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 V

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)

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!

Back-up Slides

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)

More cosmological parameters

NJP, Class. Quantum Grav. 23, 4819 (2006); Phys. Lett. B 640, 135 (2006)

Deceleration parameter slope

0|dz

dq

Jerk parameter