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

2. 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)

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

4. Energy budget of the Universe

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

6. 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

7. 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

8. 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

9. 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

10. 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)

11. 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

12. 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.

13. 1/27/ 11 Using the Einstein equations: where the average of the scalar quantities on the domain D is Integrability condition:

14. 1/27/111/27/11 = 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

15. 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 whereis the backreaction of the subdomain Scale factors for subdomains:

16. 1/27/11 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:

17. The Backreaction Framework: Einstein equations: 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)

18. 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:

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

20. 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

21. 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

22. 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:

23. 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

24. 1/27/11 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

25. 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.

26. 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