Dark Energy in f(R) Gravity Nikodem J. Popławski Indiana University 16 th Midwest Relativity...
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Transcript of Dark Energy in f(R) Gravity Nikodem J. Popławski Indiana University 16 th Midwest Relativity...
Dark Energy in f(R) Gravity
Nikodem J. Popławski
Indiana University
16th Midwest Relativity Meeting18 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
Agrees with observations
gTRgR 2
1
25210 m
Dark energyHypothetical form of energy
with strong negative pressure
EXPLANATIONS• Cosmological constant• Quintessence – dynamical field• Alternative gravity theories (talks of G. Mathews and G. J. Olmo)
NATURE OF DARK ENERGY• homogeneous• not very dense• not known to interact nongravitationally
Dark energyDark Force =
–▼Dark EnergyHypothetical form of energywith strong negative pressure
EXPLANATIONS• Cosmological constant• Quintessence – dynamical field• Alternative gravity theories
NATURE OF DARK ENERGY• homogeneous• not very dense• not known to interact nongravitationally
Variable cosmological constantCosmological 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)
energyΛ matter
Dark energy interact with matter
Current interaction rate very small
Phenomenological models of decaying Λ relate it to: t-2, a-2, H2, 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
• 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
Minimal coupling in Jordan (original) frame (JF)
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)
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
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
Variational Principles: Metric–Affine
Work presented here uses metric–affine formulation
Jordan frame
0])('[~~~
ggRf
Variation of connection
Assume action for matter is independent of connection (good for cosmology)
connection = Christoffel symbols of g{}:
Jordan frame
0])('[~~~
ggRf
Variation of connection
Variation of metric
Dynamical energy-momentum (EM) tensor generated by metric:
Assume action for matter is independent of connection (good for cosmology)
connection = Christoffel symbols of g{}
Writing ...{}~ g
and )()(
~
gRR
allows interpretation of Θ as additional source and brings EOF into GR form
:
Helmholtz LagrangianThe action in the Jordan frame is dynamically equivalent to the Helmholtz action
The scalar degree of freedom corresponding to nonlinear terms in theLagrangian is transformed into an auxiliary nondynamical scalar field p (or φ)
0)(" fprovided
GR limit and Solar System constraints under debate
Scalar – tensor gravity (STG)
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
G. Magnano, L. M. Sokołowski, Phys. Rev. D 50, 5039 (1994)
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
Equations of field and motionVariation 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
NJP, Class. Quantum Grav. 23, 2011 (2006)
V
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
Continuity equation with interaction term Q:
Interaction rate Γ=Q/εΛ
Nondimensional rate γ=Γ/H
Assume homogeneous and isotropic universe
NJP, Phys. Rev. D 74, 084032 (2006)
Cosmological parameters
NJP, Class. Quantum Grav. 23, 4819 (2006); Phys. Lett. B 640, 135 (2006)
Hubble parameter Deceleration parameter
Higher derivatives of scale factor (jerk and snap) more complicated
More nondimensional parameters: deceleration-to-acceleration transition redshift zt, dq/dz|0 etc.
Redshift H(z)Omega (L=f)
Cosmological termPalatini f(R) gravity in Einstein frame predicts (p=0)
NJP, Phys. Rev. D 74, 084032 (2006)
• Resembles simple phenomenological models of variable cosmologicalconstant• Unlike them, it arises from least-action-principle based theory
Duh! ΛCDM model says so
But: ΛCDM – constant Λ relates H and qf(R) gravity – variable Λ depends on H and q
R-1/R gravity
The simplest f(R) that produces current cosmic acceleration
25210 m
Deceleration-to-acceleration transition:
R-1/R gravity
Unification of inflation and current cosmic acceleration
T=0 2 de Sitter phases:
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)
Simplest f(R) that produces current cosmic acceleration
25210 m
Deceleration-to-acceleration transition:
R-1/R gravity
Unification of inflation and current cosmic acceleration
T=0 2 de Sitter phases:
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)
The simplest f(R) that produces current cosmic acceleration
25210 m
Deceleration-to-acceleration transition:
β/α ~10120 ?
Compatibility with observations I
A. G. Riess et al., Astrophys. J. 607, 665 (2004)
f(R) observations
Use
SNLS
X clusters
Gold
ΛCDM
j=1
Zt=-0.56+0.07-0.04
Compatibility with observations II
A. G. Riess et al., Astrophys. J. 607, 665 (2004)
f(R) observations
Use
SNLS
X clusters
Gold
ΛCDM
j=1
Zt=-0.56+0.07-0.04
Compatibility with observations III
Current interaction rate
Interaction between matter and dark energy is weak
At deceleration-to-acceleration transition
P. Wang, X. H. Meng, CQG 22, 283 (2005)
ε ~ a3-n
f(R): n=0.04
Observations n<0.1
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!
Conservation of matterBianchi 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)