\begin{center} {\Large #1 } \end{center}
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The issues of quintessence and cosmic acceleration can be discussed in the framework of higher order curvature and torsion theories of gravity. We can define effective pressure and energy density directly connected to the curvature or to the torsion fields and then ask for the conditions to get an accelerated expansion. Exact accelerated expanding solutions can be achieved for several fourth order curvature or torsion theories so that we obtain an alternative scheme to the standard quintessence scalar field, minimally coupled to gravity, usually adopted. We discuss also conformal transformations in order to see the links of quintessence between the Jordan and Einstein frames. Furthermore, we take into account a torsion fluid whose effects become relevant at large scale. Specifically, we investigate a model where a totally antisymmetric torsion field is taken into account discussing the conditions to obtain quintessence. We obtain exact solutions also in this case where dust dominated Friedmann behavior is recovered as soon as torsion effects are not relevant.
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\begin{document}

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\begin{titlepage}
\pubdate{\today}                    %fill in the date
\pubnumber{XXX-XXXXX \\ YYY-YYYYYY} %preprint number(s)

\vfill \Title{Curvature and Torsion Quintessence}

\vfill \Author{S. Capozziello\footnote{E-mail:
capozziello@sa.infn.it}}
\Address{Dipartimento di Fisica "E.R. Caianiello", Universita'  di Salerno,  \\
         I-84081 Baronissi, Salerno, Italy}
\vfill \andauth \vfill \Author{S. Carloni$^\dag $, G.
Lambiase$^\dag$, C. Stornaiolo$^*$, A. Troisi$^\dag$}
\Address{$^\dag$Dipartimento di Fisica "E.R. Caianiello", Universita'  di Salerno,  \\
         I-84081 Baronissi, Salerno, Italy}

\Address{$^*$Istituto Nazionale di Fisica Nucleare, Sez. di Napoli  \\
         Monte S. Angelo, Via Cintia, Edificio N, I-80126 Napoli, Italy}



\vfill
\begin{Abstract}
The issues of quintessence and cosmic acceleration can be
discussed in the framework of higher order curvature and torsion
theories of gravity. We can define effective pressure and energy
density directly connected to the curvature  or to the torsion
fields and then ask for the conditions to get an accelerated
expansion. Exact accelerated expanding solutions can be achieved
for several fourth order curvature or torsion theories so that we
obtain an alternative scheme to the standard quintessence scalar
field, minimally coupled to gravity, usually adopted. We discuss
also conformal transformations in order to see the links of
quintessence between the Jordan and Einstein frames. Furthermore,
we  take into account a torsion fluid whose effects become
relevant at large scale. Specifically, we investigate a model
where a totally antisymmetric torsion field is taken into account
discussing the conditions to obtain quintessence. We obtain exact
solutions also in this case where dust dominated Friedmann
behavior is recovered as soon as torsion effects are not relevant.


\end{Abstract}
\vfill
\begin{Presented}
    COSMO-01 \\
    Rovaniemi, Finland, \\
    August 29 -- September 4, 2001
\end{Presented}
\vfill
\end{titlepage}
\def\thefootnote{\fnsymbol{footnote}}
\setcounter{footnote}{0}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% The document starts here
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\section{Introduction}

One of the astonishing recent result in cosmology is the fact that
the universe is accelerating instead of decelerating along the
scheme of standard Friedmann  model as everyone has learned in
textbooks. Type Ia supernovae (SNe Ia) allow to determine
cosmological parameters probing the today values of the Hubble
constant $H_0$ and the deceleration parameter $q_0$
\cite{perlmutter}. Besides, data coming from clusters of galaxies
at low red shift  (including the mass - to - light methods,
baryon fraction and abundance evolution) \cite{cluster}, and data
coming from the CMBR investigation (e.g.
BOOMERANG)\cite{boomerang} give observational constraints from
which we deduce the picture of a spatially flat, low density
universe dominated by some kind of non-clustered dark energy.
Such an energy, which is supposed to have dynamics, should be the
origin of the cosmic acceleration.

In terms of density parameter, we have
\begin{equation}
\label{1} \Omega_{(matter)}\simeq 0.3\,,\qquad
\Omega_{\Lambda}\simeq 0.7\,, \qquad \Omega_{k}\simeq 0.0\,\,
\end{equation}
where the $matter$ is the non-relativistic baryonic and
non-baryonic (dark) matter, $\Lambda$ is the dark energy
(cosmological constant, quintessence,..), $k$ is the curvature
parameter of Friedmann-Robertson-Walker (FRW) metric of the form
\begin{equation}
ds^2=dt^2-a(t)^2\left[\frac{dr^2}{1-k r^2}+r^2d\Omega^2\right]\,,
\end{equation}
where $a(t)$ is the scale factor of the universe.

The luminosity distance can be deduced from SNe Ia used as
standard candles.

For $z\leq 1$, by the luminosity distance $d_{L}\simeq
H_0^{-1}[z+(1-q_0)z^2/2]$, the observational results indicate
\begin{equation}
-1\leq q_0 < 0
\end{equation}
which is a clear indication for the acceleration.


The deceleration parameter can be given in terms of density
parameter and then we have for FRW models
\begin{equation}
\label{2}
q_0=-\frac{\ddot{a}a}{\dot{a}^2}=\frac{1}{2}(3\gamma+1)\Omega_{(matter)}-\Omega_{\Lambda}\,,
\end{equation}
where $\gamma$ is the constant of state equation $p=\gamma\rho$.
Immediately, we realize that acceleration or deceleration also
depends on the value of $\gamma$. For standard fluid matter, it is
defined the Zeldovich range $0\leq \gamma_{(matter)} \leq 1$ where
$\gamma_{(matter)}=0$ indicates
 dust (i.e. non-relativistic matter), $\gamma_{(matter)}=1/3$ radiation
(i.e. relativistic matter). Another interesting case has been
widely considered in literature, it is $\gamma=-1$ which points
out a scalar field fluid where dynamics is dominated by
self-interaction potential or cosmological constant.  Inserting a
standard matter fluid into Friedmann--Einstein cosmological
equations gives rise to decelerated dynamics. Due to this fact,
non-standard forms of matter energy have to be taken into account
if one wants to insert observations in a theoretical frame.

Several approaches can be pursued in order to realize  this goal.
All of them can be summarized into three great families: the
cosmological constant, the variable cosmological constant and
quintessence. Essentially they are linked but, in order to match
the observations, several issue have to be satisfied. Below we
give a short summary of this three pictures.

The cosmological constant has become one of the main issue of
modern physics since by fixing its value  should provide the
gravity vacuum state \cite{weinberg}, should make to understand
the mechanism which led the early universe to the today observed
large scale structures \cite{guth},\cite{linde}, and to predict
what will be the fate of the whole universe (no--hair conjecture)
\cite{hoyle}.

From the  cosmological point of view, the main feature of
inflationary models is the presence of a finite period during
which the expansion is de Sitter (or quasi--de Sitter or power
law): this fact implies that the expansion of the scale factor
$a(t)$ is superluminal (at least $a(t)\sim t$, in general
$a(t)\sim \exp H_{0}t$ where $H_{0}$ is the Hubble parameter
nearly constant for a finite period) with respect to the comoving
proper time $t$. Such a situation arises in  presence of an
effective energy--momentum tensor which is approximately
proportional (for a certain time) to the metric tensor and takes
place in various gravitational theories: i.e. the Einstein gravity
minimally coupled with a scalar field \cite{guth},\cite{linde},
fourth or higher--order  gravity
 \cite{starobinsky,ottewill,schmidt,kluske} scalar--tensor
 gravity \cite{la,cimento}.



Several inflationary models are affected by the shortcoming of
"fine tuning" \cite{albrecht}, that is inflationary phase proceeds
from very special initial conditions, while a natural issue would
be to get inflationary solutions as attractors for a large set of
initial conditions. Furthermore, the same situation should be
achieved also in the future: if a remnant of cosmological
constant is today observed, the universe should evolve toward a
final de Sitter stage. A more precise formulation of such a
conjecture is possible for a restricted class of cosmological
models, as discussed in \cite{wald}. We have to note that the
conjecture holds when any ordinary matter field, satisfies the
dominant and strong energy conditions \cite{hawking}. However it
is possible to find models which explicitly violate such
conditions but satisfies no--hair theorem requests. Precisely,
this fact happens if extended gravity theories are involved and
matter is in the form of scalar fields, besides the ordinary
perfect fluid matter \cite{quartic}.

In any case, we need a time variation of cosmological constant to
get successful inflationary models, to be in agreement with
observations, and to obtain a  de Sitter stage
 toward the future. In other words, this means that cosmological
constant has to acquire a great value in early epoch (de Sitter
stage), has to undergo a phase transition with a graceful exit
and has to result in a small remnant toward the future
\cite{lambdastep}. The today observed  accelerated cosmological
behaviour should be the result of this dynamical process where
the value of cosmological constant is not fixed exactly at  zero.

In this context, a fundamental issue is to select the classes of
gravitational theories and the conditions which "naturally" allow
to recover an effective time--dependent cosmological constant
without considering special initial data.

The third approach is quintessence \cite{steinhardt}.
Quintessence is a time-varying, spatially inhomogeneous component
of cosmic density with negative pressure $-1\leq \gamma_Q\leq 0$.
Formally, vacuum energy density is quintessence in the limit
$\gamma_Q\rightarrow -1$ so that the three approaches present in
literature (cosmological constant, variable cosmological constant
and quintessence) are strictly linked.

However all of them claim for an {\it ingredient} which, a part a
pure cosmological constant, comes from a matter-energy counter
part. In this paper, we want to investigate if the quintessential
scheme can be achieved in  a geometrical way.

There is no {\it a priori} reason to restrict the gravitational
Lagrangian to a linear function of the Ricci scalar $R$ minimally
coupled with matter \cite{francaviglia}. Additionally, we have to
note that, recently, some authors have taken into serious
consideration the idea that there are no "exact" laws of physics
but that the Lagrangians of physical interactions are
"stochastic" functions with the property that local gauge
invariances (i.e. conservation laws) are well approximated in the
low energy limit  and physical constants can vary \cite{ottewill}.
This scheme was adopted in order to treat the quantization on
curved spacetimes and the result was that the interactions among
quantum scalar fields and background geometry or the gravitational
self--interactions yield corrective terms in the
Einstein--Hilbert Lagrangian \cite{birrell}. Futhermore, it has
been realized that such corrective terms are inescapable if we
want to obtain the effective action of quantum gravity on scales
closed to the Planck length \cite{vilkovisky}. They are
higher--order terms in curvature invariants as $R^{2}$,
$R^{\mu\nu} R_{\mu\nu}$,
$R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}$,  or nonminimally
coupled terms between scalar fields and geometry as $\phi^{2}R$.
Terms of these kinds arise also in the effective Lagrangian of
strings and Kaluza--Klein theories when the mechanism of
dimensional reduction is working \cite{veneziano}.

Beside fundamental physics motivations, all these theories have
acquired a huge interest in cosmology due to the fact that they
"naturally" exhibit inflationary behaviours and that the related
cosmological models seem very realistic \cite{starobinsky,la}.
Furthermore, it is possible to show that, via conformal
transformations, the higher--order and nonminimally coupled terms
({\it Jordan frame}) always corresponds to the Einstein gravity
plus one or more than one minimally coupled scalar fields ({\it
Einstein frame}) \cite{teyssandier,maeda,wands,conf,gottloeber} so
that these geometric contributions can always have a "matter"
interpretation.

As we will show below, quintessence can be achieved also in the
framework of higher-order theories of gravity, that is in a
geometrical way.

Alternatively,  taking into account torsion  is a straightforward
generalization to implement concepts as spin in General Relativity
\cite{hehl,trautman}. However, it was soon evident that torsion,
as considered e.g in Einstein-Cartan-Sciama-Kibble (ECKS) theory,
does not give relevant effects  in the  observed astrophysical
structures. Nevertheless it was found that for densities of the
order of $10^{47} g/cm^3$ for electrons and $10^{54} g/cm^3$ for
protons and neutrons, torsion could give observable consequences
if all the spins of the particles are aligned. These huge
densities can be reached only in the early universe so that
cosmology is the only viable approach to test torsion effects
\cite{desabbata1}. However no relevant tests confirming the
presence of torsion have been found until now and it is still an
open debate if the space-time is Riemannian or not. Considering
the cosmological point of view and, in particular the primordial
phase transitions and inflation \cite{kolb,peebles,desabbata2}, it
seems very likely that, in some regions of the early universe, the
presence of local magnetic fields could have aligned  the spins of
particles. At very high densities, this effect could influence the
evolution of primordial perturbations remaining as an imprint in
today observed large scale structures. In other words, a main goal
could be to select perturbation scales connected to the presence
of torsion in early epochs which give today-observable
cosmological effects \cite{cosimo}.

From another point of view, the presence of torsion could give
observable effects without taking into account clustered matter.

 We want to investigate if the quintessential scheme can be
achieved also by taking into account theories of gravity with
torsion.





\section{Curvature Quintessence}


A generic fourth--order theory in four dimensions can be
described by the action
 \begin{equation}\label{3}
 {\cal A}=\int d^4x \sqrt{-g} \left[f(R)+{L}_{(matter)} \right]\,{,}
 \end{equation}
 where $f(R)$ is a function of Ricci scalar $R$ and ${L}_{(matter)}$
 is the standard matter Lagrangian density.
We are using physical units $8\pi G_N=c=\hbar=1$. The field
equations are

 \begin{equation}\label{4}
 f'(R)R_{\alpha\beta}-\frac{1}{2}f(R)g_{\alpha\beta}=
 f'(R)^{;\alpha\beta}(g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\beta}g_{\mu\nu})+ \tilde{T}^{(matter)}_{\alpha\beta}\,,
 \end{equation}
 which can be recast in the more expressive form
 \begin{equation}\label{5}
 G_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}R=T^{(curv)}_{\alpha\beta}+T^{(matter)}_{\alpha\beta}\,,
 \end{equation}
 where
 \begin{equation}
 \label{6}
T^{(curv)}_{\alpha\beta}=\frac{1}{f'(R)}\left\{\frac{1}{2}g_{\alpha\beta}\left[f(R)-Rf'(R)\right]+
f'(R)^{;\alpha\beta}(g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\beta}g_{\mu\nu})
\right\}
 \end{equation}
 and
 \begin{equation}
 \label{7}
 T^{(matter)}_{\alpha\beta}=\frac{1}{f'(R)}\tilde{T}^{(matter)}_{\alpha\beta}\,,
 \end{equation}
 is the stress-energy tensor of matter where we have taken into account
 the nontrivial coupling to geometry. The prime means the derivative with respect to $R$.


However,  if $f(R)=R+2\Lambda$, the standard second--order gravity
is recovered.  Reducing the action to a point-like, FRW one, we
have to write
 \begin{equation}\label{8}
 {\cal A}_{(curv)}=\int dt {\cal L}(a, \dot{a}; R, \dot{R})\,{,}
 \end{equation}
where dot means derivative with respect to the cosmic time. The
scale factor $a$ and the Ricci scalar $R$ are the canonical
variables. This position could seem arbitrary since $R$ depends on
$a, \dot{a}, \ddot{a}$, but it is generally used in canonical
quantization \cite{vilenkin,schmidt,lambda}. The definition of
$R$ in terms of $a, \dot{a}, \ddot{a}$ introduces a constraint
which eliminates second and higher order derivatives in action
(\ref{8}), and gives a system of second order differential
equations in $\{a, R\}$. Action (\ref{8}) can be written as
 \begin{equation}\label{10}
 {\cal A}_{(curv)}=2\pi^2\int dt \left\{ a^3f(R)-\lambda\left [ R+6\left (
 \frac{\ddot{a}}{a}+\frac{\dot{a}^2}{a^2}+\frac{k}{a^2}\right)\right]\right\}\,{,}
 \end{equation}
where the Lagrange multiplier $\lambda$ is derived by varying
with respect to $R$. It is
 \begin{equation}\label{11}
 \lambda=a^3f'(R)\,{.}
 \end{equation}
 The point-like Lagrangian is then
$$
 {\cal L}={\cal L}_{(curv)}+{\cal L}_{(matter)}=a^3\left[f(R)-R
 f'(R)\right]+6a\dot{a}^2f'(R)+
$$
\begin{equation}
\label{12} \qquad +6a^2\dot{a}\dot{R}f''(R)-6ka
 f'(R)+a^3p_{(matter)}\,,
 \end{equation}
 where we have taken into account also the fluid matter
 contribution which is, essentially, a pressure term
 \cite{quartic}.

 The Euler-Lagrange equations are
\begin{equation}
\label{13}
2\left(\frac{\ddot{a}}{a}\right)+\left(\frac{\dot{a}}{a}\right)^2+
\frac{k}{a^2}=-p_{(tot)}\,,
 \end{equation}
and
\begin{equation}
\label{14}
f''(R)\left[R+6\left(\frac{\ddot{a}}{a}+\frac{\dot{a}^2}{a}^2+\frac{k}{a^2}\right)\right]=0\,.
 \end{equation}
The dynamical system is completed by energy condition
\begin{equation}
\label{15}
 \left(\frac{\dot{a}}{a}\right)^2+\frac{k}{a^2}=\frac{1}{3}\rho_{(tot)}\,.
\end{equation}


Combining Eq.(\ref{13}) and Eq.(\ref{15}), we obtain
 \begin{equation}
 \label{16}
 \left(\frac{\ddot{a}}{a}\right)=-\frac{1}{6}\left[\rho_{(tot)}+3p_{(tot)}
 \right]\,,
 \end{equation}
 where it is clear that the accelerated or decelerated behaviour
 depends on the rhs.
 However
 \begin{equation}
 \label{17}
 p_{(tot)}=p_{(curv)}+p_{(matter)}\;\;\;\;\;\rho_{(tot)}=\rho_{(curv)}+\rho_{(matter)}\,,
 \end{equation}
where we have distinguished the curvature and matter
contributions.

From the curvature-stress-energy tensor, we can define a
curvature pressure
\begin{equation}
\label{18}
p_{(curv)}=\frac{1}{f'(R)}\left\{2\left(\frac{\dot{a}}{a}\right)\dot{R}f''(R)+\ddot{R}f''(R)+\dot{R}^2f'''(R)
-\frac{1}{2}\left[f(R)-Rf'(R)\right] \right\}\,,
 \end{equation}
and a curvature density
\begin{equation}
\label{19}
\rho_{(curv)}=\frac{1}{f'(R)}\left\{\frac{1}{2}\left[f(R)-Rf'(R)\right]
-3\left(\frac{\dot{a}}{a}\right)\dot{R}f''(R) \right\}\,.
 \end{equation}

From Eq.(\ref{16}), the accelerated behaviour is achieved if
\begin{equation}
\label{20} \rho_{(tot)}+ 3p_{(tot)}< 0\,,
\end{equation}
which means
\begin{equation}
\label{21} \rho_{(curv)}> \frac{1}{3}\rho_{(tot)}\,,
\end{equation}
assuming that all matter components have non-negative pressure.

In other words, conditions to obtain acceleration  depends on the
 relation
\begin{equation}
\label{22}
 \rho_{(curv)}+3p_{(curv)}=\frac{3}{f'(R)}\left\{\dot{R}^2f'''(R)+\left(\frac{\dot{a}}{a}\right)\dot{R}f''(R)
 +\ddot{R}f''(R)-\frac{1}{3}\left[f(R)-Rf'(R)\right]\right\}\,,
 \end{equation}
 which has to be compared with matter contribution. However, it
 has to be
 \begin{equation}
 \label{23}
 \frac{p_{(curv)}}{\rho_{(curv)}}=\gamma_{(curv)}\,,
 \qquad -1\leq\gamma_{(curv)}<0\,.
 \end{equation}
 The form of $f(R)$ is the main ingredient to obtain this {\it curvature
 quintessence}.

As simple choice in order to fit the above prescriptions, we ask
for solutions of the form
 \begin{equation}
 \label{24}
 f(R)=f_0 R^n\,,\qquad
 a(t)=a_0\left(\frac{t}{t_0}\right)^{\beta}\,.
 \end{equation}
 However, the interesting cases are for $n\neq 1$ (Einstein
 gravity) and $\beta\geq 1$ (accelerated behaviour). Inserting
 Eqs.(\ref{24}) into the above dynamical system, we obtain the exact solutions
\begin{equation}
\label{25} \beta=2\,;\qquad n=-1,\;\;3/2\,;\qquad k=0\,.
\end{equation}
In both cases, the deceleration parameter is
 \begin{equation}
 \label{26}
 q_0=-\frac{1}{2}\,,
 \end{equation}
 in perfect agreement with the observational results.


The case $n=3/2$ deserves further discussion. It is interesting in
conformal transformations from Jordan frame to Einstein frame
\cite{conf,magnano} since it is possible to give explicit form of
scalar field potential. In fact, if
 \begin{equation}\label{27}
 \tilde{g}_{\alpha\beta}\equiv
 f'(R)g_{\alpha\beta}\,{,}\qquad
 \varphi=\sqrt{\frac{3}{2}}\ln f'(R)\,{,}
 \end{equation}
 we have the conformal equivalence of the Lagrangians
 \begin{equation}\label{28}
 {\cal L}=\sqrt{-g}\,f_0R^{3/2}\longleftrightarrow
 \tilde{{\cal L}}=\sqrt{-\tilde{g}}\left[-\frac{\tilde{R}}{2}+
 \frac{1}{2}\nabla_{\mu}\varphi\nabla^{\mu}\varphi-V_0\exp\left(
 \sqrt{\frac{2}{3}}\varphi\right)\right]\,{,}
 \end{equation}
 in our physical units. This is the so--called Liouville field
theory and it is one of the few cases where a fourth--order
Lagrangian can be expressed, in the Einstein frame, in terms of
elementary functions under a conformal transformation. It is
possible to obtain the general cosmological solution
\cite{lambiase} which is
 \begin{equation}\label{29}
 a(t)=a_0[c_4t^4+c_3t^3+c_2t^2+c_1t+c_0]^{1/2}\,{.}
 \end{equation}

 The constants $c_i$ are combinations of the initial conditions.
Their values determine the type of cosmological evolution. For
example, $c_4\neq 0$ gives a power law inflation while, if the
regime is dominated by the linear term in $c_1$, we get a
radiation--dominated stage.

\section{Torsion Quintessence}

In the ECSK theory, the affine connection is non-symmetric in its
lower indices and the antisymmetric part

\beq \Gamma^a_{[bc]}=S^{\phantom{bc}a}_{bc} \eeq is called
torsion. Usually, we can divide such an antisymmetric part into
three components:  one of them is
 irreducible,
while the other two can be set to zero \cite{goenner,classtor}.
This assumption yields the simplest theory containing torsion.

Furthermore, we can express by a 4-vector \beq
\sigma^a=\epsilon^{abcd}S_{bcd} \eeq the totally antisymmetric
part of torsion. If one imposes to it the symmetries of a
background which is homogeneous and isotropic, it follows that,
in comoving coordinates, only the component $\sigma^0$ survives
as a function depending only on cosmic time (see
\cite{goenner,classtor,tsamparlis}).

For a perfect fluid, the Einstein-Friedmann cosmological equations
can be written as

\beq \frac{\ddot{a}}{a}=-\frac{1}{6}(\tilde{\rho}+3\tilde{p})\,,
\eeq

and

 \beq \left(\frac{\dot{a}}{a}\right)^{2}+\frac{k}{a^{2}}=
\frac{1}{3}\tilde{\rho}\,, \eeq

(assuming $8\pi G=c=1$) where $a(t)$ is the cosmological scale
factor and $k=0,\pm 1$ is the spatial curvature constant. Energy
density and pressure can be assumed in the forms
 \cite{goenner,classtor,tsamparlis}


\beq \label{density} \tilde\rho=\rho +  f^2\,, \qquad \tilde p= p
-f^2\,, \eeq

where $f$ is a function related to $\sigma^0$, while $\rho$ and
$p$ are the usual quantities of General Relativity. This choice
can be pursued since
  we can define $S_{abc}=S_{[abc]}=f(t)$ where $f(t)$ is a
 generic function of time which we consider as the source of torsion.
 For a detailed discussion of this point see \cite{classtor}

As usual, we can define a stress-energy tensor of the form \beq
T^{tot}_{ab}=(\tilde{p}+\tilde{\rho})u_{a}u_{b}-\tilde{p}g_{ab}\,,
\eeq which, by Eqs. (\ref{density}), can be splitted as \beq
T^{(tot)}_{ab}=T^{(matter)}_{ab}+T^{(torsion)}_{ab}\,. \eeq Due to
the contracted Bianchi identity, we have \beq \label{bianchi}
T^{(tot);b}_{ab}=0\;; \eeq from which, we can assume that (cfr.
\cite{minkowski}) \beq \label{bianchi1}
T^{(matter);b}_{ab}=0\,,\qquad T^{(torsion);b}_{ab}=0\,. \eeq In
the FRW space-time, (\ref{bianchi}) becomes
 \beq
\dot{\tilde{\rho}}+3H(\tilde\rho + \tilde p)=0 \eeq which is \beq
\label{continuity} \dot{\rho} + 3H(\rho +  p) = -2f\dot{f}\,. \eeq
From Eqs.(\ref{bianchi1}), both sides of (\ref{continuity}) vanish
independently so that \beq f(t)=f_{0}=\mbox{constant.} \eeq In
other worlds, a torsion field gives rise to a constant energy
density. Taking into account standard matter which equation of
state is defined into the above Zeldovich range, we obtain

\beq \label{dense}
 \tilde{\rho}=\rho_{0}\left[\frac{a_0}{a}
\right]^{3(\gamma+1)}+f_{0}^2\,. \eeq

Inserting this result into the cosmological equations, we obtain,
in any case,  a monotonic expansion being $f_{0}^2>0$,
$\dot{a}^2>0$. The condition to obtain the accelerated behaviour
is \beq \label{acceleration} \rho+3p<2f_{0}^2\,, \eeq so that
acceleration depends on the torsion density.

In a dust-dominated universe, we have \beq \label{dust}
\tilde{\rho}=\rho_{0}\left(\frac{a_{0}}{a}\right)^{3}+f_{0}^{2}\,,
\qquad \tilde{p}=-f_{0}^2\,,\eeq and then the general solution is
\beq \label{general}
a(t)={\left(\frac{a_0^3\rho_0}{2f_0}\right)^{1/3}
\left[\cosh(f_{0}t)-1\right]}^{1/3}\,. \eeq

Obviously, if $f_{0}t\rightarrow 0$, we have $\displaystyle{\cosh
(f_{0}t)\simeq (f_{0}t)^2}$ and then $a\sim t^{2/3}$, as it has to
be.







\section{Conclusions}

In this paper, we have shown that the quintessence ''paradigm" can
be recovered in the framework of higher-order curvature and
torsion theories of gravity. In other words, as it is possible for
inflationary models, we can ask for a sort of {\it curvature or
torsion  quintessence} which can be recovered by taking into
account curvature and torsion geometric invariants. The interest
of this approach is that quintessence could be related to some
effective theory of quantum gravity where such invariants are
widely derived \cite{odintsov}.


Besides, this results tell us that quintessence can be obtained
without considering additional scalar fields in the dynamics but
only assuming that the space-time is $U_4$ instead of $V_4$. A
preminent role is played by the type of torsion we are going to
consider. In our model, we assume that torsion density is a
function of cosmic time only by imposing, as standard, the
homogeneity and isotropy of background. However, such a density
should be comparable to the observed limits of dark energy (\ie
$\Omega_{\Lambda}\sim 0.65\div 0.7$) in order to give relevant
effects. Another point is that such a {\it torsion quintessence}
should match the issues of cosmic coincidence
 \cite{steinhardt} as scalar field quintessence. This point strictly depends
on  the type of torsion since torsion can be or not  related to
the spin density \cite{classtor}. In the first case, the spin of
baryonic and non-baryonic matter would rule the dark energy
(torsion) density. In forthcoming investigations, we will face
these problems.

In any case, the models have to be improved by the comparison with
observations in order to see if it is possible to constrain the
form of $f(R)$ and torsion field without  {\it ad hoc} choices. In
this sense, some results are present in literature where the form
of $f(R)$ is selected by the CMBR constraint \cite{hwang}.





\Acknowledgements

 The authors are grateful to A.A. Starobinsky and
S.A. Bludmann for the useful discussions and comments on the
topics.



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\end{thebibliography}


\end{document}

gr-qc0111106.extracted_bib
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\bibitem{boomerang} P. de Bernardis et al. {\it Nature} {\bf 404}, 955 (2000).
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S. Weinberg, \rmp {\bf 61} (1989) 1.
\bibitem{guth}
A. Guth, \pr  {\bf D 23} (1981) 347;   \pl  {\bf 108 B} (1982)
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