KIPMU Set 1: CMB Statistics. Wayne Hu

Transcription

1 KIPMU Set 1: CMB Statistics Wayne Hu

2 CMB Blackbody COBE FIRAS spectral measurement. yellblackbody spectrum. T = 2.725K giving Ω γ h 2 = GHz B ν ( 10 5 ) error frequency (cm 1 )

3 CMB Blackbody CMB is a (nearly) perfect blackbody characterized by a phase space distribution function f = 1 e E/T 1 where the temperature T (x, ˆn, t) is observed at our position x = 0 and time t 0 to be nearly isotropic with a mean temperature of T = 2.725K Our observable then is the temperature anisotropy Θ(ˆn) T (0, ˆn, t 0) T T Given that physical processes essentially put a band limit on this function it is useful to decompose it into a complete set of harmonic coefficients

4 Laplace Eigenfunctions Spherical Harmonics 2 Y m l = [l(l + 1)]Y m l Orthogonal and complete dˆnyl m (ˆn)Yl m (ˆn) = δ ll δ mm lm Yl m (ˆn)Yl m (ˆn ) = δ(φ φ )δ(cos θ cos θ ) Generalizable to tensors on the sphere (polarization), modes on a curved FRW metric Conjugation Y m l = ( 1) m Y m l

5 Multipole Moments Decompose into multipole moments Θ(ˆn) = lm Θ lm Yl m (ˆn) So Θ lm is complex but Θ(ˆn) real: Θ (ˆn) = lm = lm Θ lmyl m (ˆn) Θ lm( 1) m Y m l (ˆn) = Θ(ˆn) = lm Θ lm Yl m (ˆn) = l m Θ l m Y m l (ˆn) so m and m are not independent Θ lm = ( 1) m Θ l m

6 N-pt correlation Since the fluctuations are random and zero mean we are interested in characterizing the N-point correlation Θ(ˆn 1 )... Θ(ˆn n ) = Θ l1 m 1... Θ ln m n Y m1 l 1 (ˆn 1 )... Y m n l n (ˆn n ) m 1...m n l 1...l n Statistical isotropy implies that we should get the same result in a rotated frame R[Y m l (ˆn)] = Dm l m(α, β, γ)y m l (ˆn) m where α, β and γ are the Euler angles of the rotation and D is the Wigner function (note Yl m is a D function) Θ l1 m 1... Θ ln m n = Θ l1 m... Θ 1 l n m n Dl 1 m 1 m... D l n mn m 1 n m 1...m n

7 N-pt correlation For any N-point function, combine rotation matrices (group multiplication; angular momentum addition) and orthogonality m ( 1) m 2 m D l 1 m 1 md l 1 m 2 m = δ m1 m 2 The simplest case is the 2pt function: Θ l1 m 1 Θ l2 m 2 = δ l1 l 2 δ m1 m 2 ( 1) m 1 C l1 where C l is the power spectrum. Check = m 1 m 2 = δ l1 l 2 C l1 δ l1 l 2 δ m 1 m 2 ( 1)m 1 Cl1 D l 1 m 1 m D l 2 1 m 2 m 2 m 1 ( 1) m 1 D l 1 m 1 m 1 D l 2 m 2 m 1 = δ l1 l 2 δ m1 m 2 ( 1) m 1 C l1

8 Using the reality of the field N-pt correlation Θ l 1 m 1 Θ l2 m 2 = δ l1 l 2 δ m1 m 2 C l1. If the statistics were Gaussian then all the N-point functions would be defined in terms of the products of two-point contractions, e.g. Θ l1 m 1 Θ l2 m 2 Θ l3 m 3 Θ l4 m 4 = δ l1 l 2 δ m1 m 2 δ l3 l 4 δ m3 m 4 C l1 C l3 + perm. More generally we can define the isotropy condition beyond Gaussianity, e.g. the bispectrum Θ l1 m 1... Θ l3 m 3 = ( l1 l 2 l 3 m 1 m 2 m 3 ) B l1 l 2 l 3

9 CMB Temperature Fluctuations quadrupole, octupole; C(θ); alignment; hemisphere Angular Power Spectrum 6000 Angular Scale l(l+1)c l/2π (µk 2 ) Model WMAP CBI ACBAR Multipole moment (l)

10 Why l 2 C l /2π? Variance of the temperature fluctuation field Θ(ˆn)Θ(ˆn) = lm = l = l Θ lm Θ l m Y l m l m C l m Y m l 2l + 1 4π C l (ˆn)Yl m (ˆn) (ˆn)Y m l (ˆn) via the angle addition formula for spherical harmonics For some range l l the contribution to the variance is Θ(ˆn)Θ(ˆn) l± l/2 l 2l + 1 4π C l l2 2π C l Conventional to use l(l + 1)/2π for reasons below

11 Cosmic Variance We only have access to our sky, not the ensemble average There are 2l + 1 m-modes of given l mode, so average Ĉ l = 1 2l + 1 m Θ lmθ lm Ĉl = C l but now there is a cosmic variance σ 2 C l = (Ĉl C l )(Ĉl C l ) C 2 l For Gaussian statistics σ 2 C l = = = ĈlĈl C 2 l C 2 l 1 Θ (2l + 1) 2 C lmθ l 2 lm Θ lm Θ lm 1 mm 1 (δ (2l + 1) 2 mm + δ m m ) = 2 2l + 1 mm

12 Cosmic Variance Note that the distribution of Ĉl is that of a sum of squares of Gaussian variates Distributed as a χ 2 of 2l + 1 degrees of freedom Approaches a Gaussian for 2l + 1 (central limit theorem) Anomalously low quadrupole is not that unlikely σ Cl is a useful quantification of errors at high l Suppose C l depends on a set of cosmological parameters c i then we can estimate errors of c i measurements by error propagation F ij = Cov 1 (c i, c j ) = ll C l c i Cov 1 (C l, C l ) C l c j = l (2l + 1) 2C 2 l C l c i C l c j

13 Idealized Statistical Errors Take a noisy estimator of the multipoles in the map ˆΘ lm = Θ lm + N lm and take the noise to be statistically isotropic N lmn l m = δ ll δ mm CNN l Construct an unbiased estimator of the power spectrum Ĉl = C l Ĉ l = 1 2l + 1 Covariance in estimator l m= l ˆΘ lm ˆΘ lm C NN l Cov(C l, C l ) = 2 2l + 1 (C l + C NN l ) 2 δ ll

14 Incomplete Sky On a small section of sky, the number of independent modes of a given l is no longer 2l + 1 As in Fourier analysis, there are two limitations: the lowest l mode that can be measured is the wavelength that fits in angular patch θ l min = 2π θ ; modes separated by l < l min cannot be measured independently Estimates of C l covary on a scale imposed by l < l min Crude approximation: account only for the loss of independent modes by rescaling the errors rather than introducing covariance Cov(C l, C l ) = 2 (2l + 1)f sky (C l + C NN l ) 2 δ ll

15 Stokes Parameters Specific intensity is related to quadratic combinations of the field. Define the intensity matrix (time averaged over oscillations) E E Hermitian matrix can be decomposed into Pauli matrices P = E E = 1 2 (Iσ 0 + Q σ 3 + U σ 1 V σ 2 ), where ( 1 0 ) ( 0 1 ) ( 0 i ) ( 1 0 ) σ 0 = 0 1, σ 1 = 1 0, σ 2 = i 0, σ 3 = 0 1 Stokes parameters recovered as Tr(σ i P)

16 Stokes Parameters Consider a general plane wave solution E(t, z) = E 1 (t, z)ê 1 + E 2 (t, z)ê 2 E 1 (t, z) = A 1 e iφ 1 e i(kz ωt) E 2 (t, z) = A 2 e iφ 2 e i(kz ωt) Explicitly: I = E 1 E 1 + E 2 E 2 = A A 2 2 Q = E 1 E 1 E 2 E 2 = A 2 1 A 2 2 U = E 1 E 2 + E 2 E 1 = 2A 1 A 2 cos(φ 2 φ 1 ) V = i E 1 E 2 E 2 E 1 = 2A 1 A 2 sin(φ 2 φ 1 ) so that the Stokes parameters define the state up to an unobservable overall phase of the wave

17 Detection. This suggests that abstractly there are two different ways to detect polarization: separate and difference orthogonal modes (bolometers I, Q) or correlate the separated components (U, V ). ε 1 ε 2 OMT g 1 g 2 Q U φ V In the correlator example the natural output would be U but one can recover V by introducing a phase lag φ = π/2 on one arm, and Q by having the OMT pick out directions rotated by π/4. Likewise, in the bolometer example, one can rotate the polarizer and also introduce a coherent front end to change V to U.

18 Detection Techniques also differ in the systematics that can convert unpolarized sky to fake polarization Differencing detectors are sensitive to relative gain fluctuations Correlation detectors are sensitive to cross coupling between the arms More generally, the intended block diagram and systematic problems map components of the polarization matrix onto others and are kept track of through Jones or instrumental response matrices E det = JE in P det = JP in J where the end result is either a differencing or a correlation of the P det.

19 Polarization Radiation field involves a directed quantity, the electric field vector, which defines the polarization Consider a general plane wave solution E(t, z) = E 1 (t, z)ê 1 + E 2 (t, z)ê 2 E 1 (t, z) = ReA 1 e iφ 1 e i(kz ωt) E 2 (t, z) = ReA 2 e iφ 2 e i(kz ωt) or at z = 0 the field vector traces out an ellipse E(t, 0) = A 1 cos(ωt φ 1 )ê 1 + A 2 cos(ωt φ 2 )ê 2 with principal axes defined by E(t, 0) = A 1 cos(ωt)ê 1 A 2 sin(ωt)ê 2 so as to trace out a clockwise rotation for A 1, A 2 > 0

20 Polarization. Define polarization angle e' 2 e 2 e' 1 Match ê 1 = cos χê 1 + sin χê 2 ê 2 = sin χê 1 + cos χê 2 E(t) χ e 1 E(t, 0) = A 1 cos ωt[cos χê 1 + sin χê 2 ] A 2 cos ωt[ sin χê 1 + cos χê 2 ] = A 1 [cos φ 1 cos ωt + sin φ 1 sin ωt]ê 1 + A 2 [cos φ 2 cos ωt + sin φ 2 sin ωt]ê 2

21 Polarization Define relative strength of two principal states A 1 = E 0 cos β A 2 = E 0 sin β Characterize the polarization by two angles A 1 cos φ 1 = E 0 cos β cos χ, A 1 sin φ 1 = E 0 sin β sin χ, A 2 cos φ 2 = E 0 cos β sin χ, Or Stokes parameters by I = E 2 0, U = E 2 0 cos 2β sin 2χ, A 2 sin φ 2 = E 0 sin β cos χ Q = E 2 0 cos 2β cos 2χ V = E 2 0 sin 2β So I 2 = Q 2 + U 2 + V 2, double angles reflect the spin 2 field or headless vector nature of polarization

22 Special cases Polarization If β = 0, π/2, π then only one principal axis, ellipse collapses to a line and V = 0 linear polarization oriented at angle χ If χ = 0, π/2, π then I = ±Q and U = 0 If χ = π/4, 3π/4... then I = ±U and Q = 0 - so U is Q in a frame rotated by 45 degrees If β = π/4, 3π/4, then principal components have equal strength and E field rotates on a circle: I = ±V and Q = U = 0 circular polarization U/Q = tan 2χ defines angle of linear polarization and V/I = sin 2β defines degree of circular polarization

23 Natural Light A monochromatic plane wave is completely polarized I 2 = Q 2 + U 2 + V 2 Polarization matrix is like a density matrix in quantum mechanics and allows for pure (coherent) states and mixed states Suppose the total E tot field is composed of different (frequency) components E tot = i E i Then components decorrelate in time average E tot E tot = ij E i E j = i E i E i

24 Natural Light So Stokes parameters of incoherent contributions add I = i I i Q = i Q i U = i U i V = i V i and since individual Q, U and V can have either sign: I 2 Q 2 + U 2 + V 2, all 4 Stokes parameters needed

25 Linear Polarization Q E 1 E 1 E 2 E 2, U E 1 E 2 + E 2 E 1. Counterclockwise rotation of axes by θ = 45 E 1 = (E 1 E 2)/ 2, E 2 = (E 1 + E 2)/ 2 U E 1E 1 E 2E 2, difference of intensities at 45 or Q More generally, P transforms as a tensor under rotations and or Q = cos(2θ)q + sin(2θ)u U = sin(2θ)q + cos(2θ)u Q ± iu = e 2iθ [Q ± iu] acquires a phase under rotation and is a spin ±2 object

26 Coordinate Independent Representation Two directions: orientation of polarization and change in amplitude, i.e. Q and U in the basis of the Fourier wavevector (pointing with angle φ l ) for small sections of sky are called E and B components E(l) ± ib(l) = dˆn[q (ˆn) ± iu (ˆn)]e il ˆn = e 2iφ l dˆn[q(ˆn) ± iu(ˆn)]e il ˆn For the B-mode to not vanish, the polarization must point in a direction not related to the wavevector - not possible for density fluctuations in linear theory Generalize to all-sky: plane waves are eigenmodes of the Laplace operator on the tensor P.

27 Laplace Eigenfunctions Spin Harmonics 2 ±2Y lm [σ 3 iσ 1 ] = [l(l + 1) 4] ±2 Y lm [σ 3 iσ 1 ] Spin s spherical harmonics: orthogonal and complete dˆn s Ylm(ˆn) s Y lm (ˆn) = δ ll δ mm sylm(ˆn) s Y lm (ˆn ) = δ(φ φ )δ(cos θ cos θ ) lm where the ordinary spherical harmonics are Y lm = 0 Y lm Given in terms of the rotation matrix 2l + 1 sy lm (βα) = ( 1) m 4π Dl ms(αβ0)

28 Statistical Representation All-sky decomposition [Q(ˆn) ± iu(ˆn)] = lm [E lm ± ib lm ] ±2 Y lm (ˆn) Power spectra E lme lm = δ ll δ mm C EE l B lmb lm = δ ll δ mm C BB l Cross correlation Θ lme lm = δ ll δ mm C ΘE l others vanish if parity is conserved

29 Thomson Scattering Polarization state of radiation in direction ˆn described by the intensity matrix E i (ˆn)E j (ˆn), where E is the electric field vector and the brackets denote time averaging. Differential cross section dσ dω = 3 8π Ê Ê 2 σ T, where σ T = 8πα 2 /3m e is the Thomson cross section, Ê and Ê denote the incoming and outgoing directions of the electric field or polarization vector. Summed over angle and incoming polarization dˆn dσ dω = σ T i=1,2

30 Polarization Generation y. E mode Heuristic: incoming radiation shakes an electron in direction of electric field vector Ê Radiates photon with polarization also in direction Ê Thomson Scattering z e Quadrupole Linear Polarization x B mode But photon cannot be longitudinally polarized so that scattering into 90 can only pass one polarization Linearly polarized radiation like polarization by reflection Unlike reflection of sunlight, incoming radiation is nearly isotropic Missing from direction orthogonal to original incoming direction Only quadrupole anisotropy generates polarization by Thomson scattering k