Georgios Tringas - uni-bonn.de · Georgios Tringas Bonn University Seminar on Non-Accelerator...
Transcript of Georgios Tringas - uni-bonn.de · Georgios Tringas Bonn University Seminar on Non-Accelerator...
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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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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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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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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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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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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
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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
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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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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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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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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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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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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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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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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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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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Comments on Gravitational waves so far
Model-independent prediction of inflation
Expansion rate H during inflation
Inflationary GWs are polarized
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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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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/