# Critical point Higgs inflation in the Palatini formulation

Vera-Maria Enckell  Sami Nurmi  Syksy Räsänen  and Eemeli Tomberg

###### Abstract

We study Higgs inflation in the Palatini formulation with the renormalisation
group improved potential in the case when loop corrections generate a feature
similar to an inflection point. Assuming that there is a threshold correction
for the Higgs quartic coupling $\lambda$ and the top Yukawa coupling $y_{t}$,
we scan the three-dimensional parameter space formed by the two jumps and the
non-minimal coupling $\xi$.

The spectral index $n_{s}$ can take any value in the observationally allowed
range. The lower limit for the running is $\alpha_{s}>-3.5\times 10^{-3}$, and
$\alpha_{s}$ can be as large as the observational upper limit. Running of the
running is small. The tensor-to-scalar ratio is $2.2\times 10^{-17}<r<2\times
10^{-5}$. We find that slow-roll can be violated near the feature, and a
possible period of ultra-slow-roll contributes to the widening of the range of
CMB predictions. Nevertheless, for the simplest tree-level action, the
Palatini formulation remains distinguishable from the metric formulation even
when quantum corrections are taken into account, because of the small tensor-
to-scalar ratio.

HIP-2020-33/TH

##  1 Introduction

#### Higgs inflation and its complications.

Higgs inflation is a conservative model that uses only the degrees of freedom
in the Standard Model (SM) of particle physics and general relativity (GR) [1]
(for reviews, see [2, 3, 4]). When the Higgs is directly coupled to the Ricci
scalar, the effective Higgs potential is exponentially flat, giving (at tree-
level and in the metric formulation of GR) a unique prediction for the
spectral index and tensor-to-scalar ratio that is in good agreement with
observations [5]. There are two complications. Quantum corrections can change
the potential [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41,
42, 43, 44, 45, 46, 47, 48, 49, 50], and the choice of the formulation of GR
can modify the relation between the particle physics potential and the
evolution of the inflaton and spacetime [51, 52].

Quantum corrections affect the mapping between the electroweak (EW) scale
parameters and the inflationary observables on the plateau. Part of this is
captured by the renormalisation group (RG) running of the Higgs quartic
coupling $\lambda$ on the inflationary plateau, where its value can be quite
different from the EW scale value. In most of the parameter space, quantum
corrections to the effective Higgs potential are not expected to generate
significant changes to the inflationary dynamics [29]. Nevertheless, in tuned
regions of parameter space where the value of $\lambda$ is of the same order
as its running, loop corrections can open new inflationary regimes, like
inflection point inflation [18, 24, 25, 39, 40, 29, 41, 42, 43, 44, 45, 46,
47] (also called critical point inflation in the case when the potential does
not necessarily have an exact inflection point), hilltop inflation [29, 43,
48] and hillclimbing inflation [53, 54].

On the gravity side, there are several different formulations of GR [55, 56,
57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73], which are
equivalent for the minimal gravity action and minimally coupled matter, but
differ for more complicated actions. The Higgs non-minimal coupling that is
central to Higgs inflation breaks the equivalence between these formulations
of GR and makes them physically distinct. The most studied alternative to the
metric formulation in the context of Higgs inflation is the Palatini
formulation [55, 71], where the metric and the connection are independent
degrees of freedom [51, 74, 43, 48, 75, 47, 76, 77, 78, 79, 49, 80, 81, 82,
83, 84].

An interesting motivation for the Palatini formulation is provided by the
unitarity problem. It is a weak point of Higgs inflation that the non-minimal
coupling spoils the SM cancellation between the EW gauge bosons and the Higgs
in the scattering amplitudes. This leads to violation of tree-level unitarity
around the EW vacuum at the scale $M_{{}_{\mathrm{Pl}}}/\xi$ in the metric
formulation of GR [10, 85, 86, 87, 88, 16, 89, 90, 21, 91, 35, 92, 26, 93, 94,
95, 96, 97]. As the inflationary scale is $M_{{}_{\mathrm{Pl}}}/\sqrt{\xi}$
and $\xi\gg 1$, this seems to imply that there is a region between the EW
vacuum and the inflationary regime where the model is not a consistent quantum
theory at the perturbative level. However, the unitarity cutoff is field-
dependent [16, 89, 92, 93, 98, 42]. When expanded around the inflationary
plateau, the cutoff is above the inflationary scale indicating that dynamics
during inflation can be studied by treating the model as a perturbative
effective theory (high-momentum modes produced during preheating could probe
the regime above the cutoff [99, 100, 101, 102]).

At the inflationary scale, the theory reduces to the chiral SM, which is under
control due to approximate shift symmetry on the inflationary plateau.
However, the behaviour between the EW scale and the inflationary scale depends
on the physics that is assumed to restore unitarity (either perturbatively,
non-perturbatively or with new degrees of freedom). The unknown threshold
corrections due to this physics can be phenomenologically modelled by adding
arbitrary jumps to the SM couplings at the unitarity violation scale [24, 39].
In the Palatini formulation, violation of perturbative unitarity and inflation
both happen around the scale $M_{{}_{\mathrm{Pl}}}/\sqrt{\xi}$, so the theory
is more reliable than in the metric formulation [74, 103, 49]. The theory is
still not guaranteed to be under control due to the vicinity of the unitarity
violation scale, although in general the scale of new physics can be higher
than the tree-level perturbative unitarity violation scale, see e.g. [104].

Higher order curvature terms can also have a large effect on the inflationary
dynamics and restore perturbative unitarity, as well as change the RG running
[105, 106, 45, 107, 108, 109, 110, 111, 112, 113, 96, 114, 115, 116, 117, 118,
119, 120, 121, 122, 102, 121, 123, 124]. In the Palatini case not all such
terms introduce new degrees of freedom, so they have a less drastic impact on
both the classical potential and loop corrections, meaning the formulation is
more robust to corrections [125]. Alternative formulations of GR introduce the
possibility of new non-minimal couplings between the Higgs and the connection,
which can qualitatively change the potential [76, 52, 82, 83]. (In addition
the Higgs kinetic terms can be non-minimally coupled, also in the metric case
[126, 127, 128, 129, 130, 131, 94, 95, 132, 133, 134, 50, 84].) We consider
only the simplest Higgs inflation action, where the Higgs doublet has a non-
minimal coupling $\xi$ to the Ricci scalar.

The SM and chiral SM RG equations augmented with threshold jumps at the
unitarity breaking scale have been applied to Higgs inflation in the hilltop
case [48] in both the metric and the Palatini formulation, and to features
similar to an inflection point in the metric formulation [41]. (See also [50]
for the New Higgs inflation case.) We extend these studies to the case when
the RG running creates an inflection point or a similar feature, often
generically called a critical point, in the Palatini formulation. We study the
effect of the choice of renormalisation scale. We check whether the slow-roll
approximation is valid in the vicinity of the feature, or whether the field
enters ultra-slow-roll.

In section 2 we set up the potential, explain how we treat the RG running,
scan the parameter space and find the predicted range of the inflationary
observables. In section 3 we discuss our results and compare to previous work,
and in section 4 we summarise our findings.

##  2 Critical point inflation

###  2.1 Classical potential

The Lagrangian of the SM coupled non-minimally to the Ricci scalar is (not
writing explicitly the non-radial part of the Higgs, which enters only via its
effect on RG running)

$S=\int d^{4}x\sqrt{-g}\quantity[\frac{1}{2}\left(M^{2}+\xi
h^{2}\right)g^{\alpha\beta}R_{\alpha\beta}(\Gamma,\partial\Gamma)-\frac{1}{2}g^{\alpha\beta}\partial_{\alpha}h\partial_{\beta}h-V(h)+\mathcal{L}_{\textrm{SM}}]\
,$ (2.1)

where $g_{\alpha\beta}$ is the metric, $R_{\alpha\beta}$ is the Ricci tensor,
$M$ is a mass scale, $h$ is the radial Higgs field, $\xi$ is the non-minimal
coupling, $V(h)=\frac{\lambda}{4}(h^{2}-v^{2})^{2}$, where $v$ is the Higgs EW
vacuum expectation value, and $\mathcal{L}_{\textrm{SM}}$ contains the rest of
the SM. The Ricci tensor is built from the connection
$\Gamma^{\gamma}_{\alpha\beta}$, and as we consider the Palatini formulation,
it is independent of the metric.

We make a conformal transformation to the Einstein frame and define the new
scalar field $\chi$ with minimal coupling to gravity and canonical kinetic
term [1, 51]:

$g_{\alpha\beta}\rightarrow(1+\xi h^{2})^{-1}g_{\alpha\beta}\
,\qquad\frac{\mathrm{d}\chi}{\mathrm{d}h}=\frac{1}{\sqrt{1+\xi h^{2}}}\ ,$
(2.2)

where we have set $M=M_{{}_{\mathrm{Pl}}}=1$ (see [76] for discussion of this
choice). We therefore have, fixing $\chi(h=0)=0$,

$\displaystyle h$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{\xi}}\sinh(\sqrt{\xi}\chi)\ .$ (2.3)

The SM Higgs potential in the original Jordan frame is

$\displaystyle V(h)$ $\displaystyle=$
$\displaystyle\frac{\lambda}{4}(h^{2}-v^{2})^{4}\approx\frac{\lambda}{4}h^{4}\
,$ (2.4)

where we have taken into account that in the inflationary region $h\gg v$. In
the Einstein frame, the potential is (again neglecting $v$)

$\displaystyle U[h(\chi)]$ $\displaystyle=$ $\displaystyle\frac{V(h)}{(1+\xi
h^{2})^{2}}=\frac{\lambda}{4}\frac{h^{4}}{(1+\xi h^{2})^{2}}$ (2.5)
$\displaystyle=$
$\displaystyle\frac{\lambda}{4\xi^{2}}\tanh^{4}(\sqrt{\xi}\chi)\equiv\frac{\lambda}{4}F(\chi)^{4}\
,$

where we have defined
$F(\chi)\equiv\frac{1}{\sqrt{\xi}}\tanh(\sqrt{\xi}\chi)$. It is convenient to
define the quantity [29]

$\delta\equiv\frac{1}{\xi h^{2}}\ ,$ (2.6)

which is small on the inflationary plateau.

###  2.2 Quantum corrections

We take into account quantum corrections to the potential in the same way as
in [48], although our choice of renormalisation scale is different. For
$\delta\ll 1$ the theory is approximated by the chiral SM, and the one-loop
correction to the Einstein frame effective potential in the $\overline{MS}$
scheme takes the form [14]

$U_{\mathrm{1-loop}}=\frac{6m_{W}^{4}}{64\pi^{2}}\left(\ln\frac{m_{W}^{2}}{\mu^{2}}-\frac{5}{6}\right)+\frac{3m_{Z}^{4}}{64\pi^{2}}\left(\ln\frac{m_{Z}^{2}}{\mu^{2}}-\frac{5}{6}\right)-\frac{3m_{t}^{4}}{16\pi^{2}}\left(\ln\frac{m_{t}^{2}}{\mu^{2}}-\frac{3}{2}\right)\
.$ (2.7)

Here the squares of effective $W$, $Z$, and top quark masses are,
respectively,

$m_{W}^{2}=\frac{g^{2}F^{2}}{4}\ ,\qquad
m_{Z}^{2}=\frac{\quantity(g^{2}+g^{\prime 2})F^{2}}{4}\ ,\qquad
m_{t}^{2}=\frac{y_{t}^{2}F^{2}}{2}\ ,$ (2.8)

and other fermions are approximated to be massless. At one-loop order, the
relevant chiral SM beta functions are given by [14, 135]:

$\displaystyle 16\pi^{2}\beta_{\lambda}$ $\displaystyle=$
$\displaystyle-6y_{t}^{4}+\frac{3}{8}\quantity[2g^{4}+(g^{\prime
2}+g^{2})^{2}]\ ,\quad
16\pi^{2}\beta_{y_{t}}=y_{t}\quantity(-\frac{17}{12}g^{\prime
2}-\frac{3}{2}g^{2}-8g_{S}^{2}+3y_{t}^{2})\ ,$ $\displaystyle
16\pi^{2}\beta_{g}$ $\displaystyle=$ $\displaystyle-\frac{13}{4}g^{3}\ ,\quad
16\pi^{2}\beta_{g^{\prime}}=\frac{27}{4}g^{\prime 3}\ ,\quad
16\pi^{2}\beta_{g_{S}}=-7g_{S}^{3}\ .$ (2.9)

In $\beta_{\lambda}$ we have omitted terms proportional to $\lambda
g^{2},\lambda g^{\prime}{}^{2}$ and $\lambda y_{t}^{2}$ as they give
subleading corrections in the limit $\lambda\simeq 0$ which is relevant for
all cases with features. For the same reason, we neglect the running of the
non-minimal coupling $\xi$. Rigorously, the theory reduces to the chiral SM
only in the limit $\delta\rightarrow 0$; in the following we however assume
the above treatment gives a decent approximation for the potential for all
$\delta<1$.

The leading order renormalisation group improved effective potential for
$\delta<1$ is then

$U(\chi)=\frac{\lambda(\mu)}{4}F(\chi)^{4}+U_{\mathrm{1-loop}}(\chi,\mu)\leavevmode\nobreak\
,$ (2.10)

where $\lambda(\mu)$ denotes the solution of (2.2) and
$U_{\mathrm{1-loop}}(\chi,\mu)$ denotes (2.7) with all the couplings replaced
by the corresponding solutions computed from (2.2). The RG improved potential
(2.10) with the one-loop beta functions satisfies the Callan–Symanzik equation
$\mathrm{d}U/\mathrm{d}{\rm ln}\mu=0$ up to next-to-leading order terms (such
as $\frac{m_{i}^{4}\beta_{j}}{16\pi^{2}c_{j}}\ln\frac{m_{i}^{2}}{\mu^{2}}$
where $c_{i}$ denotes a coupling), which would be cancelled by higher order
loop corrections. To minimise the error from neglected higher order
contributions, and the spurious dependence on $\mu$, the RG scale $\mu$ should
be chosen to minimise the loop logarithms over the $\chi$ range of interest in
the analysis. To do this, we choose $\mu$ equal to the largest mass scale
involved in the loop corrections, following e.g. [39, 41],111This corresponds
to prescription I according to the convention of [13], and is the preferred
choice when computing the quantum corrections in the Einstein frame, as we do.

$\mu(\chi)=\frac{\gamma}{\sqrt{2}}F(\chi)\leavevmode\nobreak\ .$ (2.11)

Here $\gamma\approx 0.34$, with the precise value chosen on a case by case
basis so that $\sqrt{2}\mu$ is equal to the largest SM particle mass near the
threshold scale $\delta=1$. In [48, 47] the scale $\mu$ was instead chosen so
that the loop corrections vanish at the feature, whether it is a critical
point, a local minimum or a hilltop. In principle that choice could have the
drawback that, as different pieces of the loop correction have different
signs, there could be accidental partial cancellations of large logarithms at
the feature, leading to large corrections in its vicinity. We have checked how
the results of our analysis change when using the prescription of [48, 47],
i.e. setting the loop correction to zero at the feature, instead of (2.11).
The difference in CMB predictions is negligible, showing that the analysis is
robust against variation of the renormalisation scale. We have also varied the
value of $\gamma$ in our prescription by a factor of $10$ up and down, and the
difference is typically small, less than ten percent for the scalar
perturbation amplitude $A_{s}$ and tensor-to-scalar ratio $r$, and smaller for
the spectral index $n_{s}$. The differences are larger only on the edge of the
parameter space, in particular on the small $r$ boundary in figure 1 where the
potential is very flat and sensitive to small corrections. Changing $\gamma$
moves this boundary slightly.

To connect the $\delta<1$ potential (2.10) with low-energy physics, we match
the chiral SM and SM couplings at the threshold scale $\mu_{\rm thres}$ that
corresponds to $\delta=1$, using the prescription (2.11). As $\delta=1$
corresponds to the transition between the SM region and the asymptotically
flat plateau, this ensures that the chiral SM RG equations are used only on
the plateau.222As we have two scales in the RG corrected potential – the field
value and the renormalisation scale – other criteria can lead to a situation
where the SM RG equations extend up to the plateau or the chiral SM equations
extend down from the plateau.

For $\delta>1$, i.e. $\mu<\mu_{\rm thres}$, we run the couplings using the
three-loop SM beta functions, and the central values of the observed strong
coupling constant and particle masses [136, 137]

$\frac{g_{\textrm{S}}^{2}(m_{Z})}{4\pi}=0.1179\pm 0.0010\ ,\quad
m_{H}=125.10\pm 0.14\ \mathrm{GeV}\ ,\quad m_{t}=172.9\pm 0.4\pm 0.5\
\mathrm{GeV}\ ,$ (2.12)

where the second error in $m_{t}$ is an estimate of the systematic uncertainty
in matching theory and experiment. The running is done with the code [138],
with updated matching conditions between the observables and $\overline{MS}$
scheme parameters from [139] together with input pole mass values
$m_{W}=80.385$ GeV, $m_{Z}=91.1876$ GeV, and Fermi constant
$G_{F}=1.1663787\times 10^{-5}$ GeV$-2$.

At the threshold scale, $\mu=\mu_{\rm thres}$, we let the couplings $\lambda$
and $y_{t}$ jump by $\Delta\lambda$ and $\Delta y_{t}$ to parametrise the
unknown effects of physics around the scale where perturbative unitarity is
violated. For simplicity, we omit jumps in the gauge couplings and directly
match their SM and chiral SM values at the threshold. We use the freedom in
$\Delta\lambda$ and $\Delta y_{t}$ to create a feature in the quantum-
corrected potential at scale $\delta=\delta_{0}<1$. This feature is a critical
point, where the first and second derivatives of the potential are tuned to be
small. We restrict our analysis to monotonically growing potentials; for
potentials with a local maximum or minimum, see [48, 47].

###  2.3 Equations of motion and inflationary observables

The background equations of motion in the Einstein frame read

$3H^{2}=\frac{1}{2}\dot{\chi}^{2}+U(\chi)\
,\qquad\ddot{\chi}+3H\dot{\chi}+U^{\prime}(\chi)=0\ ,$ (2.13)

where $H$ is the Hubble parameter, dot indicates derivative with respect to
cosmic time $t$, and prime indicates derivative with respect to $\chi$. We
solve these equations numerically for the quantum-corrected potential,
starting in slow-roll far above the feature scale $\delta_{0}$.

The number of e-folds from the cosmic microwave background (CMB) pivot scale
$k_{*}=0.05$ Mpc-1 to the end of inflation is

$N=56-\frac{1}{4}\ln\frac{0.067}{r}\ ,$ (2.14)

where $r$ is the tensor-to-scalar ratio. This number depends on the duration
of reheating. For the tree-level Higgs potential in the Palatini formulation,
the reheating is practically instantaneous [78], and we have assumed that this
is the case also here. We calculate $N$ numerically without using the slow-
roll approximation.

To ensure approximate scale-invariance, we demand that the field is in slow-
roll at the pivot scale, rejecting potentials where this is not the case. The
CMB observables can then be expressed in terms of the slow-roll variables

$\epsilon\equiv\frac{1}{2}\quantity(\frac{U^{\prime}}{U})^{2}\
,\qquad\eta\equiv\frac{U^{\prime\prime}}{U}\
,\qquad\zeta\equiv\frac{U^{\prime}}{U}\frac{U^{\prime\prime\prime}}{U}\
,\qquad\varpi\equiv\quantity(\frac{U^{\prime}}{U})^{2}\frac{U^{\prime\prime\prime\prime}}{U}\
.$ (2.15)

The scalar amplitude, scalar spectral index, tensor-to-scalar ratio, running,
and running of the running are, respectively,

$\begin{split}A_{s}&=\frac{U}{24\pi^{2}\epsilon}\ ,\qquad\qquad\qquad\
n_{s}=1-6\epsilon+2\eta\ ,\qquad\qquad r=16\epsilon\ ,\\\
\alpha_{s}&=16\epsilon\eta-24\epsilon^{2}-2\zeta\
,\qquad\beta_{s}=-192\epsilon^{3}+192\epsilon^{2}\eta-32\epsilon\eta^{2}-24\epsilon\zeta+2\eta\zeta+2\varpi\
.\end{split}$ (2.16)

The measured values of the CMB observables at the pivot scale 0.05 Mpc-1 from
Planck and BICEP2/Keck data are [5]

$\begin{split}A_{s}&=2.099\times 10^{-9}\,,\qquad n_{s}=0.9625\pm
0.0048\,,\qquad r<0.067\,,\\\ \alpha_{s}&=0.002\pm
0.010\,,\qquad\beta_{s}=0.010\pm 0.013\,.\end{split}$ (2.17)

We neglect the small observational uncertainty in $A_{s}$. We quote error bars
as 68% ranges and upper and lower limits as 95% ranges. The limit on $r$
assumes zero running of the running.

###  2.4 Numerical scan

We have three free parameters: $\xi$, $\Delta\lambda$, and $\Delta y_{t}$. We
fix them by the three conditions that the amplitude of scalar perturbations
$A_{s}$ matches the observed value (2.17) at the pivot scale, and the first
two slow-roll parameters take the given values $\epsilon_{0}$ and $\eta_{0}$
at the given feature scale $\delta_{0}$. The feature conditions determine
$\lambda$ and $y_{t}$ at the feature scale through a set of equations that we
solve numerically. We then run the couplings back to the threshold scale
$\mu_{\mathrm{thres}}$ to determine the jumps $\Delta\lambda$ and $\Delta
y_{t}$. The coupling $\xi$ is determined iteratively to fix $A_{s}$ to its
observed value (2.17).

We scan over all possible potentials by varying $\epsilon_{0}$, $\eta_{0}$,
and $\delta_{0}$.333In principle, we have only two free parameters after $\xi$
has been traded for $A_{s}$, but numerically it is more convenient to vary
three. We could cover all of the same possibilities if we, say, kept
$\delta_{0}$ fixed and varied $\epsilon_{0}$ and $\eta_{0}$ in a wider range,
but such a scan would be less sensitive to strong features away from the point
$\delta_{0}$. The slow-roll parameter $\eta$ is mostly negative on the
inflationary plateau, but may be positive in a narrow range of field values
near a strong feature. Hence, we do the scan in two parts. We first consider
potentials for which $\eta>0$ at some point on the plateau, and then
potentials where $\eta<0$ everywhere. For every potential, we solve the
background evolution (2.13) numerically and calculate the CMB observables in
the slow-roll approximation (2.16).

If $\eta>0$ for some field value, then there exists a field value where it
crosses zero. We center on the zero-crossing point by setting $\eta_{0}=0$. We
then perform a two-dimensional scan over $\epsilon_{0}$ and $\delta_{0}$,
using $A_{s}$ to fix $\xi$. These potentials produce the strongest features
and largest deviations from the tree-level result.

If $\eta<0$ everywhere, we have $\eta_{0}<0$. We choose $\delta_{0}$ and
perform a two-dimensional scan over $\epsilon_{0}$ and $\eta_{0}$, demanding
$\eta<0$ at all points, and again fixing $\xi$ with $A_{s}$. We repeat this
for a number of $\delta_{0}$ values to ensure full coverage of the parameter
space.

In all scans, we extend the parameter range in all directions as far as
possible. The parameter space is limited at small $\delta_{0}$ and large
$\epsilon_{0}$ by the fact that no value of $\xi$ can give the correct
amplitude $A_{s}$. At large $\delta_{0}$, we demand that the feature is in the
plateau region, $\delta_{0}<1$. Further limitations are set by our requirement
that the potential is monotonic.

In figure 1, we show the resulting possible range for observables on the
$(n_{s},r)$-plane with $\alpha_{s}$ colour coded and various regions
highlighted. In the middle dark grey area, the second slow-roll parameter
$\eta$ is always negative (second part of our scan) and there is no strong
feature in the potential. Elsewhere, there is a feature where $\eta=0$ (first
part of our scan), either above (low $r$ region) or below (high $r$ region)
the CMB pivot scale.

The feature is strongest on the blue and yellow boundaries on the left. Along
the yellow line, the feature is close to the end of inflation, and we have USR
near it. The field also enters USR in the lower dark grey area, in this case
above the CMB pivot scale. In other regions, slow-roll applies everywhere. The
CMB region and the feature region are well-separated in most of the parameter
space, but start to overlap near the upper red line and the lower USR region.
The viable inflationary region is limited from below by the black boundary,
where the USR regime reaches the pivot scale and the spectrum ceases to be
nearly scale-invariant. From above, the viable region is limited by the red
boundary that marks the line where $\alpha_{s}=0.022$, the 2$\sigma$ limit
from (2.17). The dotted boundary lines correspond to the analytical
approximations (2.27) and (2.5), discussed in the next section. The wedge
around $\log_{10}r=-8.8$ is due to the fact that for points right above the
blue edge the potential is not monotonic; instead a local maximum and an
adjacent minimum are formed around the feature scale. If such a potential
supports successful inflation, it will be of the hilltop type, covered for the
Higgs case in [48].

Our range of $n_{s}$ is much wider than the observationally allowed range
(2.17). If we restrict $n_{s}$ to the observed 2$\sigma$ range, we get
$\alpha_{s}>-3.5\times 10^{-3}$, and the upper values reach the $2\sigma$
upper limit $\alpha_{s}=0.022$ at the large $r$ edge of figure 1. Demanding
both $n_{s}$ and $\alpha_{s}$ to be within the observed $2\sigma$ range, we
get $-7.2\times 10^{-5}<\beta_{s}<6.1\times 10^{-3}$ and $2.2\times
10^{-17}<r<2.0\times 10^{-5}$. For these points, the non-minimal coupling is
$1.1\times 10^{3}<\xi<1.1\times 10^{9}$. These values of the running of the
running and the tensor-to-scalar ratio are well within the observational
bounds (2.17). The value of $r$ can be larger than the typical tree-level
Palatini result, but remains clearly below the tree-level metric result
$r\approx 4\times 10^{-3}$ [1]. The minimum value of $r$ is three orders of
magnitude smaller than the smallest value in the Palatini tree-level case,
$r=8\times 10^{-14}$.

Figure 1: Allowed points on the $(n_{s},r)$-plane with different regions of
the parameter space marked (left) and values of $\alpha_{s}$ shown (right),
see explanation in the text. The allowed region continues to larger $n_{s}$
values on the right hand side and smaller $n_{s}$ values in the upper left
corner. The dashed central line indicates the tree-level predictions with
(2.14), (2.5) and (2.20), and requiring $\lambda<1$ for perturbativity. At the
point marked with a star, the potential does not have a feature and our
results merge with the tree-level results not only for $n_{s}$ and $r$ but
also for $\alpha_{s}$ and $\beta_{s}$; see discussion around equation (2.5).

###  2.5 Analytical approximations

Let us try to understand the range of numerical predictions analytically. At
tree level, the slow-roll approximation of section 2.3 gives [51]

$\displaystyle n_{s}$ $\displaystyle=$ $\displaystyle
1-\frac{2}{N_{\mathrm{sr}}}\approx 0.96\ ,\quad\quad r=\frac{2}{\xi
N^{2}_{\mathrm{sr}}}\approx 8\times 10^{-4}\ \xi^{-1}\ ,$
$\displaystyle\alpha_{s}$ $\displaystyle=$
$\displaystyle-\frac{2}{N^{2}_{\mathrm{sr}}}\approx 8\times 10^{-4}\
,\quad\beta_{s}=-\frac{4}{N^{3}_{\mathrm{sr}}}\approx 3\times 10^{-5}\
,\quad\xi=\frac{\lambda N^{2}_{\mathrm{sr}}}{12\pi^{2}A_{s}}\approx
10^{10}\lambda\ ,$ (2.18)

where we have approximated $N_{\mathrm{sr}}\approx 50$ at the pivot scale and
used the observed value (2.17) of $A_{s}$. Note that $r$ is suppressed by
large values of $\xi$ as $r\propto 1/\xi$. Here $N_{\mathrm{sr}}$ is given by
the slow-roll approximation (also approximating that the value of $\chi$ at
the end of inflation is small)

$N_{\mathrm{sr}}\equiv\int_{0}^{\chi}\frac{d\tilde{\chi}}{\sqrt{2\epsilon(\tilde{\chi})}}\
.$ (2.19)

Due to the inaccuracy of the slow-roll approximation during the last e-folds
of inflation, this deviates somewhat from the true number of e-folds $N$ in
(2.14); numerically [78],

$N\approx N_{\mathrm{sr}}+1.8\,.$ (2.20)

This correction is of the same order as the one due to the dependence of the
energy scale of inflation on $\xi$: the number $N=50$ corresponds to
$\xi=10^{9}$, and the number of e-folds varies as $\Delta
N=-\frac{1}{4}\ln(\xi/10^{9})$. At tree level, if we take
$\lambda=10^{-6}\ldots 1$, then (2.5) gives $\xi=10^{4}\ldots 10^{10}$, so
$\Delta N=-0.6\ldots 2.9$.

To obtain some analytical understanding of the RG improved potential, we
rewrite the effective potential (2.10) (with $\mu(\chi)$ determined by (2.11))
as

$U(\chi)=\frac{\lambda_{\mathrm{eff}}(\chi)}{4}F(\chi)^{4}\ ,$ (2.21)

where the effective coupling $\lambda_{\mathrm{eff}}$ includes both the effect
of the running of the coupling $\lambda$ as well as the loop correction terms.
The amplitude of the scalar perturbation in (2.16) gives

$A_{s}=\frac{\lambda_{\mathrm{eff}}}{6\pi^{2}\xi^{2}r}\ .$ (2.22)

We want to eliminate $\lambda_{\mathrm{eff}}$ to arrive at a relation between
$r$ and $\xi$. In the potentials that have a strong feature, the effective
coupling has to be of the same order as its running, which can be
parametrically estimated from one of the contributions in (2.7):

$\lambda_{\mathrm{eff}}\sim-\frac{d\lambda_{\mathrm{eff}}}{d\ln\mu}\sim-\frac{d}{d\ln\mu}\left(\frac{3g^{4}}{128\pi^{2}}\right)=-\frac{12g^{3}\beta_{g}}{128\pi^{2}}=\frac{39g^{6}}{2048\pi^{4}}\sim
10^{-6}\ ,$ (2.23)

where we used the beta function (2.2) and set $g\approx 0.5$, a typical value
at the inflationary scale according to our numerical results. Indeed, the
numerics give $\lambda_{\mathrm{eff}}\approx 3.1\times 10^{-6}$ on average,
with only small variation around this. Combining this with equation (2.22)
gives

$r\approx\frac{25}{\xi^{2}}\ .$ (2.24)

This is accurate for most of the parameter space; in some regions the
prefactor varies between 5 and 31. The stronger suppression $r\sim 10/\xi^{2}$
explains the small values of $r$ compared to the tree-level case, where $r\sim
10^{-3}/\xi$.

Substituting (2.24) into the tree-level results (2.5) and using the adjusted
value of $N$ (2.20), we find the only point where the results of our numerical
analysis with a strong feature coincide with the predictions of a tree-level
potential for all CMB observables:

$\displaystyle n_{s}$ $\displaystyle\approx$ $\displaystyle 0.96\
,\qquad\qquad\qquad r\approx 2.5\times 10^{-8}\ ,\quad\quad\ \ N\approx 52\ ,$
$\displaystyle\alpha_{s}$ $\displaystyle\approx$ $\displaystyle-7.8\times
10^{-4}\ ,\qquad\beta_{s}\approx-3.1\times 10^{-5}\ ,\qquad\xi\approx
3.2\times 10^{4}\ .$ (2.25)

This model is marked with a star in figure 1. The other tree-level results
with a different value of $\xi$, marked with a dashed line, are of course
still possible when the quantum corrections are included, but our numerical
analysis, which is restricted to the fine-tuned cases with strong features,
does not see them.

Armed with the relation (2.24), we can make further analytical approximations
on the edges of the parameter space shown in figure 1. This is true
particularly along the yellow and blue curves, which correspond to a strong
feature with $\epsilon\approx\eta\approx 0$ at the critical point.

On the yellow line, the critical point is below the CMB pivot scale and the
slow-roll approximation fails in its vicinity. The field enters a regime of
ultra-slow-roll (USR) [140, 141, 142, 143, 144]. Above the feature, the
evolution is similar to the tree-level case. The effect of the USR period on
the spectrum on the CMB scales is to reduce the number of e-folds spent on the
plateau compared to the tree-level case (2.5), so we have

$n_{s}=1-\frac{2}{N_{\mathrm{eff}}}\ ,\qquad r=\frac{2}{\xi
N^{2}_{\mathrm{eff}}}\ ,\qquad\alpha_{s}=-\frac{2}{N_{\mathrm{eff}}^{2}}\
,\qquad\alpha_{s}=-\frac{4}{N_{\mathrm{eff}}^{3}}\ ,$ (2.26)

where $N_{\mathrm{eff}}$ varies between zero and $50$. Combining this result
with (2.24) gives

$r=\quantity(\frac{1-n_{s}}{3.1})^{4}\
,\qquad\alpha_{s}=-\frac{1}{2}\quantity(1-n_{s})^{2}\
,\qquad\beta_{s}=-\frac{1}{2}\quantity(1-n_{s})^{3}\ .$ (2.27)

This approximation is marked with a dotted line in figure 1 and agrees well
with the yellow boundary given by the numerical scan.

On the blue line, the critical point is above the CMB pivot scale. We expand
the effective coupling $\lambda_{\mathrm{eff}}$ in the potential (2.21) up to
second order in its running, fixing the coefficients so that
$\epsilon\approx\eta\approx 0$ for $\delta=\delta_{0}$. This gives

$\lambda_{\mathrm{eff}}(\delta)=\lambda_{0}-4\lambda_{0}\ln\frac{F(\delta)}{F(\delta_{0})}+8\lambda_{0}\ln^{2}\frac{F(\delta)}{F(\delta_{0})}\
,$ (2.28)

where $F(\delta)=\frac{1}{\sqrt{\xi(1+\delta)}}$ as before. Expanding
everything to leading order in the small parameters $\delta$ and $\delta_{0}$,
we get in the slow-roll approximation

$\displaystyle N_{\mathrm{sr}}$ $\displaystyle=$
$\displaystyle\frac{\delta_{0}(2\delta-\delta_{0})+2\delta(\delta-\delta_{0})\ln\quantity(1-\frac{\delta_{0}}{\delta})}{16\xi\delta_{0}^{3}\delta(\delta-\delta_{0})}\,,$
$\displaystyle n_{s}$ $\displaystyle=$ $\displaystyle
1-32\xi\delta(\delta-\delta_{0})(3\delta-\delta_{0})\,,$ $\displaystyle r$
$\displaystyle=$ $\displaystyle 512\xi\delta^{2}(\delta-\delta_{0})^{4}\ .$
(2.29)

Combining this with the value (2.24) for $r$ and the approximation $N\approx
N_{\mathrm{sr}}+2.5$ (note that the optimal shift in $N$ is different from the
tree-level case (2.20)), the result, again marked with a dotted line, agrees
with the numerical results on the blue line in figure 1. This approximation
also works for $\alpha_{s}$ and $\beta_{s}$, but we omit the lengthy
expressions which contain powers up to six and nine in $\delta$ and
$\delta_{0}$.

This approximation helps us understand the lowest allowed $r$ values in figure
1. Varying $\delta$ for a fixed potential (fixed $\delta_{0}$ and $\xi$), the
maximum value of $\eta$ turns out to be

$\eta_{\mathrm{max}}=\frac{32}{243}(10+7\sqrt{7})\delta_{0}^{3}\xi\,,$ (2.30)

located somewhat above the CMB pivot scale. This value grows with $\xi$ for
small $r$ values, until it reaches unity at $\log_{10}r\approx-16.8$. Around
this scale, slow-roll is replaced by USR which for slightly smaller $r$
reaches the pivot scale and ruins the scale-invariance of the CMB spectrum.
This corresponds to the bottom black boundary in figure 1.

###  2.6 Varying the EW scale parameters

Figure 2: Values of the jumps $\Delta\lambda$ and $\Delta y_{t}$ that lead to
a potential with a feature but with the CMB observables within the 2$\sigma$
limits (2.17). The dashed white line corresponds to the central EW parameter
values (2.12). In the coloured regions the SM parameters vary within their
$1\sigma$, $2\sigma$, and $3\sigma$ limits.

In the above analysis we have kept the SM EW scale input parameters (2.12)
fixed to their central values. The running of the Higgs quartic coupling is
well known to be sensitive to changes in these parameters. This sensitivity
does not transfer to inflationary observables, because their variation is
degenerate with changes in the jumps $\Delta\lambda$ and $\Delta y_{t}$.
Still, varying the EW parameters does affect field evolution after inflation,
as well as the size of the jumps needed to get a potential with strong
features.

For central values of the EW parameters, the $\lambda$ jumps span the range
$\Delta\lambda=0.009\dots 0.012$, as shown in figure 2. The $y_{t}$ jumps are
negative, $\Delta y_{t}=-0.07\dots-0.03$, and bring $y_{t}$ down to about
$0.40$. Note that the values of the jumps depend on the choice of the
threshold scale and the renormalisation scale. For the central values, all
cases with features are such that $\lambda$ runs negative below the threshold
scale, and $\Delta\lambda>0$ raises it to a small but still negative value in
the inflationary regime (the added quantum corrections make the potential
positive).

When we vary the EW scale input parameters (2.12) within the 2$\sigma$ range,
the ranges of the jumps change to $\Delta\lambda=-0.003\dots 0.025$ and
$\Delta y_{t}=-0.085\dots-0.017$, as shown in figure 2. Changing the top quark
mass has the largest effect. The cases $\Delta\lambda<0$ in the figure are
such that $\lambda$ stays positive all the way to the threshold scale, and
they correspond to top mass values below the central value. The figure shows
that cases with $\Delta y_{t}$ = 0 do not occur within 3$\sigma$ limits of EW
parameters. In particular, this means that there are no viable inflationary
models with a feature and both jumps set to zero. For zero jumps, the model
either exhibits an instability at the inflationary scale (the effective
potential is negative) or gives the tree-level results (2.5).444In a previous
study [48], a viable hilltop scenario with zero jumps was found within the
2$\sigma$ range, but the spectral index $n_{s}=0.94$ is now in strong tension
with updated observations. The threshold and renormalisation scales were also
chosen differently there.

Close to the $\Delta\lambda=0$ line, there are cases where a local minimum is
formed at the threshold scale. (The minimum could also be formed above the
threshold scale, but we exclude such cases in our analysis.) It would be
interesting to investigate reheating in detail in such a setting. The effect
of a second minimum below the inflationary region depends on the detailed form
of the potential, as the field may roll fast enough to pass over the second
minimum, or the minimum may be uplifted by thermal corrections during
reheating [39, 40, 41].

##  3 Discussion

###  3.1 Comparison to previous work

In [43], Higgs inflation with a critical point was studied in both the metric
and the Palatini formulation with a simple analytic approximation for the
potential, similar to (2.28), but without the linear logarithmic term, and
assuming slow-roll. We consider a wider range of possibilities, and find a
larger range of values for the spectral index, in particular we can have
$n_{s}>0.96$. Our upper range for $r$ is $2\times 10^{-5}$, compared to
$7\times 10^{-5}$ in [43]. Our ranges for $\alpha_{s}$ and $\beta_{s}$ (when
conditioned the same way) agree to within a factor of about 2 with the results
of [43]. The maximum value of $\xi$ found there was $6\times 10^{6}$, so the
analysis missed the possible strong features for large values of $\xi$ up to
$1.1\times 10^{9}$ that we find.

If we compare to metric case Higgs inflation with a critical point studied
with the RG improved potential in [41], the spectral index in both cases
covers a much wider range than the observationally allowed window, and the
possible range of values for $\alpha_{s}$ is roughly the same. In contrast,
there is no overlap between the allowed $r$ values: in the Palatini case, $r$
is always smaller. This agrees with previous analyses showing that even when
quantum corrections are taken into account, the metric and the Palatini
formulation of Higgs inflation remain distinguishable [43, 48]. This is also
true when a $R^{2}$ term is included in the action [125, 116], although a full
analysis including both the $R^{2}$ term and quantum corrections has not been
done. (If new gravitational terms are included in the action in the Palatini
case, $r$ can take values all the way to the observational upper limit [76,
82].)

In [49] it was argued that the tree-level predictions of Higgs inflation in
the Palatini case are robust to loop corrections. The threshold scale is
roughly the same as ours. It was argued that new physics there is expected to
lead at most to a jump of $\Delta\lambda=6\times 10^{-4}$; the jump of $y_{t}$
was neglected. On this basis, it was argued that the tree-level predictions
for $n_{s}$ and $r$ are unchanged, and that the non-minimal coupling is
bounded as $1.0\times 10^{6}<\xi<6.8\times 10^{7}$, which implies $1.1\times
10^{-11}<r<0.8\times 10^{-9}$. For the central values of the EW parameters, we
find that $\Delta\lambda$ has to be at least $0.009$ in order to produce a
strong feature. This agrees with the conclusion of [49] that
$\Delta\lambda=6\times 10^{-4}$ is not enough to significantly change the
inflationary predictions. However, if we move away from the mean values, the
situation changes. As discussed in section 2.6 and shown in figure 2, we can
have a strong feature even when $\Delta\lambda=0$ (if $\Delta y_{t}=-0.03$)
when we allow the EW parameters to vary within their $2\sigma$ ranges.

We conclude that the predictions of Higgs inflation in the Palatini case are
not tightly fixed by the tree-level action in the presence of threshold
corrections. However, generating a strong feature requires the threshold and
loop corrections to be highly tuned, so in a typical case there are no
features and the tree-level results are not significantly modified.

###  3.2 Unitarity

We have used the RG improved effective potential, and chosen the
renormalisation scale to be a field-dependent function $\mu(\chi)$ that
minimises the loop corrections. This follows the procedure of particle physics
scattering analysis, where the renormalisation scale is chosen to minimise the
corrections at the experimental scattering energy. We have then defined the
threshold scale $\mu_{\rm thres}$ as the scale that corresponds to
$h\sim\chi\sim 1/\sqrt{\xi}$, the cutoff of the effective theory expanded
around the EW scale, and matched the chiral SM and SM couplings at this scale
up to jumps in $\lambda$ and $y_{t}$. This choice of the threshold scale is
customary and coincides with the energy scale where tree-level unitarity is
violated in scattering processes involving the Higgs doublet, the $W$ and $Z$
bosons (and, in the Jordan frame, the graviton), computed by expanding around
the EW vacuum [10, 74, 92, 93] (see also [85, 86, 87, 88, 16, 89, 90, 21, 91,
35, 26, 94, 95, 96, 97]). Let us comment on the role of such unitary violating
scatterings for Higgs inflation.

There are a few different scales involved: the renormalisation scale
$\mu(\chi)$, the effective masses (2.8), the field value $\chi$, the energy
scale $U(\chi)^{1/4}$ of the potential, and the Hubble scale $H$, which
determines the amplitude of the power spectrum of field perturbations. In the
Palatini tree-level case, the field value $\chi\sim 10/\sqrt{\xi}$ in the CMB
region is close to the unitarity violation scale $\sim 1/\sqrt{\xi}$. So is
the energy scale of the potential $U^{1/4}\sim\lambda^{1/4}/\sqrt{\xi}$; the
Hubble scale $H\sim\lambda^{1/2}/\xi$ is much smaller. In the tree-level case
the original Jordan frame Higgs field value $h\approx 20$ is orders of
magnitude above the unitarity cutoff, unlike in the metric case where $h\sim
10/\sqrt{\xi}$. In our numerical results, $\chi=(3\dots 8)/\sqrt{\xi}$ for the
points that agree with observations, corresponding to $h=(10\dots
1148)/\sqrt{\xi}$. Values $h>1$ occur only close to the point marked with the
star in figure 1; the largest value of $h$ is 7.5, smaller than in the tree-
level case.

None of these scales is directly related to the tree-level unitarity violation
energy obtained from scattering calculations. Such scattering processes are
not expected to play any role during inflation when all particle numbers are
exponentially diluted, and the relevant momenta and amplitudes of the field
perturbations are determined by the Hubble scale, which is far below
$1/\sqrt{\xi}$. So even if perturbative unitarity is violated for the
scattering processes, meaning they cannot be reliably calculated using the
theory in the inflationary regime, this does not necessarily imply that the
theory cannot be used to reliably calculate the processes that are relevant
for inflation. (The situation during preheating can be different if modes with
momenta above the scattering unitarity violation scale are produced [99, 100,
101, 102].) Nevertheless, going beyond the tree-level, the non-renormalisable
terms in the potential (2.5) become of order unity at $h\sim\chi\sim
1/\sqrt{\xi}$, meaning that the effective theory ceases to be predictive
because of the infinite hierarchy of unsuppressed loop corrections. So
regardless of the tree-level unitarity issue, there is reason to doubt the
validity of the theory close to this scale, and thus we add jumps to the
couplings as effective threshold corrections. In contrast, in the inflationary
regime the loop corrections are suppressed by the assumed quantum extension of
the classical shift symmetry.

###  3.3 Spectral distortions

In the upper edge of figure 1, the running of the spectral index $\alpha_{s}$
and the running of the running $\beta_{s}$ are large and positive, implying
growing amplitude of perturbations towards smaller scales and an enhanced
$\mu$-distortion signal, possibly detectable by next generation spectral
distortion experiments [145]. The relation between the $\mu$-distortion signal
and the running $\alpha_{s}$ in single field models was studied in [146, 147],
and extended to $\beta_{s}$ in [148]. These general results apply also to our
setup, apart from configurations close to the red line in figure 1, where
$\alpha_{s}=0.022$. For these points, the strong feature in the potential
leads to scale-dependence of $\mathcal{P}_{\mathcal{R}}(k)$ that is not fully
described by $\alpha_{s}$ and $\beta_{s}$. Using [149, 150, 151], and assuming
instant transition between the $\mu$ and $y$ distortion at redshift $z=5\times
10^{4}$ as in [146, 147, 148], we calculate the $\mu$-signal using with the
full numerical result for $\mathcal{P}_{\mathcal{R}}(k)$. On the edge where
$\alpha_{s}=0.022$, we find $\mu\approx 9\times 10^{-8}$ for the points where
$n_{s}$ has the central value (2.17). This should be detectable with future
surveys [145]. Using the expansion in terms of $n_{s}$, $\alpha_{s}$, and
$\beta_{s}$ would instead give $\mu\approx 6\times 10^{-8}$, so the violation
of this approximation enhances the signal by $50\%$.

##  4 Conclusions

#### The predictions of critical point Palatini Higgs inflation.

We have studied the range of predictions for Palatini Higgs inflation in the
case when loop corrections generate a critical point in the potential, a
feature similar to an inflection point. We have scanned over the parameter
space formed by the non-minimal coupling $\xi$ and jumps in the quartic Higgs
coupling $\lambda$ and top Yukawa coupling $y_{t}$.

The scan shows that the spectral index $n_{s}$ can take all values in the
observationally allowed range, and beyond. Requiring $n_{s}$ to be within the
observed $2\sigma$ range, its running is limited from below as
$\alpha_{s}>-3.5\times 10^{-3}$. Requiring also the running to be below the
observed $2\sigma$ upper limit, the running of the running is $-7.2\times
10^{-5}<\beta_{s}<6.1\times 10^{-3}$. The similarly conditioned tensor-to-
scalar ratio is small, $2.2\times 10^{-17}<r<2.0\times 10^{-5}$. The lower end
is among the smallest tensor-to-scalar ratios of any inflationary model
studied so far. The upper end is below the smallest value found in the metric
formulation, even when loop corrections are taken into account. Notably, the
Palatini tree-level relation $r=2/(\xi N^{2})$ is replaced by the relation
$r\sim 10/\xi^{2}$ when we have a strong feature. The value of the non-minimal
coupling is $1.1\times 10^{3}<\xi<1.1\times 10^{9}$, so it can be smaller than
in the tree-level metric case where $\xi\approx 10^{4}$, but not quite as
small as the values $\xi\sim 400$ found in the critical point metric case [41]
or $\xi=180$ in the hilltop metric case [48].

In part of the parameter space the feature leads to a period of ultra-slow-
roll. This can’t happen too close to the CMB pivot scale, since it would lead
to too large deviations from scale-invariance. If ultra-slow-roll occurs below
the pivot scale, it reduces the number of e-folds spent on the plateau. CMB
predictions are then given by the same equations as in the tree-level case,
but with a smaller number of e-folds inserted into the equations. If ultra-
slow-roll happens above the pivot scale, it is possible for the pivot scale to
lie at a very flat point near the feature, leading to the extremely small $r$
values.

We have also studied the jumps in the couplings required to produce such
strong features. For the measured central values of the SM EW scale
parameters, we need $\Delta\lambda=0.009\dots 0.012$ and $\Delta
y_{t}=-0.07\dots-0.03$ for a feature to form. If we allow the SM parameters to
vary within $2\sigma$, we can have a strong feature even with a continuous
$\lambda$, but the jump of $y_{t}$ must be non-zero.

The CMB predictions of Higgs inflation can also be modified in other ways,
e.g. by adding an $R^{2}$ term into the action. Similar to our results, this
also widens the allowed parameter space [109, 118, 116]. However, in the case
of loop corrections, most of the predictions are accessed only with highly
tuned choices of parameters, unlike in Higgs-$R^{2}$ inflation.

We conclude that loop and threshold corrections can generate a feature with
interesting phenomenology and significant observational effects in the
Palatini formulation. Nevertheless, despite the dispersion of predictions both
in the metric and in the Palatini case, for the simplest tree-level Higgs
inflation action, the two formulations remain distinguishable due to lack of
overlap in the predicted values of the tensor-to-scalar ratio.

## Acknowledgments

This work was supported by the Estonian Research Council grants PRG803 and
MOBTT5 and by the EU through the European Regional Development Fund CoE
program TK133 “The Dark Side of the Universe”.

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