Role of Backreaction in an accelerating universe Archan S. Majumdar S. N. Bose National Centre for Basic Sciences Kolkata.

Role of Backreaction in an accelerating universe

Archan S. Majumdar

S. N. Bose National Centre for Basic SciencesKolkata

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

Energy-momentum (perfect fluid)

Dynamics (Friedmann equations)

),,,( pppdiagT

Energy budget of the Universe

Observational support for the CDM model(http:rpp.lbl.gov)

Understanding dark energy from various approaches

• Observational evidence for dark energy (present acceleration – supernovae); (weak lensing surveys); (CMB spectrum)

• Origin of dark energy: various candidates or mechanisms: cosmological constant (theoretical difficulties); dynamical dark energy models (scalar fields motivated from unification scenarios); modified gravitational dynamics [f(R), MOND, braneworld, etc..]; numerous other scenarios…

• Approach discussed here: Backreaction from inhomogeneties

Motivations:

• Observations tell us that the present Universe is inhomogeneous up to scales (< 100 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

1h

Problem of course-graining or averaging

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

TG

TgRgR 2

1

Determination of averaged cosmological metric

(hierarchical scales of course-graining)

Averaging process

does not commute with evaluating inverse metric, etc..

leading to, e.g.,

Extra term in general

Consequence: Einstein eqs. valid at local scales may not be trivially extrapolated at galactic scales

)()()()()(

2

1 locallocallocallocallocal TggRR

,)()()( lssgallocal ggg )()()( lssgallocal TTT

)()()()()()(

2

1 galgalgalgalgalgal ETggRR

0)( galE

Different approaches of averaging

Macroscopic gravity: (Zalaletdinov , GRG ‘92;’93)

(additional mathematical structure for covariant averaging scheme)

Perturbative schemes: (Clarkson et al, RPP ‘11; Kolb, CQG ‘11)

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)

TCRgRg 2

1

ggg

TGG

Perspective

• Different approaches of averaging; different interpretation of observables

• Effect of averaging: wide variety of viewpoints: (a) No effect on global cosmology, c.f., Ishibashi & Wald, CQG ‘06 (b) drastic affect: present acceleration due to backreaction, c.f., Rasanen, PRD, ‘10; Wiltshire, PRL ‘07 (c) some effect in precision cosmology, e.g., effect on BAO, c.f., Bauman et al., arXiv:1004.2488

• Spatial averaging versus lightcone averaging

• Effect of backreaction (through spatial averaging) on future evolution of the presently accelerating universe

The Buchert framework– main elements For irrotational fluid (dust), spacetime foliated into

constant time hypersurfaces

inhomogeneous

For a spatial domain D , volume

the scale factor is defined as

1/3| |

( )| |

g

gi

Da t

D D

It encodes the average stretch of all directions of the domain.

D

gg dD ||

321321)3( ),,,( dXdXdXXXXtgd g

1/27/11

Using the Einstein equations:

4 3 a

Ga

DDD

D

Q

2 1 13 8

2 2H G D DD D

R Q

0 3 t H DD D

where the average of the scalar quantities on the domain D is

Integrability condition:

1 2 3( , , , )( )

g

g

f t X X X df t

d

D

D

D

011 2

26

6 DDt

DDDt

D

Raa

Qaa

1/27/11

= local matter density

R = Ricci-scalar

aH

a D

DD

= domain dependent Hubble rate

The kinematical backreaction QD is defined as

22 222

3 D DDD

Q

where θ is the local expansion rate,2 1/ 2 ij

ij is the squared rate of shear

SEPARATION INTO ARBITRARY PARTITIONS

The “global” domain D is assumed to be separated into subregions which themselves consist of elementary space entities associated with some averaging length scale

Global averages split into averages on sub-regions is volume fraction of subregion

Based on this partitioning the expression for backreaction becomes

23 m mm

H H

DQ Q

where Q is the backreaction of the subdomain

Scale factors for subdomains:

lFlF

ll FF ll FD mlFF ml ,,0

l

FlD lff

ggll DF |||| lF

l

llD aai

33

Acceleration equation for the global domain D:

2( )

( ) m mm

a a tH H

a a t

D

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”)

Void fraction: Wall fraction:

Acceleration equation:

22 ( )aa a

H Ha a a

MD EM E M E M E

D M E

EMD EEMMD HHH

||||D

EE ||

||D

MM

1 EM

The Backreaction Framework: Einstein equations:

4 3 a

Ga

DDD

D

Q 2 1 1

3 8 2 2

H G D DD DR Q

0 3 t H DD D

where the average of the scalar quantities on the global domain D is

Global domain D is separated into sub-domains

Backreaction

backreaction in subdomain volume fraction of subdomain (dynamic)

1 2 3( , , , )( )

g

g

f t X X X df t

d

D

D

D

23 m mm

H H

DQ Q

Q

Global evolution using the Buchert framework (N. Bose, ASM ‘11;’13)

Associate scale of homogeneity with the global domain D

f(r) is function of FLRW radial coord.

Relation between global and FLRW scale factors:

Thus,

Assuming power law ansatz for void and wall,

Global acceleration:

D

Fg tarfXdgD )()(|| 33

FFgi

D aaD

rfa

3/1

||

)(

DF HH

tca EE tca MM

23

33

23

33 )1(1

)1(

ta

tg

ta

tg

a

a

DDD

D

2

3

33

3

33

12

tta

tg

a

tg

DD

Future evolution assuming present acceleration

(i) (ii)

Present wall fraction, [Weigand & Buchert, PRD ‘10]

N.Bose & ASM,GRG (2013)

5.0,995.0 66.0,02.1

09.00M

tca EE

tca MM

Multidomain model

Walls: Voids:

Ansatz: Gaussian distributions for parameters and

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

jM jE

)()( jj

jj EMD

j

jjtca MM

j

jjtca EE

j j

range of : 0.99 – 0.999 range of : 0.55 – 0.65

For narrow range of variables, acceleration increases with larger number of subdomains[N. Bose & ASM, arXiv: 1307.5022]

0

Effect of event horizon

Accelerated expansion formation of event horizon. Scale of homogeneity set by the global scale factor (earlier, tacitly !)Now, consider event horizon, explicitly in general, spatial, light cone distancesare different; however, approximation valid in the same sense as (small metric perturbations)

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:

FD aa

t D

Dh ta

tdar

)'(

'

FD aa

gi

h

gi

gD D

r

D

Da

||34

||

||3

3

h

h

D

D

r

r

a

a

Effect of event horizon…..

Void-Wall symmetry of the acceleration eq:

ensures validity of event horizon definition w.r.t any point inside global domain:

Thus, we have, two coupled equations:

I

II

Joint solution of I & II with present acceleration as “initial condition” gives future evolution of the universe with backreaction

22 ( )aa a

H Ha a a

MD EM E M E M E

D M E

23

33

23

33 )1(~1

)1(~

tr

tg

tr

tg

r

r

hhh

h

2

3

33

3

33 ~1

~2

ttr

tg

r

tg

hh

1 hD

Dh ra

ar

Two-scale (interaction-free) void-wall model M – collection of subdomains with initial overdensity (“Wall”)

E –– collection of subdomains with initial underdensity (“Void”)

(power law evolution in subdomains: acceleration equation

Effect of event horizon (Event horizon forms at the onset of acceleration)

Demarcates causally connected regions

;a c t a c t E E M M

3 3 3 3

3 2 3 2

3 3 3 3 2

3 3

( 1) ( 1)1

2 1

h h

h h

h h

h h

c t c ta

a r t r t

c t c t

r r t t

M MD

D

M M

t D

Dh ta

tdar

)'(

'

Future deceleration due to cosmic backreaction in presence of the event horizon [N. Bose, ASM; MNRAS 2011]

7.00 q(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 .

Future evolution with backreaction (Summary)[N. Bose and ASM, MNRAS Letters (2011); Gen. Rel.

Grav. (2013); arXiv: 1307.5022]

• 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 • Effect may be tested in more realistic models, e.g., multiscale models, models with no ansatz for subdomains, & other schemes of backreaction