Dark Energy in f(R) Gravity - Indiana University

Dark Energy in f(R) Gravity
Nikodem J. Popławski
Indiana University
16 th Midwest Relativity Meeting
Washington University in St. Louis,
St. Louis, MO
18 XI MMVI

Cosmic acceleration
NASA / WMAP
We are living in an accelerating universe!
References:
A. G. Riess et al., Astron. J. 116, 1009 (1998)
S. Perlmutter et al., Astrophys. J. 517, 565 (1999)
Cosmological constant
ΛCDM model
1
R
2
Rg
T g
Agrees with observations
52
2
10
m

Dark energy
Hypothetical form of energy
with strong negative pressure
NATURE OF DARK ENERGY
• homogeneous
• not very dense
• not known to interact
nongravitationally
EXPLANATIONS
• Cosmological constant
• Quintessence – dynamical field
• Alternative gravity theories
(talks of G. Mathews and G. J. Olmo)

Dark energy
Hypothetical form of energy
with strong negative pressure
NATURE OF DARK ENERGY
• homogeneous
• not very dense
• not known to interact
nongravitationally
EXPLANATIONS
• Cosmological constant
• Quintessence – dynamical field
• Alternative gravity theories
Dark Force =
– Dark Energy

Variable cosmological constant
Cosmological constant problem – why is it so small?
No known natural way to derive it from particle physics
Possible solution: dark energy decays
Cosmological constant is not constant (Bronstein, 1933)
Λ
energy
Dark energy interact with matter
Current interaction rate very small
matter
Phenomenological models of decaying Λ relate it to: t -2 , a -2 , H 2 , q, R etc.
(Berman, 1991; Ozer and Taha, 1986; Chen and Wu, 1990; Lima and Carvalho, 1994)
lack covariance and/or variational derivation

f(R) gravity
• Lagrangian – function of curvature scalar R
•R-1 or other negative powers of R → current acceleration
• Positive powers of R → inflation
Minimal coupling in Jordan (original) frame (JF)

f(R) gravity
• Lagrangian – function of curvature scalar R
•R-1 or other negative powers of R → current acceleration
• Positive powers of R → inflation
Minimal coupling in Jordan (original) frame (JF)
• Fully covariant theory based on the principle of least action
• f(R) usually polynomial in R
• Variable gravitational coupling and cosmological term
• Solar system and cosmological constraints
polynomial coefficients very small
G. J. Olmo, W. Komp, gr-qc/0403092

Variational principles I
• f(R) gravity field equations:
vary total action for both the field & matter
• Two approaches: metric and metric-affine

Variational principles I
• f(R) gravity field equations:
vary total action for both the field & matter
• Two approaches: metric and metric-affine
METRIC (Einstein–Hilbert) variational principle:
• action varied with respect to the metric
• affine connection given by Christoffel symbols (Levi-Civita connection)

Variational principles I
• f(R) gravity field equations:
vary total action for both the field & matter
• Two approaches: metric and metric-affine
METRIC (Einstein–Hilbert) variational principle:
• action varied with respect to the metric
• affine connection given by Christoffel symbols (Levi-Civita connection)
METRIC–AFFINE (Palatini) variational principle:
• action varied with respect to the metric and connection
• metric and connection are independent
• if f(R)=R metric and metric-affine give the same field equations:
variation with respect to connection connection = Christoffel symbols
E. Schrödinger, Space-time structure, Cambridge (1950)

Variational Principles: Metric
METRIC variational principle:
• connection: Christoffel symbols of metric tensor metric compatibility
• fourth-order differential field equations
• mathematically equivalent to Brans–Dicke (BD) gravity with ω=0
• 1/R gravity unstable – but instabilities disappear with additional positive
powers of R
• potential inconsistencies with cosmological evolution
• need to transform to the Einstein conformal frame to avoid violations of the
dominant energy condition (DEC) EF is physical

Variational Principles: Metric–Affine
METRIC–AFFINE variational principle:
• no a priori relation between metric and connection
• second-order differential equations of field
• mathematically equivalent to BD gravity with ω=−3/2
• field equations in vacuum reduce to GR with cosmological constant
• no instabilities
• no inconsistencies with cosmological evolution
• both the Jordan and Einstein frame obey DEC
Work presented here uses metric–affine formulation

Jordan frame
Assume action for matter is independent of connection (good for cosmology)
Variation of connection
connection = Christoffel symbols of
~
[ f '(
R)
g g
~
~
]
:
0
{} g

Jordan frame
Assume action for matter is independent of connection (good for cosmology)
Variation of connection
connection = Christoffel symbols of
Variation of metric
~
[ f '(
R)
g g
Dynamical energy-momentum (EM) tensor generated by metric:
Writing {} ~ ...
g
and
R
~
~
( ) R ( g)
~
]
0
{} g
allows interpretation of Θ as additional source and brings EOF into GR form
:

Helmholtz Lagrangian
The action in the Jordan frame is dynamically equivalent to the
Helmholtz action
provided
f " ( )
0
Scalar – tensor gravity (STG)
GR limit and Solar System constraints under debate
The scalar degree of freedom corresponding to nonlinear terms in the
Lagrangian is transformed into an auxiliary nondynamical scalar field p (or φ)
T. P. Sotiriou, Class. Quantum Grav. 23, 5117 (2006)
V. Faraoni, Phys. Rev. D 74, 023529 (2006)

Einstein frame
Conformal transformation of metric:
Effective potential
Non-minimal coupling in Einstein frame (EF)

Einstein frame
Conformal transformation of metric:
Effective potential
Non-minimal coupling in Einstein frame (EF)
• If minimal coupling in Einstein frame GR with cosmological constant
• Both JF and EF are equivalent in vacuum
• Coupling matter–gravity different in conformally related frames
• Principle of equivalence violated in EF → constraints on f(R) gravity
• Experiments should verify which frame (JF or EF) is physical
G. Magnano, L. M. Sokołowski, Phys. Rev. D 50, 5039 (1994)

Equations of field and motion
Variation of :
Variation of :
Structural equation
V

Equations of field and motion
Variation of :
Variation of :
Structural equation
• If T=0 (vacuum or radiation) algebraic equation for φ→ φ=const
GR with cosmological constant
• Gravitational coupling and cosmological term vary
• The energy-momentum tensor is not covariantly conserved
• If the EM tensor generated by the EF metric tensor is physical
constancy of V(φ) → GR with cosmological constant
V
NJP, Class. Quantum Grav. 23, 2011 (2006)

Dark energy–momentum tensor
• Non-conservation of EM tensors for matter and DE separately
• Total EM for matter + DE conserved interaction

Dark energy–momentum tensor
• Non-conservation of EM tensors for matter and DE separately
• Total EM for matter + DE conserved interaction
Assume homogeneous and isotropic universe
Continuity equation with interaction term Q:
Interaction rate Γ=Q/ε Λ
Nondimensional rate γ=Γ/H
NJP, Phys. Rev. D 74, 084032 (2006)

Cosmological parameters
Hubble parameter Deceleration parameter
Omega (L=f)
Redshift H(z)
Higher derivatives of scale factor (jerk and snap) more complicated
More nondimensional parameters:
deceleration-to-acceleration transition redshift z t, dq/dz| 0 etc.
NJP, Class. Quantum Grav. 23, 4819 (2006); Phys. Lett. B 640, 135 (2006)

Cosmological term
Palatini f(R) gravity in Einstein frame predicts (p=0)

Cosmological term
Palatini f(R) gravity in Einstein frame predicts (p=0)
Duh!
ΛCDM model says so
But:
ΛCDM – constant Λ relates H and q
f(R) gravity – variable Λ depends on H and q
• Resembles simple phenomenological models of variable cosmological
constant
• Unlike them, it arises from least-action-principle based theory
NJP, Phys. Rev. D 74, 084032 (2006)

R-1/R gravity
52
2
10 m
The simplest f(R) that produces current cosmic acceleration
Deceleration-to-acceleration transition:

R-1/R gravity
Simplest f(R) that produces current cosmic acceleration
Deceleration-to-acceleration transition:
Unification of inflation and current cosmic acceleration
T=0 2 de Sitter phases:
52
2
10 m
D. N. Vollick, Phys. Rev. D 68, 063510 (2003)
S. M. Carroll, V. Duvvuri, M. Trodden, M. S. Turner, Phys. Rev. D 70, 043528 (2004)
S. Nojiri, S. D. Odintsov, Phys. Rev. D 68, 123512 (2003); NJP, CQG 23, 2011 (2006)

R-1/R gravity
The simplest f(R) that produces current cosmic acceleration
Deceleration-to-acceleration transition:
Unification of inflation and current cosmic acceleration
T=0 2 de Sitter phases:
52
2
10 m
D. N. Vollick, Phys. Rev. D 68, 063510 (2003)
S. M. Carroll, V. Duvvuri, M. Trodden, M. S. Turner, Phys. Rev. D 70, 043528 (2004)
S. Nojiri, S. D. Odintsov, Phys. Rev. D 68, 123512 (2003); NJP, CQG 23, 2011 (2006)
β/α ~10 120 ?

Compatibility with observations I
Use
f(R) observations
A. G. Riess et al., Astrophys. J. 607, 665 (2004)
SNLS
X clusters
Gold
ΛCDM
Z t=-0.56+0.07-0.04
j=1

Compatibility with observations II
Use
f(R) observations
A. G. Riess et al., Astrophys. J. 607, 665 (2004)
SNLS
X clusters
Gold
ΛCDM
Z t=-0.56+0.07-0.04
j=1

Compatibility with observations III
Current interaction rate
At deceleration-to-acceleration transition
Interaction between matter and dark energy is weak
ε ~ a 3-n
Observations n

Conclusions
• f(R) gravity provides possible explanation for present cosmic
acceleration
• Dark energy interacts with matter in EF – decaying Λ
• R-1/R model is nice – simple, nondimensional cosmological
parameters do not depend on α
• We need stronger constraints from astronomical observations
FUTURE WORK
• Compare with JF
• Generalize to p≠0 (inflation and radiation epochs)
• Solar system constraints and Newtonian limit?
THANK YOU!

Back-up Slides

Conservation of matter
Bianchi identity
Homogeneous and isotropic universe with no pressure (comoving frame)
Time evolution of φ
NJP, Class. Quantum Grav. 23, 2011 (2006)

Dark energy density in f(R)
Matter energy density
Dark energy density
NJP, Phys. Rev. D 74, 084032 (2006)

More cosmological parameters
Deceleration parameter slope
dq
dz
| 0
Jerk parameter
NJP, Class. Quantum Grav. 23, 4819 (2006); Phys. Lett. B 640, 135 (2006)