19.01.2015 Views

Cosmological Perturbations from Non-Equilibrium Quantum Fields

Cosmological Perturbations from Non-Equilibrium Quantum Fields

Cosmological Perturbations from Non-Equilibrium Quantum Fields

SHOW MORE
SHOW LESS

Create successful ePaper yourself

Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.

<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

Hooray! Your file is uploaded and ready to be published.

Saved successfully!

Ooh no, something went wrong!