CMB beyond the power spectrum

Antony Lewis
http://cosmologist.info/Lewis arXiv:1204.5018
Lewis arXiv:1107.5431
Lewis, Challinor & Hanson arXiv:1101.2234
Pearson, Lewis & Regan arXiv:1201.1010
Planck Collaboration 2013

CMB temperature

Last scattering surface

End of inflation

gravity+pressure+diffusion

Perturbations super-horizon

Sub-horizon acoustic oscillations+ modes that are still super-horizon

z=0θ
14 000 Mpc

z~1000Observed CMB temperature power spectrum
Primordial perturbations + known physics with unknown parameters

Observations

Constrain theory of early universe+ evolution parameters and geometry

Beyond Gaussianity – general possibilities
1Flat sky approximation: Θ 𝑥 = 2𝜋 ∫ 𝑑 2 𝑙 Θ(𝑙)𝑒 𝑖𝑥⋅ 𝑙Gaussian + statistical isotropy
Θ 𝑙1 Θ 𝑙2= 𝛿 𝑙1 + 𝑙2 𝐶𝑙- power spectrum encodes all the information- modes with different wavenumber are independent
Higher-point correlations
Gaussian: can be written in terms of 𝐶𝑙
Non-Gaussian: non-zero connected 𝑛-point functions(Θ = 𝑇)

Bispectrum

𝑙3
𝑙1

𝒍𝟏 + 𝒍𝟐 + 𝒍𝟑 = 𝟎𝑙2
Flat sky approximation:〈Θ 𝑙1 Θ 𝑙2 Θ(𝑙3 )〉 =1
𝛿 𝑙 + 𝑙2 + 𝑙3 𝑏𝑙1 𝑙2𝑙3
2𝜋 1

If you know Θ 𝑙1 , Θ 𝑙2 , sign of 𝑏𝑙1 𝑙2 𝑙3 tells you which sign of Θ 𝑙3 is more likely

Trispectrum

𝑙4
𝑙1𝐿
𝑙2

𝑙3

N-spectra…

Equilateral𝑘3

𝑘1 + 𝑘2 + 𝑘3 = 0, 𝑘1 = 𝑘2 = |𝑘3 |𝑘1
𝑘2b>0+𝑇(𝑘3 )=++𝑇(𝑘1 )b<0

𝑇(𝑘2 )−𝑇(𝑘3 )

Millennium simulation

Near-equilateral to flattened:

b>0
𝑘1

𝑘3
𝑘2b<0

Local (squeezed)
𝑘1

𝑘1 + 𝑘2 + 𝑘3 = 0,𝑘1 ≪ 𝑘2 , 𝑘3𝑘2 ∼ −𝑘3

𝑘2
𝑘3b>0+𝑇(𝑘3 )+=+𝑇(𝑘1 )T(𝑘2 )

b<0−𝑇(𝑘3 )Squeezed bispectrum is a correlation of small-scale power with large-scale modes

For more pretty pictures and trispectrum see: The Real Shape of Non-Gaussianities, arXiv:1107.5431

Calculating the squeezed bispectrum‘Linear-short leg’ approximation very accurate for large scales where cosmic variance is large
𝑙1 ≪ 𝑙2 ≤ 𝑙3

with modulation field(s) 𝑋𝑖

Correlation of the modulation
with the large-scale field

Response of the small-scale power
to changes in the modulation field(non perturbative)

Note: uses only the linear short leg approximation, otherwise non-perturbatively exact

Example: local primordial non-Gaussianity
Primordial curvature perturbation is modulated as𝑙1 ≪ 100: modulation super-horizon and constant through last-scattering, 𝜁𝑔 = 𝜁0∗But even with 𝒇𝑵𝑳 = 𝟎, we observe CMB at last scattering modulated by other perturbationsz=0

θ
14 000 Mpc

Horizon size atrecombinationz~1000What is the modulating effect of large-scale super-horizon perturbations?

Single-field inflation: only one degree of freedom, e.g. everything
determined by local temperature (density) on super-horizon scales
𝑘2Cannot locally observe super-horizon perturbations (to 𝑂(𝐻 2 ))Observers in different places on LSS will see statistically exactly the same thing(at given fixed temperature/time from hot big bang)- local physics is identical in Hubble patches that differ only by super-horizon modes

BUT: a distant observer will see modulations due to the large modes <~ horizon size today- can see and compare multiple different Hubble patches•

Super-horizon modes induce linear perturbations on all scales
linear CMB anisotropies on large scales (𝑙 < 100)

• Sub-horizon perturbations are observed in perturbed universe:-small-scale perturbations are modulated by the effect large-scale modes

squeezed-shape non-Gaussianities

Linear CMB anisotropies
Linear perturbation theory with
Using the geodesic equation in the Conformal Newtonian Gauge:

All photons redshift the same way, so 𝑘𝑇 ∼ 𝐸.

Recombination fairly sharp at background time 𝜂∗ : ~ constant temperature surface.
Also add Doppler effect:⇒Temperature
perturbation atrecombination

Sachs-WolfeDoppler

ISW

Note: no scale on which Sachs-Wolfe Φ/3 result is accurate
Doppler dominates at 𝑙 ∼ 60 because other terms cancel

𝐴
Alternative𝑑𝜂𝐧𝜕Φ𝜕Φ𝑑Φ =𝑑𝜂 +𝑑𝜒𝜕𝜂
𝜕𝜒

𝜒

𝜂∗

Gauge-invariant 3-curvature on constant temperature hypersurfaces;
Redshifting from 𝛿𝑁 expansion of the beam makes the 𝛿𝑁 expansion from inflation observable(but line of sight integral is larger on large-scales: overdensity looks colder)Non-linear effect due to redshifting by large-scale modes?

Large-scale linear anisotropies are due to the linear anisotropic redshifting of theotherwise uniform (zero-order) temperature last scattering surface

𝑇 → 1 + Δ𝑇 𝑇
Also non-linear effect due to the linear anisotropic redshifting of thelinear last scattering surface
Δ𝑇small → 1 + Δ𝑇large Δ𝑇small

Reduced bispectrum

Large-scale
power spectrum(Actually very small, so not very important)

Small-scale (non-perturbative)
power spectrumLinear effects of large-scale modes- Redshifting as photons travel through perturbed universe- Transverse directions also affected:
perturbations at last scattering are distorted as well as anisotropically redshiftedJabobi map relates observed angle to physical separation of pair of rays
𝐴𝛿𝜃𝐽

𝜁𝐼

𝜆

Physical separation
vector orthogonal to ray

Jacobi map

Angular separation seen
by observer at 𝐴Evolution of Jacobi map:

Optical tidal matrix depends on the Riemann tensor:(𝑘 𝑎 is wave vector along ray, 𝐸𝐼𝑎 projects into ray-orthogonal basis)‘Riemann = Weyl + Ricci’

Non-local part (does not depend on local density):1- e.g. determined by Weyl (Newtonian) potential (Φ + Ψ)
2

differential deflection of light rays ⇒convergence and shear of beam
- (Weyl) lensing(can be modelled as transverse deflection angle)

Einstein equations relate Ricci tostress-energy tensor: depends on localdensity⇒ ray area changes due to expansion of
spacetime as the light propagates- Ricci focussing(cannot be modelled by deflection angle)

FRW background universe has Weyl=0, Ricci gives standard angular diameter distance

At radial distance 𝜒∗ , trace of Jacobi map determines physical areas:Beam propagation in a perturbed universe, e.g. Conformal Newtonian Gauge

Trace-free part of Jacobi map depends on the shear:

Area of beam determined by trace of Jacobi map:

CMB is constant temperature surface:𝜂 = 𝜂∗ + 𝛿𝜂Radial displacement(small, 𝛿𝜒 ≪ 𝜒∗ )Local aberration

Ricci focussing(Weyl) convergence𝐷≈ 𝜒∗ 𝑎∗ (1 + 𝜁𝛾 − 𝜅)2

Ricci focussing:beam contracts more leaving LSS⇒ same physical size looks smaller

underdensity

Overdensity (𝜁 larger)(Weyl lensing effect not shown and partly cancels area effect)

Gauge-invariant Ricci focussing

Observable CMB bispectrum from single-field inflation
Linear-short leg approximation for nearly-squeezed shapes:

𝐷Where 𝑋𝑖 here is 𝛿𝑇, 𝜅 and 𝜁𝛾 , with 2 ≈ 𝜒∗ 𝑎∗ 1 + 𝜁𝛾 − 𝜅 . For super-horizon adiabatic modes 𝜁𝛾 = 𝜁0 .

Weyl lensing bispectrum+ perms

Squeezed limit (𝑙1 ≪ 𝑙)

Ricci focussing bispectrum

Squeezed limit (𝑙1 ≪ 𝑙)
+ anisotropic redshifting bispectrum (from before)

𝑇𝜓

Weyl lensing depends on cross-correlation, 𝐶𝑙(note Rees-Sciama contribution is small, numerical problem with much larger result of Verde et al, Mangilli et al.;see also Junk et al. 2012 who agree with me)Does this look like squeezed non-Gaussianity 𝑓𝑁𝐿 from multi-field inflation(local modulation of small scale perturbation amplitudes in each Hubble patch)?
forecast

Dominated by lensing 𝑓𝑁𝐿 ∼ 6 − 10
Ricci is an 𝑂(1) correction
Calculation reliable for 𝑙1 < 60 where
dynamical effects suppressed by small
𝑘2: do not need fully non-linear
dynamical calculation of bispectrum ala Pitrou et al to make reliable 𝑓𝑁𝐿
constraint𝐻2

Signal easily modelled
Squeezed shape but different phase, angle and scale dependence

Lensing

𝑓𝑁𝐿

Lewis, Challinor, Hanson 1101.2234

Note: ‘Maldacena’ bispectrum

Consistency relation: 𝑓𝑁𝐿 ∼ 𝑂(𝑛𝑠 − 1)
is not an observable- cannot measure comoving curvature perturbations on scales larger than the horizon directly- 𝑑/𝑑(ln 𝑘) and CMB transfer functions do not commute: cannot get correct result from primordial 𝑓𝑁𝐿 ∼ (𝑛𝑠 − 1)

Observable CMB analogue is Ricci focussing bispectrum- larger because of acoustic oscillations, non-zero for 𝑛𝑠 = 1- different shape to 𝑓𝑁𝐿 in CMB, but projects as 𝑓𝑁𝐿 = 𝑂(1)Planck resultDiagonal squeezed trispectra

𝑘1

𝑘1 ~ 𝑘2 , 𝑘3 ~ 𝑘4 , 𝑘1 + 𝑘2 = 𝑘3 + 𝑘4 ≪ 𝑘2 , |𝑘3 |Trispectrum = power
spectrum of modulation

𝑘2

𝑘4e.g. 𝜒 = 𝜒0 1 + 𝑓𝑁𝐿 𝜒02
𝜏𝑁𝐿 ∼ 𝑓𝑁𝐿𝑘3or 𝜒 = 𝜒0 1 + 𝜙2(any correlation, 𝜏𝑁𝐿 > 𝑓𝑁𝐿)

𝑘1

𝑘2

𝑘1

𝑘2

𝑘4

𝑘3

𝑘4

𝑘3

One-leg squeezed trispectra
𝑘3

𝑘1 ≪ 𝑘2 ~ 𝑘3 ~|𝑘4 |𝑘2Correlated modulation of equilateral bispectrum
𝑘1e.g. from 𝜒 = 𝜒0 1 + 𝑔𝑁𝐿 𝜒02 [no diagonal squeezed]

𝑘4
Trispectrum>0
also from 𝜒 = 𝜒0 1 + 𝑓𝑁𝐿 𝜒0[plus diagonal squeezed]

Trispectrum<0

Reconstructing the modulation fieldMarginalized over modulation field:

𝑇 ∼ ∫ 𝑃(𝑇, 𝜓)𝑑𝜓
- Non-Gaussian statistically isotropic temperature distribution- Bispectrum + significant connected 4-point function

For a given modulation field :𝑇 ∼ 𝑃(𝑇|𝜓)- Anisotropic Gaussian temperature distribution- Re-construct the modulation fieldAnisotropy estimators – e.g. reconstruct the modulating lensing field
MagnifiedUnlensed

Demagnified

Reconstructing the modulation field
For a given (fixed) modulation field:

𝑇 ∼ 𝑃(𝑇|𝑋)- Anisotropic Gaussian temperature distribution- Modes correlated for 𝐤 2 ≠ 𝐤 3

Model-dependent function you
can calculate for 𝑋(𝐊) = 0

Can reconstruct themodulation field 𝑋!

- For small modulations can construct “optimal” QML estimator 𝑋 𝐾
by summing filtered fields appropriately over 𝑘2 , 𝑘3𝑋 𝐾 ∼ 𝑁(𝐤 2 ,𝐤3 𝐴 𝐾, 𝑘2 , 𝑘3

𝑇 𝐤 2 𝑇 𝐤 3 − mean field)
Zaldarriaga, Hu, Hanson, etc..

Planck lensing potential reconstruction (north and south galactic)

Note – about half signal, half noise, not all structures are real
map is effectively Wiener filtered
Correlation with T ⇒ Bispectrum; Power spectrum ⇒ Trispectrum

Primordial curvature modulation:

Primordial modulations?

z=0θ
14 000 Mpc

z~1000

Squeezed shape ⇒ large-scale modulation⇒Complication: Kinematic dipole modulation signalModulation𝑣

Illustrated for 𝑐 = 0.85

Aberration𝒏 → 𝒏 + 𝛻(𝒏 ⋅ 𝒗)
- just like a dipole lensing
convergence

Challinor & van Leeuwen 2002

known dipole amplitude and direction𝑣≈ 1.23 × 10−3𝑐𝑥
2Modulation 𝑓 = 𝑥 coth 2 − 1 𝒏 ⋅ 𝒗 ≡ 𝑏𝜈 𝒏 ⋅ 𝒗
Approx boost factor

𝑥𝑏𝜈 = 𝑥 coth/2 − 12Planck maps:

100 GHz143 GHz
217 GHz
353 GHz
545 GHz857 GHz

1.5235914

Map modulation
amplitude

0.18%
0.24%
0.37%
0.64%
1.1%
1.7%

Use 143, 217 only (with dust subtraction from 857)
Note: SMICA maps are a complicated mixture; modulation effect not currently included in FFP6 sims

143x217 modulation reconstruction (𝐿 ≤ 5)
- map of estimated modulation field 𝑓

Kinematics not subtracted

Kinematics subtracted
in mean field from sims

Dipole kinematic effect using appropriate quadratic estimators

Note: not included in parameter analysis- bias due to aberration average over mask ∼ 0.25𝜎

𝑓Modulation pseudo-power spectrum

Consistent with zero except for anomalous octopolePower dipole directions (𝑙 ≤ 𝑙max )Kinematic subtracted

Kinematics not subtracted(as in Doppler paper but here pure modulation estimator)

Power modulation dipole? Result for amplitude at 𝑙 ≤ 𝑙max
WMAP 5 (Hanson & Lewis 2009)

Planck 217x143 (kinematic subtracted)

1%
0.1%

Modulation < 1% for 𝑙 ≤ 𝑙max = 500

Modulation < 0.2% for 𝑙max = 1500 − 2000

Using full resolution to estimate local trispectrum
Local trispectrum often measured bymodulation power(optimal to percent level)

Conventional normalization toprimordial power

Planck 𝝉𝑵𝑳 trispectrum constraint
Estimator result
Gaussian simulations:

Consistent with Gaussian null hypothesis (octopole has small weight)

Conservative upper limit, allowing octopole to be physicalusing Bayesian posterior

Conclusions• Single field inflation predicts significant non-Gaussianity in the observed CMB- mostly due to (Weyl) lensing- total projects onto 𝑓𝑁𝐿 ∼ 7 for Planck temperature- Ricci focussing expansion of beam recovers the 𝛿𝑁 from inflation,: gives equivalent of consistency relation, but larger: small and not quite observable, projects on to 𝑓𝑁𝐿 ∼ 1• Lensing bispectrum signal important but distinctive shape- Marginally detected by Planck• Local Doppler signal predicted in trispectrum (dipole modulation)• No sign yet of primordial NG once lensing and Doppler signals subtracted..