Dark energy fluctuations
and structure formation
Rogério Rosenfeld
Instituto de Física Teórica/UNESP
I Workshop "Challenges of New Physics in Space"
Campos do Jordão, Brazil 26/04/2009
Standard Model of Cosmology
G  8GT
+ homogeneity and isotropy
FLRW model
Summary of observations
Tegmark 07
Concordance model
M  0.3,   0.7
What´s going on?
Cosmological constant
Dynamical dark energy
Modified gravity
Inhomogeneities
(Hubble bubble)
Standard Model of Cosmology
Evolution of small perturbations:
G  8GT
It is not possible to fully describe the non-linear
regime in RG:
large numerical simulations are necessary
(Millenium, MareNostrum, etc…)
Semi-analytical methods to study structure
formation (dark matter haloes) in the
non-linear regime:
• Spherical collapse model
(Gunn&Gott 1972)
• Press-Schechter formalism
(Press&Schechter 1974)
Let’s first consider
Homogeneous Dark Energy
amartine iberato and R. Rosenfeld,
JCAP 2006
Spherical collapse model
Consider a spherical region of radius r(t) with
density rc(t) constant in space (“top-hat” profile)
immersed in a homogeneous (FLRW) universe
with density r(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  in the
spherical collapse model
r c  r r


r
r
2

4 



  2H  4Gr  1    
0
3 1   
Homogeneous dark energy affects only the expansion rate!
Linear growth of perturbation 
in the spherical collapse model
linearized equation (coincides with GR)
  2H  4Gr   0
1
 a   a 
1 z
dark matter dominated universe
 a   a 0
dark energy dominated universe
Dark energy suppresses structure formation
(Weinberg’s anthropic argument)
Parametrization of dark energy
pDE  wa r DE
equation of state
Completely characterizes homogeneous dark energy
H a 
0  3
0  f  a 
  M a   DE e
2
H0
2
1
f a   3 d ln x 1  wx 
a
Linear growth of dark matter perturbations in
the presence of homogeneous dark energy
dark matter only
CDM
larger perturbations
in DE models
Non-linear growth of dark matter perturbations
in the presence of dark energy
The overdense sphere shrinks and eventually collapes (perturbation diverges!)
(we are not considering dissipative effects)
Exemple: CDM with initial conditions chosen such as the
perturbation diverges today.
non-linear evolution
linear evolution
Important quantity: c(z) is defined as the linearly extrapolated
perturbation such that the non-linear perturbation diverges at z.
c(zcol) depends on the cosmological model. Einstein-de Sitter:
 c zcol
3  3 
 0   
5 2 
2/3
 1.68647
Press-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 using
excursion set theory
This simple approximation captures main features of the
cluster mass function
Number of dark matter haloes
If DE is not a cosmological constant, its density
can (and should) also vary!
We now consider the consequences of the
existence of dark energy fluctuations
Inhomogeneous Dark Energy
. R. Abramo, R. C. Batista, . iberato e R. Rosenfeld,
JCAP 0711:012 (2007)
Parametrization of dark energy
pDE  wa r DE
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:
pDE  c
2
eff
a rDE
“effective” speed of sound (Hu 98)
Top-hat spherical collapse model
Equation of state inside perturbed region can be different from the background:
 DE
w  c  w
1   DE
2
eff
Let’s first consider the case where there is no change in w:
2
eff
c
w
Top-hat spherical collapse model
Non-linear equations for the evolution of perturbations in the 2 fluids
(dark matter and dark energy):
We showed that the same equations also arise from the so-called
2
pseudo-newtonian formalism for general ceff .
We also compared their linearized form with linearized GR recently:
L.R. Abramo, R.C. Carlotto, L. Liberato and RR
arXiv 0806.3461, Phys.Rev.D79:023516,2009
Growth of perturbations
Non-phantom case:
dark energy clusters and suppresses structure formation
Growth of perturbations
Phantom case:
dark energy becomes underdense and enhances structure formation
Growth of perturbations
non-linear regime
non-phantom
phantom
Non-linear regime
c(z) including dark energy perturbations.
Large modifications in number counts.
non-phantom case: suppresses structure formation
phantom case: enhances structure formation
Number of dark haloes
Dark energy mutation
. R. Abramo, R. C. Batista, . iberato e R. Rosenfeld,
Phys.Rev.D77:067301,2008
2
eff
c
w
Equation of state inside perturbed region can be different from the background:
 DE
w  c  w
1   DE
2
eff
small effect for DE<<1
Dark 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 = -0.8
w = -0.99
Conclusions 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:
2
ceff
• Dark energy cumpling can alter its equation of state (mutation)
• It can be possible to distinguish among different dark energy models using
future 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?
Adiabatical perturbations
 p 
 p 
 pr , S    r   S
 S 
 r 
For a two fluid combination, the perturbations are adiabatic when:
1
1  w1

2
1  w2
In the case of dark matter and dark energy:
 DE  1  wDE  m
Parametrize
M 

 M    8 
 M8 
 / 3
Viana e Liddle (1996)
M8 is the mass cointained in a sphere of radius
R8 = 8 h-1 Mpc.