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Page 1: Georgios Tringas - uni-bonn.de · Georgios Tringas Bonn University Seminar on Non-Accelerator particle physics 15/6/2016. 2/34 IntroductionHorizons again?Cosmological perturbationsQuantum

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Inflation

Georgios Tringas

Bonn UniversitySeminar on Non-Accelerator particle physics

15/6/2016

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Classical Cosmological Inflation

Metric for large scale universe: Friedmann-Robertson-Walker(FRW).

flat, spherical, hyperbolic.

Have already discussed classical dynamics of inflation.

solution to: horizon problem, flatness problem ...explains: large-scale homogeneity, isotropy and flatness of theuniverse.

Inflation with α > 0 and slowly varying H solves the problems.

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Motivation

Understand the origin of inhomogeneities.

Large scale constructions.

Is there any connection with observable quantities?

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Growing Hubble sphere

Define comoving Hubble radius throught particle horizon

χph(τ) =

∫ t

ti

dt

α=

∫ lnα

lnαi

(αH)−1dlnα ≡ τ − τi (1)

Using constant equation of state for universe dominated by fluid

χph(t) =2

(1 + 3w)(αH)−1 (2)

For αi → 0 conformal time τi → 0

For standard cosmology Hubble radius increases as theuniverse expands.

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Shrinking Hubble sphere

Assume a decreasing Hubble radius in the early Universe

d

dt(αH)−1 < 0 (3)

For αi → 0 and w < −13 , conformal time τi = −∞

d

dt(αH)−1 = − α

(α)2→ α > 0 (4)

Implies accelerated expansion!

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Hubble sphere

Figure 1 : Comoving Hubble sphere and horizon problem 1

There is negative conformal time before the standard ”big bang”1www.damtp.cam.ac.uk/user/db275/Cosmology/

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Spacetime perturbations

Initial idea: Inhomogeneities → Perturbations

gµν = gµν + δgµν , Perturbations around (flat) FRW metric (5)

Perturbation couples to matter through Einstein equations.

Perturbation to FRW spacetime.

ds2 = α2(τ)[(1 + 2A)dτ2 − 2Bi dx i dτ − (δij + hij )dx i dx j ] (6)

SVT decomposition

Page 8: Georgios Tringas - uni-bonn.de · Georgios Tringas Bonn University Seminar on Non-Accelerator particle physics 15/6/2016. 2/34 IntroductionHorizons again?Cosmological perturbationsQuantum

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Fake perturbations and gauge fixing

Metric perturbations are not uniquely defined:Old coordinates ⇔ New coordinates: Unphysical perturbations

Unphysical perturbations

?Ficticious gauge modes

?Wrong predictions for inhomogeneities

Gauge invariant perturbations

Gauge fixing → Newtonian gauge or Spatially-flat gauge

Page 9: Georgios Tringas - uni-bonn.de · Georgios Tringas Bonn University Seminar on Non-Accelerator particle physics 15/6/2016. 2/34 IntroductionHorizons again?Cosmological perturbationsQuantum

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

A conserved quantity

In arbitrary gauge for the perturbed metric the gauge invariantthree dimensional Ricci scalar is:

R = C − 1

3∇2E − H(B + υ) (7)

In Newtonian Gauge:

R =˙ρδp − ˙pδρ

3(ρ+ p)2+O(

k2

α2H2) (8)

First term vanishes for adiabatic modes δp˙p

= δρ˙ρ

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Curvature conservation

dlnRdlnα ∼ ( k

H )2

?For super-Hubble scales k H

?

R ≈ 0 does NOT evolve

Conserved on super-Hubble scales for adiabatic fluctuations.

Observational significance: The value of R computed athorizon crossing during inflation survives unaltered untillater times.

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Curvature perturbation

Figure 2 : From vacuum fluctuations to CMB anisotropies 2.

Constancy of R on superhorizon scales allows to relate CMBobservations while allowing us to be completely ignorant aboutthe high-energy physics during the early times of the universe.

2www.damtp.cam.ac.uk/user/db275/Cosmology/

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Treating Inflation quantum mechanically

Natural Fluctuations

Inflation evolution φ(t) governs the energy density of theearly universe ρ(t).

Quantum mechanical object → Uncertainty principle

δφ(t, x) = φ(t, x)− φ(t) (9)

Different regions inflate of space inflate by different amounts!

Differences in local regions δt lead to differences in the localdensities ρ(t, x) after inflation.

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Mukhanov-Sasaki Equation

S =

∫d4x√−g [

1

2gµν∂µφ∂νφ− V (φ)] (10)

In spatially flat gauge, metric perturbations are suppressedrelative to inflation fluctuations by slow-roll parameter.

Perturbed inflaton field:

φ(τ, x) = φ(τ) +f (τ, x)

α(τ)(11)

result the second order action:

S (2) ≈∫

d4x1

2[(f

′)2 − (∇f )2 +

α′′

αf 2] (12)

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Mukhanov-Sasaki Equation

Arrive to Mukhanov-Sasaki equation:

f′′

k + (k2 − α′′

α)fk = 0 (13)

with effective frequency ω2k (τ) = (k2 − α

′′

α ).

For subhorizon, k2 α′′/α

f′′

k + k2fk = 0→ fk ∝ e±ikτ (14)

Simple harmonic oscillator with ωk = k.Not quantized yet!

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Mode expansion- Quantization

We quantize canonically the field:

f (τ, x), π(τ, x)→ f (τ, x), π(τ, x)

[f (τ, x), π(τ, x′)] = iδ(x − x

′)

Variance on inflation fluctuations has non-zero quantumfluctuations:

〈|f 2|〉 = 〈0|[f †(τ, 0), f (τ, 0)]|0〉 =

∫dlnk

k3

2π2|fk (τ)|2 (15)

Recieve the power spectrum:

∆2f (k , τ) =

k3

2π2|fk (τ)|2 =⇒ ∆2

f =αδφ(k) ≈ (H

2π)2

∣∣∣∣k=aH

(16)

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

When do the fluctuations become classical?

In the superhorizon limit, kτ → 0

[f (τ, x), π(τ, x)] = [f (τ, x),−1

τf (τ, x)] = 0 (17)

Fluctuations become classical fields.

After horizon crossing we identify quantum expectation valuewith a classical ensemble average

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Variance of curvature perturbations

Treating R in spatially flat gauge C and E perturbations vanish

R = H(B + υ) (18)

and compairing to the perturbed stress tensor

R =H

φδφ −→ 〈|R|2〉 = (

H

φ)2〈|δφ|2〉 (19)

Outside the horizon: quantum expectation value →classical stochastic field

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Variance of curvature perturbations

Figure 3 : From vacuum fluctuations to CMB anisotropies 3.

3www.damtp.cam.ac.uk/user/db275/Cosmology/

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Primordial Perturbations from Inflation

At horizon crossing we switch δφ ⇔ R to conserve perturbation

∆2R(k) =

1

8π2

1

ε

H2

M2pl

∣∣∣∣k=αH

(20)

If H and ε are slow-varing functions of time the powerspectrum is scale-invariant.

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Scale Invariance

For slow varying quantities power spectrum is almost scaleinvariant:

∆2R(k) = As(

k

k?)ns−1 with ns − 1 ≡

dln∆2R

dlnk(21)

Both amplitude of scalar spectrum and k? can be measured.Recent observational value:

ns = 0.9603± 0.0073 (22)

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Gravitational waves

Tensor perturbations to spatial part of the metric:

ds2 = α2(τ)[dτ2 − (δij + 2Eij )dx i dx j ] (23)

Obtaining the second order terms from E-H action:

S =M2

pl

2

∫d4x√−gR → S (2) =

1

2

∑I =+,x

∫d4x [(f

′I )2−(∇fI )2+

α′′

αf 2I ]

(24)We got the tensor spectrum:

∆2t (k) =

2

π2

H2

M2pl

∣∣∣∣k=αH

(25)

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

Comments on Gravitational waves so far

Model-independent prediction of inflation

Expansion rate H during inflation

Inflationary GWs are polarized

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

The scale-dependence of the tensor spectrum

Scale-dependence of the tensor spectrum is defined:

∆2t (k) = At(

k

k?)nt (26)

tensor to scalar ratio:

r =At

As(27)

Tensors have not been observed yet, upper limit r ≤ 0.17.

Inflationary models can be classified according to theirpredictions for the parameters ns and r .

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Scalar index and tensor amplitude

Figure 4 : Constraints on ns and r by Planck satellite 4

4www.damtp.cam.ac.uk/user/db275/TEACHING/INFLATION/Lectures.pdf

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Temperature Anisotropies

Figure 5 : Temperature fluctuations in the CMB 5

Θ(n) =∆T (n)

T0=∑lm

αlmYlm(n) T = 2.7K (28)

5en.wikipedia.org/wiki/Cosmic microwave background

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Angular Power Spectrum

The multiple moments may be combined into the angular powerspectrum:

C TTl =

1

2l + 1

∑m

〈α∗lmαlm〉 (29)

CMB temperature fluctuations are dominated by the scalar modes

αlm = 4π(−i)l

∫d3k

(2π)3∆Tl(k)Rk Ylm(k) (30)

C TTl =

2

π

∫k2dk PR(k)︸ ︷︷ ︸

Inflation

∆2Tl (k) (31)

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Angular Power Spectrum

Figure 6 : Angular power spectrum of CMB temperature fluctuations 6, 7

6arXiv:0907.5424 [hep-th]7Wayne Hu & Martin White, arxiv.org/abs/astro-ph/9602019

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Information from Angular Power Spectrum

Inflation Plateau l < 30, third peak

Density First peak l ∼ 200 , θ= 1 degree

DM and B densities Second peak

Adiabatic perturbation Peak location

Table 1 : Locations

Ωtot ∼ 1.00

Ωb ∼ 0.04

ΩCDM ∼ 0.30

ΩΛ ∼ 0.70

As (2.196 ± 0.060) 10−9

k? 0.05 Mpc−1

ns 0.9603 ± 0.0073

H ∼ 68 kms−1 Mpc−1

Table 2 : Extracted information

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Thomson scattering

Expect the CMB to be polarized by Thomson scatteringImprints of tensor modes in CMB polarizationPolarization contains information about primordialfluctuations → Inflation!

Figure 7 : Linear polarization is generation via Thomson scattering ofradiation with a quadrupolar anisotropy 8

8cosmology.berkeley.edu/ yuki/CMBpol/

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Origin of quadruple anisotropy

Scalar Perturbation: Energy density fluctuations-Blueshift ofphotons.

Vector Perturbation: Vorticity in plasma cause Dopplershifts.

Tensor Perturbation: Gravity waves stretch and squeeze thespace.

The spin-1 polarization field can be decomposed spin-0 quantities,the so-called E and B-modes.

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Significance of E/B decomposition

Scalar density perturbations create only E-modes

Vector perturbations create mainly B-modes

Tensor (GW) perturbations create both E-modes and B-modes

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E/B Modes

Figure 8 : Patterns of E/B modes9

Figure 9 : Polarization map showsthe first E-modes detected in 2002by the Degree Angular ScaleInterferometer (DASI) telescope 10

9www.damtp.cam.ac.uk/user/db275/TEACHING/INFLATION/Lectures.pdf10cosmology.berkeley.edu/ yuki/CMBpol/

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Introduction Horizons again? Cosmological perturbations Quantum Flactuations Contact with Observations Conclusion

E/B Modes

Figure 8 : Patterns of E/B modes9

Figure 9 : Polarization map showsthe first E-modes detected in 2002by the Degree Angular ScaleInterferometer (DASI) telescope 10

9www.damtp.cam.ac.uk/user/db275/TEACHING/INFLATION/Lectures.pdf10cosmology.berkeley.edu/ yuki/CMBpol/

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Conclusion

Perturbations can lead to inhomogeneites

Inflation generates natural fluctuations

CMB observations can:

exclude modelsinitial conditionsInformation about geometry and composition

Polarization: Direct evidence for inflation

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Bibliography

V.Mukhanov, Physical Foundations of Cosmology

Daniel Baumann, Cosmol-ogy,www.damtp.cam.ac.uk/user/db275/Cosmology/Lectures.pdf

Daniel Baumann, TASI Lectures on Inflation,arxiv.org/abs/0907.5424

Kolb & Turner, The Early Universe

Scott Dodelson, Modern Cosmology

cosmology.berkeley.edu/ yuki/CMBpol/

Daniel Baumann, The Physics of Infla-tion,www.damtp.cam.ac.uk/user/db275/TEACHING/INFLATION/