Cosmic Strings A Study of the Early Universe Alexander Stameroff James O’Brien.
Finite universe and cosmic coincidences
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Transcript of Finite universe and cosmic coincidences
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