Loop Quantum Cosmology corrections to inflationary models
Michał Artymowski
Loop Quantum Cosmology holonomy corrections to inflationary models University of Warsaw With collaboration with L. Szulc and Z. Lalak
Pennstate
Michał Artymowski, University of Warsaw
25‐10‐2008
2008‐10‐25
Outline
Loop Quantum Cosmology corrections to inflationary models
Phenomenology of loop corrections •Introduction to cosmology and inflation • Background evolution in the FRW cosmology with the holonomy loop correction •Evolution of the scalar metric perturbations •Effective comoving sonic horizon • Power spectrum of the initial energy density perturbations
Michał Artymowski, University of Warsaw
2008‐10‐25
Clasical cosmology and inflation
Loop Quantum Cosmology corrections to inflationary models
Let us consider the FRW Universe with k=0. From Einstein equations one obtains Friedmann equations
ρ H = 3 2
a&& 1 = - (ρ + 3 P ) a 6
& = - 1 (ρ + P ) H 2
To solve these equations we need to know the barotropic parameter ω, where P=ωρ, or we need to know the equation of motion for matter fields Inflation appears when
a&& >0 a
and the most popular scenarios assume ω≈‐1
We need ρ≈‐P rather unnatural equation of state! Then a (t ) ≈e Ht and the comoving Hubble horizon
Michał Artymowski, University of Warsaw
1 aH
decreases rapidly
2008‐10‐25
Inflaton field
Loop Quantum Cosmology corrections to inflationary models
The most popular model: The early Universe is dominated by the scalar field φ
1 2 ρ = φ& + V(φ) 2
1 2 P = φ& - V(φ) ⇒ to get ω≈‐1 we need 2
&& + 3Hφ& + V′ φ =0
Michał Artymowski, University of Warsaw
φ& 2 << V(φ)
Equation of motion for the inflaton
2008‐10‐25
Initial perturbations
Loop Quantum Cosmology corrections to inflationary models
Now we work with the inflaton field φ(t) and its perturbation δφ(x,t)
r ρ(t ) → ρ (t ) + δρ( x , t ) r P(t ) → P ( o ) (t ) + δP( x , t ) (o)
δGμν = δTμν ⇒
After Fourier trans‐ formation we have
r r g oo (t ) → g oo (t ) + δg oo ( x , t ) = -(1 + 2Φ( x , t )) r r 2 g ij (t ) → g ij (t ) + δg ij ( x , t ) = a (1 - 2Ψ( x , t ))δ ij
We do not have any anisotropic pressure ⇒Φ = Ψ
. Δ 1 2 3 H Ψ + 3 H Ψ + 2 Ψ = - δρ 2 a
.. . & ) Ψ = 1 δP Ψ + 4 H Ψ + (3 H 2 + 2H 2
.. . . 2 Ψ + (4 + 3 c s ) H Ψ + [2 H + 3 H 2 (1 + c 2s ) - k 2 c 2s / a 2 ] Ψ = 0
For kcs<
Ψ ≈const
Perturbations are frozen outside the comoving sonic horizon 2008‐10‐25
Power spectra
Loop Quantum Cosmology corrections to inflationary models
For the strong slow‐roll approximation δφ evolves like the masless scalar field Conformal time: a(η)dη=dt
.. . δ φ + 3 Hδ φ + k 2 δφ/a 2 = 0
a (t ) ≈e For we obtain Ht
δφ = e
From the equation of motion for δφ(k,η) we get
Then we define the Power spectrum of δφ by And finally…
H 9 = Ψ + δφ φ&
η=-ikη
1 aH
ηH (i kη - 1) 2k
k3 2 7δφ (k , η) = 2 δφ(k , η) 2π
1 7δφ (k , η) = 2 H 2 (k 2 η 2 + 1) 4π
we can not measure that!
Mukhanov‐Sasaki variable. This is something like
In slow‐roll approximation Michał Artymowski, University of Warsaw
δρ ρ
H4 V 79 ≈ 2 2 ≈ 4 π φ& 24 π 2 ε 2008‐10‐25
The loop correction and Ashtekar variables
Ashtekar variables in the FRW Universe:
Loop Quantum Cosmology corrections to inflationary models
. c = γa
p = a2
{c, p} = γ
3
3N The hamiltonian gives us Friedmann equations H = − 2 p c 2 + H mat γ
The parallel transport around the loop changes the vector. If we would shrink the loop to the smallest possible size we would get the elementary correction
Michał Artymowski, University of Warsaw
2008‐10‐25
Quantum of length
Loop Quantum Cosmology corrections to inflationary models
By considering the loop quantum gravity modifications to the c we get
c → sin(cl j / p )
p lj
, where l j is the quantum of length.
The holonomy loop correction does not changes p!
l j ∝ l pl(j(j + 1 ))1 / 4
This is an extremely important variable!
No specific value of j chosen by nature!
3 ρ cr = 2 2 ∝ γ lj
M 4pl j(j+ 1)
j=
1 3 ,1, ,... 2 2
Critical (maximal) energy density of the Universe
For and big values of j we have ρ cr ~ 1/j 8πG = 1 Michał Artymowski, University of Warsaw
2008‐10‐25
Friedmann equations
Loop Quantum Cosmology corrections to inflationary models
Friedmann equations ρ⎛ ρ ⎞ ⎟ H 2 = ⎜⎜1 − 3 ⎝ ρ cr ⎟⎠
. ⎛ 1 ρ ⎞ ⎜ H = − (ρ + P )⎜1 − 2 ⎟⎟ 2 ρ cr ⎠ ⎝
ρ/ρ cr → 0 ⇒ normal FRW
Effective variables
ρ eff ρ ⎛ ρ ⎞ ⎟⎟ = ⎜⎜1 − 3 3 ⎝ ρ cr ⎠
⎛ ρ ⎞ ρ2 Peff = P⎜⎜1 − 2 ⎟⎟ − ρ cr ⎠ ρ cr ⎝ Michał Artymowski, University of Warsaw
2008‐10‐25
Perturbations in LQC
Loop Quantum Cosmology corrections to inflationary models
ds = − N dt + p(d x ) We can write Hamilton equations for , where 2
p = a 2 (1 - 2Ψ)
2
2
2
N = (1 + Φ)
We consider k 0 so perturbations are functions of time only. We do not have any anisotropic pressure ⇒ Φ = Ψ and from perturbated friedmann equations we have
. 1 2 3 H Ψ + 3 H Ψ = - δρ eff 2
.. . 1 2 & Ψ + 4 H Ψ + (3 H + 2H ) Ψ = δPeff 2
For the adiabatic perturbations we obtain
.. . . 2 Ψ + (4 + 3 cseff ) H Ψ + [2 H + 3 H 2 (1 + cs2eff )] Ψ = 0 This equation is almoust identical with the one from the standard FRW. Perturbations are frozen outside the effective sonic horizon. Michał Artymowski, University of Warsaw
2008‐10‐25
The effective speed of sound
In LQC the effective speed of ρ sound becomes infinite for ρ = cr
c
2 s eff
Loop Quantum Cosmology corrections to inflationary models
2
(ρ + P) /ρ cr δ Peff 2 = = cs - 2 δ ρ eff 1 - 2 ρ/ρ cr
No conserved information left . over from the H > 0 period
The effective Big Bang scenario!
c s2eff is not a physical velocity!!! Michał Artymowski, University of Warsaw
2008‐10‐25
The power spectrum
Loop Quantum Cosmology corrections to inflationary models
Equations for the inflaton and it’s perturbation are not changed by the loop correction
.. . δ φ + 3 Hδ φ + k 2 δφ/a 2 = 0
where δφ is the inflaton perturbation
For the slow‐roll approximation
1 V'2 ε= 2, 2(1 - ρ/ρ cr ) V
Veff = V(1 − V/ρ cr )
The power spectrum of the curvature perturbations is then in form of From the COBE normalisation we know, that if we want to avoid fine tuning we need to have
ρ cr > (1016 GeV) 4
This gives us the limit for j. For big values of j
ρ cr ∝ M 4pl /j ⇒ j < 1012
The spectral index
ns (k ) - 1 = 2η - 6 ε is changed by the LQC to
Michał Artymowski, University of Warsaw
ns ( k ) - 1 nsloop (k ) - 1 = (1 - V / ρ cr ) 2008‐10‐03
Conclusions
Loop Quantum Cosmology corrections to inflationary models
•FRW Universe in the low energy limit for LQC •No information about the scalar perturbations can cross the ρ cr energy regime untouched •We can limit the quantum of length by the COBE normalisation •More models fit’s the data for strong LQC holonomy efects
Michał Artymowski, University of Warsaw
2008‐10‐25
Nazwa rozdziału
Michał Artymowski, University of Warsaw
Loop Quantum Cosmology corrections in the inflationary models
2008‐04‐29
Nazwa rozdziału
Michał Artymowski, University of Warsaw
Loop Quantum Cosmology corrections in the inflationary models
2008‐04‐29
Nazwa rozdziału
Michał Artymowski, University of Warsaw
Loop Quantum Cosmology corrections in the inflationary models
2008‐04‐29
