A new window on primordial non-Gaussianity based on 1201.5375 and - - PowerPoint PPT Presentation

A new window on primordial non-Gaussianity

based on 1201.5375 and 1206.4479 with M. Zaldarriaga

Enrico Pajer Princeton University CMBLSS

Μ

104 0.01 1 100 104 1.0 2.0 1.5 k Mpc Rk2109

Summary

We know little about primordial perturbations outside the range 10−4 kMpc 1 µ-type spectral distortion of the CMB is a unique probe of small scales 50 kMpc 104

[Sunyaev, Zel’dovich, Silk, Peebles, Hu, Danese, de Zotti, Chluba, . . . ]

The monopole µ(ˆ n) probes the small-scale power spectrum µT cross correlation probes the primordial bispectrum in the squeezed limit floc

NL

Fisher forecast with current technology ∆floc

NL 103

Beat cosmic variance with an enormous number of modes

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 2 / 38

Outline

1 Motivations 2 Review of µ-distortion 3 A new window on primordial non-Gaussianity 4 Remarks

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 3 / 38

The golden age of cosmology

We are living in the golden age of

• bservational cosmology: COBE

goes to Stockholm, WMAP, ACT & SPT measured the CMB with 1% accuracy. now Planck and a horde of ground and ballon based experiments

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 4 / 38

Primordial perturbations

? radiation matter dark E Log a 1 H Primordial superhorizon perturbations seed the structures in our universe They teach us about the earlier stage

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 5 / 38

The non-linear regime of structure formation

Small superhorizon primordial perturbations → linear evolution Homogeneous background → no mix of different scales Perturbations re-enter the horizon → grow with time Large perturbations evolve in a complicated non-linear way → erase primordial information Linear regime: Large scale structures k < O(Mpc−1) or look back in time

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 6 / 38

The cosmic microwave background

A snapshot of the universe at 370, 000 years, z ≃ 1100 Small perturbations δρ/ρ ∼ 10−5 → linearly related to primordial perturbations For l 2000 or k > O(0.15 Mpc−1) erased by diffusing damping! Can we access smaller scales?

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 7 / 38

The CMB/LSS window

• utside

horizon CMBLSS Gaussian Scaleinv

?

107 105 0.001 0.1 10 1000 0.02 0.05 0.10 0.20 0.50 1.00 2.00 k Mpc Rk2109

k 10−4 Mpc−1 are still outside our horizon k 0.15 Mpc−1 (l 2000) have been erased by Silk damping k O(1) Mpc−1 are now contaminated by gravitational non-linearities

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 8 / 38

Primordial perturbations: What do we know?

Within the CMB/LSS window we know to the 1% precision

[Hlozek et al. ’11]

Random variable Amplitude 10−5 → can measure only few cumulants Almost scale invariant Adiabatic Gaussian Do these properties hold on smaller scales?

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 9 / 38

Outline

1 Motivations 2 Review of µ-distortion 3 A new window on primordial non-Gaussianity 4 Remarks

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 10 / 38

Photon thermodynamics: early times

Early universe: hot photon-baryon-electron plasma Before zi ≃ 2 × 106 double Compton scattering (e− + γ → e− + 2γ) and Bremsstrahlung are very efficient Perfect thermodynamical equilibrium. Any perturbations to the system is thermalized Photons can be created at low ν and re-scattered to high ν Planck spectrum n(ν) = ν2 eν/kBT − 1

0.1 0.2 0.5 1.0 2.0 5.0 10.0 ΝkT nΝ

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 11 / 38

Photon thermodynamics: µ-era

Between zi and zf ∼ 5 × 104 double Compton scattering and bremsstrahlung are not efficient enough to create photons Elastic Compton scattering (e− + γ → e− + γ) maintains kinetic equilibrium Photon number is effectively frozen (except low ν) A perturbation distort the spectrum Bose-Einstein spectrum with chemical potential µ n(ν) = ν2 eν/kBT+µ − 1

0.5 1.0 2.0 5.0 ΝkT nΝ

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 12 / 38

Photon thermodynamics: y-era

After zf ∼ 5 × 104 also elastic Compton scattering is not efficient enough No kinetic equilibrium Photon number is still effectively frozen A perturbation deforms the spectrum → y-type distortion Mixing of black bodies with different T Different ν dependence. Observationally distinguishable

[Khatri Chluba Sunyaev ’11]

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 13 / 38

Photon thermodynamics: decoupling

After zL ≃ 103 photons travel without further interactions Electrons and protons combine into neutral H (and He, . . . ) Several other astrophysical foregrounds create y-distortion (mixing

• f black bodies)

[Sunyaev Zel’dovich ’70]

µ-distortion requires re-scattering of photons (kinetic equilibrium) → few contaminations! We have now a linear probe of scales 50 < kMpc−1 < 104

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 14 / 38

Perturbations during the µ-era

Perturbations during the µ-era

[Khatri, Chluba Sunyaev ’11,’12]

5 × 104 z 2 × 106 Dissipation of acoustic modes due to diffusion damping

[Silk ’72] .

Scale invariant power spectrum ns ∼ 1 µ ≃ 2.4 × 10−8 Adiabatic cooling → Bose-Eistein condensation µBE ≃ −3 × 10−9 Non-standard physics e.g. decays of massive particles, . . . Standard scenario: µ probes the primordial power on small scales

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 15 / 38

Dissipation from random walk

Photon mean free path λmfp = (neσT )−1 ⇒ random walk λD ≃ λmfp √ N ≃ λmfp

• ∆t

λmfp Using ∆t ∼ ∆z/(zH), ne ∼ z3 and during radiation H ∼ z2 λ ∼

• 1

neσT ∆z zH ∼ z−5/2 ⇒ k ∼ a λ ∼ z3/2

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 16 / 38

Dissipation of acoustic modes

Perturbations of the adiabatic mode R re-enter the horizon,

• scillate and dissipate

δγ ∼ Rk cos(kt) e−k2/k2

D

Shear viscosity and heat conduction damp the oscillations

[Silk, Kaiser, Weinberg]

kD ≃ z3/2 4 × 10−6 Mpc−1 Now we can only see k < kD(zL) ≃ 0.2 × Mpc−1 µ-distortion can see 50 < kMpc−1 < 104 measuring only large angles! (non-linear effect)

zL zf zi 104 0.01 1 100 104 1 1 k Mpc Rk2109

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 17 / 38

µ-era: hydrodynamical derivation

For a Bose-Einstein spectrum Entropy density s ∼ T 3(1 + constµ) and photon number density nγ ∼ T 3(1 + constµ) Entropy per photon s nγ

• = 3.6
• 1 + 0.5µ + O
• µ2

⇒ ∂tµ ∝ ∂t(s/nγ) Photon number conservation and generation of entropy for dissipative fluids ∂µ(uµs) = viscosity + conduction, ∂µ(uµnγ) = conduction One finds µ with the whole spatial dependence

[EP & Zaldarriaga ’12b]

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 18 / 38

µ-era: heuristic derivation

Analytical estimate of µ

[Hu et al. 94’, Khatri et al. ’11, EP Zaldarriaga ’12b]

Energy in acoustic waves with relativistic correction δE ∼ c2

s

1 + w ¯ ρ δ2

γ

• i

f

Bose-Einstein spectrum with E → E + δE and fixed N δµ ≃ −1.4 δE ¯ ρ > 0 Integral probe of the power spectrum µ ∝

• d log k∆2

R(k)

zL

Μ

zf zi 104 0.01 1 100 104 1 1 k Mpc Rk2109

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 19 / 38

Tight coupling and free streaming

Evolution of µ between zf ≃ 5 × 104 and us

[EP & Zaldarriaga ’12b]

Tight-coupling regime zf → zLS: only monopole µ0. Dissipation erases small-scale inhomogenities µ0(q, zLS) = µ0(q, zf)e−q2/q2

D,µ(zLS)

Free streaming zLS → 1: for l ≪ 1000 simple geometric projection µl(τ, q) ≃ µ0(τf, q)jl [q (τ − τLS)] For l > 1000 there is damping from diffusion and the finite thickness of the last scattering surface qD,µ. Relevant only for µµ

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 20 / 38

Relating µ to R2 at small scales

Dissipation is smeared over a volume k−3

s

Nature takes the average for us Small cosmic variance!

[EP & Zaldarriaga ’12]

Final analytical estimate µ(x) ≃ 9 2 d3k1d3k2 (2π)6 R( k1)R( k2)ei

k+· xW

k+ ks

• ×cos (k1t) cos (k2t)p
• e−(k2

1+k2 2)/k2 D

i

f

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 21 / 38

COBE-FIRAS bounds

CMB compatible with a black body so far COBE-FIRAS puts a bound |µ| < 9 × 10−5 TRIS + COBE-FIRAS |µ| < 6 × 10−5 PIXIE could achieve ∆µ ≃ 10−8 at 1-σ

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 22 / 38

µ(ˆ n) constraints the small-scale power spectrum

The averaged µ-distortion over the whole sky (monopole)

[Hu et al. ’94]

µ(x) ≃ 9 4

• d log k ∆2

R(k)

• e−2k2/k2

D

i

f ,

Integral measure of the power spectrum at small scales For ns = 1

[Khatri Chluba Sunyaev ’12]

µ ∼ 2.4 × 10−8 Needs an absolute measurement of the CMB spectrum

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 23 / 38

Outline

1 Motivations 2 Review of µ-distortion 3 A new window on primordial non-Gaussianity 4 Remarks

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 24 / 38

Primordial non-Gaussianity

It is hard to tell by eye For a Gaussian random variable δ2n+1 = 0 , δ2n ∝ δ2n Non-vanishing odd correlation → non-Gaussianity δ ≪ 1 → δ3 is the most sensitive

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 25 / 38

Symmetries, sizes and shapes

A priori, R3 depends on k1, k2, k3: 9 real numbers Conservation of momentum: k3 = − k1 − k2 Rotational invariance: only k2

1, k2 2 and

k1 · k2 R3 depends on 3 numbers R3 ≡ (2π)3fNLF(k1, k2, k3)δ3(k1 + k2 + k3) Scale invariance: only k1/k3 and k2/k3 fNL gives the size F(k1, k2, k3) (normalized) gives the shape.

Out[1320]=

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 26 / 38

Squeezed limit

Squeezed limit k3 ≪ k1 ∼ k2. Short-scale power modulated by long scales lim

kL≪kSR(kS)2R(kL) ∼ 1

k6

S

• A

kS kL >3 + B kS kL 3,2 + C kS kL 1 Microphysical inflationary models: A = 0 ⇒ instability B = 0 ⇒ multi-field (more than one clock), e.g. curvaton, quasi-single-field, local template

[Maldacena ’02, Creminelli Zaldarriaga ’04]

C = 0 ⇒ single or multi-field, e.g. small speed of sound, equilateral template

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 27 / 38

Local non-Gaussianity

Local model peaks in the squeezed limit k3 ≪ k1 ∼ k2 Distinguishes between broad classes of models R(k1)R(k2)R(k3) ∼ ∆2

R

k3

1

∆2

R

k3

2

+ 2perm′s Local in position space, e.g.

[Komatsu Spergel ’01]

R(x) = RG(x) + floc

NL

• RG(x)2 − RG(x)2
• Enrico Pajer

(Princeton) New window on primordial NG CWR Oct 2012 28 / 38

Observations

Measure µ(ˆ n) and T(ˆ n) on the last scattering surface Decompose in spherical harmonics: aµ

lm

≡

• dˆ

nY ∗

lm(ˆ

n)µ(ˆ n) aT

lm

≡

• dˆ

nY ∗

lm(ˆ

n)t(ˆ n) Statistical isotropy, I, J = µ, T, E, B aI

lmaJ l′m′ = δll′δmm′CIJ l

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 29 / 38

µT cross correlation Μy Μx

µ ∼ R2 and T ∼ R ⇒ µT ∼ R3 µT probe the primordial bispectrum in the squeezed limit floc

NL

Straightforward computation µT ∼ CµT

l

≃ 50 ∆4

R(kp)

l(l + 1) floc

NL b ≃ 3 × 10−16

l(l + 1) floc

NL b

b ∼ ∆2

R(kD)/∆2 R(kp), if scale invariant b ∼ 1.

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 30 / 38

µµ Gaussian self correlation

µµ ∼ R4 receives both a Gaussian and a non-Gaussian

• contributions. The Gaussian is

Cµµ

l,Gauss

∼ 6 × 10−17 ∆4

R(kD,f)

∆4

R(kp)

ksr−2

L

k3

D,f

• 1.5 × 10−28

µ fluctuations are uncorrelated at distances ∆x ≫ 1/ks White noise, l-independent Very small cosmic variance! Suppressed by N−1/2

modes with

Nmodes ∼ k3

D,f

ksr−2

L

∼ 1012

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 31 / 38

Trispectrum

Trispectrum probes non-Gaussianity R4 ∼ R22

G + R4NG

Depends on 12 − 3 − 3 − 1 = 5 variables Analogous of the local template, peaks in the collinear limit k1 ∼ k2, k3 ∼ k4 R4 ∼ τNL

• ∆2

R

k2

1

∆2

R

k2

3

∆2

R

| k1 − k2|2 + perm′s

• In general τNL > (floc

NL)2 [Suyama Yamaguchi, Futamase Komatsu Sugiyama, Smith LoVerde Zaldarriaga]

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 32 / 38

µµ non-Gaussian self correlation

The non-Gaussian contribution to µµ probes the primordial trispectrum τNL One finds Cµµ

l,NG

∼ 9 × 10−23 τNL b′ l(l + 1) , b′ ∼ ∆4

R(kD)/∆4 R(kp), if scale invariant b′ ∼ 1.

Cµµ

l

is more sensitive to non-Gaussianity than CTT

l

, since there is less cosmic variance CTT

l,NG

CTT

l

≪ Cµµ

l,NG

Cµµ

l

For small non-Guassianity there is more signal in CµT

l

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 33 / 38

Fisher matrix forecast

Signal to noise for floc

NL from CµT l

S N 2 =

• l

CµT

l

CµT

l 1 2l+1CTT l

Cµµ,N

l

A figure of merit PIXIE

[Chuss et al. ’11]

S N ≃ 10−3 b floc

NL

√ 4π × 10−8 w−1/2

µ

log lmax 80 . i.e. ∆floc

NL 103 with current technology

It probes different scales than those of T anisotropy There should be (weak) limits from WMAP and Planck

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 34 / 38

How well can we do?

Nature puts a lower bound on the noise, i.e. cosmic variance We can beat it only having more modes by N−1/2

modes

For the TTT bispectrum S N ∝ N1/2

modes ∼ lmax log1/2(lmax)

Because of Silk damping one can not do better than lmax ∼ 2000. Ideal experiment ∆floc

NL 3

For µT there are many more modes. Nature beats down cosmic variance for us S N ∝ N1/2

modes ∼

• k3

D,f

ksr−2

L

∼ 106 Ideal experiment ∆floc

NL 10−3

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 35 / 38

Outline

1 Motivations 2 Review of µ-distortion 3 A new window on primordial non-Gaussianity 4 Remarks

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 36 / 38

For the future

How would a dedicated experiment look and perform? Foregrounds? Numerical analysis is needed for detail predictions How do the structure of acoustic peaks look like in CµT

l

and Cµµ

l,NG

Bounds from Planck and WMAP? Probably weak. . . Probe other properties beyond Gaussianity? Isocurvature modes?

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 37 / 38

Summary

µ-distortion probes small, otherwise unaccesible scales µT is a direct and clean probe of the primordial bispectrum in the squeezed limit, floc

NL

µµ is a direct and clean probe of the primordial trispectrum τNL Cosmic variance is very small, allowing for a large margin of improvement

Enrico Pajer (Princeton) New window on primordial NG CWR Oct 2012 38 / 38