§ INTRODUCTION
Cosmological inflation [1, 2, 3, 4, 5, 6, 7], according to which the Universe underwent a phase of quasi-de Sitter expansion, has become central to our present understanding of early Cosmology primarily because it provides a mechanism for the origin of formation of the observed large scale structure [8, 9, 10, 11, 12, 13]. A standard way to model inflation is to introduce a scalar field (the inflaton) that on the one hand provides the vacuum energy driving the expansion and on the other hand through its quantum treatment generates the spatial inhomogeneities that at presently are detected in CMB.

The common framework of all cosmological models is Einstein's General Relativity (GR) with gravity entering in the action through the standard Einstein-Hilbert term and the inflaton coupled minimally to it through the metric. Nevertheless, although gravity is treated classically, the scalar fields are expected to be sensitive to quantum corrections which are likely to generate correction terms as a non-minimal coupling of the scalar fields to the Ricci scalar $\xi\phi^2R$ or quadratic terms like $\gamma R^2$. In fact, the Starobinsky model [1] based on the latter correction has been quite successful in its predictions of inflationary observables [14, 15]. Similarly, the non-minimal coupling to the Ricci scalar is central to the Higgs inflation model [16]. The understanding is that these terms, even if they are absent in the classical action, they are bound to be generated and, therefore, they should be included in the effective action employed to model inflationary phenomena.

In the standard (metric) formulation of GR the affine connection is not an independent variable but it is constraint to depend on the metric through the Levi-Civita relation. In the case of the simple Einstein-Hilbert action with minimally coupled matter fields this is entirely equivalent to the so-called “first order" or Palatini formulation in which the connection is an independent variable [17] (see also [18, 19, 20, 21, 22, 23, 24]). Nevertheless, in the presence of non-minimal corrections, like the ones mentioned above, this no longer true and the two formulations lead to different predictions [25]. The Palatini version of the Starobinsky model does not provide any scalar degree of freedom suitable to play the role of the inflaton in contrast to its metric version. Nevertheless, Palatini inflationary models of a scalar field in the presence of an $R^2$ term have been studied with predictions compatible with existing cosmological data [26, 27] (see also [28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60] for various applications.)

The Palatini formulation of GR coupled to scalar fields falls into the framework of the so-called Metric-Affine theories of gravity in which the metric and the connection represent independent degrees of freedom [61, 62]. The Palatini formulation terminology is used when the matter fields are not coupled to the independent connection. This is usually the case of scalar fields. This generalization of GR to a Metric-Affine theory preserves general covariance and is quite natural from the geometric point of view. The difference between the independent connection and the Levi-Civita one turns out to be a tensor $\tns{C}{_\m^\l_\n}$, named the Distortion  tensor. In general, solving for the Distortion, a Metric-Affine theory can be rewritten as an “equivalent" metric theory which may or may not have the same degrees of freedom as the starting Metric-Affine theory, depending on whether the Distortion is dynamical or not. In the case that new dynamical degrees of freedom are present in the resulting theory, its phenomenology will be different and possibly interesting [63, 64].

In the present article we considered a Metric-Affine theory of gravity based on an action that includes a quadratic  Ricci scalar term ${\cal{R}}^2$ as well as linear and quadratic terms of the Holst invariant $\widetilde{\cal{R}}$, coupled non-minimally to a scalar $\phi$, and studied the inflationary behaviour of the resulting two-scalar field model.  As we have already stated above there is plenty of motivation for the study of quadratic gravitational terms, based on the fact that such terms are bound to be generated by quantum corrections. Therefore, they should be included in the effective theory of gravity to be considered as a framework for the investigation of inflationary phenomena. This is independent of whether the presence of the extra terms will provide additional dynamical degrees of freedom that could either carry out inflation or participate in it through their mixing with a fundamental inflaton. Although that in the present article we focus on the Metric-Affine formulation, the motivation applies equally well to the metric formulation as well. Apart from the theoretical interest on the models, their resulting inflationary phenomenology, although in agreement with present observational data, allows for their disapproval or favourable comparison to other models in the light of future more precise data. We have found that the model reduces to an effective single-field model with a potential analogous to the one arising in the Palatini-${\cal{R}}^2$ model with its characteristic inflationary plateau. The inflationary predictions of the model fall within the latest observational bounds for a wide range of parameters.

In Section <ref> we set up the theoretical framework of Metric-Affine theories formulated in terms of the Distortion  tensor. In Section <ref> we consider Metric-Affine theories based on an action that includes quadratic terms of the Ricci scalar ${\cal{R}}^2$ and the Holst invariant $\widetilde{\cal{R}}^2$ and derive the equivalent metric action. In Section <ref> we consider the above quadratic theory coupled non-minimally to a scalar field.  In Section <ref> we analyze in detail the inflationary predictions of the above model. Finally,
we summarize and conclude in Section <ref>.

We use natural units $\hbar=c=k_B=1$ as well as taking the reduced Planck mass $M_P=1$.  The metric signature is $(-,+,+,+)$ throughout.

§ FRAMEWORK

A Metric-Affine theory is defined in terms of a metric $g_{ \mu\nu}$ and an independent connection $\tns{\widetilde{\Gamma}}{_\m^\r_\n}$. Note that no particular symmetry is assumed for the lower indices of the connection in contrast with the Levi-Civita connection of metric theories where they are symmetric. A curvature tensor is defined as
R__^_̊ ≡ ∂_ μΓ_^_̊-∂_ νΓ_^_̊+Γ_^_̊łΓ_^ł_-Γ_^_̊łΓ_^ł_ ,The Distortion is defined as
C_^_̊ ≡ Γ_^_̊-Γ_^_̊ ,where $\tns{\Gamma}{_\m^\r_\n}$ is the Levi-Civita connection
Γ_^_̊ = 1/2g^ρσ(∂_ μg_ σν + ∂_ νg_ μσ -∂_ σg_ μν) .The Distortion can be shown to be a tensor.

The following two scalars, that are linear in the Riemann tensor, can be defined  as
R   =   R__^_̊_̣ρ^ μg^νσ = R__^^   ,

R   =   (-g)^-1/2ϵ^^_^̊ R__^_̊=(-g)^-1/2ϵ^^^^̊R____̊ ,
where $g$ is the determinant of the metric and $\epsilon^{\m\n\r\s}$ is the totally antisymmetric symbol with $\epsilon^{0123}=1$. The first scalar is the usual Ricci scalar and the second one is the so-called Holst invariant. Note that the pseudoscalar $\widetilde{\cal{R}}$ [65, 66] vanishes identically in a metric theory due to the symmetry in the lower indices of the Levi-Civita connection. Both these scalars can be written in terms of the standard (metric) Ricci scalar $R[g]$ and the Distortion $C$ as[The indices of the Distortion tensor are raised and lowered in the usual way, i.e $\tns{C}{_\a^\b^\g}=\tns{C}{_\a^\b_\d}g^{\gamma\delta}$ and $\tns{C}{_\a_\b_\g}\,=\,g_{ \beta\delta}\tns{C}{_\a^\d_\g}$.]

R   =   R+D_ μC_^^-D_ νC_^^+C_^_łC_^ł^-C_^_łC_^ł^ ,

R   =   2(-g)^-1/2ϵ^ μνρσ(D_ μC___̊+C___̊łC_^ł_) ,

where the covariant differentiation is in terms of the Levi-Civita connection.

The simplest Metric-Affine action is the analogue of the Einstein-Hilbert action
S   =   1/2∫d^4x √(-g)R
=    1/2∫d^4x √(-g){R+D_ μC_^^-D_ νC_^^+C_^_łC_^ł^-C_^_łC_^ł^} .
Variation with respect to the Distortion gives the equation of motion
δS/δC = 0   ⟹  δ_ β^ αC_ν_γ^ν+δ_ γ^ αC_ν^ν_β-C_β_γ^α-C_γ^α_β=0 ,
which has the general solution $C_{\mu\nu\rho}=U_{\mu}\,g_{\nu\rho}\,$
in terms of the arbitrary vector $U_{ \mu}$. Substituting this solution into Eq. (<ref>) the $C-$dependent terms vanish. Therefore, the Metric-Affine theory  (<ref>) is entirely equivalent to standard metric GR for any $U_{ \mu}$. Nevertheless, this is not the case for quadratic actions.

Closing this section we mention that the terms Metric-Affine and Palatini formulation are equivalent as far as they are referring to gravity theory with an independent connection, while torsion and non-metricity being an immediate consequence of that.

§ QUADRATIC METRIC-AFFINE THEORIES

Consider the following generalization of the Starobinsky model
S = ∫d^4x √(-g){α/2R + β/2 R + γ/4R^2 + δ/4R^2} .Without loss of generality we may set $\alpha=1$,  i.e. we identify the parameter $\alpha$ with the reduced Planck mass and rewrite the action in the form

S = ∫d^4x√(-g){1/2(1+γχ)R+1/2(β+δζ)R-γ/4χ^2-δ/4ζ^2 } ,introducing the two auxiliary scalars $\chi$ and $\zeta$. As we will see a dynamical scalar field can be generated in this case. Using  (<ref>) and (<ref>) of the previous section we can rewrite the action in terms of the Distortion $C$ as
S   =   ∫d^4x√(-g){1/2(1+γχ)R +1/2(1+γχ)(D_ μC_^^-D_ νC_^^+C_^_łC_^ł^-C_^_łC_^ł^).
.+(β+δζ)(-g)^-1/2ϵ^ μνρσ(D_ μC___̊+C___̊łC_^ł_) -γ/4χ^2-δ/4ζ^2} .

Substituting the covariant derivatives of the Distortion this is equivalent to
S    =    ∫d^4x√(-g){1/2(1+γχ)R-γ/4χ^2-δ/4ζ^2-1/√(-g)∂_ μ(1/2√(-g)(1+γχ))C_^^.
+1/√(-g)∂_ ν(1/2√(-g)(1+γχ))C_^^+1/2(1+γχ)(C_^_łC_^ł^-C_^_łC_^ł^.
.+Γ_^_C_^^̊+Γ_^_C_^^ -Γ_^_C_^^̊-Γ_^_C_^^ )-1/√(-g)∂_ μ(β+δζ)ϵ^ μνρσC_ νρσ
+(β+δζ)(-g)^-1/2ϵ^ μνρσC___̊łC_^ł_} .

Variation with respect to $\tns{C}{_\a^\b^\g}$ gives
Ω^2/2(δ_ β^ αC__^+δ_ γ^ αC_^_-C__̱^-C_^_)
+Ω^2/√(-g)(-ϵ^^^_C___-ϵ^^^_C___) = J_^̱_ ,

J_^̱_ = 1/2∂_ βΩ^2 δ_ γ^ α -1/2∂_ γΩ^2 δ_ β^ α+1/√(-g)∂_ μΩ^2 ϵ^^__̱.
The conformal factors $\Omega^2, \overline{\Omega}^2 $ are given by $ \Omega^2\,\equiv 1+\gamma\chi$ and $\overline{\Omega}^2\,\equiv\,\beta+\delta\zeta$.
Note also that the tensor $\tns{J}{_\b^\a_\g}$ is antisymmetric in its lower indices, i.e. $\tns{J}{_\b^\a_\g}\,=\,-\tns{J}{_\g^\a_\b}$.
Thus, we proceed by considering a trial solution for $C_{ \mu\nu\rho}$ antisymmetric in the last two indices, namely
C_ μνρ = g_ μν∂_ ρX-g_μρ∂_ νX+ϵ_ μνρσ∂^ σY .Introducing this ansatz into (<ref>) we obtain
∂X = 1/2Δ(Ω^2∂Ω^2+4Ω^2∂Ω^2) ,       ∂Y = 1/√(-g)Δ(Ω^2∂Ω^2-Ω^2∂Ω^2) ,with $\Delta\,=\,\Omega^4+4\overline{\Omega}^4$.

Substituting the solution for the Distortion into the action (<ref>) we obtain

S = ∫d^4x √(-g){ 1/2Ω^2 R + 3/4(∇Ω^2)^2/Ω^2 -3/Ω^2Δ(Ω^2∇Ω^2-Ω^2∇Ω^2)^2 -γ/4χ^2-δ/4ζ^2 } .

The final step is the passage to the Einstein-frame applying a Weyl rescaling of the form  $g_{ \mu\nu}\,=\,\bar{g}_{ \mu\nu}/\Omega^2$. Note that the standard (metric) Ricci scalar is rescaled as $R=\Omega^2 \bar{R} -6\Omega^3 \Box \Omega^{-1}$, in contrast to the Metric-Affine one which follows the rule $\mathcal{R}=\Omega^2 \bar{\mathcal{R}}$, since the curvature $\tns{\cal{R}}{_\m_\n^\r_\s}$ is metric-independent.
Under this Weyl rescaling
we obtain the Einstein-frame action
S = ∫d^4x √(-g̅){1/2R̅[g̅] -3/Ω^4Δ(Ω^2∇̅Ω^2-Ω^2∇̅Ω^2)^2 -1/4γχ^2+δζ^2/Ω^4} .Introducing the field
σ ≡ Ω^2/2Ω^2 .we obtain the scalar field Lagrangian
L = -12(∇̅σ)^2/(1+16σ^2)-1/4γ(1-Ω^-2)^2-1/4δ(2σ-βΩ^-2)^2 .Solving with respect to the non-dynamical $\Omega^2$ we get
δL/δΩ^2=0    ⟹   Ω^-2=δ+2βγσ/δ+β^2γ ,which leads to
L = -12(∇̅σ)^2/(1+16σ^2)-1/4(2σ-β)^2/(δ+β^2γ) .Thus, the model based on the action (<ref>), in addition to the standard graviton, predicts a pseudoscalar gravitational degree of freedom associated with the invariant $\widetilde{\cal{R}}$.
The model can be expressed in terms of a canonical field $s$, defined by
σ=1/4sinh(√(2/3) s) .The potential becomes
V(s) = 1/16(sinh(√(2/3) s)-2β)^2/(δ+β^2γ) .This model and its inflationary behaviour have been considered in [63, 64].
If we were to take the limit $\beta\rightarrow 0$, while $\delta\neq 0$, we would not be able to obtain an acceptable inflationary behaviour as in the case of a non-zero $\beta$.  Thus, the parameter[The parameter $1/(2\beta)$ is called Barbero-Immirzi parameter [67, 68].] $\beta$, associated with the linear Holst term, is crucial  in obtaining a suitable inflationary phenomenology, since it enhances the flatness of the potential in a certain region. It is evident that the parameters $\gamma$ and $\delta$, associated with the presence of the ${\cal{R}}^2$ and $\widetilde{\cal{R}}^2$ term, have a secondary role in the inflationary behaviour of the model, which is controled by $\beta$, while $\gamma$ and $\delta$ affect only the scale of inflation. Even without one of them, the inflationary model would still be viable, since even a single term is sufficient to achieve the desired inflationary scale. Note however, that the above pseudoscalar mode would not be present if the action were to contain only the linear Holst term. At least one of the ${\cal{R}}^2$ or $\widetilde{\cal{R}}^2$ has to be included in order to generate a dynamical scalar degree of freedom [63, 64]. In the absence of $\widetilde{\cal{R}}$, i.e. $\beta=0$, a quadratic Holst term $\widetilde{\cal{R}}^2$ can generate a dynamical scalar as well, but in this case no viable inflation arises. Note also that all this is in contrast to the Metric-Affine version of the Starobinsky model (just an ${\cal{R}}^2$ term present), where no dynamical scalar degree of freedom arises. In Table <ref> we list the possible combinations of curvature scalars up to quadratic terms in the framework of Metric-Affine gravity and indicate the presence or absence of the new dynamical scalar in the Metric equivalent theory as well as the inflationary viability of the corresponding model.

List of different combinations of the curvature scalars in Metric-Affine  gravity (upper row) and the equivalent Metric theory (lower row). VI (NVI) stands for viable (not viable) inflation and $\sigma$ is the dynamical scalar.
[1 ][c]
1.015 !

Metric-Affine Gravity      $\mathcal{R}$     $\mathcal{R}+\widetilde{\mathcal{R}}$     $\mathcal{R}+\mathcal{R}^2$     $\mathcal{R} + \widetilde{\mathcal{R}}^2$     $\mathcal{R}+ \widetilde{\mathcal{R}} +\mathcal{R}^2$     $\mathcal{R} + \widetilde{\mathcal{R}}+ \widetilde{\mathcal{R}}^2$     $\mathcal{R} +\widetilde{\mathcal{R}}^2 +\mathcal{R}^2$     $\mathcal{R} +\widetilde{\mathcal{R}} +\widetilde{\mathcal{R}}^2 +\mathcal{R}^2$

Metric Gravity     GR     GR     GR      $\sigma$ with NVI     $\sigma$ with VI     $\sigma$ with VI     $\sigma$ with NVI     $\sigma$ with VI

§ COUPLING TO A FUNDAMENTAL SCALAR

Assuming that we have a model of a scalar field $\phi$ coupled to gravity in the framework of Metric-Affine theory, as it was mentioned in the introduction non-minimal terms as the non-minimal coupling to the curvature scalars or quadratic scalar curvature terms are bound to be present in the effective theory. Therefore, we start by considering

S=∫d^4 x √(-g){1/2f(ϕ)ℛ +1/2g(ϕ)R +γ/4 ℛ^2+δ/4 R^2 + ℒ_ϕ} ,

ℒ_ϕ= -1/2g^∂_ϕ∂_ϕ-V(ϕ) .
This can be rewritten in terms of the  auxiliary scalars $\chi$ and $\zeta$ as

S=∫d^4 x √(-g){1/2(γχ+f(ϕ) ) ℛ+1/2 (ζ̣+g(ϕ))ℛ-γ/4 χ^2-δ/4 ζ^2 +ℒ_ϕ} .
Defining $\Omega^2=\g\chi + f(\phi)$, $\overline{\Omega}^2 = \d \zeta +g(\phi)$ and following the previous section we obtain

𝒮=∫d^4 x √(-g) {R/2 -
3/Ω^4 Δ(Ω^2 ∂Ω^2 -Ω^2 ∂Ω^2 )^2 -γ/4Ω^4 χ^2-δ/4Ω^4 ζ^2 -1/2(∇ϕ)^2/Ω^2-V(ϕ)/Ω^4} ,
where now we have to substitute $\d\zeta^2 = \left(\overline{\Omega}^2 -g(\phi) \right)^2/\d$ and $χ^2 = (Ω^2 -f(ϕ) )^2/$. Introducing the scalar field $σ=Ω^2/2Ω^2$ we obtain
\ba
\label{eq:action6n}
\mathcal{S}&=&\int {\rm d}^4 x \sqrt{-g} \bigg\{\frac{R}{2} -\frac{24}{1+16\s^2}\frac{(\nabla\s)^2}{2}-\frac{(\nabla\phi)^2}{2\Omega^2}-\frac{\s^2}{\d} \nonumber
\\ && -\frac {(f(\phi) - \Omega^2)^2}{4\g \Omega^4}-\frac{g(\phi)}{4\delta\Omega^4}(g(\phi)-4\sigma\Omega^2)-\frac{V(\phi)}{\Omega^4} \bigg\}\,.
\ea
Varying the action with respect to $Ω^2$ we obtain from the solution of $S/Ω̣^2=0$, that
\be
\label{eq:omega2}
\Omega^2=\frac{4\gamma V(\phi)+f^2(\phi)+\gamma g^2(\phi)/\delta}{f(\phi)+2\gamma\sigma g(\phi)/\delta-\g(\nabla\phi)^2}\,.
\ee
Substituting this back to the action~\eqref{eq:action6n} we get
\be
\label{eq:action8n}
\mathcal{S}=\int {\rm d}^4 x \sqrt{-g} \bigg\{\frac{R}{2} - K_\phi(\phi,\s)\frac{(\nabla\phi)^2}{2} +L_\phi(\phi)\frac{(\nabla\phi)^4}{4} -K_\s(\s) \frac{(\nabla\s)^2}{2}-V_{\rm eff}(\phi,\s)\bigg\}\,,
\ee
\ba
K_\phi(\phi,\s) &=& \frac{f(\phi)+2\g\s g(\phi)/\d}{\gamma g(\phi)^2/\delta+f^2(\phi) +4\g V(\phi)}\,,\nonumber
\\
L_\phi(\phi) &=&  \frac{\g}{\gamma g(\phi)^2/\delta+f^2(\phi)+4\g V(\phi)}\,, \nonumber
\\
K_\s(\s) &=& \frac{24}{1+16\s^2}\,, \nonumber
\\
V_{\rm eff}(\phi,\s) &=& \frac{V(\phi)}{f^2(\phi)+4\gamma V(\phi)}+\frac{1}{\delta}\left(\frac{f^2(\phi)+4\gamma V(\phi)}{\gamma g^2(\phi)/\delta+f^2(\phi)+4\gamma V(\phi)}\right)\left(\sigma-\sigma_0(\phi)\right)^2\,.
\label{eq:3dpot}
\ea
The potential $V_eff(ϕ,σ)$ is positive-definite\footnote{The potential $V(\phi)$ and the parameters $\gamma$ and $\delta$ are taken to be positive.} and gets minimized along the line
\be \sigma_0(\phi)\,=\,\frac{g(\phi)f(\phi)/2}{f^2(\phi)+4\gamma V(\phi)}\,.
\label{eq:sigma_0}
\ee
Note that along this line the potential coincides with that obtained in the $δ=g(ϕ)=0$ case~[26, 27, 29].
A canonical $σ$-field can readily be defined as
\be \sigma_c\,=\,\sqrt{24}\int\,\frac{ {\rm d}\sigma}{\sqrt{1+16\sigma^2}}\,\Longrightarrow\,\sigma\,=\,\frac{1}{4}\sinh\left(\sqrt{\frac{2}{3}}\sigma_c\right)\,.
\label{eq:sigma_can}
\ee
The $ϕ$-dependent mass of $σ_c$ along the minimum line is
\ba
M_{\sigma_c}^2(\phi)&\equiv&\left.\frac{\partial^2V_{eff}}{\partial\sigma_c^2}\right|_{\sigma_0}=\left.\left(\frac{\partial\sigma}{\partial\sigma_c}\right)^2\frac{\partial^2V_{eff}}{\partial\sigma^2}\right|_{\sigma_0}\,+\,{\cancel{\left.\frac{\partial^2\sigma}{\partial\sigma_c^2}\frac{\partial V_{eff}}{\partial\sigma}\right|_{\sigma_0}}} \Rightarrow \nonumber
\\ M_{\sigma_c}^2(\phi)&=&\frac{\left(f^2(\phi)+4\gamma V(\phi)\right)^2+4g^2(\phi)f^2(\phi)}{12\delta\left(f^2(\phi)+4\gamma V(\phi)\right)\left(\gamma g^2(\phi)/\delta+f^2(\phi)+4\gamma V(\phi)\right)}\,>\,0\,.
\ea
Assuming a quartic potential and a quadratic $f(ϕ)$ function, this mass tends to a constant for $ϕ→ ∞$, namely
\be \,M_{\sigma_c}^2(\phi\rightarrow \infty)\,\approx\,\left\{\bear{cc}
\frac{1}{3\gamma}\frac{f^2(\phi)}{f^2(\phi)+4\gamma V(\phi)}\,,&\,g^2(\phi)\gg\phi^4\\
\,&\,\\
\frac{1}{12\delta}\,,&\,g^2(\phi)\ll\phi^4\,.
\eear\right.\ee

Along the minimum line $σ_0(ϕ)$ the contribution of $σ$ to the potential is removed but it still has contribution to the kinetic terms. Substituting $σ_0(ϕ)$ into the $ϕ$-kinetic term, it reduces to the one obtained in the $δ=g(ϕ)=0$ case, namely
\be -\frac{1}{2}\left(\frac{f(\phi)}{f^2(\phi)+4\gamma V(\phi)}\right)(\nabla\phi)^2\,.{\label{PHI-KIN}}\ee
Substituting $σ(ϕ)$ at the minimum in the $σ$ kinetic term we obtain the correction
\ba
-K_\s(\s_0(\phi)) \frac{(\nabla\s_0(\phi))^2}{2}&=&-\frac{1}{2}\left(\frac{12}{1+\frac{4g(\phi)^2f^2(\phi)}{[f^2(\phi)+4\gamma V(\phi)]^2}}\right)(\nabla\phi)^2\left(\frac{g'(\phi)f(\phi)+g(\phi)f'(\phi)}{f^2(\phi)+4\gamma V(\phi)}\right.\nonumber
\\ && \left.-\frac{g(\phi)f(\phi)}{[f^2(\phi)+4\gamma V(\phi)]^2}\left(2f'(\phi)f(\phi)+4\gamma V'(\phi)\right)\right)^2\,.
\label{SIGMA-KIN}
\ea
Thus, the overall $ϕ$-kinetic term will be the sum of~\eqref{PHI-KIN} and~\eqref{SIGMA-KIN}.

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.8]{plot3D.pdf}
\caption{The two-field effective potential given by Eq.~\eqref{eq:3dpot}. The parameters used are $\g=10^6$,  $\d=\xi=\bar{\xi}=1$ and $\bar{\xi}' = 0$, while the quartic coupling $\lambda$ is fixed by the observed value of the amplitude of the scalar power spectrum, $A_s^\star=2.1 \times 10^{-9}$, at the pivot scale $k_\star=0.05\, {\rm Mpc^{-1}}$ in the $\s_0(\phi)$ direction (red line).}
\label{fig:3dplot}
\end{center}
\end{figure}

At this point we may proceed to specify the coupling functions\footnote{See also~[69] for a general discussion on non-minimal couplings to the torsion and non-metricity tensors.} of $ϕ$ with $R$ and $R$, namely $f(ϕ)$ and $g(ϕ)$. The standard non-minimal coupling to the Ricci scalar adopted is a quadratic function of $ϕ$ that reduces to a constant in the absence of $ϕ$. In our case this would correspond to taking
\be f(\phi) = 1+\xi\phi^2\,. \label{eq:f_func}\ee
Next, for the case of the $R$ term, since no such term is present in the minimal gravitational action and is expected to arise as a quantum correction attributed to $ϕ$, it is reasonable to assume that $g(ϕ)$ should vanish in the absence of it. In addition, since $R$ is parity-odd, it would also be reasonable to assume that $g(ϕ)$ is an odd function of $ϕ$ (see~[70, 71] for quadratic non-minimal couplings to the Holst invariant). Therefore, we shall adopt the following ansatz, namely\footnote{The parameters $\bar{\xi},\,\bar{\xi}'$, as well as the above parameter $\xi$ are assumed to be positive.}
\be g(\phi)=\bar{\xi}\phi+\bar{\xi}'\phi^3\,.
\label{eq:g_func}
\ee

In figure~\ref{fig:3dplot} we display the two-field effective potential of Eq.~\eqref{eq:3dpot} (for the canonical normalized pseudoscalar $σ_c$) using the coupling functions~\eqref{eq:f_func}-\eqref{eq:g_func} and the quartic potential $V(ϕ)=λ/4ϕ^4$. For illustrative purposes we have chosen a linear dependence on $ϕ$ for the function $g(ϕ)$, \textit{i.e.} $ξ̅'=0$. The values of the rest of the parameters are\footnote{The parameter $\delta$ will be set equal to one from now on without any further reference to it. It turns out that, along the minimum line, where we study inflation, it appears only on the higher order kinetic term which is negligible anyway.} $=10^6$ and $=̣ξ=ξ̅=1$. The parameter $λ$ is fixed by the observed value of the amplitude of the scalar power spectrum, $A_s^⋆=2.1 ×10^-9$, at the pivot scale $k_⋆=0.05  Mpc^-1$. The red curve corresponds to the minimum direction of Eq.~\eqref{eq:sigma_0}. As it is evident from this figure, in the $_c$ direction the potential grows exponentially as $_c$ gets larger, hence the pseudoscalar is doomed to fall into the valley. This will be verified in the framework of the numerical treatment of the next section.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.8]{figPotentials.pdf}
\caption{The single-field effective potential in the $\s_0(\phi)$ direction. The parameters used coincide with those in figure~\ref{fig:3dplot}. The blue dotted line corresponds to the potential $U(\phi)$ plotted against the non-canonical field $\phi$, while the red one to $U(\phi(\phi_ c))$ plotted in terms of the canonical field $\phi_c$. The dots from right to left give the field values at the pivot scale $k_\star=0.05\, {\rm Mpc^{-1}}$ and at the end of inflation. }
\label{fig:1dplot}
\end{center}
\end{figure}

It is interesting to have a look at the resulting effective Lagrangian along the $σ_0(ϕ)$ line in the large field limit. Taking the potential to be a standard renormalizable quartic potential $V(ϕ)=λ/4ϕ^4$, we obtain
\be {\cal{L}}(\phi\rightarrow\infty)\,\approx\,-\frac{1}{2}\frac{(\nabla\phi)^2}{\phi^2}\left(\frac{3}{2} + \frac{1}{\xi+\frac{\gamma\lambda}{\xi}}\right)\,-\frac{\lambda}{4(\xi^2+\gamma\lambda)}\,.{\label{EFFLANG}}\ee
The potential shows the familiar Palatini-plateau, while the kinetic term is expressed in terms of an effective $ξ$-parameter
\be \xi_{\rm eff}^{-1}\,=\,\frac{3}{2}+\frac{1}{\xi+\frac{\lambda\gamma}{\xi}}\,.\ee
Note that $ξ_eff$ is less than $1$.
We have not displayed in ({\ref{EFFLANG}}) the quartic kinetic term, anticipating that it is not expected to play any role in inflationary considerations, something that will be supported by our subsequent numerical analysis.

In figure~\ref{fig:1dplot} we have plotted the single-field effective potential results when the pseudoscalar $σ$ moves along the minimum line $σ_0(ϕ)$, \textit{i.e.} $U(ϕ)=V_eff(ϕ,_0(ϕ))$. The values of the parameters have been taken to be the same with those in figure~\ref{fig:3dplot}. Since the inversion from the non-canonical field $ϕ$ to the canonical one $ϕ_c$ cannot be achieved analytically, we plot $U(ϕ)$ (blue dotted line) and $U(ϕ(ϕ_c))$ (red line) doing the inversion to the canonical field numerically.

\section{Inflation}
In this section we analyze the inflationary behaviour of the model. We aim at restricting the parametric space, comparing its inflationary predictions to the corresponding latest observational bounds as set by  the latest
combination of Planck, BICEP/Keck and BAO data~[14, 15].
\label{sec:inflation}
\subsection{Single-field inflation}
Our system of two fields is described by the action ({\ref{eq:action8n}}), where the functions\footnote{For simplicity we have omitted the arguments of these functions.} $K_ ϕ, L_ϕ$ and $V_eff$ are given by ({\ref{eq:3dpot}}). Starting from this action the equations of motion arising from it in an FRW background are\footnote{Notice here that we use the canonical normalized pseudoscalar, given by Eq.~\eqref{eq:sigma_can}.}
%However, as we shall show in what follows, it reduces to an effective one-field system, since the pseudoscalar $\sigma$ will move along the minimum $\s_0(\phi)$ direction, given by Eq.~\eqref{eq:sigma_0}.
\ba
&&(K_{ \phi}+3L_{ \phi}\dot{\phi}^2)\ddot{\phi}+3H(K_{ \phi}+L_{ \phi}\dot{\phi}^2)\dot{\phi} + \dot{\phi} \dot{\sigma}_{c} \frac{\partial K_{ \phi}}{\partial\sigma_{c}} +\left(\frac{1}{2}\frac{\partial K_{ \phi}}{\partial\phi}+\frac{3}{4}\frac{\partial L_{ \phi}}{\partial\phi}\dot{\phi}^2\right)\dot{\phi}^2+\frac{\partial V_{eff}}{\partial\phi}=0\,, \nonumber
\\&&\ddot{\sigma}_{c}+3H\dot{\sigma}_{c}-\frac{1}{2}\frac{\partial K_{ \phi}}{\partial\sigma_{c}}\dot{\phi}^2+\frac{\partial V_{eff}}{\partial\sigma_{c}}=0\,, \nonumber
\\ && H^2=\frac{1}{3}\rho\,, \nonumber
\\ && \dot{H}\,=\,-\frac{1}{2}\left(\rho\,+\,p\right)\,,
\ea
where the total energy density and pressure are
\be
\rho =\frac{1}{2}K_{\phi}\dot{\phi}^2+\frac{3}{4}L_{\phi}\dot{\phi}^4+\frac{1}{2}\dot{\sigma}_{c}^2+V_{eff} \quad \text{and} \quad p=\frac{1}{2}K_{\phi}\dot{\phi}^2+\frac{1}{4}L_{\phi}\dot{\phi}^4+\frac{1}{2}\dot{\sigma}_{c}^2-V_{eff}
\,.
\ee

We have considered the system of differential equations for $ϕ(t)$ and $σ_c(t)$ and solved it numerically. We start with indicative initial field values $σ_c(0)=ϕ(0)=20$ and zero velocities $σ̇_c(0)=ϕ̇(0)=0$. The parameters are chosen at the characteristic values $=10^6$, $ξ=ξ̅'=1$ and $ξ̅=0$. The results are shown in figure~\ref{fig:two_field_prob}. It is evident that when $ϕ$ reaches the inflationary plateau, the $σ_c$ has already fallen into the valley defined by the minimum line~\eqref{eq:sigma_0}. This can be also understood by comparing the relevant mass of the $σ_c-$field with the inflationary Hubble scale. After a numerical calculation we obtain that $M_σ_c^2(ϕ)/H^2 ∼10^4$ (for the benchmark case of figure~\ref{fig:two_field_prob}) at times $t ≳10^5$, which reinforces further the conclusion that the model is downgraded to a \textit{single-field} inflationary model.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.541]{smindirj10.pdf}
\includegraphics[scale=0.546]{phij10.pdf}
\caption{The evolution of the scalar fields $\s_c$ (left) and $\phi$ (right). Both fields start from the initial field value $20$ with zero velocities. The black dashed line in the left plot corresponds to the minimum direction $\s_0(\phi)$. The parameters are $\g=10^6$, $\xi=\bar{\xi}'=1$ and $\bar{\xi}=0$.  }
\label{fig:two_field_prob}
\end{center}
\end{figure}

Therefore, for the inflationary period it would be sufficient to study the {\textit{single-field problem}} described by $ϕ,σ_0(ϕ)$. Then, the system is described by the single field action
\be
{\cal{S}}=\int\,{\rm d}^4x\,\sqrt{-g}\left\{\frac{1}{2}R-\frac{1}{2}\bar{K}(\phi)(\nabla\phi)^2+\frac{1}{4}L(\phi)(\nabla\phi)^4-U(\phi)\,\right\},
\label{eq:act1f}
\ee
\ba
\bar{K}(\phi) &=& \frac{f(\phi)}{f^2(\phi)+4\gamma V(\phi)}+\left(\frac{12}{1+\frac{4g(\phi)^2f^2(\phi)}{[f^2(\phi)+4\gamma V(\phi)]^2}}\right)\left(\frac{g'(\phi)f(\phi)+g(\phi)f'(\phi)}{f^2(\phi)+4\gamma V(\phi)}\right. \nonumber
\\  & &\left.-\frac{g(\phi)f(\phi)}{[f^2(\phi)+4\gamma V(\phi)]^2}\left(2f'(\phi)f(\phi)+4\gamma V'(\phi)\right)\right)^2\,, \nonumber
\\ L(\phi) &=&\frac{\gamma}{\gamma g^2(\phi)/\delta+f^2(\phi)+4\gamma V(\phi)}\,, \nonumber
\\ U(\phi) & =& \frac{V(\phi)}{f^2(\phi)+4\gamma V(\phi)}\,.
\ea
Note that both kinetic functions $K̅(ϕ)$ and $L(ϕ)$ are positive definite.

Considering an FRW background, we obtain from~\eqref{eq:act1f} the following set of equations of motion
\ba
&&\left(\bar{K}(\phi)+3L(\phi)\dot{\phi}^2\right)\ddot{\phi} + 3H\left(\bar{K}(\phi)+L(\phi)\dot{\phi}^2\right)\dot{\phi}+\frac{1}{2}\bar{K}'(\phi)\dot{\phi}^2+\frac{3}{4}L'(\phi)\dot{\phi}^{4}+U'(\phi)=0\,, \nonumber
\\ && H^2 = \frac{\rho}{3}\quad \text{and} \quad  \dot{H}=-\frac{1}{2}\left(\rho+p\right)\,,
\label{eq:eoms}
\ea
where the energy density and pressure are given by
\be
\rho=\frac{1}{2}\bar{K}(\phi)\dot{\phi}^2+\frac{3}{4}L(\phi)\dot{\phi}^4+U(\phi) \quad \text{and} \quad p=\frac{1}{2}\bar{K}(\phi)\dot{\phi}^2+\frac{1}{4}L(\phi)\dot{\phi}^4-U(\phi)\,.
\ee
Although we have displayed the single-field equations of motion for general $f(ϕ), g(ϕ), V(ϕ)$, in the analysis of inflationary observables that will follow we have focused on $f(ϕ)=1+ξϕ^2$, $g(ϕ)=ξ̅ϕ+ξ̅'ϕ^3$ and $V(ϕ)=λϕ^4/4$.

Due to the appearance of the higher order kinetic term in the equation of motion the speed of sound in general deviates from unity. This can be easily seen from the form of the speed of sound which is given by
\be
c_s^2 = \frac{1+L(\phi)\dot{\phi}^2/\bar{K}(\phi)}{1+3L(\phi)\dot{\phi}^2/\bar{K}(\phi)}\,.
\ee
Nevertheless, the deviation from unity turns out to be quite small.

\subsection{Inflationary observables}
\label{sec:observables}
As it is quite complicated to find analytical expressions for the inflationary observables, we will solve the equation of motion~\eqref{eq:eoms} numerically to calculate them. To be as precise as possible the calculation of the amplitude of the scalar power spectrum $A_s$, the spectral index $n_s$, and the tensor-to-scalar ratio $r$, is done in the framework of the adiabatic approximation as described in~[72]. This method is suitable for theories with varying speed of sound and resembles to the well known WKB method. In our model even though the speed of sound is not constant, it is very close to unity throughout the full inflationary period, resulting to insignificant corrections to the inflationary observables.

Following~[72], in order to calculate the observables we will use the Hubble flow functions given by
\be
\label{eq:hff}
\epsilon_1 = -\frac{\dot{H}}{H^2}\,, \quad \epsilon_2 = \frac{\dot{\epsilon_1}}{\epsilon_1 H}\,, \quad s_1 = \frac{\dot{c_s}}{c_s H}\,.
\ee
The scalar ($𝒫_ζ$) and tensor ($𝒫_h$) power spectra can be expanded around the arbitrary pivot scale $k_⋆= a_⋆H_⋆/c_s^⋆$, that exited the sound horizon at some time $t^⋆$. keeping the first order terms in
the Hubble flow functions~\eqref{eq:hff}, we obtain
\ba
\mathcal{P}_\zeta(k) &=& \frac{H_\star^2}{8\pi^2 \epsilon_1^\star c_s^\star}\left(1-2(D+1)\epsilon_1^\star -D\epsilon_2^\star -(2+D)s_1^\star -(2\epsilon_1^\star +\epsilon_2^\star +s_1^\star) \ln \frac{k}{k_\star}\right)\,,
\\ \mathcal{P}_h(k) &=& \frac{2H_\star^2}{\pi^2} \left(1 -2(D+1-\ln c_s^\star)\epsilon_1^\star -2\epsilon_1^\star \ln \frac{k}{k_\star} \right)\,,
\ea
where the constant $D$ is given by $D=7/19 -ln3$.
The corresponding amplitudes, for scalar and tensor perturbations, are given by,
\be
A_s^\star = \mathcal{P}_\zeta(k_\star) \quad \text{and} \quad A_t^\star = \mathcal{P}_h(k_\star)\,.
\ee
The Planck 2018 data~[14], yield a value for the amplitude of the scalar power spectrum $A_s^⋆=(2.10±0.03) ×10^-9$, at the pivot scale $k_⋆=0.05  Mpc^-1$.

The tensor-to-scalar ratio ($r$) and the spectral index of the scalar power spectrum ($n_s$) are given by
\ba
r &=& \frac{A_t^\star}{A_s^\star} = 16\epsilon_1^\star c_s^\star\left(1+2\epsilon_1^\star\ln c_s^\star  +D \epsilon_2^\star +(2+D)s_1^\star \right)\,,
\\  n_s &=&1-2\epsilon_1^\star -\epsilon_2^\star -s_1^\star\,.
\ea

The recent release of the BICEP/Keck~[15] data, imposes the bound $r_0.05 < 0.036$ at the $95%$ C.L. where the subscript denotes the pivot scale in $ Mpc^-1$. Furthermore, the combination of WMAP,
Planck and BICEP/Keck data constrains the spectral index to the range $0.958<n_s<0.975$ again at the $95%$ C.L.  for $r=0.004$.

After the end of inflation, the Universe enters to the radiation-dominated era through a reheating phase, which may last some $e-$folds or can be instantaneous. Following~[73], we can compute the number of $e-$folds during
inflation after the comoving scale, $k_⋆= a_⋆H_⋆/c_s^⋆$, crosses the sound horizon by defining $N_⋆=ln(a_end/a_⋆ )$, where $a_end /_⋆$ denote the scale factor at the end of inflation and at the scale $k_⋆$ respectively. Analyzing the  scale factor as
\be
c_s^\star k_\star = \frac{a_\star}{a_{\rm end}} \frac{a_{\rm end}}{a_{\rm reh}} \frac{a_{\rm reh}}{a_{0}} a_o H_\star
\ee
and assuming instantaneous reheating ($a_end=a_reh$) and entropy conservation, we obtain
\be
\label{eq:N_star}
N_\star=66.89-\ln c_s^\star -\ln\left( \frac{k_\star}{a_0 H_0}\right) +\frac{1}{2} \ln\left( \frac{3 H_\star^2}{\rho_{\rm end}^{1/2}}\right) -\frac{1}{12}\ln g^s_{\rm reh}.
\ee
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.8]{nsr2.pdf}
\caption{Predictions of the model using pivot scales $0.05\, {\rm Mpc^{-1}}$ for $n_s$ and $0.002\, {\rm Mpc^{-1}}$ for $r$.  Shaded regions are the allowed parameter regions
at $68\%$ and $95\%$ confidence coming from the latest
combination of Planck, BICEP/Keck and BAO data~[14, 15]. The values of the parameters are $\bar{\xi}'=0$ and $\g=10^6$, while $\xi=0.1$ (green dashed-dotted line), $\xi=1$ (red dashed line) and $\xi=10$ (black dotted line). The parameter $\bar{\xi}$ varies from $10^{-3}$ to $10^3$ in each curve in a clockwise direction indicated by the arrow. The small numbers at the edges of the curves indicate the number of $e-$folds $N_{0.05}$ calculated by~\eqref{eq:N_star}, for the extreme values of the parameter $\bar{\xi}$.}
\label{fig:ns_r_1}
\end{center}
\end{figure}
The subscripts ``reh" and ``0" denote evaluations at the end of reheating and  present epoch respectively, while $g^s_reh$ are the entropy density degrees of freedom being $106.75$ assuming the Standard Model particle content and temperatures $∼1  $ or higher. Having in mind that $V_eff^⋆≃3π^2/2A_s^⋆r_⋆$, $ρ_end≃3/2 V_eff^end$,~\footnote{This equality is exact if there are no higher order kinetic terms. In~[33] the higher-order kinetic terms have been taken into account, but as it is shown there, only an insignificant correction arises.} and omitting the speed of sound contribution we obtain for the pivot scale $k_⋆=0.05  Mpc^-1$ that

\begin{table}[h!]
\centering
\caption{Predictions of the model for the benchmark points that correspond to the maximum values of the spectral index $n_s$ of figure~\ref{fig:ns_r_1}.}
\begin{tabular}{ccccccc}
\hline  \\[-4.2mm]\hline\\[-2.2mm]
$\xi$& $\bar{\xi} $& ${n_s}_{0.05}$ & $r_{0.05}$  & $N_{0.05}$ & $r_{0.002}$ & $N_{0.002}$ \\
\hline  \\[-3.2mm]
$0.1$ & $3 $ & $0.9644$&  $6.72\times 10^{-3} $ & $56.15$ & $6.04\times 10^{-3}$ & $59.37$ \\
$1$ & $30$ &$0.9645$ & $1.50\times 10^{-3}$  & $55.39$ & $1.35\times 10^{-3}$ & $58.61$\\
$10$ & $300 $& $0.9645$ & $9.66\times 10^{-4}$  & $55.20$ & $8.66\times 10^{-4}$ & $58.42$   \\
\hline \hline
\end{tabular}
\label{tbl_1}
\end{table}

\be
N_{0.05} \simeq 55.8 +\frac{1}{4}\ln \frac{r}{0.036} -\frac{1}{4}\ln \frac{ V_{\rm eff}^{\rm end}}{V_{\rm eff}^{0.05}}.
\ee
At the pivot scale $k_⋆=0.002  Mpc^-1$, in which the tensor-to-scalar ratio is calculated in some of our figures, the numerical coefficient increases to the value $∼59.1$.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.71]{nsxib.pdf}
\includegraphics[scale=0.515]{r005xib.pdf}
\caption{Predictions of the model as a function of the parameter $\bar{\xi}$ using the pivot scale $0.05\, {\rm Mpc^{-1}}$ for both $n_s$ and  $r$.  Shaded regions in the left plot are the allowed parameter regions, $n_s=0.9649\pm 0.0042$, at $68\%$ and $95\%$ confidence coming from the Planck data~[14]. In the right plot the whole region is allowed, since $r_{0.05}< 0.036$ at $95\%$ confidence~[15]. The values of the parameters and the coloring are the same with that in figure~\ref{fig:ns_r_1}.}
\label{fig:ns_r_xibar_1}
\end{center}
\end{figure}

Let us now turn to the analysis of the parametric space of the model under consideration with respect to its predictions for the cosmological observables. For the parameter $ξ$ we choose to start from the value $ξ=0.1$, since as discussed in~[33] in the ``simple" Palatini-$ℛ^2$ model for values $ξ≳0.1$ the spectral index can be within observational limits. It is also acceptable to use lower values, but at the price of reducing the range of acceptable e-folds considerably. For vanishing $g(ϕ)$ the spectral index and the tensor-to-scalar ratio are given approximately by
\be
\label{eq:ns_r+PalR2}
n_s \simeq 1-\frac{2}{N_\star}-\frac{1}{8\xi N_\star^2}\quad \text{and} \quad r \simeq \frac{1}{N_\star^2}\frac{2\xi}{\xi^2+\g\l}\,.
\ee

In our case both of them get modified, but their analytic expressions are not available since the functions involved are too complicated. We will examine two limiting cases for the function $g(ϕ)$ for the sake of  better understanding of the behaviour of the observables.

In figure~\ref{fig:ns_r_1} we explore the case in which the linear part of the non-minimal coupling function $g(ϕ)$ of Eq.~\eqref{eq:g_func} dominates, taking the parameter $ξ̅'$ to be zero. In this figure we show the predictions of the model in the $_0(ϕ)$ direction using pivot scales $0.05  Mpc^-1$ for $n_s$ and $0.002 Mpc^-1$ for $r$. For each given set of values for the parameters, we have employed Eq.~\eqref{eq:N_star} to obtain the number of $e$-folds that complies with the constraints from reheating, while the value of quartic coupling $λ$ is fixed by the observed value of the amplitude of the scalar power spectrum, $A_s^⋆=2.1 ×10^-9$, at the pivot scale $k_⋆=0.05  Mpc^-1$. The parameter $$ is fixed to the value $10^6$, while $ξ=0.1$ (green dashed-dotted line), $ξ=1$ (red dashed line) and $ξ=10$ (black dotted line).
The parameter $ξ̅$ varies from $10^-3$ to $10^3$ in each curve in a clockwise direction as indicated in the figure. As the parameter $ξ̅$ gets larger the tensor-to-scalar ratio $r$ increases. The spectral index $n_s$ decreases up to some value of $ξ̅$ beyond which it starts to increase. A notable remark is that the predictions for $n_s$ and $r$ ``freeze" to some values (their largest values) without being affected by a further increasing of the parameter $ξ̅$. This ``freezing" occurs at the values of $ξ̅ ≃3, 30$ and $300$ for $ξ=0.1, 1$ and $10$ respectively, see also Table~\ref{tbl_1}. In figure~\ref{fig:ns_r_xibar_1} we also present the observables versus  the parameter $ξ̅$, for better understanding of the analysis above.
\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.8]{nsr.pdf}
\caption{Predictions of the model using pivot scales $0.05\, {\rm Mpc^{-1}}$ for $n_s$ and $0.002\, {\rm Mpc^{-1}}$ for $r$.  Shaded regions are the allowed parameter regions
at $68\%$ and $95\%$ confidence coming from the latest
combination of Planck, BICEP/Keck and BAO data~[14, 15]. The values of the parameters are $\bar{\xi}=0$ and $\g=10^6$, while $\xi=0.1$ (green dashed-dotted line), $\xi=1$ (red dashed line) and $\xi=10$ (black dotted line). The parameter $\bar{\xi}'$ varies from $10^{-3}$ to $10^3$ in each curve in a counterclockwise direction indicated by the arrow. The small numbers at the edges of the curves indicate the number of $e-$folds $N_{0.05}$ calculated by~\eqref{eq:N_star}, for the extreme values of the parameter $\bar{\xi}'$.}
\label{fig:ns_r_2}
\end{center}
\end{figure}

In Figs.~\ref{fig:ns_r_2} and~\ref{fig:ns_r_xibar_2} we explore the case in which the cubic  part of the non-minimal coupling function $g(ϕ)$ of Eq.~\eqref{eq:g_func} dominates, \textit{i.e.} we choose $ξ̅=0$. In this case the tensor-to-scalar ratio $r$ is constantly increasing as $ξ̅'$ gets larger (from $10^-3$ to $10^3$), while the spectral index $n_s$ initially increases, gets its maximum value and then starts to decrease and go well outside of the observational acceptable bounds~[14, 15]. The datasets that correspond to the maximum value of $n_s$ are given in Table~\ref{tbl_2}. The values of the rest parameters and the
coloring of Figs.~\ref{fig:ns_r_2} and~\ref{fig:ns_r_xibar_2} are the same with that in  Figs.~\ref{fig:ns_r_1} and~\ref{fig:ns_r_xibar_1}.

\begin{table}[h!]
\centering
\caption{Predictions of the model for the benchmark points that correspond to the maximum values of the spectral index $n_s$ of figure~\ref{fig:ns_r_2}.}
\begin{tabular}{ccccccc}
\hline  \\[-4.2mm]\hline\\[-2.2mm]
$\xi$& $\bar{\xi}' $& ${n_s}_{0.05}$ & $r_{0.05}$  & $N_{0.05}$ & $r_{0.002}$ & $N_{0.002}$ \\
\hline  \\[-3.2mm]
$0.1$ & $5\times 10^{-3} $ & $0.9643$&  $6.70\times 10^{-3} $ & $56.13$ & $6.01\times 10^{-3}$ & $59.35$ \\
$1$ & $5\times 10^{-2}$ &$0.9666$ & $9.92\times 10^{-4}$  & $55.28$ & $8.95\times 10^{-4}$ & $58.50$\\
$10$ & $6\times 10^{-1} $& $0.9701$ & $2.63\times 10^{-4}$  & $54.81$ & $2.40\times 10^{-4}$ & $58.02$   \\
\hline \hline
\end{tabular}
\label{tbl_2}
\end{table}

In all of the figures presented the parameter $γ$ has the value $10^6$, but for any value lower than $∼10^7$ the predictions will remain unaffected. This can be easily understood if we have a look at the large field limits of the effective potential and the kinetic term $K̅(ϕ)$ (see Eq.~\eqref{EFFLANG}). For both of them the limits are functions of the combination $ξ^2/ł+$, which means that if $ξ^2/ł≫$ the inflationary predictions are controlled by $ξ^2/ł$. In the parameter space under consideration the ratio $ξ^2/ł$ is always larger than $∼7×10^8$. In particular, in Figs.~\ref{fig:ns_r_2}-\ref{fig:ns_r_xibar_2} varies from $1.2×10^9 - 6.6×10^8$ (for $ξ=0.1$), $1.2×10^10 - 3.2×10^9$ (for $ξ=1$) and $1.2×10^11 - 5.5×10^9$ (for $ξ=10$). Similar values are noticed also in Figs.~\ref{fig:ns_r_1}-\ref{fig:ns_r_xibar_1}. At this point, we would like to mention that we do not consider larger values of $$ that would drive us to the regime $ξ^2/ł≪$, since the tensor-to-scalar ratio would be even smaller. The sensitivity in the measurement of the tensor-to-scalar ratio by the next-generation CMB satellites~[74, 75, 76] is expected to be $ṛ∼10^-4$, so smaller values for $r$ are only subject of academic significance. The general picture is that the presence of the pseudoscalar and its coupling to the inflaton through the term $g(ϕ)R$ has as a result an increase of the tensor-to-scalar ratio, in comparison to the Palatini-$R^2$ model, while the spectral index can either decrease or increase depending on the rest of the parameters.
In the whole range of the aforementioned parameters the number of $e$-folds is calculated to be $N_0.05≳55$, under the assumption of instantaneous reheating. A smaller number of $e$-folds can arise for larger values of $ξ$, nevertheless, this would also lead to a smaller value for the tensor-to-scalar ratio.

\begin{figure}[t!]
\begin{center}
\includegraphics[scale=0.72]{nsxibp.pdf}
\includegraphics[scale=0.515]{r005xibp.pdf}
\caption{Predictions of the model as a function of the parameter $\bar{\xi}'$ using the pivot scale $0.05\, {\rm Mpc^{-1}}$ for both $n_s$ and  $r$.  Shaded regions in the left plot are the allowed parameter regions, $n_s=0.9649\pm 0.0042$, at $68\%$ and $95\%$ confidence coming from the Planck data~[14]. In the right plot the whole region is allowed, since $r_{0.05}< 0.036$ at $95\%$ confidence~[15]. The values of the parameters and the coloring are the same with that in figure~\ref{fig:ns_r_2}.}
\label{fig:ns_r_xibar_2}
\end{center}
\end{figure}

\section{Conclusions}
\label{sec:conclusions}

In the present article we considered the framework of \textit{Metric-Affine} theories of gravity in which the metric and the connection are independent variables. We extended the Einstein-Hilbert action to include the parity-odd Holst invariant $R$ as well as quadratic terms of it and of the Ricci scalar, expected to be generated by quantum corrections. This context gives rise to a new dynamical pseudoscalar degree of freedom. Furthermore, we considered the coupling of this system to a scalar field $ϕ$ through the non-minimal terms $f(ϕ)R$ and $g(ϕ)R$, the latter corresponding to a coupling of $ϕ$ with the new pseudoscalar. The potential of the resulting Einstein-frame action of the two fields $ϕ$ and $σ$ has a minimum line along $σ_0(ϕ)$. Considering this model in an FRW background we found that the pseudoscalar $σ$ falls quickly into the valley defined by the minimum line $σ_0(ϕ)$, this transition being supported by the numerical solution of the two-field system of equations of motion. Thus, the model is reduced to an effectively single-field model.  Furthermore, along this minimum line the potential takes the form of the Palatini-$R^2$ potential with its characteristic inflationary plateau.

We have proceeded to study the resulting single-field inflationary model making a choice of the coupling functions $f(ϕ)=1+ξϕ^2$ and $g(ϕ)=ξ̅ϕ+ξ̅'ϕ^3$. A numerical calculation of the inflationary observables was undertaken in the framework of the adiabatic approximation [72]. The parametric space was constrained by the latest available bounds set by~[14, 15]. We have found
that our model complies with observations for a wide range of parameters. More precisely taking the parameter $ξ$ to be $≥0.1$, the parameters $ξ̅$ and $ξ̅'$ run in the range $10^-3-10^3$.
We examined two limiting cases for the function $g(ϕ)$ in order to better understand the behaviour of the observables: the linear case ($ξ̅'=0$) and the qubic case ($ξ̅=0$). In the linear case  the predictions for $n_s$ and $r$ ``freeze" to their largest values without being affected by a further increase of the parameter $ξ̅$. This ``freezing" occurs at the values of $ξ̅ ≃3, 30$ and $300$ for $ξ=0.1, 1$ and $10$ respectively. In the cubic case after the initial increase of the spectral index, once it gets its maximum value, it starts to decrease and go well outside of the observational acceptable bounds as $ξ̅' $ gets larger. In general we found that the coupling $g(ϕ)$ has as a result an increase of the tensor-to-scalar ratio, in comparison to the Palatini-$R^2$ model, while the spectral index can either decrease or increase depending on the position in parametric space.
The coefficient of the $ ℛ^2$ term does not modify the inflationary observables, while the one of  the $ ℛ^2$ term can have a significant result only if it is larger than $10^7$, which case is not considered here.
Finally, under the assumption of instantaneous reheating, in the whole range of the aforementioned parameters the number of $e$-folds measured from the end of inflation to the pivot scale $k_⋆=0.05  Mpc^-1$ is $N_0.05≳55$.

Summarizing, we have considered a general quadratic \textit{Metric-Affine} theory, featuring an extra dynamical degree of freedom,  and coupled it non-minimally to a scalar field. We studied inflation in the resulting two-field model and found that it effectively reduces to a single-field model, with a potential of the Palatini-$R^2$ form with its characteristic inflationary plateau and a modified kinetic term. We find that the inflationary predictions of this model fall within the latest observational bounds for a wide range of parameters. Furthermore, it allows for an increase in the tensor-to-scalar ratio.

\acknowledgments

The work of IDG was supported by the Estonian Research Council grant SJD18.
KT would like to thank the CERN Theoretical Physics Department and the Laboratoire de Physique de l’Ecole Normale Superieure, where part of this work was done, for their hospitality.

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\end{thebibliography}\endgroup
\end{document}


# On Relaxation of Dominant Sets

Max Koster

###### Abstract

In a graph $G=(V,E)$, a k-ruling set $S$ is one in which all vertices $V$ \
$S$ are at most $k$ distance from $S$. Finding a minimum k-ruling set is
intrinsically linked to the minimum dominating set problem and maximal
independent set problem, which have been extensively studied in graph theory.
This paper presents the first known algorithm for solving all k-ruling set
problems in conjunction with known minimum dominating set algorithms at only
additional polynomial time cost compared to a minimum dominating set. The
algorithm further succeeds for $(\alpha,\alpha-1)$ ruling sets in which
$\alpha>1$, for which constraints exist on the proximity of vertices v $\in
S$. This secondary application instead works in conjunction with maximal
independent set algorithms.

##  1 Introduction

A set $S\subseteq V$ of a graph $G=(V,E)$ is considered a ruling set if every
vertex $v\in V$ \ $S$ is maximum distance $k$ from $S$. A dominating set,
$D\subseteq V$ where every vertex $v\in V$ \ $D$ is adjacent to a vertex of
$D$, is thus a simplification of ruling sets where distance $k=1$. Problems
surrounding dominating sets were first discussed in the 1950s and have since
only increased in importance. An essential question in domination theory is
that of finding the minimum dominating set of a n-vertex graph, which was
first calculated in $O(2^{n}n)$ time. Recent literature from Fomin et al.
[FGKD09] has reduced time complexity to $O(1.5137^{n})$. Though ample
attention has been paid to the minimum dominating set problem, there is yet to
be a proposed algorithm to calculate minimum ruling sets beyond the dominating
set (i.e. $k=1$ instance). We will focus on calculating minimum ruling sets
regardless on distance $k$ imposed on said graphs.

The field of domination theory, and by extension ruling sets, has seen
increased focus within graph theory. The topic has many applications, the most
literal of which is likely the facility location problem. The problem seeks
out the most optimal location for emergency services among people/towns, while
also dealing with a constraint of the number of emergency service locations.
Other applications include ad-hoc computer communication networks.

All current contributions to more generalized ruling set problems neglect
basic views of the problem and instead focus on independent ruling sets. An
independent set is a set $I\subseteq V$ of a Graph $G=(V,E)$ such that every
vertex $v\in I$ is not adjacent to any other vertices in $I$. A common problem
connecting dominance and independence among graphs is that of finding a
maximal independent set (MIS) such that the MIS is not a subset of any other
independent sets in G. Informally, a MIS can not have any more vertices added
to it without becoming not independent. Though not all dominating sets are
independent, all maximal independent sets are dominant. Independent ruling
sets can be defined as a ruling set $S\subseteq V$ upon which the distance
between vertices in $V$\ $S$ is at most k and all vertices in $S$ are
independent. Thus, independent ruling sets are analogously a subset of normal
ruling sets. Discussions of ruling sets are mostly limited to independent
ruling sets; Balliu et al. [BBO20] solely focused on ruling sets in which
distance between vertices in $S$ is at minimum $\alpha$ for
$(\alpha,\beta)$-ruling sets. (The distance between any two vertices in $S$ is
at least $\alpha$, and all vertices in $V$ \ $S$ are at most $\beta$ distance
from $S$.) Even more recent literature focuses more specifically on
independent ruling sets in which $k=2$ or $k=3$ [KP12], [PP22].

We present an algorithm that transforms k-ruling set problems for graphs into
a minimum dominating set problem through modification of our original graph.
Known minimum dominating set algorithms can then be applied to the new graph
to find a set of vertices which serves as a k-ruling set on our original
graph. Our algorithm has an upper bound of $O(n^{3})$ where n is the number of
vertices in the graph. The NP-hard status of minimum dominating set algorithms
means our polynomial time complexity has little impact on speed of overall
calculations of k-ruling set problems. Our algorithm can further be applied to
specific instances of $(\alpha,\beta)$ ruling sets: particularly for when the
problem is represented as an $(\alpha,\alpha-1)$ ruling set problem, where
$\alpha>1$. In such cases, MIS algorithms are then applied to find the correct
ruling set.

##  2 Preliminaries

Let $G=(V,E)$ be an undirected and simple graph. The $distance$ between two
vertices in a graph is the edges present within the shortest path between the
two vertices. For every vertex $v\in V$ we denote the set of adjacent vertices
$N(v)$, or the neighborhood of $v$ and $N[v]=N(v)\cup\\{v\\}$. A $clique$ is a
set $S\in V$ of pairwise adjacent vertices. A $complete$ graph is a graph with
every possible edge, or a clique imposed on the entire graph.

The notation of ruling sets in past literature is not clearly established and
often changes from paper to paper. The frequent discussion of independent
ruling sets result in false equivalencies created between independent ruling
sets and all ruling sets, where the two terms are used interchangeably. Other
definitions, as in Balliu et al. [BBO20], define an $(\alpha,\beta)$ ruling
set such that the distance between vertices in $V$ \ $S$ is at most $\beta$
and every vertex in $S$ is at minimum $\alpha$ distance from every other
vertex in $S$. We will mostly avoid this definition as our paper does not
focus on independence, but return to it in later portions when discussing
applications of our algorithm in related fields. We will seek to establish a
clear definition for ruling sets, one that is quite similar to Gfeller and
Vicari [GV07]’s definition.

###### Definition 1

Given a graph $G=(V,E)$, a k-ruling set is a subset $S\subseteq V$ such that
every vertex in $V$ \ $S$ is at most distance $k$ from $S$.

##  3 Simplifying K-ruling sets to Dominating sets

Rather than view the minimum dominant set problem as a specific variant of
k-ruling sets, we seek to instead view every instance of k-ruling sets as a
relaxation of a minimum dominant set problem. Consider the relation between a
2-ruling set and an minimum dominant set.

Consider a graph with a 2-ruling set. A vertex $v\in S$ in a 2-ruling set
effectively rules every $u\in N(v)$, which we will denote as $U$, as well as
every vertex $w\in N(N(u))$ \ $v$, which we will denote as $W$. Ignoring
overlap between $U$ and $W$ (if a vertex is in $U$, it is removed from $W$),
then: $U$ is the set of vertices of distance $1$ from $v$ and $W$ is a set of
all vertices of distance $2$ from $v$. Set $S$ dominates $U\cup W$.

Similarly consider a dominant set (1-ruling) within a graph. A vertex $v\in S$
in a minimum dominant set similarly rules every vertex in $U$ (all vertices
distance $1$ away). Thus, to re-configure the 2-ruling set $S$ into a minimum
dominant set, we can add an edge between $v$ and every vertex $w\in W$. Set
$S$ would then dominate all vertices of distance $1$ away, which is
identically $W\cup U$ (even though we have decreased $k$ by 1).

Now, let us consider the instance in which the vertices $v\in S$ are currently
unknown for a 2-ruling set. We can simply construct edges from every vertex
$v\in V$ to vertices $u\in N(N(v))$. There now exist edges between vertices
two degrees from each other, so finding a solution to the minimum dominant set
problem would find a solution to the 2-ruling set problem on the original
graph we began with. An example can be seen at Figure 1.

The simplification of the 2-ruling set to a minimum dominant set problem is
not the only possible simplification to a minimum dominant set. For each
k-ruling set, an additional round of adding edges between neighbors of
neighbors of neighbors etc. is required. Thus, each k-ruling set requires k-1
rounds of iteration to reach a minimum dominant set. Our analysis leads to the
following algorithm:

Figure 1: Displays the construction of additional edges for a graph G to
convert its 2-ruling set into a minimum dominant set problem.

###  3.1 The Algorithm

In this section, we define the algorithm for transforming a k-ruling problem
into a minimum dominating set problem. We will give a proof of correctness and
further analysis shortly after.

Algorithm 1 reducekGraph

Input: G = (V,E); k = c
Output: G = (V,E); k = 1

1:$i\leftarrow k$

2:$E^{\prime}\leftarrow E$

3:while $i>1$ do

4: $E^{\prime\prime}\leftarrow E^{\prime}$

5: for all $(u,v)\in E^{\prime}$ do

6: for all $w$ s.t $(v,w)\in E^{\prime}$ do

7: $E^{\prime\prime}\leftarrow E^{\prime}\cup(u,w)$

8: end for

9: end for

10: if $E^{\prime\prime}=E^{\prime}$ then

11: $i\leftarrow 1$

12: else

13: $i\leftarrow i-1$

14: $E^{\prime}\leftarrow E^{\prime\prime}$

15: end if

16:end while

17:$G^{\prime}\leftarrow(V,E^{\prime})$

18:return $G^{\prime}$

###  3.2 Correctness and Analysis

Suppose our original graph has become complete after n iterations. Then, no
more edges will be added to set E’; thus, iterating through all edges would be
a waste of time for the remaining $(k-1)-n$ iterations. Thus, once E” = E’,
the algorithm has terminated and we can return our new graph.

###### Theorem 1

Any given vertex $v$ would, if chosen to be added to the dominating set,
dominate the same set of vertices both before and after completion of the
algorithm.

$Proof$ We prove this theorem by induction on $n$, where $n$ is the number of
iterations of the while loop. Exclusively consider $n>0$; $n=0$ is trivial. We
prove the theorem by showing it is not only true at its end but that it holds
throughout all iterations. When $n=1$ any vertices $v,w$ s.t the distance
between $v,$ is less than or equal to 2, will have an edge between them; $k$
similarly decreases to $k-1$. As discussed in the simple case of 2-ruling set
to 1-ruling set above, these graphs would have the same ruling set.

Let us prove $P(n)$ implies $P(n+1)$. Immediately apply the inductive
hypothesis; thus, every every vertex within distance $n$ of each other have an
edge between them and $k:=k-n$. An additional iteration would create an
additional edge between vertex and any other vertex of distance 2 away from
it. Thus, every vertex within $n+1$ distance of the original graph is now
adjacent and $k:=k-(n+1)$. When $n=k-1$ the algorithm will terminate and every
vertex within $k$ distance of each other will now have an edge between them.

###### Theorem 2

The algorithm reducekGraph solves our problem in $O(\frac{k\cdot
n(n-1)^{2}}{2})$ worst-case time.

$Proof$ The maximum number of edges is $\frac{n(n-1)}{2}$ upon which a
complete clique graph occurs. This results in a time complexity of
$O(\frac{n(n-1)^{2}}{2})$ per round. This round will only occur once because
the algorithm will then terminate. Thus, the entire algorithm has an upper
bound of $O(\frac{k\cdot n(n-1)^{2}}{2})$ for the entire algorithm with
constant k.

###  3.3 Independent Sets

Let us consider the applications of the algorithm to independent set problems
utilizing $(\alpha,\beta)$ notation [BBO20]. Consider a $(2,2)$ ruling set;
each vertex in set $S$ is minimum distance 2 from each other. Thus, the ruling
set is independent. Edges would be produced between every vertex within
distance 2 of each other by algorithm $reducekGraph$. Following the
algorithm’s application, running an MIS algorithm, such as Luby’s Algorithm
[Lub86], would prevent vertices which were not initially adjacent to each
other from joining the independent set due to the construction of new edges.
Similarly, applying a minimum dominating set algorithm would not work as
vertices adjacent in the original graph may be added together, and our
solution set would no longer be independent. Thus, for many $(\alpha,\beta)$
ruling sets where $\alpha\neq 1$, our algorithm will not work.

###### Theorem 3

The algorithm reducekGraph succeeds in all $(\alpha,\alpha-1)$ ruling set
problems in conjunction with MIS algorithms for $\alpha>1$.

$Proof$ Consider an $(\alpha,\alpha-1)$ ruling set. Following termination of
$reducekGraph$, all vertices of distance $\alpha-1$ will be adjacent to each
other. Then, applying a $MIS$ algorithm will ensure no adjacent vertices join
the independent set together, so only vertices of distance $\alpha$ or greater
from each other on the original graph will possibly join the ruling set, as
required. So, if $\beta=\alpha-1$, the ruling set can be found using our
algorithm.

Consider if $\beta<\alpha-1$. Like our example above of a (2,2) ruling set,
the termination of the $reducekGraph$ algorithm will result in adjacency of
all vertices within distance $\alpha-1$, but our ruling set allows for the
inclusion of more vertices; that is, those of $\beta$ distance from each
other. This does not occur, excluding said vertices from joining the ruling
set together.

Also consider if $\beta>\alpha-1$. Once again, termination the $reducekGraph$
algorithm will result in adjacency of all vertices of within distance
$\alpha-1$, but we require a minimum distance of $\beta$ between two vertices
in the ruling set; thus, running an MIS would possibly include vertices that
are not at minimum $\beta$ apart and thus do not belong in the solution set.

##  4 Future Work

A large number of open questions remain within k-ruling set problems. A
natural one is that of whether the creation of an efficient algorithm exists
for all $(\alpha,\beta)$ ruling sets, not only those in which
$\beta=\alpha-1$. One may also wonder whether one could work backwards from
dominating set problems to ruling set ones instead of traversing in the
opposite direction as we have done. One may consider the synthesis of our
universal algorithm with the examination of specific ruling sets by other
authors like [BBO20], [PP22], [KP12]. More generalized possibilities include
expanding this paper’s crux—that of simplifying an unknown problem to a known
one—into other subtopics of graph theory.

## References

  * [BBO20] Alkida Balliu, Sebastian Brandt, and Dennis Olivetti.  Distributed lower bounds for ruling sets.  2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 365–376, 2020.
  * [FGKD09] Fomin, Fedor V.; Grandoni, Fabrizio; Kratsch, and Dieter.  A measure & conquer approach for the analysis of exact algorithms.  Journal of the ACM, 56(5):1–32, 2009.
  * [GV07] Beat Gfeller and Elias Vicari.  A randomized distributed algorithm for the maximal independent set problem in growth-bounded graphs.  Proc ACM PODC, page 53–60, 2007.
  * [KP12] Kishore Kothapalli and Sriram Pemmaraju.  Super-fast 3-ruling sets.  2014 ACM Symposium on Principles of distributed computing, pages 379–81, 2012.
  * [Lub86] Michael Luby.  A simple parallel algorithm for the maximal independent set problem.  SIAM Journal on Computing, 15(4), 1986.
  * [PP22] Shreyas Pai and Sriram V. Pemmaraju.  Deterministic massively parallel algorithms for ruling sets.  2022\.