Accelerating Expansion from Inhomogeneities JeAn Gu National Taiwan

Accelerating Expansion from Inhomogeneities ? Je-An Gu (National Taiwan University) Collaborators: Chia-Hsun Chuang (astro-ph/0512651) IRGAC 2006, 2006/07/14

Accelerating Expansion Based on FRW Cosmology (homogeneous & isotropic)

Supernova data ? Cosmic Acceleration Existence of cosmic acceleration Based on FRW Cosmology Dark energy as a necessity (homogeneous & isotropic) of understanding acceleration However, apparently, our universe is NOT homogeneous & isotropic. At large scales, after averaging, the universe IS homogeneous & isotropic. But, averaging !? Is it legal ? Does it make sense ?

Einstein equations For which satisfy Einstein equations, in general DO NOT.

Dark Geometry Effects of Inhomogeneities through averaging Einstein equations Toy Model: ds 2 = dt 2 a 2 (1 + h coskx cosky coskz) (dx 2 + dy 2 + dz 2) Einstein equations after averaging in space: (perturb: h << 1) eff peff = eff / 3 peff

Questions Supernova data ? Cosmic Acceleration requires Dark Energy ? (or Inhomogeneity-induced Acceleration ? )

Cosmic Acceleration requires Dark Energy ? Normal matter attractive gravity slow down the expansion Need something abnormal : e. g. cosmological constant, dark energy -- providing anti-gravity (repulsive gravity) Is This True ? Common Intuition / Consensus

Is This True ? Intuitively, YES ! (of course !!) Normal matter attractive gravity slow down the expansion Common Intuition / Consensus ** Kolb, Matarrese, and Riotto (astro-ph/0506534) : Inhomogeneities of the universe might induce acceleration. Mission Impossible ? or Mission Difficult ? Two directions: 1. Prove NO-GO theorem. 2. Find counter-examples. This is what we did. We found counter-examples for a dust universe of spherical symmetry,

Lemaitre-Tolman-Bondi (LTB) Solution (exact solution in GR) (unit: c = 8 G = 1) Dust Fluid + Spherical Symmetry k(r) = const. , 0(r) = const. , a(t, r) = a(t) FRW cosmology Solution (parametric form with the help of ) arbitrary functions of r : k(r) , 0(r) , tb(r)

Line (Radial) Acceleration ( q. L < 0 ) Radial : Inhomogeneity Acceleration Angular : No Inhomogeneity No Acceleration

What is Accelerating Expansion ? (I) Line Acceleration L homogeneous & isotropic universe: RW metric: We found examples of q. L < 0 (acceleration) in a dust universe described by the LTB solution.

Line (Radial) Acceleration : q. L < 0 arbitrary functions of r : k(r) , 0(r) , tb(r) Inhomogeneity the less smoother, the better k(r) 1 0 kh parameters : (nk , kh , rk) , 0 , r. L , t rk r

Examples of Line (Radial) Acceleration : q. L < 0 arbitrary functions of r : k(r) , 0(r) , tb(r) 1 k(r) r rk 0 kh parameters : (nk , kh , rk) , 0 , r. L , t nk kh rk 0 r. L t q. L q. D 20 1 0. 7 1 1 1 0. 8 Acceleration Observations q ~ 1 (based on FRW cosmology)

Examples of Line (Radial) Acceleration : q. L < 0 nk kh rk 0 r. L t q. L q. D 20 1 0. 7 1 1 1 0. 8 k(r) = 0 at rk = 0. 7 Over-density Under-density

Examples of Line (Radial) Acceleration : q. L < 0 nk kh rk 0 r. L t q. L q. D 20 1 0. 7 1 1 1 0. 8 k(r) = 0 at rk = 0. 7 characterizing the accel/deceleration status of the radial line elements

Examples of Line (Radial) Acceleration : q. L < 0 nk kh rk 0 r. L t q. L q. D 20 1 0. 7 1 1 1 0. 8 k(r) = 0 at rk = 0. 7 Acceleration Deceleration

Examples of Line (Radial) Acceleration : q. L < 0

Examples of Line (Radial) Acceleration : q. L < 0 Inhomogeneity Acceleration

Examples of Line (Radial) Acceleration : q. L < 0 nk (20) kh 1 rk 0. 7 0 1 r. L 1 t 1 Deceleration nk=5 Easy to generate Acceleration larger nk larger inhomogeneity 1 0 kh k(r) r. K r

Examples of Line (Radial) Acceleration : q. L < 0 nk 20 kh 1 rk 0. 7 0 1 r. L 1 t (1) Deceleration Acceleration

Domain Acceleration ( q. D < 0 ) spherical domain r = r. D r=0

What is Accelerating Expansion ? (II) Domain Acceleration a large domain D (e. g. size ~ H 0 1) Volume VD NO-GO q. D 0 > 0 (deceleration) in a dust universe (see, e. g. , Giovannini, hep-th/0505222) We found examples of q. D < 0 (acceleration) in a dust universe described by the LTB solution. [Nambu and Tanimoto (gr-qc/0507057) : incorrect example. ]

Examples of Domain Acceleration : q. D < 0 k(r) arbitrary functions of r : k(r) , 0(r) , tb(r) parameters : (nk , kh , rk), (nt , tbh , rt), 0 , r. D , t nk kh rk nt tbh rt 0 r. D t q. D 40 40 0. 9 40 10 0. 9 105 1. 1 0. 1 1 Acceleration

Examples of Domain Acceleration : q. D < 0 nk kh rk nt tbh rt 0 r. D t q. D 40 40 0. 9 40 10 0. 9 105 1. 1 0. 1 1 k(r) = 0 at r = 0. 82 Over-density Under-density

Examples of Domain Acceleration : q. D < 0 nk kh rk nt tbh rt 0 r. D t q. D 40 40 0. 9 40 10 0. 9 105 1. 1 0. 1 1 characterizing the accel/deceleration status of the radial line elements Acceleration Deceleration

Examples of Domain Acceleration : q. D < 0 nk kh rk nt tbh rt 0 r. D t (40) 40 0. 9 40 10 0. 9 105 1. 1 0. 1 Acceleration

Examples of Domain Acceleration : q. D < 0 nk kh rk nt tbh rt 0 r. D t 40 (40) 0. 9 40 10 0. 9 105 1. 1 0. 1 Deceleration Acceleration

Examples of Domain Acceleration : q. D < 0 nk kh rk nt tbh rt 0 r. D t 40 40 (0. 9) 40 10 (0. 9) 105 1. 1 0. 1 Deceleration Acceleration

Examples of Domain Acceleration : q. D < 0 nk kh rk nt tbh rt 0 r. D t 40 40 0. 9 (40) 10 0. 9 105 1. 1 0. 1 Deceleration Acceleration larger nt larger inhomogeneity tb(r)

Examples of Domain Acceleration : q. D < 0 nk kh rk nt tbh rt 0 r. D t 40 40 0. 9 40 (10) 0. 9 105 1. 1 0. 1 Deceleration Acceleration

Examples of Domain Acceleration : q. D < 0 nk kh rk nt tbh rt 0 r. D t 40 40 0. 9 40 10 0. 9 (105) 1. 1 0. 1 Acceleration

Examples of Domain Acceleration : q. D < 0 nk kh rk nt tbh rt 0 r. D t 40 40 0. 9 40 10 0. 9 105 (1. 1) 0. 1 Deceleration Acceleration

Examples of Domain Acceleration : q. D < 0 nk kh rk nt tbh rt 0 r. D t 40 40 0. 9 40 10 0. 9 105 1. 1 (0. 1) Deceleration Acceleration

Summary and Discussions

Model: Inhomogeneous Universe (Reality? ) Toy Model: ds 2 = dt 2 a 2 (1 + h coskx cosky coskz) (dx 2 + dy 2 + dz 2) peff = eff / 3 Against the common intuition and consensus : normal matter attractive gravity deceleration, Counter-examples (acceleration) are found. These examples support : Inhomogeneity Acceleration These examples raise two issues : ? Can inhomogeneities explain cosmic acceleration ? ? How to understand these counter-intuitive examples ? (cosmology issue) (GR issue)

Can Inhomog. explain “Cosmic Acceleration”? SN Ia Data Cosmic Acceleration ? ? ? Mathematically, possible. In Reality ? ? Inhomogeneities Can Inhomogeneities explain SN Ia Data? IF YES Does Cosmic Acceleration exist?

How to understand the examples ? Normal matter attractive gravity slow down the expansion Common Intuition / Consensus Intuition from Newtonian gravity, not from GR. (valid only for … ? ) (x ) Newton? NO. GR? YES. Intuition for GR ? NO !?

Summary and Discussions GR is still not fully understood after 90 years !!

Line (Radial) Acceleration : q. L < 0 ( For constant 0 ) Sufficient and Necessary Condition: Sharp enough change in kh(r) 1 0 k(r) r. K kh Tuning/choosing the boundary condition r