Cosmology & CMB Set6: Polarisation & Secondary Anisotropies Davide Maino
Polarisation How? Polarisation is generated via Compton/Thomson scattering (angular dependence of the scattering term M) Who? Only quadrupole anisotropy is able to produce linear polarisation When? Near recombination Θ 2 is at 5-10% level of Θ 0
Polarisation Linear polarisation is described by Stokes Q and U parameters Q measures differences of intensities along the x and y axes, while U measures same differences but in a 45 rotated frame Stokes are not in general rotationally invariant Q = Qcos(2ψ) Usin(2ψ) U = Ucos(2ψ) + Qsin(2ψ) Q ± iu = e ±2iψ (Q ± iu) (Q ± iu) transform like a spin-2 tensor
Polarisation: Spin weighted spherical harmonics A general spin-2 tensor on a sphere is properly represented by spin-s spherical harmonics s Y lm Under rotation: s Y lm e ±isψ sy lm Normality and completeness dˆn s Ylm (ˆn) sy l m (ˆn) = δ ll δ mm sylm (ˆn) sy lm (ˆn ) = δ(φ φ )δ(cosθ cos(θ ) lm
Polarisation: All-sky decomposition [Q ± iu](n) = l,m a ±2 lm ±2Y lm (n) = l,m (a E,lm ± ia B,lm ) ±2 Y lm (n) E (gradient) and B (curl) modes are a E,lm = 1 2 (a+2 lm + a 2 lm ) a B,lm = i 2 (a+2 lm a 2 lm )
Polarisation: Power Spectra Four parity-independent power spectra C TT = 1 2l + 1 C EE = 1 2l + 1 a T,lm a T,lm C TE = 1 a T,lm 2l + 1 a E,lm m m a E,lm a E,lm C BB = 1 a B,lm 2l + 1 a B,lm m E-modes are generated by both scalar (density) and tensor (gw) perturbation B-modes are generated only by tensor perturbation: direct probe of inflation m
Polarisation: Power Spectra
Current Status of measurements Planck-2015 provides the best current measurements of E and TE power spectra
Secondary Anisotropies CMB photons traverse the large-scale structure of the Universe from z 1000 to the present Having nearly scale-invariant adiabatic fluctuations (from CMB observations), structures form bottom-up i.e. small scales first. Hierarchical structure formation. First objects (starts) re-ionize the Universe between z 7 30 Gravitational: Integrated Sach-Wolfe effect (ISW), gravitational lensing Scattering: peak suppression, large-angle polarisation (Compton scattering)
Gravitational Secondaries
Transfer Function We observe the distribution of matter predominantly at late epochs (now) It is possible to relate potential at these times to the primordial one Φ(k, a) = Φ p (k) {Transfer Function(k)} {Growth Factor(a)} where Φ p is the primordial value of the potential (i.e. end of inflation). The transfer function describes the evolution of perturbation from horizon crossings to radiation/matter transition The growth factor describes wavelength-independent growth at late times
Transfer Function The transfer function is usually defined as T(k) Φ(k, a late ) Φ Large Scale (k, a late ) where a late is an epoch after transfer function regime (i.e. when T(k) is constant) Φ Large Scale = 9/10Φ p The growth factor is instead Φ(a) Φ(a late ) D 1(a) a a > a late Looking at matter-dominated case where Φ is constant we have D 1 (a) = a and overdensities grows with the scale factor δ a
Growth Factor Sensible to the actual content of the Universe (dark matter, dark energy and their relative amount)
Integrated Sachs-Wolfe (ISW) effect Sensible to potential decay leading to gravitational red-shift Intrinsically a large effect on the anisotropies of the CMB Net change is integrated along a given line-of-sight Θ l (k, η 0 ) 2l + 1 = η0 0 = 2Φ(k, η MD ) [ dηe τ 2 Φ(k, ] η) j l (k(η 0 η)) η0 0 dηe τ ġ(η η 0 )j l (k(η η 0 )) On small scales where k ġ/g the integral becomes η0 dη ġ(d) j l (kd) ġ(d = l/k) 1 π k 2l 0 where we used dx j l (x) = π/2l
Integrated Sachs-Wolfe (ISW) effect On the power spectrum C l = 2 dk k 3 Θ l (k, η 0)Θ l (k, η 0 ) π k (2l + 1) 2 = 2π2 l 3 dη Dġ 2 (η) 2 Φ(l/D, η MD ) For scale-invariant potential we have l 2 C l 1/l. Of course depends on the details of ġ: particular important for dark energy models Net effect (enhancement) on CMB is much more evident at low (l < 10) multipoles Difficult to measure due to cosmic variance and local contaminations: cross-correlate CMB results with other tracers of structure formation to get hints on D 1
Integrated Sachs-Wolfe (ISW) effect
Integrated Sachs-Wolfe (ISW) effect Power spectrum sampling errors = [(l + 1/2)f sky ] 1/2 Low multipole effects severely cosmic variance limited
Gravitational Lensing Lensing is a remapping conserving surface brightness of source image reprojected via the gradient of a projected potential φ(ˆn) = 2 η0 such that fields are re-mapped as η dη (D D) DD Φ(Dˆn, η) x(ˆn) x(ˆn + φ) where x {Θ, Q, U} temperature and polarization Taylor expansion leads to product of fields and Fourier mode-coupling
Flat-sky Treatment Taylor expand Θ(ˆn) = Θ(ˆn + φ) = Θ(ˆn) + i φ(ˆn) i Θ(ˆn) + 1 2 iφ(ˆn) j φ(ˆn) i j Θ(ˆn) +... Fourier decomposition φ(ˆn) = Θ(ˆn) = d 2 l (2π) 2 φ(l)eil ˆn d 2 l il ˆn Θ(l)e (2π) 2
Flat-sky Treatment Mode coupling harmonics Θ(l) = dˆnθ(ˆn)e il ˆn = Θ(l) d 2 l 1 (2π) Θ(l 2 1 )L(l, l 1 ) where L(l, l 1 ) = φ(l l 1 )(l l 1 ) l 1 + 1 d 2 l 2 2 (2π) 2 φ(l 2)φ (l 2 + l 1 l)(l 2 l 1 )(l 2 + l 1 l) l 1 Represent a coupling between harmonics mess up the original CMB angular power spectrum
Power Spectrum Power spectra becomes C ΘΘ l Θ (l)θ(l ) = (2π) 2 δ(l l )Cl ΘΘ φ (l)φ(l ) = (2π) 2 δ(l l )C φφ l d = (1 l 2 ) C l ΘΘ 2 l 1 ΘΘ + C (2π) 2 l l 1 Cφφ l 1 [(l l 1 ) l 1 ] 2 where R = 1 dl 4π l l4 C φφ l
Smoothing of Power Spectra If C ΘΘ l const we can move it out from the integral d C l ΘΘ 2 l 1 (2π) 2 Cφφ l (l l 1 ) 2 l 2 R C ΘΘ Lensing acts to smooth features in the power spectrum Lensing generates power below the damping (last-scattering) scale l
Lensing in the Power Spectrum Lensing width l 60 10 9 ΘΘ l(l+1)c l /2π 10 10 10 11 10 12 10 13 lensed unlensed all flat error 10 11 ΘΕ l(l+1)c l /2π 10 12 10 13 10 14 10 15 10 100 1000 l
Lensing Potential Power Spectrum Planck-2015 produces best measurement of C φφ l
Polarisation Lensing Since E and B are relationships between polarisation amplitude and direction we expect lensing to create B modes Original Lensed E Lensed B
Polarisation Lensing Polarisation field harmonics lensed similarly [Q ± iu](ˆn) = Again with Taylor expansion [Q ± iu](ˆn) = [ Q ± iũ](ˆn + φ) d 2 l (2π) 2 [E ± ib] (l)e±2iφ l e l ˆn [ Q ± iũ](ˆn) + i φ(ˆn) i [ Q ± iũ](ˆn) + 1 2 iφ(ˆn) j φ(ˆn) i j [ Q ± iũ](ˆn)
Polarisation Lensing: power spectra In terms of power spectra (Zaldarriaga & Seljak 98) C TT = C TT + W l C EE = C EE + 1 [ 2 1l C TT W l 1l + Wl 2l [ W l 1l Wl 2l C BB = C BB + 1 2 C TE = C TE + W3l l TE ] C EE + 1 2 ] C EE 1 2 [ W1l 2l] l C Wl BB [ W1l 2l] l + C Wl EE Lensing mixes E and B modes: even with no primordial GW, B-modes are generated at small scales
Polarisation Lensing: power spectra
Scattering Secondaries
Scattering Secondaries Optical depth during reionization ( Ωb h 2 ) ( Ωm h 2 ) 1/2 ( ) 1 + z 3/2 τ 0.066 0.02 0.15 10 Anisotropies suppressed as e τ Θ l (k, η 0 ) η0 2l + 1 = dηe τ [2 Φ(k, η)] j l (k(η 0 η)) +... 0 Large scale fluctuations not suppressed since scattering acts on smaller scales Quadrupole from Sachs-Wolfe effect scatters into a large scale polarization bump
Thermal SZ effect Thermal velocities lead to Doppler effect but cancels out at first order due to random directions Residual effect of the order v 2 τ T e /m e τ and could be important for T e 10keV (gas within cluster of galaxies) RJ decrement and Wien enhancement described by second order collision term in Boltzmann equation: Kompaneets equation Clusters are rare objects so contribution to power spectrum suppressed but has been detected already by Planck: sensitive to (matter) power spectrum normalization σ 8
Thermal SZ effect
Thermal SZ effect
tsz & ksz effects - Unresolved sources Un-resolved sources contributes as a background noise on very small angular scales Both radio and ir-sources contributes and multi-frequency experiments can distinguish contributions
Where to go?
Where to go?