Problems of CMB Data Registration and Analysis

Abstract

In this short course, we consider some radio astronomical fundamentals and problems of radio astronomical observations. We discuss the main observational cosmological tests which are investigated with radio astronomy. The most crucial tests are connected with the Cosmic Microwave Background (CMB). Several radio telescopes for CMB study and their basic results are discussed. Some stages of the CMB data analysis pipeline are considered and examples of observational CMB anomalies are discussed. At the end of the course (Appendix 3), the short application of the GLESP package is presented for simulation of the CMB map.

Appendices

Appendix 1: Normalized Associated Legendre Polynomials

In the GLESP code, we use the normalized associated Legendre polynomials f _ℓ ^m:

$$\displaystyle{ f_{\ell}^{m}(x) = \sqrt{\frac{2\ell + 1} {2} \frac{(\ell-m)!} {(\ell+m)!}}P_{\ell}^{m}(x)\,, }$$

(30)

where x = cosθ, and θ is the polar angle. These polynomials, f _ℓ ^m(x), can be calculated using two well known recurrent relations. The first of them gives f _ℓ ^m(x) for a given m and all ℓ > m:

$$\displaystyle{f_{\ell}^{m}(x) = x\sqrt{ \frac{4\ell^{2 } - 1} {\ell^{2} - m^{2}}}f_{\ell-1}^{m}-}$$

$$\displaystyle{ -\sqrt{\frac{2\ell + 1} {2\ell - 3} \frac{(\ell-1)^{2} - m^{2}} {\ell^{2} - m^{2}}} f_{\ell-2}^{m}\,. }$$

(31)

This relation starts with

$$\displaystyle{f_{m}^{m}(x) = \frac{(-1)^{m}} {\sqrt{2}} \sqrt{\frac{(2m + 1)!!} {(2m - 1)!!}}(1 - x^{2})^{m/2},\quad }$$

$$\displaystyle{f_{m+1}^{m} = x\sqrt{2m + 3}f_{ m}^{m}\,.}$$

The second recurrent relation gives f _ℓ ^m(x) for a given ℓ and all m ≤ l:

$$\displaystyle\begin{array}{rcl} & & \sqrt{(\ell-m - 1)(\ell+m + 2)}f_{\ell}^{m+2}(x) + \frac{2x(m + 1)} {\sqrt{1 - x^{2}}} f_{\ell}^{m+1}(x) \\ & & \quad + \sqrt{(\ell-m)(\ell+m + 1)}f_{\ell}^{m}(x) = 0\,. {}\end{array}$$

(32)

This relation is started with the same f _ℓ ^ℓ(x) and f _ℓ ⁰(x) which must be found with (31).

As is discussed in Press et al. (1992, Sect. 5.5), the first recurrence relation (31) is formally unstable if the number of iteration tends to infinity. Unfortunately, there are no theoretical recommendations about the maximum iteration one can use in the quasi-stability area. However, it can be used because we are interested in the so-called dominant solution (Press et al. 1992, Sect. 5.5), which is approximately stable. The second recurrence relation (32) is stable for all ℓ and m.

Appendix 2: The GLESP Package

2.1 Structure of the GLESP Code

The code is developed in two levels of organization. The first one, which unifies F77 FORTRAN and C functions, subroutines and wrappers for C routines to be used for FORTRAN calls, consists of the main procedures: ‘signal’, which transforms given values of a _{ℓ m} to a map, ‘alm’, which transforms a map to a _{ℓ m} , ‘cl2alm’, which creates a sample of a _{ℓ m} coefficients for a given C _ℓ and ‘alm2cl’, which calculates C _ℓ for a _{ℓ m} . Procedures for code testing, parameters control, Kolmogorov-Smirnov analysis for Gaussianity of a _{ℓ m} and homogeneity of phase distribution, and others, are also included. Operation of these routines is based on a block of procedures calculating the Gauss–Legendre pixelization for a given resolution parameter, transformation of angles to pixel numbers and back.

Fig. 43

Structure of the GLESP package

The second level of the package contains the programs which are convenient for the utilization of the first level routines. In addition to the straight use of the already mentioned four main procedures, they also provide means to calculate map patterns generated by the Y ₂₀, Y ₂₁ and Y ₂₂ spherical functions, to compare two sets of a _{ℓ m} coefficients, to convert a GLESP map to a HEALPix map, to convert a HEALPix map, or other maps, to a GLESP map.

Figure 43 outlines the GLESP package. The circle defines the zone of the GLESP influence based on the pixelization library. It can include several subroutines and operating programs. The basic program ‘cl2map’ of the second level, shown as a big rectangle, interacts with the first level subroutines. These subroutines are shown by small rectangles and call external libraries for the Fourier transform and Legendre polynomial calculations. The package reads and writes data both in ASCII table and FITS formats. More than ten programs of the GLESP package operate in the GLESP zone.

The package satisfies the following principles:

• Each program is designed to be easily joined with other modules of a package. It operates both with a given file and standard output.

• Each program can operate separately.

• Each program is accessible in a command string with external parameters. It has a dialogue mode and could be tuned with a resource file in some cases.

• Output format of resulting data is organized in the standard way and is prepared in the FITS format or ASCII table accessible for other packages.

• The package programs can interact with other FADPS procedures and CATS database (http://cats.sao.ru).

2.2 Main Operations

There are four types of operations accessible in the GLESP package:

• Operations related to maps:

  • 1. Spherical harmonic decomposition of a temperature anisotropy map into a _{ℓ m} (cl2map).

  • 2. Spherical harmonic decomposition of a temperature anisotropy and Q,U-polarization maps into a _{ℓ m} and e, b _{ℓ m} -coefficients (polalm).

  • 3. Smooth a map with a Gaussian beam (cl2map).

  • 4. Sum/difference/averaging between maps (difmap).

  • 5. Scalar multiplication/division (difmap).

  • 6. Map rotation (difmap).

  • 7. Conversion from Galactic to equatorial coordinates (difmap).

  • 8. Cut temperature values in a map (mapcut).

  • 9. Cut a zone in/from a map (mapcut).

  • 10. Cut out cross-sections from a map (mapcut).

  • 11. Produce simple patterns (mappat).

  • 12. Read ASCII into binary (mappat).

  • 13. Read point sources to binary map (mappat).

  • 14. Print values in map (mapcut).

  • 15. Find min/max values in map sample per pixel (difmap).

  • 16. Simple statistic on a map (difmap).

  • 17. Correlation coefficients of two maps (difmap).

  • 18. Pixel size on a map (ntot).

  • 19. Plot figures ( f2fig).

• Operations related to a _{ℓ m} :

  • 1. Synthesize the temperature anisotropy map from given a _{ℓ m} (cl2map).

  • 2. Synthesize the temperature anisotropy and Q,U-polarization maps from given a _{ℓ m} and e, b _{ℓ m} -coefficients (polmap).

  • 3. Sum/difference (difalm).

  • 4. Scalar multiplication/division (difalm).

  • 5. Vector multiplication/division (difalm).

  • 6. Add phase to all harmonics (difalm).

  • 7. Map rotation in harmonics (difalm).

  • 8. Cut out given mode of harmonics (difalm).

  • 9. Calculate angular power spectrum C _ℓ (alm2dl).

  • 10. Calculate phases (alm2dl).

  • 11. Select the harmonics with a given phase (alm2dl).

  • 12. Compare two a _{ℓ m} samples (checkalm).

  • 13. Produce a _{ℓ m} of map derivatives (dalm).

• Operations related to angular power spectrum C _ℓ :

  • 1. Calculate power spectrum C _ℓ (alm2dl).

  • 2. Simulate a map by a given C _ℓ (cl2map).

  • 3. Simulate a _{ℓ m} by C _ℓ (createalm).

• Operations related to phases ϕ _{ℓ m} and amplitudes | a _{ℓ m}  | :

  • 1. Calculate phases ϕ _{ℓ m} (alm2dl).

  • 2. Calculate amplitudes | a _{ℓ m}  | (alm2dl).

  • 3. Simulate a _{ℓ m} by phases (createalm).

  • 4. Select harmonics with a given phase (alm2dl).

  • 5. Add a phase to all harmonics (difalm).

2.3 Main Programs

The following procedures organized as separate programs in the pixel and harmonics domain are realized now:

alm2dl :

calculates spectra and phases by a _{ℓ m} -coefficients.

checkalm :

compares different a _{ℓ m} -samples.

cmap :

converts HEALPix format maps to the GLESP package format.

cl2map :

converts a map to a _{ℓ m} -coefficients and a _{ℓ m} -coefficients to a map, simulates a map by a given C _ℓ -spectrum.

createalm :

creates a _{ℓ m} -coefficients by phases, amplitudes or/and C _ℓ -spectrum.

dalm :

calculates the first and second derivatives by a _{ℓ m} -coefficients

difalm :

calculates arithmetic operations over a _{ℓ m} -samples.

difmap :

calculates arithmetic operations over maps, produces coordinates transformations.

f2fig :

produces color pictures in GIF-images.

f2map :

converts a GLESP map to a HEALPix format map.

mapcut :

cuts amplitude and coordinates in a GLESP map, produces one-dimensional cross-sections.

mappat :

produces standard map patterns, reads ASCII data to produce a map, reads point sources position from ASCII files.

polalm :

converts temperature and Q,U-polarization anisotropy maps to a _{ℓ m} and e, b _{ℓ m} -coefficients

polmap :

converts a _{ℓ m} and e, b _{ℓ m} -coefficients tp temperature and Q,U-polarization anisotropy maps

2.4 Data Format

The GLESP data are represented in two formats describing a _{ℓ m} -coefficients and maps.

a _{ℓ m} -coefficients data contains index describing number of ℓ and m modes corresponding to the HEALPix, real and imaginary parts of a _{ℓ m} . These three parameters are described by three-fields records of the Binary Table FITS format.

Map data are described by the three-fields Binary Table FITS format containing a vector of x _i  = cosθ positions, a vector of numbers of pixels per each layer $N_{\phi _{i}}$, and set of temperature values in each pixel recorded by layers from the North Pole.

Appendix 3: Practical Work “Study of Power Spectrum”

3.1 Task

• 1. Construct map CMB within the $\Lambda$ CDM.

• 2. Generate card template with radio and its smooth Gaussian radiation pattern.

• 3. Find the power spectrum of the sum of these cards.

3.2 Necessary Resources

Packet data analysis of background radiation on the sphere GLESP⁸ (Doroshkevich et al. 2003), Library of calculating the fast Fourier transform FFTW⁹ Version 3.2 not earlier, the Internet, a computer running OS Linux (or any type of Unix).

Running Time 2 h.

3.3 Description

Highlighting the CMB, we can, on the one hand, build its power spectrum and its shape estimate cosmological parameters, and, on the other hand, research the statistics of the fluctuations. CMB power spectrum shows how much energy at any angular scales contained in the incoming radiation. For the full sphere, it is determined by as the average value of the squares of harmonic modes

$$\displaystyle{C(\ell) = \frac{1} {2\ell + 1}\left [\vert a_{\ell0}\vert ^{2} + 2\sum _{ m=1}^{\ell}\vert a_{\ell,m}\vert ^{2}\right ].}$$

This expression contains double harmonic mode m > 0, which is explained by complex conjugation of harmonics and, consequently, the equality of the squares their amplitudes. Note that in the calculation of the power spectrum, we explicitly apply the hypothesis of a random Gaussian distribution primary perturbation amplitudes which is reflected in the distribution of amplitudes of the harmonics of the CMB. This hypothesis is used in the procedure of averaging all modes m for a given multipole ℓ. Variation of harmonic mode amplitudes for a given ℓ occurs inside the confidence interval defined as the cosmic variance.

The power spectrum of C(ℓ) reflects the physical conditions in the early Universe and thus is a function of the relevant cosmological parameters (Naselsky et al. 2006),

$$\displaystyle{C(\ell) \equiv C_{\ell}(h,\Omega _{b}h^{2},\Omega _{ CDM}h^{2},\Omega _{ \Lambda },\Omega _{\nu },n,\ldots )\,.}$$

Here, in particular, it has been indicated the Hubble constant h = H ₀∕100 km/s/Mpc, the density of baryonic matter $\Omega _{b}$, hidden mass density $\Omega _{CDM}$, the density of the “dark energy” $\Omega _{\Lambda }$, the density of massive neutrinos $\Omega _{\nu }$, spectral index of adiabatic perturbations n and other parameters. The resulting current values of the main parameters are: Hubble constant h = 0. 674 pm+0. 014, matter density $\Omega _{m} = 0.314 \pm 0.020$, baryon density $100\Omega _{b}h^{2} = 2.207 \pm 0.033$, age of the universe t ₀ = 13. 813 ± 0. 058 Gyr, spectral index n _s  = 0. 9616 ± 0. 0094 at scale k = 0. 05 Mpc⁻¹ (Planck Collaboration 2014b), established in 2013 by the Planck collaboration.

The solution of the functional associated with the fitting cosmological model to the observational data, currently almost automated and executes a program CAMB (Lewis et al. 2000) CMBFast, (Seljak and Zaldarriaga 1996), receiving from its entrance cosmological parameters and outputs the result in the form of a smooth power spectrum of the microwave background radiation (Fig. 14).

In this practical work on data analysis of CMB the following stages:

• 1. Simulation maps of the CMB model for the Universe $\Lambda$ CDM.

• 2. Adding a model map radio sources with different flux density.

• 3. Smoothing card directivity pattern size selected in the field of spatial harmonics.

• 4. Calculation of the power spectrum.

3.4 Procedures in GLESP

The first step is to generate a map for a given power spectrum. Angular power spectrum can be calculated using on-line program CAMB at

http://lambda.gsfc.nasa.gov/toolbox/tb_camb_form.cfm.

For further analysis and modeling can be used FITS-file table containing encoded ASCII, or pre-calculated file available online

http://sed.sao.ru/~vo/cosmo_school/presentations/vo/LCDM.dat ,

which recorded a two-column table with the symbolic numbers of multipoles and the corresponding values power spectrum in the model of the Universe $\Lambda$ CDM ($\Omega _{\Lambda }$=0.693, $\Omega _{CDM}$=0.257, $\Omega _{b}$=0.0481) (Planck Collaboration 2014b).

To simulate the map, one can use the program ‘cl2map’ of the GLESP package, which is run with the following parameters:

cl2map -Dl LCDM.dat -r 6 -lmax -nx 1000 2001 4002 -np -ao alm.fts -o map.fts

In this command, the format contains the following flags: ‘-Dl’ indicates that the next parameter ‘LCDM.dat’ is the name of the file containing the power spectrum; the flag ‘-r’ indicates that there will be generated a random Gaussian noise corresponding to the random Gaussian fields of initial density perturbations, with some starting seed, set the following parameters string (for example, here is 6); the flag ‘-lmax 1000’ indicates that the maximum multipole for the generated map is ℓ _max = 1000; ‘-nx 2001 -np 4002’ are the resolution keys: ‘-nx’ and ‘-np’ set the pixelization grid determining the number of rings and the number of pixels in a equatorial ring respectively (the number of rings must always be less than n _x  = 2ℓ _max, and the number of pixels in the equatorial ring is 2n _x , to comply with the Nyquist theorem and make the pixels quasisquare); the flag ‘-ao’ indicates that the following parameter ‘alm.fts’ is the name of output file containing generated spherical harmonic coefficients a  _ellm in the form of FITS-file; the flag ‘-o’ indicates that the following parameter ‘map.fts’ is the file of the generated output map of CMB temperature anisotropy.

The map can be visualized by placing in the GIF-image and displayed, for example, by using the program ‘xv’:

f2fig map.fts -o map.gif; xv map.gif

To see the coefficients a  _ellm , one can move them from a binary representation in ASCII-table using the ‘alm2dl’:

alm2dl -g alm.fts ¿ alm.dat; less alm.dat

In the second phase of work, one should create a file a list of radio sources, which then should be applied to the sky. The file contains information written in the ASCII-format:

hh:mm:ss1 dd:mm:ss1 amplitude1

hh:mm:ss2 dd:mm:ss2 amplitude2

hh:mm:ss3 dd:mm:ss3 amplitude3,

wherein the first and second columns show the equatorial coordinates of radio sources, and the third contains the flux density in mJy. Lists of real sources can be obtained using a database CATS.¹⁰ For example, one can construct a sample of radio sources from the NVSS survey (Condon et al. 1998) (Fig. 44), conducted on radio interferometer VLA (USA).

Fig. 44

Map of radio sources in the survey NVSS (Condon et al. 1998)

To add sources to the CMB map, one should first generate a source map with the following command:

mappat -fp src.dat -o src.fts -nx 2001 -np 4002 

Resolution of both maps (the number of rings and pixels in the equatorial ring) must be identical. The flag ‘–fp’ suggests that the next parameter ‘src.dat’ is the file with a list of radio sources. The flag ‘-o’ is used to enter the name of the output file ‘src.fts’.

Further, both maps: with CMB and sources can be stacked with the command ‘difmap’:

difmap -sum src.fts map.fts -o map_src.fts .

In the third stage of the workshop, must be decomposed into spherical harmonics new map ‘map_src.fts’ and smooth it Gauss diagram orientation. To calculate the expansion in spherical harmonics applying the used earlier procedure ‘cl2map’, but with a new flag:

cl2map -map map_src.fts -lmax 1000 -ao alm_src.fts ¿ /dev/null

Here, the flag ‘–map’ indicates that the map is used to enter ‘map_src.fts’ for the harmonic analysis.

Smoothing is done with the procedure ‘rsalm’:

rsalm alm_src.fits -fw 10 -o alm_sm.fits

Here, the glag ‘-fw 10’ suggests that the smoothing is done with a Gaussian beam pattern of 10^′. The resulting harmonics are written into the FITS-file ‘alm_sm.fits’.

Using the program ‘alm2dl’, one can calculate the power spectrum of C(ℓ):

alm2dl -lmax 1000 alm_sm.fits -cl ¿ cl.dat,

which is visualized as a two-column ASCII-table, for example, using the ‘xmgr’. Maps ‘alm_sm.fits’ can be constructed from harmonics using the procedure ‘cl2map’:

cl2map -lmax -nx 1000 2001 4002 -np -ai alm_sm.fts -o map_sm.fts

f2fig map_sm.fts -o map_sm.gif