Multipole fine structure of the cosmic microwave background: reconstruction of the temperature power spectrum
Article first published online: 19 NOV 2012
DOI: 10.1111/j.1365-2966.2012.22024.x
© 2012 The Author Monthly Notices of the Royal Astronomical Society © 2012 RAS
Monthly Notices of the Royal Astronomical Society
Volume 427, Issue 2, pages 1363–1383, 1 December 2012
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Tomaschitz, R. (2012), Multipole fine structure of the cosmic microwave background: reconstruction of the temperature power spectrum. Monthly Notices of the Royal Astronomical Society, 427: 1363–1383. doi: 10.1111/j.1365-2966.2012.22024.x
Publication History
- Issue published online: 5 NOV 2012
- Article first published online: 19 NOV 2012
- Manuscript Accepted: 31 AUG 2012
- Manuscript Received: 27 AUG 2012
Keywords:
- cosmic background radiation;
- cosmology: theory
ABSTRACT
- Top of page
- ABSTRACT
- 1 INTRODUCTION
- 2 CMB TEMPERATURE FLUCTUATIONS: GENERAL SETTING
- 3 MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE
- 4 RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM
- 5 MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS
- 6 OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS
- 7 CONCLUSION
- REFERENCES
- Appendix A
- Appendix B
The fine structure of the temperature power spectrum of the cosmic microwave background (CMB) radiation is investigated in the presently accessible multipole range up to l ∼ 104. The temperature fluctuations are reproduced by an isotropic Gaussian random field on the unit sphere, whose Green function is defined by a Hermitian matrix kernel inferred from the data sets by way of spectral fits. The reconstruction of the temperature autocorrelation function from the measured multipole moments Cl is a classical inverse problem, which does not require specification of cosmic evolution equations for the photon density. The scale-invariant correlation function admits a multipole expansion in zonal spherical harmonics. The multipole coefficients are obtained as averages over Hermitian spectral matrices determining the angular power spectrum of the spherical random field. The low-l multipole regime of the CMB temperature fluctuations is composed of overlapping Gaussian peaks, followed by an intermediate oscillatory regime manifested by a modulated exponentially decaying Cl slope. The high-l regime above l ∼ 4000 comprises a power-law ascent with exponential cut-off. The fine structure of the Gaussian, oscillatory and high-l regimes is reproduced by zooming into the respective l intervals on linear and logarithmic scales.
1 INTRODUCTION
- Top of page
- ABSTRACT
- 1 INTRODUCTION
- 2 CMB TEMPERATURE FLUCTUATIONS: GENERAL SETTING
- 3 MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE
- 4 RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM
- 5 MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS
- 6 OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS
- 7 CONCLUSION
- REFERENCES
- Appendix A
- Appendix B
The goal is the reconstruction of the temperature multipole spectrum of the cosmic microwave background (CMB) from the measured data sets. There are now quite precise measurements available, stretching over a multipole range of up to l ∼ 104 (Jones et al. 2006; Brown et al. 2009; Reichardt et al. 2009, 2012; Sievers et al. 2009; Nakamura et al. 2010; Das et al. 2011; Jarosik et al. 2011; Keisler et al. 2011; Larson et al. 2011). This makes it worthwhile to have a closer look at the multipole fine structure of the temperature autocorrelation function throughout this range, by zooming into subintervals, and to figure out ways of modelling it. We will find an isotropic Gaussian random field on the unit sphere with an analytically tractable Green function capable of reproducing the observed fine structure of the CMB temperature power spectrum over the complete multipole range accessible today.
In Section 'CMB TEMPERATURE FLUCTUATIONS: GENERAL SETTING', we sketch the general setting, the Planckian photon distribution with a fluctuating temperature variable treated as spherical Gaussian random field. We develop the formalism of random fields on the two-sphere to the extent needed to model fluctuating CMB power spectra. We study spherical Green functions with Hermitian matrix kernels, and perform the multipole expansion thereof. In Appendix Appendix, we sketch orthogonality and completeness relations of Legendre expansions in zonal spherical harmonics, appropriate for isotropic scalar random fields. The Gaussian random field is completely determined by specifying the positive-definite Hermitian kernel of the two-point correlation function, which can be inferred from a multipole spectral fit.
This approach to CMB fluctuations deviates from Green function techniques traditionally used in field theory, which are based on evolution equations derived from a Hamiltonian or Lagrangian. Here, we reconstruct the spectral kernel of the spherical Gaussian random field from a fit of the temperature power spectrum, now available in good accuracy over an extended multipole range. Spectral fits of multipole moments Cl are usually presented on compressed linear or logarithmic multipole scales, which tend to conceal the fine structure of the data sets. Here, we employ an analytic method based on Hermitian spectral matrices, which is quite explicit and capable of reproducing the fine structure of the 〈TT〉 autocorrelation in the resolution observable today. This reconstruction of the Green function of the spherical random field from the actual data sets is particularly attractive with regard to CMB power fluctuations, as it does not require specification of cosmic interaction mechanisms of the photon density.
In Section 'MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE' and Appendix Appendix, we study the Hermitian spectral matrices in the integral kernels of the multipole coefficients. We use an Euler-type representation assembled from SU(N) subgroups and diagonal matrices with Gaussian power-law densities defining the spectral amplitudes. As for CMB correlations, it suffices to consider two-dimensional unitary groups, which generate the modulations seen in the intermediate multipole range.
In Section 'RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM', we derive scaling relations for the multipole coefficients Cl of the spherical random field; the CMB temperature power fits are performed in the scale-invariant limit. The multipole coefficients are obtained by averaging products of spherical Bessel functions and derivatives thereof with Hermitian spectral matrices. In the scale-invariant CMB fits, all Bessel derivatives drop out, so that the spectral average only involves squares of spherical Bessel functions.
In Section 'MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS', we perform the multipole fit of the CMB temperature fluctuations. Figs 1-5 give an overview: the spectral map in Fig. 1 covers the complete multipole range investigated (1 ≤ l ≤ 105). The low-l region comprises a precursory Gaussian regime composed of merged peaks of roughly equal height, followed by a main peak, which is likewise Gaussian, cf. Fig. 2. This is followed by a transitional regime of two non-Gaussian peaks, terminating in an oscillatory descending slope, cf. Fig. 3. The high-l regime consists of a slowly rising power-law slope, cf. Fig. 4, terminating in exponential decay, and producing a peak at about l ∼ 15 400, cf. Fig. 5. The quality of the depicted data sets allows us to zoom into the enumerated multipole regimes and to reconstruct the Cl fine structure, cf. Figs 6-13. In Section 'OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS', we sketch multicomponent spherical random fields, and discuss cosmic variance in the context of this reconstruction. In Section 'CONCLUSION', we present our conclusions.
2 CMB TEMPERATURE FLUCTUATIONS: GENERAL SETTING
- Top of page
- ABSTRACT
- 1 INTRODUCTION
- 2 CMB TEMPERATURE FLUCTUATIONS: GENERAL SETTING
- 3 MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE
- 4 RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM
- 5 MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS
- 6 OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS
- 7 CONCLUSION
- REFERENCES
- Appendix A
- Appendix B
We study angular fluctuations of the temperature in the Planckian photon distribution, , where kB is the Boltzmann constant, and we have put ℏ = c = 1. The angular dependent background temperature is denoted by Tb(p 0), where the angular variable p 0 is the unit vector of the photon momentum . It is convenient to factorize the temperature field as , where T0 ≈ 2.7 K is the present-day mean background temperature and is the fluctuating field with zero mean. We conformally rescale Tb with the cosmic expansion factor, , where T(p 0) stands for the angular fluctuations δTb/T0. This determines the cosmic time dependence of the distribution function f(p); at the present epoch τ0, the expansion factor can be chosen as a(τ0) = 1. To preserve the conformal time scaling, we do not assume a time dependence of the fluctuating temperature variable , which will be treated as Gaussian random field (scalar and isotropic) on the unit sphere .
The following reconstruction of the Green function of the CMB temperature fluctuations does not require any specific cosmological model; it applies irrespectively of the expansion factor, curvature sign and topology of the cosmic 3-space. In this section, we give a self-contained derivation of the two-point autocorrelation function 〈TT〉 of the spherical random field employed in the multipole fit of the temperature power spectrum in Section 'MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS'.
2.1 Legendre expansion of delta distributions on the unit sphere
We start with a Gaussian random field T(p) in Euclidean space, and then restrict to the unit sphere . The conventions for three-dimensional Fourier transforms are
- (2.1)
Reality of T(p) requires . Fourier transforms are denoted by a hat. The exponential admits a standard expansion in Legendre polynomials (Newton 1982):
- (2.2)
Unit vectors are denoted by a subscript zero, k = k k 0 and p = p p 0. The jl(x) are spherical Bessel functions, , where l = 0, 1, 2, … , and the Pl(x) are Legendre polynomials. We introduce polar coordinates with k or p as polar axis and as polar angle, substitute into equation (2.2), and differentiate n times with respect to kp to find
- (2.3)
where . The superscript (n) denotes the nth derivative and . The Poisson integral representation of the spherical Bessel functions in equations (2.2) and (2.3) reads (Magnus, Oberhettinger & Soni 1966)
- (2.4)
We consider the distributions
- (2.5)
where and q = q q 0. The parameters p, q, k and k′ are non-negative real numbers. The solid-angle increment indicates integration over the unit sphere; , in polar coordinates with polar axis k 0, so that in equation (2.1). In equation (2.5), we substitute series (2.3) for and . The angular integration can readily be carried out by using the orthogonality relation of Legendre polynomials on the unit sphere, cf. (A3) and (Landau & Lifshitz 1991)
- (2.6)
In this way, we obtain the expansion of Dm, n in Legendre polynomials,
- (2.7)
These distributions are real, and their symmetry properties with regard to a simultaneous interchange of indices and arguments are evident from this expansion. Dm, n can be obtained from by multiple differentiation,
- (2.8)
We perform a radial integration of Dm, n, which defines the kernel
- (2.9)
This can also be written as, cf. (2.8),
- (2.10)
Employing the series expansion in equation (2.7), we obtain
- (2.11)
Here, the Bessel integral is a representation of the Dirac function (Jackson 1999)
- (2.12)
valid for integer l ≥ 0 and positive p and q. We use the Legendre representation of the delta function on the unit sphere, cf. (A2) and (A7),
- (2.13)
to factorize kernel (2.9),
- (2.14)
The delta function in Euclidean 3-space can be split as, cf. Appendix Appendix,
- (2.15)
so that
- (2.16)
This Cartesian representation of kernel (2.9) can directly be recovered from equations (2.5) and (2.10).
2.2 Temperature autocorrelation function
We define the correlation function of the Fourier components in equation (2.1) as
- (2.17)
Isotropy requires the power spectrum g00(k) to depend only on , and the delta function reflects homogeneity in Euclidean 3-space, so that the Fourier transform of only depends on the distance , cf. (2.21) and (2.25).
In polar coordinates, the Euclidean delta function factorizes as in equation (2.15), so that
- (2.18)
where denotes the delta function on the unit sphere, cf. Appendix Appendix and equation (2.13). Isotropy is ensured by , which is the only angular dependent factor. We abandon homogeneity (since the random field will ultimately be restricted to the unit sphere) by replacing the singular radial factor g00(k)δ(k − k′)/k2 by a more general kernel function,
- (2.19)
where k = k k 0, and
- (2.20)
Here, gmn(k) is an N-dimensional matrix, which will be chosen as positive-definite or semidefinite Hermitian. At this stage, we do not impose any symmetry requirements on gmn(k), which is thus an arbitrary complex N × N matrix. The homogeneous case (2.18) corresponds to N = 1.
The Fourier transform of the two-point function (2.19) is defined as
- (2.21)
where p = p p 0 and . Zero subscripts denote unit vectors. We may write this as
- (2.22)
One of the angular integrations can readily be carried out by virtue of the delta function, cf. (2.5),
- (2.23)
We substitute Δ(k, k′) in equation (2.20), perform the partial integrations, use equation (2.8) and perform one integration by means of the delta function, to find the representation
- (2.24)
There are several ways to proceed. First, we may substitute
- (2.25)
Alternatively, we may use equation (2.8) to write (2.24) as
- (2.26)
Finally, we may substitute the Legendre expansion (2.7) of D0, 0 into equation (2.24) to find
- (2.27)
where we identified the multipole moments as
- (2.28)
One of the radial integrations in equation (2.28) is carried out by means of the delta function in Δ(k, k′), cf. (2.20), and we find, by multiple partial integration,
- (2.29)
This representation (equations (2.27), (2.28), (2.29)) of the Green function can be recovered by substituting the Legendre expansion (2.7) of Dm, n into (2.26).
We denote the two-point function (2.27) by , regarding it as an isotropic kernel on the unit sphere depending on the angle and two arbitrary positive scale-parameters p and p′, cf. (2.19) and (2.21). The symmetry properties of with respect to p and p′ depend on the coefficients Cl(p, p′) in equation (2.29). The Cl(p, p′) are symmetric in p and p′ if the matrix gmn(k) is symmetric, and they are real if gmn(k) is real. If the matrix gmn(k) is Hermitian, we find . These symmetries of Cl(p, p′) are inherited by the Green function .
If p′ = p (which will be assumed in the sequel), we write Cl(p) or simply Cl for Cl(p, p) in equation (2.29) and or for the correlation function in equation (2.27). The coefficients Cl(p) are real if gmn(k) is Hermitian or an arbitrary real matrix. Positivity of Cl(p) is ensured if gmn(k) is a positive-definite Hermitian (or real symmetric) matrix. The condition Cl(p) ≥ 0 is satisfied if gmn(k) is semidefinite. Thus, is a positive (semi)definite kernel on the unit sphere if gmn(k) is a positive (semi)definite Hermitian matrix. Reality, symmetry and positive definiteness are the requirements for to be a Gaussian correlation function, cf. Section 'OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS'.
3 MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE
- Top of page
- ABSTRACT
- 1 INTRODUCTION
- 2 CMB TEMPERATURE FLUCTUATIONS: GENERAL SETTING
- 3 MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE
- 4 RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM
- 5 MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS
- 6 OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS
- 7 CONCLUSION
- REFERENCES
- Appendix A
- Appendix B
3.1 Hermitian spectral matrices defining angular power spectra
The Green function (2.27) is defined by multipole moments depending on a positive-definite or semidefinite Hermitian N × N matrix gmn(k), m, n = 0, … , N − 1, and a scale parameter p,
- (3.1)
Multiple derivatives of the spherical Bessel functions jl(x) are indicated by superscripts (m) and (n). We will also occasionally use a superscript (N) for the matrix dimension, mainly N = 1 and 2 in this paper. The case N = 1 can readily be settled; we write P(k) for density to obtain the moments (3.1) as
- (3.2)
As for N = 2, we factorize the 2 × 2 matrix gmn(k) as
- (3.3)
where the diagonal matrix is defined by real constants A ≥ 0, B ≥ 0, and is a unitary matrix of determinant 1, parametrized as
- (3.4)
so that . The Hermitian gmn(k) thus reads
- (3.5)
with m, n = 0, 1 and . This parametrization covers all two-dimensional positive semidefinite Hermitian matrices. gmn(k) depends on four independent real parameters (A, B, θ, ϕ), where ϕ = (χ − ψ)/2. These four parameters can be arbitrary real functions of the spectral variable k. The case ϕ = 0 is just a rotation in the Euclidean plane, resulting in a real symmetric matrix. In higher dimensions, N ≥ 3, we use subgroups Ui(θi, ψi, χi) as in equation (3.4) to obtain an Euler-type parametrization of the Hermitian matrix, cf. Appendix Appendix. In the case that gmn(k) factorizes as , we can substitute in equation (3.1)
- (3.6)
More generally, we can always split this quadratic form into a sum of N squares by diagonalization as in equation (3.3). In two dimensions, cf. (3.5),
- (3.7)
where we have put χ − ψ = 2ϕ. Here, the vectors hm(k) defining the squares in equation (3.5) are just the rows of matrix in equation (3.4), multiplied with a convenient phase factor and the root of the respective coefficient in the diagonal matrix in equation (3.3); the same holds for higher dimensions. A diagonal results in a series of squared derivatives . Positive definiteness of requires A > 0 as well as B > 0.
According to equation (3.7), the multipole coefficients in equation (3.1) can be decomposed as
- (3.8)
- (3.9)
where
- (3.10)
The argument of the Bessel functions is kp, and the amplitudes and angles A, B, θ and ϕ depend on the spectral variable k as indicated in equation (3.7). If we put in matrix (3.5), the mixed terms drop out.
When performing the integrations (3.9), it is convenient to write the spectral functions (3.10) linear in the harmonics,
- (3.11)
We note that and only differ by a change of sign of the harmonics, apart from the amplitudes A and B, and do not depend on the sign of the angle ϕ. The amplitudes A(k) and B(k) as well as the angles θ(k) and ϕ(k) will be specified in equations (3.15) and (3.16).
We consider linear combinations of the Green functions in equation (3.1),
- (3.12)
where the summation is taken over a set of one- and two-dimensional matrices gmn(k). The multipole coefficients Cl(p) in equation (3.12) are obtained by adding the coefficients of the respective components , cf. (3.1). For instance, on adding in equation (3.2) and in equations (3.8) and (3.9), we find
- (3.13)
As for the integral kernels, P(k) is a density specified in equation (3.14), and the spectral functions and are stated in equation (3.11), with angular parametrization (3.15) and amplitudes (3.16). More generally, any Green function (3.12) obtained by summation over a finite set of positive (semi)definite Hermitian matrices (of the same or varying dimension N) is a positive-definite or semidefinite Hermitian kernel, and the same holds for linear combinations with positive coefficients. We may also use different scale parameters p in each of the component functions in equation (3.12). In the following, we will perform a summation over one- and two-dimensional matrices as in equation (3.13), using the same scale parameter p in each component .
3.2 Spectral parametrization of multipole moments: Gaussian power-law densities and Kummer distributions
As for the coefficients in equation (3.2), we parametrize the kernel P(k) with a series of Gaussian power-law densities,
- (3.14)
with amplitudes a0, i > 0 and real exponents μ0, i, β0, i, and α0, i > 0. This series corresponds to a summation over a set of one-dimensional matrices in equation (3.12).
Regarding the coefficients in equation (3.8), we need to specify the k dependence of the angles and amplitudes in the 2 × 2 matrix (3.5) and the associated spectral functions in (3.11). We use a linear k parametrization of the angles,
- (3.15)
where ω, θ0, ω0 and ϕ0 are real constants. When performing the CMB temperature fit, it suffices to put θ0 = 0 and ϕ(k) = 0 from the outset. The amplitudes are Gaussian power laws like in equation (3.14),
- (3.16)
with a ≥ 0, b ≥ 0 and real exponents μ1, 2, β1, 2 and α1, 2. In the CMB temperature fit, we use α1, 2 = 0 and β1, 2 > 0, that is, power-law densities with exponential cut-off. If the amplitudes a and b in equation (3.16) have opposite sign, the Hermitian spectral matrix (3.5) is indefinite, but the multipole coefficients in equation (3.13) can still be positive for all l. Similarly, if some of the Gaussian amplitudes a0, i in equation (3.14) are negative, the total moments Cl(p) in equation (3.13) can still be positive. Positivity of the amplitudes is sufficient but not necessary to ensure a positive-definite kernel (3.12).
On substituting the angles (3.15) and amplitudes (3.16) into the spectral functions in equation (3.11), we find
- (3.17)
where and denote the terms depending on in equation (3.11),
- (3.18)
The terms and in equation (3.17) contain the squared derivatives as factor,
- (3.19)
The contributions and to the spectral functions (3.17) stem from the mixed terms in equation (3.11),
- (3.20)
and
- (3.21)
The multipole coefficients in equation (3.13) are obtained by integration of the spectral functions (3.17), (3.18), (3.19), (3.20), (3.21), cf. Section 'RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM'. We have written the harmonics depending on the spectral variable k as real and imaginary parts of exponentials to facilitate this integration.
4 RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM
- Top of page
- ABSTRACT
- 1 INTRODUCTION
- 2 CMB TEMPERATURE FLUCTUATIONS: GENERAL SETTING
- 3 MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE
- 4 RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM
- 5 MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS
- 6 OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS
- 7 CONCLUSION
- REFERENCES
- Appendix A
- Appendix B
4.1 Assembling the multipole coefficients: integrated spectral functions
We start with the Bessel integral
- (4.1)
with real exponents α ≥ 0, β, ω and μ. If α = 0, we assume a positive exponent β. The multipole component in equation (3.13), generated by density P(k) in equation (3.14), reads
- (4.2)
Occasionally, we will indicate the parameter dependence, . In the figures, we label the contribution of the components to the total multipole coefficients Cl by Pi. The low-l region of the CMB temperature power spectrum, the main peak, as well as the crossover to the modulated decaying slope is an additive combination of eight Gaussian peaks, cf. Figs 1 and 2, so that the summation index in equation (4.2) runs from i = 1 to 8, cf. Table 1. A detailed description of the multipole fit in the Gaussian regime, in particular of the fitting parameters of the Gaussian peaks recorded in Table 1, is given in Section 'Gaussian multipole moments in the low-l region', after having discussed the scaling relations and the scale-invariant limit of the Gaussian and oscillatory multipole components in Section 'Scaling relations for the multipole moments'.
i | a0, i | α0, i | β0, i | r0, i = −α0, i/β0, i |
---|---|---|---|---|
1 | 2.5 × 102 | 6.25 × 10−2 | −0.25 | 0.25 |
2 | 1.5 | 1.47 × 10−2 | −0.3 | 4.9 × 10−2 |
3 | 5.0 × 10−5 | 7.0 × 10−3 | −0.5 | 1.4 × 10−2 |
4 | 2.1 × 10−2 | 6.6 × 10−4 | −0.06 | 1.1 × 10−2 |
5 | 3.6 × 10−5 | 4.5 × 10−4 | −0.1 | 4.5 × 10−3 |
6 | 3.05 × 10−3 | 6.26 × 10−5 | −0.02 | 3.13 × 10−3 |
7 | 1.0 × 10−16 | 1.067 × 10−4 | −0.11 | 9.7 × 10−4 |
8 | 9.5 × 10−19 | 5.04 × 10−5 | −0.08 | 6.3 × 10−4 |
As for the oscillatory multipole component in equation (3.13), generated by the Hermitian kernel (3.5), we replace the squared spherical Bessel function in integral (4.1) by a product of derivatives,
- (4.3)
so that . These integrals are convergent for μ > −3 and or and β > 0. The coefficients can be split, according to equations (3.13) and (3.17), (3.18), (3.19), (3.20), (3.21), as
- (4.4)
The superscript (0,0) indicates the -dependent multipole component (stemming from the spectral functions and in equation (3.18)), calculated as linear combination of the averages in equation (4.3):
- (4.5)
and
- (4.6)
The parameter dependence of these moments is and . The frequency ω and the angle θ0 stem from the parametrization (3.15) of the matrix kernel (3.5). We also note that μ1, 2 > −3 is a requirement for the coefficients and to be convergent, and similarly for in equation (4.2), where μ0, i > −3 is required. The zeroth multipole moment Cl= 0 of the CMB temperature fit depicted in the figures is safely finite (and positive), but does not show due to the l(l + 1) normalization of the moments. A preferable though less customary normalization of the Cl plots is (l + 1/2)2.
The superscript (1,1) in equation (4.4) indicates the -dependent contribution to the multipole moments, defined by and in equation (3.19), and calculated by means of the integrals in equation (4.3):
- (4.7)
and
- (4.8)
The parameter dependence is the same as of and , cf. the text after equation (4.6).
The superscript (0,1) in equation (4.4) labels the multipole contribution of the mixed coefficients , stemming from in equation (3.20) and in equation (3.21). We find, by means of the integrals in equation (4.3),
- (4.9)
and
- (4.10)
The angles θ0 and ϕ0 and the frequencies ω and ω0 are arbitrary constants, fitting parameters in the angular parametrization (3.15) of the matrix kernel. If we put ω0 = 0 and , the coefficients and vanish, cf. the text after equation (3.10). Otherwise, their parameter dependence is and , and the same holds for the total coefficients and in equation (4.4). As in equation (4.2), we may perform a summation over a set of 2 × 2 matrices, cf. (3.12),
- (4.11)
The index i labels the parameter sets defining the two-dimensional matrices, cf. (3.5), (3.15) and (3.16), and each component function is compiled as stated in equations (4.4), (4.5), (4.6), (4.7), (4.8), (4.9), (4.10). In the CMB temperature fit, we put θ0, i = ω0, i = ϕ0, i = 0 as well as α1, i = α2, i = 0 from the outset, so that the Bessel derivatives in equation (4.3) are averaged with a power-law density with modulated exponential cut-off (Kummer distribution). In the figures, we use the shortcut to label the multipole component and for , as well as for their sum , cf. the dotted curves in Figs 3 and 4. The fit of the CMB power spectrum is performed with two two-dimensional matrices, which suffice to adequately reproduce the oscillatory and high-l regimes, so that the summation in equation (4.11) is over i = 1, 2, cf. Table 2. The multipole fit in these regimes and the fitting parameters in Table 2 are explained in Section 'Oscillatory multipole spectrum generated by Kummer distributions in the transitional and high-l regimes'.
i | ai | μ1, i | β1, i | bi | μ2, i | β2, i | ωi |
---|---|---|---|---|---|---|---|
1 | 2.6 × 10−6 | 1 | 4.6 × 10−3 | 5.4 × 10−5 | 0 | 2.3 × 10−3 | 2.15 × 10−2 |
2 | 6.6 × 10−13 | 1 | 2.1 × 10−4 | 0 | – | – | 0 |
The total multipole coefficients Cl are obtained by adding the contribution of the one- and two-dimensional matrix kernels in equations (4.2) and (4.11),
- (4.12)
The component is a Gaussian average which dominates the low-l regime including the main peak, cf. the text after equation (4.2). The oscillatory component generated by Kummer distributions reproduces the decaying modulated slope and the subsequent power-law ascent with exponential cut-off, cf. Figs 4 and 5. The crossover between the Gaussian main peak and the modulated slope consists of two secondary peaks of nearly equal height, to which the Gaussian and oscillatory multipole components in equation (4.12) contribute in equal measure, cf. Figs 9-11.
4.2 Scaling relations for the multipole moments
The Bessel integrals in equations (4.1) and (4.3) satisfy the scaling relation
- (4.13)
Applying this to the Gaussian multipole components in equation (4.2), we find
- (4.14)
where the parameter p has been scaled into the arguments
- (4.15)
In particular, the scale factor p−μ − 3 in equation (4.13) is absorbed by the indicated rescaling of the amplitude a0, i of the , cf. (4.2). Thus, the p dependence of the Gaussian coefficients can be completely absorbed in the fitting parameters, resulting in scale invariance. In effect, we can put p = 1 and use in the CMB temperature fit, with the indicated variables as independent fitting parameters, cf. Table 1.
We turn to the p scaling of the oscillatory multipole moments in equations (4.4) and (4.11), generated by the Hermitian kernel (3.5). The component functions and in equations (4.5) and (4.6) are scale invariant,
- (4.16)
as we can absorb the scaling parameter p in the fitting parameters,
- (4.17)
The coefficients and in equations (4.7) and (4.8) scale like and in equation (4.16). The mixed components and in equations (4.9) and (4.10) are likewise scale invariant,
- (4.18)
with the rescaled parameters listed in equation (4.17) and .
Thus, the coefficients and in equations (4.4) and (4.11) reduce to second-order polynomials in p if we use the rescaled parameters (4.17) and as independent fitting parameters:
- (4.19)
and
- (4.20)
In the Gaussian multipole components (4.14), the scale parameter p can be absorbed in the fitting parameters, as done in equation (4.15). In contrast, in the oscillatory components compiled in equations (4.11), (4.19) and (4.20), there remains an explicit p dependence breaking the scale invariance. The scale parameter p enters as an additional fitting parameter, as a weight factor determining the contribution of the Bessel products , and (weighted by 1, p and p2, respectively) to the multipole moments. There is no other p dependence, as the rescaled variables indicated by a tilde in equations (4.19) and (4.20) are independent fitting parameters. Scale invariance is attained in the limit p → 0, where the coefficients and coincide with and , respectively. In effect, this means to discard the linear and quadratic p terms in equation (4.4) (which give the multipole contributions of the and products), and to put p = 1 in the explicit expressions for and in equations (4.5) and (4.6). The rescaled variables in and can be renamed to the original ones, cf. (4.17), to arrive at
- (4.21)
The multipole components (4.19) and (4.20) thus reduce to (4.21) in the scale-invariant limit p = 0 adopted in the CMB temperature power fit. The fitting parameters indicated as arguments in (4.21) are listed in Table 2.
5 MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS
- Top of page
- ABSTRACT
- 1 INTRODUCTION
- 2 CMB TEMPERATURE FLUCTUATIONS: GENERAL SETTING
- 3 MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE
- 4 RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM
- 5 MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS
- 6 OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS
- 7 CONCLUSION
- REFERENCES
- Appendix A
- Appendix B
5.1 Gaussian multipole moments in the low-l region
To summarize, the CMB temperature multipole moments,
- (5.1)
plotted in Figs 1-13, are assembled from Gaussian and oscillatory components. The Gaussian moments, cf. (4.14),
- (5.2)
consist of Gaussian averages defined by the Bessel integrals in equations (4.1) and (4.2). In the figures, the plots of the individual components , i = 1, … , 8, are labelled by Pi, which stands for the Gaussian density (3.14) generating the respective coefficients . In equation (5.2), the power-law exponents μ0, i have been put to zero from the outset; the remaining fitting parameters α0, i, β0, i and a0, i determining the location, width and amplitude of the peaks Pi = 1, … , 8 are listed in Table 1.
The peaks labelled Pi in the figures (dashed curves) are the l plots of the Gaussian multipole components , cf. (4.2), where DssB is the Bessel integral (4.1). As a rule of thumb, the ratio r0, i = −α0, i/β0, i listed in Table 1 determines the location of the peak Pi; a smaller r0, i shifts the peak to the right, towards higher l values. The negative exponent β0, i determines the width of the peak, a smaller |β0, i| resulting in a larger width. These qualitative features of the averages (4.1) hold particularly well for peaks at moderate and high l, such as P7 and P8 in the first transitional regime, cf. Fig. 10; the high-l asymptotics of the Bessel integrals in equations (4.1) and (4.3) and their numerical evaluation will be discussed elsewhere. The Gaussian component (5.2) dominates the CMB temperature fit at low l, up to about l ∼ 100, cf. Figs 6 and 7. In this regime, the fit Cl is obtained by adding the Gaussian peaks Pi and a tiny admixture of the oscillatory component in equation (5.1), emerging at the lower edge of Fig. 6 as dotted curve . The main peak shown in Figs 8 and 9 is essentially generated by the Gaussian peak P6, with admixtures of smaller adjacent Gaussian peaks and the mentioned oscillatory component (discussed in Section 'Oscillatory multipole spectrum generated by Kummer distributions in the transitional and high-l regimes'), which becomes more dominant with increasing l. The main peak covers the multipole region 100 ≤ l ≤ 400.
5.2 Oscillatory multipole spectrum generated by Kummer distributions in the transitional and high-l regimes
The oscillatory moments in equation (5.1) are compiled as, cf. (4.11) and (4.21),
- (5.3)
where we use the shortcuts
- (5.4)
for the component functions and , which are explicitly stated in equations (4.5) and (4.6) as linear combinations of the Bessel averages (4.3). In equation (5.4), we have put the exponents α1, i and α2, i to zero from the outset, which means to drop the quadratic term in the exponentials in equation (4.3). We have also equated the angle θ0, i to zero, which appears in the angular parametrization (3.15) of the Hermitian spectral matrices generating the oscillatory moments. The CMB temperature fit in Figs 1-13 is performed with the total moments Cl in equation (5.1), obtained by adding the Gaussian moments (5.2) specified in Table 1 to the oscillatory moments listed in equations (5.3) and (5.4) and Table 2.
The decaying modulated slope in Figs 11 and 12 and the subsequent power-law rise of Cl in Fig. 13 are generated by the multipole component (5.3); the Gaussian peaks (5.2) do not affect multipoles beyond l ∼ 1000. The plots of the individual components in equation (5.4) are labelled by in the figures, the plot of by and the plot of the sum in equation (5.3) by . The component (labelled suffices to model the high-l regime, so that we have put . , and stand for Kummer distributions in the Bessel averages (4.3) defining the moments , and in equation (5.4), cf. Table 2.
In the first row of Table 2, we have listed the fitting parameters of the moments and , which constitute the oscillatory multipole component generating the decaying intermediate-l slope in Figs 11 and 12. In the interval 1000 ≤ l ≤ 2500, the multipole fit essentially consists of these two components, , whose plots (dotted curves) are labelled and in the figures. The moments Cl are obtained by adding these two curves, cf. Figs 11 and 12; the contributions of the Gaussian peak P8 and of the ascending slope (both indicated at the lower edge of Fig. 12) are negligible in this interval. The second row of Table 2 contains the fitting parameters of the moments (depicted as dotted curve in Figs 12 and 13), which dominate the fit above l ∼ 4000, . This component generates the extended non-Gaussian peak at l ≈ 15 400 in Fig. 5.
In Section 'Gaussian multipole moments in the low-l region', we have studied the Gaussian regime 0 ≤ l ≤ 400, cf. Figs 6-9. In this section, we discuss the intermediate oscillatory regime, the interval 1000 ≤ l ≤ 2500 containing the modulated decaying slope of Cl, cf. Figs 11 and 12, as well as the high-l regime above l ∼ 4000, cf. Fig. 13. There are two transitional regimes. The first, 400 ≤ l ≤ 1000, is the crossover region from the Gaussian to the oscillatory regime depicted in Figs 8-11. The crossover consists of two secondary peaks of nearly the same height following the main peak. These peaks are the result of pronounced modulations in the oscillatory component (comprising the moments and discussed above and depicted as dotted curves and and of two Gaussian peaks P7 and P8 located in this transitional region. The fit in the crossover interval 400 ≤ l ≤ 1000 is thus obtained as , with a small admixture from the main peak P6, cf. Fig. 10. The second transitional regime is the interval 2500 ≤ l ≤ 4000, cf. Fig. 12, joining the oscillatory multipole component to the ascending power-law slope of the high-l component . The fit in this crossover region is obtained by adding the exponentially damped tail to the emerging rising slope , cf. Figs 12 and 13, the latter dominating the multipole spectrum above l ∼ 4000.
6 OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS
- Top of page
- ABSTRACT
- 1 INTRODUCTION
- 2 CMB TEMPERATURE FLUCTUATIONS: GENERAL SETTING
- 3 MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE
- 4 RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM
- 5 MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS
- 6 OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS
- 7 CONCLUSION
- REFERENCES
- Appendix A
- Appendix B
We consider T(p) as a component of a multicomponent scalar field on the sphere, where the index i = T, E, B, … labels, for instance, temperature, E and B polarization, circular polarization if detectable, an angular galaxy distribution, etc. The temperature field T(p) reads in this notation . The counterpart to the Green function in equation (2.27) is
- (6.1)
The multipole coefficients Cij; l are symmetric in i and j, assembled by averaging squared spherical Bessel functions with Gaussian power laws and Kummer distributions as explained in Sections 'MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE' and 'RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM'. The Hermitian spectral kernels defining the off-diagonal elements need not be positive definite, so that the diagonal matrices in the decomposition (3.3) and (B1) can have negative coefficients, also see the text after equation (3.16). The inverse is defined by the same series with the real symmetric multipole coefficients Cij; l replaced by the inverse matrices . By making use of the orthogonality relation (2.6) of Legendre polynomials and the series representation (2.13) of the delta function on the unit sphere, we find
- (6.2)
The kernels are real and symmetric, depending only on and the scale parameter p. Due to the assumed isotropy, it suffices to expand the Green function in Legendre polynomials rather than in products of spherical harmonics. Isotropy is crucial if the CMB rest frame is to define an absolute frame of reference (Tomaschitz 2012).
The matrices Cij; l(p) can be regarded as positive definite, as the positive CTT; l(p) component usually overpowers all others, so that the inverse correlation function (6.1) is a Gaussian kernel. [Even if the Cij; l are not invertible, one can try the characteristic functional (6.6) as a starting point, instead of the Gaussian density (6.5), to generate n-point correlations.] As we will only use unit vectors in this section, we drop the zero subscripts as well as the scale parameter as argument, writing or for the Green function (6.1) and its inverse on the unit sphere . Spherical integrations are denoted by the solid-angle element , and the real random variables by . The multipole expansion (6.1) can be inverted by way of the orthogonality relation (2.6) and Pl(1) = 1,
- (6.3)
The second solid-angle integration has been added for symmetry reasons.
The observationally determined Green function is found by specifying the multipole coefficients in equations (6.1) and (6.3) as , where are coefficients extracted from the two-dimensional CMB maps, by replacing the average in equation (6.3) by the product of the measured field components. We may substitute the integral representation of the coefficients (i.e. equation (6.3) without angle brackets) into the Legendre series (6.1), interchange integration and summation, and use the completeness relation (A4) of the Legendre polynomials. In this way, we arrive at an integral representation of the Green function equivalent to the Legendre series (6.1) (with as coefficients),
- (6.4)
In Section 'RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM', we derived an analytic approximation to the temperature autocorrelation based on the observed field configuration , and found an integral approximation of the measured coefficients which is uniform in l, covering the multipole range depicted in the figures.
The inverse of the Green function (6.4) (identical to the Legendre series (6.1) with expansion coefficients is the kernel of a Gaussian density,
- (6.5)
The Fourier transform of the normalized density reads
- (6.6)
obtained by introducing a new integration variable in the Fourier integral via the shift and by employing identity (6.2). The n-point correlations are generated by multiple differentiation of , using functional derivatives on the unit sphere, , with the spherical delta function . In this way, we find the Wick expansion of the four-point function,
- (6.7)
in products of the Green functions , ia, b = T, E, B, …. If there is an odd number of factors, the expectation value vanishes. The 2n-point function is obtained by summing a product of n factors σab over all unordered index pairs according to the pattern (6.7); there are (2n)!/(n!2n) summands.
We consider the Legendre-weighted average of the four-point correlation (6.7),
- (6.8)
which can be evaluated as
- (6.9)
Here, we used equations (6.3), (6.7) and the series representation (6.1) as well as the integral
- (6.10)
where the variables can be interchanged in pairs, p i rightarrow p j. The integrations over the four unit spheres in equation (6.10) are readily done by repeated application of the orthogonality relation for Legendre polynomials in equation (2.6). Weighted higher order correlations such as are defined according to patterns (6.3) and (6.8), and can be expressed as linear combinations of products of multipole coefficients analogous to equation (6.9).
We rename the indices in equation (6.9) to i1, 2, 3, 4 = i, j, m, n. The covariance matrix 〈ΔCij; lΔCmn; k〉, , thus reads
- (6.11)
which gives the cosmic variance . The angle brackets refer to the Gaussian functional (6.5) and can formally be imagined as ensemble average over multiple copies of the sky, with root mean squares σij; l quantifying the fluctuations of the coefficients Cij; l. This ensemble interpretation is borrowed from statistical mechanics, but is less appealing here, as the ensemble and the average to which the σij; l refer are not realizable. The large error bars at low l, cf. Figs 6 and 7, are almost entirely due to cosmic variance (calculated via equation (6.11) with , as the measurement errors are very small at low l as compared to σij; l (Jarosik et al. 2011; Larson et al. 2011). The depicted error bars obscure the fine structure of the low-l power spectrum of the measured field configuration, the only accessible one of the envisaged ensemble. Here, we have found an analytic Green function for the temperature autocorrelation, which fits the power spectrum within the actual measurement errors (rather than within the error bars defined by the variance of a hypothetical ensemble average over independent universes).
By identifying the pairs (ia, p a) in the four-point function (6.7), one obtains contractions such as , where σ11 and σ22 are constants and is the squared two-point function (6.1) depending on . Correlations on large angular scales, e.g. the weak temperature autocorrelation at (Bennett et al. 2011), are quantified by truncated angular averages such as , which can be evaluated by means of the Legendre series (6.1) and the uniform analytic approximation of the multipole coefficients stated in Section 'RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM'.
7 CONCLUSION
- Top of page
- ABSTRACT
- 1 INTRODUCTION
- 2 CMB TEMPERATURE FLUCTUATIONS: GENERAL SETTING
- 3 MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE
- 4 RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM
- 5 MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS
- 6 OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS
- 7 CONCLUSION
- REFERENCES
- Appendix A
- Appendix B
We designed an analytic method to reconstruct correlation functions of spherical Gaussian random fields from measured power spectra. The isotropic correlations are defined by a Hermitian matrix kernel, composed of Gaussian power-law densities and Kummer distributions. We obtained a closed analytic expression for the CMB temperature autocorrelation function, fitted its kernel to the measured multipole spectrum and tested the quality of the fit in various intervals over an extended multipole range, cf. Figs 6-13. The multipole coefficients are obtained by averaging squared spherical Bessel functions with the matrix kernel of the Green function, cf. Section 'MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE'. In Tables 1 and 2, we list the fitting parameters of the distributions defining the spectral kernel. Once the kernel of the Green function is specified, so are all higher n-point correlations of the Gaussian random field.
The fine structure of the CMB temperature power spectrum in the low-l regime suggests that the Sachs–Wolfe ‘plateau’ precursory to the main peak is a superposition of overlapping Gaussian peaks, shown in Figs 6 and 7 as dashed curves. In contrast, the high-l power spectrum above l ∼ 1000 consists of two non-Gaussian oscillatory components (depicted in Figs 9-13 as dotted curves and generated by Kummer distributions, cf. Table 2.
The fine structure of CMB power spectra can be hidden in compressed spectral plots as shown in Figs 1-5, which give no more than an overview of the basic features of the spectral map, and make it even difficult to identify the Gaussian and transitional peaks and to discern them from periodic modulations. The low-l regime up to l ∼ 400 is composed of Gaussian peaks, cf. Figs 6 and 7. The two transitional peaks in the crossover region 400 ≤ l ≤ 1000, cf. Figs 8 and 9, are mixtures of Gaussian peaks and large-amplitude modulations of the Kummer distributions. The intermediate and high-l multipole regimes in Figs 12 and 13 comprise a modulated decaying slope and a rising power-law slope stemming from the Kummer distributions in the spectral kernel of the correlation function.
Hermitian spectral matrices are an efficient analytic tool to reconstruct correlation functions of spherical Gaussian random fields from angular power spectra. Here, we studied the CMB temperature autocorrelation, based on a multipole spectrum measured up to l ∼ 104, cf. Fig. 1. Other applications are CMB polarization correlations and temperature–polarization cross-correlations, or galaxy angular correlations. In all these cases, the multipole expansion of the spherical Green function is a Legendre series in zonal spherical harmonics due to isotropy, cf. Section 'OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS', so that the correlation functions can be reconstructed from the measured (cross-)power spectra as described in Sections 'MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE' and 'RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM'.
REFERENCES
- Top of page
- ABSTRACT
- 1 INTRODUCTION
- 2 CMB TEMPERATURE FLUCTUATIONS: GENERAL SETTING
- 3 MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE
- 4 RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM
- 5 MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS
- 6 OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS
- 7 CONCLUSION
- REFERENCES
- Appendix A
- Appendix B
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Appendix A
- Top of page
- ABSTRACT
- 1 INTRODUCTION
- 2 CMB TEMPERATURE FLUCTUATIONS: GENERAL SETTING
- 3 MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE
- 4 RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM
- 5 MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS
- 6 OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS
- 7 CONCLUSION
- REFERENCES
- Appendix A
- Appendix B
DELTA FUNCTION AND LEGENDRE EXPANSION ON THE UNIT SPHERE
The delta function on the unit sphere can be defined by , where is the solid-angle element with polar axis n, and f(n) is an arbitrary smooth and square-integrable function on the unit sphere. Thus, , with δ(cos θ − cos θ′) = δ(θ − θ′)/sin θ, where the angles (θ, ϕ) and (θ′, ϕ′) are the polar coordinates of the unit vectors n and .
Spherical harmonics Ylm(θ, ϕ) are denoted by Ylm(n), where is a unit vector in polar coordinates. The Ylm(θ, ϕ), l = 0, 1, 2…, −l ≤ m ≤ l, are complete on the unit sphere; their orthogonality and completeness relations (Olver et al. 2010) read in this notation
- (A1)
- (A2)
and we use the normalizations and . Isotropic spherical random fields can be expanded in Legendre polynomials or zonal harmonics, which constitute a complete orthogonal set over the interval −1 ≤ x ≤ 1, with the orthogonality and completeness relations
- (A3)
- (A4)
A square-integrable function f(x) on the interval [−1, 1] admits the expansion
- (A5)
If we put , we find the Legendre expansion of the isotropic field f(n k 0) on the sphere as
- (A6)
The multipole coefficients Cl are independent of the arbitrarily chosen unit vector k 0. We also note P0(x) = 1, Pl(1) = 1, as well as the reflection symmetry Pl(x) = (−1)lPl(−x). This expansion can be traced back to spherical harmonics via the addition theorem (Newton 1982; Jackson 1999)
- (A7)
where we may interchange k 0 and n. Combining equation (A7) with the completeness relation for spherical harmonics (A2), we find the Legendre series of the delta function on the unit sphere as stated in equation (2.13).
We consider isotropic spherical random fields (depending only on the polar angle θ), so that Legendre expansions of type (A6) in zonal harmonics suffice. We note , where is the Laplace–Beltrami operator on the unit sphere (Landau & Lifshitz 1991). The high-l asymptotics
- (A8)
can be used to identify the wavelength on the unit sphere (Jeans 1923). This gives an estimate of the angular resolution achieved by high-order multipole moments in the Legendre expansion (2.27) of the two-point function , where we put .
Appendix B
- Top of page
- ABSTRACT
- 1 INTRODUCTION
- 2 CMB TEMPERATURE FLUCTUATIONS: GENERAL SETTING
- 3 MULTIPOLE MOMENTS OF ISOTROPIC GAUSSIAN RANDOM FIELDS ON THE UNIT SPHERE
- 4 RECONSTRUCTION OF THE CMB TEMPERATURE POWER SPECTRUM
- 5 MULTIPOLE FINE STRUCTURE OF CMB TEMPERATURE FLUCTUATIONS
- 6 OUTLOOK: MULTICOMPONENT SPHERICAL RANDOM FIELDS
- 7 CONCLUSION
- REFERENCES
- Appendix A
- Appendix B
EULER PARAMETRIZATION OF HERMITIAN SPECTRAL MATRICES
We consider a positive semidefinite Hermitian 3 × 3 matrix gmn, cf. (3.1), which we decompose as
- (B1)
with real diagonal coefficients aj ≥ 0, analogous to the two-dimensional case in equations (3.3) and (3.4). Positive definiteness requires aj > 0. The matrices Ri(θi, ψi, χi), i = 1, 2, 3, in equation (B1) denote SU(3) subgroups composed of the SU(2) matrices
- (B2)
in the following way:
- (B3)
The inverse matrices in the decomposition (B1) are found as
- (B4)
If we put ψi = χi = 0, i = 1, 2, 3, then the product R1(θ1)R2(−θ2)R3(θ3) is a classical Euler parametrization of the rotation group SO(3), with and . In this case, the Hermitian gmn in equation (B1) is real and symmetric. Here, we do not impose any restrictions on the angles θi, χi and ψi other than reality.
In N dimensions, the Euler parametrization of a Hermitian matrix gmn, m, n = 0, … , N − 1, is performed in like manner,
- (B5)
withthree-parameter subgroups Ri(θi, ψi, χi), i = 1, … , N(N − 1)/2, of SU(N), each of them defined by an SU(2) matrix inserted into an N × N identity matrix according to the pattern (B3). In this way, the matrices gmn are parametrized by N2 independent real parameters, (aj ≥ 0, θi, ϕi), j = 1, … , N and i = 1, … , N(N − 1)/2, where ϕi can be chosen as linear combinations of the N(N − 1) angles ψi and χi of the SU(N) subgroups Ri(θi, ψi, χi). For the amplitudes and angles, we use a spectral parametrization (aj(k), θi(k), ϕi(k)) analogous to equations (3.15) and (3.16), that is, a linear or quadratic k dependence of the angles, and Gaussian power-law densities as amplitudes.