Multipole fine structure of the cosmic microwave background: reconstruction of the temperature power spectrum

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

Reality of T(p) requires . Fourier transforms are denoted by a hat. The exponential admits a standard expansion in Legendre polynomials (Newton 1982):

Unit vectors are denoted by a subscript zero, k = k k ₀ and p = p p ₀. The j_l(x) are spherical Bessel functions, , where l = 0, 1, 2, … , and the P_l(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

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)

The expansion (2.3) thus follows from equation (A5).

We consider the distributions

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)

In this way, we obtain the expansion of D_{m, n} in Legendre polynomials,

These distributions are real, and their symmetry properties with regard to a simultaneous interchange of indices and arguments are evident from this expansion. D_{m, n} can be obtained from by multiple differentiation,

We perform a radial integration of D_{m, n}, which defines the kernel

This can also be written as, cf. (2.8),

Employing the series expansion in equation (2.7), we obtain

Here, the Bessel integral is a representation of the Dirac function (Jackson 1999)

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),

to factorize kernel (2.9),

The delta function in Euclidean 3-space can be split as, cf. Appendix Appendix,

so that

This Cartesian representation of kernel (2.9) can directly be recovered from equations (2.5) and (2.10).

We define the correlation function of the Fourier components in equation (2.1) as

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

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,

where k = k k ₀, and

Here, g_mn(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 g_mn(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

where p = p p ₀ and . Zero subscripts denote unit vectors. We may write this as

One of the angular integrations can readily be carried out by virtue of the delta function, cf. (2.5),

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

There are several ways to proceed. First, we may substitute

Alternatively, we may use equation (2.8) to write (2.24) as

Finally, we may substitute the Legendre expansion (2.7) of D_{0, 0} into equation (2.24) to find

where we identified the multipole moments as

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,

This representation (equations (2.27), (2.28), (2.29)) of the Green function can be recovered by substituting the Legendre expansion (2.7) of D_{m, 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 C_l(p, p′) in equation (2.29). The C_l(p, p′) are symmetric in p and p′ if the matrix g_mn(k) is symmetric, and they are real if g_mn(k) is real. If the matrix g_mn(k) is Hermitian, we find . These symmetries of C_l(p, p′) are inherited by the Green function .

If p′ = p (which will be assumed in the sequel), we write C_l(p) or simply C_l for C_l(p, p) in equation (2.29) and or for the correlation function in equation (2.27). The coefficients C_l(p) are real if g_mn(k) is Hermitian or an arbitrary real matrix. Positivity of C_l(p) is ensured if g_mn(k) is a positive-definite Hermitian (or real symmetric) matrix. The condition C_l(p) ≥ 0 is satisfied if g_mn(k) is semidefinite. Thus, is a positive (semi)definite kernel on the unit sphere if g_mn(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'.

The Green function (2.27) is defined by multipole moments depending on a positive-definite or semidefinite Hermitian N × N matrix g_mn(k), m, n = 0, … , N − 1, and a scale parameter p,

Multiple derivatives of the spherical Bessel functions j_l(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

As for N = 2, we factorize the 2 × 2 matrix g_mn(k) as

where the diagonal matrix is defined by real constants A ≥ 0, B ≥ 0, and is a unitary matrix of determinant 1, parametrized as

so that . The Hermitian g_mn(k) thus reads

with m, n = 0, 1 and . This parametrization covers all two-dimensional positive semidefinite Hermitian matrices. g_mn(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 U_i(θ_i, ψ_i, χ_i) as in equation (3.4) to obtain an Euler-type parametrization of the Hermitian matrix, cf. Appendix Appendix. In the case that g_mn(k) factorizes as , we can substitute in equation (3.1)

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),

where we have put χ − ψ = 2ϕ. Here, the vectors h_m(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

where

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,

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),

where the summation is taken over a set of one- and two-dimensional matrices g_mn(k). The multipole coefficients C_l(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

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 .

As for the coefficients in equation (3.2), we parametrize the kernel P(k) with a series of Gaussian power-law densities,

with amplitudes a_{0, i} > 0 and real exponents μ_{0, i}, β_{0, i}, and α_{0, 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,

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),

with a ≥ 0, b ≥ 0 and real exponents μ_{1, 2}, β_{1, 2} and α_{1, 2}. In the CMB temperature fit, we use α_{1, 2} = 0 and β_{1, 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 a_{0, i} in equation (3.14) are negative, the total moments C_l(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

where and denote the terms depending on in equation (3.11),

The terms and in equation (3.17) contain the squared derivatives as factor,

The contributions and to the spectral functions (3.17) stem from the mixed terms in equation (3.11),

and

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.

We start with the Bessel integral

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

Occasionally, we will indicate the parameter dependence, . In the figures, we label the contribution of the components to the total multipole coefficients C_l by P_i. 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'.

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,

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

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):

and

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 μ_{1, 2} > −3 is a requirement for the coefficients and to be convergent, and similarly for in equation (4.2), where μ_{0, i} > −3 is required. The zeroth multipole moment C_{l= 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 C_l 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):

and

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),

and

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),

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 θ_{0, i} = ω_{0, i} = ϕ_{0, i} = 0 as well as α_{1, i} = α_{2, 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'.

The total multipole coefficients C_l are obtained by adding the contribution of the one- and two-dimensional matrix kernels in equations (4.2) and (4.11),

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.

The Bessel integrals in equations (4.1) and (4.3) satisfy the scaling relation

Applying this to the Gaussian multipole components in equation (4.2), we find

where the parameter p has been scaled into the arguments

In particular, the scale factor p^{−μ − 3} in equation (4.13) is absorbed by the indicated rescaling of the amplitude a_{0, 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,

as we can absorb the scaling parameter p in the fitting parameters,

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,

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:

and

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

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.

To summarize, the CMB temperature multipole moments,

plotted in Figs 1-13, are assembled from Gaussian and oscillatory components. The Gaussian moments, cf. (4.14),

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 P_i, which stands for the Gaussian density (3.14) generating the respective coefficients . In equation (5.2), the power-law exponents μ_{0, i} have been put to zero from the outset; the remaining fitting parameters α_{0, i}, β_{0, i} and a_{0, i} determining the location, width and amplitude of the peaks P_{i = 1, … , 8} are listed in Table 1.

The peaks labelled P_i in the figures (dashed curves) are the l plots of the Gaussian multipole components , cf. (4.2), where D_ssB is the Bessel integral (4.1). As a rule of thumb, the ratio r_{0, i} = −α_{0, i}/β_{0, i} listed in Table 1 determines the location of the peak P_i; a smaller r_{0, i} shifts the peak to the right, towards higher l values. The negative exponent β_{0, i} determines the width of the peak, a smaller |β_{0, 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 C_l is obtained by adding the Gaussian peaks P_i 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.

The oscillatory moments in equation (5.1) are compiled as, cf. (4.11) and (4.21),

where we use the shortcuts

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 α_{1, i} and α_{2, 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 θ_{0, 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 C_l 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 C_l 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 C_l 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 C_l, 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.