1. Finite universe and cosmic coincidences Kari Enqvist, University of Helsinki COSMO 05 Bonn, Germany, August 28 - September 01, 2005

2. cosmic coincidences dark energy –why now:   ~ (H 0 M P ) 2 ? CMB –why supression at largest scales: k ~1/H 0 ? UV problem IR problem

3. Do we live in a finite Universe? large box: closed universe   1 → L >> 1/H small box – periodic boundary conditions non-trivial topology: R > few  1/H – non-periodic boundary conditions does this make sense at all? maybe – if QFT is not the full story (not interesting)

4. CMB & multiply connected manifolds discrete spectrum with an IR cutoff along a given direction (”topological scale”)  suppression at low l geometric patterns encrypted in spatial correlators (”topological lensing” – rings etc.) correlators depend on the location of the observer and the orientation of the manifold (increased uncertainty for C l ) See e.g. Levin, Phys.Rept.365,2002

5. a pair of matched circles, Weeks topology (Cornish) - many possible multiple connected spaces - typically size of the topological domain restricted to be > 1/H 0 explains the suppression of low multipoles with another coincidence

6. spherical box  IR cutoff L ground state wave function j 0 ~ sin(kr)/kr for r < r B radius of the box which boundary conditions? 1)Dirichlet wave function vanishes at r = r B → max. wavelength c = 2r B = 2L → allowed wave numbers k nl = (l  /2 + n  )/r B 2) Neumann derivative of wave function vanishes allowed modes given through j l (kr B ) l/kr B – j l+1 (kr B ) = 0 for each l, a discrete set of k no current out of U. KE, Sloth, Hannestad

7. Power spectrum: continuous → discrete IR cutoff shows up in the Sachs-Wolfe effect C l = N  k  k c j l (k nl r) P R (k nl ) / k nl CMB spectrum depends on: - IR cutoff L ( ~ r B ) - boundary conditions - note: no geometric patterns IR cutoff → oscillations of power in CMB at low l

8. Sachs-Wolfe with IR cutoff at l = 10

9. WHY A FINITE UNIVERSE? - observations: suppression, features in CMB at low l - cosmological horizon: effectively finite universe  holography?

10. HOLOGRAPHY Black hole thermodynamics  Bekenstein bound on entropy classical black hole: dA  0, suggests that S BH ~ A  generalized 2nd law dS total = d(S matter + A BH /4)  0 R matter with energy E, S ~ volume spherical collapse S ~ area either give up: 1) unitarity (information loss) 2) locality violation of 2nd law unless S matter  2  ER Bekenstein bound

11. QFT: dofs ~ Volume; gravitating system: dofs ~ Area  QFT with gravity overcounts the true dofs QFT breaks down in a large enough V  QFT as an effective theory: must incorporate (non-local) constraints to remove overcounting  QFT as an effective theory: must incorporate (non-local) constraints to remove overcounting Cohen et al; M. Li; Hsu; ’t Hooft; Susskind argue: locally, in the UV, QFT should be OK  constraint should manifest itself in the IR argue:

12. WHAT IS THE SIZE OF THE INFRARED CUT-OFF L? - maximum energy density in the effective theory:  4  Require that the energy of the system confined to box L 3 should be less than the energy of a black hole of the same size: (4  /3)L 3  4 <  LM P 2 Cohen, Kaplan, Nelson - assume: L defines the volume that a given observer can ever observe future event horizon R H = a  t  dt/a R H ~ 1/H in a Universe dominated by dark energy ’causal patch’ more restrictive than Bekenstein: S max ~ (S BH ) 3/4 Li Susskind, Banks

13. the effectively finite size of the observable Universe constrains dark energy:  4 < 1/L 2 dark energy = zero point quantum fluctuation ~

14. for phenomenological purposes, assume: 1) IR cutoff is related to future event horizon: R H = cL, c is constant 2) the energy bound is saturated:   = 3c 2 (M P /R H ) 2  a relation between IR and UV cut-offs = a relation between dark energy equation of state and CMB power spectrum at low l Friedmann eq. +  = 1: R H = c / (   H) now ½

15. dark energy equation of state w = -1/3 - 2/(3c)   ½ predicts a time dependent w with -(1+2/c) < 3w < -1 Note: if c < 1, then w < -1  phantom; OK? - e.g. for Dirichlet the smallest allowed wave number k c = 1.2/(  H 0 ) - the distance to last scattering depends on w, hence the relative position of cut-off in CMB spectrum depends on w

16. translating k into multipoles: l = k l (  0 -   ) comoving distance to last scattering  0 -   =  dz/H(z) 0 z*z* H(z) 2 = H 0 2 [   (1+z) (3+3w) +(1-   )(1+z) 3 ] 0 0 w = w(c,   ) l c = l c (c)

17. Parameter Prior Distribution Ω = Ω m + Ω X 1 Fixed h 0.72 ± 0.08 Gaussian Ω b h 2 0.014-0.040 Top hat n s 0.6-1.4 Top hat  0-1 Top hat Q - Free b - Free strategy: 1) choose a boundary condition: 2) calculate  2 for each set of c and k cut, marginalising over all other cosmological parameters fits to data: we do not fix k c but take it instead as a free parameter k cut

18. Neumann

19. Dirichlet fits to WMAP + SDSS data 95% CL68% CL  2 = 1444.8  2 = 1441.4 Best fit  CDM:  2 = 1447.5

20. 95% CL68% CL Likelihood contours for SNI data WMAP, SDSS + SNI bad fit, SNI favours w ~ -1

21. other fits: Zhang and Wu, SN+CMB+LSS: c = 0.81  w 0 = - 1.03 but: fit to some features of CMB, not the full spectrum; no discretization

22. conclusions ’cosmic coincidences’ might exist both in the UV (dark energy) and IR (low l CMB features) finite universe  suppression of low l holographic ideas  connection between UV and IR toy model: CMB+LSS favours, SN data disfavours – but is c constant? very speculative, but worth watching! E.g. time dependence of w