Extended theories of gravity
and the acceleration of the Universe I
Francisco S. N. Lobo
Centro de Astronomia e Astrofísica
da Universidade de Lisboa
http://cosmo.fis.fc.ul.pt/~flobo/
Modified Gravity Theories:
Beyond Einstein's Legacy
16th September 2013
Lisbon
• On the 13th March 1998, “The Astronomical Journal”
received the paper “Observational Evidence from
Supernovae for an Accelerating Universe and a
Cosmological Constant”
• Finally published in the September issue, 1998
• On the 8th September 1998, “The Astrophysical Journal”
received a paper entitled “Measurements of Ω and Λ
from 42 High-Redshift Supernovae”
• Finally published on 1st June 1999
The 2011 Nobel Prize in Physics was
awarded to Adam Riess, Brian Schmidt,
and Saul Perlmutter
(for their discovery of the accelerating expansion of the universe)
The Supernova Cosmology Project & The High-z Supernova Search Team
The Nobel Prize was attributed to
the intriguing discovery that the
expansion of the universe was not
decelerating, but accelerating!!
Forced theorists and experimentalists to pose the question:
Is General Relativity (GR) the correct
relativistic theory of gravitation?
GR facing so many challenges indicative of a need for new
gravitational physics!
Outline of the lectures:
• First Lecture:
•
•
•
•
•
Cosmology in nutshell;
Alternative approaches to the ΛCDM model;
Basic criteria for the viability of a Gravitation Theory;
Foundations of Gravitation Theory;
Modified theories of gravity:
• f(R) gravity;
• (Modified) Gauss-Bonnet gravity;
• Dark gravity from braneworlds;
• Etc,etc;
• Revision of recent work.
• Second Lecture (18th Sep.):
• More detailed and technical analysis of recent work.
Introduction
• Indeed, a central theme in modern Cosmology is the perplexing fact
that the Universe is undergoing an accelerating expansion.
• This represents a new imbalance in the governing gravitational
equations.
Cause? Remains an open and tantalizing question
• Historically, physics has addressed such imbalances by either
identifying sources that were previously unaccounted for, or by
altering the governing equations.
• The standard model of cosmology has favored the first route to
addressing the imbalance by a missing stress-energy component.
• One may also explore the alternative viewpoint, namely, through a
modified gravity approach.
1. Cosmology in a nutshell
Assume the Universe to be spatially homogeneous and
isotropic, governed by the FLRW metric:
2

dr
2
2
2
2
2
2
2 
ds  dt  a (t ) 
 r d  sin  d 
2
1  kr


k = -1,0,+1 (open, flat,
or closed Universe)

1. Cosmology in a nutshell
Assume the Universe to be spatially homogeneous and
isotropic, governed by the FLRW metric:
2

dr
2
2
2
2
2
2
2 
ds  dt  a (t ) 
 r d  sin  d 
2
1  kr


k = -1,0,+1 (open, flat,
or closed Universe)
Matter content described
by a perfect fluid:
T     p U U   pg 
  energy density,
p  uniform pressure

Einstein field equation:
G
Friedmann equations:
8G
k
 a 
 2,
  
3
a
a

a
4G
  3 p 

a
3
Accelerated expansion:
Violates the SEC!!
2
1
 R  Rg   8GT
2
Ordinary matter obeys
Energy Conditions:
1. NEC :   p  0
2. WEC :   0,   p  0
3. SEC :   p  0,   3 p  0
4. DEC :   p
a  0
  3p  0
Introduce cosmological constant  vacuum energy:
G  g   8GT
( vac )
T


g  CDM model
8G
Problems with the
ΛCDM model




Simplest model
Compatible with all data so far
No other model is a better fit
However, theory cannot explain it!
Problems with the
ΛCDM model




Simplest model
Compatible with all data so far
No other model is a better fit
However, theory cannot explain it!

~ H 02 M p2 ~ (10 33 eV) 2 (10 28 eV) 2 ~ (10 3 eV) 4
8G
  theory  vacuum energy ~ (10 27 eV) 4    obs
  obs 


Why so small? (note a difference of 120-orders-ofmagnitude. EMBARRASSING!!!)
Why so fine-tuned?
2. Alternatives to ΛCDM?
or Dark
Gravity?
“Modified matter models” – Dynamical Dark Energy
in General Relativity (GR)
Gmn = 8p GTmn + 8p GTmndark
Tmndark = time-varying DE field
w=
pDE
rDE
1
<3
“Modified matter models” – Dynamical Dark Energy
in General Relativity (GR)
Gmn = 8p GTmn + 8p GTmndark
Tmndark = time-varying DE field
w=
pDE
rDE
1
<3
“Modified Gravity models” – Modify GR on
large scales
dark
G  G
 8GT
dark
G
 new scalar
to modify expansion
Modified (dark) gravity
Is GR wrong on large scales ?
Example from history:
Mercury perihelion
– Newton + ‘dark’ planet ?
no – modified gravity!
Today:
Modified Friedman equations
(schematic)
8G
K
H (1  Adark ) 
 2
3
a
1 
K

H (1  Adark )  HAdark  4G (   p )  2
2
a
2
  3H (   p)  0
Adark  modificati on to GR
General Relativity (GR):
Hilbert-Einstein action
S=


ò
é 1
ù
mn
d x -g ê
R + Lm (g , y )ú
ë 2k
û
4
GR is a classical theory, therefore no reference to
an action is required;
However, the Lagrangian formulation is elegant,
and has merits:
1.
2.
3.
Quantum level: the action acquires a physical meaning,
and a more fundamental theory of gravity will provide
an effective gravitational action at a suitable limit;
Easier to compare alternative gravitational theories
through their actions rather than by their field
equations (the latter are more complicated);
In many cases one has a better grasp of the physics as
described through the action, i.e., couplings, kinetic
and dynamical terms, etc
General Relativity (GR):
Hilbert-Einstein action
Consider a 4-dimensional manifold with a
connection,  , and a symmetric metric g  .

The affine connection is the Levi-Civita connection.
This requires that:
1.
2.



The metric to be covariantly conserved;
The connection to be symmetric.
No fields other than the metric mediate the
gravitational interaction;
Field equations should be 2nd order partial
differential equations;
Field equations should be covariant (or the action be
diffeomorphism invariant).
Experimental tests of GR




Accounts for the perihelion advance of Mercury;
Eddington’s measurement of light deflection, in 1919;
Pound and Rebka successfully measure the
gravitational redshift of light, in 1960;
Other experimental tests:






Lense-Thirring effect (an orbital perturbation due to the rotation
of a body);
De Sitter effect (a secular motion of the perigee and node of the
lunar orbit);
Nordtvedt effect (lunar motion; effect would be observed if the
gravitational self-energy of a body contributed to the gravitational
mass and not its inertial mass);
Shapiro time delay (radar signals passing near a massive
object take slightly longer to travel to a target);
Discovery of the binary pulsar, by Taylor and Hulse, leading to
the first indirect evidence of gravitational waves.
Etc, etc, etc (C.M. Will)
3. Foundations of Gravitation Theory

Schiff and Dicke: gravitational experiments do not
necessarily test GR, i.e., do not test the validity of
specific field equations, experiments test the validity
of principles;

Triggered the development of powerful tools for
distinguishing and testing theories, such as the
Parametrized Post-Newtonian (PPN) expansion
(pioneered by Nordvedt; extended by Nordvedt and Will)

Indeed, the idea that experiments test principles and
not specific theories, implies the need of exploring
the conceptual basis of a gravitational theory.
3. Foundations of Gravitation Theory

Dicke Framework. Probably the most unbiased
assumptions to start with, in developing a gravitation
theory:



Spacetime is a 4-dim manifold, with each point in the
manifold corresponding to a physical event (note that a
metric and affine connection is not necessary at this stage);
The equations of gravity and the mathematical entities in
them are to be expressed in a form that is independent of
the coordinates used, i.e., in a covariant form.
It is common to think of GR, or any other gravitation
theory, as a set of field equations (or an action).


However, a complete and coherent axiomatic formulation of
GR, or any other gravitation theory, is still lacking.
One needs to formulate “principles”; an important one is the
covariance principal, present in the Dicke Framework
Basic criteria for the viability
of a Gravitation Theory (following C.Will)

It must be complete: The theory should be able to
analyse from “first principles” the outcome of any
experiment.

It must be self-consistent: Predictions should be
unique and independent of the calculation method.

It must be relativistic: Should reduce to Special
Relativity when gravity is “turned off”.

It must have the correct Newtonian limit: In the limit of
weak gravitational fields and slow motion it should
reproduce Newton’s laws.

Equivalence Principle (current forms, C. Will):

Weak Equivalence Principle (WEP):
 If an uncharged test body is placed in an initial event in spacetime
and given an initial velocity there, then its subsequent trajectory will
be independent of its internal structure and composition.

Equivalence Principle (current forms, C. Will):

Weak Equivalence Principle (WEP):
 If an uncharged test body is placed in an initial event in spacetime
and given an initial velocity there, then its subsequent trajectory will
be independent of its internal structure and composition.

Einstein Equivalence Principle (EEP):
 (i) The WEP is valid;
 (ii) The outcome of any local non-gravitational test experiment is
independent of the velocity of the freely-falling apparatus (Local
Lorentz Invariance, LLI);
 (iii) The outcome of any local non-gravitational test experimental is
independent of where and when in the universe it is performed
(Local Position Invariance, LPI).

Equivalence Principle (current forms, C. Will):

Weak Equivalence Principle (WEP):
 If an uncharged test body is placed in an initial event in spacetime
and given an initial velocity there, then its subsequent trajectory will
be independent of its internal structure and composition.

Einstein Equivalence Principle (EEP):
 (i) The WEP is valid;
 (ii) The outcome of any local non-gravitational test experiment is
independent of the velocity of the freely-falling apparatus (Local
Lorentz Invariance, LLI);
 (iii) The outcome of any local non-gravitational test experimental is
independent of where and when in the universe it is performed
(Local Position Invariance, LPI).

Strong Equivalence Principle (SEP):
 (i) The WEP is valid for self-gravitating bodies, and for test bodies;
 (ii) The outcome of any local test experiment is independent of the
velocity of the freely-falling apparatus (LLI);
 (iii) The outcome of any local test is independent of where and when
in the universe it is performed (LPI).

Equivalence Principle (current forms, C. Will):

Einstein Equivalence Principle (EEP): The EEP is at the heart of
gravitation theory. Thus if the EEP is valid, then gravitation must be a
curved spacetime phenomenon, i.e., it must obey the postulates of
Metric Theories of Gravity:
 (i) Spacetime is endowed with a metric (second rank nondegenerate tensor);
 (ii) The world lines of test bodies are geodesics of that metric;
 (iii) In local freely falling frames, Lorentz frames, the nongravitational laws of physics are those of Special Relativity.

Equivalence Principle (current forms, C. Will):

Einstein Equivalence Principle (EEP): The EEP is at the heart of
gravitation theory. Thus if the EEP is valid, then gravitation must be a
curved spacetime phenomenon, i.e., it must obey the postulates of
Metric Theories of Gravity:
 (i) Spacetime is endowed with a metric (second rank nondegenerate tensor);
 (ii) The world lines of test bodies are geodesics of that metric;
 (iii) In local freely falling frames, Lorentz frames, the nongravitational laws of physics are those of Special Relativity.

Strong Equivalence Principle (SEP):



Essentially extends the validity of the WEP to self-gravitating bodies;
and the validity of the EEP to local gravitational experiments.
GR is the only theory that satisfies the SEP;
Can be shown that if the field equations for the metric contains
derivatives of the metric no higher than 2nd order, the outcome of
any experiment, gravitational or non-gravitational, is independent of
the surroundings, and therefore independent of the position and the
velocity of the frame.

Therefore, theories that include higher order derivatives of the meric, such as 4th
order gravity, do not satisfy the SEP (Sotoriou, 2007)
4. Extended theories of gravity
Higher order actions may include various curvature
invariants, such as:
R 2 , R R  , R R  , etc.
Consider f(R) gravity, for simplicity:
 1

S  d x  g
f ( R)  Lm ( g  , )
 2

4
Appealing feature: combines mathematical
simplicity and a fair amount of generality!
4.1. f(R) gravity:
The equivalence of theories
f(R) gravity
Action:
 1


S  d x  g
f ( R)  Lm ( g , )
 2

4
Gravitational field equations (vary action with g):
FR
1
df

(m)
 f g      F  g    F  T , F 
2
dR
f(R) gravity
Action:
 1


S  d x  g
f ( R)  Lm ( g , )
 2

4
Gravitational field equations (vary action with g):
FR
1
df

(m)
 f g      F  g    F  T , F 
2
dR
eff
Effective Einstein equation: G  T
with Teff  T(c )  T 
( m)
(m)
T   T( m ) / F , and T( c ) 
Conservation law:
1
F


1





F

g
RF



F


T


  

4
 T( c ) 
1 ( m) 
T  F
2 
F
Ricci scalar is a dynamical degree of freedom:
FR  2 f  3     F   T
Introduces a new light scalar degree of freedom.
Consider:
f ( R)  R 
4
R
,
 ~ H0
at low curvatures,
1/R dominates
This produces late-time self-acceleration
 But the light scalar strongly violates solar system
constraints.
 All f(R) models have this problem
 Way out: ‘chameleon’ mechanism, i.e. the scalar
becomes massive in the solar system
- very contrived!!
Late-time cosmic acceleration
f(R) gravity may lead to an effective dark energy.
Consider the FLRW metric, and a perfect fluid,
the generalised Friedmann equations:

 a  
 a 
   tot ,     tot  3 ptot 
3
3
a
a
2
tot     ( c ) ,
ptot  p  p( c )
Where the curvature terms are given by:
(c) 
p( c )

1 1
 a  


f
(
R
)

RF
(
R
)

3
R
F
'
(
R
)
 


F ( R)  2
a
 

1   a  
F ' ( R)  R 2 F ' ' ( R)  1  f ( R)  RF ( R)

2  RF ' ( R)  R
F ( R)   a 
2

Late-time acceleration:
tot  3 ptot  0
!!
Late-time cosmic acceleration
e.g. vacuum   p  0 , effective EOS
eff  p( c ) /  ( c )

A. Consider f ( R)  R n , generic power law a(t )  a0 t / t0 
Results:
eff
6n 2  7 n  1 ,
 2
6n  9n  3
 2n 2  3n  1

n2
Suitable choice of n leads to eff  1 / 3 and late-time
acceleration
B. Another example
eff
f ( R)  R   2( n1) / R n :
2n  2
 1 
32n  1n  1
4.2 Gauss-Bonnet gravity
o
Considering higher-order gravity, motivation
consistent with several “quantum gravity” candidates
o
String/M-theory predict unusual gravity-matter
couplings
o
Couple a scalar field with higher order invariants
o
String/M-theory predict scalar field couplings with the
Gauss-Bonnet invariant important in the appearance
of non-singular early time cosmologies
o
Apply these motivations to the late-time Universe!
4.2.1 Action of Gauss-Bonnet gravity:

 R

4

S  d x  g
      V ( )  f ( )G   S m
 2 2

  1
; phantom field
Gauss-Bonnet invariant:
G  R  4R R
Canonical field:
2
  1

 R R

4.2.1 Action of Gauss-Bonnet gravity:

 R

4

S  d x  g
      V ( )  f ( )G   S m
 2 2

  1
; phantom field
Gauss-Bonnet invariant:
G  R  4R R
Canonical field:
2
  1

 R R
FLRW metric, gravitational field equations:
 GB 
 GB 
pGB
3

2
H ,
pGB  

3H

1
2
 2 H

 2
  V ( )  24f ' ( ) H 3 ,
2
 2

   V ( )  8 H 2 f   16f ' ( ) H 3
2
t

Equation of motion for the scalar field:
   3H  V ' ( )  24 f ' ( ) H 2 H  H 2  0



Define an effective equation of state:

eff 
pGB
GB
2 H
 1 
3H 2
- Exponential scalar potential and scalar-GB coupling:
V ( )  V0 e 2 / 0 , f ( )  f 0 e 2 / 0
- Scale factor:
- Scalar field:
EOS:
eff  1 
h0

a
t
,
for h0  0
 0
a (t )  
h0

a
(
t

t
)
, for h0  0
 0 s
for h0  0
0 ln( t / t1 ),
 (t )  
h0

ln
[(
t

t
)
/
t
]
, for h0  0
s
1
 0
2
3h0
, if
h0 < 0, weff > -1 ,
h0 > 0, weff < -1
4.2.2 Modified Gauss-Bonnet gravity (interesting
alternative gravitational theory):
 R

4
S  d x  g
 f (G )
 2

Introduce two auxiliary scalar fields, A and B:
R

S   d 4 x  g   BG  A  f ( A)
 2

1. Vary with B  A=G
2. Vary with A  B=f´(G)
4.2.2 Modified Gauss-Bonnet gravity (interesting
alternative gravitational theory):
 R

4
S  d x  g
 f (G )
 2

Introduce two auxiliary scalar fields, A and B:
R

S   d 4 x  g   BG  A  f ( A)
 2

1. Vary with B  A=G
2. Vary with A  B=f´(G)
Taking into account the following definitions:
  A,
Final action:
and
V   Af ' ( A)  f ( A)
 R

S  d x  g
 V    f ( )G 
 2

4
Gauss-Bonnet action, without the kinetic term!
4.3. Dark gravity from braneworlds?
String theory - our 4D universe may be moving in 10D
spacetime
ST unifies the
4 interactions
DGP – the simplest example
4D brane universe in 5D bulk
Friedman on the
brane
H2 
H 8G


rc
3
late time :   0  H 
1
1
rc
early time : H  rc  H 2 
8G

3
early universe – recover GR dynamics
late universe – acceleration without DE
gravity “leaks” off the brane
therefore gravity on the brane weakens
passes the solar system test: DGP
GR
The background is very simple – like LCDM
… too good to be true
5D analysis of perturbations shows
- there is a ghost in the scalar sector of the
gravitational field
- ghost  negative sign in kinetic term (negative
energy densities)
The ghost makes the quantum vacuum unstable
This ghost is from 5D gravity
* It is not apparent in the background
… too good to be true
5D analysis of perturbations shows
- there is a ghost in the scalar sector of the
gravitational field
- ghost  negative sign in kinetic term (negative
energy densities)
The ghost makes the quantum vacuum unstable
This ghost is from 5D gravity
* It is not apparent in the background
Recent work: claimed that a higher codimension
generalization of the DGP model is free of ghost
instabilities
Recent work: DGP ruled out due to structure
formation(?!)
5. Some recent work
• C-theories: Unification of Einstein and Palatini gravities
(Amendola, Enqvist, Koivisto, PRD 2011).
• Horava gravity (Horava, PRD 2009): attempt of a quantum
field theory of gravity; breaks Lorentz invariance at ultrahigh energies.
• Entropic gravity (Verlinde, 2010): space is emergent
through a holographic scenario. Gravity is explained as an
entropic force caused by changes in the information
associated with the positions of material bodies.
• Gravity's rainbow: distortion of the spacetime metric at
energies comparable to the Planck energy (see Remo
Garattini’s talk)
5. Some recent work
Horava gravity:
1. Recent (Jan 2009) idea in theoretical physics for trying to
develop a quantum field theory of gravity;
2. It is not a string theory, nor loop quantum gravity, but is
instead a quantum field theory that breaks Lorentz
invariance at ultra-high (trans-Planckian) energies, while
retaining Lorentz invariance at low and medium energies.
3. Interesting applications:
• Passes the solar system tests;
• Predicts a bouncing universe scenario, rather than a
Big Bang, etc, etc.
4. Presence of a spin-0 scalar graviton is potentially
dangerous. Ghosts and instabilities!!
5. Still relatively active field, although the initial frenzy has
subsided.
5. Some recent work
•
f ( R, Lm ) theory: Generalization of all previous f(R) gravitational
models.
(Harko, FL, EPJC 2010);
(Harko, FL, IJMPD 2012; Honorable Mention in the Gravity Research
Foundation Essay Contest 2012, IJMPD 2012)
• Specific application: f (R,T ) gravity
(Harko, FL, Odintsov, Nojiri, PRD 2011).
mn
f
(R,T,
R
T
) gravity
mn
• Generalization:
(Haghani, Harko, FL, Sepangi, Shahidi, PRD 2013)
• Hybrid metric-Palatini theory
(Harko, Koivisto, FL, Olmo, PRD 2012).
(Capozziello, Harko, FL, Olmo, arXiv:1305.3756; Honorable Mention in
the Gravity Research Foundation Essay Contest 2013, IJMPD 2013).
Hybrid metric-Palatini gravity
(Harko, Koivisto, FL, Olmo, PRD 2012)

Interesting features:



Dark matter problem in hybrid metric-Palatini gravity:



Virial theorem (Capozziello, Harko, Koivisto, FL, Olmo,
JCAP,2013)
Galactic rotation curves (Capozziello, Harko, Koivisto, FL, Olmo,
arXiv:1307.0752 [gr-qc])
Cosmological applications:


Predicts the existence of a long-range scalar field, that explains the
late-time cosmic acceleration;
Passes the local tests, even in the presence of a light scalar field.
See (Capozziello, Harko, Koivisto, FL, Olmo, JCAP 2013)
Wormhole geometries:

See (Capozziello, Harko, Koivisto, FL, Olmo, PRD 2012)
Conclusions


Observations imply a late-time cosmic acceleration. But, theory cannot
satisfactorily explain it;
Generalizations of the Einstein-Hilbert (EH) action not such a
straightforward procedure:
- Two distinct classes, metric variational principle and Palatini
approach (Both approaches lead to GR for the EH action);
- Metric-affine theories of gravity: independent connection
coupled to matter. Several aspects still completely obscure,
such as: exact solutions, post-Newtonian expansions and Solar
System tests, cosmological phenomenology, structure structure,
application to particle physics, etc.
Conclusions






Observations imply a late-time cosmic acceleration. But, theory cannot
satisfactorily explain it;
Generalizations of the Einstein-Hilbert (EH) action not such a
straightforward procedure:
- Two distinct classes, metric variational principle and Palatini
approach (Both approaches lead to GR for the EH action);
- Metric-affine theories of gravity: independent connection
coupled to matter. Several aspects still completely obscure,
such as: exact solutions, post-Newtonian expansions and Solar
System tests, cosmological phenomenology, structure structure,
application to particle physics, etc.
Confrontation with cosmological, astrophysical and Solar System
constraints clarified the difficulty of constructing simple viable models in
modified gravity:
- Viable models need to account for all the cosmological epochs;
- Solar system tests: problematic issue of “chameleon” mechanism!
Even if the theory is tailored to fit cosmological observations and pass
local tests, problems related with stability arise;
Consider “first principles” in order to construct gravitation theories.
Theorists need to keep exploring: better models, better observational tests
Thank you!