Extended Cosmologies

Extended Cosmologies\star\star$\star$This paper is a contribution to the
Special Issue on Analytical Mechanics and Differential Geometry in honour of
Sergio Benenti. The full collection is available at
http://www.emis.de/journals/SIGMA/Benenti.html

Salvatore CAPOZZIELLO ${}^{{\dagger}^{1}{\dagger}^{2}{\dagger}^{3}}$,
Mariafelicia F. DE LAURENTIS
${}^{{\dagger}^{3}{\dagger}^{4}{\dagger}^{5}{\dagger}^{6}}$,
Lorenzo FATIBENE ${}^{{\dagger}^{7}{\dagger}^{8}}$, Marco FERRARIS
${}^{{\dagger}^{7}}$ and Simon GARRUTO ${}^{{\dagger}^{7}{\dagger}^{8}}$

S. Capozziello, M.F. De Laurentis, L. Fatibene, M. Ferraris and S. Garruto

${}^{{\dagger}^{1}}$ Dipartimento di Fisica, University of Napoli “Federico
II”, Italy<EMAIL_ADDRESS>
${}^{{\dagger}^{2}}$ INFN Sezione Napoli – Iniziativa Specifica QGSKY, Italy
${}^{{\dagger}^{3}}$ Gran Sasso Science Institute (INFN), L’Aquila, Italy
<EMAIL_ADDRESS>${}^{{\dagger}^{4}}$ Tomsk State
Pedagogical University, Russia ${}^{{\dagger}^{5}}$ Laboratory of Theoretical
Cosmology, Tomsk State University of Control Systems
${}^{{\dagger}^{5}}$ and Radioelectronics (TUSUR), Russia ${}^{{\dagger}^{6}}$
Institut für Theoretische Physik, Goethe-Universität, Max-von-Laue-Str. 1,
${}^{{\dagger}^{6}}$ 60438 Frankfurt, Germany ${}^{{\dagger}^{7}}$
Dipartimento di Matematica, University of Torino, Italy
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>${}^{{\dagger}^{8}}$ INFN Sezione Torino – Iniziativa Specifica QGSKY, Italy

Received September 29, 2015, in final form January 16, 2016; Published online
January 20, 2016

We shall discuss cosmological models in extended theories of gravitation. We
shall define a surface, called the model surface, in the space of observable
parameters which characterises families of theories. We also show how this
surface can be used to compare with observations. The model surface can
potentially be used to falsify whole families of models instead reasoning on a
single model basis as it is usually done by best fit arguments with
observations.

cosmology; extended theories of gravitation

83F05; 83D05; 37N20

##  1 Introduction

The aim of this paper is to provide a detailed framework for interpreting
extended cosmologies in the family of Palatini $f({\cal R})$-models. The
interpretation of extended theories is trickier than usually recognised in the
literature, since the dynamics is not sufficient to constrain all details of
the interpretation or the relation between the cosmographic parameters and the
corresponding model quantities. All interpretation issues are generated by the
fact that in Palatini $f({\cal R})$-theories one has two metrics which play a
natural role, the original metric $g$ and a conformal metric $\tilde{g}$ which
naturally appears when solving the model.

From the geometric point of view, while in standard general relativity (GR)
the geometry of spacetime is described by a Lorentzian structure, in $f({\cal
R})$-theories spacetime is described by a Weyl geometry, i.e., by a conformal
structure $[g]$ together with a compatible connection $\tilde{\Gamma}$ which,
in view of field equations, turns out to be the Levi-Civita connection of the
conformal metric $\tilde{g}{=\varphi\cdot g}$. The scalar field $\varphi$ is
called the conformal factor and it parametrises conformal transformations.
Weyl geometries have been considered in the past in gravitational theories;
see, for example, [4, 23, 24, 26, 27, 28].

Unlike in standard GR, where one has only one metric which does everything, in
Palatini $f({\cal R})$-theories one has to declare which metric is used to do
what; see, for example, [1, 2, 7]. For example, the dynamics of field
equations does not determine the free fall. If one couples to a matter field
of course one can compute the characteristics of matter field equations and
argue that characteristic curves describe free fall. In particular, if one
couples a Klein–Gordon scalar field $\phi$ to the metric $g$ then
characteristics are determined as the geodesics of $g$. Unlike lightlike
geodesics, the timelike geodesics of $g$ are not geodesics of $\tilde{g}$.
Accordingly, it seems that the free fall is determined by the variational
principle assumed for gravity and matter.

However, one can show that a conformal transformation can be extended to act
on the matter field $\phi$ as well and define a new matter field
$\tilde{\phi}=\varphi^{\alpha}\cdot\phi$ (with the parameter
$\alpha\in{\mathbb{R}}$ to be determined) so that the new field $\tilde{\phi}$
obeys new (different) field equations (since conformal transformations are not
symmetries of the matter field equations). The exponent $\alpha$ in the action
of the conformal transformation on the Klein–Gordon field can be chosen so
that the characteristics of the new field $\tilde{\phi}$ are geodesics of the
conformal metric $\tilde{g}$; see [18]. Something similar can be done for
Dirac fields.

Hence arguing that the variational principle determines the free fall is
correct, provided one assumes what is observed as a matter field. Depending on
the choice of $\phi$ or $\tilde{\phi}$ as the observed field, then free fall
is determined to be described by $g$ or $\tilde{g}$, respectively. In this
paper we shall assume that free fall is given by $\tilde{g}$.

Other examples are the issues of whether the dynamics determines the metric
which is used for defining spatial lengths, minimal coupling to matter,
standard clocks. For example, in standard GR it is natural assuming that
physical lengths are associated to the geometric length measured with respect
to the only metric one has around, namely $g$. Having more that one metric in
the model makes the choice much more problematic.

Let us consider two parallel mirrors at a constant distance and a light ray
bouncing between them, which is called a gravitational clock. If one considers
an atomic clock (which is in fact built with some quantum harmonic oscillator)
and compares its rate to the rate of a gravitational clock, the assumption of
standard GR is pretty neat: the two clocks (at the same point) are
synchronised forever.

It is quite obvious that the discrepancy between the two clocks, if any,
cannot be too big otherwise we shall see some effect at the solar system
scales. However, there is no strong constraint on what happens at bigger
scales. One should not assume that there is no discrepancy. On the contrary,
the correct procedure would be to assume there is, derive observational
consequences, and constrain the effects by observations.

We shall hereafter follow this second guideline. In this paper we shall assume
that lengths are related to $g$, not to $\tilde{g}$. Accordingly, we shall
assume that matter is coupled to $g$, and not to $\tilde{\Gamma}$.

These choices are also motivated by the Ehlers–Pirani–Schild (EPS) framework
for gravitational theories; see [14]. In the early seventies EPS proposed a
framework able to derive the geometry of spacetime based on basic properties
of light rays and (free falling) particles. In this framework one does not
assume a Lorentzian structure on spacetime, rather a Lorentzian metric
(actually a Lorentzian conformal structure) on spacetime can be defined due to
the properties assumed for light rays. As a consequence, in the EPS framework
there is a tight link between the geometric structure of spacetime and
observations. One could say that the EPS framework instructs an observer to
establish observational protocols and to define distances and time lapses.
Once this is done EPS considered the properties of particles and find they are
described by a connection $\tilde{\Gamma}$ (actually by a projective structure
on the spacetime) which describes, by construction, the free fall.

The two structures $([g],\tilde{\Gamma})$ on spacetime are not arbitrary
though. Since one knows that lightlike directions are an upper bound for the
speed of particles, one can show that the connection $\tilde{\Gamma}$ must be
in the form

$\displaystyle\tilde{\Gamma}^{\alpha}_{\beta\mu}=\\{g\\}^{\alpha}_{\beta\mu}+\hbox{$\frac{1}{2}$}\big{(}g^{\alpha\epsilon}g_{\beta\mu}-2\delta^{\alpha}_{(\beta}\delta^{\epsilon}_{\mu)}\big{)}A_{\epsilon}$
(1.1)

for some covector $A_{\epsilon}$. This is quite similar to the original
proposal by Weyl for a unified theory of gravity and electromagnetism (then
proven wrong); see [28]. However, we are not interpreting here $A$ as an
electromagnetic potential and for us the covector $A_{\epsilon}$ will be
related to the possible shift in synchronisation between atomic and
gravitational clocks.

On the basis of the EPS framework we shall consider gravitational theories for
the fields $(g,\tilde{\Gamma})$ to be extended theories of gravitation (ETG)
when their field equations enforce as a consequence the relation (1.1); see
[10, 11]. We shall call such theories integrable when field equations also
imply that $A$ is an exact $1$-form and as a consequence $\tilde{\Gamma}$ is a
metric connection; see [13, 16].

In principle one can consider any invariant of the curvature to define a
dynamics. We shall hereafter restrict to Palatini $f({\cal R})$-theories in
which the dynamics is given in terms of a (regular) function of the scalar
curvature ${\cal R}$. One can show that Palatini $f({\cal R})$-gravity is an
integrable extended theory of gravitation (iETG). In view of the EPS framework
we shall interpret $\tilde{\Gamma}$ as defining the free fall, while $g$ is
related to distances and time lapses.

Standard GR is then a particular case of Palatini $f({\cal R})$-theory.
However, the standard Hilbert–Einstein dynamics imposes that the conformal
factor $\varphi$ is exactly and everywhere equal to 1. In this special case
the two metrics $g$ and $\tilde{g}$ do coincide and all issues related to
choosing which metric is used to do what are overcome.

It is particularly worth noticing that the EPS project was originally started
to show that standard GR follows from assumptions and it almost reached the
goal. Instead of finding a Lorentzian structure on the spacetime as assumed in
standard GR, EPS find something less (just a conformal structure of Lorentzian
metrics) and something more (a projective structure of connections which is
not uniquely determined as the Levi-Civita connection of the metric
structure).

EPS asked in their analysis what one needs to add to the framework in order to
force the structure to be Lorentzian. They showed that in fact if one requires
that atomic and gravitational clocks have the same rate then standard GR
follows.

In view of this, Palatini $f({\cal R})$-theories can be seen to generalise
standard GR in two respects. In the first place, the modified dynamics induces
(effective) dark sources which can be used to model dark matter and dark
energy. In the second place, the conformal factor does account for the
possibility that gravitational clocks shift with respect to atomic clocks, the
effect being measured by the field $\varphi$, i.e., the conformal factor; see
[9, 17].

These two effects (dark sources and time shifts) are quite independently
assumed in the theory and in our opinion one should not assume they are not
there. Instead, they should be analysed by considering them in a model, one
should make predictions and test the results, possibly ruling them out.

The material is organised as follows. In Section 2 we shall review Palatini
$f({\cal R})$-theories. The conservation of stress tensors and their relation
with conformal transformations is discussed in Appendix A. In Section 3 we
shall discuss cosmological models based on Palatini $f({\cal R})$-theories. In
Section 4 we shall consider the example $f({\cal R})={\cal
R}-\frac{\epsilon}{2}{\cal R}^{2}$, the solutions of which are briefly
discussed in the Appendix B.

##  2 Palatini $\boldsymbol{f({\cal R})}$-theories of gravitation

In view of the EPS formalism one should consider a Palatini formulation. For
the sake of generality, one can consider a general dynamics, not restricting a
priori to standard GR. For the sake of simplicity let us consider a Palatini
$f({\cal R})$-dynamics, and we shall assume for it the interpretation
described above.

Thus our theory is described by two fields
$(g_{\mu\nu},\tilde{\Gamma}^{\alpha}_{\beta\mu})$ a metric $g$ which is
associated with rulers and clocks and a (torsionless) connection
$\tilde{\Gamma}$ which is associated to free fall of test particles.

Let us define the scalar curvature ${\cal
R}=g^{\mu\nu}\tilde{R}_{\mu\nu}(\tilde{\Gamma})$ and the Lagrangian

$\displaystyle\hbox{\bf L}=f({\cal R})\sqrt{g}d\sigma+\hbox{\bf L}_{\rm mat},$

where $d\sigma$ denotes the standard local pointwise basis of $4$-forms on a
$4$-dimensional spacetime $M$ induced by coordinates. The field equations turn
out to be

$\displaystyle f^{\prime}({\cal
R})\tilde{R}_{(\mu\nu)}-\hbox{$\frac{1}{2}$}f({\cal R})g_{\mu\nu}=T_{\mu\nu},$
$\displaystyle\tilde{\nabla}_{\alpha}\big{(}\sqrt{g}f^{\prime}({\cal
R})g^{\mu\nu}\big{)}=0,$

where the stress tensor $T_{\mu\nu}$ is defined by variation of matter
Lagrangian with respect to the metric $g$. We also assumed that the matter
does not couple to the connection $\tilde{\Gamma}$ but only to the metric $g$
and we understand the matter field equations.

The second equation does determine the connection $\tilde{\Gamma}$ as the
Levi-Civita connection of a conformal metric

$\displaystyle\tilde{g}_{\mu\nu}:=\varphi\cdot
g_{\mu\nu},\qquad\varphi=\frac{f^{\prime}({\cal
R})}{\varphi_{0}}\qquad\Rightarrow\qquad\tilde{\Gamma}=\\{\tilde{g}\\}$

for some constant $\varphi_{0}$ chosen so that one has $\varphi(x_{0})=1$ here
and now. Let us assume that the conformal factor $\varphi$ is positive and the
metric $\tilde{g}$ is also Lorentzian. Using this information in the first
equation, the Ricci tensor $\tilde{R}_{\mu\nu}$ becomes the (symmetric) Ricci
tensor of the conformal metric $\tilde{g}$, i.e., one obtains

$\displaystyle f^{\prime}({\cal
R})\tilde{R}_{\mu\nu}-\hbox{$\frac{1}{2}$}f({\cal R})g_{\mu\nu}=T_{\mu\nu}.$

In order to solve this equation let us consider its trace by $g^{\mu\nu}$
first

$\displaystyle f^{\prime}({\cal R}){\cal R}-2f({\cal R})=T,$ (2.1)

which is called the master equation. It is an algebraic (i.e., not
differential) equation in ${\cal R}$ and $T=g^{\mu\nu}T_{\mu\nu}$; except some
degenerate cases, for a generic (analytic) $f({\cal R})$ one can solve it for
${\cal R}$ obtaining

$\displaystyle{\cal R}={\cal R}(T).$

Accordingly, one can express the conformal factor as a function of the trace
$T$, i.e.,

$\displaystyle\varphi(T)=\frac{f^{\prime}({\cal R}(T))}{\varphi_{0}}.$

Using again this information to write the first field equation in terms of the
new conformal metric $\tilde{g}$ one has

$\displaystyle\tilde{R}_{\mu\nu}-\hbox{$\frac{1}{2}$}\tilde{R}\tilde{g}_{\mu\nu}=\tilde{T}_{\mu\nu}:=\hbox{$\frac{1}{\varphi}$}T_{\mu\nu}+\hbox{$\frac{1}{2\varphi}$}f({\cal
R})g_{\mu\nu}-\hbox{$\frac{1}{2}$}{\cal R}g_{\mu\nu},$

where $\tilde{R}=\varphi^{-1}{\cal R}$ is the scalar curvature of the
conformal metric. In other words, the conformal metric $\tilde{g}$ is
determined by a Einstein equation though with a modified source stress tensor

$\displaystyle\tilde{T}_{\mu\nu}:=\hbox{$\frac{1}{\varphi}$}T_{\mu\nu}+\hbox{$\frac{1}{2\varphi}$}f({\cal
R})g_{\mu\nu}-\hbox{$\frac{1}{2}$}{\cal R}g_{\mu\nu},$ (2.2)

which can in fact be considered as a function of $\tilde{g}$ and the matter
fields (included the conformal factor $\varphi$).

As a minimal check one can show that if the matter energy-momentum stress
tensor $T_{\mu\nu}$ is conserved (with respect to $g$) then the effective
stress tensor $\tilde{T}_{\mu\nu}$ is automatically conserved as well, with
respect to $\tilde{g}$, though. See Appendix A for details.

##  3 Palatini $\boldsymbol{f({\cal R})}$-cosmologies

We can use this theory of gravitation for producing cosmological models. For,
one has to state cosmological principle, i.e., restricting to spacetimes with
an isometry group which enforces spatial homogeneity and isotropy (or,
equivalently, spatial isotropy with respect to any point). As Benenti pointed
out, the cosmological principle freezes dynamics since there is only one
dynamics, namely the one dictated by Friedman–Robertson–Walker (FRW)
equations, which is compatible with homogeneity and isotropy; see [5].

However, if, as in this case, a theory can be described in terms of more than
one metric, one has to state for which metric isometries are considered in the
cosmological principle. Luckily, in Palatini $f({\cal R})$-theories this is
not an issue, since the conformal factor $\varphi(t)$ itself is expected to be
a function of time only, there are two metrics which are conformal to each
other by a conformal factor which depends on time only, and if a metric $g$ is
spatially homogeneous and isotropic then the conformal metric $\tilde{g}$ is
spatially homogeneous and isotropic as well, though in a particular and
precise sense which we need to make explicit for later convenience.

A metric $g$ obeys cosmological principle if and only if there exists a
coordinate system $(t,r,\theta,\phi)$ in which the metric is in the form

$\displaystyle
g=-dt^{2}+a^{2}(t)\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right),\qquad
d\Omega^{2}:=d\theta^{2}+\sin^{2}(\theta)d\phi^{2}$ (3.1)

for some function of time $a(t)$ which is called the scale factor. A metric in
this form is called a FRW metric. One can (and we will) always rescale the
radial coordinate so to have $a(t_{0})=1$ at a time $t_{0}$ denoting today
(which for convenience can be set to $t_{0}=0$).

If the conformal factor is a function of time only the conformal
transformation sends a FRW metric into a FRW metric, though with another time
and scale factor. If $g$ is in the form (3.1) with radial coordinate rescaled
to obtain $a(t_{0})=1$, then the conformal metric $\tilde{g}=\varphi(t)\cdot
g$ is in the form

$\displaystyle\tilde{g}=-d{\tilde{t}}{}^{2}+\tilde{a}^{2}(\tilde{t})\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\Omega^{2}\right)\qquad\Rightarrow\qquad\begin{cases}\tilde{a}(\tilde{t})=\sqrt{\varphi(t)}a(t),\\\
d{\tilde{t}}=\sqrt{\varphi(t)}dt,\end{cases}$ (3.2)

and one still has $\tilde{a}(\tilde{t}_{0})=1$ if and only if the conformal
factor $\varphi(t)$ is chosen so that $\varphi(t_{0})=1$.

Most of what follows relies on having a metric in FRW form and, consequently,
it applies to both $g$ and $\tilde{g}$. We shall hereafter briefly review what
is routinely done for $g$ in standard cosmology with the aim of then applying
it to $\tilde{g}$.

Once one has a FRW metric $g_{\mu\nu}$, one can compute its Einstein tensor
$G_{\mu\nu}$ and find it in the form of a stress tensor of a perfect fluid
$T_{\mu\nu}$, i.e., there exists a vector field $u$ which is a timelike unit
vector for the metric $g$ orthogonal to the submanifolds at $t={\rm const}$
and two functions $p(t)$ (the pressure) and $\rho(t)$ (the energy density)
such that

$\displaystyle
T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}\qquad\Rightarrow\qquad
T=-(\rho+p)+4p=3p-\rho.$ (3.3)

This has not much to do with dynamics since the stress tensors for a perfect
fluid are the only tensors which are compatible with spatial homogeneity and
isotropy. In some sense the cosmological principle freezes the dynamics and
reduces it to a purely kinematical fact: any FRW metric obeys Einstein
equations with a perfect fluid source for a suitable pressure and energy
density functions. The dynamics only specifies the relation of density and
pressure with the matter fields.

Einstein equations are $10$ equations, though in this case only two of them
are independent. They are the so-called Friedman equations

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}+\frac{kc^{2}}{a^{2}}=\frac{1}{3}\rho,\qquad\frac{\ddot{a}}{a}=-\frac{1}{6}(\rho+3p).$
(3.4)

These are two equations for the three functions $a(t)$, $\rho(t)$, $p(t)$
which then cannot be determined unless a further equation is provided. In
principle one should obtain that extra equation from the matter field
equations. In practice, it is customary in cosmology to describe matter by
means of an (algebraic) equation of state (EoS) which constrains $\rho$ and
$p$. The most general EoS $\sigma(p,\rho)=0$ usually is chosen in the form
$p=p(\rho)$ and often linearised simply to $p=w\rho$, where $w$ is called the
parameter of the EoS, the value of which determines the kind of matter we are
dealing with. Different cases will be analysed below.

Equations (3.4) are Lagrange equation for a suitable Lagrangian. If we
consider the matter described by an energy-density function $\rho(a)$, then we
can define the so-called point-like Lagrangian

$\displaystyle L=\big{(}{-}a\dot{a}^{2}+kc^{2}a-\lambda\rho(a)a^{3}\big{)}dt,$
(3.5)

which is obtained by integrating on space the Hilbert Lagrangian for a FRW
metric (and neglecting a total time-derivative).

This Lagrangian is time-independent so that its “total energy”

$\displaystyle{\cal
H}=-2a\dot{a}^{2}+a\dot{a}^{2}-kc^{2}a+\lambda\rho(a)a^{3}=-\left(\frac{\dot{a}^{2}}{a^{2}}+\frac{kc^{2}}{a^{2}}-\lambda\rho(a)\right)a^{3}$

is conserved. Notice that by imposing this quantity to be zero one obtains

$\displaystyle\frac{\dot{a}^{2}}{a^{2}}+\frac{kc^{2}}{a^{2}}=\lambda\rho,$

which agrees with the first FRW equation by setting
$\lambda=\hbox{$\frac{1}{3}$}$. The equation of motion for the Lagrangian
(3.5) is

$\displaystyle
2\ddot{a}=\frac{1}{3}(2\rho+\rho^{\prime}a)a\qquad\Rightarrow\qquad\frac{\ddot{a}}{a}=\frac{1}{6}(2\rho+\rho^{\prime}a),$

which becomes the second FRW equation provided one has the pressure defined to
be

$\displaystyle-(\rho+3p)=2\rho+\rho^{\prime}a\qquad\Rightarrow\qquad
p=-\big{(}\rho+\hbox{$\frac{1}{3}$}\rho^{\prime}a\big{)}.$

This condition is equivalent to the conservation of the energy-momentum stress
tensor, namely $\nabla_{\mu}T^{\mu\nu}=0$ for a fluid tensor in the form
(3.3).

Thus we select the type of matter fluid by choosing the function $\rho(a)$
which in turn determines the matter equation of state (EoS) $p=p(\rho)$. The
definition of the pressure above is in fact a form for the EoS parametrised by
the scale factor $a$

$\displaystyle\rho=\rho(a),\qquad
p=p(a):=-\big{(}\rho(a)+\hbox{$\frac{1}{3}$}\rho^{\prime}(a)a\big{)}.$

For example, dust is determined by setting $p=0$, i.e.,
$3\rho+\rho^{\prime}a=0$ from which $\rho=\rho_{0}a^{-3}$ follows. More
generally, assuming an EoS in the form $p=w\rho$ one has

$\displaystyle\rho=\rho_{0}a^{-3(w+1)}.$

Radiation is defined to be $p=\hbox{$\frac{1}{3}$}\rho$, thence
$\rho=\rho_{0}a^{-4}$. Cosmological constant or (quintessence) is defined to
be $p=-\rho$, so that one has $\rho=\rho_{0}$; see [9].

One can also assume the so-called polytropic equation of state
$p=\rho^{\gamma}$ and obtain

$\displaystyle\rho(a)=\left(\frac{3}{(ca)^{3(\gamma-1)}-3}\right)^{\hbox{$\frac{1}{\gamma-1}$}}.$

In view of the point-like Lagrangian, FRW cosmology is captured by first FRW
equation which is a Weierstrass equation:

$\displaystyle\frac{\dot{a}^{2}}{a^{2}}+\frac{k}{a^{2}}-\hbox{$\frac{1}{3}$}\rho(a)=0\qquad\Rightarrow\qquad\dot{a}^{2}=-k+\hbox{$\frac{1}{3}$}\rho(a)a^{2}=:\Phi(a).$
(3.6)

This function $\Phi(a)$ is essentially the so-called effective potential. The
quantity

$\displaystyle H=\frac{\dot{a}}{a}$

is called the Hubble parameter. The value of the Hubble parameter today is
denoted by $H_{0}$.

One can observe that if $k=0$ in equation (3.6) then $\rho_{0}^{\rm
cr}=3H_{0}^{2}$ which is called the critical density (today). Hence if one has
different matter species each with its own density
$\rho_{i}=\rho_{i}^{0}a^{n_{i}}$ and pressure $p_{i}$, and its own EoS
$p_{i}=p_{i}(\rho_{i})$, the equation (3.6) specifies to

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}=\frac{1}{3}\left(\rho-\frac{3k}{a^{2}}\right)=\frac{1}{3}\left(\sum_{i=1}^{k}\rho_{i}+\rho_{0}\right)=\frac{1}{3}\boldsymbol{\rho},$

where we introduced one more species, called the curvature species, with
density $\rho_{k}=-3ka^{-2}$. The total density
$\boldsymbol{\rho}=\sum\limits_{i=0}^{k}\rho_{i}$ must be critical today.

Then one can recast the FRW equation as

$\displaystyle\frac{\dot{a}^{2}}{a^{2}}=\hbox{$\frac{1}{3}$}\rho_{0}^{\rm
cr}\sum_{i}\frac{\rho_{i}}{\rho_{0}^{\rm
cr}}=H_{0}\sum_{i=0}^{k}\Omega_{i}^{0}a^{n_{i}},$

where we set

$\displaystyle\Omega_{i}^{0}=\frac{\rho^{0}_{i}}{\rho_{0}^{\rm
cr}},\qquad\sum_{i=0}^{k}\Omega_{i}^{0}=1,$

which are called abundances (today) and are one-dimensional parameter for each
species. Since a set of abundances are constrained by $\sum\Omega^{0}_{i}=1$,
they can be interpreted as percentages of each species in the total budget
corresponding to the critical density. A set of such percentages are called a
cosmic pie. Then in this more general framework, the FRW equation reads as

$\displaystyle\dot{a}^{2}=a^{2}H_{0}\sum_{i}\Omega_{i}^{0}a^{n_{i}}=\Phi\big{(}a;H_{0},\Omega_{i}^{0}\big{)}$

and solutions are uniquely determined by initial condition $a(0)=1$ and today
abundances $\Omega_{i}^{0}$.

###  3.1 Einstein frame

In Palatini $f({\cal R})$-cosmologies, also the conformal metric $\tilde{g}$
is a FRW metric and as such it obeys Friedman equations for some suitable
pressure and density as well. We already showed that the conformal metric can
be recast in FRW form by a suitable rescaling.

Thus we have the metric in the from (3.2). Since it is a FRW metric, it obeys
some sort of Friedman equations which means that the modified stress tensor
$\tilde{T}_{\mu\nu}$ must be in the form of a perfect fluid stress tensor.

In fact the timelike unit vector $\tilde{u}$ for the metric $\tilde{g}$ is

$\displaystyle{\tilde{u}}^{\mu}\tilde{g}_{\mu\nu}{\tilde{u}}^{\nu}=-1,\qquad
u^{\mu}=\sqrt{\varphi}\tilde{u}^{\mu}\quad\iff\quad
u^{\mu}g_{\mu\nu}u^{\nu}=-1.$

The corresponding covector is

$\displaystyle\tilde{u}_{\mu}=\tilde{g}_{\mu\nu}\tilde{u}^{\nu}=\varphi
g_{\mu\nu}\tilde{u}^{\nu}=\sqrt{\varphi}g_{\mu\nu}u^{\nu}=\sqrt{\varphi}u_{\mu}$

and the modified stress tensor (2.2) is in the form

$\displaystyle\tilde{T}_{\mu\nu}=\frac{p+\rho}{\varphi^{2}}\tilde{u}_{\mu}\tilde{u}_{\nu}+\frac{1}{4\varphi^{2}}(p+\rho-\varphi{\cal
R})\tilde{g}_{\mu\nu},$

where the master equation has been used to eliminate the dependence on
$f({\cal R})$.

Thus also $\tilde{T}_{\mu\nu}$ is in canonical form provided one defines the
effective pressure $\tilde{p}$ and energy density $\tilde{\rho}$ as

$\displaystyle\tilde{p}+\tilde{\rho}=\frac{p+\rho}{\varphi^{2}}\qquad\Rightarrow\qquad\tilde{\rho}=\frac{3(p+\rho)}{4\varphi^{2}}+\frac{{\cal
R}}{4\varphi},\qquad\tilde{p}=\frac{p+\rho}{4\varphi^{2}}-\frac{{\cal
R}}{4\varphi}.$

By replacing ${\cal R}={\cal R}(T)={\cal R}(T)={\cal R}(3p-\rho)$ and
$\varphi=\varphi_{0}^{-1}f^{\prime}({\cal
R}(T))=\varphi_{0}^{-1}f^{\prime}({\cal R}(3p-\rho))$ and using the EoS, one
obtains the effective pressure and energy density as functions of $\rho$,
i.e.,

$\displaystyle\tilde{\rho}=\tilde{\rho}(\rho),\qquad\tilde{p}=\tilde{p}(\rho),$

which is an implicit form of the EoS for the effective matter (parametrised by
the matter energy density $\rho$).

##  4 An example

Let us consider the simple case (see [22])

$\displaystyle f(R)={\cal R}-\hbox{$\frac{\epsilon}{2}$}{\cal
R}^{2},\qquad\epsilon\geq 0.$

The stress tensor $T_{\mu\nu}$ is assumed to be a fluid stress tensor. Visible
matter will be described by this tensor and, for the sake of simplicity, we
will assume visible matter to be dust. The master equation reads as

$\displaystyle{\cal R}-\epsilon{\cal R}^{2}-2{\cal R}+\epsilon{\cal
R}^{2}=-{\cal R}=-\rho\qquad\Rightarrow\qquad{\cal R}=\rho$

and the conformal factor is

$\displaystyle\varphi=\frac{1-\epsilon{\cal
R}}{\varphi_{0}}=\frac{1-{\epsilon}\rho}{1-{\epsilon}\rho_{0}},\qquad\rho=\rho_{0}a^{-3}.$

Let us stress that there are two allowed regions for the scale factor, one for
$a\geq a_{m}=(\epsilon\rho_{0})^{1/3}>0$ and one for negative values of $a$.
The effective pressure and energy density are

$\displaystyle\tilde{\rho}=\frac{3+\varphi}{4\varphi^{2}}\rho=\frac{1-\epsilon\rho_{0}}{(1-{\epsilon}\rho)^{2}}\big{(}1-\hbox{$\frac{3}{4}$}\epsilon\rho_{0}-\hbox{$\frac{1}{4}$}\epsilon\rho\big{)}\rho,$
$\displaystyle\tilde{p}=\frac{1-\varphi}{4\varphi^{2}}\rho=\frac{\epsilon}{4}\frac{1-\epsilon\rho_{0}}{(1-\epsilon\rho)^{2}}(\rho-\rho_{0})\rho.$

This corresponds to an effective EoS; see Fig. 1a.

Figure 1: a) Effective EoS $\tilde{p}(\tilde{\rho})$. (The dash line
corresponds to a negative energy density $\rho$ for visible matter, which may
be considered unphysical.) b) Scale factors $\tilde{a}(a)$. Dash line is
standard GR.

The relation between the scale factors is (see Fig. 1b)

$\displaystyle\tilde{a}(a)=\sqrt{\frac{1-\epsilon\rho}{1-\epsilon\rho_{0}}}a.$

At this point we can expand $\tilde{\rho}$ and $\tilde{p}$ as functions of $a$
and check that

$\displaystyle\tilde{p}=-\left(\tilde{\rho}+\hbox{$\frac{1}{3}$}\hbox{$\frac{d\tilde{\rho}}{d\tilde{a}}$}\tilde{a}\right),\qquad\frac{d\tilde{\rho}}{d\tilde{a}}=\frac{d\tilde{\rho}}{d\rho}\frac{d\rho}{da}\frac{da}{d\tilde{a}}=-3\rho_{0}a^{-4}\frac{d\tilde{\rho}}{d\rho}\left(\frac{d\tilde{a}}{da}\right)^{-1}.$
(4.1)

The condition (4.1) is in fact generally equivalent to conservation laws for
the effective energy-momentum stress tensor $\tilde{T}_{\mu\nu}$, which is
general follows from conservation of the original energy-momentum stress
tensor $T_{\mu\nu}$ (see Appendix A), which in turn follows from EoS imposed
for matter, or, equivalently, from covariance of the matter Lagrangian.

Then one decides what kind of effective matter is needed in the model. This is
not something decided within cosmology, it can be a choice of ours or an input
from other physics (e.g., constraints from baryogenesis or structures
formation).

For example, let us assume we want a model with cosmological constant, dust
and radiation. This corresponds to a total density in the form

$\displaystyle\hat{\rho}(\tilde{a})=\tilde{\rho}{}_{\Lambda}^{0}+\frac{\tilde{\rho}{}^{0}_{d}}{\tilde{a}^{3}}+\frac{\tilde{\rho}{}^{0}_{r}}{\tilde{a}^{4}}$

with the parameters
$(\tilde{\rho}{}_{\Lambda}^{0},\tilde{\rho}{}^{0}_{d},\tilde{\rho}{}^{0}_{r})$
to be determined so that $\hat{\rho}$ best approximates $\tilde{\rho}$.

Best fit can be obtained simply by imposing that the value, first and second
derivatives of $\tilde{\rho}$ and $\hat{\rho}$ coincides at $\tilde{a}=1$,
i.e., today. The best approximation takes the form of

$\displaystyle\tilde{\rho}{}_{\Lambda}^{0}=\tilde{\rho}{}_{\Lambda}^{0}(\epsilon,\rho_{0}),\qquad\tilde{\rho}{}^{0}_{d}=\tilde{\rho}{}^{0}_{d}(\epsilon,\rho_{0}),\qquad\tilde{\rho}{}^{0}_{r}=\tilde{\rho}{}^{0}_{r}(\epsilon,\rho_{0}).$
(4.2)

This functions are quite complicated functions, also for a simple model as the
one we are considering here. However, one can have an idea of their shape (and
the effective physics they describe) by expanding them for small $\epsilon$
(i.e., small modification of standard GR). In this case we have:

$\displaystyle\tilde{\rho}{}_{\Lambda}^{0}\simeq\frac{\rho_{0}^{2}}{8}\epsilon-\frac{\rho_{0}^{3}}{4}\epsilon^{2},\qquad\tilde{\rho}{}^{0}_{d}\simeq\rho_{0}-\frac{5\rho_{0}^{2}}{4}\epsilon+\frac{11\rho_{0}^{3}}{8}\epsilon^{2},\qquad\tilde{\rho}{}^{0}_{r}\simeq\frac{9\rho_{0}^{2}}{8}\epsilon-\frac{9\rho_{0}^{3}}{8}\epsilon^{2}.$

Let us remark that those reduce to visible matter (dust only) for $\epsilon=0$
and that all energy densities are positive for a small positive $\epsilon$.

This means that Palatini $f({\cal R})$-cosmology with $f({\cal R})={\cal
R}-\hbox{$\frac{\epsilon}{2}$}{\cal R}^{2}$ and purely dust visible matter is
well approximated today by standard cosmology with cosmological constant, dust
and radiation. Of course, far in the past and future the approximation fails
to be accurate.

We can also compute the abundances of different kinds of effective matter by
dividing by the critical density today $\tilde{\rho}_{0}^{\rm
cr}=3\tilde{H}_{0}=\rho_{0}$ to obtain the effective abundance parameters
today

$\displaystyle\tilde{\Omega}{}_{\Lambda}^{0}\simeq\frac{\rho_{0}}{8}\epsilon-\frac{\rho_{0}^{2}}{4}\epsilon^{2},\qquad\tilde{\Omega}{}^{0}_{d}\simeq
1-\frac{5\rho_{0}}{4}\epsilon+\frac{11\rho_{0}^{2}}{8}\epsilon^{2},\qquad\tilde{\Omega}{}^{0}_{r}\simeq\frac{9\rho_{0}}{8}\epsilon-\frac{9\rho_{0}^{2}}{8}\epsilon^{2}.$

In principle, one can try and determine $\epsilon$ so that, e.g.,
$\Omega_{\Lambda}\simeq 0.75$ as observed, e.g., by WMAP. However, there are
numerical evidences that the (exact) abundance of effective dark energy in
$f({\cal R})={\cal R}-\hbox{$\frac{\epsilon}{2}$}{\cal R}^{2}$ models cannot
exceed a value of about 0.62. In this case the whole family of models $f({\cal
R})={\cal R}-\hbox{$\frac{\epsilon}{2}$}{\cal R}^{2}$ can be rejected since it
is not compatible with observations.

###  4.1 The model surface

What we got above as a description of the cosmological model is the abundance
parameters
$\big{(}\Omega^{0}_{k},\Omega^{0}_{\Lambda},\Omega^{0}_{d},\Omega^{0}_{r},\dots\big{)}$
as functions of the parameter $\epsilon$ which in fact parametrises the family
of models under consideration, namely in the simple example of the function
$f({\cal R})={\cal R}-\hbox{$\frac{\epsilon}{2}$}{\cal R}^{2}$.

Abundances are particularly interesting parameters since they can be measured
quite directly observing the Cosmic Microwave Background radiation.
Accordingly, one has independent effective abundances $\Omega^{0}_{i}$
($i=1,\dots,n$) which result from observations (the curvature $\Omega^{0}_{k}$
is eliminated by the constraint $\sum\Omega_{i}^{0}=1$). More generally, a
family of models is represented by a number of parameters $\epsilon^{\alpha}$
(the constants determining the function $f({\cal R})$) and the independent
abundances

$\displaystyle\Omega^{0}_{i}=\Omega^{0}_{i}(\epsilon^{\alpha})$ (4.3)

define a map from the set of models into the space of cosmic pies.

By letting the parameters $\epsilon^{\alpha}$ vary, the equations (4.3)
describe a parametrised subset $\Sigma\subset{\mathbb{R}}^{n}$. Each point on
$\Sigma$ represents the predictions of a model in the family. In this setting
one can translate physical issues in geometrical claims, e.g., whether an
observed cosmic pie $(\Omega^{0}_{i})\in{\mathbb{R}}^{n}$ belongs to the image
$\Sigma\subset{\mathbb{R}}^{n}$.

The subset $\Sigma$ is called the model surface and it captures all the
predictions of the models in terms of observable quantities. Thus if the
observed cosmic pie is not on $\Sigma$ then the family cannot account for
observations and must be abandoned.

In the model above, one has one parameter $\epsilon$ and two independent
abundances
$\big{(}\Omega^{0}_{\Lambda},\Omega^{0}_{d}\big{)}\in{\mathbb{R}}^{2}$. The
corresponding line representing the model is shown in Fig. 2.

Figure 2: Line model for $f({\cal R})={\cal
R}-\hbox{$\frac{\epsilon}{2}$}{\cal R}^{2}$. Dash line is the constraint
$\sum_{i}\Omega^{0}_{i}=1$. The dot is the observered cosmic pie (e.g., WMAP).

Since observations are considered to support
$\big{(}\Omega^{0}_{\Lambda},\Omega^{0}_{d}\big{)}\simeq(0.75,0.25)$ the
family is discarded.

More generally, one can consider further observable quantities as the
cosmographic parameters

$\displaystyle q=-\frac{1}{H^{2}}\frac{\ddot{a}}{a},\qquad
j=\frac{1}{H^{3}}\frac{\dot{\ddot{a}}}{a},\qquad
j=\frac{1}{H^{4}}\frac{\ddot{\ddot{a}}}{a},\qquad\dots$

in the target space. On the other hand, ideally, one can expand the function
$f({\cal R})$ in series (for example in Laurent series) and consider the
infinitely many coefficients as the family parameters $\epsilon^{\alpha}$.
Although, this becomes mathematically elaborated (having an infinite-
dimensional surface embedded into an infinite-dimensional space) one can
regard this as a universal setting to represent theories and observations as
geometric intrinsic claims even when one may simplify it projecting to finite-
dimensional representations in some particular context. For example, in this
context is relatively easy and clear to argue if relations among observables
hold in general or are valid within a specific family of theories when they
emerge on the surface.

##  5 Conclusions and perspectives

In the literature, Palatini $f({\cal R})$-theories have not been studied much
because of a number of shortcomings they are believed to have; see, for
example, [8] and [25] where it was noticed that Palatini variation is not
generally equivalent to metric variation. Since we totally agree on this
point, we did not consider metric variations which are different theories. For
us Palatini theories are motivated by the EPS formalism and they are, in our
opinion, more natural than the metric ones.

In the second place, one can perform a conformal on-shell transformation to
show dynamical equivalence with purely metric Brans–Dicke theories (with a
suitable potential); see [12, 15]. Since Brans–Dicke models (without a
potential) are ruled out by solar system observations (e.g., precession of
Mercury’s perihelia) it has been claimed that then any Palatini $f({\cal
R})$-theory should be ruled out as well. This is false due to two issues.
First, one should consider the role of the potential. Second, and more
importantly, although field equations of the two theories are equivalent that
does not imply a complete physical equivalence. In particular, in the case of
precession of perihelia, the test on Brans–Dicke models is performed by
assuming that planets go along timelike geodesics of $g$ while in Palatini
$f({\cal R})$-theories they go along geodesics of $\tilde{g}$. One can show by
a detailed analysis that there is a huge family of Palatini $f({\cal
R})$-theories which in fact pass the test of perihelia; see [19].

Moreover, the conformal transformation needed to recast Palatini-$f({\cal R})$
theories into Brans–Dicke form is quite peculiar. One starts with a Palatini
theory in which conformal transformations act on the metric (which do not
touch the connection which is independent) and they are symmetries. Then one
solves the field equation related to the connection learning that the
connection is in fact the Levi-Civita connection of a suitably defined
conformal metric $\tilde{g}$. One is left with the field equation depending on
the metric (and the conformal factor) alone since the connection has been
eliminated. At this point by performing the inverse conformal transformation
on the metric (which now affects the connection as well) the theory is recast
into a Brans–Dicke form. There should be no surprise that the two theories are
inequivalent since the latter transformation is not a symmetry.

Another, issue is about star models. It was argued that assuming polytropic
equation of state (EoS) one cannot match the inner solution with the outer
solution for $g$. This is in fact true due to the fact that the conformal
factor is continuous but not ${\cal C}^{1}$ on the surface of the star. If one
matches the conformal metric $\tilde{g}$ (how is natural to do in view of the
interpretation above) this is possible and the discontinuity appears only in
the original metric $g$. Moreover, it has been noted that here the issue is
due to the assumption about the EoS which is not a fundamental principle.
Allowing a slight deformation of EoS one can in fact match the metrics at the
surface of the star. The same result can be obtained by changing the external
solution from a vacuum solution to an almost-vacuum one as it is reasonable
and physically sound; see [3, 20, 21].

We showed how to classify extended cosmological models with respect to their
predictions about the cosmic pie observations and to their predictions of the
evolution of the scale factor.

We showed that, the species revealed in the cosmic pie are essentially a
choice in cosmology, or, in other words, an input from other physics
(structure formation, baryogenesis, …). However, once this input is given,
extended gravity models produce quantitative predictions which allow in
principle to falsify or corroborate models.

The choice of effective species is not forced by mathematics essentially
because of the approximation method used. We chose to expand $\tilde{\rho}$ in
negative powers of $\tilde{a}$ around the value $\tilde{a}=1$. Even if we had
$\tilde{\rho}=3\tilde{a}^{-2}$ and decide we want to see only dust and
cosmological factor we would find

$\displaystyle\tilde{\rho}\sim 1+\frac{2}{{\tilde{a}}{}^{3}},$

which approximates the actual effective density around today $\tilde{a}=1$.

One clearly sees that if one has a direct observation of $a(t)$ in the past
and assumes the approximated value then one can predict at what time in the
past the deviation is first observed.

Let us stress that, although it may not be computationally simple, the
predictions of the models for the cosmic pie are extracted before solving the
model.

Figure 3: Approximation of a curvature density by cosmological constant and
dust (dash line is the approximated density).

Future investigations are to be devoted to models in which the conformal
factor stays positive everywhere, so that signature changes are prevented.

Another research perspective will be investigating how one can encode physical
properties (e.g., late time acceleration) as properties of the model surface
and viceversa reading physical properties of the theories from the geometry of
the surface. Currently, the main difficulty to be overcome is to control with
enough generality the inversion of the master equation which plays a prominent
role in expressing the embedding map (4.2). On the other hand, the model
surface may be a good tool exactly because it shows that the properties are
algebraically encoded in the model precisely through the master equation of
the theory.

##  Appendix A Conservation laws

In a $f({\cal R})$-theory, the effective stress tensor $\tilde{T}_{\mu\nu}$ is
defined by equation (2.2). Its conservation with respect to the conformal
metric $\tilde{g}$ reads as $\tilde{\nabla}_{\mu}\tilde{T}^{\mu\nu}=0$, where
indices have been raised by $\tilde{g}$. The relations between controvariant
stress tensors is then

$\displaystyle\varphi^{3}\tilde{T}^{\mu\nu}=T^{\mu\nu}-(\varphi{\cal
R}-f({\cal R}))g^{\mu\nu}.$

Let us stress that indices of $\tilde{T}^{\mu\nu}$ are here raised by
$\tilde{g}$, while indices of $T^{\mu\nu}$ are raised by $g$.

Let us introduce the tensor

$\displaystyle
K^{\alpha}_{\beta\mu}=\\{\tilde{g}\\}^{\alpha}_{\beta\mu}-\\{g\\}^{\alpha}_{\beta\mu}=-\hbox{$\frac{1}{2}$}\big{(}g^{\alpha\epsilon}g_{\beta\mu}-2\delta^{\alpha}_{(\beta}\delta^{\epsilon}_{\mu)}\big{)}{\buildrel\ast\over{\nabla}}_{\epsilon}\ln\varphi,$

which accounts for the difference between the connections of the metrics
$\tilde{g}$ and $g$. Hereafter the covariant derivatives with respect to $g$
will be denoted by $\nabla_{\epsilon}$ while the ones with respect to
$\tilde{g}$ will be denoted by $\tilde{\nabla}_{\epsilon}$. The symbol
${\buildrel\ast\over{\nabla}}_{\epsilon}$ means a covariant derivative when it
is independent of any connection (in this case being applied to the scalar
$\ln\varphi$).

One can easily show that

$\displaystyle
K^{\mu}_{\epsilon\mu}\tilde{T}^{\epsilon\nu}=2\tilde{T}^{\epsilon\nu}{\buildrel\ast\over{\nabla}}_{\epsilon}\ln\varphi$

and

$\displaystyle
K^{\nu}_{\epsilon\mu}\tilde{T}^{\epsilon\mu}=\tilde{T}^{\nu\epsilon}{\buildrel\ast\over{\nabla}}_{\epsilon}\ln\varphi-\hbox{$\frac{1}{2}$}g^{\nu\alpha}g_{\epsilon\mu}\tilde{T}^{\epsilon\mu}{\buildrel\ast\over{\nabla}}_{\epsilon}\ln\varphi.$

Then one has directly that

$\displaystyle\varphi^{3}\tilde{\nabla}_{\mu}\tilde{T}^{\mu\nu}=\varphi^{3}\big{(}\nabla_{\mu}\tilde{T}^{\mu\nu}+K^{\mu}_{\epsilon\mu}\tilde{T}^{\epsilon\nu}+K^{\nu}_{\epsilon\mu}\tilde{T}^{\epsilon\mu}\big{)}$
$\displaystyle\hphantom{\varphi^{3}\tilde{\nabla}_{\mu}\tilde{T}^{\mu\nu}}{}=\nabla_{\mu}T^{\mu\nu}-\hbox{$\frac{1}{2}$}g^{\nu\alpha}{\buildrel\ast\over{\nabla}}_{\alpha}\ln\varphi\big{(}T^{\mu\epsilon}-f({\cal
R})g^{\mu\epsilon}\\!-\hbox{$\frac{1}{2}$}Tg^{\mu\epsilon}\big{)}g_{\mu\epsilon}-g^{\nu\alpha}{\buildrel\ast\over{\nabla}}_{\alpha}f({\cal
R})-\hbox{$\frac{1}{2}$}g^{\nu\alpha}{\buildrel\ast\over{\nabla}}_{\alpha}T$
$\displaystyle\hphantom{\varphi^{3}\tilde{\nabla}_{\mu}\tilde{T}^{\mu\nu}}{}=\nabla_{\mu}T^{\mu\nu}+g^{\nu\alpha}\big{(}f^{\prime\prime}({\cal
R}){\cal R}{\buildrel\ast\over{\nabla}}_{\alpha}{\cal R}-f^{\prime}({\cal
R}){\buildrel\ast\over{\nabla}}_{\alpha}{\cal
R}-\hbox{$\frac{1}{2}$}{\buildrel\ast\over{\nabla}}_{\alpha}T\big{)}.$

However, by taking the covariant derivative of the master equation (2.1) one
has

$\displaystyle f^{\prime\prime}({\cal R}){\cal
R}{\buildrel\ast\over{\nabla}}_{\alpha}{\cal R}+f^{\prime}({\cal
R}){\buildrel\ast\over{\nabla}}_{\alpha}{\cal R}-2f^{\prime}({\cal
R}){\buildrel\ast\over{\nabla}}_{\alpha}{\cal
R}=\hbox{$\frac{1}{2}$}{\buildrel\ast\over{\nabla}}_{\alpha}T,$

so that one simply has

$\displaystyle\varphi^{3}\tilde{\nabla}_{\mu}\tilde{T}^{\mu\nu}=\nabla_{\mu}T^{\mu\nu},$

so that $\tilde{T}^{\mu\nu}$ is conserved with respect to $\tilde{g}$ iff
$T^{\mu\nu}$ is conserved with respect to $g$.

##  Appendix B Model solution

Solving the FRW equations may be difficult though in a sense trivial since, in
the end, one has to solve the Weierstrass equations with

$\displaystyle\tilde{\Phi}(\tilde{a})=-k+\frac{1}{3}\tilde{\rho}(a(\tilde{a}))\tilde{a}^{2},$

i.e., computing the integral

$\displaystyle\tilde{t}=\tilde{t}_{0}+\int_{\tilde{a}_{0}}^{\tilde{a}}\frac{d\tilde{a}}{\sqrt{\tilde{\Phi}(\tilde{a})}}.$

Maple can make it (with some effort) numerically (which means one can obtain a
plot of $\tilde{a}=\tilde{a}(\tilde{t})$).

An easier way is to change variable back to $(t,a)$, solve the equation and
then compute the conformal factor, the new time and the conformal scale
factor. This can be done by considering that

$\displaystyle\frac{d\tilde{a}}{d\tilde{t}}=\frac{d\tilde{a}}{da}\frac{da}{dt}\frac{dt}{d\tilde{t}}\qquad\Rightarrow\qquad\left(\frac{da}{dt}\right)^{2}=\left({\frac{d\tilde{t}}{dt}}\right)^{2}\left({\frac{d\tilde{a}}{da}}\right)^{-2}\tilde{\Phi}(\tilde{a})=:\Phi(a).$

Since we know that $d\tilde{t}=\sqrt{\varphi}dt$ and
$\tilde{a}(a)=a\sqrt{\varphi}$ one has for $f({\cal R})={\cal
R}-\hbox{$\frac{\epsilon}{2}$}{\cal R}^{2}$

$\displaystyle\Phi(a):=\left(\frac{3-\epsilon\rho(a)}{3-\epsilon\rho_{0}}\right)\left({\frac{d\tilde{a}}{da}(a)}\right)^{-2}\tilde{\Phi}(\tilde{\rho}(a)).$

Once we have $\Phi(a)$ we obtain the time coordinate as

$\displaystyle t(a)=t_{0}+\int_{a_{0}}^{a}\frac{da}{\sqrt{\Phi(a)}}$

and the conformal time as

$\displaystyle\tilde{t}(a)=\tilde{t}_{0}+\int_{t_{0}}^{t}\sqrt{\varphi(a(t))}dt=\tilde{t}_{0}+\int_{a_{0}}^{a}\sqrt{\varphi(a)}\frac{da}{\sqrt{\Phi(a)}}.$

Finally the original density $\rho$ is given as $\rho(a)$ by EoS as well as
the conformal density and pressure $(\tilde{\rho}(a),\tilde{p}(a))$.

Figure 4: Weierstrass function $\Phi(a)$. (Dash line is standard GR.)

In our special case $f({\cal R})={\cal R}-\hbox{$\frac{\epsilon}{2}$}{\cal
R}^{2}$, setting $\epsilon=\hbox{$\frac{1}{2}$}$ and $\rho_{0}=1$ , one
obtains (see Fig. 4)

$\displaystyle\Phi(a)=\frac{1}{3a}\frac{(2a^{3}-1)(5a^{3}-1)}{(4a^{3}+1)^{2}}$

with two allowed regions, one being $a\in[0,5^{-1/3}]$ the other being $a\geq
2^{-1/3}$. The first region corresponds to an oscillating behaviour, hence to
a closed universe. The second region corresponds to an expanding universe. Let
us consider the second region.

Figure 5: a) Scale factor $a$ as a function of $t$ (dash line standard GR). b)
Energy density $\rho$ for visible matter as a function of $t$ (dash line
standard GR).

The original scale factor hence appears as a quite natural parameter to (more)
easily integrate equations for $\tilde{a}$.

In this way one is able to plot parametrically everything; see Fig. 5. Notice
how the energy density of the visible matter $\rho(t)$ stays bounded and the
scale factor $a(t)$ stays non-zero when standard GR has a singularity.

This means that the solution can be extended to negative $t$ and the big bang
singularity is avoided (as some model of quantum gravity predicts; see [6]).

### Acknowledgements

We are grateful to A. Borowiec for comments and discussion. We acknowledge the
contribution of INFN (Iniziativa Specifica QGSKY), the local research project
Metodi Geometrici in Fisica Matematica e Applicazioni (2015) of Dipartimento
di Matematica of University of Torino (Italy). This paper is also supported by
INdAM-GNFM.

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