# 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

Additional Information(Show All)(Show All)

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* ∼ 10^{4}.
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 *C*_{l}
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 *C*_{l}
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* ∼ 10^{4}
(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 *C*_{l}
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 *C*_{l}
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*
≤ 10^{5}). 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 *C*_{l}
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 *k*_{B}
is the Boltzmann constant, and we have put ℏ = *c*
= 1. The angular dependent background temperature is denoted by *T*_{b}(*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 *T*_{0}
≈ 2.7 K is the present-day mean background temperature and is the fluctuating field with zero
mean. We conformally rescale *T*_{b}
with the cosmic expansion factor, , where *T*(*p*_{0}) stands for the angular fluctuations δ*T*_{b}/*T*_{0}.
This determines the cosmic time dependence of the distribution function
*f*(** p**); at
the present epoch τ

_{0}, the expansion factor can be chosen as

*a*(τ

_{0}) = 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

*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

**or**

*k***as polar axis and as polar angle, substitute into equation (2.2), and differentiate**

*p**n*times with respect to

*kp*to find

- (2.3)

where . The superscript (*n*)
denotes the *n*th 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 *D*_{m,
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. *D*_{m, n}
can be obtained from by multiple differentiation,

- (2.8)

We perform a radial integration of *D*_{m,
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 *g*_{00}(*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 *g*_{00}(*k*)δ(*k*
− *k*′)/*k*^{2}
by a more general kernel function,

- (2.19)

where ** k** =

*k*

*k*_{0}, and

- (2.20)

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

- (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 *D*_{0, 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 *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'.

### 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 *g*_{mn}(*k*),
*m*, *n* = 0, … , *N*
− 1, and a scale parameter *p*,

- (3.1)

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

- (3.2)

As for *N* = 2, we factorize the 2 × 2
matrix *g*_{mn}(*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 *g*_{mn}(*k*)
thus reads

- (3.5)

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)

- (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 *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

- (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 *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

- (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 *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,

- (3.15)

where ω, θ_{0}, ω_{0}
and ϕ_{0} are real constants. When performing the
CMB temperature fit, it suffices to put θ_{0} = 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 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

- (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
*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'.

i |
a_{0,
i} |
α_{0,
i} |
β_{0,
i} |
r_{0,
i} = −α_{0, i}/β_{0,
i} |
---|---|---|---|---|

1 | 2.5
× 10^{2} |
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 θ_{0}
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):

- (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 θ_{0} and ϕ_{0}
and the frequencies ω and ω_{0} are arbitrary
constants, fitting parameters in the angular parametrization (3.15)
of the matrix kernel. If we put ω_{0} = 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 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'.

i |
a_{i} |
μ_{1, i} |
β_{1, i} |
b_{i} |
μ_{2, i} |
β_{2, i} |
ω_{i} |
---|---|---|---|---|---|---|---|

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 *C*_{l}
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 *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,

- (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 *p*^{2}, 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 *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 *P*_{7} and *P*_{8}
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 *P*_{6},
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 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 *P*_{8}
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 *P*_{7}
and *P*_{8} 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 *P*_{6}, 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*(

**) reads in this notation . The counterpart to the Green function in equation (2.27) is**

*p*- (6.1)

The multipole coefficients *C*_{ij;
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 *C*_{ij;
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 *C*_{ij;
l}(*p*) can be
regarded as positive definite, as the positive *C*_{TT;
l}(*p*) component
usually overpowers all others, so that the inverse correlation function
(6.1)
is a Gaussian kernel. [Even if the *C*_{ij;
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 *P*_{l}(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 , *i*_{a,
b} = *T*, *E*,
*B*, …. If there is an odd number of factors, the
expectation value vanishes. The 2*n*-point function
is obtained by summing a product of *n* factors σ_{ab}
over all unordered index pairs according to the pattern (6.7);
there are (2*n*)!/(*n*!2^{n})
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 *i*_{1, 2, 3, 4} = *i*,
*j*, *m*, *n*.
The covariance matrix 〈Δ*C*_{ij;
l}Δ*C*_{mn;
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 σ_{ij;
l} quantifying the fluctuations of the
coefficients *C*_{ij;
l}.
This ensemble interpretation is borrowed from statistical mechanics,
but is less appealing here, as the ensemble and the average to which
the σ_{ij; 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 σ_{ij;
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 (*i _{a}*,

*p*_{a}) in the four-point function (6.7), one obtains contractions such as , where σ

_{11}and σ

_{22}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* ∼
10^{4}, 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*(

**) 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 .**

*n*Spherical harmonics *Y*_{lm}(θ,
ϕ) are denoted by *Y _{lm}*(

**), where is a unit vector in polar coordinates. The**

*n**Y*

_{lm}(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 *C*_{l}
are independent of the arbitrarily chosen unit vector *k*_{0}. We also note *P*_{0}(*x*)
= 1, *P*_{l}(1)
= 1, as well as the reflection symmetry *P*_{l}(*x*)
= (−1)^{l}*P*_{l}(−*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 *g*_{mn},
cf. (3.1),
which we decompose as

- (B1)

with real diagonal coefficients *a*_{j}
≥ 0, analogous to the two-dimensional case in equations (3.3)
and (3.4).
Positive definiteness requires *a*_{j}
> 0. The matrices *R*_{i}(θ_{i},
ψ_{i}, χ_{i}),
*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 ψ_{i} = χ_{i}
= 0, *i* = 1, 2, 3, then the product *R*_{1}(θ_{1})*R*_{2}(−θ_{2})*R*_{3}(θ_{3})
is a classical Euler parametrization of the rotation group SO(3), with and . In this case, the Hermitian *g*_{mn}
in equation (B1)
is real and symmetric. Here, we do not impose any restrictions on the
angles θ_{i}, χ_{i}
and ψ_{i} other than reality.

In *N* dimensions, the Euler
parametrization of a Hermitian matrix *g*_{mn},
*m*, *n* = 0, … , *N*
− 1, is performed in like manner,

- (B5)

withthree-parameter subgroups *R*_{i}(θ_{i},
ψ_{i}, χ_{i}),
*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 *g*_{mn}
are parametrized by *N*^{2}
independent real parameters, (*a*_{j}
≥ 0, θ_{i}, ϕ_{i}),
*j* = 1, … , *N* and *i*
= 1, … , *N*(*N* − 1)/2, where ϕ_{i}
can be chosen as linear combinations of the *N*(*N*
− 1) angles ψ_{i} and χ_{i}
of the SU(*N*) subgroups *R*_{i}(θ_{i},
ψ_{i}, χ_{i}).
For the amplitudes and angles, we use a spectral parametrization (*a*_{j}(*k*),
θ_{i}(*k*),
ϕ_{i}(*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.

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