Dark energy Dark energy fluctuations and fluctuations and

structure formationstructure formationRogério RosenfeldRogério Rosenfeld

Instituto de Física Instituto de Física Teórica/UNESPTeórica/UNESP

I Workshop "Challenges of New Physics in Space" Campos do Jordão, Brazil 26/04/2009

Standard Model of Standard Model of CosmologyCosmology

TGG 8+ homogeneity and isotropy+ homogeneity and isotropy

FLRW modelFLRW model

Summary of Summary of observationsobservationsConcordance model

7.0,3.0 M

Tegmark 07

What´s going on?What´s going on?

Cosmological constantCosmological constant

Dynamical dark energyDynamical dark energy

Modified gravityModified gravity

Inhomogeneities Inhomogeneities

(Hubble bubble)(Hubble bubble)

Standard Model of Standard Model of CosmologyCosmology

Evolution of small perturbations:Evolution of small perturbations:

TGG 8

It is not possible to fully describe the non-linear It is not possible to fully describe the non-linear regime in RG:regime in RG:large large numerical simulationsnumerical simulations are necessary are necessary(Millenium, MareNostrum, etc…)(Millenium, MareNostrum, etc…)

Semi-analytical methods to study structure formation (dark matter haloes) in the non-linear regime:

• Spherical collapse modelSpherical collapse model (Gunn&Gott (Gunn&Gott 1972)1972)

• Press-Schechter formalismPress-Schechter formalism (Press&Schechter (Press&Schechter 1974)1974)

Homogeneous Dark Energy Homogeneous Dark Energy

Let’s first considerLet’s first consider

amartine iberato and R. Rosenfed, JCAP 2006

Spherical collapse modelSpherical collapse model

Consider a spherical region of radius r(t) withdensity c(t) constant in space (“top-hat” profile) immersed in a homogeneous (FLRW) universe with density (t)

This region first expands with an expansion rate a bit smaller than the Hubble expansion. The density contrast increases and eventually this region detaches from the expansion of the universe and starts to contract (turn around).

Spherical collapse model

Spherical collapse model

Spherical collapse model

Spherical collapse model

Modelo de colapso esférico

Growth of perturbations Growth of perturbations in the in the spherical collapse modelspherical collapse model

c

013

4142

2

GH

Homogeneous dark energy affects onlyonly the expansion rate!

LinearLinear growth of perturbation growth of perturbation in the spherical collapse modelin the spherical collapse model

042 GH linearized equation (coincides with GR)

dark matter dominated universe

dark energy dominated universe

z

aa

1

1

0aa

Dark energy suppresses structure formation (Weinberg’s anthropic argument)

Parametrization of dark energyParametrization of dark energy

DEDE awp

equation of state

afDEM ea

H

aH 03020

2

1

1ln3a

xwxdaf

Completely characterizes homogeneous dark energy

Linear growth of dark matter perturbations inLinear growth of dark matter perturbations inthe presence of homogeneous dark energythe presence of homogeneous dark energy

CDMdark matter only

larger perturbations in DE models

Non-linearNon-linear growth of dark matter perturbations growth of dark matter perturbations in the presence of dark energyin the presence of dark energy

The overdense sphere shrinks and eventually collapes (perturbation diverges!) (we are not considering dissipative effects)

ExempleCDM with initial conditions chosen such as theperturbation diverges today.

non-linear evolution

linear evolution

Important quantityImportant quantitycc(z) is defined as the (z) is defined as the linearlylinearly extrapolated extrapolated perturbation such that the perturbation such that the non-linearnon-linear perturbation diverges at z. perturbation diverges at z.

c(zcol) depends on the cosmological model. Einstein-de Sitter:

68647.12

3

5

30

3/2

colc z

Press-Schechter formalismPress-Schechter formalism

Estimate the number density of dark matter haloes with mass M at a redshift z.

P&S hypothesis: fluctuations of linear density contrast are gaussian. Structure form in regions where c .

Critical density is computed in the spherical approx.

P&S mass function can be derived rigorously usingexcursion set theory

This simple approximation captures main features of the cluster mass function

Number of dark matter haloesNumber of dark matter haloes

Inhomogeneous Dark EnergyInhomogeneous Dark Energy

If DE is not a cosmological constant, its density If DE is not a cosmological constant, its density can (and should) also vary! can (and should) also vary!

We now consider the consequences of the We now consider the consequences of the existence of existence of dark energy fluctuationsdark energy fluctuations

. R. Abramo, R. C. Batista,. iberato e R. Rosenfed, JCAP 0711:012 (2007)

Parametrization of dark energyParametrization of dark energy

DEDE awp

background equation of state

Characterizes completely the dark energy background

In order to characterize the pressure perturbations of dark energy in the context of a simplified assumption it is convenient to introduce:

DEeffDE acp 2

“effective” speed of sound (Hu 98)

Top-hat spherical collapse modelTop-hat spherical collapse model

DE

DEeff wcw

1

2

Equation of state inside perturbed region can be different from the background:Equation of state inside perturbed region can be different from the background:

wceff 2

Let’s first consider the case where there is no change in w:Let’s first consider the case where there is no change in w:

Non-linear equations for the evolution of perturbations in the 2 fluidsNon-linear equations for the evolution of perturbations in the 2 fluids(dark matter and dark energy):(dark matter and dark energy):

We showed that the same equations also arise from the so-called We showed that the same equations also arise from the so-called pseudo-newtonian formalism for general .pseudo-newtonian formalism for general .We also compared their linearized form with linearized GR recently:We also compared their linearized form with linearized GR recently: L.R. Abramo, R.C. Carlotto, L. Liberato and RRL.R. Abramo, R.C. Carlotto, L. Liberato and RR arXiv 0806.3461, Phys.Rev.D79:023516,2009arXiv 0806.3461, Phys.Rev.D79:023516,2009

Top-hat spherical collapse modelTop-hat spherical collapse model

2effc

Growth of perturbationsGrowth of perturbations

Non-phantom case: dark energy clusters and suppresses structure formation

Phantom case: dark energy becomes underdense and enhances structure formation

Growth of perturbationsGrowth of perturbations

Growth of perturbationsGrowth of perturbations non-linear regimenon-linear regime

phantom

non-phantom

Non-linear regimeNon-linear regimecc(z) including dark energy perturbations. (z) including dark energy perturbations. Large modifications in number counts.Large modifications in number counts.

phantom case: enhances structure formation phantom case: enhances structure formation

non-phantom case: suppresses structure formation non-phantom case: suppresses structure formation

Number of dark haloesNumber of dark haloes

Dark energy mutationDark energy mutation

DE

DEeff wcw

1

2

Equation of state inside perturbed region can be different from the background:

wceff 2

. R. Abramo, R. C. Batista,. iberato e R. Rosenfed, Phys.Rev.D77:067301,2008 

small effect for DE<<1

Dark energy mutationDark energy mutation

w can be large in the non-linear regime, DE~1 Effect was already seen in Mota & van de Bruck (2004) in the context of a scalar field

w = -0.8w = -0.8 w = -0.99w = -0.99

Conclusions and ChallengesConclusions and Challenges• Dark energy has a large impact on the structure formation in the universe (used in the 1987 Weinberg’s anthropic argument)

• Effects of homogeneous dark energy is completely characterized by its equation of state

• Effects of inhomogenous dark energy needs at least one extra function:

• Dark energy cumpling can alter its equation of state (mutation)

• It can be possible to distinguish among different dark energy models usingfuture cluster number counts data. Errors in parameter estimations using Fisher matrix and characteristics of a given experiment (SPT+DES, LSST, EUCLID, ...). See Abramo’s talk!

• It would be important to have a more precise study of a scalar field in GR with spherical symmetry (LTB) to confirm (or not) the approximations. see Ronaldo’s talk!

• N-body simulations including DE fluctuations?

2effc

Adiabatical perturbationsAdiabatical perturbations

For a two fluid combination, the perturbations are adiabatic when: For a two fluid combination, the perturbations are adiabatic when:

In the case of dark matter and dark energy:

SS

ppSp

,

2

2

1

1

11 ww

mDEDE w 1

M8 is the mass cointained in a sphere of radius R8 = 8 h-1 Mpc.

Parametrize

3/

88

M

MM Viana e Liddle (1996)