1. 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: [astro-ph]
A. Ali & ASM, JCAP 01, 054 (2017)

2. 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

3. 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)

4. Standard cosmological model Cosmological principle (Isotropy and homogeneity at large scales) Friedmann-Lemaitre-Robertson-Walker metric Energy-momentum (perfect fluid) Dynamics (Friedmann equations)

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

6. Motivations:
Observations tell us that the present Universe is inhomogeneous
up to scales (< 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

7. 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

8. 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

9. 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)

10. 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)

11. 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
It encodes the average stretch of all directions of the domain.

12. Using the Einstein equations:
where the average of the scalar quantities on the domain D is
Integrability condition:
1/27/11

13. = domain dependent Hubble rate
= local matter density
= Ricci-scalar
= domain dependent Hubble rate
The kinematical backreaction QD is defined as
where θ is the local expansion rate,
is the squared rate of shear
1/27/11

14. 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
where
is the backreaction of the subdomain
Scale factors for subdomains:

15. Acceleration equation for the global domain 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:
(various possibilities of study)
1/27/11

16. Global evolution using the Buchert framework (N
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, Toy Model: Assuming power law ansatz for void and wall, Global acceleration:

17. Future evolution assuming present acceleration (i) (ii) Present wall fraction, [Weigand & Buchert, PRD ‘10] N.Bose & ASM, GRG (2013)

18. Global evolution using the Buchert framework (A
Global evolution using the Buchert framework (A. Ali, ASM ‘16) Associate scale of homogeneity with the global domain D Assuming power law ansatz for void Wall: Closed domain with positive curvature Global acceleration:

19. Future evolution of the global domain As time evolves, falls off more rapidly compared to Even though the wall occupies a tiny fraction of the total volume, the decrease of is more than compensated by the comparative evolution of and

20. Analogous scalar field cosmology Effective perfect fluid E-M tensor in the Backreaction formalism: Buchert equations recast in standard Friedman form: corresponding to energy density and pressure of effective global scalar field at scales much larger than the scale of inhomogeneities

21. 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)

22. Analogous scalar field cosmology [A
Analogous scalar field cosmology [A. Ali, ASM, JCAP’17] (quintessence) Field (morphon) potential: Equation of motion:

23. Scalar field dynamics Thawing quintessence

24. 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

25. Observational analysis (Combined SNIa, BAO, CMB data) [A
Observational analysis (Combined SNIa, BAO, CMB data) [A. Ali, ASM, JCAP ‘17] Baysian (Akaike) Information criteria : L: maximum likelihood; k: no. of parameters N: no. of data points Difference from model: (still compares favorably to some modified gravity models ! ) q .6 8 13 .7 9 .8 10 14 .9 15 24 28 26 31 33

26. Other directions : (i) Consideration of event horizon (ii) multidomain models

27. 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 distances are 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:

28. Effect of event horizon…
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

29. Demarcates causally connected regions
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
1/27/11

30. 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 =

31. 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

32. range of : 0.99 – range of : – For narrow range of variables, acceleration increases with larger number of subdomains [N. Bose & ASM, arXiv: ]

33. Future evolution with backreaction (Summary) [N
Future evolution with backreaction (Summary) [N. Bose and ASM, MNRAS Letters (2011); Gen. Rel. Grav. (2013); arXiv: ; 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