Foundations of (modified) gravitation theory and the late ...non-gravitational laws of physics are those of Special Relativity. Francisco S.N. Lobo Centre of Astronomy and Astrophysics

Introduction Foundations of gravitation Curvature-matter coupling Hybrid metric-Palatini gravity Conclusions

Foundations of (modified) gravitation theory andthe late-time cosmic acceleration

Francisco S.N. LoboCentre of Astronomy and Astrophysics of the University of

Lisbon

Dark Side of the Universe 2013IX International Workshop,

17th October 2013, SISSA (Trieste, Italy)

Francisco S.N. Lobo Centre of Astronomy and Astrophysics of the University of Lisbon

Foundations of (modified) gravitation theory and the late-time cosmic acceleration


Introduction

Introduction

A central theme in modern Cosmology is the perplexing factthat the Universe is undergoing an accelerating expansion.This represents a new imbalance in the governing gravitationalequations.Cause? Remains an open and tantalizing question.Historically, physics has addressed such imbalances by eitheridentifying sources that were previously unaccounted for, or byaltering the governing equations.The standard model of cosmology has favored the first routeto addressing the imbalance by a missing stress-energycomponent.One may also explore the alternative viewpoint, namely,through a modified gravity approach.




Motivations

Motivations

A promising way to explain the major problems facing GeneralRelativity (GR) is to assume that at large scales GR breaksdown, and a more general action describes the gravitationalfield.

Generalizations of the Einstein-Hilbert Lagrangian have beenproposed, involving second order curvature invariants such asR2, RµνR

µν , RαβµνRαβµν , εαβµνRαβγδR

γδµν , CαβµνC

αβµν , etc.

Physical motivations for these modifications of gravity:possibility of a more realistic representation of thegravitational fields near curvature singularities and to createsome first order approximation for the quantum theory ofgravitational fields.




Lagrangian formulation

General Relativity (GR): Hilbert-Einstein action

GR is a classical theory, therefore no reference to an action isrequired. Consider the Hilbert-Einstein action:

S =

∫d4x√−g[

R

2κ2+ Lm(gµν , ψ)

]. (1)

But, the Lagrangian formulation is elegant, and has merits:

Quantum level: the action acquires a physical meaning, and amore fundamental theory of gravity will provide an effectivegravitational action at a suitable limit;Easier to compare alternative gravitational theories throughtheir actions rather than by their field equations;In many cases one has a better grasp of the physics asdescribed through the action, i.e., couplings, kinetic anddynamical terms, etc




Lagrangian formulation

Consider a 4-dimensional manifold with a connection, Γαµν ,and a symmetric metric gµν .

The affine connection is the Levi-Civita connection.This requires that:

The metric to be covariantly conserved;The connection to be symmetric.No fields other than the metric mediate the gravitationalinteraction;Field equations should be 2nd order partial differentialequations;Field equations should be covariant (or the action bediffeomorphism invariant).




Foundations of Gravitation Theory

Experimental tests of GR

Accounts for the perihelion advance of Mercury;Eddington’s measurement of light deflection, in 1919;Pound and Rebka measure the gravitational redshift (in 1960);Other experimental tests:

Lense-Thirring effect (an orbital perturbation due to therotation of a body);De Sitter effect (secular motion of the perigee of lunar orbit);Nordtvedt effect (lunar motion; effect would be observed if thegravitational self-energy of a body contributed to thegravitational mass and not its inertial mass);Shapiro time delay (radar signals passing near a massive objecttake slightly longer to travel to a target);Discovery of the binary pulsar (Taylor and Hulse) leading to thefirst indirect evidence of gravitational waves. Etc, (C.M. Will).






Schiff and Dicke: gravitational experiments do not necessarilytest GR, i.e., do not test the validity of specific fieldequations, experiments test the validity of principles;

Triggered the development of powerful tools for distinguishingand testing theories, such as the ParametrizedPost-Newtonian (PPN) expansion (pioneered by Nordvedt;extended by Nordvedt and Will)

Indeed, the idea that experiments test principles and notspecific theories, implies the need of exploring the conceptualbasis of a gravitational theory.





Dicke framework

Dicke Framework. Probably the most unbiased assumptionsto start with, in developing a gravitation theory:

Spacetime is a 4-dim manifold, with each point in the manifoldcorresponding to a physical event (note that a metric andaffine connection is not necessary at this stage);The equations of gravity and the mathematical entities inthem are to be expressed in a form that is independent of thecoordinates 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) (Sotiriou, 2007).

However, a complete and coherent axiomatic formulation ofGR, or any other gravitation theory, is still lacking.One needs to formulate “principles”; an important one is thecovariance principal, present in the Dicke Framework





Basic criteria for the viability of a Gravitation Theory

It must be complete: The theory should be able to analysefrom “first principles” the outcome of any experiment.

It must be self-consistent: Predictions should be unique andindependent of the calculation method.

It must be relativistic: Should reduce to Special Relativitywhen gravity is “turned off”.

It must have the correct Newtonian limit: In the limit of weakgravitational fields and slow motion it should reproduceNewton’s laws.





Einstein Equivalence Principle (C. Will)

Einstein Equivalence Principle (EEP): The EEP is at the heartof gravitation theory.

Thus if the EEP is valid, then gravitation must be a curvedspacetime phenomenon, i.e., it must obey the postulates ofMetric Theories of Gravity:

Spacetime is endowed with a metric (second ranknon-degenerate tensor);The world lines of test bodies are geodesics of that metric;In local freely falling frames, Lorentz frames, thenon-gravitational laws of physics are those of Special Relativity.





Strong Equivalence Principle (C. Will)

Strong Equivalence Principle (SEP):

Essentially extends the validity of the WEP to self-gravitatingbodies; and the validity of the EEP to local gravitationalexperiments.GR is the only theory that satisfies the SEP;Can be shown that if the field equations for the metric containsderivatives of the metric no higher than 2nd order, theoutcome of any experiment, gravitational or non-gravitational,is independent of the surroundings, and therefore independentof the position and the velocity of the frame.

Therefore, theories that include higher order derivatives of themetric, such as 4th order gravity, do not not satisfy the SEP(Sotiriou 2007).




Nonminimal curvature-matter coupling


For instance, consider f (R) gravity, for simplicity.Appealing feature: combines mathematical simplicity and afair amount of generality!

Generalization of f (R) gravity: include a nonminimalcurvature-matter coupling

S =

∫ {1

2f1(R) + [1 + λf2(R)] Lm

}√−g d4x , (2)

fi (R) (with i = 1, 2) arbitrary functions of the Ricci scalar R(Bertolami, Boehmer, Harko, FL, PRD, 2007).

Generates an extra force: Connections with MOND and withthe Pioneer anomaly are further discussed.

Applications to dark energy, dark matter, etc.






Varying the action with respect to the metric gµν yields thefield equations, given by

F1(R)Rµν −1

2f1(R)gµν −∇µ∇ν F1(R) + gµν�F1(R)

= −2λF2(R)LmRµν + 2λ(∇µ∇ν − gµν�)LmF2(R)

+ [1 + λf2(R)]T (m)µν , (3)

where we have denoted Fi (R) = f ′i (R), and the primerepresents the derivative with respect to the scalar curvature.

The matter energy-momentum tensor is defined as

T (m)µν = − 2√

−gδ(√−g Lm)

δ(gµν). (4)





Now, taking into account the covariant derivative of the fieldequation, the Bianchi identities, ∇µGµν = 0, and the identity

(�∇ν −∇ν�)Fi = Rµν ∇µFi , (5)

one finally deduces the relationship

∇µT (m)µν =

λF2

1 + λf2

[gµνLm − T (m)

µν

]∇µR . (6)

Thus, the coupling between the matter and the higherderivative curvature terms describes an exchange of energyand momentum between both.Analogous couplings arise after a conformal transformation inscalar-tensor theories of gravity (and string theory).In the absence of the coupling, one verifies the conservation ofthe energy-momentum, which can also be verified from thediffeomorphism invariance of the matter part of the action.





In order to test the motion in the model, consider a perfectfluid.

Equation of motion for a fluid element

Duα

ds≡ duα

ds+ Γαµνu

µuν = f α , (7)

where f α is an extra-force given by

f α =1

ρ+ p

[λF2

1 + λf2(Lm − p)∇νR +∇νp

]hαν . (8)

hµλ = gµλ + uµuλ is the projection operator;the extra force f α is orthogonal to the four-velocity of theparticle, f αuα = 0.

Intriguing feature!! Consider Lm = p or Lm = −ρ.(Bertolami, FL, Paramos, PRD 2008)





Open questions

Way out?

Rule out these models due to the Equivalence Principle?However, it has been recently reported, from data of the AbellCluster A586, that interaction of dark matter and dark energydoes imply the violation of the equivalence principle.Notice that the violation of the equivalence principle is alsofound as a low-energy feature of some compactified version ofhigher-dimensional theories.

Consider anisotropic pressures?But, what is the correct Lagrangian for this?




f (R, Lm) gravity

Action field equations of f (R , Lm) gravity

Action of f (R, Lm) gravity (Harko, FL, EPJC 2010):

S =

∫f (R, Lm)

√−g d4x . (9)

Gravitational field equation is given by:

fR (R, Lm)Rµν + (gµν∇µ∇µ −∇µ∇ν) fR (R, Lm)

−12 [f (R, Lm)− fLm (R, Lm) Lm] gµν = 1

2 fLm (R, Lm)Tµν .(10)

Hilbert-Einstein Lagrangian: f (R, Lm) = R/2κ2 + Lm,we recover the standard Einstein field equations.

For f (R, Lm) = f1(R) + [1 + λf2(R)] Lm, recover the fieldequations with an arbitrary curvature-matter coupling.




f (R, Lm) gravity

Covariant divergence of the energy-momentum tensor

These models possess extremely interesting properties. First,the covariant divergence of the energy-momentum tensor isnon-zero, and is given by

∇µTµν = 2∇µ ln [fLm (R, Lm)]∂Lm∂gµν

. (11)

The requirement of the conservation of the energy-momentumtensor of matter, ∇µTµν = 0, yields an effective functionalrelation between the matter Lagrangian density and thefunction fLm (R, Lm), given by

∇µ ln [fLm (R, Lm)] ∂Lm/∂gµν = 0 (12)




f (R,T ) gravity

Action of f (R ,T ) gravity

Action (Harko, FL, Odintsov, Nojiri, PRD 2011):

S =1

16π

∫f (R,T )

√−g d4x +

∫Lm√−g d4x . (13)

f (R,T ) is an arbitrary function of the Ricci scalar, R, and ofthe trace T of the energy-momentum tensor of the matter,Tµν . Lm is the matter Lagrangian density.

Note that the dependence from T may be induced by exoticimperfect fluids or quantum effects (conformal anomaly).

May be considered a relativistically covariant model ofinteracting dark energy.




f (R,T ) gravity

Further generalization: f (R ,T ,RµνTµν) gravity

“Further matters in space-time geometry: f (R,T ,RµνTµν)

gravity”(Haghani, Harko, FL, Sepangi, Shahidi, PRD 2013)

“f (R,T ,RµνTµν) gravity phenomenology and ΛCDM

universe”(Odintsov, Gomez, PLB 2013)




Action of field equations

Action of hybrid metric-Palatini gravity

Action (Harko, Koivisto, FL, Olmo, PRD 2011):

S =1

2κ2

∫d4x√−g [R + f (R)] + Sm , (14)

where Sm is the matter action, κ2 ≡ 8πG ,R is the Einstein-Hilbert term, R ≡ gµνRµν is the Palatinicurvature,and Rµν is defined in terms of an independent connection Γαµνas Rµν ≡ Γαµν,α − Γαµα,ν + ΓααλΓλµν − ΓαµλΓλαν .





May be expressed as the following scalar-tensor theory

S =

∫d4x√−g

2κ2

[(1 + φ)R +

3

2φ∂µφ∂

µφ− V (φ)

]+ Sm.

(15)This action differs from the w = −3/2 Brans-Dicke theory inthe coupling of the scalar to the curvature, which in thew = −3/2 theory is of the form φR.

This simple modification will have important physicalconsequences.





The gravitational field equation can finally be written as

(1 + φ)Rµν = κ2

(Tµν −

1

2gµνT

)+

1

2gµν (V + �φ)

+ ∇µ∇νφ−3

2φ∂µφ∂νφ . (16)

The scalar field is governed by the by the second-orderevolution equation (modified Klein-Gordon equation):

−�φ+1

2φ∂µφ∂

µφ+φ[2V − (1 + φ)Vφ]

3=φκ2

3T . (17)

Thus, unlike in the Palatini case, the scalar field is dynamicaland not affected by the microscopic instabilities found inPalatini models with infrared corrections.




Weak-field behaviour

Weak-field, slow-motion behaviour

The Solar System dynamics can be determined by studyingthe weak-field and slow-motion limit of field equations.

Consider an expansion of the metric and the scalar field abouta cosmological solution, which sets the asymptotic boundaryvalues, using a quasi-Minkowskian coordinate system, inwhich gµν ≈ ηµν + hµν , with |hµν | � 1.





Denoting the asymptotic value of φ as φ0 and the localperturbation as ϕ(x), to linear order Eq. (17) becomes

(~∇2 −m2ϕ)ϕ =

φ0κ2

3ρ , (18)

where m2ϕ ≡ (2V − Vφ − φ(1 + φ)Vφφ)/3|φ=φ0

, and we have

neglected the time derivatives of ϕ (slow-motion regime).

Imposing standard gauge conditions, the perturbationshµν = gµν − ηµν satisfy the following equation

−1

2~∇2hµν =

1

1 + φ0

(Tµν −

1

2Tηµν

)+V0 + ~∇2ϕ

2(1 + φ0)ηµν , (19)

where to this order T00 = ρ, Tij = 0, T = −ρ.





Far from the sources and assuming spherical symmetry, thesolution to Eqs. (18) and (19) lead to (M =

∫d3xρ(x))

ϕ(r) =2G

3

φ0M

re−mϕr (20)

h(2)00 (r) =

2GeffM

r+

V0

1 + φ0

r2

6(21)

h(2)ij (r) =

(2γGeffM

r− V0

1 + φ0

r2

6

)δij . (22)





We have defined the effective Newton constant Geff and thepost-Newtonian parameter γ as

Geff ≡ G

1 + φ0

[1− (φ0/3) e−mϕr

], (23)

γ ≡ 1 + (φ0/3) e−mϕr

1− (φ0/3) e−mϕr. (24)

As is clear from the above expressions, the coupling of thescalar field to the local system depends on the amplitude ofthe background value φ0.

If φ0 is small, then Geff ≈ G and γ ≈ 1 regardless of the valueof the effective mass m2

ϕ.





This contrasts with the result obtained in the metric versionof f (R) theories:

ϕ =2GM

3re−mf r , (25)

Geff ≡ G(1 + e−mf r/3

)/φ0 (26)

γ ≡(

1− e−mf r

3

)/(1 +

e−mf r

3

). (27)

which requires a large mass m2f ≡ (φVφφ − Vφ)/3 to make the

Yukawa-type corrections negligible in local experiments.




Late-time cosmic acceleration

Late-time cosmic speedup

Consider the FRW metric (k = 0): ds2 = −dt2 + a2(t)dx2.

Modified Friedmann equations:

3H2 =1

1 + φ

[κ2ρ+

V

2− 3φ

(H +

φ

4φ

)], (28)

2H =1

1 + φ

[−κ2(ρ+ P) + Hφ+

3

2

φ2

φ− φ

]. (29)

Scalar field equation:

φ+ 3Hφ− φ2

2φ+φ

3[2V − (1 +φ)Vφ] = −φκ

2

3(ρ− 3P) . (30)





Consistency at Solar System and cosmological scales

Consider for mathematical simplicity:

V (φ) = V0 + V1φ2 . (31)

Trace of the field eq. automatically implies R = −κ2T + 2V0.

As T → 0 with the cosmic expansion, naturally evolves into ade Sitter phase (V0 ∼ Λ) for consistency with observations.

If V1 is positive, the de Sitter regime represents the minimumof the potential.

The effective mass for local experiments,m2ϕ = 2(V0 − 2V1φ)/3, is positive if φ < V0/V1.

For V1 � V0, the amplitude small enough to pass SS tests.

The exact de Sitter solution is compatible with dynamics ofthe scalar field in this model.





Consistency at the galactic scales

“The virial theorem and the dark matter problem in hybridmetric-Palatini gravity,”(Capozziello, Harko, Koivisto, FL, Olmo, JCAP 2013)

“Galactic rotation curves in hybrid metric-Palatini gravity,”(Capozziello, Harko, Koivisto, FL, Olmo, Astroparticle Physics2013)




Some Conclusions

Some Conclusions

1 Observations imply acceleration.2 Theory cannot satisfactorily explain it.3 GR with dynamical dark energy: no natural model.4 Modifications to GR − dark gravity: theory gives no natural

model.5 Essential to find viable modified theories of gravity:

That explain the Solar System tests and the cosmologicalepochs.

6 Consider “first principles” in order to construct gravitationtheories.

7 Theorists need to keep exploring:* better models* better observational tests



Documents

Foundations of (modified) gravitation theory and the late ...non-gravitational laws of physics are those of Special Relativity. Francisco S.N. Lobo Centre of Astronomy and Astrophysics