Álvaro de la Cruz-Dombriz Theoretical Physics Department Complutense University of Madrid in collaboration with Antonio L. Maroto & Antonio Dobado Different aspects of f(R) gravity theories

2 On this talk 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 in  CDM. - 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 independent Therefore, 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 a 0 (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=t 0, 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) DETERMINATION Rewrite equation [1] as a second order differential equation in R. Initial conditions: f(0) = 0 df(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 H 0 (t). Standard Friedmann equation should be recovered. Cosmological constant is negligible compared with dust. R by that time eV 2 implies eV 4 and eV 4 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 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: , gr-qc/

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 equationsand

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) ] k2k2

 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/ Starobinsky astro-ph/ Zhang astro-ph/

Uff… at last!!! Thank you !!! It was impressive !!