Álvaro de la Cruz-DombrizTheoretical Physics Department

Complutense University of Madrid

in collaboration with Antonio L. Maroto & Antonio Dobado

Different aspects

of f(R) gravity

theories

2On this talk...

1 . Outlook

2 . Tensorial field equations● - CDM vs. f(R) gravities.● - Approach, ansatz and solution.● - Tests for f(R) in radiation dominated universe.

3 . Scalar perturbations

- Usual analysis inCDM.

- Conventions and recent data

- General procedure for f(R) gravities.

- General expressions.

- Some models and solutions.

3

1. OUTLOOK One of the most important last years discoveries has been the accelerated expansion of the universe,

- usually explained through a cosmological constant . - more generically through a Dark Energy contribution.

Quintessence, braneworlds, Scalar-tensor theories.

Metric formalism vs. Palatini formalism.

but its nature

remains ignored

{g, } {g}

- Fourth order equations- Levi-Civita dependence

- Second order equations- g and are independentTherefore, within Metric formalism, the only

variation to perform in the ACTION will be with respect to the

metric.

CONVENTIONS AND RECENT DATA

• Metric signature: (+ , - , - , -)• Riemann tensor:

• Ricci Tensor and Scalar Curvature:

• Gravitational and Total Action:

• Energy-momentum Tensors:

X = Matter or Dark Energy.

• Recent cosmological data WMAP, March 2006 plus other experiments.

• Flat FLRW metric

• Cosmological constant .

• Einstein field equations in metric formalism.

• Perfect and barotropic cosmological fluid P=P() =

• Non relativistic matter =0, including dark matter.

k = 0 (Inflation)

2. TENSORIAL FIELD EQUATIONS

In CDM, of course !!

I said !

U.C.M.

√ The FIELD TENSORIAL EQUATIONS obtained from the variation of the action.

√ In metric formalism:

Motion equations

tt Field Tensorial Equation• CDM

• f(R) theories

Third order differential

equation in a(t)

[1]

9

…and our ansatz will be…

mmm…!

Find a function f(R) such that solution a(t) for Field Tensorial Equations will be exactly THE SAME as solution a0(t) obtained by using Standard Cosmology (CDM). In other words we want to find a f(R) such that:

for the same initial (or better present, i.e. t=t0, conditions).

If it were possible to find this function f(R) then it would be possible to avoid the necessity for introducing any cosmological constant just by modifying the gravitational sector of the action.

f(R) DETERMINATIONRewrite equation [1] as a second order differential equation in R.

Initial conditions:

f(0) = 0df(0)/dR

= 0

-No cosmological constant.

- Restore standard EH gravity at ordinary curvatures.

Particular solution: Homogeneus solution:

Hypergeometric functions

Initial conditions

Tests for f(R) in radiation universe

Cosmological Standard Model fits correctly primordial light elements abundances during BBN with a 10 % relative error for H0(t).

Standard Friedmann equation

should be recovered.

• Cosmological constant is negligible compared with dust.

• R by that time 10-39 eV2 implies 1016 eV4 and 1021 eV4 for dust and radiaton densities respectively.

• To reproduce the light elements abundances • This fact requires a strict fine tuning in the precision of

the c critical value. The found c value is about 0.065.

Is our f(R) function well behaved during that period, ie. M = 1/3, and in particular, during the Big Bang Nucleosynthesis (BBN)?

Conclusions• f(R) function which exactly reproduces the same evolution of

the Universe, from BBN to present times, as standard CDM cosmology.

• The gravitational lagrangian is analytical at the origin.

• R = 0 is a vacuum solution of the field equations, therefore Minkowski & Schwarzschild are also solutions for this f(R).

• Classical Actions are real but Effective Quantum Actions usually have a complex structure coming from loops and related to unitarity.

• Published in Phys.Rev.D74: 087501, 2006. gr-qc/0607118.

I’m tired…!!!

But I want more !!!

First order

perturbed

equations in dust matter

universe

CDM MODEL

• Pure Einstein Hilbert action

• CDM action

• SubHubble modes

Linder’s suggestion

= 6/11

= a

Tensorial equations

and

f(R) GRAVITIES

• Is still valid the process to reproduce an exact differential equation for decoupled from the rest of perturbed quantities?

• Is differential equation for second order?

• Does it present some singularity?

• If it does, does it depend on the model?

Background Tensorial Equations (combining density & pressure equations)

Perturbed Motion equations

Notation

[ (A) ]

k2

Perturbed Tensorial equations

[ (00) ]

[ (0i) ]

[ (ii) ]

[ (ij) ] = [ (R) ] NOT EQUAL TO

DIFFERENTIAL EQUATION [()]

• We separate coefficients in-EH part: From linear part in gravitational action (´s)

-f theory part: From non-linear part in gravitational action (´s).

involves terms with f´0p and f´´0p .

• iv and ´´´ coefficients DO NOT have EH part.

• If f theory part is removed, usual expression for CDM or EH is recovered.

Some coefficients

Coefficients for iv term

For SubHubble modes, an expansion in parameter can be performed.

Non-dimensional parameters

Coefficients for ´´´

term

Coefficients for ´´

term

Very rapid f(R) part

dominance !!

Coefficients for ´

term

Coefficients for term

• v

Conclusions

• Only

• A completely general equation for have been obtained decoupled from other perturbed quantities { v }.

• For CDM & EH theories, f(R) = - 2 or 0 respectively and coefficients ´s are the usual ones. In extreme SubHubble limit well-known equations are recovered.

is not always equal to .

and

are equal to .

• For a general f(R) theory neglecting iv & ´´´ terms and extreme SubHubble modes,

• Integrating [ () ] equation we get density contrast growth behavior for a particular f(R) theory, so comparing with experimental data such that

-Weak lensing & Type Ia Supernova data (SNAP experiment).

- CMB (Planck satellite).

some f(R) theories may be ruled out.

Tsujikawa astro-ph/07051032

Starobinsky astro-ph/07062041

Zhang astro-ph/0511218

Uff… at last!!!

Thank you !!!It was

impressive !!