Finite universe and cosmic coincidences

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

• dark energy
• why now ?? (H0MP)2 ?
• CMB
• why supression at largest scales k 1/H0 ?

UV problem
IR problem
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Do we live in a finite Universe?

• large box closed universe
• ? ? 1 ? L gtgt 1/H
• small box
• periodic boundary conditions
• non-trivial topology R gt few ? 1/H
• non-periodic boundary conditions
• does this make sense at all?
• maybe if QFT is not the full story

(not interesting)
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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
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a pair of matched circles, Weeks
topology (Cornish)

• many possible multiple connected spaces
• - typically size of the topological domain
  restricted to be gt 1/H0

explains the suppression of low multipoles with
another coincidence
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KE, Sloth, Hannestad
spherical box ? IR cutoff L
ground state wave function j0 sin(kr)/kr for
r lt rB radius of the box
which boundary conditions?

• 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
  jl1(krB ) 0

for each l, a discrete set of k
no current out of U.
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Power spectrum continuous ? discrete IR
cutoff shows up in the Sachs-Wolfe effect
Cl N ?k?kc 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
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Sachs-Wolfe with IR cutoff at l 10
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WHY A FINITE UNIVERSE?

• observations suppression, features in CMB at
  low l
• cosmological horizon effectively finite
  universe
• ? holography?

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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
spherical collapse
S area
violation of 2nd law unless Smatter ? 2? ER
matter with energy E, S volume
Bekenstein bound
either give up 1) unitarity (information loss)
2) locality
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argue
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

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
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WHAT IS THE SIZE OF THE INFRARED CUT-OFF L?
- assume L defines the volume that a given
observer can ever observe
RH a ?t? dt/a
causal patch
future event horizon
Susskind, Banks
Li
RH 1/H in a Universe dominated by dark energy
- 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

Cohen, Kaplan, Nelson
more restrictive than Bekenstein Smax (SBH)3/4
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the effectively finite size of the
observable Universe constrains dark energy
?4 lt 1/L2

dark energy zero point quantum fluctuation
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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
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w -1/3 - 2/(3c) ??
dark energy equation of state
½
predicts a time dependent w with
-(12/c) lt 3w lt -1
Note if c lt 1, then w lt -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 position
• of cut-off in CMB spectrum depends on w

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translating k into multipoles
l kl (?0 - ?? )
comoving distance to last scattering
z
?0 - ?? ? dz/H(z)
0
H(z)2 H02 ??(1z)(33w)(1-?? )(1z)3
0
0
w w(c,?? )
lc lc(c)
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fits to data we do not fix kc but take it
instead as a free parameter kcut
strategy 1) choose a boundary condition 2)
calculate ?2 for each set of c and kcut,
marginalising over all other cosmological
parameters
Parameter Prior Distribution O Om OX 1
Fixed h 0.72 0.08 Gaussian Obh2
0.014-0.040 Top hat ns 0.6-1.4 Top hat ?
0-1 Top hat Q - Free b - Free
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Neumann
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fits to WMAP SDSS data
95 CL
68 CL
Neumann
Dirichlet
?2 1444.8
?2 1441.4
Best fit ?CDM ?2 1447.5
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95 CL
68 CL
Likelihood contours for SNI data
WMAP, SDSS SNI
bad fit, SNI favours w -1
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other fits
Zhang and Wu, SNCMBLSS c 0.81 ? w0 -
1.03
but fit to some features of CMB, not the full
spectrum no discretization
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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 CMBLSS favours, SN data disfavours
  but is c constant?
• very speculative, but worth watching! E.g. time
  dependence of w