rightarrow\tilde{K}$ $\displaystyle=$ $\displaystyle
K-3F(\Sigma_{1}^{2}+\Sigma_{2}^{2}-\Sigma_{3}^{2}+4\Sigma_{1}\Sigma_{2}+4\Sigma_{1}\Sigma_{2}+2\Sigma_{2}\Sigma_{3}+4\Sigma_{3}\Sigma_{1})$
$\displaystyle-2A_{1}(5\Sigma_{1}+7\Sigma_{2}+3\Sigma_{3})-4A_{2}(\Sigma_{1}+5\Sigma_{2}-3\Sigma_{3})$
$\displaystyle-4B_{1}(5\Sigma_{1}^{2}+17\Sigma_{2}^{2}+9\Sigma_{3}^{2}+14\Sigma_{1}\Sigma_{2}-6\Sigma_{2}\Sigma_{3}+6\Sigma_{3}\Sigma_{1})$
$\displaystyle-2B_{2}(7\Sigma_{1}^{2}+31\Sigma_{2}^{2}-9\Sigma_{3}^{2}+34\Sigma_{1}\Sigma_{2}+6\Sigma_{2}\Sigma_{3}-6\Sigma_{3}\Sigma_{1})$
$\displaystyle-2B_{3}(5\Sigma_{1}+7\Sigma_{2}+3\Sigma_{3})^{2}-8B_{4}(\Sigma_{1}+5\Sigma_{2}-3\Sigma_{3})^{2}$
$\displaystyle-4B_{5}(5\Sigma_{1}^{2}+35\Sigma_{2}^{2}-9\Sigma_{3}^{2}+32\Sigma_{1}\Sigma_{2}-6\Sigma_{2}\Sigma_{3}-12\Sigma_{3}\Sigma_{1})$
$\displaystyle A_{1}\rightarrow\tilde{A}_{1}$ $\displaystyle=$ $\displaystyle
A_{1}+\frac{1}{2}F(\Sigma_{1}+3\Sigma_{2}+3\Sigma_{3})+4B_{1}(\Sigma_{1}+\Sigma_{2}+\Sigma_{3})+2B_{2}(\Sigma_{1}+3\Sigma_{2}-\Sigma_{3})$
$\displaystyle+2B_{3}(5\Sigma_{1}+7\Sigma_{2}+3\Sigma_{3})+2B_{5}(\Sigma_{1}+5\Sigma_{2}-3\Sigma_{3})$
$\displaystyle A_{2}\rightarrow\tilde{A}_{2}$ $\displaystyle=$ $\displaystyle
A_{2}+\frac{1}{4}F(\Sigma_{1}-3\Sigma_{2}-3\Sigma_{3})+4B_{1}(\Sigma_{2}-\Sigma_{3})+2B_{2}(\Sigma_{1}+\Sigma_{2}+\Sigma_{3})$
(25)
$\displaystyle+4B_{4}(\Sigma_{1}+5\Sigma_{2}-3\Sigma_{3})+B_{5}(5\Sigma_{1}+7\Sigma_{2}+3\Sigma_{3})\
.$

#### Unconstrained case.

In the case when the torsion is left free to be determined by the field
equations and we have no $T^{\alpha}\partial_{\alpha}h$ terms in the original
action, we have $\Sigma_{3}=\Sigma_{1}-\Sigma_{2}$. Using (2.3) and setting
$\tilde{A}_{i}=0$ then fixes $\Sigma_{1},\Sigma_{2}$ as

$\displaystyle\Sigma_{1}$ $\displaystyle=$
$\displaystyle\frac{-A_{1}(2B_{1}+8B_{4}+B_{5})+2A_{2}(B_{2}+B_{3}+2B_{5})}{M}$
$\displaystyle\Sigma_{2}$ $\displaystyle=$
$\displaystyle\frac{A_{1}(-8B_{1}+8B_{2}-16B_{4}+16B_{5}-F)-4A_{2}(4B_{1}+8B_{3}-2B_{5}+F)}{8M}\
,$ (26)

where $M$ is defined in (2.2).

We can now take $\Omega=F$ to set $\tilde{F}=M_{{}_{\mathrm{Pl}}}^{2}$ (and
choose units such that $M_{{}_{\mathrm{Pl}}}=1$) and then choose
$\Sigma_{1},\Sigma_{2}$ to set $\tilde{A}_{i}=0$. According to (2.3) and (2.3)
we get

$\displaystyle\Sigma_{1}$ $\displaystyle=$
$\displaystyle\\{-[A_{1}+(2B_{2}+2B_{3}+4B_{5})\omega^{\prime}](2B_{1}+8B_{4}+B_{5})$
$\displaystyle+2[A_{2}+(2B_{1}+8B_{4}+B_{5})\omega^{\prime}](B_{2}+B_{3}+2B_{5})\\}M^{-1}$
$\displaystyle\Sigma_{2}$ $\displaystyle=$
$\displaystyle\\{[A_{1}+(2B_{2}+2B_{3}+4B_{5})\omega^{\prime}](-8B_{1}+8B_{2}-16B_{4}+16B_{5}-F)$
(27)
$\displaystyle-4[A_{2}+(2B_{1}+8B_{4}+B_{5})\omega^{\prime}](4B_{1}+8B_{3}-2B_{5}+F)\\}(8M)^{-1}\
.$

It is straightforward to check that this shift gives the same result for the
connection as solving for $Q_{\alpha\beta\gamma}$ directly from (13) and
applying (5), up to terms proportional to $F^{\prime}$, which have to be
accounted for by a conformal transformation of the metric. The correspondence
is

$\displaystyle\Sigma_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{2}q_{1}-q_{2}-t+\frac{F^{\prime}}{2F}$
$\displaystyle\Sigma_{2}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}q_{1}+\frac{1}{2}t-\frac{F^{\prime}}{4F}\ .$ (28)

The overall transformation of the kinetic term is, according to (2.3) and
(2.3) and taking into account $\Sigma_{3}=\Sigma_{1}-\Sigma_{2}$,

$\displaystyle K$ $\displaystyle\rightarrow$ $\displaystyle
F^{-1}\\{K-2(A_{1}+4A_{2})\omega^{\prime}-2(4B_{1}+B_{2}+B_{3}+16B_{4}+4B_{5})\omega^{\prime}{}^{2}$
(29)
$\displaystyle+F(-12\Sigma_{1}^{2}+6\Sigma_{2}^{2}-24\Sigma_{1}\Sigma_{2})-8[A_{1}+(2B_{2}+2B_{3}+4B_{5})\omega^{\prime}](2\Sigma_{1}+\Sigma_{2})$
$\displaystyle+[A_{2}+(2B_{1}+8B_{4}+B_{5})\omega^{\prime}](8\Sigma_{1}-32\Sigma_{2})+B_{1}(-80\Sigma_{1}^{2}-128\Sigma_{2}^{2}+64\Sigma_{1}\Sigma_{2})$
$\displaystyle+B_{2}(16\Sigma_{1}^{2}-32\Sigma_{2}^{2}-128\Sigma_{1}\Sigma_{2})-B_{3}(128\Sigma_{1}^{2}+32\Sigma_{2}^{2}+128\Sigma_{1}\Sigma_{2})$
$\displaystyle+B_{4}(-32\Sigma_{1}^{2}-512\Sigma_{2}^{2}+256\Sigma_{1}\Sigma_{2})+B_{5}(64\Sigma_{1}^{2}-128\Sigma_{2}^{2}-224\Sigma_{1}\Sigma_{2})\\}\
,$

with $\Sigma_{1},\Sigma_{2}$ given by (2.3).

As we have discussed, with $\tilde{F}^{\prime}=0$ and $\tilde{A}_{i}=0$, the
connection equation of motion (2.2) gives
$Q_{\alpha\beta\gamma}=T_{\alpha\beta\gamma}=0$ (because the $B_{i}$ terms
break the projective invariance), so $\Gamma^{\gamma}_{\alpha\beta}$ becomes
the Levi–Civita connection, and the system reduces to metric gravity with a
minimally coupled scalar field $h$. We can then choose $\chi$ to set
$\tilde{K}=1$, though in some cases it may be more convenient to keep using
$h$ and retain a non-canonical kinetic term, if
$\frac{\mathrm{d}\chi}{\mathrm{d}h}=\pm\sqrt{K(h)}$ cannot be integrated in
closed form. In terms of the action, we have

$\displaystyle S$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2}g^{\alpha\beta}R_{\alpha\beta}(\Gamma,\partial\Gamma)-\frac{1}{2}K(h)g^{\alpha\beta}\nabla_{\alpha}h\nabla_{\beta}h-\frac{V(h)}{F(h)^{2}}\right]$
(30) $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\left[\frac{1}{2}g^{\alpha\beta}R_{\alpha\beta}(\Gamma,\partial\Gamma)-\frac{1}{2}g^{\alpha\beta}\nabla_{\alpha}\chi\nabla_{\beta}\chi-U(\chi)\right]\
,$

where $K(h)$ is given by (29) and $U(\chi)\equiv V[h(\chi)]/F[h(\chi)]^{2}$.

#### Zero torsion.

When torsion is put to zero a priori, we have $\Sigma_{3}=0$, so according to
(2.3) setting $\tilde{A}_{i}=0$ fixes $\Sigma_{1},\Sigma_{2}$ as

$\displaystyle\Sigma_{1}$ $\displaystyle=$
$\displaystyle\frac{-A_{1}(16B_{1}+8B_{2}+80B_{4}+28B_{5}-3F)+2A_{2}(8B_{1}+12B_{2}+28B_{3}+20B_{5}+3F)}{N}$
$\displaystyle\Sigma_{2}$ $\displaystyle=$
$\displaystyle\frac{A_{1}(8B_{2}+16B_{4}+20B_{5}+F)-2A_{2}(8B_{1}+4B_{2}+20B_{3}+4B_{5}+F)}{N}\
,$ (31)

with $N$ given in (2.2). It is again straightforward to verify that this shift
gives the same result for the connection as solving for
$Q_{\alpha\beta\gamma}$ directly from (13) and applying (5), up to terms
proportional to $F^{\prime}$.

Again first transforming the metric to get $\tilde{F}=1$ and then adjusting
the connection to get $\tilde{A}_{i}=0$ we have, from (2.3) and (2.3),

$\displaystyle\Sigma_{1}$ $\displaystyle=$
$\displaystyle\\{-[A_{1}+(2B_{2}+2B_{3}+4B_{5})\omega^{\prime}](16B_{1}+8B_{2}+80B_{5}+28B_{5}-3F)$
$\displaystyle+2[A_{2}+(2B_{1}+8B_{4}+B_{5})\omega^{\prime}](8B_{1}+12B_{2}+28B_{3}+20B_{5}+3F)\\}N^{-1}$
$\displaystyle\Sigma_{2}$ $\displaystyle=$
$\displaystyle\\{[A_{2}+(2B_{1}+8B_{4}+B_{5})\omega^{\prime}](8B_{2}+16B_{4}+20B_{5}+F)$
(32)
$\displaystyle-2[A_{2}+(2B_{1}+8B_{4}+B_{5})\omega^{\prime}](8B_{1}+4B_{2}+20B_{3}+4B_{5}+F)\\}N^{-1}\
,$

Again, in the chosen field coordinates the connection is Levi–Civita, and all
effects of the original non-minimal coupling and non-metricity are shifted to
the kinetic term and potential of the scalar field, and the action has the
form (30). The kinetic term reads, from (18) and (19),

$\displaystyle K$ $\displaystyle\rightarrow$ $\displaystyle
F^{-1}\\{K-2(A_{1}+4A_{2})\omega^{\prime}-2(4B_{1}+B_{2}+B_{3}+16B_{4}+4B_{5})\omega^{\prime}{}^{2}$
(33)
$\displaystyle-3F(\Sigma_{1}^{2}+\Sigma_{2}^{2}+4\Sigma_{1}\Sigma_{2}+4\Sigma_{1}\Sigma_{2})-2[A_{1}+(2B_{2}+2B_{3}+4B_{5})\omega^{\prime}](5\Sigma_{1}+7\Sigma_{2})$
$\displaystyle-4[A_{2}+(2B_{1}+8B_{4}+B_{5})\omega^{\prime}](\Sigma_{1}+5\Sigma_{2})$
$\displaystyle-4B_{1}(5\Sigma_{1}^{2}+17\Sigma_{2}^{2}+14\Sigma_{1}\Sigma_{2})-2B_{2}(7\Sigma_{1}^{2}+31\Sigma_{2}^{2}+34\Sigma_{1}\Sigma_{2})-2B_{3}(5\Sigma_{1}+7\Sigma_{2})^{2}$
$\displaystyle-2B_{3}(5\Sigma_{1}+7\Sigma_{2})^{2}-8B_{4}(\Sigma_{1}+5\Sigma_{2})^{2}-4B_{5}(5\Sigma_{1}^{2}+35\Sigma_{2}^{2}+32\Sigma_{1}\Sigma_{2})\\}\
,$

with $\Sigma_{1},\Sigma_{2}$ given by (2.3).

##  3 The Higgs case

###  3.1 The potential and CMB observables

#### Structure of the potential and the kinetic term.

We have fully mapped the effects of the non-minimal coupling and non-metricity
term to the kinetic term and the potential in the scalar sector, so their
effect can be analysed with the usual inflationary vocabulary. Let us now
specialise to the case of Higgs inflation and consider what kind of
phenomenology the terms can lead to.

We identify $h$ with the SM background Higgs field. As the Higgs is part of a
doublet, it only appears in even powers in the action. Restricting to terms of
up to dimension 4, we have

$\displaystyle K=k\ ,\qquad F=f+\xi h^{2}\ ,\qquad A_{i}=a_{i}h\ ,\qquad
B_{i}=b_{i0}+b_{i1}h^{2}\ ,$ (34)

where $k,f,\xi,a_{i},b_{i0},b_{i1}$ are constants. Note that $f$ does not have
to be close to $M_{{}_{\mathrm{Pl}}}^{2}=1$, since the Planck scale is defined
in terms of gravity analysed in the Einstein frame. The SM Higgs potential is

$\displaystyle V(h)=\frac{1}{4}\lambda(h^{2}-v^{2})^{2}\ .$ (35)

The only thing that changes the potential as a function of $h$ is the non-
minimal coupling $F$, which (for $\xi\neq 0$) flattens the transformed
potential for large values of $h$ as usual in Higgs inflation,

$\displaystyle
U(\chi)\equiv\frac{V[h(\chi)]}{\Omega[h(\chi)]^{2}}=\frac{V[h(\chi)]}{F[h(\chi)]^{2}}=\frac{\lambda}{4}\frac{(h^{2}-v^{2})^{2}}{(f+\xi
h^{2})^{2}}\simeq\frac{\lambda}{4\xi^{2}}\left(1-\frac{2f}{\xi h^{2}}\right)\
.$ (36)

However, the potential as a function of $\chi$ is affected also by the
redefinition of the kinetic term. The kinetic function has the same
qualitative structure in the unconstrained case (29) and in the zero torsion
case (33). With the functions (34), the kinetic term in both cases reads

$\displaystyle K(h)$ $\displaystyle=$ $\displaystyle\frac{k}{f+\xi
h^{2}}+\frac{h^{2}}{(f+\xi
h^{2})^{3}(O_{0}+O_{1}h^{2}+O_{2}h^{4})^{2}}\sum_{n=0}^{5}K_{n}h^{2n}\ ,$ (37)

where $K_{n}$ and the coefficients of $O(h)\equiv O_{0}+O_{1}h^{2}+O_{2}h^{4}$
are complicated polynomials of the constants defined in (34). The function
$K(h)$ is determined by and (2.3) and (29) (with torsion) or (2.3) and (33)
(no torsion). In the case with torsion, $O(h)=M(h)$ given in (2.2), and in the
case with no torsion, $O(h)=N(h)$ given in (2.2).

The field transformation to the canonically normalised field $\chi$ is given
by

$\displaystyle\frac{\mathrm{d}\chi}{\mathrm{d}h}=\pm\sqrt{K(h)}\ .$ (38)

In the small field limit $h\ll 1$, the second term in (37) does not
contribute, so (38) gives $\chi=\sqrt{k/f}h$, and the potential reduces to
$U(\chi)=V(\sqrt{f/k}\chi)/f^{2}$. The parameters of the Higgs potential (35)
written in terms of the canonical field become
$\lambda\rightarrow\lambda/k^{2},v\rightarrow\sqrt{k/f}v$. The Higgs mass
correspondingly changes as $m_{h}=\sqrt{2\lambda}v\rightarrow
m_{h}/\sqrt{kf}$. Starting from the original action, we can therefore scale
the quartic coupling and the Higgs mass at will. In particular, a large Higgs
mass can be brought down to the EW scale by choosing a large $f$. Restoring
dimensions, we have $F=M^{2}+\xi h^{2}$ and
$\Omega=\frac{M_{{}_{\mathrm{Pl}}}^{2}}{M^{2}+\xi h^{2}}$. In the limit of
small $h$, the transformation to the minimally coupled canonical field reduces
to the field-independent redefinition
$m_{h}\rightarrow\frac{1}{\sqrt{k}}\frac{M}{M_{{}_{\mathrm{Pl}}}}m_{h}$, so
the hierachy between $m_{h}$ and $M_{{}_{\mathrm{Pl}}}$ can be transformed
into the question why $M$ is much smaller than $M_{{}_{\mathrm{Pl}}}$.

#### CMB observables.

The observational constraints from cosmic microwave background (CMB) data on
the scalar perturbation amplitude $A_{s}$, scalar spectral index $n_{s}$ and
tensor-to-scalar ratio $r$ are (assuming no running of the spectral index)
[20]

$\displaystyle 24\pi^{2}A_{s}$ $\displaystyle=$
$\displaystyle\frac{U}{\epsilon}=(4.97\pm 0.07)\times 10^{-7}$ $\displaystyle
n_{s}$ $\displaystyle=$ $\displaystyle 1+6\epsilon-2\eta=0.9653\pm 0.0041$
$\displaystyle r$ $\displaystyle=$ $\displaystyle 16\epsilon<0.07\ ,$ (39)

where the slow-roll parameters are
$\epsilon\equiv\frac{1}{2}(U^{\prime}/U)^{2},\eta\equiv U^{\prime\prime}/U$.

The inflationary behaviour is determined by the shape of $U(h)$ and $K(h)$.
Let us consider some possibilities and compare to the observed values (3.1).

###  3.2 Types of inflationary potentials

#### Large-field case with $\xi\neq 0$.

If the non-minimal coupling is non-zero, the potential $U(h)$ is
asymptotically flat in the limit of large $h$, and the behaviour of $K(h)$ in
this limit determines how rapidly the plateau is approached.888How large $h$
has to be depends on the value of $\xi$ and the coefficients appearing in
$K(h)$. As there are many terms, the numerical factors multiplying $h$ can be
large, so large field does not necessarily mean $h\gg 1$. The function $K(h)$
is the ratio of two polynomials, the numerator being of order 12 and the
denominator of order 14. Thus, in the general case when neither of the leading
order coefficients is tiny, $K(h)$ is proportional to $h^{-2}$ in the large
field limit. This leads to an exponential potential for $\chi$ as in usual
Higgs inflation. In the case $A_{i}=B_{i}=0$, we have $K=k/(\xi h^{2})$, and
for $k=1$ we get $h=\frac{1}{2\sqrt{\xi}}e^{\sqrt{\xi}\chi}$ and
$U=\frac{\lambda}{4\xi^{2}}(1-2e^{-2\xi\chi})$ [140]. In the general case, the
coefficient of the leading contribution is a complicated polynomial of the
constants defined in (34). In the limit $h\gg 1,\xi\gg 1$ the kinetic function
reduces to
$K\simeq(k-4a_{1}-16a_{2}-32b_{11}-8b_{21}-8b_{31}-128b_{41}-32b_{51})/(\xi
h^{2})$ both in the unconstrained case and in the zero torsion case. (The
kinetic functions in the two cases differ at order $1/(\xi^{2}h^{2})$.) This
corresponds to the replacement
$\xi\rightarrow\xi/(k-4a_{1}-16a_{2}-32b_{11}-8b_{21}-8b_{31}-128b_{41}-32b_{51})$
in the exponential potential. In the tree-level Palatini case with the
exponential potential the normalisation of the perturbations gives
$\lambda/\xi=10^{-10}$, requiring $\xi=10^{9}$ for $\lambda=0.1$. (When loop
corrections change the potential, $\xi$ can be orders of magnitude smaller
[59, 60, 68].) If the coefficients dividing $\xi$ are of order unity, $\xi$ is
reduced by at most two orders of magnitude, unless they conspire so that the
sum is much smaller than the individual terms. Thus, absent such tuning, when
the leading terms of the polynomials in the numerator and denominator of
$K(h)$ dominate, the situation is not much changed from the case with
$A_{i}=B_{i}=0$.

Different behaviour can be obtained by choosing the coefficients so that the
leading term of the numerator or the denominator vanishes. For example, with
the choice $k=1,f=1,\xi=2\times 10^{5}$ and $a_{i}=10,b_{i0}=b_{i1}=-0.96$
(with torsion) or $a_{i}=1.463314\times 10^{5},b_{i0}=b_{i1}=-1.39$ (no
torsion), the $h^{12}$ term in the numerator in suppressed relative to the
$h^{10}$ term, so for large $h$ we have $K(h)\simeq\kappa_{1}h^{-4}$ with
$\kappa_{1}=10^{-3}$. The relation (38) then gives
$\chi=\sqrt{\kappa_{1}}h^{-1}$, and the potential becomes

$\displaystyle
U(\chi)\simeq\frac{\lambda}{4\xi^{2}}\left(1-\frac{2}{\xi\kappa_{1}}\chi^{2}\right)\
,$ (40)

with $\chi\ll 1$. This is a small-field potential of the hilltop type. We have
chosen the parameters (taking again $\lambda=0.1$) to reproduce the amplitude
$A_{s}$ and the spectral index $n_{s}$ in agreement with the observations
(3.1), taking the CMB pivot scale to correspond to $\chi=0.06$. We did not
check the constraint of obtaining the right number of e-folds. For this
example, the tensor-to-scalar ratio is $r=1\times 10^{-5}$. The value of $\xi$
is four orders of magnitude smaller than in the case with only the non-minimal
coupling $F$, because now $U(\chi)$ is less flat, so the slow-roll parameter
$\epsilon$ is larger; recall that $24\pi^{2}A_{s}=U/\epsilon=16U/r$. Hilltop-
type Higgs inflation has previously been considered with loop corrections [56,
59, 60], here we get similar behaviour at the classical level. As the
polynomials in $K(h)$ are of high order and the coefficients are large,
canceling the leading terms requires a lot of tuning. This is also true for an
observationally viable hilltop generated by quantum corrections.

Alternatively, we can choose the coefficients (34) so that $O_{2}=0$ and the
two leading terms of the denominator in (37) vanish.999It is not possible to
make just the leading term to vanish identically, although we could try to
tune $O_{1}$ and $O_{2}$ so that the leading term is strongly subdominant to
the next-to-leading term. This would lead to a potential of the form
$U(\chi)\propto 1-\kappa_{2}\chi^{-2}$, which gives a viable inflationary
model. In this case $K(h)\simeq\kappa_{3}h^{2}$ for large $h$, so (38) gives
$\chi=\frac{\sqrt{\kappa_{3}}}{3}h^{3}$, and the potential becomes

$\displaystyle
U(\chi)\simeq\frac{\lambda}{4\xi^{2}}\left(1-\frac{2f}{\xi\kappa_{3}}\chi^{-2/3}\right)\
.$ (41)

In the case with torsion, this is not possible, as all of the leading terms of
the numerator are proportional to $M^{2}$, so they will also be suppressed. In
the case without torsion, this case can be realised for example by choosing
$a_{1}=315,a_{2}=289,b_{i0}=1,b_{i1}=1.4$, and $\chi=8$ at the pivot scale
gives $A_{s}$ and $n_{s}$ in agreement with the central values (3.1) and
$r=0.04$. Again, we did not check the number of e-folds. For these
coefficients, the $1/h^{2}$ term and other higher order terms can contribute
to $K(h)$, and might have to be suppressed by further tuning of coefficients.

#### Large-field case with $\xi=0$.

Let us now consider the case without non-minimal coupling.101010In general,
$\xi$ is generated by quantum corrections [160], and it will run with scale,
and thus cannot be zero on all scales. However, it can be negligibly small in
the inflationary region. Now $U$ is not asymptotically flat. We have
$K_{4}=K_{5}=0$, so the numerator and denominator of $K(h)$ are both
polynomials of order 8, with complicated coefficients. If coefficients are not
tuned, in the large field limit the field transformation (38) thus simply
rescales the field by a constant, so we have $U\propto\chi^{4}$. The rescaling
could be used to make the value of the effective quartic coupling compatible
with the constraint from the perturbation amplitude, but $n_{s}$ would anyway
be below and $r$ above the observational bound (3.1) [20].

If we choose the leading term of the numerator to vanish, we have
$K(h)=\kappa_{4}h^{-2}$ for large $h$. This gives $h\propto
e^{\chi/\sqrt{\kappa_{4}}}$, so $U\propto e^{4\chi/\sqrt{\kappa_{4}}}$. Such a
potential gives $r$ in excess of the observational upper bound.

If we instead choose the two leading terms of the denominator to vanish, we
have $K(h)\propto h^{4}$ for large $h$, leading to $\chi\propto h^{3}$ and
$U\propto\chi^{4/3}$, which gives $r\gtrsim 0.1$, taking into account the
constraint (3.1) on $n_{s}$, and is thus excluded. Suppressing only the first
leading term would give $K(h)\simeq h^{2}$, $\chi\propto h^{2}$ and
$U\propto\chi^{2}$, which would also give to a too large $r$.

####  $\alpha$-attractor.

We can have inflation at intermediate values of $h$ if the potential is very
flat there. There are two possibilities. We can have an $\alpha$-attractor,
where the denominator of $K(h)$ is zero [161, 162, 163],111111We can also have
an $\alpha$-attractor at large field values. Usual Higgs inflation in the
large $h$ limit can be understood as an $\alpha$-attractor model [163, 24]. or
we can have an inflection point (or a near-inflection point) where $K(h)$
remains regular. In the context of Higgs inflation an inflection point is
usually also an extremum (or nearly so), and it is then called a critical
point [44, 51, 50, 52, 53, 56, 58, 57, 59, 67, 68].

Let us first consider the $\alpha$-attractor case. The kinetic function $K(h)$
has a zero in the denominator at $h_{0}$ if the functions $F,B_{i}$ are chosen
so that $O(h_{0})=0$. This produces an inflationary plateau if the numerator
is positive at $h_{0}$. In the case of Higgs inflation (34), the parameters
$f,\xi,b_{i0},b_{i1}$ can be chosen to make the pole either of order
$(h-h_{0})^{-2}$ or $(h-h_{0})^{-4}$. In order to have a viable model, we must
also demand that $K(h)>0$ for $0\leq h<h_{0}$. These conditions are easy to
satisfy. For example, in the case with torsion the choice of parameters
$k=1,f=1,\xi=0,a_{1}=-a_{2}=a,b_{10}=1,b_{50}=-1,b_{j0}=-1,b_{11}=1,b_{51}=5000,b_{j1}=0$
(with $j=2,3,4$) with arbitrary $a$, gives a second order pole,
$K(h)\simeq\kappa_{5}(h-h_{0})^{-2}$ at $h_{0}=0.01$ with $\kappa_{5}=2\times
10^{-9}a^{2}$. In the case with no torsion we can choose
$k=1,f=1,\xi=0,a_{i}=a,b_{10}=b_{50}=1,b_{j0}=-1,b_{11}=1,b_{51}=5000,b_{j1}=0$
(with $j=2,3,4$) to get a second order pole at $h_{0}=0.01$ with
$\kappa_{5}=2\times 10^{-8}a^{2}$. In the limit of large number of e-folds
$N$, this gives [163]

$\displaystyle n_{s}=1-\frac{2}{N}\ ,\quad r=\frac{8\kappa_{5}}{N^{2}}\ .$
(42)

The spectral index is the same as in the usual Higgs inflation case (and thus
agrees with observations for $N=50$, the usual number of e-folds for the case
without the extra terms [59, 164, 165, 166] – though see [167, 168]), whereas
the tensor-to-scalar ratio can be adjusted at will. The normalisation of the
perturbations (3.1) implies $\lambda=10^{-7}rh_{0}^{-4}$, which for the above
examples gives $r=0.08\lambda$. However, the parameters could be adjusted to
change $h_{0}$ to make $r$ smaller or larger.

#### Inflection point.

We can also choose parameters to produce an inflection point, where
$\mathrm{d}^{2}U/\mathrm{d}\chi^{2}=0$ and $K(h)$ remains regular and
positive. The important difference to an $\alpha$-attractor is that the field
can roll through an inflection point. In the context of Higgs inflation,
inflection points from quantum corrections have been considered to get large
values of $r$ [44, 51, 50, 52, 53, 56, 58, 57, 59] or an enhancement in the
spectrum to produce primordial black holes as dark matter candidates [67, 68].

It is easy to tune the parameters to make a regular inflection point. For
example, in the case with torsion,
$k=1,f=1,\xi=0,a_{1}=1,a_{2}=-100,b_{20}=-1,b_{j0}=1,b_{i1}=-1$, (with
$j=1,3,4,5$) gives an inflection point at $h=0.77$. In the case without
torsion, $k=1,f=1,\xi=0,a_{1}=1,a_{2}=100,b_{20}=-1,b_{j0}=1,b_{i1}=1$ (with
$j=1,3,4,5$) gives an inflection point at $h=0.04$.

We cannot get $\mathrm{d}U/\mathrm{d}\chi=0$ exactly (unless $\xi<0$), because
$U$ is monotonic as a function of $h$, and while field redefinitions can
change the curvature of the potential, they do not affect the existence or
locations of its extrema. Therefore we cannot obtain hillclimbing inflation
[61, 62], nor have an exact hilltop [56, 59, 60] or an exact critical point.
However, it may be possible to suppress $\mathrm{d}U/\mathrm{d}\chi$ by tuning
the coefficients so as to get a near-critical point or a potential that looks
similar to the vicinity of a hilltop. We leave investigation of such
possibilities and detailed comparison to observations, including checking the
right number of e-folds, for future work.

##  4 Discussion

#### Fermions and gauge bosons.

Let us now look at fermions and gauge bosons. Considering only terms with up
to a total of two derivatives, the only new fermionic contributions are

$\displaystyle S_{\mathrm{f}}$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\left[D_{1}(h)\bar{\psi}\gamma^{\alpha}Q_{\alpha}\psi+D_{2}(h)\bar{\psi}\gamma^{\alpha}\hat{Q}_{\alpha}\psi\right]\
.$ (43)

With the solution (2.2) for $Q_{\alpha\beta\gamma}$, we will end up with a
term of the form $g(h)\bar{\psi}\gamma^{\alpha}\partial_{\alpha}h\psi$. In the
SM, where we have a $U(1)$ gauge field, such a term corresponds to a gauge
transformation, and thus does not lead to any physical effect such as
generation of gravitational waves, unlike the corresponding pseudoscalar
term.121212Thanks to Lorenzo Sorbo for pointing this out. Fermions also
contribute to the equation of motion of $\Gamma^{\gamma}_{\alpha\beta}$ by
generating a spin current via their coupling to the connection [96, 98, 114].
This is not important during inflation when the scalar field dominates, and
while fermions become significant during reheating, there is no overall spin
order, so the spin current is expected to be highly suppressed.

As for gauge fields, if we restrict to a total of two derivatives, there are
no new terms involving $Q_{\alpha\beta\gamma}$. If we allow terms with more
than two derivatives in total (as long as they are not acting on the same
field), at leading order in the derivatives we have

$\displaystyle S_{\mathrm{g}}$ $\displaystyle=$
$\displaystyle\int\mathrm{d}^{4}x\sqrt{-g}\left[E_{1}(h)F_{\alpha\beta}Q^{\alpha\beta\gamma}Q_{\gamma}+E_{2}(h)F_{\alpha\beta}Q^{\alpha\beta\gamma}\hat{Q}_{\gamma}+E_{3}(h)F_{\alpha\beta}Q^{\alpha}\hat{Q}^{\beta}\right.$
(44)
$\displaystyle\left.+E_{4}(h)\epsilon^{\alpha\beta\gamma\delta}F_{\alpha\beta}Q_{\gamma\delta}^{\
\ \
\epsilon}Q_{\epsilon}+E_{5}(h)\epsilon^{\alpha\beta\gamma\delta}F_{\alpha\beta}Q_{\gamma\delta}^{\
\ \
\epsilon}\hat{Q}_{\epsilon}+E_{6}(h)\epsilon^{\alpha\beta\gamma\delta}F_{\alpha\beta}Q_{\gamma}\hat{Q}_{\delta}\right]\
,$

where $F_{\alpha\beta}$ is an Abelian field strength. In the case of the SM,
$F_{\alpha\beta}$ is the $U(1)_{Y}$ gauge field strength, and these terms
terms break the conformal symmetry of the gauge field, so they might at first
sight seem like candidates for inflationary magnetogenesis [169, 170].
However, when $Q_{\alpha\beta\gamma}$ is sourced only by a single scalar
field, all of these contributions vanish in the equations of motion due to the
symmetry of the $Q_{\alpha\beta\gamma}$ contributions paired with the
antisymmetry of $F_{\alpha\beta}$ or $\epsilon_{\alpha\beta\gamma\delta}$.
When fermions and gauge bosons contribute significantly during preheating,
this could change. The situation would also be different in models where more
than one scalar field contributes, as then $Q^{\alpha}$ and $\hat{Q}^{\alpha}$
are in general not aligned.

#### Higher order $Q_{\alpha\beta\gamma}$ terms.

We have allowed at most two derivatives in the action, but as the partial
derivatives of the metric vanish (by pairing with the Levi–Civita connection
in the equations of motion), higher powers of $Q_{\alpha\beta\gamma}$ do not
lead to higher order of the equations of motion. The restriction therefore
seems arbitrary. Keeping to terms of up to dimension 4, there are no new terms
mixing the derivatives the metric with the scalar field, fermions or gauge
bosons, nor are there any new terms of order $(\nabla g)^{3}$. The only new
terms are of order $(\nabla g)^{4}$, which do not couple directly to the
scalar field. There are dozens of different contractions, but they all
contribute in essentially the same way to the connection equation of motion
(2.2), changing the equation from first to third order in
$Q_{\alpha\beta\gamma}$. The tensor structure of the solution (2.2) will
remain the same, but the functions $q_{i}$ and $t$ will depend on
$g^{\alpha\beta}\partial_{\alpha}h\partial_{\beta}h$ in addition to $h$ via
equations of the form (denoting $t\equiv q_{3}$)
$q_{k}[P_{k}^{(1)}(h)+\sum_{i,j=1}^{3}P^{(2)}_{kij}(h)q_{i}q_{j}g^{\alpha\beta}\partial_{\alpha}h\partial_{\beta}h]=P^{(3)}_{k}(h)$,
where $P_{k}^{(1)},P^{(2)}_{kij},P_{k}^{(3)}$ are combinations of the
functions of $h$ appearing in the action. The functions $\Sigma_{i}$ will then
depend on $g^{\alpha\beta}\partial_{\alpha}h\partial_{\beta}h$, and so will
the kinetic function $K$. Therefore, the equation that relates $h$ to the
minimally coupled scalar field $\chi$ with a canonical kinetic term will also
depend on $g^{\alpha\beta}\partial_{\alpha}h\partial_{\beta}h$. During slow-
roll inflation and in the long-wavelength limit, the derivative contribution
is suppressed, so this is expected to have little effect on potential-driven
slow-roll inflation. However, such terms could enable inflation driven by
kinetic terms [171]. One issue is that as the equations are third order, they
could have homogeneous solutions, meaning that the equations of motion do not
completely determine the connection. Avoiding such ambiguity would translate
into a constraint on the terms allowed in the action.

##  5 Conclusions

#### The gravity track of Higgs inflation.

We have considered scalar field inflation in the Palatini formulation of
general relativity, where the connection is an independent variable. The
covariant derivative of the metric (i.e. the non-metricity) is then non-zero,
and it can couple to the scalar field. Assuming that the connection enters
only via the Riemann tensor and the covariant derivative, we have written down
all coupling terms with at most two derivatives such that the equations of
motion follow from the bulk action without needing to add boundary terms. We
have considered both the unconstrained case when the antisymmetric part of the
connection (the torsion) is determined by the field equations and the case
when it set to zero a priori.

Terms that are quadratic in the derivative of the metric are motivated by the
fact that they break the projective invariance of the action and thus allow
the equations of motion to fully determine the degrees of freedom in the case
when the torsion is not taken to be zero. If the non-minimal coupling of the
scalar field and the mixing between the kinetic terms of the metric and the
scalar field are zero, the quadratic terms simply set the non-metricity and
torsion to zero and make the connection Levi–Civita. When non-minimal coupling
and/or the kinetic mixing between the metric and the scalar field are non-
zero, they source non-metricity and torsion. We transform to the Einstein
frame, where the connection becomes Levi–Civita and the scalar field is
minimally coupled to gravity, and the effects of non-metricity and torsion are
mapped to the kinetic term and the potential of the scalar field.

Specialising to the Standard Model Higgs field $h$ as the inflaton (meaning
that for large field values the potential is quartic and the field appears
only in even powers), we study the range of inflationary models made possible
by the new terms. Including only terms of up to dimension 4, the kinetic term
is the ratio of two complicated polynomials of $h$, the numerator being of
order 10 and the denominator of order 14. In general, if the leading terms of
the polynomials dominate at large field values, then the kinetic function is
$\propto h^{-2}$ and the qualitative behaviour is the same as in usual Higgs
inflation, the new terms just effectively change the value of the non-minimal
coupling $\xi$. If the leading terms of either the numerator or the
denominator are zero, we can get a variety of effective inflationary
potentials for the canonically normalised scalar field $\chi$, such as
$U\propto 1-\alpha\chi^{2}$, $U\propto 1-\alpha\chi^{-2/3}$,
$U\propto\chi^{2}$, $U\propto\chi^{4/3}$, an $\alpha$-attractor and a
potential with an inflection point. Some of these can give inflation in
agreement with the observations even if the non-minimal coupling is zero. They
can produce a range of values for the tensor-to-scalar ratio $r$, from too
small to be detected in near-future experiments to values in excess of the
current observational limits. This shows that Higgs inflation in the Palatini
formulation can produce a large $r$, which has not been obtained earlier with
quantum corrections nor with the inclusion of a $R^{2}$ term in the action
[140, 59, 60, 63, 68]. However, we did not check the constraint on the total
number of e-folds nor investigate the full parameter space, which would
require a numerical study.

Getting leading terms of the polynomials to vanish requires fine-tuning
(absent a symmetry that would relate the coefficients in the action to each
other), just like obtaining a critical point [44, 51, 50, 52, 53, 56, 58, 57,
59, 65, 66, 67, 68], hilltop [56, 59, 60] or a degenerate vacuum [61, 62]
using quantum corrections. Tuning of the classical terms is also presumably
unstable to quantum corrections. The new terms could contribute to addressing
the unitarity issue of Higgs inflation [84, 38, 85, 86, 42, 87, 48, 88, 89,
90, 91, 92, 93, 77], given that the non-minimal coupling can be smaller than
in the usual case or even zero. They could also change renormalisation group
running and affect reheating.

Extension of the original tree-level Higgs inflation proposal has been widely
discussed from the point of view of quantum corrections [34, 35, 38, 41, 36,
37, 39, 40, 42, 43, 44, 45, 46, 47, 48, 51, 52, 53, 54, 55, 56, 49, 57, 58,
50, 59, 60, 64, 63, 65, 66, 67, 68, 62, 84, 85, 87, 88, 89, 90, 91, 92, 93,
86, 172, 173, 174, 175, 176, 154, 155, 177, 178, 179, 180], including the
generation of higher order terms in the Riemann tensor [69, 70, 66, 71, 72,
73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83].

The inclusion of kinetic terms for the metric extends the study on the
parallel track of the choice of gravitational degrees of freedom. Earlier work
has looked at the differences between the metric and the Palatini formulation
for the same action [140, 86, 59, 60, 63, 68], whereas we have analysed the
effect of terms that arise in the Palatini case and have no equivalent in the
metric case. It would be interesting to extend the frame-covariant formalism
developed to deal with conformal transformations of the metric [142] and
partially translations of the connection [153, 156, 154, 155, 157, 158, 159]
to fully cover the transformations in the space of the scalar field, the
metric and the connection, and make more transparent the frame-dependent role
of the non-metricity and the connection. This connects to formulating the
quantum theory in terms of frame-invariant variables to settle the issue of
possible non-commutation of quantisation and frame transformations [88, 172,
173, 91, 174, 175, 56, 176, 154, 155, 37, 39, 42, 44, 47, 177, 89, 178, 179,
180].

From the effective theory point of view, if gravity is described by the
Palatini formulation, the terms we have considered are required for not just
for Higgs inflation, but for any scalar field model of inflation, and (like
$R^{2}$ terms [80, 81, 83]) can significantly modify inflationary predictions.

## Acknowledgements

I thank Daniel Figueroa and Lorenzo Sorbo for helpful correspondence, Lumi-
Pyry Wahlman for useful discussions and the Institute for Theoretical Physics
at KU Leuven for hospitality.

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