Cosmological Perturbations from Non-Equilibrium Quantum Fields
Cosmological Perturbations from Non-Equilibrium Quantum Fields
Cosmological Perturbations from Non-Equilibrium Quantum Fields
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<strong>Cosmological</strong> <strong>Perturbations</strong><br />
<strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong><br />
Arttu Rajantie<br />
13 July 2012<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Particle Physics ↔ Cosmology<br />
Explain cosmological observations:<br />
Baryon asymmetry (Huber,Tranberg,Saffin)<br />
Dark matter (Cirelli)<br />
Dark energy<br />
Inflation<br />
Test particle physics theories:<br />
Topological defects (monopoles, strings)<br />
Gravity waves<br />
<strong>Perturbations</strong> (CMB, LSS)<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Cosmic Microwave Background<br />
Penzias&Wilson 1964<br />
Thermal radiation <strong>from</strong> recombination t ≈ 300000yrs<br />
Redshifted <strong>from</strong> T ≈ 4000K to T ≈ 2.7K<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Cosmic Microwave Background<br />
Temperature anisotropies δT ∼ 10 −5 K<br />
COBE 1992, WMAP 2003–2010<br />
The best source of information about the early universe<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Cosmic Microwave Background<br />
Primordial density perturbation: δT/T = δρ/ρ<br />
Origin of large-scale structure:<br />
Galaxies, galaxy clusters<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Cosmic Microwave Background<br />
Highly Gaussian −10 < f NL < 74 (i.e. ζ NG 10 −8 )<br />
Nearly scale-invariant spectrum n s = 0.963 ± 0.012<br />
Just as inflation predicts!<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Planck Satellite<br />
High angular resolution, polarisation<br />
Very sensitive to non-Gaussianity (ζ NG ∼ 10 −9 )<br />
Data released in early 2013<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Particle Physics in the CMB<br />
Adiabatic evolution ⇒ Perturbation conserved<br />
⇒ We need non-equilibrium dynamics<br />
Horizon then ≪ Horizon now<br />
⇒ We need correlations over many, many horizon lengths<br />
Causality: Correlations cannot appear after inflation<br />
⇒ We need an isocurvature perturbation<br />
Simplest possibility: Light (m < H) scalar field<br />
Since last week we know such fields actually exist!<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Separate Universes<br />
Each horizon volume ∼ separate FRW universe (Salopek&Bond 1990)<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Separate Universes<br />
Each horizon volume ∼ separate<br />
FRW universe (Salopek&Bond 1990)<br />
Described by a separate<br />
Friedmann equation<br />
(ȧ ) 2<br />
H 2 = = ρ<br />
a<br />
3M 2 P l<br />
Curvature perturbation<br />
ζ = δT/T = δ ln a| ρ=ρ∗<br />
Valid at distances d ≫ 1/H<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Conservation of ζ<br />
Curvature perturbation<br />
ζ = δT/T = δ ln a| ρ=ρ∗<br />
If the equation of state P = P (ρ)<br />
is the same everywhere,<br />
d ln a<br />
dρ = − 1<br />
3(ρ + P (ρ)) ⇒ dζ<br />
dρ ∗<br />
= 0<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Another Light Scalar χ<br />
Light scalar m χ < H<br />
⇒ Scale-invariant perturbations<br />
P χ (k) ≈ H2<br />
4π 2<br />
Each separate ”universe” has<br />
different initial value χ 0<br />
Affects expansion<br />
⇒ Scale factor depends on χ 0<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Another Light Scalar χ<br />
Each separate ”universe” has<br />
different initial value χ 0<br />
Affects expansion<br />
⇒ Scale factor depends on χ 0<br />
Thermalisation erases<br />
memory of χ 0 ⇒ P = P (ρ)<br />
Curvature perturbation<br />
determined at thermalisation:<br />
ζ rec = ζ therm = δ ln a| ρ=ρtherm<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Calculating the Perturbation<br />
Solve Friedmann+field eqs for each separate universe<br />
(ȧ ) 2<br />
H 2 = =<br />
a<br />
ρ(φ, χ)<br />
3M 2 P l<br />
Gives a(t), ρ(t)<br />
<strong>Non</strong>-linear, includes gravity, valid at d ≫ H −1<br />
Pick ρ ∗ < ρ therm , and calculate ζ = δ ln a| ρ=ρ∗<br />
⇒ Gives curvature perturbation for given φ 0 , χ 0<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Calculating the Perturbation<br />
Solve Friedmann+field eqs for each separate universe<br />
Pick ρ ∗ < ρ therm , and calculate ζ = δ ln a| ρ=ρ∗<br />
Two fields: ζ = ζ(φ 0 , χ 0 )<br />
δφ 0 ⇒ Usual inflationary perturbations<br />
δχ 0 ⇒ New contribution<br />
Need to calculate N(χ 0 ) = ln a(ρ ∗ , χ 0 )<br />
Perturbative approach: δN formalism<br />
Fully non-linear:<br />
Solve field and Friedmann eqs numerically<br />
for many different initial values χ 0<br />
⇒ Whole non-linear function N(χ 0 ) (Bassett&Tanaka 2003, Suyama&Yokoyama 2006)<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Describe each “universe”<br />
as a lattice (Chambers&AR 2007)<br />
Inhomogeneous fields,<br />
FRW metric<br />
Lattice size L ∼ 1/H<br />
Lattice Calculation<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Lattice Calculation<br />
Solve field + Friedmann eqs<br />
numerically on lattice<br />
Initial conditions<br />
χ(x) = χ 0 + δχ(x):<br />
δχ(x): subhorizon quantum<br />
fluctuations<br />
χ 0 : mean value over horizon<br />
Find curvature perturbation as<br />
ζ(χ 0 ) = δ ln a(χ 0 )| ρ=ρ∗<br />
Function of χ 0 , not x<br />
χ<br />
χ 0<br />
δχ<br />
L<br />
x<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Applications<br />
Light scalar χ + nonequilibrium dynamics<br />
⇒ <strong>Perturbations</strong><br />
Examples:<br />
Massless preheating<br />
Resonant curvaton<br />
Electrically charged curvaton<br />
Future: Concrete particle physics models<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Massless Preheating<br />
Chaotic inflation + massless scalar χ (Prokopec&Roos 1997)<br />
V (φ, χ) = 1 4 λφ4 + 1 2 g2 φ 2 χ 2 ,<br />
λ ∼ 7 × 10 −14<br />
Inflation: φ > 2 √ 3M Pl<br />
V<br />
φ<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Massless Preheating<br />
Chaotic inflation + massless scalar χ (Prokopec&Roos 1997)<br />
V (φ, χ) = 1 4 λφ4 + 1 2 g2 φ 2 χ 2 ,<br />
λ ∼ 7 × 10 −14<br />
After inflation: Radiation domination a ∝ t 1/2<br />
V<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012<br />
φ
Massless Preheating<br />
Chaotic inflation + massless scalar χ (Prokopec&Roos 1997)<br />
V (φ, χ) = 1 4 λφ4 + 1 2 g2 φ 2 χ 2 ,<br />
λ ∼ 7 × 10 −14<br />
After inflation: Radiation domination a ∝ t 1/2<br />
Inhomogeneous χ modes χ k = a −1 ˜χ k<br />
˜χ ′′<br />
k +<br />
[κ 2 + g2<br />
λ cn2 (τ; 1/ √ ]<br />
2) ˜χ k = 0, κ 2 = k2<br />
λ ˜φ 2 ini<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Parametric Resonance<br />
Floquet theorem:<br />
˜χ k (τ) = e µτ f(τ) with periodic f(τ)<br />
Real µ: Exponential growth = resonance<br />
Resonance transfers energy to χ<br />
When χ grows enough,<br />
evolution becomes non-linear<br />
⇒ Resonance ends – Thermalisation<br />
Amount of growth needed depends on initial χ<br />
⇒ Curvature perturbation (Chambers&AR,2007)<br />
Κ 2<br />
3.0<br />
2.5<br />
2.0<br />
1.5<br />
1.0<br />
0.5<br />
0.0<br />
0 2 4 6 8 10<br />
Effect strongest when κ = 0 dominates the resonance<br />
g 2<br />
<br />
Λ<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Massless Preheating<br />
δN = ln(a end /a ref ) * 10 5<br />
12.0<br />
10.0<br />
8.0<br />
6.0<br />
4.0<br />
2.0<br />
0.0<br />
broadened transfer function<br />
quadratic fit for small χ values<br />
raw Monte-Carlo samples<br />
µ 0 T periodicity and its harmonics<br />
fluctuations<br />
-2.0<br />
0.1 1 10 100<br />
(χ ini /M P ) * 10 7<br />
(Bond et al 2009)<br />
Chaotic behaviour: Highly non-Gaussian (Chambers&AR 2007)<br />
Not well approximated by a quadratic f NL<br />
Amplitude ∼ 10 −4 – Observable!<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Curvaton<br />
Scalar field σ (Mollerach 1990; Linde&Mukhanov 1997; Enqvist&Sloth 2002; Lyth&Wands 2002)<br />
V (φ, σ) = V (φ) + 1 2 m2 σ 2 + hσ ¯ψψ<br />
During inflation H > m: Scale-invariant fluctuations<br />
After inflation m > H: Behaves like matter<br />
P = 1 − r<br />
3 ρ tot, r = ρ χ<br />
ρ tot<br />
Decays perturbatively to ψ particles ⇒ ζ conserved<br />
ζ = N(χ + δχ) = r decay<br />
2<br />
δχ<br />
χ + r decay<br />
4<br />
( δχ<br />
χ<br />
) 2<br />
+ . . .<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Curvaton Resonance<br />
1.5×10 -5<br />
1.0×10 -5<br />
5.0×10 -6<br />
0.0<br />
10 18 10 19 10 20 10 21 10 22 10 23<br />
1/ρ [M Pl<br />
-4 ]<br />
(Chambers,Nurmi&AR, 2009)<br />
Curvaton σ coupled to another scalar (Enqvist et al 2009)<br />
Decays through parametric resonance<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
ζ<br />
ζ<br />
ζ<br />
Curvaton Resonance<br />
3×10 -5<br />
2×10 -5<br />
1×10 -5<br />
0<br />
-1×10 -5<br />
-2×10 -5<br />
1×10 -4<br />
0.0004996 0.0004998 0.0005000 0.0005002 0.0005004<br />
-1×10 -4 0<br />
-2×10 -4<br />
0.0009996 0.0009998 0.0010000 0.0010002 0.0010004<br />
2×10 -4<br />
-2×10 -4 0<br />
-4×10 -4<br />
0.0019996 0.0019998 0.0020000 0.0020002 0.0020004<br />
σ 0 [M Pl ]<br />
<strong>Non</strong>linear N(σ 0 ) ⇒ Highly non-Gaussian (Chambers,Nurmi&AR, 2009)<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Electrically Charged Curvaton<br />
Charged under electroweak U(1) (D’Onofrio,Lerner&AR 2012)<br />
Coleman-Weinberg potential<br />
V eff (σ) = m 2 |σ| 2 + 3g′4<br />
64π 2 |σ|4 ln<br />
( |σ|<br />
2<br />
µ 2 )<br />
Known coupling g ′ ≈ 0.357 (or integer multiple)<br />
During inflation:<br />
<strong>Perturbations</strong> mainly neutral (“Higgs mode”)<br />
Charge density screened by other charged particles<br />
After inflation:<br />
Perturbative decay (as usual)<br />
Interaction with thermal bath<br />
Parametric resonance with the gauge field<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Qualitative Constraints<br />
100<br />
90<br />
80<br />
max<br />
µ/H * = 100<br />
µ/H * = 10<br />
µ/H * = 10 -1<br />
µ/H * = 10 -5<br />
70<br />
60<br />
σ * /H *<br />
50<br />
40<br />
30<br />
20<br />
10<br />
0<br />
10 -3 10 -2 10 -1 10 0<br />
m/H *<br />
Light curvaton, stable vacuum, linear evolution<br />
Shaded=allowed<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
100<br />
90<br />
80<br />
70<br />
60<br />
Resonance Parameter<br />
q * = 10 4<br />
q * = 10 3<br />
q * = 10 2<br />
q * = 10<br />
σ * /H *<br />
50<br />
40<br />
30<br />
20<br />
10<br />
10 -3 10 -2 10 -1 10 0<br />
Strong resonance (q ∗ ≫ 1):<br />
Lattice simulations needed<br />
m/H *<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
100<br />
90<br />
80<br />
70<br />
60<br />
Quantitative Constraints<br />
H * > 3 x 10 9 GeV<br />
H * > 1 x 10 9 GeV<br />
H * > 2 x 10 8 GeV<br />
σ * /H *<br />
50<br />
40<br />
30<br />
20<br />
10<br />
0<br />
10 -3 10 -2 10 -1 10 0<br />
Assume the curvaton is the only source of perturbations:<br />
Has to have the observed amplitude ζ ∼ 10 −5<br />
m/H *<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Caveat with FRW<br />
Finite-volume error when long wavelengths dominate<br />
Dynamical metric with Einstein eq (Bastero-Gil,AR&Szmigiel, in progress)<br />
No limitation on lattice size, in principle exact<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012
Summary<br />
Light scalar + non-equilibrium physics ⇒ <strong>Perturbations</strong><br />
Separate universes + lattice field theory<br />
Massless preheating, Curvaton resonance, Electrically<br />
charged curvaton:<br />
Possibly observable effects ζ ∼ 10 −5<br />
Highly non-Gaussian<br />
Realistic particle physics models<br />
Higgs, SUSY scalars<br />
Potentially testing physics far beyond the Standard Model<br />
Arttu Rajantie<br />
<strong>Cosmological</strong> <strong>Perturbations</strong> <strong>from</strong> <strong>Non</strong>-<strong>Equilibrium</strong> <strong>Quantum</strong> <strong>Fields</strong> – SEWM 2012