Backreaction from inhomogeneities
and late time cosmological evolution
Archan S. Majumdar
S. N. Bose National Centre for Basic Sciences
Kolkata
N. Bose & ASM, MNRAS 418, L45 (2011)
N. Bose & ASM, Gen. Rel. Grav. 45, 1971 (2013)
N. Bose & ASM, arXiv: 1307.5022 [astro-ph]
A. Ali & ASM, JCAP 01, 054 (2017)
Plan
• Late time (present era) cosmology plagued with dark
energy problem
• Backreaction formalism offers prospect of study
without invoking non-standard gravity
• Observational viability investigated in the context of
analogous scalar field cosmology
Structure of the universe at very large scales
• Homogeneous: no preferred location; every point
appears to be the same as any other point; matter is
uniformly distributed
• Isotropic: no preferred direction; no flow of matter
in any particular direction
(Cosmological Principle)
Standard cosmological model
Cosmological principle (Isotropy and homogeneity at large scales)
Friedmann-Lemaitre-Robertson-Walker metric
2

dr
2
2
2
2
2
2
2
2
d  dt  a (t ) 
 r d  r sin d 
2
1  kr

Energy-momentum (perfect fluid)
T  diag (  , p, p, p)
Dynamics (Friedmann equations)
k

 a  8G
2
H  2
 2
3
a
3
a 

a
4G
  3 p   

a
3
3
Observational support for the CDM model
(http:rpp.lbl.gov)
Motivations:
• Observations tell us that the present Universe is inhomogeneous
up to scales (< 100 h 1 Mpc)
[Features: Spatial volume is
dominated by voids; peculiar structures at very large scales]
• Cosmology is very well described by spatially homogeneous and isotropic
FLRW model
• Observational concordance comes with a price: more that 90% of the
energy budget of the present universe comes in forms that have never
been directly observed (DM & DE); DE not even theoretically understood
• Scope for alternative thinking without modifying GR or extending SM;
application of GR needs to be more precisely specified on large scales
• Backreaction from inhomogeneities could modify the evolution of the
Universe. Averaging over inhomogeneities to obtain global metric
Problem of course-graining or averaging
R
1
 Rg   g    T
2
G   T 
Einstein’s equations: nonlinear
Einstein tensor constructed from average metric tensor will not be
same in general as the average of the Einstein tensor of the actual
metrics
Determination of averaged cosmological metric
(hierarchical scales of course-graining)
R
( local)

1 (local) (local)
( local)
( local)
 R
g
  g
   T

2
Averaging process (local)
g 
( gal )
( lss)
(local)
( gal)
( lss)
 g 
 g 
, T
 T
 T
does not commute with evaluating inverse metric, etc..
( gal )
leading to, e.g., R
1 ( gal) ( gal)
( gal )
( gal )
( gal )
 R g   g 
  T
 E
2
( gal )
Extra term E
 0 in general
Consequence: Einstein eqs. valid at local scales may not be trivially
extrapolated at galactic scales
Different approaches of averaging
Macroscopic gravity: (Zalaletdinov , GRG ‘92;’93)
g

R
1  
  g
2
R  C   T
(additional mathematical structure for covariant averaging scheme)
Perturbative schemes: (Clarkson et al, RPP ‘11; Kolb, CQG ‘11)
g   g    g 
G   G   T
Spatial averages : (Buchert, GRG ‘00; ‘01)
Lightcone averages: (Gasperini et al., JCAP ’09;’11)
Bottom-up approach [discrete cosmological models]: (Tavakol , PRD’12; JCAP’13)
Perspective
• Different approaches of averaging; different interpretation of
observables
• Effect of backreaction (through spatial averaging of
inhomogeneities observed as structure in the present universe)
on future evolution of the presently accelerating universe
•
(Buchert & Co-workers)
The Buchert framework– main elements
For irrotational fluid (dust), spacetime foliated into constant time
hypersurfaces
inhomogeneous
For a spatial domain D , volume
| D |g 
 d
g
D
d g 
( 3)
g (t , X 1 , X 2 , X 3 )dX 1dX 2 dX 3
1
/3
the scale factor is defined as
|D
|g
a
t 
D()
|D| 

i
g


It encodes the average stretch of all directions of the domain.
Using the Einstein equations:

a
D
3


4
G

Q

D D
a
D
1
1
2
3
H

8
G

RQ



D
D
D
D
2
2
0



3
H
t
D
D
D



where the average of the scalar quantities on the domain D is

1 2 3
f
(
t
,
X
,
X
,
X
)
d
g

D
fD
(
t
)

g
d

D
Integrability condition:



1
1
6
2

a
Q


a
t
D
D
t
D RD
6
2
aD
aD
 0
11
 = local matter density
R = Ricci-scalar
aD
HD 
= domain dependent Hubble rate
aD
The kinematical backreaction QD
is defined as
22
2


2
Q





2

D
D
3 D D
where θ is the local expansion rate,
ij
2 1/2
is the squared rate of shear
ij
1
/
2
SEPARATION INTO ARBITRARY
PARTITIONS
The “global” domain D is assumed to be separated into subregions Fl
which themselves consist of elementary space entities Fl  associated with
some averaging length scale
Fl   Fl

Fl  Fm  0,     , l  m
D  l Fl
f D 
Global averages split into averages on sub-regions
l  | Fl | g | D | g is volume fraction of subregion Fl

l
l
Based on this partitioning the expression for backreaction becomes



Q


Q

3

H

H


D
m
where
Q

m
is the backreaction of the subdomain
Scale factors for subdomains:
2
m
a D3   li al3
l
f
Fl
Acceleration equation for the global domain D:



a
a
(
t
)
2
D


 m
H

H


m

m
a
a
(
t
)
D
2-scale interaction-free model (Weigand & Buchert, PRD ‘10):
M – those parts that have initial overdensity (“Wall”)
E – those parts that have initial underdensity (“Void”)
H D  M H M  E H E
D  M E
Void fraction:
E  | E | | D |
Wall fraction:
M  | M | | D |
M  E  1
Acceleration equation:
 

a
a
2
M a
D
E



2
(
H

H
)
M
E
M
EM
E
a
a
a
D
M E
(various possibilities of study)
Global evolution using the Buchert framework
(N. Bose, ASM ‘11;’13)
Associate scale of homogeneity with the global domain D
| D |g  
 g d 3 X  f (r )aF3 (t )
D
f(r) is function of FLRW radial coord.
Relation between global and FLRW scale factors:
HF  HD
Thus,
Toy Model:
Assuming power law ansatz for void and wall,
Global acceleration:
 f (r )
aD  
| D |
i g

a E  cE t 
D
a
g 3t 3  (   1) 
g 3t 3

 1 
3
2
3
aD
aD
t
a
D

2
3 3
g t
3
aD
1/ 3

g 3t 3 
1 
3
a
D





aF  aF
a M  cM t 
  (  1)

2
t

   
  
t 
 t
2
Future evolution assuming present acceleration
(i)
  0.995,   0.5
Present wall fraction,
a M  cM t 
a E  cE t 
N.Bose & ASM,
GRG (2013)
(ii)
  1.02,   0.66
M  0.09 [Weigand & Buchert, PRD ‘10]
0
Global evolution using the Buchert framework
(A. Ali, ASM ‘16)
Associate scale of homogeneity with the global domain D
| D |g  
 g d 3 X  f (r )aF3 (t )
D
a E  cE t 
Assuming power law ansatz for void
q
1  cos  
2q  1
q
  sin  
t
2q  1
aM 
Wall: Closed domain with positive curvature
Global acceleration:
D  ct 3  
a
ct 3    1
2
 1  3   qH M   3
aD 
aD 
aD
t2
ct 3
2 3
aD
 ct 3
1  3
aD



H


 M
t



2
Future evolution of the global domain
q  0.6,   0.7
a E  cE t 
q
1  cos  
2q  1
q
  sin  
t
2q  1
aM 
H D  M H M  E H E
As time evolves, H E falls off more rapidly
compared to H M
Even though the wall occupies a tiny fraction of
the total volume, the decrease of M is more
than compensated by the comparative evolution
of H E and H M
Analogous scalar field cosmology
Effective perfect fluid E-M tensor in the Backreaction formalism:
1
1
Q

R
D
D
16G
16G
1
1

QD 
R D
16G
16G
D
 eff
 
PeffD

D
Buchert equations recast in standard Friedman form:


a D
D
3
 4G  eff
 3PeffD  
aD
D
3H D2  8G eff

D
D
 eff
 3H D  eff
 PeffD   0
corresponding to energy density and pressure of effective global
scalar field at scales much larger than the scale of inhomogeneities
Cosmology with effective scalar field -- advantages
• Buchert framework: backreaction fluid: modelled as an effective scalar field.
• At scales much larger that the scale of inhomogeneities: dynamics is analogous to
that of a two fluid universe (matter & scalar field).
• Scalar field cosmologies widely studied in the context of understanding dark
energy (various classes of models; also unified DM-DE models).
• Effective scalar field has justified status in the present framework; no extraneous
source for this field is required.
• Inhomogeneities encoded in the backreaction term fix the field potential; no
phenomenological parametrization required.
• This correspondence allows a realistic interpretation of a variety of potentials in
phenomenological models involving scalar fields. (Observations constrain
backreaction model parameters)
Analogous scalar field cosmology [A. Ali, ASM, JCAP’17]
D
 eff
 
(quintessence)
1
2
D
 D
D   D2  VD
PeffD  PD
PD 
1 2
 D  VD
2
Field (morphon) potential:
3 m0
 a D (t ) 
D (t )
a
 
V  (t )  
 2
3
a D (t )
a
(
t
)
2
a
D (t )
 D

2
Equation of motion:
(t )  3H D (t ) (t ) 

dV ( )
0
d
Scalar field dynamics
Thawing quintessence
Data analysis of effective scalar field backreaction model
In this framework the two parameters determining evolution are:
(i)
: present matter density; (ii)
: initial value of scalar field
• Supernovae (SnIa) data:
collective model parameters
• Large scale structure data for BAO :
• CMB shift parameter:
Combining all three data sets
Observational analysis (Combined SNIa, BAO, CMB data)
[A. Ali, ASM, JCAP ‘17]
AIC  2 ln L  2k
Baysian (Akaike) Information criteria :
L: maximum likelihood; k: no. of parameters
N: no. of data points
Difference from CDM model:
(still compares favorably to some modified
gravity models ! )
BIC  2 ln L  k ln N
q

.6
.6
8
13
.6
.7
9
13
.6
.8
10
14
.6
.9
10
15
.7
.8
24
28
.8
.8
26
31
.9
.8
28
33
AIC BIC
Other directions :
(i) Consideration of event horizon
(ii) multidomain models
Effect of event horizon
Accelerated expansion
formation of event horizon.
a D  aF
Scale of homogeneity set by the global scale factor
(earlier, tacitly !)

Now, consider event horizon, explicitly
d t'
rh  aD
in general, spatial, light cone distances
aD (t ' )
t
are different; however, approximation
valid in the same sense as
a  aF
(small metric perturbations) D

Event horizon: observer dependent (defined w.r.t either void or wall).
Assumption: scale of global homogeneity lies within horizon volume
(Physics is translationally invariant over such scales)
Volume scale factor:
a
3
D

| D |g
| Di | g
4
 rh3
 3
| Di | g
r
D
a
 h
aD
rh
Effect of event horizon…..
 

Void-Wall symmetry of the acceleration eq:
a
a
2
M a
D
E

2
(
H

H
)
M
E
M
E
M
E
a
a
D
M a
E
ensures validity of event horizon definition w.r.t any point inside global domain:
Thus, we have, two coupled equations:
I
~ 3t 3   (   1) 
~ 3t 3 
rh
g
g


3
2
1  r 3
rh
rh
t
h

~ 3t 3 
g
2
rh3
II
rh 
~ 3t 3 

g

1  r 3
h

  (  1)

2

t

   

 

t 
 t
2
a D
rh  1
aD
Joint solution of I & II with present acceleration as “initial condition” gives
future evolution of the universe with backreaction
Two-scale (interaction-free) void-wall model
M – collection of subdomains with initial overdensity (“Wall”)
E –– collection of subdomains with initial underdensity (“Void”)


a

c
t
;
a

ct
E
E
M M
(power law evolution in subdomains:
acceleration equation


(


1
)  ct 

(


1
)

1



3 3

c
t
a
M
D
 h3
a
r
D
h
3 3
M
h
3
h

r 


3 3
 c
3 3

2
c
t
t


M
M

2 h3 
1
 h3 
  


r
r
t
t
h
h


Effect of event horizon

rh  aD

t
2
t
2
t

(Event horizon forms at the onset of acceleration)
d t'
aD (t ' )
Demarcates causally connected regions
Future deceleration due to cosmic backreaction in
presence of the event horizon [N. Bose, ASM; MNRAS 2011]
(i) α = 0.995, β = 0.5 , (ii) α = 0.999, β = 0.6 ,
(iii) α = 1.0, β = 0.5 , (iv) α = 1.02, β = 0.66 .
q0  0.7
Multidomain model
Walls: M
j
Voids:
D  ( j M j )  ( j E j )
Ej
a M j  cM j t
j
a E j  cE j t
Ansatz: Gaussian distributions for parameters  j
j
and
j
2
é
ù
x
m
(
)
1
ú
exp ê 2
2s
ê
ú
s 2p
ë
û
similar gaussian distributions for the volume fractions
Motivations:
(i) to study how global acceleration on the width of the distributions
(ii) to determine if acceleration increases with more sub-domains
a
b
0
 : 0.99 – 0.999
range of
range of  : 0.55 – 0.65
12
(i)
(ii)
(iii)
(iv)
10
äD
aD H02
8
6
4
2
0
0.5
1
1.5
2
2.5
t/ t0
For narrow range of variables, acceleration increases
with larger number of subdomains
[N. Bose & ASM, arXiv: 1307.5022]
3
Future evolution with backreaction (Summary)
[N. Bose and ASM, MNRAS Letters (2011); Gen. Rel. Grav. (2013);
arXiv: 1307.5022; A. Ali and ASM, arXiv: JCAP (2017)]
• Effect of backreaction due to inhomogeneities on the future evolution of
the accelerating universe (Spatial averaging in the Buchert framework)
• The global homogeneity scale (or cosmic event horizon) impacts the role of
inhomogeneities on the evolution, causing the acceleration to slow down
significantly with time.
• Backreaction could be responsible for a decelerated era in the future.
(Avoidance of big rip !) Possible within a small region of parameter space
• Analogous scalar field cosmology: Form of potential fixed by backreaction
model; Observational constraints from data analysis
• Effect may be tested in more realistic models, e.g., multiscale models,
models with no ansatz for subdomains, & other schemes of backreaction