Dark Energy and Dark Matter
as
Curvature Effects
Salvatore Capozziello
Università di Napoli Federico II
and INFN, Sez. di Napoli, Italy
Università di Camerino, May 11, 2011
Summary of the talk
Observational status of art
Ø 
Ø 
Why f(R)-gravity?
Theoretical Motivations
Ø 
Dark Energy as a Curvature Effect
Ø 
Dark Matter as a Curvature Effect
Ø 
The PPN limit, Solar System Experiments and Satellites Ø 
Conclusions and Perspectives
Ø 
v 
v 
The content of the universe is, up today, absolutely
unknown for its largest part. The situation is very DARK
while the observations are extremely good!
Precision Cosmology without theoretical foundations??!
: DE and DM come out from
the Observations!
Status of Art
95%!
Unknown!!
Main observational evidences for DE & DM
After 1998, more and more data have been obtained confirming these results
SNe Ia
LSS
CMB(WMAP)
The Observed Universe Evolution
- Universe evolution seems characterized by different phases of
expansion
Dark Matter
Ordinary
Matter
Radiation
Dark Energy
EINSTEN'S MODEL
The universe expands
more gradually, in
balance with gravity
As dark energy
weakens, gravity
causes the
universe
to collapse
Strengthening dark
energy speeds up
the universe,
causing it to
break apart
suddenly
A plethora of theoretical models!
DARK MATTER
DARK ENERGY
ü 
Neutrinos
Ø 
Cosmological constant
ü 
WIMPs
Ø 
Scalar field Quintessence
Ø 
Phantom fields
Ø 
String-Dilaton scalar field
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Wimpzillas, Axions, the
particle forest .....
ü 
MOND
Ø 
Braneworlds
ü 
MACHOS
Ø 
Unified theories
ü 
Black Holes
Ø 
........
ü 
......
“…there are the ones that invent OCCULT FLUIDS to
understand the Laws of Nature.
They will come to conclusions, but they now run out into
DREAMS and CHIMERAS
neglecting the true constitution of things…..
…however there are those that from the simplest
observation of Nature, they reproduce New Forces (i.e.
New Theories)… ”
From the Preface of PRINCIPIA (II Edition)
1687 by Isaac Newton, written by
Mr. Roger Cotes
there is a fundamental issue:
Are extragalactic observations and cosmology probing the
breakdown of General Relativity at large (IR) scales?
The problem could be reversed
We are able to observe only
baryons, radiation, neutrinos
and gravity
Dark Energy and Dark Matter
as “shortcomings” of GR.
Results of flawed physics?
The “correct” theory of gravity could
be derived by matching the largest
number of observations at ALL SCALES!
Accelerating behaviour (DE) and dynamical phenomena (DM)
as CURVATURE EFFECTS
f(R)-gravity
ü 
Generalization of the
Hilbert-Einstein action to
a generic (unknown) f(R)
theory of gravity
ü 
ü 
ü 
ü 
ü 
ü 
ü 
Theoretical motivations and features:
Quantization on curved space-time needs higher-order invariants
corrections to the Hilbert-Einstein Action.
These corrections are also predicted by several unification schemes as
String/M-theory, Kaluza-Klein, etc.
A generic action is
We can consider only fourth order terms f(R) which give the main
contributions at large scales.
DE under the standard of FOG inflationary cosmology (Starobinsky) but
at different scales and late times.
This scheme allows to obtain an Einstein two fluid model in which
one component has a geometric origin
Some results:
Ø 
Curvature Quintessence from f(R)-gravity [SC 2002]
Ø 
SNeIa Hubble Diagram in f(R)-gravity [SC, Cardone, Troisi 2003]
Ø 
Ø 
f(R)-gravity in Palatini approach vs. Dark Energy [Francaviglia et al. 2004]
PPN limit in f(R)-gravity in metric and Palatini approaches [SC, Stabile,
Troisi 2005, Olmo]
Ø 
Ø 
Lensing in f(R)-gravity [Mendoza, Rosas-Guevara 2007]
Unstable FRW matter-dominated solutions evolving into accelerated
solutions at late times [ S.C., Nojiri, Odintsov, Troisi 2006]
Ø 
The Chaucy Problem in f(R)-gravity [SC, Cianci,Vignolo 2009]
Ø 
Stochastic background of GW in f(R )-gravity [SC, De Laurentis 2007]
Ø 
Open problems in f(R) [Sotiriou, Faraoni 2008]
Ø 
LSS in f(R)-gravity [Hu, Sawicki 2007]
Ø 
Quantum Phenomena from ETG (entanglement) ? [Cafaro, SC, Mancini]
In particular, the f(R)-gravity:
Superstring
Theory
Higher Order Theories of Gravity
[
]
A = ∫ − g Λ + c0 R + c1R2 + c2 Rµν R µν + .... + Lmat d 4 x
Generalizations
of Einstein gravity
at higher dimensions
(Lovelock gravity)
Fourth Order Gravity
f(R)-gravity
[
A = ∫ d 4 x - g − χ Rn + Lmat
Renormalization of the
matter stress energy
tensor in QFT
]
Conservation Properties of Higher Order Theories
The Hamiltonian derivative of any
fundamental invariant is
divergence free!!
ρ curv + 3H ( ρcurv + pcurv ) = 0
Ø  A key point in the above
approach is related to the
conservation equations.
Eddington proved that every
higher order invariant
correction to the HilbertEinstein Lagrangian produces
terms that are divergence free:
the curvature fluid is
conserved on its own and the
matter follows the standard
conservation equations
f(R) – Gravity
Ideal fluid
description
Realistic cosmology
can be realized
via modified gravity
Scalar tensor
Gravity
The theoretical building:
Fourth order field equations
This scheme allows to obtain an
Einstein two fluid picture in which a
component has a geometric origin
(FRW)
Dark Energy as a curvature effect
Starting from the above considerations, it is possible
to write a curvature pressure and a curvature
energy density in the FRW metric (curvature EoS)
As a simple toy model, we can assume a power law
function for f(R) and for the scale factor a(t)
Matching with data
SNeIa data [SC, Cardone, Carloni, Troisi 2003,2004,2005,2006]
The test with the SNeIa by the distance modulus
Theoretical solutions have been compared to observations
The luminosity distance dL(z) is defined in relation to the cosmological model.
The analysis is performed minimizing:
f(R) solutions fitted against SNeIa
Very promising results in metric and Palatini approaches!
We used GOODS survey (S.C., Cardone, Troisi 2006)
R^n – Gravity as Dark Energy source (SNeIa fitted with n=3.42 (No
DM), n= 3.52 (with DM)) [SC, Cardone, Troisi 2006]
[dL – dL (DM)] vs z
11 Gyr
Hubble
diagram of 20
radio-galaxies
+ SNeIa Gold
sample
WMAP 1σ
region
16 Gyr
The Age test
The age of the universe can be theoretically calculated if one is capable of
measuring the Hubble parameter. We have:
For the experimental value, we have considered both data coming from
globular cluster observations and WMAP measurements of CMBR
(this last estimate is very precise 13.7 ± 0.02 Gyr) [SC, Cardone, Troisi 2005]
Result
A fourth order theory
is able to fit SNeIa
data and WMAP age prediction within these ranges
Is f (R) gravity (as the simplest extensions of Einstein Gravity) in good
agreement
with DARK ENERGY?
Inverse scattering procedure: observational H(z) gives f(R)
[SC, Cardone, Troisi 2005]
From field equations
Fourth order equation for a(t)
Considering the relation between R and H (FRW-metric) and changing time
variabile t with redshift z, we get a third order differential equation for H(z)
and, as a direct consequence, for f(z)
with Hi functions defined in term of R(H(z),z) and its z derivatives
Two level approach:
1) H(z) can be inferred by observations and phenomenological
considerations. 2) One can deal with the Hubble flow of a certain
cosmological model and deduce the corresponding f(R)-gravity
model capable of providing the same cosmic dynamics.
Q-essence,
Unified Models
Chaplygin,
Exponential Quintessence
In general, we can reconstruct curvature invariants from luminosity distance
H(z) is inferred from cosmological data. The goal is numerically
finding f(z) and then back tranforming this thanks to z=z(R) in f(R). A
similar approach can be pursued using also deceleration, snap and jerk
parameters (SC, Cardone, Salzano 2008).
LESSON:
In principle, any DE model can be reproduced by f(R)-gravity!
Can f(R)-theories reproduce also Dark
Matter dynamics?
Main research interests:
1)  Galactic dynamics (rotation curves of spiral galaxies)
2)  DM in the Ellipticals
3) Galaxy cluster dynamics
The problem: we search for f (R)-solutions capable of fitting consistently
the data. A nice feature could be that the same f(R) – model
works for Dark Energy (very large, unclustered scales) and Dark Matter
(small and clustered scales).
DM in the galaxies as a curvature effect
Theoretical issues
Spiral galaxies are ideal candidates to test DM models. In particular, LSB
galaxies are DM-dominated so that fitting their rotation curves
WITHOUT DM is a good test for any alternative gravity theory
A further test is trying to fit Milky Way rotation curve WITHOUT DM. In
this case,
the approach could be problematic
In general, is it possible to define an effective dark matter halo induced by
cuvature effects?
What about baryonic Tully-Fischer relation without DM?
DM as a curvature effect
Let us consider again:
Ø In the low energy limit, the spacetime metric can be written as
ds 2 = A(r )dt 2 − B(r )dr 2 − r 2 dΩ2
Ø A physically motivated hypothesis is A(r) = B-1 as potentials
Ø An exact solution is
2
with
Ø For n=1, β = 0 . The approach is consistent with GR!
SC,Cardone,Troisi MNRAS 2007
Ø The pointlike rotation curve is
2
Ø The galaxy can be modeled considering a thin disk and a bulge
component.
Ø The potential can be splitted in a Newtonian and a correction part
GM (r )
Φ(r ) = Φ N (r ) + Φ c (r ) =
+ Φ c (r )
r
Ø Integration can be performed considering a spherical symmetry for the
bulge and a cylindrical symmetry for the disk.
~
r
2π
+z
0
0
−z
Φ i (~
r , z) = ∫ ~
r ' d~
r ' ∫ dφ ∫ dz ' ρ (~
r ' , z ' ) r ηi
Ø The rotation curve is given by
vcirc
2
⎧− 1
ηi = ⎨
⎩β (n)
∂Φ i
~
~
(r ) = r ~
∂r
z =0
i=N
i=c
DM in LSB galaxies as a curvature effect: results of fits
on a sample of 15 galaxies [sample from de Blok, Bosma 2002]
Experimental points vs. Fourth Order Gravity induced
theoretical curves
Data:
● 
● 
fit
Rotation curves
Mass surface density
● 
Gas component
● 
Disk photometry
● 
● 
● 
M/L ratio
Results
ß =universal parameter
rc = characteristic
parameter related to the
total mass of every
galaxy
No DARK component has been used!
Best Fit
Constraints on log rc, fg (gas fraction), M/L obtained from the LSB galaxies with the corrected Newton potential.
The parameter Beta is fixed β =0.817 corresponding to n=3.5 which is the best fit value with respect to SNeIa.
Varying beta into the corresponding range 2.5 <n<4.5 (according to the SNeIa fit) log rc and fg do not change
within the errors.
Log rc vs log Md: Log rc=(0.55±0.04) x logMd+(-7.46 ± 0.38) ; σ=0.18
Further evidences by Frigerio Martins & Salucci (2007) which used the following sample:
At this point, it is worth wondering whether a link can be found
between f(R) Gravity and the standard approach, based on effective
dark matter haloes, since both approaches fit equally well the data
Considering
and
due to modified gravity
due to effective DM halo
being
Equating the two expressions, we get
where
and
this means that the mass profile of an effective spherically
symmetric DM halo, which provides corrected disk rotation curves,
can be reproduced. For
Burkert models are
exactly reproduced! [Burkert 1995]
Baryonic Tully-Fisher relation
From Virial theorem:
where
is the disk mass. For
empirical baryonic Tully-Fisher relation is reproduced!
[SC, Cardone, Troisi 2006, 2007]
Baryonic Tully-Fisher using HSB+LSB galaxies by Mc Gaugh. Dashed curve is our model, continuous is the
Mc Gaugh model.a=3.37± 0.32 b=2.77± 0.54 σ=0.26. Md is the disk mass, Vflat is the circular velocity in the
flat part of rotation curve. Theoretical values are obtained assuming the same best fit β while rc is given
according to the flat rotation curve.
Tully-Fisher using HSB+LSB galaxies by Mc Gaugh. Dashed curve is our model, continuous is the Mc Gaugh model.
a=2.96± 0.30 b=3.66±0.49 σ=0.29 (theoretical relation)
a=3.34 ± 0.17 b=2.81±0.37 σ=0.22 (observed relation)
Md is the disk mass, Vflat is the circular velocity in the flat part of rotation curve.
Theoretical values are obtained assuming the same best fit β while rc is given according to the flat rotation curve.
Clusters of galaxies
Point like potential:
• 
Gravity action :
• 
General requirement: Taylor expandable
Interaction length:
Lagrangian
Purpose: Fit clusters mass profiles
Effective actions from quantum field
theory on curved space-time
If:
If:
Our potential:
Build in a Self consistent theory
r >>L Gravitational coupling G
r<< L Gravitational coupling
Clusters of galaxies model
Cluster model: spherical mass distribution in hydrostatic equilibrium
- Boltzmann equation:
- Newton classical approach:
- f(R) approach:
- Rearranging Boltzmann equation:
Fitting Mass Profiles
DATA:
-  Sample: 12 clusters from Chandra (Vikhlinin 2005, 2006)
-  Temperature profile from spectroscopy
-  Gas density: modified beta-model
- Galaxy density:
Fitting Mass Profiles
METHOD:
- Minimization of chi-square:
- Markov Chain Monte Carlo:
Reject min < 1:
new point out of prior
new point with greater chi-square
Accept min = 1: new point in prior and less chi-square
Sample of accepted points
Sampling from underlying probability distribution
-  Power spectrum test convergence:
Discrete power spectrum from samples
Convergence = flat spectrum
Results: gravitational length
- Differences between theoretical and observed fit less than 5%
- Typical scale in [100; 150] kpc range where is a turning-point:
•  Break in the state of hydrostatic equilibrium
•  Limits in the expansion series of f(R):
in the range [19;200] kpc
Individual proper gravitational scale (the same for galaxies, SC et al 2007)
•  Similar issues in Metric-Skew-Tensor-Gravity (Brownstein, 2006): we are
better and more detailed
Results
Results
Results: expectations
- First derivative, a1 : very well constrained
It scales with the system size
a1 à3/4
- Newtonian limit:
Newtonian limit
Solar System
Galaxies
Cluster
s
Point like potential:
Clusters
Galaxies
Solar system
Newton
Results
Results
Results: expectations
- Gravitational length:
Strong characterization of
Gravitational potential
- Mean length:
-  Strongly related
to virial mass
(the same for gas mass):
- Strongly related
to average temperature:
Results: expectations
- Gravitational length:
Strong characterization of
Gravitational potential
- Mean length:
- Strongly related
to virial mass
(the same for gas mass):
-  Strongly related
to average temperature:
In summary: Dark Matter dynamics can be
reproduced by f(R)-gravity
Results:
1) Galactic dynamics (rotation curves of LSB spiral galaxies)
2) Milky Way rotation curve
3) Burkert haloes
4) Baryonic Tully-Fisher relation
5) Galaxy Clusters
Further issues: HSB galaxies, Elliptical and Dwarf galaxies.
A nice feature, as said, could be that the same f(R) – theory
works for Dark Energy (very large scales, unclustered) and
Dark Matter (small and medium scales, clustered structures)
Last issue: Parametrized Post-Newtonian limit
Considering the post-Newtonian approximation of the metric
One can attempt to evaluate deviations from Newtonian gravity (and
then GR) via astrophysical experiments or ground based tests
GR prescriptions: γ = 1 ; β = 1 !!!
Parametrized Post-Newtonian limit of fourth order gravity
inspired by scalar-tensor gravity
- Exploiting the fourth order gravity – scalar tensor gravity analogy,
the previous PPN formalism can be generalized to f(R) Lagrangians
- The PPN parameters are recovered through R dependent quantities
via the relations
R
(S.C., Troisi 2005)
- Considering the above definitions of PPN parameters as two
differential equations and recasting β in terms of γ, one obtains the
general solution
- which is a cubic function where deviations with respect to GR
emerge if γ is different from 1. NOTE: FOR γ=1, GR IS
RESTORED!
- If γ fulfils the condition |γ-1|<10^(-4) given by the experimental
bounds, deviations from GR are possible inside Solar System and
could be tested by ground based experiments!
(SC, Stabile, Troisi 2006)
In order to test the theory, we need extremely accurate measurements:
• GAIA, RAMOD, GREM, GAME to address this issue
•  PPN γ as a by-product of the global sphere reconstruction
•  GAIA could reach a sensitivity for the estimation of γ≈3 x 10-7
at the 3σ level
•  One can use up to ≈10 6 stars, V=15, well behaved
•  Monte Carlo simulations confirm these predictions
•  f(R)≠R models could be fully discriminated from GR by this accuracy!
Main science topics of Solar System Satellites
Measurement of Weak Field Gravity in the Solar System
Spacetime curvature induced by Sun: γ to a few 10^-8
> Classical test: deflection of starlight (Dyson et al. 1919)
Gravity superposition linearity:
β to a few 10^-6
> Classical test: precession of Mercury perihelion
Mass distribution of major planets (Jupiter, Saturn)
> Quadrupole moment to a few 10^-5 (Sun: ~2e-7)
Spacetime curvature around massive objects
Deflection angle [arcsec]
1.5
1".74 at Solar limb ≅ 8.4 µrad
GM
δψ = (1 + γ ) 2
c d
1
1 + cosψ
1 − cosψ
0.5
0
0
G: Newton’s gravitational
constant
d: distance Sun-observer
M: solar mass
c: speed of light
1
2
3
4
5
Distance from Sun centre [degs]
ψ: angular distance of the
source to the Sun
6
Light deflection ⇔ Apparent variation of star position, related
the gravitational field of the Sun
⇔ γ in Parametrised Post-Newtonian (PPN) formulation
to
Dyson-Eddington-Davidson experiment (1919) - I
Moon
Real (curved) photon
trajectories
Negative of Eddington's
photographs, presented in
1920 paper
First test of General Relativity by light deflection close to the Sun
Epoch (1): unperturbed direction of stars S1, S2 (dashed lines)
Epoch (2): apparent direction as seen by observer (dotted line)
Dyson-Eddington-Davidson experiment (1919) - II
Repeated throughout XX century
Precision achieved: ~10%
[A. Vecchiato et al., MGM 11 2006]
Limiting factors:
Authors
Dyson & al.
Dodwell & al.
Freundlich & al.
Mikhailov
van Biesbroeck
van Biesbroeck
Schmeidler
Schmeidler
TMET
Year
1920
1922
1929
1936
1947
1952
1959
1961
1973
Deflection ["]
1.98 ± 0.16
1.77 ± 0.40
2.24 ± 0.10
2.73 ± 0.31
2.01 ± 0.27
1.70 ± 0.10
2.17 ± 0.34
1.98 ± 0.46
1.66 ± 0.19
•  Need for natural eclipses
Short exposures, high background
•  Atmospheric turbulence
Large astrometric noise
•  Portable instruments
Limited resolution, collecting area
Current experimental results on γ…
Hipparcos
Different observing conditions: global astrometry,
estimate of full sky deflection on survey sample
Earth
Sun
Precision achieved: 3e-3
Cassini
Radio link delay timing, δν/ν~1e-14
(similarly for Viking, VLBI: Shapiro delay effect,
“temporal” component)
[B. Bertotti et al., Nature 2003]
Precision achieved: 2e-5
Cassini
Complementarity of GR tests and PPN parameters
GAME: measurement of
β from Mercury orbit +
high eccentricity
asteroids
2 + 2γ − β
Same instrument ⇒ crosscalibration
The GAME
concept
A space mission in the visible range to achieve
- long permanent artificial eclipses
- removal of atmospheric disturbances
Experimental approach:
Repeated observation of fields close to the Ecliptic
Measurement of angular separation of stars between fields
Track separation with epoch ⇔ distance to Sun
Implementation methods: Astrometry + Coronagraphy
Mission profile
Sun-synchronous orbit, 1500
km elevation ⇒ no eclipse
100% nominal
observing time
Stable solar power supply and
thermal environment ⇒
instrument stability
Soyuz Fregat launcher: high
transfer capacity to low orbit
The results of simulations considering 105 stars (Vecchiato et al 2003) and 106 stars can be compared. Left panel, old simulation,
Right panel, new simulation. In each panel, the first column gives the number of stars used in Monte Carlo, the second column is
the result for the σ of the simulation, the third column is the result of the fit considering only the data on the left panel, the fourth
column is the result considering all the simulations. The last row of the right panel gives the foreseen σ for one million stars.
Results of the Monte Carlo simulations using the GAIA best case. The Red Circles are the points by Vecchiato et al. 2003,
the Black Line is the fit of the original set of simulations, the Blue Diamonds are the results considering up to 105 stars.
SC, Crosta, Lattanzi, Vecchiato 2011 in preparation
Summarizing the results
Ø  Higher Order Curvature Theories give solutions consistent with
Dark Energy dynamics in metric and Palatini approaches [SC (2002),
SC, Cardone, Troisi (2003, 2004); SC, Cardone, Francaviglia (2004,2006)].
Ø Solutions agree with SNeIa e WMAP data (distance measurements)
[ SC, Cardone, Troisi (2003,2004), S.C., Cardone, Francaviglia (2006)].
Ø Lookback time methods [SC, Cardone, Funaro, Andreon (2004)]
Ø Cosmological constant and f(R) [SC, Garattini (2007)]
Ø Cosmological Noether solutions in f(R) [SC, De Felice (2007)]
Ø f(R) stability analysis [Carloni, SC, Dunsby, Troisi (2005)]
Ø Rotation curves of LSB galaxies [SC, Cardone, Troisi (2006,2007)]
Ø Galaxy cluster masses [SC, De Filippis, Salzano (2006)]
Ø Baryonic Tully-Fisher, Burkert haloes [SC, Cardone, Troisi (2006)]
Ø  Gravitational Lensing in f(R) gravity [SC, Cardone, Troisi (2006)]
Ø Room for f(R) in Solar System? [SC, Stabile, Troisi (2006)]
Ø Stochastich background of GW in f(R) [SC, De Laurentis,Francaviglia (2007)]
Conclusions (1)-DE
ü 
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ü 
ü 
Higher Order Gravity seems a viable approach to describe the Dark Side of
the Universe. It is based on a straightforward generalization of Einstein
Gravity and does not account for exotic fluids, fine-tuning problems or
unknown scalar fields (following Starobinsky, R can be considered a
“geometric” scalar field!).
Comfortable results are obtained by matching the theory with data (SNeIa,
Radio-galaxies, Age of the Universe, CMBR).
Transient dust-like Friedman solutions (LSS) evolving in de Sitter- like
expansion (DE) at late times are particularly interesting (debated issue).
Generic quintessential and DE models can be easily “mimicked” by f(R)
through an inverse scattering procedure where H(z) is phenomenologically
given by observations. Deatailed models can be achieved using also
deceleration, jerk and snap parameters.
Conformal transformations should be carefully considered (it seems that
physical solutions in Jordan Frame match the data).
A comprehensive cosmological model from early to late epochs should be
achieved by f(R). LSS issues have to be carefully addressed.I.N
PROGRESS!!!
Conclusions (2)-DM
Ø 
Ø 
Ø 
Ø 
Ø 
Rotation curves of galaxies can be naturally reproduced, without
huge amounts of DM, thanks to the corrections to the Newton
potential, which come out in the low energy limit.
The baryonic Tully- Fisher relation has a natural explanation in
the framework of f(R) theories.
Effective haloes of spiral galaxies are reproduced by the same
mechanism. Also in this case, no need of DM.
Good evidences also for galaxy clusters
The PPN limit of these models falls into the experimental bounds
of Solar System experiments. It seems that there is room for
alternative theories also at small scales. Final answer from VLBI
and GAIA, GAME?
Perspectives:
DE & DM as curvature effects
Ø  Matching other DE models
Ø Jordan Frame and Einstein Frame
Ø  Systematic studies of rotation curves for other galaxies
Ø  Galaxy cluster dynamics (virial theorem, SZE, etc.)
Ø Luminosity profiles of galaxies in f(R).
Ø Faber-Jackson & Tully-Fisher, Bullet Cluster
Weak Fields, GW,
Further results
Ø  Systematic studies of PPN formalism
Ø  Relativistic Experimental Tests in f(R)
Ø  Gravitational waves and lensing
Ø  Birkhoff ‘s Theorem in f (R)-gravity
Ø  f(R) with torsion
WORK in PROGRESS! (suggestions are welcome!)
A detailed summary of the results in the book…
•  Beyond Einstein Gravity
•  A Survey of Gravitational Theories
for Cosmology and Astrophysics
Fundamental Theories of Physics
SPRINGER Vol. 170
•  S. Capozziello & V. Faraoni
•  1st Edition., 2011, XIX, 448 p.,
Hardcover
•  ISBN: 978-94-007-0164-9
• 
November 2010