$\displaystyle\partial_{y}\tilde{N}(k)=$
$\displaystyle\left(I-\tilde{K}\right)^{-1}(\partial_{y}\tilde{N}_{0}+(\partial_{y}\tilde{K})\tilde{N})(k),$

where

$\displaystyle\langle
y\rangle^{j}\|\partial_{y}\tilde{N}_{0}+(\partial_{y}\tilde{K})\tilde{N}\|_{L^{2}}\mathchar
13366\relax C(\|R\|_{Y}),$

for $j=0,1$. Therefore, same as the above estimates, the following norm of the
variable $k$ is bounded with

$\displaystyle\langle
y\rangle^{j}(\parallel\partial_{y}\tilde{\mu}-(\partial_{y}\eta,0)\parallel_{L^{2}}+\sum_{j=1}^{{N_{0}}}\big{|}\partial_{y}\tilde{\mu}_{1}(k_{j})\partial_{y}\eta\big{|}+\sum_{j=1}^{{N_{0}}}\big{|}\partial_{y}\tilde{\mu}_{2}(\bar{k}_{j})\big{|})\mathchar
13366\relax C(\|R\|_{Y}).$ (5.87)

And for $y\in\mathbb{R}^{-}$, $M^{(2)}_{r}$ admits the properties same as
$M^{(2)}_{l}$.

Because the estimate in the above section is under the scaling transformation
of $y$, the equivalency between the integral norm in $\mathbb{R}_{x}$ and
$\mathbb{R}_{y}$ is first given to arrive at the $L^{2}$-estimate with respect
to the variable $x$.

######  Lemma 5.4.

For any function $h(x(\cdot))\in L^{p}(\mathbb{R}_{y})$, $1\mathchar
13366\relax p\mathchar 13366\relax\infty$, there appears that $h\in
L^{p}(\mathbb{R}_{x})$.

###### Proof.

When $p=\infty$, the result is obvious. On the other hand, it follows from
(2.128) that $dx=\left(1+2\frac{\partial_{y}\alpha_{1}}{\alpha_{1}}\right)dy.$
For $1\mathchar 13366\relax p<\infty$, from Lemma 4.2 it is ascertained that

$\displaystyle y+2-2\|1/\alpha_{1}\|_{L^{\infty}}\mathchar 13366\relax
x\mathchar 13366\relax y+2\|\alpha_{1}\|_{L^{\infty}}.$

Therefore,

$\displaystyle\int_{\mathbb{R}}|h(x)|dx\mathchar
13366\relax\int_{\mathbb{R}}|h(x(y))|\left(1+\Big{|}2\frac{\frac{\partial}{\partial
y}\alpha_{1}}{\alpha_{1}}\Big{|}\right)dy<\infty.$ (5.88)

∎

We now establish the $L^{2}$-estimate on the solution $m$ by using the
reconstruction formula in (5.57).

######  Lemma 5.5.

Assume that $m_{0}(x)>0,\forall x\in\mathbb{R}$ such that $m_{0}-1\in
H^{2,1}(\mathbb{R})\cap H^{1,2}(\mathbb{R})$. Then for every $T>0,t\in[0,T)$,
we have the following estimates

$\displaystyle\parallel m_{x}\parallel_{L^{2,2}(\mathbb{R})},\ \parallel
m_{xx}\parallel_{L^{2,1}(\mathbb{R})},\ \parallel
m-1\parallel_{L^{2,2}(\mathbb{R})}\mathchar 13366\relax C(T,\|R(0)\|_{Y}).$
(5.89)

###### Proof.

It is known from Lemma 5.4 that the estimates in (5.89) are equivalent to
those of integral norms on the variable $y$. Applying Lemma 2.1 and Lemma 3.1
gives that $R(t)\in Y$ for every $t\in[0,T)$. Then from the RH problem 4 and
RH problem 5, their solutions $M^{(2)}_{l}(z;t,y)$ on $y\in\mathbb{R}^{+}$ and
$M^{(2)}_{r}(z;t,y)$ on $y\in\mathbb{R}^{-}$ exist respectively. Resembling
the procedure in Section 3.3, the reconstruction formula (5.57) is also
practicable for every $t\in[0,T)$.

The first step is to prove $m_{x},\ m_{xx},\ b_{+}\in L^{2,2}(\mathbb{R})$. We
only give the details of the case $L^{2,2}(\mathbb{R}^{+}),$ since the case
$y\in\mathbb{R}^{-}$ can be shown in a similar way. For $y\in\mathbb{R}^{+}$,
by using Lemma 5.2 and (5.57), it holds that

$\displaystyle m_{x}$
$\displaystyle=\lim_{k\to\infty}k(M_{l,1}^{(2)}-(\eta-2))$
$\displaystyle=\frac{1}{2\pi
i}\int_{\mathbb{R}}(\tilde{\mu}_{2}(s)-1)r_{2}(s)e^{2\theta}ds+\frac{1}{2\pi
i}\int_{\mathbb{R}}r_{2}(s)e^{2\theta}ds+\sum_{j=1}^{{N_{0}}}\tilde{d}_{j}e^{y\text{Im}z_{j}}\tilde{\mu}_{2}(\bar{k}_{j}).$
(5.90)

For the first integral, it is inferred from (5.69) that

$\displaystyle\int_{\mathbb{R}}(\tilde{\mu}_{2}(s)-1)r_{2}(s)e^{2\theta}ds=\int_{\mathbb{R}}r_{2}(s)e^{isy/2}\mathcal{C}_{+}(-r_{1}e^{iky/2})(s)ds$
$\displaystyle+\int_{\mathbb{R}}r_{2}(s)e^{isy/2}\mathcal{C}_{+}((\tilde{\mu}_{2}(s;y)-1)r_{1}e^{-iky/2})(s)ds+\int_{\mathbb{R}}\frac{d_{j}e^{y\text{Im}z_{j}}}{s-k_{j}}\tilde{\mu}_{1}(k_{j})r_{2}(k)e^{2\theta}ds$
$\displaystyle\mathchar 13356\relax I_{1}(y)+I_{2}(y)+I_{3}(y).$

Note that

$\displaystyle|I_{3}|\mathchar
13366\relax|d_{j}|e^{y\text{Im}z_{j}}|\tilde{\mu}_{1}(k_{j})|\int_{\mathbb{R}}\frac{|r_{2}(k)|}{\sqrt{s^{2}+\text{Im}k_{j}^{2}}}ds\mathchar
13366\relax C(T,\|R(0)\|_{Y})e^{y\text{Im}z_{j}},$ (5.91)

which leads to $I_{3}(\cdot)\in L^{2,2}(\mathbb{R}^{+})$ immediately. It also
implies from Lemma 5.3 that $I_{1}(\cdot)\in L^{2,2}(\mathbb{R}^{+})$. From
Lemma 5.3, Lemma 5.1 and (5.86), applying the Hölder inequality yields

$\displaystyle\Big{|}\langle y\rangle^{4}I_{2}(y)\Big{|}$
$\displaystyle=\Big{|}\langle
y\rangle^{2}\int_{\mathbb{R}}(\tilde{\mu}_{2}(s;y)-1)r_{1}(s)e^{isy/2}\mathcal{C}_{+}(r_{2}e^{-iky/2})(s)ds\Big{|}$
$\displaystyle\mathchar 13366\relax\parallel\langle
y\rangle^{2}(\tilde{\mu}_{2}(\cdot;y)-1)r_{1}(\cdot)\parallel_{L^{2}}\Big{\|}\langle
y\rangle^{2}\mathcal{C}_{+}(r_{2}e^{-i\cdot
y/2})(\cdot)\Big{\|}_{L^{2}}\mathchar 13366\relax C(T,\|R(0)\|_{Y}),$

which in turn implies that $I_{2}(\cdot)\in L^{2,2}(\mathbb{R}^{+})$. The
second integral in (5.90) can be controlled by $\|r\|_{H^{1,2}\cap H^{2,1}}$
with the Fourier transformation and the third integral is bounded. It is then
concluded that $m_{x}\in L^{2,2}(\mathbb{R}^{+})$ and

$\displaystyle\|m_{x}\|_{L^{2,2}(\mathbb{R}^{+})}\leq C(T,\|R(0)\|_{Y}).$

Taking $y$-derivative of the first reconstruction formula (5.57) leads to

$\displaystyle\partial_{y}\lim_{k\to\infty}k(M_{l,1}^{(2)}-(\eta-2))=-\frac{1}{4\pi}\int_{\mathbb{R}}s(\tilde{\mu}_{2}(s)-1)r_{2}(s)e^{2\theta}ds+\frac{1}{2\pi
i}\int_{\mathbb{R}}\partial_{y}\tilde{\mu}_{2}(s)r_{2}(s)e^{2\theta}ds$
$\displaystyle-\frac{1}{4\pi}\int_{\mathbb{R}}sr_{2}(s)e^{2\theta}ds+\sum_{j=1}^{{N_{0}}}\tilde{d}_{j}e^{y\text{Im}z_{j}}\left(\text{Im}z_{j}\tilde{\mu}_{2}(\bar{k}_{j})+\partial_{y}\tilde{\mu}_{2}(\bar{k}_{j})\right).$

Similar to the above calculation, it appears from $kr_{2}\in
W^{1,2}(\mathbb{R})$ that

$\displaystyle\Big{|}\langle
y\rangle^{2}\partial_{y}\lim_{k\to\infty}k(M_{l,1}^{(2)}-(\eta-2))\Big{|}\mathchar
13366\relax C(T,\|R(0)\|_{Y}).$ (5.92)

Then from

$\displaystyle
m_{xx}=\alpha_{1}^{-1}(\alpha_{1}+2\partial_{y}\alpha_{1})\partial_{y}m_{x},$

it is found that $m_{xx}\in L^{2,1}(\mathbb{R}^{+})$ with

$\displaystyle\|m_{xx}\|_{L^{2,2}(\mathbb{R}^{+})}\leq C(T,\|R(0)\|_{Y}).$

On account of the reconstruction formula (5.57) and Lemma 5.2, it is deduced
that

$\displaystyle b_{+}=$ $\displaystyle-\frac{1}{2\pi
i}\int_{\mathbb{R}}(\tilde{\mu}_{1}(k)-(\eta-2))r_{1}(k)e^{-2\theta}ds$
$\displaystyle-\frac{1}{2\pi
i}\int_{\mathbb{R}}(\eta-2)r_{1}(k)e^{-2\theta}ds+\sum_{j=1}^{{N_{0}}}d_{j}e^{y\text{Im}z_{j}}\tilde{\mu}_{1}(k_{j}).$
(5.93)

The estimate of $|b_{+}|\mathchar 13366\relax C(T,\|R(0)\|_{Y})$ is obviously
a result from Lemmas 5.1 and 5.2. This thus implies that $b_{+}$ is uniformly
bounded with respect to $y$. For its $L^{2}$-integrability, similar to the
above step, combining the jump condition and the integral equation of
$M^{(2)}$, it is deduced that

$\displaystyle\parallel b_{+}\parallel_{L^{2,2}(\mathbb{R}^{+})}\mathchar
13366\relax C(T,\|R(0)\|_{Y}).$ (5.94)

The next step is to obtain $m-1\in L^{2,2}.$ Since

$\displaystyle\partial_{x}b_{+}=\frac{m^{2}-1}{m},\ \quad{\rm
and}\quad\partial_{x}^{2}b_{+}=\frac{m^{2}+1}{m^{2}}m_{x}.$

Together with the boundedness of $\eta=1-\frac{1}{m}$ shown in Lemma 4.2 and
boundedness of $m$ shown in Lemma 4.1, it is accomplished that
$\partial_{x}^{2}b_{+}\in L^{2}(\mathbb{R})$. Then applying the Sobolev-
Gagliardo-Nirenberg inequality yields $\frac{\partial}{\partial x}b_{+}\in
L^{2}(\mathbb{R})$, which means $m-1\in L^{2}(\mathbb{R})$. To arrive at the
weighted integrable property, one needs to consider the $x$-derivative for the
function $xb_{+}$,

$\displaystyle\partial_{x}(xb_{+})=x\frac{m^{2}-1}{m}+b_{+},\
\partial_{x}^{2}(xb_{+})=2\frac{m^{2}-1}{m}+x\frac{m^{2}+1}{m^{2}}m_{x}.$

Then $xb_{+}$, $\partial_{x}^{2}(xb_{+})\in L^{2}(\mathbb{R})$. Analogously,
by the Sobolev-Gagliardo-Nirenberg inequality, it is ascertained that
$\partial_{x}(xb_{+})\in L^{2}(\mathbb{R})$, which implies
$x\frac{m^{2}-1}{m}\in L^{2}(\mathbb{R})$. Again we analyze $x^{2}b_{+}$,

$\displaystyle\partial_{x}(x^{2}b_{+})=x^{2}\frac{m^{2}-1}{m}+2xb_{+},\
\partial_{x}^{2}(x^{2}b_{+})=2b_{+}+4x\frac{m^{2}-1}{m}+x^{2}\frac{m^{2}+1}{m^{2}}m_{x},$

and obtain that $x\frac{m^{2}-1}{m}\in L^{2}(\mathbb{R})$. Therefore,
$m(t,\cdot)-1\in L^{2,2}(\mathbb{R})$ with

$\displaystyle\|m(t,\cdot)-1\|_{L^{2,2}(\mathbb{R})}\leq C(T,\|R(0)\|_{Y}).$

This completes the proof of Lemma 5.5. ∎

###  5.3. Proof of the main theorem

We are now in the position to give a proof of Theorem 1.1.

###### Proof of Theorem 1.1.

First, the estimate (5.89) in Lemma 5.5 implies that

$\displaystyle m(t,\cdot)-1\in H^{2,1}(\mathbb{R})\cap
H^{1,2}(\mathbb{R}),\;\forall t>0.$ (5.95)

Next, the continuity of $m$ on $t>0$ follows from the same step in Lemma 4.1.
In fact, the operator $\tilde{\mathcal{K}}(t)$ defined by (5.69) and function
$\tilde{N}_{0}$ defined by (5.70) corresponding to scattering data $R(t)$ are
continuous on $t>0$. It also gives that $M^{(2)}_{l}$ and $M^{(2)}_{r}$ are
continuous on $t>0$. This in turn implies that all terms on the right of
(5.90) and (5.93) are continuous on $t>0$. This is concluded that the global
solution $m$ in the sense of norm $\|m(t,\cdot)-1\|_{H^{2,1}\cap H^{1,2}}$ is
continuous on any $t>0$ , which together (5.95) implies that

$m-1\in C([0,+\infty);H^{2,1}(\mathbb{R})\cap H^{1,2}(\mathbb{R})).$

This completes the proof of Theorem 1.1. ∎

Acknowledgements The work of Fan and Yang is partially supported by the
National Science Foundation of China under grants 12271104, 51879045 and
12247182. The work of Liu is partially supported by the Simons Foundation
under grant 499875.

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DESY 19-058

CTPU-PTC-19-10

OCHA-PP-354

OU-HET-1005

LDU2019-03

Higgs inflation in metric and Palatini formalisms:

Required suppression of higher dimensional operators

Ryusuke Jinno,a,b Mio Kubota,c Kin-ya Oda,d and Seong Chan Parke

a | Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany
---|---
b | Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS),
| Daejeon 34126, Korea
c | Department of Physics, Ochanomizu University, Tokyo 112-8610, Japan
d | Department of Physics, Osaka University, Osaka 560-0043, Japan
e | Department of Physics & IPAP, Yonsei University, Seoul 03722, Korea

We investigate the sensitivity of Higgs(-like) inflation to higher dimensional
operators in the nonminimal couplings and in the potential, both in the metric
and Palatini formalisms. We find that, while inflationary predictions are
relatively stable against the higher dimensional operators around the
attractor point in the metric formalism, they are extremely sensitive in the
Palatini one: for the latter, inflationary predictions are spoiled by
$\left|\xi_{4}\right|\gtrsim 10^{-6}$ in the nonminimal couplings
$\left(\xi_{2}\phi^{2}+\xi_{4}\phi^{4}+\cdots\right)R$, or by
$\left|\lambda_{6}\right|\gtrsim 10^{-16}$ in the Jordan-frame potential
$\lambda_{4}\phi^{4}+\lambda_{6}\phi^{6}+\cdots$ (both in Planck units). This
extreme sensitivity results from the absence of attractor in the Palatini
formalism. Our study underscores the challenge of realizing inflationary
models with the nonminimal coupling in the Palatini formalism.

###### Contents

  1. 1 Introduction
  2. 2 Setup
    1. 2.1 Basic ingredients: metric and Palatini formalisms
    2. 2.2 Standard predictions without higher dimensional operators
  3. 3 Sensitivity to corrections in the Weyl-rescaling factor
    1. 3.1 Predictions
    2. 3.2 Results
    3. 3.3 Interpretation
  4. 4 Sensitivity to corrections in the potential
    1. 4.1 Predictions
    2. 4.2 Results
    3. 4.3 Interpretation
  5. 5 Summary and discussion
  6. A Equations
    1. A.1 Correction to the nonminimal coupling
    2. A.2 Correction to the potential
  7. B Difference from $f(R)$ Palatini theories

##  1 Introduction

The inflationary paradigm [1, 2, 3] is one of the fundamental elements of
modern cosmology, providing an elegant solution to the horizon and flatness
problems [3] and the dilution of possible unwanted relics [2]. Furthermore,
the slow-roll inflation [4, 5, 6] successfully produces the primordial density
perturbations by quantum fluctuations of the inflaton field [7, 8]. The
inflation has become precision science, in which a few parameters of the model
beautifully fits hundreds of data points in the observational data [9]. We
expect further experimental improvements in near future, with the tensor-to-
scalar ratio $r$ being explored by the LiteBIRD [10], POLARBEAR-2 [11], and
CORE [12] down to $r\sim 10^{-3}$.\diamondsuit1\diamondsuit1$\diamondsuit$1
The observation of $r$ is important as it will be an indirect evidence of the
quantum graviton fluctuation during the inflation, namely the first evidence
of quantum gravity. Even direct observation of the cosmic graviton background
might be possible in the future space interferometers such as the (Ultimate)
DECIGO experiment [13, 14].

The Higgs field is the only elementary scalar field in the Standard Model (SM)
of particle physics, and it is tempting to consider a possibility that it
plays the role of the inflaton field [15, 16, 17, 18]. In these early studies,
the possibility of induced gravity has been mainly pursued where the Einstein-
Hilbert action $S_{\text{EH}}$ is absent, and the nonminimal coupling
$\xi_{2}$ between the Higgs field and the Ricci scalar is required to be of
order $\xi_{2}\sim 10^{34}$.\diamondsuit2\diamondsuit2$\diamondsuit$2 See also
Refs. [19, 20] for inflation with the nonminimal coupling.  The modern version
of the Higgs inflation [21] allows $S_{\text{EH}}$, which is not prohibited by
any symmetry, and the nonminimal coupling becomes of order $\xi_{2}\sim
10^{5\text{--}6}$.\diamondsuit3\diamondsuit3$\diamondsuit$3 This possibility
has also been commented in the earlier Ref. [15] with essentially the same
parameters: $\xi_{2}\sim 10^{4}$ and the Higgs quartic coupling (around the
Planck scale) $\lambda_{4}\sim\left(\xi_{2}/10^{5}\right)^{2}\sim 10^{-2}$.
This modern version is indeed one of the best fit models of the current
observational data [9].\diamondsuit4\diamondsuit4$\diamondsuit$4 We note that
the value of the running Higgs quartic coupling $\lambda_{4}$ has to be
positive near the Planck scale for the successful Higgs inflation (see e.g.
Refs. [22, 23] for an uncertainty due to the lack of our knowledge of the
ultraviolet completion), requiring the pole mass of the top quark to be
$m_{t}^{\text{pole}}\lesssim 171.3\,\text{GeV}$ (see e.g. Ref. [24]). The
frequently quoted constraint $m_{t}^{\text{MC}}=173.0\pm 0.4$ GeV [25] is on
the Monte Carlo mass, a parameter in the Monte Carlo code, whose relation to
the pole mass is unknown. Currently the pole mass is best deduced from the
cross-section measurements, and a simple combination of experiments gives
$m_{t}^{\text{pole}}=173.1\pm 0.9$ GeV [25], which is 2$\sigma$-consistent
with the Higgs inflation. Note that this bound may be too stringent since it
assumes totally uncorrelated systematic errors among experiments. We also note
that the inclusion of the Higgs-portal dark matter greatly relaxes the
constraint even under the existence of the right-handed neutrinos [26].  There
are also variations in addition to this vanilla model: the critical [27, 28,
24, 26], the hill-climbing [29, 30], and the hill-top [31] Higgs inflations.
Sharing the virtue of the vanilla model, these variations predict different
values of $r$ along with other observables such as $dn_{s}/d\ln k$,
$d^{2}n_{s}/{d\ln k}^{2}$ as well as $n_{t}$, $dn_{t}/d\ln k$, which can
possibly be used to distinguish between these models in near-future
experiments.

However, such various Higgs inflations have been studied mainly in the so-
called metric formalism for past decades. It is known that, even in the
Einstein gravity, there are two different ways to formulate gravity: metric
and Palatini formalisms.\diamondsuit5\diamondsuit5$\diamondsuit$5 It has been
noted [32] that the original Palatini’s paper [33] ‘was rather far from what
is usually meant by “Palatini’s method,” which was instead formulated … by
Einstein’ [34]. Here we follow the convention and call it Palatini anyway.
The former assumes the Levi-Civita connection from the beginning by imposing
the metricity and torsion free conditions, while the latter regards the metric
and connection as independent variables, with respect to which we take the
variation of the action.\diamondsuit6\diamondsuit6$\diamondsuit$6 Throughout
this paper we assume vanishing torsion. In stringy context, this corresponds
to assuming the trivial background for any higher-rank gauge fields. We note
that while a three-form gauge-field background can be treated as a torsion,
five- and higher-form gauge-field background cannot.  Though these two
formalisms give the same dynamics within the Einstein gravity, their
predictions differ once we introduce nontrivial couplings with matter and
gravity. This indeed happens in the Higgs inflation, and their difference has
been attracting considerable interest recently [35, 36, 37, 38, 39, 40, 41,
42, 31, 43, 44, 45, 46, 47, 48, 49, 50, 51].

One may worry about the theoretical issues arising in the Palatini formalism,
namely, strong coupling and uniqueness, which are pointed out in Ref. [65] for
$f(R)$ theories. However, we found that such issues do not directly apply to
Palatini formalism in the case of Higgs inflation, in contrast to f(R)
theories (see Appendix B).

In this paper, we study the (vanilla) Higgs inflation both in metric and
Palatini formalisms. From the viewpoint of effective theory, the nonminimal
coupling in the Higgs inflation $\xi_{2}\phi^{2}R$ should be regarded as a
truncation in the infinite series
$(1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}+\cdots)R$. The potential term
$\lambda_{4}\phi^{4}$ should also be regarded as a truncation in
$\lambda_{4}\phi^{4}+\lambda_{6}\phi^{6}+\cdots$. While there are extensive
discussion on the scale at which these higher-order couplings appear [52, 53,
54, 55, 56],\diamondsuit7\diamondsuit7$\diamondsuit$7 Note that the cutoff-
scale issue has turned out to be more severe at the preheating regime in the
metric formalism [57] compared with earlier estimations [58, 59, 60]. This is
due to explosive production of the longitudinal gauge bosons (see also Ref.
[61]). On the other hand, such an effect is reported to be absent in the
Palatini formalism [62].  we at least expect that new physics effects appear
at the Planck scale $M_{\text{P}}$. Therefore, if the inflationary predictions
significantly depend on the higher-order terms suppressed by the Planck scale,
it means that the model construction is challenging in such setups. This is
indeed what we find for the Higgs inflation with the Palatini formalism.

The organization of the paper is as follows. In Sec. 2 we first introduce
basic ingredients and review the inflationary predictions in the standard
setup. In Sec. 3, we investigate the sensitivity of the Higgs-like inflation
to the higher-order corrections in the Weyl rescaling factor. In Sec. 4, we
investigate the sensitivity to the potential. Finally, we give summary and
discussion in Sec. 5. In Appendix A, we show the full expressions for the
relevant formulae.

##  2 Setup

In this section, we summarize two independent formalisms of Higgs-like
inflation, namely the metric and Palatini formalisms, and review the standard
predictions without higher order corrections for each case.

###  2.1 Basic ingredients: metric and Palatini formalisms

We start from the action in the Jordan frame:

$\displaystyle S$ $\displaystyle=\int
d^{4}x\sqrt{-g_{\text{J}}}\left[\frac{1}{2}\Omega^{2}\\!\left(\phi\right)g_{\text{J}}^{\mu\nu}R_{\text{J}\mu\nu}-\frac{1}{2}g_{\text{J}}^{\mu\nu}\partial_{\mu}\phi\,\partial_{\nu}\phi-
V_{\text{J}}(\phi)\right],$ (2.1)

where $\phi$ is the (Jordan-frame) inflaton and

$\displaystyle\Omega^{2}\\!\left(\phi\right)$
$\displaystyle=1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}+\cdots,~{}~{}~{}~{}~{}~{}V_{\text{J}}\\!\left(\phi\right)=\lambda_{4}\phi^{4}+\lambda_{6}\phi^{6}+\cdots,$
(2.2)

are the Weyl-rescaling factor and the Jordan-frame potential, respectively;
here we have assumed the $Z_{2}$ symmetry:
$\phi\to-\phi$.\diamondsuit8\diamondsuit8$\diamondsuit$8 For the Higgs field,
this is a natural assumption since the gauge invariance allows only a pair
$H^{\dagger}H$ as a basic building block. If we relax this condition, there
are other potentially interesting cases without the $Z_{2}$ symmetry such as
$\Omega^{2}=1+\xi_{2}\phi^{2}+\xi_{3}\phi^{3}$ and
$V_{\text{J}}=\lambda_{4}\phi^{4}+\lambda_{6}\phi^{6}$. In this case, the
potential in Einstein frame has a plateau at a large field limit,
$\phi\gg\max\left({\xi_{2}\over\xi_{3}},\sqrt{\lambda_{4}\over\lambda_{6}}\right)$
as
$V\sim\frac{V_{\text{J}}}{\Omega^{4}}\sim\frac{\lambda_{6}}{\xi_{3}^{2}}\left(1-\frac{\xi_{2}}{\xi_{3}\phi}+\frac{\lambda_{4}}{\lambda_{6}\phi^{2}}\right),$
which is universal as long as
$\lim_{\phi\to\infty}{V_{\text{J}}\over\Omega^{4}}\to\text{const.}$ [63]. In
this case inflationary dynamics would happen at a lower regime
$\phi<{\xi_{2}\over\xi_{3}},\sqrt{\lambda_{4}\over\lambda_{6}}$.  The Ricci
tensor is $R_{\text{J}\mu\nu}=R_{\text{J}\mu\nu}(g_{\text{J}})$ in the metric
formalism while $R_{\text{J}\mu\nu}=R_{\text{J}\mu\nu}(\Gamma_{\text{J}})$ in
the Palatini formalism, with the arguments $g_{\text{J}}$ and
$\Gamma_{\text{J}}$ symbolically denoting the metric and connection in the
Jordan frame, respectively. The subscript J refers to the Jordan frame. We use
the Planck unit $M_{\text{P}}=1$ throughout the paper, with
$M_{\text{P}}\equiv 1/\sqrt{8\pi G}=2.4\times 10^{18}$ GeV being the reduced
Planck mass.

A metric redefinition by a Weyl transformation
$g_{\mu\nu}=\Omega^{2}g_{\text{J}\mu\nu}$ gives the transformation of the
Ricci scalar as

$\displaystyle
R_{\text{J}}=\left\\{\begin{matrix}\displaystyle\Omega^{2}\left[R+3\Box\ln\Omega^{2}-\frac{3}{2}(\partial\ln\Omega^{2})^{2}\right]~{}~{}~{}~{}&{\rm(metric)},\\\\[8.61108pt]
\Omega^{2}R(\Gamma)~{}~{}~{}~{}&{\rm(Palatini)},\end{matrix}\right.$ (2.3)

where the result for the Palatini formalism is trivially obtained as the Ricci
tensor is independent of the metric transformation. After this, we obtain the
Einstein-frame action

$\displaystyle S$ $\displaystyle=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}g^{\mu\nu}R_{\mu\nu}-\frac{1}{2}(\partial\chi)^{2}-V(\phi)\right],$
(2.4)

where $V$ is the Einstein-frame potential $V=V_{\text{J}}/\Omega^{4}$ and
$\chi$ is the canonical inflaton field in this frame. The relation between
$\chi$ and the Jordan-frame inflaton $\phi$ depends on the formalism:

$\displaystyle\frac{d\chi}{d\phi}$
$\displaystyle=\left\\{\begin{matrix}\displaystyle\sqrt{\frac{1}{\Omega^{2}}+\frac{3}{2}\left(\frac{d\ln\Omega^{2}}{d\phi}\right)^{2}}~{}~{}~{}~{}&{\rm(metric)},\\\\[8.61108pt]
\displaystyle\frac{1}{\Omega}~{}~{}~{}~{}&{\rm(Palatini)}.\end{matrix}\right.$
(2.5)

The slow-roll parameters and the $e$-folding are calculated through

$\displaystyle\epsilon$
$\displaystyle=\frac{1}{2}\left(\frac{dV/d\chi}{V}\right)^{2}=\frac{1}{2}\left(\frac{dV/d\phi}{V}\right)^{2}\frac{1}{(d\chi/d\phi)^{2}},$
(2.6) $\displaystyle\eta$
$\displaystyle=\frac{d^{2}V/d\chi^{2}}{V}=\left[\frac{d^{2}V/d\phi^{2}}{V}\frac{1}{(d\chi/d\phi)^{2}}-\frac{dV/d\phi}{V}\frac{d^{2}\chi/d\phi^{2}}{(d\chi/d\phi)^{3}}\right],$
(2.7) $\displaystyle N$
$\displaystyle=\int\frac{d\chi}{\sqrt{2\epsilon}}=\int\frac{d\phi}{d\ln
V/d\phi}\left(\frac{d\chi}{d\phi}\right)^{2},$ (2.8)

and the resulting inflationary predictions are expressed by

$\displaystyle A_{s}$
$\displaystyle=\frac{1}{24\pi^{2}}\frac{V}{\epsilon},~{}~{}~{}~{}~{}~{}n_{s}=1-6\epsilon+2\eta,~{}~{}~{}~{}~{}~{}r=16\epsilon.$
(2.9)

We use $A_{s}=2.1\times 10^{-9}$ (Planck TT,TE,EE+lowE+lensing+BK14+BAO [9])
throughout our analysis.

###  2.2 Standard predictions without higher dimensional operators

Before including higher dimensional operators in Secs. 3 and 4, let us review
the inflationary predictions in the simplest setup. We truncate the Weyl-
rescaling factor and the Jordan-frame potential as

$\displaystyle\Omega^{2}$
$\displaystyle=1+\xi_{2}\phi^{2},~{}~{}~{}~{}~{}~{}V_{\text{J}}=\lambda_{4}\phi^{4},$
(2.10)

for both metric and Palatini formalisms. The Einstein-frame potential becomes

$\displaystyle V$
$\displaystyle=\frac{\lambda_{4}\phi^{4}}{\left(1+\xi_{2}\phi^{2}\right)^{2}}.$
(2.11)

#### Metric formalism

In the metric formalism, the relation (2.5) between $\chi$ and $\phi$ becomes

$\displaystyle\frac{d\chi}{d\phi}$
$\displaystyle=\frac{\sqrt{1+\xi_{2}\left(1+6\xi_{2}\right)\phi^{2}}}{1+\xi_{2}\phi^{2}},$
(2.12)

and the integral of the $e$-folding (2.8) can be exactly performed:

$\displaystyle N$
$\displaystyle=\frac{1}{8}\left[\left(1+6\xi_{2}\right)\phi^{2}-6\ln\left(1+\xi_{2}\phi^{2}\right)\right],$
(2.13)

where we have defined $N$ as the number of $e$-folding from $\phi=0$. Note
that generically the difference between this definition and the usual one
measured from $\max\\!\left(\left|\epsilon\right|,\left|\eta\right|\right)=1$
is suppressed by $1/N$ in the expression of $\phi(N)$.

Eq. (2.12) can be solved as\diamondsuit9\diamondsuit9$\diamondsuit$9 The
following formulae are useful: for $-1<x<1$,
$\displaystyle\operatorname{arcsinh}x$
$\displaystyle=\ln\left(x+\sqrt{1+x^{2}}\right),~{}~{}~{}~{}~{}~{}\operatorname{arctanh}x=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right).$
(2.14)

$\displaystyle\chi(\phi)$
$\displaystyle=\sqrt{\frac{1+6\xi_{2}}{\xi_{2}}}~{}\operatorname{arcsinh}\left[\sqrt{\xi_{2}\left(1+6\xi_{2}\right)}\phi\right]-\sqrt{6}~{}\operatorname{arctanh}\left[\frac{\sqrt{6}\xi_{2}\phi}{\sqrt{1+\xi_{2}\left(1+6\xi_{2}\right)\phi^{2}}}\right].$
(2.15)

Although this is an exact formula, it is hardly inverted in general. However,
in the limit $\xi_{2}\gg 1$, the new kinetic term
$\sim\left(\partial\ln\Omega^{2}\right)^{2}$ originating from the Ricci scalar
dominates over the original kinetic term $\sim\left(\partial\phi\right)^{2}$
of the inflaton field, and thus the canonical inflaton field is well
approximated as

$\displaystyle\chi$
$\displaystyle\simeq\sqrt{\frac{3}{2}}\ln\Omega^{2}=\sqrt{\frac{3}{2}}\ln\left(1+\xi_{2}\phi^{2}\right).$
(2.16)

Substituting this back to the potential, we get

$\displaystyle V$
$\displaystyle\simeq\frac{\lambda_{4}}{\xi_{2}^{2}}\left(1-e^{-\sqrt{\frac{2}{3}}\chi}\right)^{2}.$
(2.17)

The slow-roll parameters with this potential are calculated as

$\displaystyle\epsilon$
$\displaystyle\simeq\frac{4}{3}e^{-2\sqrt{\frac{2}{3}}\chi}\simeq\frac{3}{4N^{2}},~{}~{}~{}~{}~{}~{}\eta\simeq-\frac{4}{3}e^{-\sqrt{\frac{2}{3}}\chi}\simeq-\frac{1}{N},$
(2.18)

with the $e$-folding given by

$\displaystyle N$ $\displaystyle\simeq\frac{3}{4}e^{\sqrt{\frac{2}{3}}\chi}.$
(2.19)

The inflationary predictions become

$\displaystyle A_{s}$
$\displaystyle\simeq\frac{N^{2}}{32\pi^{2}}\frac{\lambda_{4}}{\xi_{2}^{2}},~{}~{}~{}~{}~{}~{}n_{s}\simeq
1-\frac{2}{N},~{}~{}~{}~{}~{}~{}r\simeq\frac{12}{N^{2}}.~{}~{}~{}~{}~{}~{}\text{(Metric)}$
(2.20)

In the expression of $n_{s}$, the deviation from unity is dominated by the
contribution from $\eta$ in Eq. (2.9).

Calculations above are performed in terms of the canonical inflaton $\chi$ in
the Einstein frame. From Eq. (2.13), we can read off the behavior of $\phi$
for $\xi_{2}\gg 1$ as

$\displaystyle\phi$ $\displaystyle\simeq\sqrt{\frac{4N}{3\xi_{2}}},$ (2.21)

which is consistent with Eqs. (2.16) and (2.19). We see that the value of
$\phi$ for a fixed $e$-folding scales as $\propto
1/\sqrt{\xi_{2}}$.\diamondsuit10\diamondsuit10$\diamondsuit$10In general, for
monomial functions $\Omega^{2}=1+\xi_{m}\phi^{m}$ and
$V_{\text{J}}\propto\phi^{2m}$, the scaling behavior is
$\phi\simeq\left(\frac{4N}{3\xi_{m}}\right)^{1/m}$ with $m\geq 2$ [63]. This
scaling behavior turns out to be important in interpreting our results later.

#### Palatini formalism

In Palatini formalism, the relation (2.5) becomes

$\displaystyle\frac{d\chi}{d\phi}$
$\displaystyle=\frac{1}{\sqrt{1+\xi_{2}\phi^{2}}},$ (2.22)

and the $e$-folding (2.8) is obtained as

$\displaystyle
N=\frac{\phi^{2}}{8}~{}~{}~{}~{}~{}~{}\leftrightarrow~{}~{}~{}~{}~{}~{}\phi=\sqrt{8N},$
(2.23)

where we have again defined $N$ as the number of $e$-folding from $\phi=0$. We
see that the relation is independent of $\xi_{2}$ in contrast to the metric
formalism. We also see that the value of $\phi_{\rm Palatini}\sim\sqrt{N}$ at
a given $e$-folding $N$ is significantly larger than that in metric one,
$\phi_{\rm metric}\sim\sqrt{N/\xi_{2}}$ for $\xi_{2}\gg 1$ in Eq. (2.21).

Again one can exactly solve Eq. (2.22):

$\displaystyle\chi=\frac{1}{\sqrt{\xi_{2}}}\operatorname{arcsinh}\left[\sqrt{\xi_{2}}\phi\right]=\frac{1}{\sqrt{\xi_{2}}}\ln\left[\sqrt{\xi_{2}}\phi+\sqrt{1+\xi_{2}\phi^{2}}\right]~{}~{}~{}~{}~{}~{}\leftrightarrow~{}~{}~{}~{}~{}~{}\phi=\frac{1}{\sqrt{\xi_{2}}}\sinh\left[\sqrt{\xi_{2}}\chi\right],$
(2.24)

and the Einstein-frame potential becomes

$\displaystyle V$
$\displaystyle=\frac{\lambda_{4}}{\xi_{2}^{2}}\tanh^{4}\left[\sqrt{\xi_{2}}\chi\right],$
(2.25)

In the limit $\sqrt{\xi_{2}}\chi\gg 1$, we see

$\displaystyle\phi$ $\displaystyle\simeq{1\over
2\sqrt{\xi_{2}}}e^{\sqrt{\xi_{2}}\chi},~{}~{}~{}~{}~{}~{}N\simeq\frac{1}{32\xi_{2}}e^{2\sqrt{\xi_{2}}\chi},$
(2.26)

and the slow-roll parameters become

$\displaystyle\epsilon$ $\displaystyle\simeq
128\xi_{2}e^{-4\sqrt{\xi_{2}}\chi}\simeq\frac{1}{8N^{2}\xi_{2}},~{}~{}~{}~{}~{}~{}\eta\simeq-32\xi_{2}e^{-2\sqrt{\xi_{2}}\chi}\simeq-\frac{1}{N}.$
(2.27)

From the expression of $\eta$ we see that the inflation ends at
$\chi\simeq{\ln\left(32\xi_{2}\right)\over 2\sqrt{\xi_{2}}}\ll 1$ for
$\xi_{2}\gg 1$, which corresponds to $\phi\simeq 2\sqrt{2}$. The curvature
perturbation and other inflationary observables become

$\displaystyle A_{s}$
$\displaystyle\simeq\frac{N^{2}}{3\pi^{2}}\frac{\lambda_{4}}{\xi_{2}},~{}~{}~{}~{}~{}~{}n_{s}\simeq
1-\frac{2}{N},~{}~{}~{}~{}~{}~{}r\simeq\frac{2}{N^{2}\xi_{2}}.~{}~{}~{}~{}~{}~{}\text{(Palatini)}$
(2.28)

In the expression of $n_{s}$, the deviation from unity is dominated by the
contribution from $\eta$ in Eq. (2.9), similarly to the metric formalism. Note
that the scaling $A_{s}\propto 1/\xi_{2}$ in Palatini formalism is different
from $A_{s}\propto 1/\xi_{2}^{2}$ in metric one. Also, the tensor-to-scalar
ratio $r$ is multiplicatively suppressed by a factor of $1/\xi_{2}\ll 1$ in
contrast to the metric formalism.

##  3 Sensitivity to corrections in the Weyl-rescaling factor

The original truncation (2.10) takes into account up to the next-to-leading
and leading terms in $V_{\text{J}}$ and $\Omega^{2}$, respectively. We
naturally expect further terms arising from radiative corrections and from new
physics effects. Without knowing all sources of corrections, we
phenomenologically parametrize the corrections by introducing higher
dimensional operators as in Eq. (2.2).

To be specific, we first consider the correction to the Weyl rescaling factor
$\Omega^{2}$ in both metric and Palatini formalisms in detail and see their
effects to the inflationary dynamics and observables. Correction to the
Jordan-frame potential will be discussed in Sec. 4. Throughout both the
analysis, we assume that $|\xi_{p>2}|\ll\xi_{2}$ and
$|\lambda_{q>4}|\ll\lambda_{4}$ so that the success of the original Higgs-like
inflation with nonminimal coupling $\xi_{2}\gg 1$ is maintained.

The first case we consider is a correction to the Weyl rescaling factor:

$\displaystyle\Omega^{2}$
$\displaystyle=1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4},~{}~{}~{}~{}~{}~{}V_{\text{J}}=\lambda_{4}\phi^{4}.$
(3.29)

Here we take into account the correction to $\Omega^{2}$ only, and leave the
correction to $V_{\text{J}}$ for the later section. We fix their signs of
$\xi_{2}$ and $\lambda_{4}$ to be positive, while we allow both positive and
negative values for $\xi_{4}$. Now the Einstein-frame potential becomes

$\displaystyle V$
$\displaystyle=\frac{\lambda_{4}\phi^{4}}{\left(1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}\right)^{2}}.$
(3.30)

In this setup we have three parameters

  * $(\lambda_{4},\xi_{2},\xi_{4})$,

but we can eliminate one of them by fixing the observed value of $A_{s}$,
which we take to be $A_{s}=2.1\times 10^{-9}$ (Planck
TT,TE,EE+lowE+lensing+BK14+BAO [9]) as mentioned above.

###  3.1 Predictions

#### Metric formalism

In metric formalism, the relation (2.5) between $\chi$ and $\phi$ becomes

$\displaystyle\frac{d\chi}{d\phi}$
$\displaystyle=\frac{\sqrt{1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}+6\left(\xi_{2}\phi+2\xi_{4}\phi^{3}\right)^{2}}}{1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}}.$
(3.31)

The slow-roll parameters and $e$-folding are calculated through Eqs.
(2.6)–(2.8):

$\displaystyle\epsilon$
$\displaystyle=\frac{8(1-\xi_{4}\phi^{4})^{2}}{\phi^{2}\left[1+\left(\xi_{2}+6\xi_{2}^{2}\right)\phi^{2}+\left(\xi_{4}+24\xi_{2}\xi_{4}\right)\phi^{4}+24\xi_{4}^{2}\phi^{6}\right]},$
(3.32) $\displaystyle\eta$
$\displaystyle=\frac{4\left[3+\left(\xi_{2}+12\xi_{2}^{2}\right)\phi^{2}+(-2\xi_{2}^{2}-12\xi_{2}^{3}+24\xi_{2}\xi_{4}-11\xi_{4})\phi^{4}+\cdots+96\xi_{4}^{4}\phi^{14}\right]}{\phi^{2}\left[1+\left(\xi_{2}+6\xi_{2}^{2}\right)\phi^{2}+\left(\xi_{4}+24\xi_{2}\xi_{4}\right)\phi^{4}+24\xi_{4}^{2}\phi^{6}\right]^{2}},$
(3.33) $\displaystyle N$
$\displaystyle=\frac{1+6\xi_{2}}{16\sqrt{\xi_{4}}}\ln\left[\frac{1+\sqrt{\xi_{4}}\phi^{2}}{1-\sqrt{\xi_{4}}\phi^{2}}\right]-\frac{3}{4}\ln\left[\left(1-\xi_{4}\phi^{4}\right)\left(1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}\right)\right].$
(3.34)

Here we do not show a complete expression for $\eta$ just to avoid
complications; see Appendix A for it. Note that the $e$-folding diverges for
$\xi_{4}>0$ at $\phi=\xi_{4}^{-1/4}$, which corresponds to the point where the
potential derivative vanishes. The scalar power spectrum amplitude reads

$\displaystyle A_{s}$ $\displaystyle={\lambda_{4}\phi^{6}\over
192\pi^{2}}{{1+\xi_{2}\left(1+6\xi_{2}\right)\phi^{2}+\left(1+24\xi_{2}\right)\xi_{4}\phi^{4}+24\xi_{4}^{2}\phi^{6}}\over\left(1-\xi_{4}\phi^{4}\right)^{2}\left(1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}\right)^{2}}.$
(3.35)

#### Palatini formalism

In Palatini formalism, the relation (2.5) becomes

$\displaystyle\frac{d\chi}{d\phi}$
$\displaystyle=\frac{1}{\sqrt{1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}}},$ (3.36)

which can be explicitly solved as shown in Appendix A. Also, from Eqs.
(2.6)–(2.8), the slow-roll parameters and $e$-folding become

$\displaystyle\epsilon$
$\displaystyle=\frac{8(1-\xi_{4}\phi^{4})^{2}}{\phi^{2}\left[1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}\right]},$
(3.37) $\displaystyle\eta$
$\displaystyle=\frac{4(3-2\xi_{2}\phi^{2}-14\xi_{4}\phi^{4}-2\xi_{2}\xi_{4}\phi^{6}+3\xi_{4}^{2}\phi^{8})}{\phi^{2}\left[1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}\right]},$
(3.38) $\displaystyle N$
$\displaystyle=\frac{1}{16\sqrt{\xi_{4}}}\ln\left[\frac{1+\sqrt{\xi_{4}}\phi^{2}}{1-\sqrt{\xi_{4}}\phi^{2}}\right]~{}~{}~{}~{}~{}~{}\leftrightarrow~{}~{}~{}~{}~{}~{}\phi=\xi_{4}^{-1/4}\sqrt{\tanh\left[8\sqrt{\xi_{4}}N\right]},$
(3.39)

where we inverted the relation in the last equality. Note that the $e$-folding
diverges at the same point $\phi=\xi_{4}^{-1/4}$ as the metric formalism for
$\xi_{4}>0$ because the potential derivative vanishes there. The scalar power
spectrum amplitude reads

$\displaystyle A_{s}$ $\displaystyle={\lambda_{4}\phi^{6}\over
192\pi^{2}}{1\over\left(1-\xi_{4}\phi^{4}\right)^{2}\left(1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}\right)}.$
(3.40)

Figure 1: Effects of the higher dimensional operator
$\xi_{4}\phi^{4}R_{\text{J}}$ in the metric (left) and Palatini (right)
formalisms with $N=60$ (top) and 50 (bottom). We plot contours of fixed
$\xi_{2}$ (blue, horizontal) and of fixed $\xi_{4}$ (red, vertical) in the
$n_{s}$-$r$ plane. The value of $\lambda_{4}$ is also shown as a density plot.
Allowed regions of $1\sigma$ and $2\sigma$ from the Planck experiment [9]
(TT,TE,EE+lowE+lensing+BK14+BAO) are also shown in the center.

Metric ($N=60$)

Palatini ($N=60$)

Figure 2: Behavior of $\lambda_{4}$ along constant $\xi_{2}$ slices in Fig. 1
for $N=60$. We show $\lambda_{4}$ as a function of $\left|\xi_{4}\right|$ for
$\xi_{4}<0$ (left) and $\xi_{4}>0$ (right). The lines are $\xi_{2}=1$ (blue),
$10^{2}$ (red), and $10^{4}$ (green) for the metric case, while
$\xi_{2}=10^{6}$ (blue), $10^{8}$ (red), and $10^{10}$ (green) for the
Palatini case.

Metric ($N=60$)

Palatini ($N=60$)

Figure 3: Behavior of $n_{s}$ (solid, left axis) and $r$ (dashed, right axis)
along constant $\xi_{2}$ slices in Fig. 1 for $N=60$. All the solid lines are
degenerate for the Palatini case.

Metric ($N=50$)

Palatini ($N=50$)

Figure 4: The same as in Fig. 2 except for $N=50$.

Metric ($N=50$)

Palatini ($N=50$)

Figure 5: The same as in Fig. 3 except for $N=50$.

###  3.2 Results

In Fig. 1, we show predictions of the metric (left) and Palatini (right)
formalisms in the $n_{s}$-$r$ plane for $N=60$ (top) and $50$ (bottom). The
blue contours show constant values of $\xi_{2}$, while the red ones show
constant values of $\xi_{4}$. The value of the quartic coupling $\lambda_{4}$
is also presented as a density plot. The smiley marker is the prediction of
the quartic chaotic inflation, while the star denotes the attractor point for
$\xi_{2}\gg 1$. For $\xi_{2}<10^{-3}$ and $\xi_{2}>1$, the corresponding blue
lines are almost degenerate with the upper and lower boundaries with the gray
regions, respectively. We see the following:

  * •

For relatively small $\xi_{2}$ ($\lesssim 10^{-1}$), even a small injection of
$\left|\xi_{4}\right|\gtrsim 10^{-5}$ drastically changes the inflationary
predictions in both metric and Palatini formalisms.

  * •

For relatively large $\xi_{2}$ ($\gtrsim 1$), the inflationary predictions are
stable against the injection of $\xi_{4}$ for the metric formalism (left): we
see that all the red lines converge to the attractor point. On the other hand,
the predictions are no more stable against the injection of
$\left|\xi_{4}\right|\gtrsim 10^{-5}$ for the Palatini formalism (right).

In Fig. 2, we plot the behavior of $\lambda_{4}$ as a function of
$\left|\xi_{4}\right|$ for $\xi_{4}<0$ (left panel) and $\xi_{4}>0$ (right
panel) with $N=60$ for fixed values of $\xi_{2}$. In Fig. 3, we plot $n_{s}$
(solid, left axis) and $r$ (dashed, right axis) similarly. Figs. 4 and 5 are
the corresponding ones with $N=50$. One immediately sees that the deviation
from the attractor occurs around $\left|\xi_{4}\right|\sim 10^{-4}\xi_{2}^{2}$
and around $\left|\xi_{4}\right|\sim 10^{-5}$ for the metric and Palatini
cases, respectively.

###  3.3 Interpretation

Let us interpret our results, especially the threshold value of $\xi_{4}$
which gives deviation from the observationally allowed region. In the
following we use $N\gg 1$ and $\xi_{2}\gg 1$ and keep the leading contribution
for each order in $\xi_{4}$ expansion when necessary.

In the metric formalism, let us first expand Eq. (3.34) by small $\xi_{4}$
around $\phi\simeq\phi_{\xi_{4}=0}\simeq\sqrt{4N/3\xi_{2}}$ (see Eq. (2.21)):

$\displaystyle N$
$\displaystyle=\left[\frac{1+6\xi_{2}}{8}\phi^{2}-\frac{3}{4}\ln(1+\xi_{2}\phi^{2})\right]+\xi_{4}\left[\frac{1+6\xi_{2}}{24}\phi^{6}-\frac{3}{4}\frac{\xi_{2}\phi^{6}}{1+\xi_{2}\phi^{2}}\right]+\cdots.$
(3.41)

Substituting $\phi=\phi_{\xi_{4}=0}(1+c\xi_{4})$ and comparing leading terms
in $\xi_{4}$ we find that the deviation of $\phi$ is given by

$\displaystyle\phi$
$\displaystyle\simeq\phi_{\xi_{4}=0}\left(1-\frac{8N^{2}}{27\xi_{2}^{2}}\xi_{4}\right).$
(3.42)

We next expand $\epsilon$ (3.32) and $\eta$ (3.33) by small $\xi_{4}$ around
the same point of $\phi$, and then substitute Eq. (3.42):

$\displaystyle\epsilon$
$\displaystyle\simeq\frac{4}{3\xi_{2}^{2}\phi^{4}}\left(1-2\xi_{4}\phi^{4}\right)\simeq\frac{3}{4N^{2}}\left(1-\frac{64N^{2}}{27\xi_{2}^{2}}\xi_{4}\right),$
(3.43) $\displaystyle\eta$
$\displaystyle\simeq-\frac{4}{3\xi_{2}\phi^{2}}\left(1+\xi_{4}\phi^{4}\right)\simeq-\frac{1}{N}\left(1+\frac{64N^{2}}{27\xi_{2}^{2}}\xi_{4}\right).$
(3.44)

We see that the deviation becomes sizable for
$\left|\xi_{4}\right|\gtrsim\xi_{2}^{2}/N^{2}$. Since $\epsilon$ and $\eta$
corresponds to the $r$ (vertical) and $n_{s}$ (horizontal) axes in Fig. 1,
respectively, the direction of deviation for positive and negative $\xi_{4}$
is also explained.

In the Palatini formalism, let us first expand Eq. (3.39) by small $\xi_{4}$
around $\phi\simeq\phi_{\xi_{4}=0}\simeq\sqrt{8N}$ (see Eq. (2.23)):

$\displaystyle N$
$\displaystyle=\frac{1}{8}\phi^{2}+\frac{1}{24}\xi_{4}\phi^{6}+\cdots.$ (3.45)

Substituting $\phi=\phi_{\xi_{4}=0}(1+c\xi_{4})$ and comparing leading terms
in $\xi_{4}$, we find

$\displaystyle\phi$
$\displaystyle\simeq\phi_{\xi_{4}=0}\left(1-\frac{32N^{2}}{3}\xi_{4}\right).$
(3.46)

We next expand $\epsilon$ (3.37) and $\eta$ (3.38) by small $\xi_{4}$ around
the same point of $\phi$, and then substitute Eq. (3.46):

$\displaystyle\epsilon$
$\displaystyle\simeq\frac{8}{\xi_{2}\phi^{4}}\left(1-2\xi_{4}\phi^{4}\right)\simeq\frac{1}{8N^{2}\xi_{2}}\left(1-\frac{256N^{2}}{3}\xi_{4}\right),$
(3.47) $\displaystyle\eta$
$\displaystyle\simeq-\frac{8}{\phi^{2}}\left(1+\xi_{4}\phi^{4}\right)\simeq-\frac{1}{N}\left(1+\frac{256N^{2}}{3}\xi_{4}\right).$
(3.48)

We see that the deviation becomes sizable for $\left|\xi_{4}\right|\gtrsim
10^{-2}/N^{2}$ and that the direction of the deviation is the same as in the
metric formalism.

To summarize, the expressions for the slow-roll parameters, especially for
$\eta$, imply that the inflationary predictions (especially on $n_{s}$) are
significantly affected for

$\displaystyle\left|\xi_{4}\right|$
$\displaystyle\gtrsim\begin{cases}\displaystyle\frac{\xi_{2}^{2}}{N^{2}}&~{}~{}\text{(metric)},\\\\[8.53581pt]
\displaystyle\frac{10^{-2}}{N^{2}}&~{}~{}\text{(Palatini)}.\end{cases}$ (3.49)

This explains the behavior in Figs. 2–5. After taking the curvature
normalization into account, these values read

$\displaystyle\left|\xi_{4}\right|$
$\displaystyle\gtrsim\begin{cases}\displaystyle\frac{\xi_{2}^{2}}{N^{2}}\sim\frac{\lambda_{4}}{32\pi^{2}A_{s}}\sim
10^{6}\lambda_{4}&~{}~{}{\rm(metric)},\\\\[8.53581pt]
\displaystyle\frac{10^{-2}}{N^{2}}&~{}~{}{\rm(Palatini)}.\end{cases}$ (3.50)

Therefore, we see that the inflationary predictions in the metric formalism
become stable for $\xi_{2}\gg 1$ (unless $\lambda_{4}\ll 10^{-6}$), while they
are spoiled by $\left|\xi_{4}\right|\sim 10^{-5}$ independently of
$\lambda_{4}$ or $\xi_{2}$ in the Palatini formalism.

In the case $\xi_{4}>0$, we note that in both the formalisms $\epsilon$
becomes zero at $\xi_{4}\phi^{4}=1$ and hence the $e$-folding diverges in the
limit $\phi\to\xi_{4}^{-1/4}$. Then the necessary condition
$\phi<\xi_{4}^{-1/4}$ at $\phi\sim\sqrt{4N/3\xi_{2}}$ (metric) and
$\phi\sim\sqrt{8N}$ (Palatini) reads

$\displaystyle\xi_{4}$
$\displaystyle\lesssim\begin{cases}\displaystyle{9\xi_{2}^{2}\over
16N^{2}}&~{}~{}\text{(metric)},\\\\[8.53581pt] \displaystyle{1\over
64N^{2}}&~{}~{}\text{(Palatini)}.\end{cases}$ (3.51)

Note that we used the relation between $\phi$ and $N$ for $\xi_{4}=0$ (and for
$\xi_{2}\gg 1$ in metric formalism), and therefore these conditions are only
approximate. We see that these conditions give comparable threshold values to
Eq. (3.49).

As we increase $\left|\xi_{4}\right|$ from zero, the inflationary prediction
starts to deviate at around the value in the right-hand side of Eq. (3.49). We
note that this occurs when the higher dimensional operator
$\left|\xi_{4}\phi^{4}\right|$ is still much smaller than the lower
dimensional one $\xi_{2}\phi^{2}$. Indeed, by substituting the approximate
values $\phi\simeq\sqrt{4N/3\xi_{2}}$ (metric) and $\sqrt{8N}$ (Palatini), the
condition

$\displaystyle\left|\xi_{4}\phi^{4}\right|$
$\displaystyle\lesssim\xi_{2}\phi^{2}$ (3.52)

becomes

$\displaystyle\left|\xi_{4}\right|$
$\displaystyle\lesssim\begin{cases}\displaystyle{\xi_{2}^{2}\over
N}&\text{(metric)},\\\\[8.53581pt] \displaystyle 10^{-1}{\xi_{2}\over
N}&\text{(Palatini)},\end{cases}$ (3.53)

which is well satisfied at the value in the right-hand side of Eq. (3.49).

##  4 Sensitivity to corrections in the potential

Second, we consider a correction to the Jordan-frame potential:

$\displaystyle\Omega^{2}$
$\displaystyle=1+\xi_{2}\phi^{2},~{}~{}~{}~{}~{}~{}V_{\text{J}}=\lambda_{4}\phi^{4}+\lambda_{6}\phi^{6}.$
(4.1)

Again we fix the signs of $\xi_{2}$ and $\lambda_{4}$ to be positive, while we
allow both positive and negative signs for $\lambda_{6}$. The Einstein-frame
potential becomes

$\displaystyle V$
$\displaystyle=\frac{\lambda_{4}\phi^{4}+\lambda_{6}\phi^{6}}{(1+\xi_{2}\phi^{2})^{2}}.$
(4.2)

In this setup we have three parameters

  * $(\lambda_{4},\lambda_{6},\xi_{2})$,

but we can eliminate one since $A_{s}$ is fixed by the observation.

###  4.1 Predictions

Before discussing predictions in each formalism, we note that the dependence
of $\epsilon$, $\eta$, and $N$ on $\lambda_{4}$ and $\lambda_{6}$ shown below
appears only through the ratio $\lambda_{6}/\lambda_{4}$. In the following
analysis, we fix the quartic coupling $\lambda_{4}$ by the overall
normalization $A_{s}$ and adopt $\lambda_{6}/\lambda_{4}$ as an independent
variable instead of $\lambda_{6}$.

#### Metric formalism

In the metric formalism, the relation (2.5) between $\chi$ and $\phi$ is the
same as Eqs. (2.12)–(2.15). The slow-roll parameters and $e$-folding are
calculated through Eqs. (2.6)–(2.8):

$\displaystyle\epsilon$
$\displaystyle=\frac{2\left(2\lambda_{4}+3\lambda_{6}\phi^{2}+\xi_{2}\lambda_{6}\phi^{4}\right)^{2}}{\phi^{2}\left[1+\left(\xi_{2}+6\xi_{2}^{2}\right)\phi^{2}\right]\left(\lambda_{4}+\lambda_{6}\phi^{2}\right)^{2}},$
(4.3) $\displaystyle\eta$
$\displaystyle=\frac{2\left[6\lambda_{4}+\left(2\xi_{2}\lambda_{4}+24\xi_{2}^{2}\lambda_{4}+15\lambda_{6}\right)\phi^{2}+\cdots+\left(2\xi_{2}^{3}\lambda_{6}+12\xi_{2}^{4}\lambda_{6}\right)\phi^{8}\right]}{\phi^{2}\left[1+\left(\xi_{2}+6\xi_{2}^{2}\right)\phi^{2}\right]^{2}\left(\lambda_{4}+\lambda_{6}\phi^{2}\right)},$
(4.4) $\displaystyle N$ $\displaystyle=\int
d\phi~{}\frac{\phi\left[1+(\xi_{2}+6\xi_{2}^{2})\phi^{2}\right]\left(\lambda_{4}+\lambda_{6}\phi^{2}\right)}{2\left(1+\xi_{2}\phi^{2}\right)\left(2\lambda_{4}+3\lambda_{6}\phi^{2}+\xi_{2}\lambda_{6}\phi^{4}\right)}.$
(4.5)

Here we do not show a complete expression for $\eta$ to avoid complications;
see Appendix A for it. Note that for $\lambda_{6}<0$ the $e$-folding diverges
at the point where the potential derivative vanishes, namely, at
$\phi=\sqrt{-3+\sqrt{9-8\xi_{2}\lambda_{4}/\lambda_{6}}/2\xi_{2}}$. The scalar
power spectrum amplitude reads

$\displaystyle A_{s}$ $\displaystyle={\lambda_{4}\phi^{6}\over
192\pi^{2}}{\displaystyle\left[1+\xi_{2}\phi^{2}\left(1+6\xi_{2}\right)\right]\left(1+{\lambda_{6}\over\lambda_{4}}\phi^{2}\right)^{3}\over\displaystyle\left(1+\xi_{2}\phi^{2}\right)\left[1+{3\lambda_{6}\over
2\lambda_{4}}\phi^{2}\left(1+{\xi_{2}\over 3}\phi^{2}\right)\right]^{2}}.$
(4.6)

#### Palatini formalism

In the Palatini formalism, the relation (2.5) between $\chi$ and $\phi$ is the
same as Eqs. (2.22)–(2.24). The slow-roll parameters and $e$-folding are
calculated through Eqs. (2.6)–(2.8):

$\displaystyle\epsilon$
$\displaystyle=\frac{2\left(2\lambda_{4}+3\lambda_{6}\phi^{2}+\xi_{2}\lambda_{6}\phi^{4}\right)^{2}}{\phi^{2}\left(1+\xi_{2}\phi^{2}\right)\left(\lambda_{4}+\lambda_{6}\phi^{2}\right)^{2}},$
(4.7) $\displaystyle\eta$
$\displaystyle=\frac{2\left[6\lambda_{4}+\left(-4\xi_{2}\lambda_{4}+15\lambda_{6}\right)\phi^{2}+7\xi_{2}\lambda_{6}\phi^{4}+2\xi_{2}^{2}\lambda_{6}\phi^{6}\right]}{\phi^{2}\left(1+\xi_{2}\phi^{2}\right)\left(\lambda_{4}+\lambda_{6}\phi^{2}\right)},$
(4.8) $\displaystyle N$ $\displaystyle=\int
d\phi~{}\frac{\phi\left(\lambda_{4}+\lambda_{6}\phi^{2}\right)}{2\left(2\lambda_{4}+3\lambda_{6}\phi^{2}+\xi_{2}\lambda_{6}\phi^{4}\right)}.$
(4.9)

For $\lambda_{6}<0$, the $e$-folding diverges at the same point as the metric
formalism, where the potential derivative vanishes. The scalar power spectrum
amplitude reads

$\displaystyle A_{s}$ $\displaystyle={\lambda_{4}\phi^{6}\over
192\pi^{2}}{\displaystyle\left(1+{\lambda_{6}\over\lambda_{4}}\phi^{2}\right)^{3}\over\displaystyle\left(1+\xi_{2}\phi^{2}\right)\left[1+{3\lambda_{6}\over
2\lambda_{4}}\phi^{2}\left(1+{\xi_{2}\over 3}\phi^{2}\right)\right]^{2}}.$
(4.10)

Figure 6: Effects of the higher dimensional operator $\lambda_{6}\phi^{6}$ in
the metric (left) and Palatini (right) formalisms with $N=60$ (top) and 50
(bottom). We plot contours of fixed $\xi_{2}$ (blue, horizontal) and of fixed
$\lambda_{6}/\lambda_{4}$ (red, vertical) in the $n_{s}$-$r$ plane. The value
of $\lambda_{4}$ is also shown as a density plot. Allowed regions of $1\sigma$
and $2\sigma$ from the Planck experiment [9] (TT,TE,EE+lowE+lensing+BK14+BAO)
are also shown in the center. Dashed lines in the right panels correspond to
$\lambda_{6}=-10^{-16.25,\,-16.5}$ and
$10^{-16.5,\,-16.25,\,-16,\,-15.75,\,-15.5}$ from left to right.

Metric ($N=60$)

Palatini ($N=60$)

Figure 7: Behavior of $\lambda_{4}$ along constant $\xi_{2}$ slices in Fig. 6
for $N=60$. We plot $\lambda_{4}$ as a function of
$\left|\lambda_{6}/\lambda_{4}\right|$ for $\lambda_{6}/\lambda_{4}<0$ (left)
and $\lambda_{6}/\lambda_{4}>0$ (right). The lines are $\xi_{2}=1$ (blue),
$10^{2}$ (red), and $10^{4}$ (green) for the metric case, while
$\xi_{2}=10^{6}$ (blue), $10^{8}$ (red), and $10^{10}$ (green) for the
Palatini case.

Metric ($N=60$)

Palatini ($N=60$)

Figure 8: Behavior of $n_{s}$ (solid, left axis) and $r$ (dashed, right axis)
along constant $\xi_{2}$ slices in Fig. 6 for $N=60$.

Metric ($N=50$)

Palatini ($N=50$)

Figure 9: The same as in Fig. 7 except for $N=50$.

Metric ($N=50$)

Palatini ($N=50$)

Figure 10: The same as in Fig. 8 except for $N=50$.

###  4.2 Results

First in Fig. 6, we show predictions of the metric (left) and Palatini (right)
formalisms at $N=60$ (top) and 50 (bottom) in the $n_{s}$-$r$ plane. The blue
contours show constant values of $\xi_{2}$, while the red ones show constant
values of $\lambda_{6}/\lambda_{4}$. The value of $\lambda_{4}$ is also shown
as a density plot. The smiley marker is the prediction of the quartic chaotic
inflation, while the star denotes the attractor point $\xi_{2}\gg 1$. For
$\xi_{2}<10^{-3}$ and $\xi_{2}>1$, the corresponding blue lines are almost
degenerate with the upper and lower boundaries with the gray regions,
respectively. We see the following:

  * •

For relatively small $\xi_{2}$ ($\lesssim 10^{-1}$), even a small injection of
$\left|\lambda_{6}/\lambda_{4}\right|$ drastically changes the inflationary
predictions in both formalisms.

  * •

For relatively large $\xi_{2}$ ($\gtrsim 1$), the inflationary predictions are
stable against the injection of $\lambda_{6}/\lambda_{4}$ for the metric
formalism (left), just as Sec. 3. On the other hand, the predictions are no
more stable against the injection of
$\left|\lambda_{6}/\lambda_{4}\right|\gtrsim 10^{-5}\xi_{2}^{-1}$ for the
Palatini formalism (right).

In Fig. 7, we plot the behavior of $\lambda_{4}$ for $N=60$ as a function of
$\left|\lambda_{6}/\lambda_{4}\right|$ for $\lambda_{6}/\lambda_{4}<0$ (left
panel) and $\lambda_{6}/\lambda_{4}>0$ (right panel) while fixing the overall
normalization $A_{s}$ as mentioned above. Similarly, in Fig. 8 we plot $n_{s}$
(left axis) and $r$ (right axis). Figs. 9 and 10 are the corresponding ones at
$N=50$. We note the following:

  * •

In the metric formalism (upper panels of Figs. 7–10), deviation from the
attractor occurs at the threshold $\left|\lambda_{6}/\lambda_{4}\right|\sim
10^{-3}\xi_{2}$. For $\lambda_{6}<0$ the behavior of $n_{s}$ and $r$ is
relatively simple (upper left panels of Figs. 8 and 10): as
$\left|\lambda_{6}/\lambda_{4}\right|$ increases, both $n_{s}$ and $r$ start
to decrease at this threshold value. From the relation $n_{s}\simeq
1-6\epsilon+2\eta$ and $r\simeq 16\epsilon$, we see that both $\epsilon$ and
$\eta$ start to decrease at this threshold value while keeping the relation
$\left|\epsilon\right|\ll\left|\eta\right|$. On the other hand, for
$\lambda_{6}>0$ the behavior of $n_{s}$ and $r$ is more complicated (upper
right panels of Figs. 8 and 10): as $\left|\lambda_{6}/\lambda_{4}\right|$
increases, $r$ reaches to ${\mathcal{O}}(1)$ values while $n_{s}$ first
increases and then decreases. From the relation $n_{s}\simeq
1-6\epsilon+2\eta$ and $r\simeq 16\epsilon$, we see that as
$\left|\lambda_{6}/\lambda_{4}\right|$ increases $\epsilon$ reaches to
${\mathcal{O}}(0.1)$ values and exceeds $\eta$, causing the decrease in
$n_{s}$ observed in the upper right panels of Figs. 8 and 10.

  * •

In the Palatini formalism (lower panels of Figs. 7–10), there is no attractor
for $\xi_{2}\gg 1$ and the deviation from the observationally allowed region
occurs around the threshold $\left|\lambda_{6}/\lambda_{4}\right|\sim
10^{-5}\xi_{2}^{-1}$.

###  4.3 Interpretation

Let us interpret our results and estimate the threshold value of $\lambda_{6}$
which causes significant deviation from the observationally allowed region. In
the following we use $N\gg 1$ and $\xi_{2}\gg 1$ and keep the leading
contribution for each order in $\xi_{4}$ expansion when necessary.

In the metric formalism, let us first expand Eq. (4.5) by small
$\lambda_{6}/\lambda_{4}$ around
$\phi\simeq\phi_{\lambda_{6}=0}\simeq\sqrt{4N/3\xi_{2}}$ (see Eq. (2.21)):

$\displaystyle N$ $\displaystyle=\int d\phi~{}\frac{\phi\cdot
6\xi_{2}^{2}\phi^{2}\cdot\left(\lambda_{4}+\lambda_{6}\phi^{2}\right)}{2\cdot\xi_{2}\phi^{2}\cdot\left(2\lambda_{4}+\xi_{2}\lambda_{6}\phi^{4}\right)}+\cdots=\frac{3}{4}\xi_{2}\phi^{2}-\frac{1}{8}\frac{\lambda_{6}}{\lambda_{4}}\xi_{2}^{2}\phi^{6}+\cdots.$
(4.11)

Substituting $\phi=\phi_{\lambda_{6}=0}(1+c(\lambda_{6}/\lambda_{4}))$ and
comparing leading terms in $\lambda_{6}/\lambda_{4}$, we find that the
deviation of $\phi$ is given by

$\displaystyle\phi$
$\displaystyle\simeq\phi_{\lambda_{6}=0}\left(1+\frac{4N^{2}}{27\xi_{2}}\frac{\lambda_{6}}{\lambda_{4}}\right).$
(4.12)

We next expand $\epsilon$ (4.3) and $\eta$ (4.4) by small
$\lambda_{6}/\lambda_{4}$ around the same point of $\phi$, and then substitute
Eq. (4.12):

$\displaystyle\epsilon$
$\displaystyle\simeq\frac{4}{3\xi_{2}^{2}\phi^{4}}\left(1+\frac{\lambda_{6}}{\lambda_{4}}\xi_{2}\phi^{4}\right)\simeq\frac{3}{4N^{2}}\left(1+\frac{32N^{2}}{27\xi_{2}}\frac{\lambda_{6}}{\lambda_{4}}\right),$
(4.13) $\displaystyle\eta$
$\displaystyle\simeq-\frac{4}{3\xi_{2}\phi^{2}}\left(1-\frac{1}{2}\frac{\lambda_{6}}{\lambda_{4}}\xi_{2}\phi^{4}\right)\simeq-\frac{1}{N}\left(1-\frac{32N^{2}}{27\xi_{2}}\frac{\lambda_{6}}{\lambda_{4}}\right).$
(4.14)

We see that the deviation of $\eta$ (and hence of $n_{s}$) becomes sizable for
$\left|\lambda_{6}/\lambda_{4}\right|\gtrsim\xi_{2}/N^{2}$. From the relation
$n_{s}\simeq 1-6\epsilon+2\eta$ and $r\simeq 16\epsilon$, the direction of
deviation in Fig. 6 for positive and negative $\lambda_{6}$ with a fixed
$\xi_{2}$ can also be explained.

In the Palatini formalism, let us first expand Eq. (4.9) by small
$\lambda_{6}/\lambda_{4}$ around
$\phi\simeq\phi_{\lambda_{6}=0}\simeq\sqrt{8N}$ (see Eq. (2.23)):

$\displaystyle N$ $\displaystyle=\int
d\phi~{}\frac{\phi\cdot\left(\lambda_{4}+\lambda_{6}\phi^{2}\right)}{2\cdot\left(2\lambda_{4}+\xi_{2}\lambda_{6}\phi^{4}\right)}+\cdots=\frac{1}{8}\phi^{2}-\frac{1}{48}\frac{\lambda_{6}}{\lambda_{4}}\xi_{2}\phi^{6}+\cdots.$
(4.15)

Substituting $\phi=\phi_{\lambda_{6}=0}(1+c(\lambda_{6}/\lambda_{4}))$ and
comparing leading terms in $\lambda_{6}/\lambda_{4}$, we find

$\displaystyle\phi$
$\displaystyle\simeq\phi_{\lambda_{6}=0}\left(1+\frac{16N^{2}\xi_{2}}{3}\frac{\lambda_{6}}{\lambda_{4}}\right).$
(4.16)

We next expand $\epsilon$ (4.7) and $\eta$ (4.8) by small
$\lambda_{6}/\lambda_{4}$ around the same point of $\phi$, and then substitute
Eq. (4.16):

$\displaystyle\epsilon$
$\displaystyle\simeq\frac{8}{\xi_{2}\phi^{4}}\left(1+\frac{\lambda_{6}}{\lambda_{4}}\xi_{2}\phi^{4}\right)\simeq\frac{1}{8N^{2}\xi_{2}}\left(1+\frac{128N^{2}\xi_{2}}{3}\frac{\lambda_{6}}{\lambda_{4}}\right),$
(4.17) $\displaystyle\eta$
$\displaystyle\simeq-\frac{8}{\phi^{2}}\left(1-\frac{1}{2}\frac{\lambda_{6}}{\lambda_{4}}\xi_{2}\phi^{4}\right)\simeq-\frac{1}{N}\left(1-\frac{128N^{2}\xi_{2}}{3}\frac{\lambda_{6}}{\lambda_{4}}\right).$
(4.18)

We see that the deviation of $\eta$ (and hence of $n_{s}$) becomes sizable for
$\left|\lambda_{6}/\lambda_{4}\right|\gtrsim 10^{-1}/N^{2}\xi_{2}$ and that
the direction of the deviation is the same as in the metric formalism.

To summarize, the deviation of $\eta$ (and hence of $n_{s}$) becomes sizable
for

$\displaystyle\left|\lambda_{6}\right|$
$\displaystyle\gtrsim\begin{cases}\displaystyle{\lambda_{4}\xi_{2}\over
N^{2}}&~{}~{}\text{(metric)},\\\\[8.53581pt] \displaystyle
10^{-1}{\lambda_{4}\over N^{2}\xi_{2}}&~{}~{}\text{(Palatini)},\end{cases}$
(4.19)

while $\epsilon$ (and hence $r$) is rather insensitive compared to $\eta$. If
we substitute $\lambda_{4}$ with $A_{s}$, we get

$\displaystyle\left|\lambda_{6}\right|$
$\displaystyle\gtrsim\begin{cases}\displaystyle 10^{2}{A_{s}\xi_{2}^{3}\over
N^{4}}\sim 10^{-6}{\xi_{2}^{3}\over
N^{4}}&~{}~{}\text{(metric)},\\\\[8.53581pt] \displaystyle{A_{s}\over
N^{4}}\sim 10^{-9}{1\over N^{4}}&~{}~{}\text{(Palatini)}.\end{cases}$ (4.20)

We see that near the attractor $\xi_{2}\gg 1$ the metric formalism is stable
against injection of $\lambda_{6}$ (unless $\xi_{2}\gg 10^{5}$), while it is
extremely sensitive to even tiny injection of $\lambda_{6}$ of order
$10^{-16}$ in the Palatini formalism, regardless of $\xi_{2}$.

As we increase $\left|\lambda_{6}\right|$ from zero, the inflationary
prediction starts to deviate at around the value in the right-hand side of Eq.
(4.19). We note that this occurs when the higher dimensional operator
$\lambda_{6}\phi^{6}$ is still much smaller than the lower dimensional one
$\lambda_{4}\phi^{4}$. Indeed, by substituting the approximate values
$\phi\simeq\sqrt{4N/3\xi_{2}}$ (metric) and $\sqrt{8N}$ (Palatini), the
condition

$\displaystyle\left|\lambda_{6}\phi^{6}\right|$
$\displaystyle\lesssim\lambda_{4}\phi^{4}$ (4.21)

becomes

$\displaystyle\left|\lambda_{6}\right|$
$\displaystyle\lesssim\begin{cases}\displaystyle{\lambda_{4}\xi_{2}\over
N}&~{}~{}\text{(metric)},\\\\[8.53581pt] \displaystyle
10^{-1}{\lambda_{4}\over N}&~{}~{}\text{(Palatini)},\end{cases}$ (4.22)

which is well satisfied at the value in the right-hand side of Eq. (4.19).

##  5 Summary and discussion

The Higgs(-like) inflation with the quartic coupling $\lambda_{4}$ and the
non-minimal coupling $\xi_{2}$ is one of the best models from the viewpoint of
cosmic microwave background observations. We have investigated the sensitivity
of this model to higher dimensional operators in the Weyl rescaling factor
($\xi_{4}\phi^{4}$) or in the potential ($\lambda_{6}\phi^{6}$) both in the
metric and Palatini formalisms.

We have found that, in the metric formalism, injection of order
$\left|\xi_{4}\right|\sim 10^{-6}$ or $\left|\lambda_{6}\right|\sim 10^{-16}$
makes the value of $n_{s}$ out of the observationally allowed region for
$\xi_{2}\lesssim 1$, while the inflationary predictions are relatively stable
against $\lambda_{6}$ or $\xi_{4}$ for $\xi_{2}\gg 1$ because of the existence
of attractor. On the other hand, in the Palatini formalism, we have found that
large $\xi_{2}\gg 1$ does not help the stability: injection of order
$\left|\xi_{4}\right|\sim 10^{-6}$ or $\left|\lambda_{6}\right|\sim 10^{-16}$
spoils the successful inflationary predictions regardless of the value of
$\xi_{2}$. We have also pointed out that these threshold values cannot be
estimated from a naïve comparison between $\xi_{2}\phi^{2}$ and
$\xi_{4}\phi^{4}$ or between $\lambda_{4}\phi^{4}$ and $\lambda_{6}\phi^{6}$
at the $\phi$ value corresponding to $e$-folding $N\sim 50-60$. Our study
underscores the theoretical challenge in realizing inflationary models with
the nonminimal coupling in the Palatini formalism.

Our result shows that large $\xi_{2}$ is necessary not only for the
realization of inflation itself but also for the robustness of the prediction
against the injection of the tiny higher dimensional operators in the metric
formalism. This is interesting from theoretical point of view because it is
unlikely to have such a large $\xi_{2}$ only in a single term in the field
expansion in the low energy effective field theory. This may indicate a
principle beyond.\diamondsuit11\diamondsuit11$\diamondsuit$11 See e.g. Ref.
[64]. \diamondsuit12\diamondsuit12$\diamondsuit$12 Even in the vanilla model
with only $\lambda_{4}$ and $\xi_{2}$, it is theoretically intriguing why we
have $\lambda_{2}\ll\lambda_{4}$ and $1\ll\xi_{2}$; the former is nothing but
the hierarchy problem for the Higgs mass-squred.

It is known that the radiative corrections to $\lambda_{4}$ becomes important
for $\lambda_{4}\ll 1$, namely for the critical Higgs inflation. This will be
studied in a separate publication.

### Acknowledgments

The authors are grateful to Satoshi Yamaguchi for useful comments. The work of
RJ was supported by Grants-in-Aid for JSPS Overseas Research Fellow (No.
201960698). The work of RJ was supported by IBS under the project code,
IBS-R018-D1. The work of SP was supported in part by the National Research
Foundation of Korea (NRF) grant funded by the Korean government (MSIP)
(NRF-2018R1A4A1025334) and (NRF-2019R1A2C1089334).

##  Appendix A Equations

In this appendix we summarize expressions for the $\phi$-$\chi$ relations,
slow-roll parameters, and $e$-folding. We consider $\xi_{2}>0$,
$\lambda_{4}>0$, and $\phi>0$ for Appendix A.1, while $\xi_{2}>0$,
$\lambda_{4}>0$, and $\phi>0$ for Appendix A.2.

###  A.1 Correction to the nonminimal coupling

#### Metric formalism

$\phi$-$\chi$ relation:

$\displaystyle\frac{d\chi}{d\phi}$
$\displaystyle=\frac{\sqrt{1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}+6\left(\xi_{2}\phi+2\xi_{4}\phi^{3}\right)^{2}}}{1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}}.$
(A.1)

Slow-roll parameters and $e$-folding:

$\displaystyle\epsilon$
$\displaystyle=\frac{8(1-\xi_{4}\phi^{4})^{2}}{\phi^{2}\left[1+\left(\xi_{2}+6\xi_{2}^{2}\right)\phi^{2}+\left(\xi_{4}+24\xi_{2}\xi_{4}\right)\phi^{4}+24\xi_{4}^{2}\phi^{6}\right]},$
(A.2) $\displaystyle\eta$
$\displaystyle=\frac{4\left[\begin{matrix}3+\left(\xi_{2}+12\xi_{2}^{2}\right)\phi^{2}+(-2\xi_{2}^{2}-12\xi_{2}^{3}+24\xi_{2}\xi_{4}-11\xi_{4})\phi^{4}\\\\[2.58334pt]
-\left(18\xi_{2}\xi_{4}+144\xi_{2}^{2}\xi_{4}\right)\phi^{6}-\left(2\xi_{2}^{2}\xi_{4}+12\xi_{2}^{3}\xi_{4}+11\xi_{4}^{2}+384\xi_{2}\xi_{4}^{2}\right)\phi^{8}\\\\[2.58334pt]
+\left(\xi_{2}\xi_{4}^{2}-12\xi_{2}^{2}\xi_{4}^{2}-288\xi_{4}^{3}\right)\phi^{10}+\left(3\xi_{4}^{3}+72\xi_{2}\xi_{4}^{3}\right)\phi^{12}+96\xi_{4}^{4}\phi^{14}\end{matrix}\right]}{\phi^{2}\left[1+\left(\xi_{2}+6\xi_{2}^{2}\right)\phi^{2}+\left(\xi_{4}+24\xi_{2}\xi_{4}\right)\phi^{4}+24\xi_{4}^{2}\phi^{6}\right]^{2}},$
(A.3) $\displaystyle N$ $\displaystyle=\int
d\phi~{}\frac{\phi\left[1+\left(\xi_{2}+6\xi_{2}^{2}\right)\phi^{2}+\left(\xi_{4}+24\xi_{2}\xi_{4}\right)\phi^{4}+24\xi_{4}^{2}\phi^{6}\right]}{4(1-\xi_{4}\phi^{4})(1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4})}$
$\displaystyle=\left\\{\begin{matrix}\displaystyle\frac{1+6\xi_{2}}{8\sqrt{\xi_{4}}}~{}\operatorname{arctanh}\left[\sqrt{\xi_{4}}\phi^{2}\right]-\frac{3}{4}\ln\left[\left(1-\xi_{4}\phi^{4}\right)\left(1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}\right)\right]\\\\[10.33327pt]
=\displaystyle\frac{1+6\xi_{2}}{16\sqrt{\xi_{4}}}\ln\left[\frac{1+\sqrt{\xi_{4}}\phi^{2}}{1-\sqrt{\xi_{4}}\phi^{2}}\right]-\frac{3}{4}\ln\left[\left(1-\xi_{4}\phi^{4}\right)\left(1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}\right)\right]&~{}~{}~{}~{}(\xi_{4}>0~{}\&~{}\phi<\xi_{4}^{1/4}),\\\\[10.33327pt]
\displaystyle\frac{1+6\xi_{2}}{8\sqrt{-\xi_{4}}}~{}{\rm
arctan}\left[\sqrt{-\xi_{4}}\phi^{2}\right]-\frac{3}{4}\ln\left[\left(1-\xi_{4}\phi^{4}\right)\left(1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}\right)\right]\\\\[10.33327pt]
=\displaystyle\frac{1+6\xi_{2}}{16\sqrt{\xi_{4}}}\ln\left[\frac{1+\sqrt{\xi_{4}}\phi^{2}}{1-\sqrt{\xi_{4}}\phi^{2}}\right]-\frac{3}{4}\ln\left[\left(1-\xi_{4}\phi^{4}\right)\left(1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}\right)\right]&~{}~{}~{}~{}(\xi_{4}<0).\end{matrix}\right.$
(A.4)

#### Palatini formalism

$\phi$-$\chi$ relation:

$\displaystyle\frac{d\chi}{d\phi}$
$\displaystyle=\frac{1}{\sqrt{1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4}}},$ (A.5)
$\displaystyle\chi$
$\displaystyle=-i\sqrt{\frac{\xi_{2}+\sqrt{\xi_{2}^{2}-4\xi_{4}}}{2\xi_{4}}}~{}F\\!\left(i~{}\operatorname{arcsinh}\left[\sqrt{\frac{2\xi_{4}}{\xi_{2}+\sqrt{\xi_{2}^{2}-4\xi_{4}}}}~{}\phi\right]\,\Bigg{|}\,-1+\frac{\xi_{2}(\xi_{2}+\sqrt{\xi_{2}^{2}-4\xi_{4}})}{2\xi_{4}}\right),$
(A.6)

where $F\\!\left(\varphi\,|\,m\right)$ is the elliptic integral of the first
kind.
Slow-roll parameters and $e$-folding:

$\displaystyle\epsilon$
$\displaystyle=\frac{8(1-\xi_{4}\phi^{4})^{2}}{\phi^{2}(1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4})},$
(A.7) $\displaystyle\eta$
$\displaystyle=\frac{4(3-2\xi_{2}\phi^{2}-14\xi_{4}\phi^{4}-2\xi_{2}\xi_{4}\phi^{6}+3\xi_{4}^{2}\phi^{8})}{\phi^{2}(1+\xi_{2}\phi^{2}+\xi_{4}\phi^{4})},$
(A.8) $\displaystyle N$ $\displaystyle=\int
d\phi~{}\frac{\phi}{4(1-\xi_{4}\phi^{4})}$
$\displaystyle=\left\\{\begin{matrix}\displaystyle\frac{1}{8\sqrt{\xi_{4}}}~{}\operatorname{arctanh}\left[\sqrt{\xi_{4}}\phi^{2}\right]=\frac{1}{16\sqrt{\xi_{4}}}\ln\left[\frac{1+\sqrt{\xi_{4}}\phi^{2}}{1-\sqrt{\xi_{4}}\phi^{2}}\right]&~{}~{}~{}~{}(\xi_{4}>0~{}\&~{}\phi<\xi_{4}^{1/4}),\\\\[10.33327pt]
\displaystyle\frac{1}{8\sqrt{-\xi_{4}}}~{}{\rm
arctan}\left[\sqrt{-\xi_{4}}\phi^{2}\right]=\frac{1}{16\sqrt{\xi_{4}}}\ln\left[\frac{1+\sqrt{\xi_{4}}\phi^{2}}{1-\sqrt{\xi_{4}}\phi^{2}}\right]&~{}~{}~{}~{}(\xi_{4}<0).\end{matrix}\right.$
(A.9)

###  A.2 Correction to the potential

#### Metric formalism

$\phi$-$\chi$ relation:

$\displaystyle\frac{d\chi}{d\phi}$
$\displaystyle=\frac{\sqrt{1+\xi_{2}(1+6\xi_{2})\phi^{2}}}{1+\xi_{2}\phi^{2}},$
(A.10) $\displaystyle\chi$
$\displaystyle=\sqrt{\frac{1+6\xi_{2}}{\xi_{2}}}\operatorname{arcsinh}\left[\sqrt{\xi_{2}\left(1+6\xi_{2}\right)}\phi\right]-\sqrt{6}\operatorname{arctanh}\left[\frac{\sqrt{6}\xi_{2}\phi}{\sqrt{1+\xi_{2}\left(1+6\xi_{2}\right)\phi^{2}}}\right]$
$\displaystyle=\sqrt{\frac{1+6\xi_{2}}{\xi_{2}}}\ln\left[\sqrt{\xi_{2}\left(1+6\xi_{2}\right)}\phi+\sqrt{1+\xi_{2}\left(1+6\xi_{2}\right)\phi^{2}}\right]$
$\displaystyle\quad-\frac{\sqrt{6}}{2}\ln\left[\frac{\sqrt{1+\xi_{2}\left(1+6\xi_{2}\right)\phi^{2}}+\sqrt{6}\xi_{2}\phi}{\sqrt{1+\xi_{2}\left(1+6\xi_{2}\right)\phi^{2}}-\sqrt{6}\xi_{2}\phi}\right].$
(A.11)

Slow-roll parameters and $e$-folding:

$\displaystyle\epsilon$
$\displaystyle=\frac{2(2+3\lambda_{64}\phi^{2}+\xi_{2}\lambda_{64}\phi^{4})^{2}}{\phi^{2}\left[1+(\xi_{2}+6\xi_{2}^{2})\phi^{2}\right](1+\lambda_{64}\phi^{2})^{2}},$
(A.12) $\displaystyle\eta$
$\displaystyle=\frac{2\left[\begin{matrix}6+(2\xi_{2}+24\xi_{2}^{2}+15\lambda_{64})\phi^{2}+(-4\xi_{2}^{2}-24\xi_{2}^{3}+2\xi_{2}\lambda_{64}+72\xi_{2}^{2}\lambda_{64})\phi^{4}\\\\[2.58334pt]
+(9\xi_{2}^{2}\lambda_{64}+36\xi_{2}^{3}\lambda_{64})\phi^{6}+(2\xi_{2}^{3}\lambda_{64}+12\xi_{2}^{4}\lambda_{64})\phi^{8}\end{matrix}\right]}{\phi^{2}\left[1+(\xi_{2}+6\xi_{2}^{2})\phi^{2}\right]^{2}(1+\lambda_{64}\phi^{2})},$
(A.13) $\displaystyle N$ $\displaystyle=\int
d\phi~{}\frac{\phi\left[1+(\xi_{2}+6\xi_{2}^{2})\phi^{2}\right](1+\lambda_{64}\phi^{2})}{2(1+\xi_{2}\phi^{2})(2+3\lambda_{64}\phi^{2}+\xi_{2}\lambda_{64}\phi^{4})},$
(A.14)

where we use

$\displaystyle\lambda_{64}\equiv{\lambda_{6}/\lambda_{4}}$ (A.15)

for abbreviation.

#### Palatini formalism

$\phi$-$\chi$ relation:

$\displaystyle\frac{d\chi}{d\phi}$
$\displaystyle=\frac{1}{\sqrt{1+\xi_{2}\phi^{2}}},$ (A.16) $\displaystyle\chi$
$\displaystyle=\frac{1}{\sqrt{\xi_{2}}}~{}\operatorname{arcsinh}\left[\sqrt{\xi_{2}}\phi\right]=\frac{1}{\sqrt{\xi_{2}}}\ln\left[\sqrt{\xi_{2}}\phi+\sqrt{1+\xi_{2}\phi^{2}}\right].$
(A.17)

Slow-roll parameters and $e$-folding:

$\displaystyle\epsilon$
$\displaystyle=\frac{2(2+3\lambda_{64}\phi^{2}+\xi_{2}\lambda_{64}\phi^{4})^{2}}{\phi^{2}(1+\xi_{2}\phi^{2})(1+\lambda_{64}\phi^{2})^{2}},$
(A.18) $\displaystyle\eta$
$\displaystyle=\frac{2\left[6+(-4\xi_{2}+15\lambda_{64})\phi^{2}+7\xi_{2}\lambda_{64}\phi^{4}+2\xi_{2}^{2}\lambda_{64}\phi^{6}\right]}{\phi^{2}(1+\xi_{2}\phi^{2})(1+\lambda_{64}\phi^{2})},$
(A.19) $\displaystyle N$ $\displaystyle=\int
d\phi~{}\frac{\phi(1+\lambda_{64}\phi^{2})}{2(2+3\lambda_{64}\phi^{2}+\xi_{2}\lambda_{64}\phi^{4})}.$
(A.20)

##  Appendix B Difference from $f(R)$ Palatini theories

It has been pointed out in Ref. [65] that $f(R)$ theories in Palatini
formalism are equivalent to the strong coupling limit of the Blans-Dicke
theory. This can be seen by rewriting the $f(R)$ action as

$\displaystyle S$ $\displaystyle=\int
d^{4}x~{}\sqrt{-g}f(R)~{}~{}~{}~{}\to~{}~{}~{}~{}S=\int
d^{4}x~{}\sqrt{-g}\left[f(\chi)+f^{\prime}(\chi)(R-\chi)\right].$ (B.1)

This action is equivalent to the $w\to 0$ limit of the Brans-Dicke theory

$\displaystyle S$ $\displaystyle=\int d^{4}x~{}\sqrt{-g}\left[\Phi
R-\frac{w}{\Phi}g^{\mu\nu}\nabla_{\mu}\Phi\nabla_{\nu}\Phi-V(\Phi)\right],$
(B.2)

with the identification $f^{\prime}(\chi)=\Phi$ and $V(\Phi)=\chi
f^{\prime}(\chi)-f(\chi)$. The point here is that there is no kinetic term in
the auxiliary field $\chi$. In the Palatini formalism, no additional kinetic
term of $\chi$ appears when moving to the Einstein frame. The canonical field
becomes $\Phi_{\rm canoncal}\sim\sqrt{w}\ln\Phi$ in that frame and causes
strong coupling problems in the $w\to 0$ limit. This argument does not
directly apply to our setup, since our starting action (2.1) has a kinetic
term in the original frame.

Another issue mentioned in Ref. [65] is about picking up a particular theory
from infinitely many possibilities. In this reference, the relation between
the connection and metric is imposed explicitly by a Lagrange multiplier. If
this multiplier is such that the connection becomes the Levi-Civita type in a
frame different from the original one (e.g. the Einstein frame), a question
would arise why a multiplier that gives the Levi-Civita connection in this
specific frame is chosen despite infinitely many other possibilities. However,
in our setup we regard the connection as a variable which we take variation
with. Then the connection is uniquely determined by the Levi-Civita form made
from the Einstein-frame metric as a result of the variation principle, and
therefore no such an ambiguity arises.

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