text:
\section{Introduction}

Extra dimensions appear in Kaluza-Klein (KK) field theory and gravity, supersymmetry and supergravity, string theory and brane world, mainly in the context of unification of particles and fields. It is, therefore, natural to study multi-dimensional cosmological models also, and relate them to the observed Universe. However, the progress in this direction was limited in the literature, because the observed Universe is four-dimensional, so that any multi-dimensional cosmological model has to end up with the effective four-dimensional theory that fits the FLRW framework and is consistent with observations. Extra dimensions unavoidably   lead to extra scalar fields (called {\it moduli}) that must be stabilized.  In addition, the mass hierarchy has to be satisfied as follows:
\begin{equation} \label{hier}
M_{\rm inf.} \ll M_{\rm KK} \ll M_{\rm Pl}~~.
\end{equation}
Moreover, extra dimensions usually open a lot of possibilities that should be constrained both theoretically and experimentally. This would imply interesting relations between the four-dimensional cosmological quantities and their higher dimensional counterparts, and offer a possible multi-dimensional origin of our Universe.

One of the well studied approaches in this direction is based on the modified $f(R)$ gravity actions in higher dimensions with a warped product geometry, where $R$ stands for the Ricci
scalar in $D>4$ spacetime dimensions.  However, as was demonstrated in
Refs.~\cite{zhuk1,zhuk2,zhuk3,zhuk4}, the higher-dimensional $(R+\gamma R^n-2\Lambda)$ gravity models together with their spontaneous compactification to four dimensions  {\it do not} lead to a successful phenomenological description of dark energy, because of a necessarily negative (induced) cosmological constant in four dimensions. These models also fail to describe the early universe inflation because of low values of the scalar index $n_s$ and the e-foldings number $N_e$. Adding extra (matter) $p$-form fields with a Freund-Rubin-like compactification ansatz \cite{frr} can stabilize extra dimensions for a certain range of parameters \cite{zhuk2}, but still does not lead to a successful phenomenology. In particular, the four-dimensional inflationary models based on the  compactified $(R+\gamma R^n-2\Lambda)$ gravity in dimensions $D<8$ were found to be not viable \cite{zhuk4}.  It raises a question about whether this situation can be improved by changing or relaxing some of the assumptions used in Refs.~\cite{zhuk1,zhuk2,zhuk3,zhuk4}. It is also desirable to get the constraints restricitng the values of a higher dimension $D>4$, the power $n$ of the scalar curvature $R$ in the modified gravity action, the value of the higher-dimensional cosmological constant $\Lambda$, and the rank $p$ of a $p$-form field, if any.

In our paper \cite{kn2}, we proposed a derivation of the viable inflaton scalar potential from the
higher $(D)$ dimensional  $(R+\gamma R^n-2\Lambda)$ gravity, by giving up the condition of
spontaneous compactificaton of extra dimensions and ignoring the moduli, i.e. just assuming that
the compactification happened before inflation and it can be made spontaneous by adding some more fields.  As a result, the inflaton scalar potential in four spacetime dimensions turns out to be dependent upon the parameters $(\gamma,\Lambda ,D,n)$, while the viable
inflationary phenomenology requires
\begin{equation} \label{nD}
n=D/2~~,
\end{equation}
with the dimension $D$ being a multiple of four. The condition (\ref{nD}) arises by demanding the existence of a {\it plateau} with a positive height for the inflationary scalar potential, as
is apparently favoured by the Planck mission observations
\cite{Ade:2015xua,Ade:2015lrj,Array:2015xqh}, and is the case in the famous Starobinsky inflationary model \cite{Starobinsky:1980te}, though is in contrast to
Refs.~\cite{zhuk1,zhuk2,zhuk3,zhuk4} where the scalar potential was demanded to vanish before the onset of inflation. Our results  were significantly enhanced in Ref.~\cite{nad} where
a spontaneous compactification and stabilization of the volume of extra dimensions was achieved
by adding a single $(p-1)$-form gauge field having a non-vanishing flux in compact dimensions and  obeying the condition
\begin{equation} \label{np}
p=n~.
\end{equation}

In this paper we extend this analysis in the first relevant higher dimension $D=8$, and  consider an embedding of the $D=8$ modified gravity model into a (modified) $D=8$ supergravity.

Our paper is organized as follows. Our modified gravity model in $D=8$ is formulated in Sec.~2.
Also in Sec.~2 we consider the Freund-Rubin-type compactification of our model on a 4-sphere down to four spacetime dimensions, derive the scalar potential, and stabilize the volume modulus of  the compact dimensions described by the 4-sphere. In Sec.~3 we apply our model to a description of cosmological inflation in the early Universe. In Sec.~4 we speculate about a possible embedding of our model into a modified $D=8$ supergravity. Sec.~5 is our conclusion. We collect all technical details into four appendices: appendix A is devoted to the Legendre-Weyl transform of the modified gravity model to the Einstein frame in 8 dimensions; appendix B describes the Freund-Rubin-type compactification of the transformed action to 4 dimensions on a 4-sphere, it includes a derivation of the two-field scalar potential in four dimensions; appendix C is devoted to a detailed study of the scalar potential in 4 dimensions; appendix D is devoted to a (partial) derivation of the (modified and gauged) $D=8$ supergravity from a modified $D=11$ supergravity by compactifying the latter on a 3-sphere.

\section{The $D=8$ model and its $D=4$ compactification}

The $f(R)$ gravity in {\it four} spacetime dimensions is the standard theoretical approach in modern cosmology, capable of describing both cosmological inflation in the early Universe and dark energy in the present Universe --- see e.g., the reviews \cite{svrev,tsu,clrev,myrev,starp} and references therein. The
basic idea is to replace the scalar curvature $R$ in the Einstein-Hilbert (EH) gravity action by a function $f(R)$ obeying certain physical requirements in the relevant range of its argument $R$, such as the absence of ghosts and tachyons, in order to fit the (observed) accelerating Universe.

The distinguished property of $f(R)$ gravity theories is their classical equivalence (duality) to the
scalar-tensor gravity theories \cite{fmbook}, which is known for the long time --- see e.g.,
Refs.~\cite{wag,bick,whitt,jk,bc,maeda,kkw1,kw1,kw2}. The existence of this (Legendre-Weyl) transformation relating these apparently different gravity theories is guaranteed by the physical conditions on the $f(R)$-function, namely, positivity of its first and second derivatives (in the proper notation, and in the relevant range of the scalar curvature values).

The simplest and, perhaps, most famous $f(R)$ gravity model of Starobinsky \cite{Starobinsky:1980te}
is defined by the action~\footnote{We use the natural units $\hbar=c=1$ with the reduced Planck mass $M_{\rm Pl}=1$, and the $D=4$ spacetime signature $(-,+,+,+)$.}
 \begin{equation} \label{star}
S_{\rm Starobinsky} = \fracmm{1}{2} \int \mathrm{d}^4x\sqrt{-g} \left[ R +\fracmm{1}{6M^2}R^2\right]~.
\end{equation}
The Starobinsky model is known as an excellent model of inflation, in very good agreement with the Planck data \cite{Ade:2015xua,Ade:2015lrj,Array:2015xqh}. On the one hand, {\it any} viable inflationary model with $f(R)=R+\tilde{f}(R)$ gravity must be close to the Starobinsky model (\ref{star}) in the sense of having
 $\tilde{f}(R)=R^2 A(R)$ with a slowly-varying function  $A(R)$. The Starobinsky model is also known as an {\it attractor} for inflation \cite{gklr}. On the other hand, {\it any} $(R+\gamma R^n)$ gravity model in $D=4$
 with an integer power $n$ higher than two is {\it not} viable for
 inflation \cite{kkw2}.~\footnote{Having $n$ to be non-integer and close to 2 is still possible for inflation  in $D=4$ \cite{moto}.}

The only real parameter $M$ of the Starobinsky model can be identified with the inflaton mass, whose value is fixed by the observational Cosmic  Microwave Background (CMB) data as $M=(3.0 \times10^{-6})(\frac{50}{N_e})$ where $N_e$ is the e-foldings number.  The corresponding scalar potential of the (canonically normalized) inflaton field $\phi$ (dubbed {\it scalaron} in the given context) in the dual (scalar-tensor gravity) picture reads
\cite{jdb}
\begin{equation} \label{starpot}
V(\phi) = \fracmm{3}{4} M^2\left( 1- e^{-\sqrt{\frac{2}{3}}\phi }\right)^2~.
\end{equation}
This scalar potential is bounded from below, has a Minkowski vacuum and a {\it  plateau} of a positive height for slow roll inflation. During the inflation the scalar potential (\ref{starpot}) is simplified to
\begin{equation} \label{starpot2}
V(\phi) \approx V_0\left( 1- 2e^{-\alpha_s\phi }\right)~,
\end{equation}
where we have kept only the leading (exponentially small) correction to the emergent cosmological constant $V_0=\frac{3}{4} M^2$, and have introduced the notation $ \alpha_s=\sqrt{\frac{2}{3}}$~.

It is the demand of having a plateau for the scalar potential in higher $D$ dimensions that results in
the condition (\ref{nD}) \cite{bc,kn2}. But it is still insufficient for moduli stabilization that requires
at least one $p$-form field obeying the condition (\ref{np}) \cite{nad}.

Therefore, our minimal model in $D=8$ is defined by the action
\begin{equation} \label{act1}
S= \fracmm{M_8^6}{2} \int d^8X\sqrt{-g_8} \left[
R_8+\gamma_8 R_8^4-2\Lambda_8-g^{A_1B_1}g^{A_2B_2}g^{A_3B_3} g^{A_4B_4}F_{A_1A_2A_3A_4}F_{B_1B_2B_3B_4}\right].
\end{equation}
It depends upon two fields, a metric $g_{AB}$ and a 3-form gauge potential $A_{ABC}$, whose
field strength 4-form is $F=dA$, and has three parameters: the gravitational mass scale $M_8$,
 the (modified gravity) coupling constant $\gamma_8>0$ and the cosmological constant $\Lambda_8>0$,
 all in $D=8$ dimensions --- see Appendix A for more details.

Applying the Legendre-Weyl transform to the action (\ref{act1}) in $D=8$ results in the dual (classically equivalent) action (see Appendix A for its derivation)
\begin{equation} \label{act2}
\eqalign{
S_{\rm dual} = & \fracmm{M_8^6}{2} \int d^8X\sqrt{-\tilde{g}_8} \left[
\tilde{R}_8-42\tilde{g}^{AB}\partial_Af\partial_Bf  -M^2_8\tilde{V}(f) \right. \cr
 & \left. -\tilde{g}^{A_1B_1}\tilde{g}^{A_2B_2}\tilde{g}^{A_3B_3}\tilde{g}^{A_4B_4}F_{A_1A_2A_3A_4}F_{B_1B_2B_3B_4}\right]~~,\cr}
\end{equation}
depending upon three fields, the Weyl transformed (new) metric $\tilde{g}_{AB}$, the 4-form $F=dA$,
and the real scalaron  $f(X)$ having the scalar potential
\begin{equation} \label{p1}
\tilde{V}(f)=  a^{-2}(1-e^{-6f})^{\frac{4}{3}}+2e^{-8f}\tilde{\Lambda}_8~~,
\end{equation}
in terms of the (dimensionless) coupling constants
\begin{equation} \label{rdf}
\gamma_8 \equiv M_8^{-6}\tilde{\gamma}_8~~, \quad \Lambda_8\equiv M_8^{2}\tilde{\Lambda}_8~~,
\quad \fracmm{3}{4} \left(\fracmm{1}{4\tilde{\gamma}_8}\right)^{\frac{1}{3}}\equiv a^{-2}~~.
\end{equation}
The dual action ({\ref{act2}) has the standard form of Einstein's gravity coupled to the matter fields $(f,A)$ and
having the scalar potential (\ref{p1}) in $D=8$. This scalar potential has a plateau of the positive height $a^{-2}$ for large positive values of $f$.

Let us consider a compactification of the $D=8$ theory (\ref{act2}) on a 4-sphere $S^4$ with the warp factor  $\chi$, down to {\it four} spacetime dimensions, i.e. in a curved spacetime with the local structure
\begin{equation} \label{compactify}
{\cal M}_8 = {\cal M}_4\times e^{2\chi}S^4~~.
\end{equation}
The 8-dimensional coordinates $(X^A)$ can then be decomposed into the 4-dimensional spacetime coordinates $(x^{\alpha})$ with $\alpha=0,1,2,3$, and the coordinates $(y^a)$ of four compact dimensions of
$S^4$, with  $a,b,\ldots=1,2,3,4$. The compactification ansatz reads
\begin{equation} \label{cansatz}
ds^2_8=\tilde{g}_{AB}dX^AdX^B=g_{\alpha\beta}dx^{\alpha}dx^{\beta}+e^{2\chi}g_{ab}dy^ady^b~~,
\end{equation}
where $g_{\alpha\beta}=g_{\alpha\beta}(x)$, $g_{ab}=g_{ab}(y)$ and $\chi=\chi(x)$.

This compactification results in the following $D=4$ action (see Appendix B for its derivation):
\begin{align} \label{4tot}
S_4 [\hat{g}_{AB},\chi, f]
&=\fracmm{M_{\rm Pl}^2}{2}\int d^4x\sqrt{-g}\left[ \hat{R}-12\hat{g}^{\alpha\beta}\partial_{\alpha}\chi\partial_{\beta}\chi \right. \nonumber\\
  &\left. - 42\hat{g}^{\alpha\beta}\partial_{\alpha}f\partial_{\beta}f
 - M_{\rm Pl}^2\left( e^{-4\chi}\tilde{V}(f)-2e^{-6\chi}-e^{-12\chi}F^2\right) \right]~~,
\end{align}
of three fields: a metric $\hat{g}_{\alpha\beta}(x)$, the scalaron $f(x)$ and the $S^4$ (volume) modulus
$\chi(x)$, with the scalar potential depending upon the parameters $(a,\tilde{\Lambda}_8)$ {\it and}
 the 4-form gauge field strength {\it flux\/} $F$ defined by  the integration
 \begin{equation} \label{fpar}
\int d^4y\sqrt{g_y}\,g^{a_1b_1}\cdots g^{a_4b_4}F_{a_1...a_4}F_{b_1...b_4} = M_8^{-2}F^2
\end{equation}
over the $S^4$. The full two-scalar potential in $D=4$ thus reads
\begin{equation} \label{2p}
M_{\rm Pl}^{-4}V(\chi,f)=\left[ a^{-2}(1-e^{-6f})^{\frac{4}{3}}+2\tilde{\Lambda}_8 e^{-8f}\right]e^{-4\chi}-2e^{-6\chi}+F^2e^{-12\chi}~~.
\end{equation}
We have restored the reduced Planck scale $M_{\rm Pl}$ in Eqs.~(\ref{4tot}) and (\ref{2p}) for reader's convenience.

The scalar potential (\ref{2p}}) is investigated in detail in Appendix C. It has a stable Minkowski vacuum
and a plateau with a positive height provided that
\begin{equation} \label{Frange}
1<F^2/(16\tilde{\gamma}_8)\equiv 1+\delta<(\frac{3}{2})^3~~,
\end{equation}
where the inequality on the right hand side is also needed to ensure a positive mass squared of the modulus $\chi$ at the onset of inflation --- see Eq.~(\ref{posh}).

For generic values of $\delta$ in Eq.~(\ref{Frange}) one gets a two-field inflation. However, the modulus
$\chi$ is strongly stabilized when $\delta \ll 1$ that implies only a small shift of the minimum of $\chi$ during
inflation, from  $\chi_c$ to $\chi_0$, as
\begin{equation} \label{dds}
 0<\chi_c-\chi_0 \approx \fracmm{1}{12}\delta \ll 1~~,
\end{equation}
and results in a {\it single-field inflation} driven by the inflaton (scalaron) $f$ in $D=4$.

The physical {\it hierarchy} of scales in Eq.~(\ref{hier}) can be satisfied provided that
\begin{equation} \label{Fr}
F^2\gg 1~~,
\end{equation}
where we have used the KK scale
\begin{equation} \label{KKs}
M_{\rm KK}\approx e^{-\chi_0}M_{\rm Pl}
\end{equation}
with the warp factor due to the compactification ansatz (\ref{compactify}) and (\ref{cansatz}).

The mass condition  $m\low{\hat{f}_0} <  m\low{\hat{\chi}_0}$ implies $F^2/\tilde{\gamma}_8 <72$ that is
already satisfied due to (\ref{Frange}). However, it is not possible to get a stronger condition
 $m\low{\hat{f}_0} \ll  m\low{\hat{\chi}_0}$.

A profile of the scalar potential in $D=4$ is given in Fig.~1. It should be mentioned that the cosmological constant
in $D=8$ is given by Eq.~(\ref{fl8}) that implies
\begin{equation}\label{cc8}
\tilde{\Lambda}_8 = \fracmm{\delta^{-1/3}}{2a^2}~~,
\end{equation}
where we have used Eq.~(\ref{rdf}) also. In particular, it means that $\delta$ cannot vanish.

\begin{figure}[htbp]
\begin{center}
\includegraphics[clip,width=7cm,height=5cm]{fig1.pdf}
\caption{The profile of the scalar potential (\ref{2p}) for the numerical input $F^2=10^6$, $\tilde{\gamma}_8= 6\cdot 10^4$
and $\tilde{\Lambda}_8\approx  0.0174$. The bottom line shows the inflationary trajectory.}
\label{fig:one}
\end{center}
\end{figure}

\section{Towards a supergravity embedding of our model}

In this Section we explore a possibility of embedding our 8-dimensional model (\ref{act1}) into a $D=8$ supergravity. First, supergravity may be the natural origin of the $p$-form field because higher-dimensional supergravities usually include such fields. Second, the supergravity extensions of modified gravity certainly exist  in $D=4$ \cite{myrev,kter}, and they should also exist in higher dimensions $D\leq 11$.

Unfortunately, to the best of our knowledge, no fully supersymmetric extension of any $(R+R^4)$ gravity
in higher $(8\leq D \leq 11)$  dimensions was ever derived, so that our investigation in this Section cannot be fully consistent and compelling, unlike that in the previous Sections. Moreover, any standard (two-derivative) supergravity theory does not allow a positive cosmological constant in its action (it would break supersymmetry), so that the origin of the cosmological consant in $D=8$ can only be either due to a spontaneous supersymmetry breaking or/and some nonperturbative effects. So, this Section ends up with a conjecture.

A good starting point of this investigation is the maximally supersymmetric $D=11$ supergravity, because
of its uniqueness. It can be modified by the quartic scalar curvature term and then compactified down to
$D=8$ on a compact manifold (3-sphere $S^3$) --- see Appendix D for details. Moreover, the $SO(3)$  non-abelian isometries of the $S^3$ can be gauged, thus producing the non-abelian gauge fields and a scalar potential in $D=8$. Taken together, it leads to the bosonic part of the (modified and gauged) $D=8$ supergravity action, having the form (\ref{8saction}),
\begin{gather}
S_8=\int d^8x\fracmm{e}{2\kappa^2}[R+\tilde{\gamma}e^{2\kappa\phi}R^4-\kappa^2e^{2\kappa\phi}F_{\mu\nu}^\alpha F^{\mu\nu}_\alpha-2\kappa^2\partial_\mu\phi\partial^\mu\phi-V(T)-P_{\mu ij}P^{\mu ij}\nonumber\\
-\fracmm{1}{2}\kappa^2e^{-4\kappa\phi}\partial_\mu B\partial^\mu B-\fracmm{\tilde{\kappa}^2}{12}e^{2\kappa\phi}G_{\mu\nu\rho\sigma}G^{\mu\nu\rho\sigma}-\fracmm{\kappa^3}{432}e^{-1}\varepsilon^{\mu_1...\mu_8}G_{\mu_1...\mu_4}G_{\mu_5...\mu_8}B]+...~, \label{finale}
\end{gather}
in terms of the following $D=8$ fields: a metric $g_{\mu\nu}$, dilaton $\phi$, the $SO(3)$ gauge field strength
$F^{\alpha}_{\mu\nu}$, the vector fields  $P_{\mu ij}$, the 4-form gauge field strength  $G_{\mu\nu\rho\sigma}$ and $(5+1)$ scalars $(T,B)$ whose scalar potential is
\begin{equation} \label{spt8}
V(T)=\fracmm{g^2}{4\kappa^2}e^{-2\kappa\phi}(T_{ij}T^{ij}-\frac{1}{2}T^2)~.
\end{equation}
The supergravity (\ref{finale})  has the required {\it quartic} scalar curvature term and the gauge 3-form kinetic term given by the gauge field strength {\it 4-form} squared, while the abelian vector fields  $P_{\mu ij}$ are merely the spectators here. Hence, (\ref{finale}) could be the supersymmetric extension of our action (\ref{act1}) provided that (i) the dilaton $\phi$ is stabiized, and (ii) a positive cosmological constant is generated. One usually assumes in the literature that the dilaton potential is generated by quantum non-perturbative corrections beyond the supergravity level. And the cosmological constant may be generated by the non-perturbative vacuum expectation value
\begin{equation} \label{vevF}
\VEV{\kappa^2e^{2\kappa\phi}F_{\mu\nu}^\alpha F^{\mu\nu}_{\alpha}}=2\Lambda_8~~.
\end{equation}
Unfortunately, we do not have means to compute the dilaton vacuum expectation value and the
gluon condensate (\ref{vevF}) in $D=8$.

\section{Inflation}

Once the modulus $\chi$ is strongly stabilized (Sec.~2), the inflaton potential (\ref{2p}) takes the form
($M_{\rm Pl}=1$)
\begin{equation} \label{effsp}
e^{4\chi_0}a^2V(f)=\left(1-e^{-6f}\right)^{\frac{4}{3}}+\lambda e^{-8f} - \lambda(1+\lambda^3)^{-\frac{1}{3}}~~.
\end{equation}
with $\lambda=2a^2\tilde{\Lambda}_8=\delta^{-1/3}$. This potential has the absolute minimum at
\begin{equation} \label{min8}
f_0=\frac{1}{6}\ln\left(1+\lambda^3\right)~~,
\end{equation}
where it vanishes in the Minkowski vacuum. A profile of the scalar potential (\ref{effsp}) is given in Fig.~2.

\begin{figure}[htbp]
\begin{center}
\includegraphics[clip,width=7cm,height=5cm]{fig2.pdf}
\caption{The profile of the scalar potential (\ref{effsp}) for $\lambda=1$ (green), $\lambda=2$ (red) and
$\lambda=2.88$ (blue).}
\label{fig:two}
\end{center}
\end{figure}

During inflationary slow roll along the plateau, the scalar potential  (\ref{effsp})  can be approximated as
\begin{equation} \label{pclass}
V(\phi) = V_0 - V_1e^{-\alpha\phi}~~,
\end{equation}
with
\begin{equation} \label{aval}
\alpha = \sqrt{\fracmm{6}{7}}~~.
\end{equation}
This value  of $\alpha$ determines the key observational parameter $r$ related to primordial gravity waves and known as the tensor-to-scalar ratio,
\begin{equation} \label{rbound}
r =\fracmm{8}{\alpha^2N_e^2}=\fracmm{28}{3N_e^2}~~.
\end{equation}
The Planck data \cite{Ade:2015lrj} sets the upper bound on $r$ (with 95\% of CL) as $r<0.08$. It implies
\begin{equation}\label{abound}
\alpha >\fracmm{10}{N_e}= 0.2 \left(\fracmm{50}{N_e}\right)~,
\end{equation}
while our result (\ref{aval}) is clearly  above this bound.

As regards the other CMB spectral tilts (the inflationary observables), the scalar spectral index $n_s$
and its running $dn_s/d{\ln}k$, their values derived from the potential  (\ref{pclass}) are
\begin{equation} \label{otilts}
n_s\approx 1 - \fracmm{2}{N_e} \qquad {\rm and} \qquad \fracmm{dn_s}{d \ln k} \approx -\fracmm{(1-n_s)^2}{2}\approx -\fracmm{2}{N_e^2}~~,
\end{equation}
i.e. they are the same as in the Starobinsky model (\ref{star}) and (\ref{starpot}).

The microscopic parameters of our model can be easily tuned to get the {\it same} inflaton mass $M$, so that
our effective inflationary model obtained from the higher ($D=8$) dimensions is almost indistinguishable from
the orginal Starobinsky model having $\alpha_s=\sqrt{2/3}$.

 When a conventional matter action is added to our gravity model, Weyl rescalings of the metric result in the {\it universal} couplings (via the GR covariant derivatives) of inflaton $f$ to all matter fields with powers of $\exp{(-\alpha\kappa_4 f)}$. The value (\ref{aval}) of $\alpha$ derived from $D=8$ is only slightly different from the Starobinsky value $\alpha_s=\sqrt{2/3}$, while all the matter couplings to the scalaron are suppressed by the Planck mass. Therefore, the impact of higher dimensions on the inflationary observables and reheating is negligible in
 our approach.

\section{Conclusion}

Our main results are summarized in the Abstract.

We used the Starobinsky inflationary model of the $(R+R^2)$ gravity (\ref{star}) in four dimensions as the prototype for deriving the new inflationary models from higher dimensions. Among the advantages of this approach are (i) its geometrical nature, as only gravitational interactions are used, (ii) consistency with the current astronomical observations of CMB, (iii)
the clear physical nature of inflaton (scalaron) as the spin-0 part of metric. In this paper we focused on $D=8$ dimensions only. In our scenario, the Universe was born multi-dimensional, and then four spacetime dimensions became infinite, while the others curled up by some unknown mechanism before inflation. The inflation happened after the compactification and the moduli stabilization.

In higher-dimensions, it turned out to be necessary to include a cosmological constant
and a gauge (form) field, with the strong conditions on the higher dimension, the power $n$ of the scalar curvature
and the rank of the form, see Eqs.~(\ref{nD}) and (\ref{np}). The moduli stabilization and the scale hierarchy are also possible to achieve, while both are non-trivial in the present context. It may also be possible to embed our $D=8$ modified gravity model into the modified $D=8$ supergravity and, ultimately, into the modified $D=11$ supergravity.

As regards the observational predictions of our model, it leads to the certain value (\ref{rbound}) of the CMB tensor-to-scalar ratio that is, however, only slightly different from that of the original Starobinsky model.

Our results may be used for studying inflation and moduli stabilization in more general frameworks, such as unification of
fields and forces, KK theories of gravity, supergravity and superstrings, and braneworld.~\footnote{In particular, as was found in \cite{nk1}, the modified $(R+R^2)$ gravity in the Randall-Sundrum (RSII) braneworld \cite{rs} does {\it not} destabilize the
famous Randall-Sundrum solution to the hierarchy problem in particle physics.}
\vglue.2in

\section*{Acknowledgements}

S.V.K. was supported by a Grant-in-Aid of the Japanese Society for Promotion of Science (JSPS) under No.~26400252, the Competitiveness Enhancement Program of Tomsk Polytechnic University in Russia, and the World Premier International Research Centre Initiative (WPI Initiative), MEXT, Japan. S.V.K. is grateful to A. Zhuk and Y. Aldabergenov for discussions and correspondence.
\vglue.2in

\section*{Appendix A: Legendre-Weyl transform in $D=8$}

We denote spacetime vector indices in eight dimensions by capital latin letters $A,B,\ldots=0,1,\ldots, 7$, and use the spacetime signature $(-,+,\ldots,+)$.

Let us begin with the following gravitational action in an 8-dimensional curved spacetime:
\begin{equation} \label{action1}
 S_{8,{\rm grav.}}=\fracmm{1}{2\kappa_8^2}\int d^8X\sqrt{-g_8}(R_8+\gamma_8 R_8^4-2\Lambda_8)~~,
\end{equation}
where we have introduced the gravitational coupling constant $\kappa_8$ of (mass) dimension
$(-3)$, the (modified gravity) coupling constant $\gamma_8>0$ of (mass) dimension $(-6)$, and
and the cosmological constant $\Lambda_8>0$ of (mass) dimension $(+2)$, all in $8$ dimensions.~\footnote{The results of this Appendix are obtained by specifying the more general
results (for any $D$ and $n$) of Ref.~\cite{kn2} to $D=8$ and $n=4$.}

The action (\ref{action1}) can be rewritten to the form
\begin{equation}  \label{action2}
S_{8,{\rm grav.}} =\fracmm{1}{2\kappa^2_8}\int d^8X\sqrt{-g_8}\left[(1+
 B)R_8 - \fracmm{3}{4}
 \left(\fracmm{B^4}{4\gamma_8}\right)^{\frac{1}{3}}  -2\Lambda_8\right]~~,
\end{equation}
where we have introduced the new scalar field $B$. The field $B$ enters the action
(\ref{action2})  algebraically, while its "equation of motion" reads
\begin{equation}  \label{eom}
B=4\gamma_8 R^{3}_8~~.
\end{equation}
Substituting it back into the action (\ref{action2})  yields the original action (\ref{action1}). Hence, the actions (\ref{action1}) and (\ref{action2}) are classically equivalent.

In order to transform the action (\ref{action2}) to Einstein frame, we apply a Weyl transformation of the metric with the spacetime-dependent parameter $\Omega(X)$ in $8$ dimensions,
\begin{equation} \label{weylt}
 g_{AB}=\Omega^{-2}\tilde{g}_{ AB},\ \ \ \sqrt{-g_8}=\Omega^{-8}\sqrt{-\tilde{g}_8}~~,
\end{equation}
where we have introduced the new spacetime metric $\tilde{g}_{ AB}$. As a result of this transformation, the (Ricci) scalar curvature gets transformed as
\begin{equation} \label{weylr}
 R_8=\Omega^2[\tilde{R}_8+14\Tilde{\square}_8f-42\tilde{g}^{AB}f_{A}f_{B}]~~,
\end{equation}
where we have introduced the notation
\begin{equation}
 f=\ln \Omega~,\qquad f_{A}=\fracmm{\partial_A\Omega}{\Omega}~~,
\end{equation}
and the covariant wave operator $\Tilde{\square}_8=\Tilde{D}^A\Tilde{D}_A$ in $8$ dimensions.

The Weyl-transformed (and also equivalent by the field-redefinition (\ref{weylt})) action reads
\begin{align} \label{action3}
 S_{8,{\rm grav.}}=&\fracmm{1}{2\kappa_8^2}\int d^8X\sqrt{-\tilde{g}_{ 8}}\Omega^{-8}\left[ (1+B)\Omega^2(\tilde{R}_8+14\Tilde{\square}_8f \right. \nonumber\\
 &\left. -42\tilde{g}^{AB}f_Af_B)- \fracmm{3}{4} \left(\fracmm{B^4}{4\gamma_8}\right)^{\frac{1}{3}}
 -2\Lambda_8 \right]~.
\end{align}

Hence, the action in the Einstein frame is obtained by choosing the local parameter $\Omega$ as
\begin{equation} \label{omegaf}
\Omega^{6}=e^{6f}=1+B~~.
\end{equation}
After ignoring the total derivative in the Lagrangian, it yields
\begin{align} \label{action4}
 S_{8,{\rm grav.}}[\tilde{g}_{AB},f] =&\fracmm{1}{2\kappa^2_8} \int d^8X \sqrt{-\tilde{g}_{8}} \left[ \tilde{R}_8
 -42\tilde{g}^{AB}\partial_Af\partial_Bf \right. \nonumber\\
 - & \left. \fracmm{3}{4} \left(\fracmm{1}{4\gamma_8}\right)^{\frac{1}{3}}\left(1-e^{-6f}\right)^{4/3} -
 2e^{-8f}\Lambda_8 \right]~~.
\end{align}

Let us redefine the coupling constants in $8$ dimensions as
\begin{equation} \label{ccrdf}
\kappa_8\equiv M_8^{-3}~~,\quad \gamma_8 \equiv M_8^{-6}\tilde{\gamma}_8~~,
\quad \Lambda_8\equiv M_8^{2}\tilde{\Lambda}_8~~,\quad
\fracmm{3}{4} \left(\fracmm{1}{4\tilde{\gamma}_8}\right)^{\frac{1}{3}}\equiv a^{-2}~~,
\end{equation}
in terms of the new (mass) parameter $M_8>0$ of dimension $(+1)$, and the dimensionless
parameters $\tilde{\Lambda}_8$ and $a>0$. Then the action (\ref{action4}) takes the form
\begin{equation} \label{action5}
  S_{8,{\rm grav.}}[\tilde{g}_{AB},f] =\fracmm{M_8^6}{2} \int
  d^8X\sqrt{-\tilde{g}_{8}}\left[ \tilde{R}_8-42\tilde{g}^{AB}\partial_Af\partial_Bf
  -M^2_8\tilde{V}(f)\right]
\end{equation}
with the (dimensionless) scalar potential
\begin{equation} \label{pot1}
\tilde{V}(f)=  a^{-2}(1-e^{-6f})^{\frac{4}{3}}+2e^{-8f}\tilde{\Lambda}_8~~.
\end{equation}

Given an 8-dimensional action of the 4-form $F$ (the totally antisymmetric gauge field strength $F=dA$ of a gauge 3-form potential $A$) in the form
\begin{equation} \label{4faction1}
 S_{8} [g_{AB},F_4] =- \fracmm{M_8^6}{2}\int d^8X\sqrt{-g_8}\,g^{A_1B_1}g^{A_2B_2}g^{A_3B_3} g^{A_4B_4}F_{A_1A_2A_3A_4}F_{B_1B_2B_3B_4}~~,
\end{equation}
the $F$ has (mass) dimension $(+1)$, and the $A$ is dimensionless.

Under the Weyl transform (\ref{weylt}), the $\Omega$-factors are {\it cancelled} against
each other, so that the action (\ref{4faction1}) remains unchanged,
\begin{equation} \label{4faction2}
 S_{8} [\tilde{g}_{AB},F_4] =-\fracmm{M_8^6}{2}\int d^8X\sqrt{-\tilde{g}_8}\,\tilde{g}^{A_1B_1}\cdots  \tilde{g}^{A_4B_4}F_{A_1...A_4}F_{B_1...B_4}~~.
\end{equation}

The action of our model in 8 dimensions is defined by
\begin{equation} \label{action6}
S_8 [\tilde{g}_{AB},f,F_4] = S_{8,{\rm grav.}}[\tilde{g}_{AB},f] +  S_{8} [\tilde{g}_{AB},F_4] ~~.
\end{equation}

\section*{Appendix B: Freund-Rubin-type compactificaton}

In this Appendix we consider the compactification of the theory (\ref{action6}) on a 4-sphere $S^4$ with the warp factor $\chi$, down to 4 spacetime dimensions. We separate the 8-dimensional coordinates $(X^A)$ into the 4-dimensional spacetime coordinates $(x^{\alpha})$
with $\alpha=0,1,2,3$, and the coordinates $(y^a)$ of four extra (compact) dimensions with
$a,b,\ldots=1,2,3,4$.~\footnote{Our results in this Appendix fully agree with Ref.~\cite{nad}.}

We use the standard compactification ansatz
\begin{equation} \label{mansatz}
ds^2_8=\tilde{g}_{AB}dX^AdX^B=g_{\alpha\beta}dx^{\alpha}dx^{\beta}+e^{2\chi}g_{ab}dy^ady^b~~,
\end{equation}
where $g_{\alpha\beta}=g_{\alpha\beta}(x)$, $g_{ab}=g_{ab}(y)$ and $\chi=\chi(x)$, with the
normalization
\begin{equation} \label{norm4s}
  \int d^4y\sqrt{g_y}=M_8^{-4}~~.
\end{equation}
Taking into account the  $S^4$ Euler number equal to 2, yields
 \begin{equation} \label{eulers}
 \int d^4y\sqrt{g_y}R_y=2M_8^{-2}~~,
\end{equation}
where $R_y$ is the scalar curvature of the sphere $S^4$. The decomposition (\ref{mansatz}) also implies
\begin{equation} \label{measured}
\sqrt{-\tilde{g}_8}=e^{4\chi}\sqrt{-g_4}\sqrt{g_{y}}
\end{equation}
and
\begin{equation} \label{riccideco}
\tilde{R}_8 =R+e^{-2\chi}R_{y}-8e^{-\chi} \Tilde{\square}e^{\chi}
- 12 e^{-2\chi}g^{\alpha\beta} \partial_{\alpha}e^{\chi}\partial_{\beta}e^{\chi}~~,
\end{equation}
where we have introduced the Ricci scalar $R$ and the generally covariant wave operator
$\Tilde{\square}=g^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}$ in four spacetime dimensions.

The volume ${\cal V}$ of four (compact) extra dimensions is given by
\begin{equation} \label{4smod}
{\cal V} =   \int d^4y\sqrt{ \det(e^{2\chi} g_{ab})} = e^{4\chi}M_8^4~~,
\end{equation}
so that the warp factor $\chi$ can be identified with the volume modulus of the sphere $S^4$.

A substitution of  Eqs.~(\ref{mansatz}), (\ref{measured}) and (\ref{riccideco}) into the action (\ref{action5}), and an integration over the compact dimensions by using Eqs.~(\ref{norm4s}) and
(\ref{eulers}), lead to the action
\begin{align}\label{4action1}
 S_4[g_{\alpha\beta},f,\chi]
 &=\fracmm{M_8^2e^{4\chi_0}}{2}\int
 d^4x\sqrt{-g}\left(\fracmm{e^{\chi}}{e^{\chi_0}}\right)^4\left[ R+2M_8^2e^{-2\chi}\right. \nonumber \\
 &\left. +12g^{\alpha\beta}\partial_{\alpha}\chi\partial_{\beta}\chi-42g^{\alpha\beta}\partial_{\alpha}f\partial_{\beta}f-M_8^2\tilde{V}(f)\right]~~,
\end{align}
where we have introduced the vacuum expectation value $\VEV{\chi}_0=\chi_0=const$.

The action (\ref{4action1}) is still in a Jordan frame, so that the wrong sign of the kinetic term of the field $\chi$ is not necessarily a problem. The Weyl transformation with the parameter
$h(x)$ to the Einstein frame is given by
\begin{equation} \label{weyl2a}
 g_{\alpha\beta}=e^{-2h}\hat{g}_{\alpha\beta},\quad
 h=2(\chi-\chi_0)~~.
\end{equation}
It implies
\begin{equation} \label{weyl2b}
 g^{\alpha\beta}=e^{2h}\hat{g}^{\alpha\beta},\quad \sqrt{-g}=e^{-4h}\sqrt{-\hat{g}}~,
\end{equation}
and
\begin{equation} \label{weyl2c}
 R=e^{2h}\left[\hat{R}+6\hat{g}^{\alpha\beta}\nabla_{\alpha}\nabla_{\beta}h-6\hat{g}^{\alpha\beta}\partial_{\alpha}h\partial_{\beta}h\right]~~.
\end{equation}
Accordingly, the action (\ref{4action1}) gets transformed to
 \begin{align} \label{4action2}
  S_4 [\hat{g}_{\alpha\beta},f,\chi]  &=\fracmm{M_8^2e^{4\chi_0}}{2}\int
  d^4x\sqrt{-\hat{g}_4}\left\{ \hat{R}-12\hat{g}^{\alpha\beta}\partial_{\alpha}\chi\partial_{\beta}\chi
  \right. \nonumber\\
  &\left. -42\hat{g}^{\alpha\beta}\partial_{\alpha}f\partial_{\beta}f
  -\left(\fracmm{e^{\chi}}{e^{\chi_0}}\right)^{-4}M_8^2\left[ \tilde{V}(f)-2e^{-2\chi}\right]\right\}~,
 \end{align}
with the {\it physical} signs in front of all the kinetic terms. This also fixes the four-dimensional (reduced) Planck mass as
\begin{equation}\label{planckm}
M_{\rm Pl}^2\equiv \kappa^{-2}= M_8^2e^{4\chi_0}~.
\end{equation}
Therefore, we have
\begin{align}  \label{4action3}
 S_4  [\hat{g}_{\alpha\beta},f,\chi]
 &= \fracmm{M^2_{\rm Pl}}{2}\int
 d^4x\sqrt{-\hat{g}}\left[ \hat{R}-12\hat{g}^{\alpha\beta}\partial_{\alpha}\chi\partial_{\beta}\chi
 \right. \nonumber \\
 -&\left. 42\hat{g}^{\alpha\beta}\partial_{\alpha}f\partial_{\beta}f
 -e^{-4\chi} M^2_{\rm Pl} \left( \tilde{V}(f)-2e^{-2\chi}\right)\right]~~.
\end{align}

Similarly, applying the compactification ansatz (\ref{mansatz}) to the 4-form action
(\ref{4faction2}) in 8 dimensions yields
 \begin{equation} \label{8actionF}
 S_{8,F} [\tilde{g}_{AB},F]=-\fracmm{M_8^6}{2}\int d^4x\sqrt{-g}\int d^4y\sqrt{g_y}\,e^{-4\chi}g^{a_1b_1}\cdots g^{a_4b_4}F_{a_1...a_4}F_{b_1...b_4}~~.
\end{equation}
After defining the (dimensionless) {\it flux\/} parameter $F^2$ as
\begin{equation} \label{fluxp}
\int d^4y\sqrt{g_y}\,g^{a_1b_1}\cdots g^{a_4b_4}F_{a_1...a_4}F_{b_1...b_4} = M_8^{-2}F^2=const.~~,
\end{equation}
and using the Weyl transformation (\ref{weyl2a}), the action (\ref{8actionF}) reduces to
\begin{align} \label{4actionF}
S_{4,F}[\hat{g}_{AB},\chi]
&=-\fracmm{M_8^2e^{4\chi_0}}{2}\int d^4x\sqrt{-g}\left(\fracmm{e^{\chi}}{e^{\chi_0}}\right)^4e^{-8\chi}M_8^2F^2\nonumber \\
&=-\fracmm{M_8^2e^{4\chi_0}}{2}\int d^4x\sqrt{-\hat{g}}\,e^{-4h}\left(\fracmm{e^{\chi}}{e^{\chi_0}}\right)^4e^{-8\chi}M^2F^2\nonumber \\
&=-\fracmm{M^4_{\rm Pl}}{2}\int d^4x\sqrt{-\hat{g}}\,e^{-12\chi}F^2~~.
\end{align}

The total action in 4 dimensions is given by a sum of Eqs.~(\ref{4action3}) and (\ref{4actionF}),
\begin{align} \label{4total}
S_4 [\hat{g}_{AB},\chi, f]
&=\fracmm{M_{\rm Pl}^2}{2}\int d^4x\sqrt{-g}\left[ \hat{R}-12\hat{g}^{\alpha\beta}\partial_{\alpha}\chi\partial_{\beta}\chi \right. \nonumber\\
  &\left. - 42\hat{g}^{\alpha\beta}\partial_{\alpha}f\partial_{\beta}f
 - M_{\rm Pl}^2\left( e^{-4\chi}\tilde{V}(f)-2e^{-6\chi}-e^{-12\chi}F^2\right) \right]~~.
\end{align}

The canonical scalar fields $\hat{\chi}$ and $\hat{f}$ are thus given by
\begin{equation} \label{norm2s}
\hat{\chi}=2\sqrt{3}M_{\rm Pl}\chi \quad {\rm and} \quad  \hat{f}=\sqrt{42}M_{\rm Pl}f~~,
\end{equation}
and the two-scalar potential in 4 dimensions reads
\begin{equation} \label{2pot}
M_{\rm Pl}^{-4}V(\chi,f)=\left[ a^{-2}(1-e^{-6f})^{\frac{4}{3}}+2\tilde{\Lambda}_8 e^{-8f}\right]e^{-4\chi}-2e^{-6\chi}+F^2e^{-12\chi}~~.
\end{equation}

\section*{Appendix C : study of the scalar potential}

In this Appendix we investigate the scalar potential (\ref{2pot}) in four dimensions. It depends upon
two fields, the inflaton $f$ and the modulus $\chi$, and has three parameters $(a^{-2},F^2,\tilde{\Lambda}_8)$ originating from eight dimensions (see appendices A and B).~\footnote{A partial analysis of generic potentials arising in the
same way from any dimension $D=2n$ was done in Ref.~\cite{nad}. We get more results for $D=8$.}

The potential (\ref{2pot}) has a Minkowski vacuum $(f_0,\chi_0)$ defined by the equations
\begin{equation} \label{mvac}
\left. \fracmm{\pa V}{\pa f} \right|_{f=f_0}=\left. \fracmm{\pa V}{\pa \chi} \right|_{\chi=\chi_0}=\left.
V\right|_{f=f_0,~\chi=\chi_0}=0~~.
\end{equation}
The solution to these three equations is given by
\begin{equation} \label{smvac}
e^{6f_0}= 1 + (2\tilde{\Lambda}_8a^2)^3 \quad {\rm and} \quad  e^{6\chi_0}= 2F^2~,
\end{equation}
together with a condition of the parameters,
\begin{equation} \label{fl8}
\frac{2}{3}\tilde{\Lambda}_8=\left( \fracmm{1}{16F^2 - 256 \tilde{\gamma}_8}\right)^{1/3}~~,
\end{equation}
where we have used the third relation (\ref{ccrdf}) between $\tilde{\gamma}_8$ and $a$.

The second derivatives of the scalar potential (\ref{2pot}) at the critical point (\ref{smvac}) determine the masses of the canonically normalized scalars  (\ref{norm2s}) as
\begin{equation} \label{massf}
m^2_{\hat{f}_0}=\left. \fracmm{\pa^2 V}{\pa f^2} \right|_{f=f_0}\fracmm{1}{42M^2_{\rm Pl}} =
\fracmm{M^2_{\rm Pl}}{56F^2}\left( \fracmm{F^2}{\tilde{\gamma}_8} - 16\right)~~,
\end{equation}
and
\begin{equation} \label{massh}
m^2_{\hat{\chi}_0}=\left. \fracmm{\pa^2 V}{\pa \chi^2} \right|_{\chi=\chi_0}\fracmm{1}{12M^2_{\rm Pl}} =
\fracmm{M^2_{\rm Pl}}{F^2}~~,
\end{equation}
where we have used (\ref{fl8}) also. Equations (\ref{fl8}) and (\ref{massf}) imply the same condition
\begin{equation} \label{stab}
\fracmm{F^2}{\tilde{\gamma}_8}>16
\end{equation}
for both the existence of a Minkowski vacuum and its stability, respectively.

At the onset of inflation ($f=+\infty$), the scalar potential of the modulus $\chi$ is given by
\begin{equation} \label{hpot}
M_{\rm Pl}^{-4}V(\chi)=a^{-2}e^{-4\chi}-2e^{-6\chi}+F^2e^{-12\chi}
\end{equation}
that only depends upon two (free) parameters $(a^{-2},F^2)$.

The critical points of the potential (\ref{hpot}) are determined by the condition
\begin{equation} \label{hpotcr}
a^{-2}-3e^{2\chi_c}+3F^2e^{-8\chi_c}=0
\end{equation}
that has the form of the {\it depressed quartic} equation
\begin{equation} \label{dqua}
z^4 + qz +r =0
\end{equation}
in terms of
\begin{equation} \label{quadef}
z= e^{-2\chi_c}~~,\qquad q=\fracmm{-1}{F^2}<0~,\qquad r = \fracmm{1}{3a^2F^2}>0~~.
\end{equation}
The {\it quartic} discriminant is given by
\begin{equation} \label{quadis}
\fracmm{\Delta_4}{27\cdot 256} = (r/3)^3 - (q/4)^4 ~~,
\end{equation}
while writing down an explicit solution to (\ref{dqua}) depends upon the sign of $\Delta_4$.

By using the auxiliary (Ferrari's) resolvent cubic equation
\begin{equation} \label{ferc}
m^3 - rm - q^2/8=0~~,
\end{equation}
we can factorize the left-hand-side of the quartic equation (\ref{dqua}) as
\begin{equation} \label{quafact}
\left(z^2+m +\sqrt{2m} z - \fracmm{q}{2\sqrt{2m}}\right)\left(z^2+m -\sqrt{2m} z + \fracmm{q}{2\sqrt{2m}}\right)=0~.
\end{equation}
Because each term in the first factor is positive in our case, we get a {\it quadratic}
equation from the vanishing second factor whose two roots are given by
\begin{equation} \label{qurt2}
z_{1,2} = \sqrt{\fracmm{m}{2}}\left[ 1 \pm \sqrt{-\fracmm{q}{m}-\sqrt{2m} }\right]~.
\end{equation}
These two roots precisely correspond to the existence of a local (meta-stable) minimum and a local maximum of the potential (\ref{hpot}), with $-\infty<\chi_{\rm min.}<\chi_{\rm max.}<+\infty$.

The {\it cubic} discriminant $\Delta_3=4r^3-27(q^2/8)^2$ of the {\it depressed} cubic equation (\ref{ferc}) is
simply related to $\Delta_4$ as
\begin{equation} \label{qiscrrel}
\fracmm{\Delta_3}{4\cdot 27} = (r/3)^3 - (q/4)^4 = \fracmm{\Delta_4}{27\cdot 256}~~~.
\end{equation}

When $\Delta_{3,4}\geq 0$, three real solutions to the cubic equation (\ref{ferc}) are given by the Viet\'e formula
\begin{equation} \label{viete}
m_{k}=2\sqrt{r/3}\cos\theta_k~~,\qquad k=0,1,2~,
\end{equation}
where
\begin{equation} \label{viete2}
\theta_k = \fracmm{1}{3}\arccos \left( \fracmm{3q^2}{16r}\sqrt{3/r}\right)-\fracmm{2\pi k}{3}~,
\end{equation}
while we should choose the highest (positive) root. The condition $\Delta_{3,4}\geq 0$ implies
\begin{equation} \label{dicsp}
\fracmm{F^2}{\tilde{\gamma}_8}\geq 27~~.
\end{equation}

When $\Delta_{3,4}\leq 0$ or, equivalently, $F^2/\tilde{\gamma}_8\leq 27$, the angle (\ref{viete2}) does not
exist. Instead, we should use the Viet\'e's substitution in Ferrari's equation with
\begin{equation} \label{vietesub}
m = w +\fracmm{r}{3w}~,\qquad r>0~~,
\end{equation}
that yields a {\it quadratic} equation for $w^3$,
\begin{equation} \label{quadrvi}
(w^3)^2 - \fracmm{q^2}{8}w^3 +\fracmm{r^3}{27}=0~~,
\end{equation}
whose roots are
\begin{equation} \label{qvieter}
w^3_{1,2} = (q/4)^2 \left[ 1 \pm \sqrt{ 1 - \fracmm{(r/3)^3}{(q/4)^4} } \right] ~~.
\end{equation}

Going back to the critical condition (\ref{hpotcr}) in the form
\begin{equation} \label{hpotcr2}
F^2 = e^{6\chi_c}\left[  1 - \fracmm{1}{3}a^{-2}e^{2\chi_c}\right]~,
\end{equation}
and inserting it into the potential (\ref{hpot}) yields the {\it height} of the inflationary potential
$V_{\rm plateau}$ at the onset of inflation,
\begin{equation} \label{plateau}
M_{\rm Pl}^{-4}V_{\rm plateau}=e^{-6\chi_c}\left[  \fracmm{2}{3}a^{-2}e^{2\chi_c}-1\right]~.
\end{equation}
Demanding its positivity, $V_{\rm plateau}>0$, gives us the restriction  (\ref{stab}) again.

The second derivative of the potential (\ref{hpot}) at the critical point  (\ref{hpotcr}) is given by
\begin{equation} \label{secdc}
\left. \fracmm{\pa^2 V}{\pa \chi^2} \right|_{\chi=\chi_c}=8e^{-6\chi_c}\left( 9 -4a^{-2} e^{2\chi_c}\right)~.
\end{equation}
Its positivity (stability) implies
\begin{equation} \label{posh}
\fracmm{F^2}{\tilde{\gamma}_8}<54~~.
\end{equation}
Taken together with (\ref{stab}) and (\ref{dicsp}), this implies that the values of the ratio $F^2/\tilde{\gamma}_8$ have to be restricted as follows:
\begin{equation} \label{rangerat}
\eqalign{
16<\fracmm{F^2}{\tilde{\gamma}_8}\leq 27~~, & \qquad \Delta_{3,4} \leq 0~~,\cr
27\leq \fracmm{F^2}{\tilde{\gamma}_8}< 54~~, & \qquad \Delta_{3,4} \geq 0~~.\cr}
\end{equation}

Because of $1<F^2/(16\tilde{\gamma}_8)\equiv 1+\delta<(\frac{3}{2})^3$, it is instructive to investigate the case of $0<\delta\ll 1$ describing the strong stabilization of the modulus $\chi$.  In this case,  (\ref{smvac}) and (\ref{hpotcr2})  give rise to
\begin{equation} \label{difdelta}
 0<\chi_c-\chi_0 \approx \fracmm{1}{12}\delta \ll 1~~,
\end{equation}
leading to a {\it single-field inflation} driven by inflaton (scalaron) $f$ indeed.

The physical {\it hierarchy} of scales ({\it cf.} Eq.~(\ref{hier})) reads
\begin{equation} \label{hier2}
 m\low{\hat{f}_0} <  m\low{\hat{\chi}_0} \ll M\low{\rm KK} \ll  M\low{\rm Pl}~~.
\end{equation}
The KK scale in our case is given by $M_{\rm KK}\approx e^{-\chi_0}M_{\rm Pl}$, where the presence
of the warp factor is dictated by the compactification ansatz (\ref{mansatz}).

The condition $M\low{\rm KK} \ll  M\low{\rm Pl}$ implies
\begin{equation} \label{firstc}
2F^2 \gg 1
\end{equation}
because of (\ref{smvac}). The condition  $m_{\hat{\chi}_0} \ll M\low{\rm KK}$ implies
\begin{equation} \label{secondc}
F^2 \gg \sqrt{2}
\end{equation}
that is slightly stronger than (\ref{firstc}). Both conditions can be easily satisfied by taking
 $F^2\gg 1$.

The remaining condition  $m\low{\hat{f}_0} <  m\low{\hat{\chi}_0}$ implies $F^2/\tilde{\gamma}_8 <72$ that is
already satisfied under the conditions (\ref{rangerat}). However, it is not possible to get
 $m\low{\hat{f}_0} \ll  m\low{\hat{\chi}_0}$ here.

\section*{Appendix D : $D=8$ gauged supergravity}

The $D=8$ supergravity (with 16 supercharges) received relatively little attention in the literature versus the supergravities in $D=10$ and $D=11$. For our purposes, we need a $D=8$ supergravity modified by the quartic scalar curvature term and having a scalar potential. In this Appendix we recall the $SU(2)$ gauged $N=2$ supergravity in $D=8$, which was derived by Salam and Sezgin \cite{ss8} by using the
Scherk-Schwarz-type dimensional reduction \cite{ssgred} of the 11-dimensional supergravity \cite{cjs}.

The 11-dimensional supergravity \cite{cjs} is {\it unique}, so that it is the good point to start with. Its standard action is well known, while its existence can be related to the existence of the 11-dimensional supermultiplet containing the 11-dimensional spacetime scalar curvature $\mathscr{R}$ among its field components. Therefore, there is little doubt that the $(\mathscr{R}+\mathscr{R}^4)$ {\it supergravity} action in $D=11$ also exist, though (to the best of our knowledge) it was never constructed in the literature. So, assuming its existence, we write down the relevant part of its bosonic terms as~\footnote{We use the spacetime signature $(-,+,+,...,+)$ in $D=11$.}
\begin{equation}
S_{11}=\int d^{11}X\fracmm{E}{2\tilde{\kappa}^2}\left(\mathscr{R}+\tilde{\gamma}\mathscr{R}^4-\fracmm{\tilde{\kappa}^2}{12}G_{ABCD}G^{ABCD}+\fracmm{8\tilde{\kappa}^3}{144^2}\varepsilon^{A_1...A_{11}}G_{A_1...}G_{A_5...}V_{...A_{11}}\right)~,\label{S11}
\end{equation}
where we have simply added the quartic curvature term (with the coupling constant $\tilde{\gamma}$) to the standard bosonic action of the 11-dimensional supergravity. Of course, adding the $\mathscr{R}^4$ term also requires adding its supersymmetric completion that is going to result in more bosonic terms in the action. However, because all extra terms are going to be the higher-derivative couplings of the bosonic 3-form field, also non-minimally coupled to gravity, we assume that these extra couplings are irrelevant for the {\it scalar} sector of the theory (see below).~\footnote{It is worth mentioning that our approach is apparently {\it different} from M-theory, because we treat the $\mathscr{R}^4$ term nonperturbatively, so
that its presence leads to the new physical degrees of freedom in $D=11$, which are absent in the
standard $D=11$ supergravity, similarly to the $(R+R^4)$ gravity in lower dimensions.}

As regards our notation, we denote $E\equiv\det {E_M}^A$ in terms of an {\it elfbein}
${E_M}^A$ in $D=11$. Here we denote  the 11-dimensional Lorentz indices by early capital latin letters as $A,B,C,...$, while the middle capital latin letters $M,N,P,...$ are used for  the 11-dimensional Einstein (curved) indices. The $\tilde{\kappa}$ is the gravitational constant in $D=11$. The scalar curvature $\mathscr{R}$ is defined in terms of the spin connection
\begin{gather}
\omega_{ABC}\equiv E^M_A\omega_{MBC}=\frac{1}{2}\eta_{CE}(E^M_AE^N_B-E^M_BE^N_A)\partial_ME_N^E-\frac{1}{2}\eta_{AE}(E^M_BE^N_C-E^M_CE^N_B)\partial_ME_N^E\nonumber\\+\frac{1}{2}\eta_{BE}(E^M_CE^N_A-E^M_AE^N_C)\partial_ME_N^E \label{spinc}
\end{gather}
as
\begin{equation} \label{sc11}
\mathscr{R}=\omega_{ABC}\omega^{CAB}+\omega_A\omega^A-2E^{-1}\partial_M(E{E^M}_A\omega^A)~,
\end{equation}
where $\omega_A\equiv \eta^{BC}\omega_{BCA}$ and $\eta_{AB}$ is Minkowski metric in $D=11$. The
4-form field strength $G_{ABCD}$ is defined in terms of the 3-form gauge potential $V_{ABC}$ as
\begin{equation} \label{gp11}
G_{ABCD}=4\partial_{[A}V_{BCD]}+12{\omega_{[AB}}^E V_{CD]E}~.
\end{equation}

To dimensionally reduce the modified $D=11$ supergravity to eight dimensions on a sphere $S^3$, we use the ansatz \cite{ss8}
\begin{equation}
{E_M}^A=
 \begin{pmatrix}
  e^{-\tilde{\kappa}\phi/3}e^a_\mu & 0 \\
  2\tilde{\kappa} e^{2\tilde{\kappa}\phi/3}A^\alpha_\mu L^i_\alpha & e^{2\tilde{\kappa}\phi/3}L^i_\alpha
 \end{pmatrix}~,\quad
{E^M}_A=
 \begin{pmatrix}
  e^{\tilde{\kappa}\phi/3}e_a^\mu & -2\tilde{\kappa} e^{\tilde{\kappa}\phi/3}e^\mu_a A^\alpha_\mu\\
  0 & e^{-2\tilde{\kappa}\phi/3}L_i^\alpha
 \end{pmatrix}~,\label{ansatz8}
\end{equation}
where we have introduced the 8-dimensional Lorentz indices $a,b,c,...$ and the 8-dimensional Einstein indices
 $\mu,\nu,\rho,...$, as well as the 3-dimensional (compact) Lorentz and Einstein indices, $i,j,k,...$ and $\alpha,\beta,\gamma,...$, respectively. The dilaton $\phi$ represents the volume modulus of the 3-sphere, the  $e_\mu^a$ is an 8-dimensional {\it achtbein}, the $L^i_\alpha$ is the unimodular matrix ($\det L^i_\alpha=1$) having 5 scalars, and the $A_\mu^\alpha$ is a set of 8-dimensional vectors.

The Scherk-Schwarz dimensional reduction is used to gauge symmetries of a compact manifold in the reduced theory by allowing the fields to depend on the compact coordinates
\cite{ssgred}. Let us denote the non-compact coordinates by $\{x\}$, and the compact  coordinates by
 $\{y\}$, and then factorize the $y$-dependence as
\begin{equation} \label{factxy}
e^a_\mu(x,y)=e^a_\mu(x)~,\quad
A^\alpha_\mu(x,y)={U^{-1\alpha}}_\beta(y)A^\beta_\mu(x)~,\quad
L^i_\alpha(x,y)={U_\alpha}^\beta(y)L^i_\beta(x)~,
\end{equation}
where ${U_\alpha}^\beta(y)$ are elements of the gauge group $SU(2)$ in our case. The $SU(2)$ structure constants
\begin{equation}
f^\gamma_{\alpha\beta}\equiv {U^{-1}_\alpha}^{\alpha'}{U^{-1}_\beta}^{\beta'}(\partial_{\beta'}{U_{\alpha'}}^\gamma-\partial_{\alpha'}{U_{\beta'}}^\gamma)=-\fracmm{g}{2\tilde{\kappa}}\varepsilon_{\alpha\beta\delta}g^{\delta\beta}
\end{equation}
are $y$-{\it independent}, where we have introduced the $SU(2)$ gauge coupling constant $g$ and
the 3-dimensional Levi-Civita tensor $\varepsilon_{\alpha\beta\gamma}$.

Substituting the ansatz \eqref{ansatz8} into \eqref{spinc} reduces the spin connection components as
\cite{ss8}
\begin{align}
\begin{aligned}
&\omega_{abc}=e^{\tilde{\kappa}\phi/3}(\tilde{\omega}_{abc}-\frac{1}{3}\tilde{\kappa}\eta_{ab}\partial_c\phi+\frac{1}{3}\tilde{\kappa}\eta_{ac}\partial_b\phi)~,\\
&\omega_{abi}=\tilde{\kappa} e^{4\tilde{\kappa}\phi/3}F_{abi}~,\\
&\omega_{aij}=e^{\tilde{\kappa}\phi/3}Q_{aij}~,\\
&\omega_{iab}=-\tilde{\kappa}e^{4\tilde{\kappa}\phi/3}F_{abi}~,\\
&\omega_{ija}=e^{\tilde{\kappa}\phi/3}(P_{aij}+\frac{2}{3}\tilde{\kappa}\delta_{ij}\partial_a\phi)~,\\
&\omega_{ijk}=-\frac{g}{4\tilde{\kappa}}e^{-2\tilde{\kappa}\phi/3}(\varepsilon_{jkl}{T_i}^l+\varepsilon_{kli}{T_j}^l-\varepsilon_{lij}{T_k}^l)~,\label{spincc}
\end{aligned}
\end{align}
where we have used the notation
\begin{gather}
F^\alpha_{\mu\nu}=\partial_\mu A_\nu^\alpha-\partial_\nu A_\mu^\alpha+g\varepsilon_{\alpha\beta\gamma}A^\beta_\mu A^\gamma_\nu~,\nonumber \\
P_{\mu ij}+Q_{\mu ij}=L_i^\alpha(\delta^\beta_\alpha\partial_\mu-g{{\varepsilon_\alpha}^\beta}_\gamma A^\gamma_\mu)L_{\beta j}~,\label{PQ} \\
T^{ij}=L^i_\alpha L^j_\beta \delta^{\alpha\beta}~,\nonumber
\end{gather}
with $P_{\mu ij}$ representing the symmetric part of the r.h.s. of \eqref{PQ}, and $Q_{\mu ij}$ representing the antisymmetric part. The fields $L_\alpha^i$ are subject to the relations \cite{ss8}
\begin{equation}
L^i_\alpha L^j_\beta\delta_{ij}=g_{\alpha\beta},\;\;L^i_\alpha L^j_\beta g^{\alpha\beta}=\delta^{ij}~,
\end{equation}
where $g_{\alpha\beta}$ is the metric of the compact manifold $(S^3)$.

As regards $V_{ABC}$ and $G_{ABCD}$, their relevant components are
\begin{gather}
\begin{gathered}
\varepsilon_{\alpha\beta\gamma}B\equiv e^{2\tilde{\kappa}\phi}L^i_\alpha L^j_\beta L^k_\gamma V_{ijk}~,\\
\varepsilon_{\alpha\beta\gamma}\partial_\mu B\equiv e^{5\tilde{\kappa}\phi/3}e^a_\mu L^i_\alpha L^j_\beta L^k_\gamma G_{aijk}~,\label{GV}
\end{gathered}
\end{gather}
where $B$ is another {\it scalar} field.

Equations \eqref{ansatz8}, \eqref{spincc}, and \eqref{GV} allow us to rewrite the 11-dimensional action \eqref{S11} as
\begin{equation} \label{11act2}
\eqalign{
S_{11}= & \int d^8x d^3y\;U(y)\fracmm{e}{2\tilde{\kappa}^2}[R+\tilde{\gamma}e^{2\tilde{\kappa}\phi}R^4-\tilde{\kappa}^2e^{2\tilde{\kappa}\phi}F_{\mu\nu}^\alpha F^{\mu\nu}_\alpha-2\tilde{\kappa}^2\partial_\mu\phi\partial^\mu\phi \cr
&  -\fracmm{g^2}{4\tilde{\kappa}^2}e^{-2\tilde{\kappa}\phi}(T_{ij}T^{ij}-\frac{1}{2}T^2)\\
-P_{\mu ij}P^{\mu ij}-\frac{1}{2}\tilde{\kappa}^2e^{-4\tilde{\kappa}\phi}\partial_\mu B\partial^\mu B \cr
& -\fracmm{\tilde{\kappa}^2}{12}e^{2\tilde{\kappa}\phi}G_{\mu\nu\rho\sigma}G^{\mu\nu\rho\sigma}-\fracmm{\tilde{\kappa}^3}{432}e^{-1}\varepsilon^{\mu_1...\mu_8}G_{\mu_1...\mu_4}G_{\mu_5...\mu_8}B]+\ldots~, \cr}
\end{equation}
where $U(y)\equiv \det {U_\alpha}^\beta(y)$, $T\equiv {T_i}^i$, $R$ is the $8$-dimensional scalar curvature and  the dots stand for irrelevant terms. Since the only $y$-dependent function is $U(y)$, one can perform $y$-integration with
\begin{equation}
\int d^3y U(y)=V_0~,
\end{equation}
defining the invariant volume $V_0$ of the compact manifold $(S^3)$. With the gravitational coupling
$\kappa=\tilde{\kappa}/\sqrt{V_0}$ in $D=8$, rescaling dilaton as $\phi\rightarrow\phi/\sqrt{V_0}$ (and similarly for the other fields $A_\mu^\alpha$, $B$ and $V_{\mu\nu\rho}$) and rescaling the gauge coupling
as $g\rightarrow g\sqrt{V_0}$ leads to the action
\begin{gather}
S_8=\int d^8x\fracmm{e}{2\kappa^2}[R+\tilde{\gamma}e^{2\kappa\phi}R^4-\kappa^2e^{2\kappa\phi}F_{\mu\nu}^\alpha F^{\mu\nu}_\alpha-2\kappa^2\partial_\mu\phi\partial^\mu\phi-V(T)-P_{\mu ij}P^{\mu ij}\nonumber\\
-\fracmm{1}{2}\kappa^2e^{-4\kappa\phi}\partial_\mu B\partial^\mu B-\fracmm{\tilde{\kappa}^2}{12}e^{2\kappa\phi}G_{\mu\nu\rho\sigma}G^{\mu\nu\rho\sigma}-\fracmm{\kappa^3}{432}e^{-1}\varepsilon^{\mu_1...\mu_8}G_{\mu_1...\mu_4}G_{\mu_5...\mu_8}B]+...~,\label{8saction}
\end{gather}
whose scalar poitential is given by \cite{ss8}
\begin{equation} \label{spot8}
V(T)=\fracmm{g^2}{4\kappa^2}e^{-2\kappa\phi}(T_{ij}T^{ij}-\frac{1}{2}T^2)~.
\end{equation}

\vglue.2in

redpajama_set_name: RedPajamaArXiv


text:
EarthGang Are Strip Club-Loving Puppets In The New Video For "Top Down": Watch
The track is from their 2019 Dreamville Records debut Mirrorland.
by Eli Enis
PUP perform a Christmas version of "Kids" alongside puppets on CBC Kids: Watch
A holiday spin on the Morbid Stuff track.
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Michael Jackson musical blames sexual abuse allegations on alien glove
For the Love of Glove opens in January in Los Angeles.
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on December 05, 2019, 12:30pm
Comedy Central to reboot Crank Yankers
Comedy Central is bringing back the early aughts TV show about potty-mouthed puppets.
Slash Ft. Myles Kennedy and The Conspirators reveal live-action animated video for "Driving Rain"
The band's new album, Living the Dream, was recently released.
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on September 25, 2018, 11:45am
Portugal. The Man share bizarre video for anthemic single "Live In the Moment": Watch
The band will also take a victory lap across North America in February.
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on November 01, 2017, 1:20pm
Parquet Courts are low-rent puppets in their video for "Human Performance" — watch
Featuring puppet-on-puppet sex and violence.
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A puppet Aesop Rock plays with his real life cat in video for "Kirby" — watch
Aesop's ode to his kitten is the most adorable thing you'll see today.
Puppet versions of Tame Impala star in "'Cause I'm a Man" video — watch
The Dominic Simper puppet is pretty spot on.
on May 29, 2015, 12:45am

redpajama_set_name: RedPajamaCommonCrawl


Aim of the game is to clear the screen in each level and get as many points as possible. There is a 120 second time limit in each level. Get 3 balls or more of the same colour together to remove them from the screen and to get points. There are 8 colours as well as 3 bonus balls types. Controls: Move the mouse to position the shooter. Hold down the mouse to aim. Let go of the mouse to shoot.


When to consider Mediation and when to duke it out in court | Cipriano Law Offices, P.C.
In every case, it's advisable to speak with your attorney first about negotiating with the other party. Once you know what you want, you can commence negotiations If the other party doesn't respond, then you may have no choice but to go to arbitration or trial and have a judge determine the outcome of your divorce. That's just one circumstance where mediation might not work.
It depends on the circumstances, but if violence was a part of your relationship, mediation usually isn't viable. It pits victims against their abusers, and usually is too great of a risk. Abusive partners may not be able to see your side of things, either, so mediation doesn't work well. Arbitration, on the other hand, is a possible option.
It should go without saying, but if you're not willing to negotiate and settle with your spouse, then mediation won't work. Mediation requires both parties to work together to create a settlement on which they can both agree. If you won't be happy no matter what happens, then mediation won't work for you.
Mediation has many more benefits than downsides. It can speed up the divorce process and help couples work through some of the problems that came up during the marriage. You can learn to handle your problems differently and to communicate better with your ex. If you two can come together to work out a settlement with some support from a mediator, it's a good idea to try to do so.
Related Posts: Mediation and property division: Maintain more control, Handle custody issues in mediation to protect your children, Grandparents often have visitation rights in New Jersey, Will an affair affect New Jersey asset division or alimony?


text:
Pearce Makes Her Case
Well, we didn't get our first public face-off between State Treasurer Beth Pearce and the public sector unions on Thursday. But we did get a better sense of Pearce's argument for cutting benefits in the face of growing unfunded liabilities in the state's pension funds.
The Senate Government Operations Committee had set aside an hour and 45 minutes to hear from Pearce and the teachers' and state employees' unions. But Pearce's presentation lasted almost an hour and a half. At that point, GovOps chair Jeanette White declared that there was "no time today to hear from the unions." They'll be back in the virtual witness chair as soon as next week.
That should be interesting. The unions haven't exactly welcomed pension cuts in their public reactions, but they'd be well advised to come to the committee with some ideas of their own. Because the state of the pension funds — especially the teachers' fund — is not good.
(Pearce's PowerPoint presentation to the committee, and her full report on the state of the pensions, can be found on the GovOps website.)
And the unions ought to be prepared for this. According to Pearce, she's been meeting with them "at least weekly since mid-December" to discuss what to do. She also held virtual town meetings with roughly 1,000 members of the Vermont-National Education Association and around 350 members of the Vermont State Employees Association. She told the committee she wanted the unions to be involved throughout the process.
Until recently, Pearce had been adamant about not cutting benefits. She changed her tune after an outside actuarial review determined that the pension plans' unfunded liabilities continue to grow, and have reached a critical level. "We need to do something now," she told the panel. "You cannot say 'We're going to do a study and decide next year.'"
How big is the problem? The state employees' fund is more than a billion dollars short; the teachers' fund is almost two billion short. Both unfunded liabilities have steadily increased since the Great Recession of 2008-09, and took a big jump last year. That led Pearce to reluctantly call for cuts.
There are two main reasons for a decade-plus of shortfalls, Pearce said. The first is demographic changes; the workforces are getting older and retirees are living longer. (Teacher retirements hit an all-time record in 2020, due mostly to Covid-19.) The second is that the funds have failed to meet earnings projections.
Pearce recommends a reduction in cost-of-living adjustments for future VSEA retirees, and an elimination of COLA increases for retiring teachers. (Current retirees would see no changes in their pensions.) Members of both unions would see their required contribution increase — teachers more than state workers. She also suggests some other adjustments, but those are the two biggies.
Pearce continues to oppose a switch from defined benefit pensions to defined contribution (401K style). She asserted that the latter system would actually cost more. Sen. White is not quite convinced. "The voices for changing to defined contribution are swelling up," she said. "We need to see studies that show it doesn't save any money."
Although Pearce has been consulting with the unions, she told the committee "there is no buy-in yet from NEA or VSEA. I'm committed to continuing the dialogue."
The unions' initial public reactions were not positive, but we have yet to hear their full viewpoints. In previous public statements, they expressed a desire to keep pensions whole, but also acknowledged the need for some kind of action.
Pearce indicated one path that would avoid cuts: Identifying a "dedicated revenue source" that restore the plans' fiscal health without massive infusions of General Fund money. One possibility: The next federal Covid relief bill. "If that has money with no strings attached, we could use it to address the [pension] system," Pearce said.
Of course, Congress has yet to take action on a third Covid bill. Plus, the last two bills included restrictions on how the money could be spent.
Not to mention that if Vermont does get another influx of federal cash, there will be many eager hands reaching into the pot. And I'm skeptical that Gov. Phil Scott would agree to the idea.
We've seen just the start of what's sure to be a thorny process. Pressure will be on lawmakers to come up with a sustainable pension plan that's the least offensive possible to all concerned. Especially the unions, which are a core constituency for the majority Democrats.
This entry was posted in Labor and tagged Beth Pearce, Jeanette White, Vermont National Education Association, Vermont State Employees Association on January 29, 2021 by John S. Walters.
← In Which I Discover the Limits of Phil Scott's Generosity to Business Hey, Another Trumpster Running for Office! And This One is Barely Literate! →
3 thoughts on "Pearce Makes Her Case"
walter h moses January 29, 2021 at 6:32 pm
Really. No mention in Pearce's report of Howard Dean cutting the necessary legislative payments into the system, followed by Douglas in lock step. Goodness, the legislators had no hand in the the downfall of the system either. It was all demographic and the lack of investment returns that will now make retirees do with less. I read that Dean said he would do it all again if he had the opportunity. I would vote against him again if I had the opportunity. What a mess of tripe. And, Senator White is the committee chair? John, get ready for more Shaw's chocolate cake and hot coffee.
John S. Walters Post author January 29, 2021 at 7:47 pm
She has made reference to Howard Dean's role and Jim Douglas' as well, and I've written about it. And of course the Legislature went along. (Dean started the underfunding early in his administration, Douglas continued it until he and the Legislature agreed to fully fund the pensions.) That's part of the problem; the new information is about what's happened since 2007. That's what brought about Pearce's change of heart. If we were only dealing with the Dean/Douglas underfunding, we could pay it down over time. The more recent issues with demographics and rates of return are what pushed the issue over the edge.
If the report deals only with issues since 2007 then I understand. However, anyone who expected reasonable returns on investment after the banking collapse of 2007-2008 had to be dreaming. You did not have to be an economist to know that expecting decent returns for at least the next five years was a bit foolish.


text:
THE AMERICAN CRIMINALIZATION OF POVERTY
by ch@nce Saturday, Aug. 12, 2000 at 6:51 PM
coh@sfo.com
"Homeless" laws specifically target homeless individuals and their activities. "Status" laws punish people for their economic condition, rather than behavior.
errorTHE AMERICAN CRIMINALIZATION OF POVERTY
by Dave Oehl
What do you get when you cross a booming national economy with homeless people? Less homeless people, right?
What you get, apparently, is not only greater homelessness, but also enactment and greater enforcement of laws and policies that criminalize poverty or homelessness. Efforts in many cities are now focused on excluding the homeless from downtown areas and places where they congregate. It is important to examine public policy as a means of social control instead of social change or improvement. To this end, this article will provide and discuss some examples of anti-poor policies, and solutions being proposed or utilized.
Anti-poor and homeless policies
According to the Atlanta Task Force for the Homeless (ATFH) Criminalization of Poverty report, published in 1993, there are two kinds of laws that discriminate against the poor and homeless. "Homeless" laws specifically target homeless individuals and their activities. "Status" laws punish people for their economic condition, rather than behavior.
Many cities across the country have homeless laws, but they generally fall into the general categories. Panhandling is restricted or banned altogether; Massachusetts passed a law of this kind several years ago but it was struck down by the state courts as unconstitutional. A lawsuit in Los Angeles is challenging its anti-panhandling law. Anti-camping ordinances prohibit sleeping on streets or in parks at all or after curfew, such as in Austin, Texas. Many cities have no-standing zones where people may not linger, or no-sitting areas. Austin and Los Angeles have laws such as these. There also exist such arcane laws as the prohibition of public parking lot crossing, in Atlanta.
In the Arizona city of Tucson, a special zone was created in which it is a crime to simply be homeless. Police were arresting homeless people without cause and releasing them only when they agreed to stay out of the area for a certain period of time. Alan Mason, arrested under this law, was banned from an area that covered just about all of downtown, including his lawyer, all the courthouses, the voter registration office, and places of worship.
The line between homeless laws and status laws blurs. Most disorderly conduct laws are considered status laws by the ATFH, since many homeless are mentally ill, or predisposed to erratic behavior, caused by, or the cause of, their homelessness. Criminal trespassing, public urination, and public drinking are also considered status ordinances.
An example of a recent status ordinance is an ordinance proposed, not passed, in September by Ray Suarez of the Chicago City Council to prohibit sleeping in cars. Mr. Suarez said that some residents did not feel safe because people were sleeping in cars near their homes. Why do they not feel safe? Activists are now making sure that Councilor Suarez is educated on this issue and is dealing constructively with it and discussing affordable housing.
Many administrations choose to selectively enforce laws to punish homeless people, laws that were not originally meant to do so. The laws involved vary from stolen property to general trespassing to general sanitation (laws that prohibit dumping in vacant lots or blocking entrances or alleys)
New York Mayor Rudy Giuliani is creatively sculpting NYC laws to punish the homeless and clear the streets, and making up new policies as he goes along.
Giuliani has ordered massive street sweeps to clear out New York's homeless people since a tourist was seriously injured when attacked by an allegedly homeless man last November. One commentator remarked that this is the first time punishment has been meted out before there is even a suspect. If you won't work, you'll be kicked out of a shelter, and then you'll be arrested for sleeping on the street. The Mayor is also opposed to any minimum wage increases that might help the poor pay for housing. Meanwhile he is promoting New York as an urban Disneyland; as a result, rents are going up, and the city and state are not creating affordable housing. After a while, one begins to suspect his motives.
A New York City police manual for carrying out street sweeps called "Quality of Life enforcement Options: A Police Reference Guide" lists 35 offenses for which one can be arrested. A homeless man challenging these policies in court was arrested in 1997 on an obscure sanitation code violation. He was strip-searched and held for 27 hours; the ticket turned out to be invalid.
In Los Angeles, Downtown business improvement districts hired private security forces to patrol the streets. They have been charged in a lawsuit with coercive detention, invasion of privacy, and assault and battery. These security officers routinely interview people on the street and keep files on the people they interview. They photograph and search the belongings of people they think don't belong. This policy does nothing to address these people's needs or the causes of their situation.
The effects of criminalization
One of my friends described the worst feeling in the world as "when someone doesn't want you anymore." These policies increase the alienation and what Michael Sullivan from Bread and Jams calls "paranoia." It is frightening to live knowing that by virtue of who you are, or at least what condition you find yourself in, you could be arrested. The man who started the suit against New York City now avoids contact with service workers, fearful of being arrested again. He never sleeps in the same place twice. Homeless people are less likely to seek help and shelter if they think they may get harassed or arrested. This can lead to more deaths or destructive behavior.
Homeless artist and writer Robert Lederman says that these policies may cause more homeless to be shot by police while resisting arrest or acting "suspiciously," not only hurting people but also inviting lawsuits.
These policies make it difficult for poor and homeless people to find and hold jobs. Employers are reluctant to hire poor and homeless people when they have a criminal record. Furthermore, if people are arrested for "quality of life" violations and miss work, they may lose their jobs.
The costs of litigation, police activity, and jailing homeless people can be substantial. The Atlanta Task Force for the Homeless suggested that jailing, for three days, half of the 9000 homeless people arrested in 1995 would cost 742,500 dollars. That doesn't include court and administrative costs.
These policies also feed negative public opinion, distracting it away from positive, long-term solutions and focusing it on the people themselves, not their condition.
Challenges to this Discrimination
Challenges to these policies are coming in several forms. Many court cases have been won or are pending, as noted above. Then there are activist groups and democratic government action.
The National Law Center on Homelessness and Poverty is an organization that documents legal abuses and aids homeless defendants and plaintiffs. They provide legal advise and often file Friend of the Court briefs on behalf of plaintiffs. They have supported many homeless people in successful lawsuits.
For example, in 1997, a settlement agreement was arrived at in the Pottinger v. City of Miami (FL), whereby the city agreed to implement a training program to ensure that homeless people's rights are not violated. The police may not destroy the property of homeless people. An advisory committee was created to monitor police contact with homeless people, and monetary compensation was provided to the plaintiffs. This case was a "landmark development" and is provisions are being reviewed for possible use all over the country.
The National Coalition for the Homeless is a nationwide advocacy organization with programs and resources. They are currently conducting a National Homeless Civil Rights Organizing Project. The purpose is to link grassroots organizations in a national network in order to "fortify those local efforts and to strengthen cooperation. This project helps individuals or groups in resource poor locales to organize an immediate response by providing knowledge, experience and resources. The NCH recognizes that to avoid simply returning to the old status quo, the campaign must work both to protect civil rights and to stop the causes of homelessness once the discrimination stops.
A third method of countering this discrimination is as a quasi-governmental body, such as the Multi-disciplinary Working Group (MWG) convened to address issues of Homelessness, Public Intoxication, and Nuisance Behaviors in Cambridge MA. Rather than enforce nuisance ordinances in Cambridge, the MWG was formed. It is composed of a range of individuals, including homeless or formerly homeless people, police officers, various public service agencies, and city government agencies. The MWG focused on how to help "problem" homeless people, those who used the city's services the most. The MWG's report states that their task "was to formulate recommendations for responding to the public nuisance behaviors of these individuals." The MWG discussed ways to streamline service provision and cut down on redundancy, fill in gaps in services, how to improve relations between the housed and homeless populations, and community involvement.
This is a constructive, educational and cooperative method for the solution of the problem.
Social control is at work here. The vast majority of these policies are lauded by businesses who, as one might expect, are only concerned with profit margins. These policies are enacted by elected representatives responding to constituents or to campaign contributors. However, there is a significant effort to organize in opposition of this control. Most larger American cities have organizations of homeless and poor people; some are stronger than others. All can be brought together to combat discrimination.
oehlda@hotmail.com
HOMELESS PEOPLE'S NETWORK
7,000+ POSTS by or via homeless & ex-homeless people.
Nothing About Us Without Us - Democratize Public Policy
aspin.asu.edu/hpn
some notes to the article above dave oehl Thursday, Feb. 10, 2005 at 4:09 PM


LUNAFEST is a program of short films about women created by women. It's about giving women the opportunity to tell their stories and opening the door for filmmakers who might not otherwise be given the chance to show their art. LUNAFEST proceeds are donated to charity. Various London agencies will be presenting information about their services to the London Community at this event.


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