Finite universe and cosmic coincidences

Kari Enqvist, University of Helsinki

COSMO 05

Bonn, Germany, August 28 - September 01, 2005

cosmic coincidences

• dark energy– why now: ~ (H0MP)2 ?

• CMB – why supression at largest scales: k ~1/H0 ?

UV problem

IR problem

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)

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 Cl )

See e.g. Levin, Phys.Rept.365,2002

a pair of matchedcircles, Weeks topology(Cornish)

- many possible multiple connected spaces

- typically size of the topological domain restricted to be > 1/H0

explains the suppression of low multipoles with another coincidence

spherical box IR cutoff L spherical box IR cutoff L

ground state wave function j0 ~ sin(kr)/kr for r < rB radius of the box

which boundary conditions?

1) Dirichlet

wave function vanishes at r = rB → max. wavelength c = 2rB = 2L

→ allowed wave numbers knl = (l/2 + n )/rB

2) Neumann

derivative of wave function vanishes

allowed modes given through jl(krB ) l/krB – jl+1(krB ) = 0

for each l, a discreteset of k

no current out of U.

KE, Sloth, Hannestad

Power spectrum: continuous → discrete

IR cutoff shows up in the Sachs-Wolfe effect

Cl = N kkc jl(knl r) PR(knl ) / knl

CMB spectrum depends on:

- IR cutoff L ( ~ rB ) - boundary conditions- note: no geometric patterns

IR cutoff → oscillations of power in CMB at low l

Sachs-Wolfe with IR cutoff at l = 10

WHY A FINITE UNIVERSE?

- observations: suppression, features in CMB at low l

- cosmological horizon: effectively finite universe

holography?

HOLOGRAPHY

Black hole thermodynamics Bekenstein bound on entropy

classical black hole: dA 0, suggests that SBH ~ A

generalized 2nd law dStotal = d(Smatter + ABH/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 Smatter 2 ER

Bekenstein bound

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:

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 L3 should be less than the energy of a black hole of the same size:

(4/3)L34 < LMP2 Cohen, Kaplan, Nelson

- assume: L defines the volume that a given observer can ever observe

future event horizon RH = a t

dt/a

RH ~ 1/H in a Universe dominated by dark energy

’causal patch’

more restrictive than Bekenstein: Smax ~ (SBH)3/4

Li

Susskind, Banks

the effectively finite size of the observableUniverse constrains dark energy:

4 < 1/L2

dark energy = zero point quantum fluctuation

~

for phenomenological purposes, assume:

1) IR cutoff is related to future event horizon:RH = cL, c is constant

2) the energy bound is saturated: = 3c2(MP /RH )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:

RH = c / (H)now

½

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 kc = 1.2/(H0 )

- the distance to last scattering depends on w, hence the relative positionof cut-off in CMB spectrum depends on w

translating k into multipoles:

l = kl (0 - )

comoving distance to last scattering

0 - = dz/H(z)

0

z*

H(z)2 = H02 [(1+z)(3+3w)+(1- )(1+z)3]0 0

w = w(c, )

lc = lc(c)

Parameter Prior Distribution

Ω = Ωm + ΩX 1 Fixed

h 0.72 ± 0.08 Gaussian

Ωbh2 0.014-0.040 Top hat

ns 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 setof c and kcut, marginalising over all other cosmological parameters

fits to data: we do not fix kc but take it instead as a free parameter kcut

Neumann

Neumann Dirichlet

fits to WMAP + SDSS data 95% CL 68% CL

2 = 1444.82 = 1441.4

Best fit CDM: 2 = 1447.5

95% CL 68% CL

Likelihood contours for SNI data WMAP, SDSS + SNI

bad fit, SNI favours w ~ -1

other fits:

Zhang and Wu, SN+CMB+LSS:

c = 0.81 w0 = - 1.03

but: fit to some features of CMB, not the full spectrum; no discretization

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