# Palatini $F(R,X)$: a new framework for inflationary attractors

Christian Dioguardi  Antonio Racioppi

###### Abstract

Palatini $F(R)$ gravity proved to be a powerful tool in order to realize
asymptotically flat inflaton potentials. Unfortunately, it also inevitably
implies higher-order inflaton kinetic terms in the Einstein frame that might
jeopardize the evolution of the system out of the slow-roll regime. We prove
that a $F(R+X)$ gravity, where $X$ is the inflaton kinetic term, solves the
issue. Moreover, when $F$ is a quadratic function such a choice easily leads
to a new class of inflationary attractors, fractional attractors, that
generalizes the already well-known polynomial $\alpha$-attractors.

###### keywords:

inflation, attractors, Palatini

[Taltech]organization=Tallinn University of Technology, addressline=Akadeemia
tee 23, city=Tallinn, postcode=12618, country=Estonia
[KBFI]organization=National Institute of Chemical Physics and Biophysics,
addressline=Rävala 10, city=Tallinn, postcode=10143, country=Estonia

##  1 Introduction

The observation of the cosmic microwave background radiation (CMB) supports
the cosmological principle. In order to explain the observed flatness and
homogeneity a period of accelerated expansion (inflation) is required in the
very early universe [1, 2, 3, 4]. With the inflationary mechanism we also
achieve to produce primordial inhomogeneities that seed the current large-
scale structure of the universe. The most simple model requires a scalar field
embedded in Einstein gravity which drives the initial expansion. However, more
sophisticated models have shown to achieve interesting results in predicting
the CMB observables (e.g. [5] and refs. therein). The Palatini formulation of
gravity (e.g. [6, 7, 8] and refs. therein), in which the Levi-Civita
connection is considered to be independent from the metric, shows many
appealing features. In particular $F(R)$ models, for which the gravity sector
is taken to be a general function of the Ricci scalar, generate asymptotically
flat potentials that can be used to describe experimentally viable slow-roll
inflation (e.g. [9, 10, 11] and refs. therein). However, models that diverge
faster than $R^{2}$, are not well behaved beyond slow-roll [11], hence they
could only be used as effective theories during the slow-roll phase. In this
paper we extend the class of $F(R)$ theories to the class of $F(R,X)$ where
$X$ is the inflaton kinetic term. Models with non minimal couplings involving
$R$ and $X$ have been already explored in the past (e.g. [12, 13] and refs.
therein) but in the metric formulation of gravity. According to our knowledge,
this is the first time where such kind of study is performed in the Palatini
approach.

In particular we focus on the quadratic models $F(R_{X})=2\Lambda+\omega
R_{X}+\alpha R_{X}^{2}$ where $R_{X}=R+X$. We classify those models and derive
their general predictions for inflation introducing two new inflationary
attractors that we called $canonical$ and $tailed$ fractional attractors.

##  2 Palatini $F(R,X)$

Consider the action

$S=\int d^{4}x\sqrt{-g^{J}}\quantity(\frac{1}{2}F(R_{X})-V(\phi))$ (1)

where we assumed Planck units, $M_{\text{P}}=1$, and a space-like metric
signature. $V(\phi)$ is the inflaton scalar potential and $F(R_{X})$ is an
arbitrary function of its argument. We define111In a similar way, this setup
can be easily generalized to $S=\frac{1}{2}\int
d^{4}x\sqrt{-g^{J}}F(R_{\mathcal{L}})$, where
$R_{\mathcal{L}}=R_{J}+\mathcal{L}(\phi^{i},\psi^{j},A_{\mu}^{k})$ with
$\mathcal{L}(\phi^{i},\psi^{j},A_{\mu}^{k})$ representing the Lagrangian for
all the scalars, fermions and vectors of the theory. After introducing the
auxiliary field $\zeta$, this would imply having all Lagragian
$\mathcal{L}(\phi^{i},\psi^{j},A_{\mu}^{k})$ with a $F^{\prime}(\zeta)$
prefactor (cf. eq. (2)). The corresponding model building and phenomenology
are beyond the scope of this paper and postponed to a future work.
$R_{X}=R_{J}+X$ with $X=-g^{\mu\nu}_{J}\partial_{\mu}\phi\partial_{\nu}\phi$
denoting the inflaton kinetic term and
$R_{J}=g^{\mu\nu}_{J}R_{\mu\nu}(\Gamma)$ where $R_{\mu\nu}(\Gamma)$ is the
Ricci tensor built from the independent connection i.e. we are operating in
the Palatini formulation. We can rewrite the action in the following way by
introducing an auxiliary field $\zeta$:

$S=\int
d^{4}x\sqrt{-g^{J}}\quantity(\frac{F(\zeta)+F^{\prime}(\zeta)(R_{X}-\zeta)}{2}-V(\phi))\,,$
(2)

where the symbol ′ indicates differentiation with respect to the argument of
the function. It is easy to check that action (1) is obtained from action (2)
by taking the solution of the equation of motion for $\zeta$ i.e.
$\zeta=R_{X}$. By means of a conformal transformation
$g_{\mu\nu}^{E}=F^{\prime}(\zeta)g_{\mu\nu}^{J}$ we can rewrite the action in
the Einstein frame where the theory is linear in $R$ and minimally coupled to
the metric $g_{\mu\nu}^{E}$. The action in the Einstein frame reads:

$S=\int
d^{4}x\sqrt{-g^{E}}\quantity(\frac{R_{E}}{2}-\frac{1}{2}g^{\mu\nu}_{E}\partial_{\mu}\phi\partial_{\nu}\phi-U(\zeta,\phi))\,,$
(3)

with

$U(\zeta,\phi)=\frac{V(\phi)}{F^{\prime}(\zeta)^{2}}-\frac{F(\zeta)}{2F^{\prime}(\zeta)^{2}}+\frac{\zeta}{2F^{\prime}(\zeta)}\,.$
(4)

First of all we notice that the scalar field $\phi$ has a canonical kinetic
term in the Einstein frame while $\zeta$ has no kinetic term at all, therefore
it is still auxiliary also in the Einstein frame, as typical of the Palatini
formulation222See A and B for further details.. As we will discuss shortly,
this is a crucial difference with respect to [10, 11]. By taking the variation
with respect to $\zeta$ we get the equation of motion for the auxiliary field
(and the consistency condition $F^{\prime\prime}\neq 0$):

$\frac{1}{4}\quantity(2F(\zeta)-\zeta F^{\prime}(\zeta))=V(\phi)\,,$ (5)

which in principle can be solved in $\zeta$ to get $\zeta(\phi)$. By using (5)
into (4) we get the potential in terms of the auxiliary field only

$U(\zeta)=\frac{\zeta}{4F^{\prime}(\zeta)}\,.$ (6)

Such a result has an immediate consequence on the allowed values of $\zeta$.
When written in the form of eq. (2), it is easy to understand which
consistency constraints are imposed on the theory and its parameters:
$F^{\prime}(\zeta)>0$ is necessary to achieve a low-energy GR limit and a well
defined conformal transformation. Hence, in order to have a stable positive
potential suitable for inflation (see action (3) and eq. (6)) $\zeta\geq 0$
must hold.

Before concluding we remark that eq. (5) was already introduced in [10] where
it holds as an approximation valid in the slow-roll regime333See A for further
details.. However, in this case eq. (5) is exact and valid even in presence of
arbitrary large kinetic terms for the scalar field $\phi$. In other words, the
auxiliary field $\zeta=\zeta(\phi)$ is a function of the scalar field only and
not of its derivatives. This happens because the inflaton kinetic term in the
Einstein frame does not have anymore any $F^{\prime}$ contribution. This
allows us to find an explicit form for $U(\zeta(\phi))$ in terms of the
canonical scalar field whenever it is possible to solve (5) explicitly for
$\zeta=\zeta(\phi)$.

##  3 Quadratic $F(R,X)$

Now we focus on a quadratic $F$, parametrized as

$F(R_{X})=2\Lambda+\omega R_{X}+\alpha{R_{X}}^{2}\,,$ (7)

with $\Lambda,\omega,\alpha$ real constants. This is the most general form for
a quadratic $F(R_{X})$.

In such a case eq. (5) takes the simple form

$\Lambda+\frac{\omega}{4}\zeta=V(\phi)\,,$ (8)

with exact solution

$\zeta=\frac{4}{\omega}\left(V(\phi)-\Lambda\right)=\frac{4\bar{V}(\phi)}{\omega}\,,$
(9)

where $\bar{V}(\phi)=V(\phi)-\Lambda$. This leads to the Einstein frame
potential

$U(\phi)=\frac{V(\phi)-\Lambda}{8\alpha(V(\phi)-\Lambda)+\omega^{2}}=\frac{\bar{V}(\phi)}{8\alpha\bar{V}(\phi)+\omega^{2}}\,.$
(10)

The corresponding vacuum (i.e. $V(\phi)=0$) solutions are

$\zeta_{0}=-4\Lambda/\omega,\qquad
U_{\Lambda}=\frac{\Lambda}{8\alpha\Lambda-\omega^{2}}\,.$ (11)

Moreover, the large potential configuration ( $V(\phi)\to\pm\infty$) presents
a plateau at (see Figs. 1, 3)

$U\approx U_{\alpha}=\frac{1}{8\alpha}\,.$ (12)

Requiring the positivity of $\zeta$ and $U(\phi)$ at any $\phi$ values, vacuum
configuration included (i.e. $\zeta_{0},U_{\Lambda}\geq 0$) leads to two
possible scenarios:

$\displaystyle 1)\quad\omega>0,\quad\Lambda\leq 0,\quad\bar{V}\geq 0\,,$ (13)
$\displaystyle 2)\quad\omega<0,\quad\Lambda>0,\quad\bar{V}<0\,,$ (14)

with the common constraint

$\alpha>\frac{\omega^{2}}{8\Lambda}\,$ (15)

only in case $\Lambda\neq 0$. Notice that an exactly null $\Lambda$ (or
alternatively $\zeta_{0}=0$) is allowed only for $\omega>0$. The sign of
$\omega$ affects the behaviour of the Einstein frame potential (10) in a way
that will be clarified later on. In the following we will study the two
different sign choices separately.

###  3.1 The standard case: $\omega>0$

First of all we notice that, if $V(\phi)$ has an absolute minimum (or
alternatively a horizontal asymptote), we can set it to be null via a
redefinition of $\Lambda$ as long as (13) is satisfied. Without loss of
generality, in this subsection we work in this configuration. Morever, eq.
(13) would in principle allow for negative values for $\alpha$. However,
unless some _ad hoc_ choice of $V(\phi)$, negative values for $\alpha$ would
also imply the insurgence of a pole in $U(\phi)$. In order to avoid such a
case, we restrict ourselves to $\alpha>0$. In such a case, given the
constraints (13), we can easily check that $U_{\alpha}>U_{\Lambda}$ and
$U_{\Lambda}$ becomes the cosmological constant $U_{\text{CC}}$. Consistency
with data [14] then requires $U_{\Lambda}=U_{\text{CC}}\sim
10^{-47}\text{GeV}^{4}$. Treating $\alpha$ and $\omega$ as free parameters,
such a constraint can be solved in function of $\Lambda$, giving

$\Lambda=\frac{\omega^{2}\,U_{\text{CC}}}{8\alpha U_{\text{CC}}-1}.$ (16)

Unless we choose ad hoc gigantic values for $\alpha$ and/or $\omega$, it is
easy to check that $|\Lambda|$ must be approximately of the same order of
magnitude as $U_{\text{CC}}$. Therefore this setup cannot actually solve the
cosmological constant problem, but only reproduce its value by tuning the free
parameters of the model.

Figure 1: Reference plots for $\bar{V}(\phi)$ (continuous) and $U(\phi)$
(dashed) vs. $\phi$ when $\omega>0$.

Comparing our result (10) with the one in [9], we notice that we retain the
same form of the potential (i.e. a plateau for big enough $\bar{V}(\phi)$, see
Fig. 1). In particular, at large potential configuration
$V(\phi)\rightarrow\infty$, this plateau takes the form:

$U(\phi)\approx U_{\alpha}\quantity(1-\frac{\omega^{2}U_{\alpha}}{V(\phi)})$
(17)

Hence, we can provide an asymptotically flat potential independently of the
original choice of $V(\phi)$ as in the standard $F(R)$ case [9] but with the
advantage of having a canonically normalized scalar field $\phi$ and no higher
order kinetic term.

In order to describe the inflationary predictions, we need first of all the
slow roll-parameters, defined as:

$\displaystyle\epsilon(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{2}\quantity(\frac{U^{\prime}(\phi)}{U(\phi)})^{2}\,,$
(18) $\displaystyle\eta(\phi)$ $\displaystyle=$
$\displaystyle\frac{U^{\prime\prime}(\phi)}{U(\phi)}\,.$ (19)

The first slow-roll parameter $\epsilon$ allows us to compute the number of
e-folds of expansion of the Universe as

$N_{e}=\int_{\phi_{\textrm{end}}}^{\phi_{N}}{\rm
d}\phi\,\frac{U(\phi)}{U^{\prime}(\phi)},$ (20)

where the field value at the end of inflation444One might argue that also
$|\eta|=1$ could trigger the end of slow-roll before reaching $\epsilon=1$.
However, this is never the case in our example scenarios, at least for the
parameter space that we considered. is given by
$\epsilon(\phi_{\textrm{end}})=1$, while the field value $\phi_{N}$ at the
time a given scale left the horizon is given by the corresponding $N_{e}$. The
spectral index $n_{\textrm{s}}$ and the tensor-to-scalar ratio $r$ are:

$\displaystyle n_{\textrm{s}}$ $\displaystyle=$ $\displaystyle
1+2\eta(\phi_{N})-6\epsilon(\phi_{N})\,,$ (21) $\displaystyle r$
$\displaystyle=$ $\displaystyle 16\epsilon(\phi_{N})\,,$ (22)

and the amplitude of the scalar power spectrum [15] is

$A_{\textrm{s}}=\frac{1}{24\pi^{2}}\frac{U(\phi_{N})}{\epsilon(\phi_{N})}\simeq
2.1\times 10^{-9}\,.$ (23)

(a)  (b)

(c)  (d)

Figure 2: $r$ vs. $n_{s}$ (a), $r$ vs. $\alpha$ (b), $\alpha$ vs. $n_{s}$ (c),
$\lambda$ vs. $\alpha$ (d) for $\omega=1$ and $V(\phi)=\lambda_{k}\phi^{k}$
with $k=1$ (purple), $k=2$ (blue), $k=3$ (green), $k=4$ (red) and $k\to\infty$
(black) for $N_{e}=50$ (thin, dot-dashed line) and $N_{e}=60$ (thick, dot-
dashed line). In the same color code we show the limiting values $\alpha=0$
(continuous line, bullets) and $\alpha\rightarrow\infty$ (triangles). In (a)
the dashed lines represent the prediction of the standard Palatini $R^{2}$
model [9] for the corresponding potential and the squares the limiting values
for $\alpha\rightarrow\infty$. The gray areas represent the 1,2$\sigma$
allowed regions coming from the latest combination of Planck, BICEP/Keck and
BAO data [16].

In the standard quadratic $F(R)$ case [9], at the leading order, only $r$ is
affected by the Palatini $R^{2}$ term, while all the other observables stay
unchanged. In contrast to that, here we cannot find a compact straightforward
expression for the phenomenological parameters in eqs. (21), (22) and (23) in
terms of the corresponding ones computed for $V(\phi)$, because now the
inflaton field $\phi$ is already canonically normalized in the Einstein frame
action (3). However we can still find a compact expression for $r$ in the
strong coupling limit $\alpha\to\infty$. In such a case, the Einstein frame
potential asymptotically approaches the plateau (12). Therefore, inserting
this value in (23) we can easily prove that at the leading order

$r\approx\frac{1}{12\pi^{2}A_{s}\alpha}\,.$ (24)

This means that we can arbitrarily lower $r$ by increasing $\alpha$, exactly
as found in [9]. On the contrary, we cannot make a model independent
prediction for $n_{s}$.

For illustrative purposes, we consider a monomial potential of the form

$V(\phi)=\lambda_{k}\phi^{k},\qquad\lambda_{k}=\frac{\lambda^{k}}{k!}$ (25)

where the unusual prefactor, $\lambda_{k}$, is chosen for numerical
convenience. In such a case, we have:

$\displaystyle N_{e}$ $\displaystyle=$
$\displaystyle\int^{\phi_{N}}_{\phi_{end}}d\phi\frac{1}{\sqrt{2}\sqrt{\epsilon}_{+}(\phi)}$
(26) $\displaystyle r$ $\displaystyle=$ $\displaystyle
16\epsilon_{+}(\phi_{N})$ (27) $\displaystyle n_{s}$ $\displaystyle=$
$\displaystyle 1-6\epsilon_{+}(\phi_{N})+2\eta_{+}(\phi_{N})$ (28)
$\displaystyle A_{s}$ $\displaystyle=$
$\displaystyle\frac{\omega^{2}\bar{V}(\phi_{N})}{24\pi^{2}\epsilon_{+}(\phi_{N})\left(8\alpha\bar{V}(\phi_{N})+\omega^{2}\right)}$
(29)

where

$\displaystyle\epsilon_{+}(\phi)=\frac{\lambda_{k}^{2}k^{2}\omega^{4}\phi^{2k-2}}{2\bar{V}(\phi)^{2}(8\alpha\bar{V}(\phi)+\omega^{2})^{2}}$
(30)
$\displaystyle\eta_{+}(\phi)=\frac{2\lambda_{k}\omega^{2}\phi^{k-2}}{\bar{V}(\phi)(8\alpha\bar{V}(\phi)+\omega^{2})^{2}}\times$
(31)
$\displaystyle\quantity(8\alpha(k+1)\lambda_{k}\phi^{k}+(k-1)(8\alpha\Lambda-\omega^{2}))$
(32)

In the strong coupling limit $\alpha\rightarrow\infty$, we recover the well-
known polynomial $\alpha$-attractors [17], whose predictions are:

$\displaystyle r$ $\displaystyle\sim$ $\displaystyle 0$ (33) $\displaystyle
n_{s}$ $\displaystyle=$ $\displaystyle 1-\frac{k+1}{k+2}\frac{2}{N_{e}}$ (34)

A numerical analysis for the reference value $\omega=1$ with $\Lambda$ and
$\lambda$ fixed using $\lambda$ eqs. (16) and (23) is showed in Fig. 2, where
we plot the observables $r$ vs. $n_{s}$ (a), $r$ vs. $\alpha$ (b), $\alpha$
vs. $n_{s}$ (c), $\lambda$ vs. $\alpha$ (d) for $V(\phi)=\lambda_{k}\phi^{k}$
with $k=1$ (purple), $k=2$ (blue), $k=3$ (green), $k=4$ (red) and $k\to\infty$
(equivalent to $V(\phi)=e^{\lambda\phi}$) (black) for $N_{e}=50$ (thin, dot-
dashed line) and $N_{e}=60$ (thick, dot-dashed line). In the same color code
we show the limiting values555The case $k\to\infty$ for $\alpha=0$ is not
visible in Fig. 2(a) because far away from the allowed region. $\alpha=0$
(continuous line, bullets) and $\alpha\rightarrow\infty$ (triangles). In (a)
the dashed lines represent the prediction of the standard Palatini $R^{2}$
model [9] for the corresponding potential and the squares the limiting values
for $\alpha\rightarrow\infty$. The gray areas represent the 1,2$\sigma$
allowed regions coming from the latest combination of Planck, BICEP/Keck and
BAO data [16]. Comparing the results of the quadratic $R_{X}$ model with the
standard one [9], we notice that, as expected, we still have a suppression in
$r$. On the other hand, while in [9] $n_{s}$ stays essentially unchanged, now
it increases by increasing $\alpha$. In this way, for any $k\geq 1$, it is
possible to find a $N_{e}$ value so that the predicted $r$ and $n_{s}$ lay in
the experimentally allowed region [16]. Finally, in (d) we notice that the
coupling $\lambda$ has to increase along with $\alpha$ in order to satisfy the
constraint (23). Notice in particular that the value of $\lambda$ does not
change substantially between $N_{e}=50$ and $N_{e}=60$ so the two lines appear
almost superimposed in the plot.

To conclude we stress that the form of (17) is the same as in [17] but our
construction is certainly more immediate as it does not require any field
redefinition, and more general because it can be applied to any positive $V$,
not just monomial choices. Since the potential in (10) with the setup (13)
predicts asymptotically flat potentials whenever $\alpha\gg\omega$, we label
such a choice as _canonical fractional attractors_ , to be counterposed to a
more _exotic_ configuration which will be discussed in the next subsection.

###  3.2 The alternative case: $\omega<0$

The first peculiar feature of this configuration is the wrong sign for the
linear term in $R$ in the Jordan frame action (2). At first glance it might
seem that such a configuration is forbidden, coming with issues in the Jordan
frame like repulsive gravity or a negative inflaton kinetic energy. However,
as explained after eq. (6), the condition $F^{\prime}>0$ ensures that gravity
stays attractive. The same holds for the inflaton kinetic term, which receives
not only contributions from the linear term in $\omega$ but also from the
quadratic term $\alpha R_{X}^{2}$ (see eq. (7)). When all the contributions
are taken into account, it is easy to prove that the positivity of
$F^{\prime}$ ensures also the positivity of the inflaton kinetic energy (see
also eq. (64)). Therefore such a setup is allowed as long as the consistency
constraint of $F^{\prime}>0$ is satisfied (provided that $F^{\prime\prime}\neq
0$) (e.g. [10, 11] and refs. therein). This happens when all the conditions in
eq. (14) are respected. Particularly relevant is the role of $\Lambda>0$ and
of the lower bound on $\alpha$ in eq. (15) that ensure the positivity of
$F^{\prime}$ wherever eq. (9) admits a solution. The simplest way666Another
possible, but more tuned, choice is to have $V(\phi)$ positive but bounded
from above in such a way that $\bar{V}$ remains negative. to achieve it is to
have the Jordan frame scalar potential $V(\phi)$ (semi)definite negative.
Analogously to the previous case, also now $\alpha$ needs to be constrained to
positive values. It is convenient to introduce

$\underline{V}(\phi)=-\bar{V}(\phi)=-V(\phi)+\Lambda\
>\frac{\omega^{2}}{8\alpha}\,,$ (35)

which is strictly positive and bounded from below because of (14). In this way
we can rewrite the Einstein frame potential as

$U(\phi)=\frac{\underline{V}(\phi)}{8\alpha\underline{V}(\phi)-\omega^{2}}\,.$
(36)

Comparing with eq. (10), now $\omega^{2}$ contributes with a negative sign to
the denominator. However it can be easily proven that the Einstein frame
potential (36) is positive and bounded from above (without any appearance of a
pole) thanks to (14) and (35). A reference plot for $U(\phi)$ is given in Fig.
3. We notice that now the potential exhibits two plateaus, the usual
$U_{\alpha}$ in (12), when $\underline{V}\to\infty$, but also another one,
$U\approx U_{\Lambda}$ in (11), when $\underline{V}\to\Lambda$ i.e. $V\to 0$.
Notice that in opposition to the previous $\omega>0$ case, given (14) and
(35), now $U_{\Lambda}>U_{\alpha}$. The height of the plateaus is dictated
only by the $F$’s parameters $\Lambda,\omega,\alpha$ regardless777If we would
consider the Einstein frame action (3) with $U(\phi)$ given by (36) as a
starting point, then all the dependence on the original parameter $\Lambda$
would be moved to the existence of a strictly positive minimum value for
$\underline{V}$. of the initial choice of $V(\phi)$. We label such class of
potentials as _tailed-fractional attractors_. Since we have two plateaus, it
is in principle possible to use one single potential to explain both early
universe inflation, which happens close to the $U_{\Lambda}$ plateau, and late
universe acceleration, where the value $U_{\alpha}$ sets the value of the
cosmological constant observed today. We will discuss such a topic later on.
Now we focus on the phenomenology around the first plateau. In its proximity
(in the strong coupling regime), the scalar potential can be approximated as:

$U(\phi)=U_{\Lambda}\quantity(1-\frac{U_{\Lambda}}{\Lambda
U_{\alpha}}\absolutevalue{V(\phi)})\,,$ (37)

which has the form of a generalized hilltop potential.

Figure 3: Reference plots for $\underline{V}(\phi)$ (continuous) and $U(\phi)$
(dashed) vs. $\phi$ when $\omega<0$.

Also in this case we can get a universal limit for the tensor-to-scalar ratio
$r$ only

$r\approx\frac{2}{3\pi^{2}A_{s}}\frac{\Lambda}{8\Lambda\alpha-\omega^{2}}\,,$
(38)

which holds around the first plateau, that is for $V\sim 0$. When $\alpha\gg
1$, the most trivial configuration is the one where $\omega$ can be neglected
and eq. (38) reduces to eq. (24).

Before moving to a more quantitative analysis, it is important to stress that
an additional constraint needs to be imposed in order to achieve the end of
slow-roll. A qualitative plot for the first slow-roll parameter $\epsilon$ is
shown in Fig. 4. As expected, for $V(\phi)\to 0$ (or $-\infty$), $\epsilon\to
0$ since $U(\phi)$ approaches the corresponding plateaus. However, in between
the two plateaus, $\epsilon$ does not diverge to $+\infty$, ensuring a certain
end of slow-roll, but it reaches a local maximum. This is because the absolute
minimum of $U(\phi)$ is not null. Therefore, in order to end slow-roll, we
need to ensure that the local maximum in $\epsilon$ is bigger than one. We can
intuitively understand (cf. eq. (18)) that this is achieved by lowering the
second plateau $U\approx\frac{1}{8\alpha}$ i.e. by increasing $\alpha$.
Therefore this sets a minimum value $\alpha>\alpha_{min}$. Its numerical value
is model dependent and must be calculated for the specific choice of the
scalar potential $V(\phi)$. However, the existence of the minimum value
$\alpha_{min}$ is model independent.

Figure 4: Reference plot for $\epsilon(\phi)$ (dashed) vs. $\phi$ when
$\omega<0$.

(a)  (b)

(c)  (d)

Figure 5: $r$ vs. $n_{s}$ (a), $r$ vs. $\alpha$ (b), $\alpha$ vs. $n_{s}$ (c),
$\lambda$ vs. $\alpha$ (d) for $\omega=-1$ and $V(\phi)=-\lambda_{k}\phi^{k}$
with $k=4$ (blue), $k=6$ (red), $k=8$ (green) and $k\to\infty$ (black) for
$N_{e}=50$ (dashed), and $N_{e}=60$ (continuous). In the same color code the
squares represent the value $\alpha_{min}$ for which we can have a graceful
exit from inflation. The grey areas represent the 1,2$\sigma$ allowed regions
coming from the latest combination of Planck, BICEP/Keck and BAO data [16].

In order to perform a more quantitative comparison with the canonical
fractional attractors case, we consider now a negative monomial potential
$V(\phi)=-\lambda_{k}\phi^{k}$ with $k\geq 3$ and $\lambda_{k}$ defined in eq.
(25). In this case the inflationary observables are:

$\displaystyle N_{e}$ $\displaystyle=$
$\displaystyle\int_{\phi_{N}}^{\phi_{end}}d\phi\frac{1}{\sqrt{2}\sqrt{\epsilon}_{-}(\phi)}$
(39) $\displaystyle r$ $\displaystyle=$ $\displaystyle
16\epsilon_{-}(\phi_{N})$ (40) $\displaystyle n_{s}$ $\displaystyle=$
$\displaystyle 1-6\epsilon_{-}(\phi_{N})+2\eta_{-}(\phi_{N})$ (41)
$\displaystyle A_{s}$ $\displaystyle=$
$\displaystyle\frac{\omega^{2}\underline{V}(\phi_{N})}{24\pi^{2}\epsilon_{-}(\phi_{N})\left(8\alpha\underline{V}(\phi_{N})-\omega^{2}\right)}$
(42)

where

$\displaystyle\epsilon_{-}(\phi)=\frac{\lambda_{k}^{2}k^{2}\omega^{4}\phi^{2k-2}}{2\underline{V}(\phi)^{2}(8\alpha\underline{V}(\phi)-\omega^{2})^{2}}$
(43)
$\displaystyle\eta_{-}(\phi)=\frac{2\lambda_{k}\omega^{2}\phi^{k-2}}{\underline{V}(\phi)(8\alpha\underline{V}(\phi)-\omega^{2})^{2}}\times$
(44)
$\displaystyle\quantity(8\alpha(k+1)\lambda_{k}\phi^{k}+(k-1)(8\alpha\Lambda-\omega^{2}))$
(45)

The corresponding strong coupling results are:

$\displaystyle r$ $\displaystyle\sim$ $\displaystyle 0$ (46) $\displaystyle
n_{s}$ $\displaystyle=$ $\displaystyle 1-\frac{k-1}{k-2}\frac{2}{N_{e}}\,,$
(47)

which are the hilltop inflationary predictions for small $r$, as expected.

A numerical study for the reference value $\omega=-1$ with
$\Lambda\simeq\frac{6}{5}\frac{1}{8\alpha}$, ensuring that eq. (15) holds, and
$\lambda$ fixed using eq. (23) is showed in Fig. 5, where we plot the
observables $r$ vs. $n_{s}$ (a), $r$ vs. $\alpha$ (b), $\alpha$ vs. $n_{s}$
(c), $\lambda$ vs. $\alpha$ (d) for $V(\phi)=-\lambda_{k}\phi^{k}$ with $k=4$
(blue), $k=6$ (red), $k=8$ (green) and $k\to\infty$ (black) for $N_{e}=50$
(dashed), and $N_{e}=60$ (continuous). In the same color code the squares
represent the value $\alpha_{min}$ for which we can have a graceful exit from
inflation. The grey areas represent the 1,2$\sigma$ allowed regions coming
from the latest combination of Planck, BICEP/Keck and BAO data [16]. In this
case we see that agreement with data requires $k\geq 6$. As in the previous
case, from (d) we notice that the coupling $\lambda$ has to increase along
with $\alpha$ in order to satisfy the constraint (23). Notice in particular
that the value of $\lambda$ does not change substantially between $N_{e}=50$
and $N_{e}=60$ so the two lines appear almost superimposed in the plot.
Moreover, we notice that for $k\to\infty$ (equivalent to replacing
$\lambda_{k}\phi^{k}$ with $e^{\lambda\phi}$), (34) and (47) converge to the
same limit. Before concluding, we stress that the model predicts $r\lesssim
10^{-4}$, implying that the predicted tensor-to-scalar ratio would not be
measurable even by more futuristic high-resolution satellite mission, such as
PICO [18]. This happens because the requirement $\alpha>\alpha_{min}$, needed
to achieve the end of slow-roll, automatically forces $\alpha$ in the strong
coupling regime.

Moreover, when $\alpha>\alpha_{min}$, not only slow-roll inflation ends but
also restarts at a later time, allowing the possibility for a joined
explanation of the earlier and later accelerated expansion of the Universe, as
already mentioned before. In order to understand the corresponding
phenomenology, it is convenient to introduce the parameter $\delta$ so that

$\alpha\Lambda=\frac{\omega^{2}}{8}(1+\delta).$ (48)

Using such a definition in (38), we obtain

$r\approx\frac{1}{12\pi^{2}A_{s}\alpha}\frac{1+\delta}{\delta}\,.$ (49)

in the limit of $\alpha\gg 1$. Also, as long as $\delta\sim O(1)$ (in Fig. 5
we used $\delta\simeq 0.2$), the exact numerical value for $r$ depends on the
actual value of $\delta$ (unless $\delta\ll 1$), but it is still suppressed
and of the same order of (24) in the big $\alpha$ limit. In this case,
however, it is immediate to see that:

$U_{\Lambda}=\frac{\Lambda}{8\alpha\Lambda-\omega^{2}}=\frac{1}{8\alpha}\frac{1+\delta}{\delta}\gtrsim\frac{1}{8\alpha}\,.$
(50)

Therefore, adjusting $U_{\alpha}=\frac{1}{8\alpha}$ to the observed value of
the vacuum energy density [14] also lowers the inflationary plateau making its
value too low to be phenomenologically consistent with the evolution of the
universe. An alternative option is to take $\delta\ll 1$, but $\alpha\delta
A_{s}\gg 1$ so that we still get a small value

$r\sim\frac{1}{12\pi^{2}\alpha\delta A_{s}}\,.$ (51)

For instance, if we consider $U_{\alpha}=\frac{1}{8\alpha}=U_{\text{CC}}\sim
10^{-47}\text{GeV}^{4}$ we have that $\alpha\sim 10^{122}$. In order to get a
value of $r\sim 10^{-6}$ (which still corresponds to a high enough energy
scale for inflation around $10^{15}\text{GeV}$) we would need an extremely
fine-tuned $\delta\sim 10^{-110}$. Unfortunately, even though mathematically
possible, such a tuning of the parameters, looks more like a confirmation
rather than a solution of the cosmological constant problem.

On the other hand, it has been proven that quadratic $F$’s with negative
$\omega$, are the strong coupling limit configuration for $F$’s of order
higher than quadratic [11]. In such cases, independently of the chosen $F$,
the energy potential tail automatically approaches zero by construction,
resembling the feature of the quintessential inflation models (e.g. [19] and
refs. therein), implying a possible more natural solution of the cosmological
constant problem. However, the search for a successful quintessential
$F(R_{X})$ model is beyond the scope of the present article and will be
postponed to a future work.

##  4 Conclusions

We studied single-field inflation embedded in $F(R_{X})$ Palatini gravity. We
showed that such a choice solves several of the issues araising from the usual
Palatini $F(R)$ models while still providing flat inflaton potentials. In
particular we focused on quadratic $F(R_{X})$’s showing that only two possible
configurations are possible. Those configurations both lead to flat inflaton
potentials that can be classified as attractors given their general prediction
on the inflationary observables. We named those two classes canonical
fractional attractors, which generalize the well known polynomial
$\alpha$-attractors, and tailed fractional-attractors which generalize the
predictions of hilltop potentials. Moreover, we showed that the class of
quadratic theories with $\omega<0$ can in principle explain both early and
late universe accelerated expansion with one single potential. Even though
fine tuned, this is a quite intriguing finding. In light of the results of
[11], such an idea deserves further studies, going to appear in a future work.

## Acknowledgments

This work was supported by the Estonian Research Council grants PRG1055,
RVTT3, RVTT7 and the CoE program TK202 “Fundamental Universe”.

##  Appendix A Palatini $F(R)$ and its issues

In this appendix we give a brief summary of the results and issues found in
the context of slow-roll inflation embedded in Palatini $F(R)$ gravity. More
details can be found in [10, 11]. We start by taking the action

$S=\int
d^{4}x\sqrt{-g^{J}}\quantity(\frac{1}{2}F(R_{J})-\frac{1}{2}g^{\mu\nu}_{J}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi))$
(52)

where we assumed Planck units, $M_{\text{P}}=1$, a space-like metric
signature. $V(\phi)$ is the inflaton scalar potential and $F(R)$ is an
arbitrary function of its argument with
$R_{J}=g^{\mu\nu}_{J}R_{\mu\nu}(\Gamma)$ where $R_{\mu\nu}(\Gamma)$ is the
Ricci tensor built from the independent connection i.e. we are operating in
the Palatini formulation. We can rewrite the action by introducing an
auxiliary field $\zeta$:

$\displaystyle S=\int
d^{4}x\sqrt{-g^{J}}\Bigg{(}\frac{F(\zeta)+F^{\prime}(\zeta)(R_{J}-\zeta)}{2}$
$\displaystyle-\frac{1}{2}g^{\mu\nu}_{J}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\Bigg{)}\,,$
(53)

where the symbol ′ indicates differentiation with respect to the argument of
the function. It is easy to check that the action (52) is obtained from the
action (53) by using the solution of the equation of motion for $\zeta$ i.e.
$\zeta=R_{J}$. By means of a conformal transformation
$g_{\mu\nu}^{E}=F^{\prime}(\zeta)g_{\mu\nu}^{J}$ we can rewrite the action in
the Einstein frame where the theory is linear in $R$ and minimally coupled to
the metric $g_{\mu\nu}^{E}$. The action in the Einstein frame reads:

$S=\int
d^{4}x\sqrt{-g^{E}}\quantity(\frac{R_{E}}{2}-\frac{g^{\mu\nu}_{E}\partial_{\mu}\phi\partial_{\nu}\phi}{2F^{\prime}(\zeta)}-U(\zeta,\phi))\,,$
(54)

with

$U(\zeta,\phi)=\frac{V(\phi)}{F^{\prime}(\zeta)^{2}}-\frac{F(\zeta)}{2F^{\prime}(\zeta)^{2}}+\frac{\zeta}{2F^{\prime}(\zeta)}\,.$
(55)

Note that $U(\zeta,\phi)$ as the same functional form as for the $F(R_{X})$
case shown in eq. (4). Provided the consistency condition
$F^{\prime\prime}\neq 0$, the equations of motion of the system are (in the
approximation of exactly homogeneous and isotropic Universe):

$\displaystyle\ddot{\phi}+3H\dot{\phi}+\frac{V^{\prime}(\phi)}{F^{\prime}(\zeta)}=\frac{\dot{\phi}\dot{\zeta}F^{\prime\prime}(\zeta)}{F^{\prime}(\zeta)}\,,$
(56) $\displaystyle
3H^{2}=\frac{1}{2}\frac{\dot{\phi}^{2}}{F^{\prime}(\zeta)}+U(\phi,\zeta)\,,$
(57)
$\displaystyle-\frac{1}{2}\dot{\phi}^{2}F^{\prime}(\zeta)+2V(\phi)-G(\zeta)=0\,,$
(58)

with

$G(\zeta)=\frac{1}{4}\quantity(2F(\zeta)-\zeta F^{\prime}(\zeta))\,.$ (59)

It can be easily checked that eq. (58), when plugged back into action (54)
induces non trivial higher order inflaton kinetic terms which complicate a lot
the dynamics of the system. Moreover, solving exactly the equation (58) proves
to be hard for a generic $F(R)$, therefore we replace it with an equation for
the time derivative of $\zeta$ [10],

$\dot{\zeta}=\frac{3H\dot{\phi}^{2}F^{\prime}(\zeta)+3V^{\prime}(\phi)\dot{\phi}}{2G^{\prime}(\zeta)+\frac{3}{2}\dot{\phi}^{2}F^{\prime\prime}(\zeta)}\,.$
(60)

As long as $\zeta$ solves (58) initially, equation (60) guarantees that (58)
holds at all times. Eq. (60) also shows an issue that might arise in generic
$F(R)$ models: if $G^{\prime}(\zeta)$ and $F^{\prime\prime}(\zeta)$ have
opposite signs, then $\dot{\zeta}$ presents a pole. Such a pole is a direct
consequence of the $1/F^{\prime}$ prefactor in front of the inflaton kinetic
term in the Einstein frame (see eq. (54)) and it might bring to catastrophic
consequences for the dynamics of the system [11]. For instance, this happens
when $F$ is a function of higher order than quadratic. Such kind of
$F^{\prime}s$ provide model independently an Einstein frame scalar potential
featuring a plateau at early times and a tail at late times, allowing the
possibility for a joined explanation of the earlier and later accelerated
expansion of the Universe [11]. However the insurgence of $\dot{\zeta}$ pole
before the end of inflation [11], makes such an idea unviable. As shown in
Section 2, a possible solution is to upgrade to a $F(R_{X})$ model.

##  Appendix B Metric vs. Palatini

In this appendix we give more details about the differences that arise when
using the metric or Palatini formulation of gravity in the context of $F(R)$
and $F(R_{X})$ models. The starting point is the following action:

$S\\!=\\!\\!\int\\!d^{4}x\sqrt{-g^{J}}\quantity(\frac{F(R_{J}+\sigma
X)+(1-\sigma)X}{2}-V(\phi)),$ (61)

where, again, we assumed Planck units, $M_{\text{P}}=1$ and a space-like
metric signature. $V(\phi)$ is the inflaton scalar potential, $F$ is an
arbitrary function of its argument,
$X=-g^{\mu\nu}_{J}\partial_{\mu}\phi\partial_{\nu}\phi$ denotes the inflaton
kinetic term and $R_{J}=g^{\mu\nu}_{J}R_{\mu\nu}(\Gamma)$ where
$R_{\mu\nu}(\Gamma)$ is the Ricci tensor built from the connection
$\Gamma_{J,\mu\nu}^{\rho}$. Setting $\sigma=0,1$ allows to describe
respectively the $F(R)$ and the $F(R_{X})$ scenario. It is easy to check that
when $\sigma=0$ action (61) becomes action (52), while when $\sigma=1$ action
(61) becomes action (1). Regardless of the gravity formulation, we can rewrite
action (61) in terms of the auxiliary field $\zeta$:

$S_{J}=S_{J}(R_{J})+S_{J}(\zeta,\phi)\,,$ (62)

where

$\displaystyle
S_{J}(R_{J})=\int\differential^{4}x\sqrt{-g}_{J}\left[\frac{1}{2}F^{\prime}(\zeta)R_{J}\right]\,,$
(63) $\displaystyle
S_{J}(\zeta,\phi)=\int\differential^{4}x\sqrt{-g}_{J}\mathcal{L}_{J}(\zeta,\phi)$
(64)

with

$\mathcal{L}_{J}(\zeta,\phi)=-\frac{1}{2}\left[\sigma
F^{\prime}(\zeta)+(1-\sigma)\right]\partial_{\mu}\phi\partial^{\mu}\phi-V(\zeta,\phi)$
(65)

and

$V(\zeta,\phi)=V(\phi)-\frac{F(\zeta)-\zeta F^{\prime}(\zeta)}{2}\,.$ (66)

Using the EoM $\zeta=R_{J}+\sigma X$ into (62), we restore eq. (61).
Performing the Weyl rescaling

$g_{\mu\nu}^{E}=F^{\prime}(\zeta)g_{\mu\nu}^{J}\,,$ (67)

needed to move the theory to the Einstein frame, the term (64) becomes

$S_{E}(\zeta,\phi)=\int d^{4}x\sqrt{-g^{E}}\mathcal{L}_{E}(\zeta,\phi)\,,$
(68)

with

$\mathcal{L}_{E}(\zeta,\phi)=-\frac{1}{2}\left[\sigma+\frac{(1-\sigma)}{F^{\prime}(\zeta)}\right]g^{\mu\nu}_{E}\partial_{\mu}\phi\partial_{\nu}\phi-U(\zeta,\phi)\,,$
(69)

and

$U(\zeta,\phi)=\frac{V(\phi)}{F^{\prime}(\zeta)^{2}}-\frac{F(\zeta)}{2F^{\prime}(\zeta)^{2}}+\frac{\zeta}{2F^{\prime}(\zeta)}\,,$
(70)

both in the metric and Palatini formulation (cf. the second and third terms in
actions (54) and (3)). The difference between the two types of gravity arises
in the transformation of the term (63). Performing the rescaling (67) under
Palatini gravity, action (63) becomes simply the Einstein-Hilbert action:

$S^{EH}_{\text{Palatini}}=\int d^{4}x\sqrt{-g^{E}}\frac{R_{E}}{2}\,.$ (71)

On the other hand, in case of metric gravity, an additional contribution is
generated, giving

$S^{EH}_{\text{metric}}=\int
d^{4}x\sqrt{-g^{E}}\quantity(\frac{R_{E}}{2}-\frac{3}{2}\frac{\partial_{\mu}F^{\prime}(\zeta)\partial^{\mu}F^{\prime}(\zeta)}{\left(F^{\prime}(\zeta)\right)^{2}})\,.$
(72)

The second term of eq. (72) represents a kinetic term for $\zeta$. This is the
main difference between the two gravity formulations. Starting from a metric
$F(R)$ or $F(R_{X})$ theory, $\zeta$ becomes a dynamical degree of freedom
after moving to the Einstein frame, implying an eventual double-field
inflationary setup. On the other hand, starting from a Palatini $F(R)$ or
$F(R_{X})$ theory, $\zeta$ remains an auxiliary field after moving the theory
to the Einstein frame, implying just a single-field inflationary setup.

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