1<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN"> 2 3<!--Converted with LaTeX2HTML 2018.3 (Released July 19, 2018) --> 4<HTML lang="EN"> 5<HEAD> 6<TITLE>HEALPix conventions</TITLE> 7<META NAME="description" CONTENT="HEALPix conventions"> 8<META NAME="keywords" CONTENT="intro"> 9<META NAME="resource-type" CONTENT="document"> 10<META NAME="distribution" CONTENT="global"> 11 12<META HTTP-EQUIV="Content-Type" CONTENT="text/html; charset=utf-8"> 13<META NAME="viewport" CONTENT="width=device-width, initial-scale=1.0"> 14<META NAME="Generator" CONTENT="LaTeX2HTML v2018.3"> 15 <link rel='apple-touch-icon' sizes='180x180' href='images/favicons/apple-touch-icon.png?v=2017'> 16 <link rel='icon' type='image/png' sizes='32x32' href='images/favicons/favicon-32x32.png?v=2017'> 17 <link rel='icon' type='image/png' sizes='16x16' href='images/favicons/favicon-16x16.png?v=2017'> 18 <link rel='manifest' href='images/favicons/manifest.json?v=2017'> 19 <link rel='mask-icon' href='images/favicons/safari-pinned-tab.svg?v=2017' color='#5bbad5'> 20 <link rel='shortcut icon' href='images/favicons/favicon.ico?v=2017'> 21 <meta name='apple-mobile-web-app-title' content='HEALPix'> 22 <meta name='application-name' content='HEALPix'> 23 <meta name='msapplication-config' content='images/favicons/browserconfig.xml?v=2017'> 24 <meta name='theme-color' content='#ffffff'> 25 26<LINK REL="STYLESHEET" HREF="intro.css"> 27 28<LINK REL="next" HREF="intro_Pixel_window_functions.htm"> 29<LINK REL="previous" HREF="intro_HEALPix_Software_Package.htm"> 30<LINK REL="next" HREF="intro_Pixel_window_functions.htm"> 31</HEAD> 32 33<body text="#000000" bgcolor="#FFFFFA"> 34 35<DIV CLASS="navigation"><!--Navigation Panel--> 36<A 37 HREF="intro_HEALPix_Software_Package.htm"> 38<IMG WIDTH="63" HEIGHT="24" ALT="previous" SRC="prev.png"></A> 39<A 40 HREF="intro_Introduction_HEALPix.htm"> 41<IMG WIDTH="26" HEIGHT="24" ALT="up" SRC="up.png"></A> 42<A 43 HREF="intro_Pixel_window_functions.htm"> 44<IMG WIDTH="37" HEIGHT="24" ALT="next" SRC="next.png"></A> 45<A ID="tex2html97" 46 HREF="intro_TABLE_CONTENTS.htm"> 47<IMG WIDTH="65" HEIGHT="24" ALT="contents" SRC="contents.png"></A> 48<BR> 49<B> Previous:</B> <A 50 HREF="intro_HEALPix_Software_Package.htm">The HEALPix Software Package</A> 51 52<B>Up:</B> <A 53 HREF="intro_Introduction_HEALPix.htm">Introduction to HEALPix</A> 54 55<B> Next:</B> <A 56 HREF="intro_Pixel_window_functions.htm">Pixel window functions</A> 57<B> Top:</B> <a href="main.htm">Main Page</a></DIV> 58<!--End of Navigation Panel--> 59<!--Table of Child-Links--> 60<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A> 61 62<UL CLASS="ChildLinks"> 63<LI><A ID="tex2html99" 64 HREF="intro_HEALPix_conventions.htm#SECTION610">Angular power spectrum conventions</A> 65<LI><A ID="tex2html100" 66 HREF="intro_HEALPix_conventions.htm#SECTION620"><b>HEALPix</b> and Boltzmann codes</A> 67<UL> 68<LI><A ID="tex2html101" 69 HREF="intro_HEALPix_conventions.htm#SECTION621">CMBFAST</A> 70<LI><A ID="tex2html102" 71 HREF="intro_HEALPix_conventions.htm#SECTION622">CAMB and CLASS</A> 72</UL> 73<BR> 74<LI><A ID="tex2html103" 75 HREF="intro_HEALPix_conventions.htm#SECTION630">Polarisation convention</A> 76<UL> 77<LI><A ID="tex2html104" 78 HREF="intro_HEALPix_conventions.htm#SECTION631">Internal convention</A> 79<LI><A ID="tex2html105" 80 HREF="intro_HEALPix_conventions.htm#SECTION632">Relation to previous releases</A> 81<LI><A ID="tex2html106" 82 HREF="intro_HEALPix_conventions.htm#SECTION633">Relation with IAU convention</A> 83<LI><A ID="tex2html107" 84 HREF="intro_HEALPix_conventions.htm#SECTION634">How <b>HEALPix</b> deals with these discrepancies: <SPAN CLASS="texttt">POLCCONV</SPAN> keyword</A> 85</UL> 86<BR> 87<LI><A ID="tex2html108" 88 HREF="intro_HEALPix_conventions.htm#SECTION640">Spherical harmonic conventions</A> 89</UL> 90<!--End of Table of Child-Links--> 91<HR> 92 93<H1><A ID="SECTION600"></A> 94<A ID="sec:conventions"></A> 95<BR> 96<b>HEALPix</b> conventions 97</H1>A bandlimited function <SPAN CLASS="MATH"><I>f</I></SPAN> on the sphere can 98be expanded in spherical harmonics, <SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img11.png" 99 ALT="$Y_{\ell m}$"></SPAN>, 100as 101<BR> 102<DIV ALIGN="CENTER"><A ID="eq:alms"></A> 103<!-- MATH 104 \begin{eqnarray} 105f({ \gamma})&\myequal &\sum_{\ell =0}^{\ell_{\mathrm{max}}}\sum_{m}a_{\ell m}Y_{\ell m}(\gamma), 106\end{eqnarray} 107 --> 108<SPAN CLASS="MATH"> 109<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 110<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img12.png" 111 ALT="$\displaystyle f({ \gamma})$"></TD> 112<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 113 ALT="$\textstyle \myequal $"></TD> 114<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 6.06ex; vertical-align: -2.66ex; " SRC="intro_img14.png" 115 ALT="$\displaystyle \sum_{\ell =0}^{\ell_{\mathrm{max}}}\sum_{m}a_{\ell m}Y_{\ell m}(\gamma),$"></TD> 116<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 117(<SPAN CLASS="arabic">4</SPAN>)</TD></TR> 118</TABLE> 119</SPAN></DIV> 120<BR CLEAR="ALL"><P></P> 121where <SPAN CLASS="MATH"><IMG STYLE="height: 1.57ex; vertical-align: -0.55ex; " SRC="intro_img15.png" 122 ALT="${{\gamma}}$"></SPAN> denotes a unit vector pointing at polar angle <!-- MATH 123 $\theta\in[0,\pi]$ 124 --> 125<SPAN CLASS="MATH"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img16.png" 126 ALT="$\theta\in[0,\pi]$"></SPAN> and 127azimuth <!-- MATH 128 $\phi\in[0,2\pi)$ 129 --> 130<SPAN CLASS="MATH"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img17.png" 131 ALT="$\phi\in[0,2\pi)$"></SPAN>. Here we have assumed that there is insignificant signal power in modes 132with <!-- MATH 133 $\ell>\ell_{\mathrm{max}}$ 134 --> 135<SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img18.png" 136 ALT="$\ell>\ell_{\mathrm{max}}$"></SPAN> and introduce the notation that all sums over <SPAN CLASS="MATH"><I>m</I></SPAN> run from 137<!-- MATH 138 $-\ell_{\mathrm{max}}$ 139 --> 140<SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img19.png" 141 ALT="$-\ell_{\mathrm{max}}$"></SPAN> to <!-- MATH 142 $\ell_{\mathrm{max}}$ 143 --> 144<SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img20.png" 145 ALT="$\ell_{\mathrm{max}}$"></SPAN> but all quantities with index <SPAN CLASS="MATH"><IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="intro_img21.png" 146 ALT="${\ell m}$"></SPAN> vanish 147for <SPAN CLASS="MATH"><IMG STYLE="height: 1.69ex; vertical-align: -0.16ex; " SRC="intro_img22.png" 148 ALT="$m>\ell$"></SPAN>. Our conventions for <SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img11.png" 149 ALT="$Y_{\ell m}$"></SPAN> are defined in subsection 150<A HREF="#sphericalstuff">A.4</A> below. 151 152<P> 153Pixelating <SPAN CLASS="MATH"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img23.png" 154 ALT="$f({\gamma})$"></SPAN> corresponds to sampling it at <!-- MATH 155 $N_{\mathrm{pix}}$ 156 --> 157<SPAN CLASS="MATH"><I>N</I><SUB>pix</SUB></SPAN> 158 locations <SPAN CLASS="MATH"><IMG STYLE="height: 1.75ex; vertical-align: -0.75ex; " SRC="intro_img24.png" 159 ALT="$\gamma_{p}$"></SPAN>, <!-- MATH 160 $p\in[0,N_{\mathrm{pix}}-1]$ 161 --> 162<SPAN CLASS="MATH"><IMG STYLE="height: 2.45ex; vertical-align: -0.75ex; " SRC="intro_img25.png" 163 ALT="$p\in[0,N_{\mathrm{pix}}-1]$"></SPAN>. The sample 164function values <SPAN CLASS="MATH"><I>f</I><SUB><i>p</i></SUB></SPAN> can then be used 165to estimate <SPAN CLASS="MATH"><IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="intro_img26.png" 166 ALT="$a_{\ell m}$"></SPAN>. A straightforward estimator is 167<BR> 168<DIV ALIGN="CENTER"><A ID="eq:hata"></A> 169<!-- MATH 170 \begin{eqnarray} 171\hat{a}_{\ell m}&\myequal & \frac{4\pi}{N_{\mathrm{pix}}}\sum_{p=0}^{N_{\mathrm{pix}}-1} 172 Y^\ast_{\ell m}(\gamma_p) f(\gamma_p), 173\end{eqnarray} 174 --> 175<SPAN CLASS="MATH"> 176<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 177<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img27.png" 178 ALT="$\displaystyle \hat{a}_{\ell m}$"></TD> 179<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 180 ALT="$\textstyle \myequal $"></TD> 181<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 6.47ex; vertical-align: -2.88ex; " SRC="intro_img28.png" 182 ALT="$\displaystyle \frac{4\pi}{N_{\mathrm{pix}}}\sum_{p=0}^{N_{\mathrm{pix}}-1} 183Y^\ast_{\ell m}(\gamma_p) f(\gamma_p),$"></TD> 184<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 185(<SPAN CLASS="arabic">5</SPAN>)</TD></TR> 186</TABLE> 187</SPAN></DIV> 188<BR CLEAR="ALL"><P></P> 189where the superscript star denotes complex conjugation, and an equal weight was assumed for each pixel. This 190zeroth order estimator, as well as higher order estimators, are implemented in the 191Fortran90 facility <A HREF="./fac_anafast.htm#fac:anafast"><SPAN CLASS="texttt">anafast</SPAN></A>, included in the 192package. 193 194<P> 195 196<H2><A ID="SECTION610"> 197Angular power spectrum conventions</A> 198</H2> 199These <!-- MATH 200 $\hat{a}_{\ell m}$ 201 --> 202<SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img29.png" 203 ALT="$\hat{a}_{\ell m}$"></SPAN> can be used to compute estimates of the angular power spectrum 204 <SPAN CLASS="MATH"><IMG STYLE="height: 2.62ex; vertical-align: -0.45ex; " SRC="intro_img30.png" 205 ALT="$\hat{C}_\ell$"></SPAN> as 206<BR> 207<DIV ALIGN="CENTER"><A ID="eq:hatC"></A> 208<!-- MATH 209 \begin{eqnarray} 210\hat{C}_\ell&\myequal &\frac{1}{2\ell +1}\sum_{m} \vert\hat{a}_{\ell m}\vert^2. 211\end{eqnarray} 212 --> 213<SPAN CLASS="MATH"> 214<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 215<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.62ex; vertical-align: -0.45ex; " SRC="intro_img31.png" 216 ALT="$\displaystyle \hat{C}_\ell$"></TD> 217<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 218 ALT="$\textstyle \myequal $"></TD> 219<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 5.30ex; vertical-align: -2.42ex; " SRC="intro_img32.png" 220 ALT="$\displaystyle \frac{1}{2\ell +1}\sum_{m} \vert\hat{a}_{\ell m}\vert^2.$"></TD> 221<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 222(<SPAN CLASS="arabic">6</SPAN>)</TD></TR> 223</TABLE> 224</SPAN></DIV> 225<BR CLEAR="ALL"><P></P> 226Equations (<A HREF="#eq:hata">5</A>) and (<A HREF="#eq:hatC">6</A>) above do not consider the impact of a pixel masking or weighting 227<!-- MATH 228 $f(\gamma_p) \longrightarrow f(\gamma_p) w(\gamma_p)$ 229 --> 230<SPAN CLASS="MATH"><IMG STYLE="height: 2.45ex; vertical-align: -0.75ex; " SRC="intro_img33.png" 231 ALT="$f(\gamma_p) \longrightarrow f(\gamma_p) w(\gamma_p)$"></SPAN> 232on the power spectrum estimation of <SPAN CLASS="MATH"><I>f</I></SPAN>, which is described in 233<A 234 HREF="intro_Bibliography.htm#whg2001">Wandelt, Hivon & Górski (2001)</A> 235and addressed in 236<A 237 HREF="intro_Bibliography.htm#master">Hivon et al. (2002)</A>, <A 238 HREF="intro_Bibliography.htm#polspice">Chon et al. (2004)</A>, <A 239 HREF="intro_Bibliography.htm#xspect">Tristram et al. (2005)</A>, <A 240 HREF="intro_Bibliography.htm#xfaster">Rocha et al. (2009)</A> and 241<A 242 HREF="intro_Bibliography.htm#planck2015-11">Planck 2015-XI (2015)</A> 243among others. 244 245<P> 246The <b>HEALPix</b> package contains the Fortran90 facility 247<A HREF="./fac_synfast.htm#fac:synfast"><SPAN CLASS="texttt">synfast</SPAN></A>, 248which takes as input a power spectrum <SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img34.png" 249 ALT="$C_\ell$"></SPAN> and generates a realisation of 250<SPAN CLASS="MATH"><IMG STYLE="height: 2.45ex; vertical-align: -0.75ex; " SRC="intro_img35.png" 251 ALT="$f(\gamma_p)$"></SPAN> 252on the <b>HEALPix</b> grid. The convention for power spectrum input into 253<SPAN CLASS="texttt">synfast</SPAN> is straightforward: each <SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img34.png" 254 ALT="$C_\ell$"></SPAN> is just the expected 255variance of the <SPAN CLASS="MATH"><IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="intro_img26.png" 256 ALT="$a_{\ell m}$"></SPAN> at that <SPAN CLASS="MATH"><IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="intro_img36.png" 257 ALT="$\ell$"></SPAN>. 258 259<P> 260<BLOCKQUOTE> 261<SPAN CLASS="textbf">Example</SPAN>: The spherical harmonic coefficient <SPAN CLASS="MATH"><I>a</I><SUB>00</SUB></SPAN> is the 262integral of the <!-- MATH 263 $f(\gamma)/\sqrt{4 \pi}$ 264 --> 265<SPAN CLASS="MATH"><IMG STYLE="height: 2.56ex; vertical-align: -0.68ex; " SRC="intro_img37.png" 266 ALT="$f(\gamma)/\sqrt{4 \pi}$"></SPAN> over the sphere. To 267obtain realisations of functions which have <SPAN CLASS="MATH"><I>a</I><SUB>00</SUB></SPAN> distributed as a Gaussian 268with zero mean and variance 1, set <SPAN CLASS="MATH"><I>C</I><SUB><i>0</i></SUB></SPAN> to 1. The value of the 269synthesised function at each pixel will 270be Gaussian distributed with mean zero and variance <SPAN CLASS="MATH"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img38.png" 271 ALT="$1/(4\pi)$"></SPAN>. 272As required, the integral of <SPAN CLASS="MATH"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img39.png" 273 ALT="$f(\gamma)$"></SPAN> over the full <SPAN CLASS="MATH"><IMG STYLE="height: 1.57ex; vertical-align: -0.10ex; " SRC="intro_img40.png" 274 ALT="$4\pi$"></SPAN> 275solid angle of the sphere has zero mean and variance <SPAN CLASS="MATH"><IMG STYLE="height: 1.57ex; vertical-align: -0.10ex; " SRC="intro_img40.png" 276 ALT="$4\pi$"></SPAN>. 277 278</BLOCKQUOTE> 279Note that this definition implies the standard result that the total power 280at the angular wavenumber <SPAN CLASS="MATH"><IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="intro_img36.png" 281 ALT="$\ell$"></SPAN> is <!-- MATH 282 $(2\ell+1)C_\ell$ 283 --> 284<SPAN CLASS="MATH"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img41.png" 285 ALT="$(2\ell+1)C_\ell$"></SPAN>, because there are 286<SPAN CLASS="MATH"><IMG STYLE="height: 1.81ex; vertical-align: -0.28ex; " SRC="intro_img42.png" 287 ALT="$2\ell+1$"></SPAN> modes for each <SPAN CLASS="MATH"><IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="intro_img36.png" 288 ALT="$\ell$"></SPAN>. 289 290<P> 291This defines unambiguously how the <SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img34.png" 292 ALT="$C_\ell$"></SPAN> have to be defined given the 293units of the physical quantity <SPAN CLASS="MATH"><I>f</I></SPAN>. In cosmic 294microwave background research, 295popular choices for simulated maps are 296<DL COMPACT><DT><DD><IMG WIDTH="14" HEIGHT="14" SRC="greenball.png" ALT="*"> 297 <SPAN CLASS="MATH"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img43.png" 298 ALT="$\Delta T/T $"></SPAN>, a dimensionless quantity measuring relative 299fluctuations about the average CMB temperature. 300<DT><DD><IMG WIDTH="14" HEIGHT="14" SRC="greenball.png" ALT="*"> 301 The absolute quantity <SPAN CLASS="MATH"><IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="intro_img44.png" 302 ALT="$\Delta T$"></SPAN> in <SPAN CLASS="MATH"><IMG STYLE="height: 2.16ex; vertical-align: -0.55ex; " SRC="intro_img45.png" 303 ALT="$\mu K$"></SPAN> or <SPAN CLASS="MATH"><I>K</I></SPAN>. 304</DL> 305 306<P> 307 308<H2><A ID="SECTION620"> 309<b>HEALPix</b> and Boltzmann codes</A> 310</H2> 311 312<H3><A ID="SECTION621"></A> 313<A ID="subsec:cmbfast"></A> 314<BR> 315CMBFAST 316</H3> 317A widely used solver of the Boltzmann equations for the computation 318of theoretical predictions of the spectrum of CMB anisotropy used to be CMBFAST 319(<A ID="tex2html13" 320 HREF="https://lambda.gsfc.nasa.gov/toolbox/tb_cmbfast_ov.cfm"><SPAN CLASS="texttt">https://lambda.gsfc.nasa.gov/toolbox/tb_cmbfast_ov.cfm</SPAN></A>). 321 322<P> 323CMBFAST made its outputs in ASCII files, which instead 324of <SPAN CLASS="MATH"><IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="intro_img46.png" 325 ALT="$C_{X,\ell}$"></SPAN> contain quantities defined as 326<BR> 327<DIV ALIGN="CENTER"> 328 329<!-- MATH 330 \begin{eqnarray} 331D_{X,\ell}&\myequal &\frac{\ell(\ell+1)}{(2\pi)T_{CMB}^2}C_{X,\ell}, 332\end{eqnarray} 333 --> 334<SPAN CLASS="MATH"> 335<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 336<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="intro_img47.png" 337 ALT="$\displaystyle D_{X,\ell}$"></TD> 338<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 339 ALT="$\textstyle \myequal $"></TD> 340<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 5.54ex; vertical-align: -2.27ex; " SRC="intro_img48.png" 341 ALT="$\displaystyle \frac{\ell(\ell+1)}{(2\pi)T_{CMB}^2}C_{X,\ell},$"></TD> 342<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 343(<SPAN CLASS="arabic">7</SPAN>)</TD></TR> 344</TABLE> 345</SPAN></DIV> 346<BR CLEAR="ALL"><P></P> 347where <!-- MATH 348 $T_{CMB}=2.726K$ 349 --> 350<SPAN CLASS="MATH"><I>T</I><SUB><I>CMB</I></SUB>=2.726<I>K</I></SPAN> is the temperature of the CMB today and <SPAN CLASS="MATH"><I>X</I></SPAN> stands for T, 351E, B or C (see § <A HREF="#subsec:pol">A.3</A>). 352 353<P> 354The version 4.0 of CMBFAST also created a FITS file containing the power spectra 355<SPAN CLASS="MATH"><IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="intro_img46.png" 356 ALT="$C_{X,\ell}$"></SPAN>, designed for interface with <b>HEALPix</b>. The spectra for polarization were renormalized to match the 357normalization used in <b>HEALPix</b> 1.1, which was different from the one used by 358CMBFAST and by <b>HEALPix</b> 1.2 (see § <A HREF="#subsub:relatoldversion">A.3.2</A> for details). 359 360<P> 361A later version of CMBFAST (4.2, released in Feb. 2003) generated FITS files containing 362<SPAN CLASS="MATH"><IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="intro_img46.png" 363 ALT="$C_{X,\ell}$"></SPAN>, with the same convention for polarization as the one used 364internally. It therefore matches the convention adopted by <b>HEALPix</b> in its 365version 1.2. 366 367<P> 368For backward compatibility, we provide an IDL code 369(<A HREF="./idl_convert_oldhpx2cmbfast.htm#idl:convert_oldhpx2cmbfast"><SPAN CLASS="texttt">convert_oldhpx2cmbfast</SPAN></A>) 370to change the normalization of existing FITS files created with CMBFAST 4.0. 371When created with the correct normalization (with CMBFAST 4.2) 372or set to the correct normalization (using <SPAN CLASS="texttt">convert_oldhpx2cmbfast</SPAN>), the FITS file will include a 373specific keyword (<SPAN CLASS="texttt">POLNORM = CMBFAST</SPAN>) in their header to identify them. 374The map simulation code 375<A HREF="./fac_synfast.htm#fac:synfast"><SPAN CLASS="texttt">synfast</SPAN></A> 376will issue a warning if the input power 377spectrum file does not contain the keyword <SPAN CLASS="texttt">POLNORM</SPAN>, but no attempt will 378be made to renormalize the power spectrum. If the keyword is present, it will be 379inherited by the simulated map. 380 381<P> 382 383<H3><A ID="SECTION622"> 384CAMB and CLASS</A> 385</H3> 386Newer and actively maintained Boltzmann codes currently include 387<SPAN CLASS="texttt">camb</SPAN> and <SPAN CLASS="texttt">class</SPAN>: 388<DL COMPACT><DT><DD><IMG WIDTH="14" HEIGHT="14" SRC="greenball.png" ALT="*"> 389 <SPAN CLASS="texttt">camb</SPAN> (<A ID="tex2html14" 390 HREF="https://camb.info"><SPAN CLASS="texttt">https://camb.info</SPAN></A>) 391is written in Fortran 90 with a python wrapper, and can optionally output into FITS files 392the <SPAN CLASS="MATH"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img49.png" 393 ALT="$C_X(\ell)$"></SPAN> power spectra in [K]<SPAN CLASS="MATH"><SUP><i>2</i></SUP></SPAN> in a format directly usable by <b>HEALPix</b>; 394<BR><DT><DD><IMG WIDTH="14" HEIGHT="14" SRC="greenball.png" ALT="*"> 395 <SPAN CLASS="texttt">class</SPAN> (<A ID="tex2html15" 396 HREF="http://class-code.net"><SPAN CLASS="texttt">http://class-code.net</SPAN></A>) 397is written in C and C++, and only outputs <!-- MATH 398 $\frac{\ell(\ell+1)}{2\pi}C_X(\ell)$ 399 --> 400<SPAN CLASS="MATH"><IMG STYLE="height: 3.09ex; vertical-align: -0.90ex; " SRC="intro_img50.png" 401 ALT="$\frac{\ell(\ell+1)}{2\pi}C_X(\ell)$"></SPAN> in plain text files 402(optionally in [<SPAN CLASS="MATH"><IMG STYLE="height: 1.57ex; vertical-align: -0.55ex; " SRC="intro_img51.png" 403 ALT="$\mu$"></SPAN>K]<SPAN CLASS="MATH"><SUP><i>2</i></SUP></SPAN> and in a order of columns for polarized spectra matching the one of <SPAN CLASS="texttt">camb</SPAN>). 404</DL> 405Both codes are parallelized for faster computations and provide fine control of the output accuracy. 406 407<P> 408 409<H2><A ID="SECTION630"></A> 410<A ID="subsec:pol"></A> 411<BR> 412Polarisation convention 413</H2> 414 415<P> 416 417<DIV class="CENTER"><A ID="fig:orthpol"></A><A ID="1058"></A> 418<TABLE> 419<CAPTION class="BOTTOM"><STRONG>Figure 4:</STRONG> 420Orthographic projection of a fake full sky for temperature (color 421coded) and polarization (represented by the rods). All the input Spherical 422Harmonics coefficients are set to 0, except for 423<!-- MATH 424 $a_{21}^{TEMP}=\ -\ a_{2-1}^{TEMP}=1$ 425 --> 426<SPAN CLASS="MATH"><IMG STYLE="height: 2.51ex; vertical-align: -0.83ex; " SRC="intro_img52.png" 427 ALT="$a_{21}^{TEMP}=\ -\ a_{2-1}^{TEMP}=1$"></SPAN> and 428<!-- MATH 429 $a_{21}^{GRAD}=\ -\ a_{2-1}^{GRAD}=1$ 430 --> 431<SPAN CLASS="MATH"><IMG STYLE="height: 2.56ex; vertical-align: -0.83ex; " SRC="intro_img53.png" 432 ALT="$a_{21}^{GRAD}=\ -\ a_{2-1}^{GRAD}=1$"></SPAN></CAPTION> 433<TR><TD> 434<DIV CLASS="centerline" ID="par4244" ALIGN="CENTER"> 435<IMG STYLE="" 436 SRC="./plot_orthpolrot.png" 437 ALT="Image plot_orthpolrot"></DIV> 438 439<P></TD></TR> 440</TABLE> 441</DIV> 442 443<P> 444 445<H3><A ID="SECTION631"> 446Internal convention</A> 447</H3> 448Starting with version 1.20 (released in Feb 2003),<b>HEALPix</b> uses the same 449conventions as CMBFAST for the sign and normalization of the polarization power 450spectra, as exposed below (adapted from <A 451 HREF="intro_Bibliography.htm#zalda">Zaldarriaga (1998)</A>). How this relates to 452what was used in previous releases is exposed in <A HREF="#subsub:relatoldversion">A.3.2</A>. 453 454<P> 455<BLOCKQUOTE> 456<SMALL CLASS="FOOTNOTESIZE">The CMB radiation field is described by a <!-- MATH 457 $2\, \times \, 2$ 458 --> 459<SPAN CLASS="MATH"><IMG STYLE="height: 1.57ex; vertical-align: -0.29ex; " SRC="intro_img54.png" 460 ALT="$2\, \times \, 2$"></SPAN> 461intensity tensor 462<SPAN CLASS="MATH"><I>I</I><SUB><I>ij</I></SUB></SPAN> 463(<A 464 HREF="intro_Bibliography.htm#chandra">Chandrasekhar, 1960</A>). The Stokes parameters <SPAN CLASS="MATH"><I>Q</I></SPAN> and <SPAN CLASS="MATH"><I>U</I></SPAN> are defined as 465<!-- MATH 466 $Q=(I_{11}-I_{22})/4$ 467 --> 468<SPAN CLASS="MATH"><I>Q</I>=(<I>I</I><SUB>11</SUB>-<I>I</I><SUB>22</SUB>)/4</SPAN> and <SPAN CLASS="MATH"><I>U</I>=<I>I</I><SUB>12</SUB>/2</SPAN>, while the temperature anisotropy 469is given by <!-- MATH 470 $T=(I_{11}+I_{22})/4$ 471 --> 472<SPAN CLASS="MATH"><I>T</I>=(<I>I</I><SUB>11</SUB>+<I>I</I><SUB>22</SUB>)/4</SPAN>. The fourth Stokes parameter <SPAN CLASS="MATH"><I>V</I></SPAN> that 473describes circular polarization is not necessary in standard cosmological 474models because it cannot be generated through the process of Thomson 475scattering. While the temperature is a scalar quantity <SPAN CLASS="MATH"><I>Q</I></SPAN> and <SPAN CLASS="MATH"><I>U</I></SPAN> are 476not. They depend on the direction of observation <SPAN CLASS="MATH"><B>n</B></SPAN> 477and on the two axis <!-- MATH 478 $(\textbf{e}_{1}, \textbf{e}_{2})$ 479 --> 480<SPAN CLASS="MATH">(<B>e</B><SUB>1</SUB>, <B>e</B><SUB>2</SUB>)</SPAN> 481perpendicular to <SPAN CLASS="MATH"><B>n</B></SPAN> used to define them. If for a given 482<SPAN CLASS="MATH"><B>n</B></SPAN> the axes <!-- MATH 483 $(\textbf{e}_{1}, \textbf{e}_{2})$ 484 --> 485<SPAN CLASS="MATH">(<B>e</B><SUB>1</SUB>, <B>e</B><SUB>2</SUB>)</SPAN> are rotated by an angle 486<SPAN CLASS="MATH"><IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="intro_img55.png" 487 ALT="$\psi$"></SPAN> such that 488<!-- MATH 489 ${\textbf{e}_{1}}^{\prime}=\cos \psi \ {\textbf{e}_{1}}+\sin\psi \ {\textbf{e}_{2}}$ 490 --> 491<SPAN CLASS="MATH"><IMG STYLE="height: 2.21ex; vertical-align: -0.55ex; " SRC="intro_img56.png" 492 ALT="${\textbf{e}_{1}}^{\prime}=\cos \psi \ {\textbf{e}_{1}}+\sin\psi \ {\textbf{e}_{2}} $"></SPAN> 493and <!-- MATH 494 ${\textbf{e}_{2}}^{\prime}=-\sin \psi \ {\textbf{e}_{1}}+\cos\psi \ {\textbf{e}_{2}}$ 495 --> 496<SPAN CLASS="MATH"><IMG STYLE="height: 2.21ex; vertical-align: -0.55ex; " SRC="intro_img57.png" 497 ALT="${\textbf{e}_{2}}^{\prime}=-\sin \psi \ {\textbf{e}_{1}}+\cos\psi \ {\textbf{e}_{2}} $"></SPAN> 498the Stokes parameters change as 499</SMALL></BLOCKQUOTE> 500<BR> 501<DIV ALIGN="CENTER"><A ID="QUtrans"></A> 502<!-- MATH 503 \begin{eqnarray} 504Q^{\prime}&\myequal &\cos 2\psi \ Q + \sin 2\psi \ U \nonumber \\ 505 U^{\prime}&\myequal &-\sin 2\psi \ Q + \cos 2\psi \ U 506\end{eqnarray} 507 --> 508<SPAN CLASS="MATH"> 509<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 510<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.55ex; " SRC="intro_img58.png" 511 ALT="$\displaystyle Q^{\prime}$"></TD> 512<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 513 ALT="$\textstyle \myequal $"></TD> 514<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="intro_img59.png" 515 ALT="$\displaystyle \cos 2\psi \ Q + \sin 2\psi \ U$"></TD> 516<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 517 </TD></TR> 518<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 1.87ex; vertical-align: -0.10ex; " SRC="intro_img60.png" 519 ALT="$\displaystyle U^{\prime}$"></TD> 520<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 521 ALT="$\textstyle \myequal $"></TD> 522<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="intro_img61.png" 523 ALT="$\displaystyle -\sin 2\psi \ Q + \cos 2\psi \ U$"></TD> 524<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 525(<SPAN CLASS="arabic">8</SPAN>)</TD></TR> 526</TABLE> 527</SPAN></DIV> 528<BR CLEAR="ALL"><P></P><BLOCKQUOTE></BLOCKQUOTE> 529<P> 530<BLOCKQUOTE><SMALL CLASS="FOOTNOTESIZE">To analyze the CMB temperature on the sky, it is natural to 531expand it in spherical harmonics. These are not appropriate 532for polarization, because 533the two combinations <SPAN CLASS="MATH"><IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="intro_img62.png" 534 ALT="$Q\pm iU$"></SPAN> are quantities of spin <SPAN CLASS="MATH"><IMG STYLE="height: 1.57ex; vertical-align: -0.29ex; " SRC="intro_img63.png" 535 ALT="$\pm 2$"></SPAN> 536(<A 537 HREF="intro_Bibliography.htm#goldberg">Goldberg, 1967</A>). They 538should be expanded in spin-weighted harmonics <!-- MATH 539 $\, _{\pm2}Y_l^m$ 540 --> 541<SPAN CLASS="MATH"><IMG STYLE="height: 2.27ex; vertical-align: -0.67ex; " SRC="intro_img64.png" 542 ALT="$\, _{\pm2}Y_l^m$"></SPAN> 543(<A ID="tex2html109" target="contents" 544 HREF="intro_Bibliography.htm#longspin">Seljak & Zaldarriaga, 1997</A>; <A ID="tex2html110" target="contents" 545 HREF="intro_Bibliography.htm#spinlong">Zaldarriaga & Seljak, 1997</A>), 546</SMALL></BLOCKQUOTE> 547<BR> 548<DIV ALIGN="CENTER"><A ID="Pexpansion"></A> 549<!-- MATH 550 \begin{eqnarray} 551T(\textbf{n})&\myequal &\sum_{lm} a_{T,lm} Y_{lm}(\textbf{n}) \nonumber \\ 552 (Q+iU)(\textbf{n})&\myequal &\sum_{lm} 553 a_{2,lm}\;_2Y_{lm}(\textbf{n}) \nonumber \\ 554 (Q-iU)(\textbf{n})&\myequal &\sum_{lm} 555 a_{-2,lm}\;_{-2}Y_{lm}(\textbf{n}). 556\end{eqnarray} 557 --> 558<SPAN CLASS="MATH"> 559<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 560<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><I>T</I>(<B>n</B>)</TD> 561<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 562 ALT="$\textstyle \myequal $"></TD> 563<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 4.37ex; vertical-align: -2.66ex; " SRC="intro_img65.png" 564 ALT="$\displaystyle \sum_{lm} a_{T,lm} Y_{lm}(\textbf{n})$"></TD> 565<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 566 </TD></TR> 567<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT">(<I>Q</I>+<I>iU</I>)(<B>n</B>)</TD> 568<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 569 ALT="$\textstyle \myequal $"></TD> 570<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 4.37ex; vertical-align: -2.66ex; " SRC="intro_img66.png" 571 ALT="$\displaystyle \sum_{lm} 572a_{2,lm}\;_2Y_{lm}(\textbf{n})$"></TD> 573<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 574 </TD></TR> 575<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT">(<I>Q</I>-<I>iU</I>)(<B>n</B>)</TD> 576<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 577 ALT="$\textstyle \myequal $"></TD> 578<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 4.37ex; vertical-align: -2.66ex; " SRC="intro_img67.png" 579 ALT="$\displaystyle \sum_{lm} 580a_{-2,lm}\;_{-2}Y_{lm}(\textbf{n}).$"></TD> 581<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 582(<SPAN CLASS="arabic">9</SPAN>)</TD></TR> 583</TABLE> 584</SPAN></DIV> 585<BR CLEAR="ALL"><P></P><BLOCKQUOTE><SMALL CLASS="FOOTNOTESIZE"> 586To perform this expansion, <SPAN CLASS="MATH"><I>Q</I></SPAN> and <SPAN CLASS="MATH"><I>U</I></SPAN> in equation (<A HREF="#Pexpansion">9</A>) 587are measured relative to <!-- MATH 588 $(\textbf{e}_{1}, \textbf{e}_{2})=(\textbf{e}_\theta , \textbf{e}_\phi )$ 589 --> 590<SPAN CLASS="MATH"><IMG STYLE="height: 2.45ex; vertical-align: -0.75ex; " SRC="intro_img68.png" 591 ALT="$(\textbf{e}_{1}, \textbf{e}_{2})=(\textbf{e}_\theta , \textbf{e}_\phi )$"></SPAN>, the unit vectors of the spherical coordinate system. 592Where <!-- MATH 593 $\textbf{e}_\theta$ 594 --> 595<SPAN CLASS="MATH"><IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="intro_img69.png" 596 ALT="$\textbf{e}_\theta $"></SPAN> is tangent to the local meridian and directed from North 597to South, and <!-- MATH 598 $\textbf{e}_\phi$ 599 --> 600<SPAN CLASS="MATH"><IMG STYLE="height: 1.75ex; vertical-align: -0.75ex; " SRC="intro_img70.png" 601 ALT="$\textbf{e}_\phi $"></SPAN> is tangent to the local parallel, and directed from 602West to East. 603The coefficients <!-- MATH 604 $_{\pm 2}a_{lm}$ 605 --> 606<SPAN CLASS="MATH"><IMG STYLE="height: 1.46ex; vertical-align: -0.61ex; " SRC="intro_img71.png" 607 ALT="$_{\pm 2}a_{lm}$"></SPAN> 608are observable on the sky and their power spectra 609can be 610predicted for different cosmological models. Instead of <!-- MATH 611 $_{\pm 2}a_{lm}$ 612 --> 613<SPAN CLASS="MATH"><IMG STYLE="height: 1.46ex; vertical-align: -0.61ex; " SRC="intro_img71.png" 614 ALT="$_{\pm 2}a_{lm}$"></SPAN> 615it is convenient 616to use their linear combinations 617</SMALL></BLOCKQUOTE> 618<BR> 619<DIV ALIGN="CENTER"> 620 621<!-- MATH 622 \begin{eqnarray} 623a_{E,lm}&\myequal &-(a_{2,lm}+a_{-2,lm})/2 \nonumber \\ 624a_{B,lm}&\myequal &-(a_{2,lm}-a_{-2,lm})/2i, 625\end{eqnarray} 626 --> 627<SPAN CLASS="MATH"> 628<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 629<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><I>a</I><SUB><I>E</I>,<I>lm</I></SUB></TD> 630<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 631 ALT="$\textstyle \myequal $"></TD> 632<TD ALIGN="LEFT" WIDTH="50%" NOWRAP>-(<I>a</I><SUB>2,<I>lm</I></SUB>+<I>a</I><SUB>-2,<I>lm</I></SUB>)/2</TD> 633<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 634 </TD></TR> 635<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><I>a</I><SUB><I>B</I>,<I>lm</I></SUB></TD> 636<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 637 ALT="$\textstyle \myequal $"></TD> 638<TD ALIGN="LEFT" WIDTH="50%" NOWRAP>-(<I>a</I><SUB>2,<I>lm</I></SUB>-<I>a</I><SUB>-2,<I>lm</I></SUB>)/2<I>i</I>,</TD> 639<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 640(<SPAN CLASS="arabic">10</SPAN>)</TD></TR> 641</TABLE> 642</SPAN></DIV> 643<BR CLEAR="ALL"><P></P><BLOCKQUOTE><SMALL CLASS="FOOTNOTESIZE"> 644which transform differently 645under parity. 646Four power spectra are needed to 647characterize fluctuations in a gaussian theory, 648the autocorrelation between 649<SPAN CLASS="MATH"><I>T</I></SPAN>, <SPAN CLASS="MATH"><I>E</I></SPAN> and <SPAN CLASS="MATH"><I>B</I></SPAN> and the cross correlation of <SPAN CLASS="MATH"><I>E</I></SPAN> and <SPAN CLASS="MATH"><I>T</I></SPAN>. 650Because of parity considerations the cross-correlations 651between <SPAN CLASS="MATH"><I>B</I></SPAN> and the 652other quantities vanish and one is left with 653</SMALL></BLOCKQUOTE> 654<BR> 655<DIV ALIGN="CENTER"><A ID="Cls"></A> 656<!-- MATH 657 \begin{eqnarray} 658\langle a_{X,lm}^{*} 659 a_{X,lm^\prime}\rangle &\myequal & \delta_{m,m^\prime}C_{Xl} 660 \quad 661 \langle a_{T,lm}^{*}a_{E,lm}\rangle=\delta_{m,m^\prime}C_{Cl}, 662\end{eqnarray} 663 --> 664<SPAN CLASS="MATH"> 665<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 666<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.62ex; vertical-align: -0.97ex; " SRC="intro_img72.png" 667 ALT="$\displaystyle \langle a_{X,lm}^{*} 668a_{X,lm^\prime}\rangle$"></TD> 669<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 670 ALT="$\textstyle \myequal $"></TD> 671<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 2.62ex; vertical-align: -0.97ex; " SRC="intro_img73.png" 672 ALT="$\displaystyle \delta_{m,m^\prime}C_{Xl} 673\quad 674\langle a_{T,lm}^{*}a_{E,lm}\rangle=\delta_{m,m^\prime}C_{Cl},$"></TD> 675<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 676(<SPAN CLASS="arabic">11</SPAN>)</TD></TR> 677</TABLE> 678</SPAN></DIV> 679<BR CLEAR="ALL"><P></P><BLOCKQUOTE><SMALL CLASS="FOOTNOTESIZE"> 680where 681<SPAN CLASS="MATH"><I>X</I></SPAN> stands for <SPAN CLASS="MATH"><I>T</I></SPAN>, <SPAN CLASS="MATH"><I>E</I></SPAN> or <SPAN CLASS="MATH"><I>B</I></SPAN>, <!-- MATH 682 $\langle\cdots \rangle$ 683 --> 684<SPAN CLASS="MATH"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img74.png" 685 ALT="$\langle\cdots \rangle$"></SPAN> 686means ensemble average and <SPAN CLASS="MATH"><IMG STYLE="height: 2.39ex; vertical-align: -0.75ex; " SRC="intro_img75.png" 687 ALT="$\delta_{i,j}$"></SPAN> is the Kronecker delta. 688</SMALL></BLOCKQUOTE> 689<P> 690<BLOCKQUOTE><SMALL CLASS="FOOTNOTESIZE">We can rewrite 691equation (<A HREF="#Pexpansion">9</A>) as 692</SMALL></BLOCKQUOTE> 693<BR> 694<DIV ALIGN="CENTER"><A ID="Pexpansion2"></A> 695<!-- MATH 696 \begin{eqnarray} 697T(\textbf{n})&\myequal &\sum_{lm} a_{T,lm} Y_{lm}(\textbf{n}) \nonumber \\ 698 Q(\textbf{n})&\myequal &-\sum_{lm} a_{E,lm} X_{1,lm} 699 +i a_{B,lm}X_{2,lm} \nonumber \\ 700 U(\textbf{n})&\myequal &-\sum_{lm} a_{B,lm} X_{1,lm}-i a_{E,lm} X_{2,lm} 701 702\end{eqnarray} 703 --> 704<SPAN CLASS="MATH"> 705<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 706<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><I>T</I>(<B>n</B>)</TD> 707<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 708 ALT="$\textstyle \myequal $"></TD> 709<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 4.37ex; vertical-align: -2.66ex; " SRC="intro_img65.png" 710 ALT="$\displaystyle \sum_{lm} a_{T,lm} Y_{lm}(\textbf{n})$"></TD> 711<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 712 </TD></TR> 713<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><I>Q</I>(<B>n</B>)</TD> 714<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 715 ALT="$\textstyle \myequal $"></TD> 716<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 4.37ex; vertical-align: -2.66ex; " SRC="intro_img76.png" 717 ALT="$\displaystyle -\sum_{lm} a_{E,lm} X_{1,lm} 718+i a_{B,lm}X_{2,lm}$"></TD> 719<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 720 </TD></TR> 721<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><I>U</I>(<B>n</B>)</TD> 722<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 723 ALT="$\textstyle \myequal $"></TD> 724<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 4.37ex; vertical-align: -2.66ex; " SRC="intro_img77.png" 725 ALT="$\displaystyle -\sum_{lm} a_{B,lm} X_{1,lm}-i a_{E,lm} X_{2,lm}$"></TD> 726<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 727(<SPAN CLASS="arabic">12</SPAN>)</TD></TR> 728</TABLE> 729</SPAN></DIV> 730<BR CLEAR="ALL"><P></P><BLOCKQUOTE><SMALL CLASS="FOOTNOTESIZE"> 731where we have introduced 732<!-- MATH 733 $X_{1,lm}(\textbf{n})=(\;_2Y_{lm}+\;_{-2}Y_{lm})/2$ 734 --> 735<SPAN CLASS="MATH"><IMG STYLE="height: 2.39ex; vertical-align: -0.75ex; " SRC="intro_img78.png" 736 ALT="$X_{1,lm}(\textbf{n})=(\;_2Y_{lm}+\;_{-2}Y_{lm})/2$"></SPAN> 737and <!-- MATH 738 $X_{2,lm}(\textbf{n})=(\;_2Y_{lm}-\;_{-2}Y_{lm})/ 2$ 739 --> 740<SPAN CLASS="MATH"><IMG STYLE="height: 2.39ex; vertical-align: -0.75ex; " SRC="intro_img79.png" 741 ALT="$X_{2,lm}(\textbf{n})=(\;_2Y_{lm}-\;_{-2}Y_{lm})/ 2$"></SPAN>. 742They satisfy <!-- MATH 743 $Y^{*}_{lm} = (-1)^m Y_{l-m}$ 744 --> 745<SPAN CLASS="MATH"><I>Y</I><SUP>*</SUP><SUB><I>lm</I></SUB> = (-1)<SUP><i>m</i></SUP> <I>Y</I><SUB><I>l</I>-<I>m</I></SUB></SPAN>, 746<!-- MATH 747 $X^{*}_{1,lm}=(-1)^m X_{1,l-m}$ 748 --> 749<SPAN CLASS="MATH"><I>X</I><SUP>*</SUP><SUB>1,<I>lm</I></SUB>=(-1)<SUP><i>m</i></SUP> <I>X</I><SUB>1,<I>l</I>-<I>m</I></SUB></SPAN> and 750<!-- MATH 751 $X^*_{2,lm}=(-1)^{m+1}X_{2,l-m}$ 752 --> 753<SPAN CLASS="MATH"><I>X</I><SUP><i>*</i></SUP><SUB>2,<I>lm</I></SUB>=(-1)<SUP><I>m</I>+1</SUP><I>X</I><SUB>2,<I>l</I>-<I>m</I></SUB></SPAN> which 754together with <!-- MATH 755 $a_{T,lm}=(-1)^m a_{T,l-m}^*$ 756 --> 757<SPAN CLASS="MATH"><I>a</I><SUB><I>T</I>,<I>lm</I></SUB>=(-1)<SUP><i>m</i></SUP> <I>a</I><SUB><I>T</I>,<I>l</I>-<I>m</I></SUB><SUP><i>*</i></SUP></SPAN>, <!-- MATH 758 $a_{E,lm}=(-1)^m a_{E,l-m}^*$ 759 --> 760<SPAN CLASS="MATH"><I>a</I><SUB><I>E</I>,<I>lm</I></SUB>=(-1)<SUP><i>m</i></SUP> <I>a</I><SUB><I>E</I>,<I>l</I>-<I>m</I></SUB><SUP><i>*</i></SUP></SPAN> and 761<!-- MATH 762 $a_{B,lm}=(-1)^m a_{B,l-m}^*$ 763 --> 764<SPAN CLASS="MATH"><I>a</I><SUB><I>B</I>,<I>lm</I></SUB>=(-1)<SUP><i>m</i></SUP> <I>a</I><SUB><I>B</I>,<I>l</I>-<I>m</I></SUB><SUP><i>*</i></SUP></SPAN> make <SPAN CLASS="MATH"><I>T</I></SPAN>, <SPAN CLASS="MATH"><I>Q</I></SPAN> and <SPAN CLASS="MATH"><I>U</I></SPAN> real. 765</SMALL></BLOCKQUOTE> 766<P> 767<BLOCKQUOTE><SMALL CLASS="FOOTNOTESIZE">In fact <!-- MATH 768 $X_{1,lm}(\textbf{n})$ 769 --> 770<SPAN CLASS="MATH"><I>X</I><SUB>1,<I>lm</I></SUB>(<B>n</B>)</SPAN> and <!-- MATH 771 $X_{2,lm}(\textbf{n})$ 772 --> 773<SPAN CLASS="MATH"><I>X</I><SUB>2,<I>lm</I></SUB>(<B>n</B>)</SPAN> have the form, 774<!-- MATH 775 ${X_{1,lm}(\textbf{n})=\sqrt{(2l+1) / 4\pi} F_{1,lm}(\theta)\ e^{im\phi}}$ 776 --> 777<SPAN CLASS="MATH"><IMG STYLE="height: 3.56ex; vertical-align: -1.17ex; " SRC="intro_img80.png" 778 ALT="${X_{1,lm}(\textbf{n})=\sqrt{(2l+1) / 4\pi} F_{1,lm}(\theta)\ e^{im\phi}}$"></SPAN> 779and <!-- MATH 780 ${X_{2,lm}(\textbf{n})=\sqrt{(2l+1) / 4\pi} F_{2,lm}(\theta)\ e^{im\phi}}$ 781 --> 782<SPAN CLASS="MATH"><IMG STYLE="height: 3.56ex; vertical-align: -1.17ex; " SRC="intro_img81.png" 783 ALT="${X_{2,lm}(\textbf{n})=\sqrt{(2l+1) / 4\pi} F_{2,lm}(\theta)\ e^{im\phi}}$"></SPAN>, 784<!-- MATH 785 ${F_{(1,2),lm}(\theta)}$ 786 --> 787<SPAN CLASS="MATH"><IMG STYLE="height: 2.56ex; vertical-align: -0.85ex; " SRC="intro_img82.png" 788 ALT="${F_{(1,2),lm}(\theta)}$"></SPAN> can be calculated in terms of Legendre 789polynomials (<A 790 HREF="intro_Bibliography.htm#kks">Kamionkowski et al., 1997</A>) 791</SMALL></BLOCKQUOTE> 792<BR> 793<DIV ALIGN="CENTER"><A ID="def:basis"></A> 794<!-- MATH 795 \begin{eqnarray} 796F_{1,lm}(\theta)&\myequal & N_{lm} 797\left[ -\left({l-m^2 \over \sin^2\theta} 798+{1 \over 2}l(l-1)\right)P_l^m(\cos \theta) 799+(l+m) {\cos \theta \over \sin^2 \theta} 800P_{l-1}^m(\cos\theta)\right] \nonumber \\ 801F_{2,lm}(\theta)&\myequal & N_{lm}{m \over 802\sin^2 \theta} 803[ -(l-1)\cos \theta P_l^m(\cos \theta)+(l+m) P_{l-1}^m(\cos\theta)], 804\end{eqnarray} 805 --> 806<SPAN CLASS="MATH"> 807<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 808<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.39ex; vertical-align: -0.75ex; " SRC="intro_img83.png" 809 ALT="$\displaystyle F_{1,lm}(\theta)$"></TD> 810<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 811 ALT="$\textstyle \myequal $"></TD> 812<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="intro_img84.png" 813 ALT="$\displaystyle N_{lm} 814\left[ -\left({l-m^2 \over \sin^2\theta} 815+{1 \over 2}l(l-... 816... \theta) 817+(l+m) {\cos \theta \over \sin^2 \theta} 818P_{l-1}^m(\cos\theta)\right]$"></TD> 819<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 820 </TD></TR> 821<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.39ex; vertical-align: -0.75ex; " SRC="intro_img85.png" 822 ALT="$\displaystyle F_{2,lm}(\theta)$"></TD> 823<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 824 ALT="$\textstyle \myequal $"></TD> 825<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 4.25ex; vertical-align: -1.69ex; " SRC="intro_img86.png" 826 ALT="$\displaystyle N_{lm}{m \over 827\sin^2 \theta} 828[ -(l-1)\cos \theta P_l^m(\cos \theta)+(l+m) P_{l-1}^m(\cos\theta)],$"></TD> 829<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 830(<SPAN CLASS="arabic">13</SPAN>)</TD></TR> 831</TABLE> 832</SPAN></DIV> 833<BR CLEAR="ALL"><P></P><BLOCKQUOTE><SMALL CLASS="FOOTNOTESIZE"> 834where 835</SMALL></BLOCKQUOTE> 836<BR> 837<DIV ALIGN="CENTER"> 838 839<!-- MATH 840 \begin{eqnarray} 841N_{lm}(\theta)&\myequal & 2 \sqrt{(l-2)!(l-m)! \over (l+2)!(l+m)!}. 842\end{eqnarray} 843 --> 844<SPAN CLASS="MATH"> 845<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 846<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img87.png" 847 ALT="$\displaystyle N_{lm}(\theta)$"></TD> 848<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 849 ALT="$\textstyle \myequal $"></TD> 850<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 7.05ex; vertical-align: -2.81ex; " SRC="intro_img88.png" 851 ALT="$\displaystyle 2 \sqrt{(l-2)!(l-m)! \over (l+2)!(l+m)!}.$"></TD> 852<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 853(<SPAN CLASS="arabic">14</SPAN>)</TD></TR> 854</TABLE> 855</SPAN></DIV> 856<BR CLEAR="ALL"><P></P><BLOCKQUOTE><SMALL CLASS="FOOTNOTESIZE"> 857Note that <!-- MATH 858 $F_{2,lm}(\theta)=0$ 859 --> 860<SPAN CLASS="MATH"><IMG STYLE="height: 2.39ex; vertical-align: -0.75ex; " SRC="intro_img89.png" 861 ALT="$F_{2,lm}(\theta)=0$"></SPAN> if <SPAN CLASS="MATH"><I>m</I>=0</SPAN>, as it must to make the 862Stokes parameters real. 863</SMALL></BLOCKQUOTE> 864<P> 865<BLOCKQUOTE><SMALL CLASS="FOOTNOTESIZE">The correlation functions between 2 points on the sky (noted 1 and 2) separated 866by an angle <SPAN CLASS="MATH"><IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="intro_img90.png" 867 ALT="$\beta$"></SPAN> 868can be calculated using equations (<A HREF="#Cls">11</A>) 869and (<A HREF="#Pexpansion2">12</A>). However, as pointed out in <A 870 HREF="intro_Bibliography.htm#kks">Kamionkowski et al. (1997)</A>, the 871natural coordinate system to express the correlations is one in which 872<!-- MATH 873 $\textbf{e}_{1}$ 874 --> 875<SPAN CLASS="MATH"><B>e</B><SUB>1</SUB></SPAN> vectors at each point are tangent to the great circle 876connecting these 2 points, with the <!-- MATH 877 $\textbf{e}_{2}$ 878 --> 879<SPAN CLASS="MATH"><B>e</B><SUB>2</SUB></SPAN> vectors being perpendicular to 880the <!-- MATH 881 $\textbf{e}_{1}$ 882 --> 883<SPAN CLASS="MATH"><B>e</B><SUB>1</SUB></SPAN> vectors. With this choice of reference frames, and using 884the addition theorem for the spin harmonics (<A 885 HREF="intro_Bibliography.htm#primer">Hu & White, 1997</A>), 886</SMALL></BLOCKQUOTE> 887<BR> 888<DIV ALIGN="CENTER"><A ID="addtheo"></A> 889<!-- MATH 890 \begin{eqnarray} 891\sum_m \;_{s_1} Y_{lm}^*(\textbf{n}_1) 892\;_{s_2} Y_{lm}(\textbf{n}_2)&\myequal &\sqrt{2l+1 \over 4 \pi} 893\;_{s_2} Y_{l-s_1}(\beta,\psi_1)e^{-is_2\psi_2} 894\end{eqnarray} 895 --> 896<SPAN CLASS="MATH"> 897<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 898<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 4.14ex; vertical-align: -2.42ex; " SRC="intro_img91.png" 899 ALT="$\displaystyle \sum_m \;_{s_1} Y_{lm}^*(\textbf{n}_1) 900\;_{s_2} Y_{lm}(\textbf{n}_2)$"></TD> 901<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 902 ALT="$\textstyle \myequal $"></TD> 903<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 5.83ex; vertical-align: -2.00ex; " SRC="intro_img92.png" 904 ALT="$\displaystyle \sqrt{2l+1 \over 4 \pi} 905\;_{s_2} Y_{l-s_1}(\beta,\psi_1)e^{-is_2\psi_2}$"></TD> 906<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 907(<SPAN CLASS="arabic">15</SPAN>)</TD></TR> 908</TABLE> 909</SPAN></DIV> 910<BR CLEAR="ALL"><P></P><BLOCKQUOTE></BLOCKQUOTE> 911<P> 912<BLOCKQUOTE><SMALL CLASS="FOOTNOTESIZE">we have (<A 913 HREF="intro_Bibliography.htm#kks">Kamionkowski et al., 1997</A>) 914</SMALL></BLOCKQUOTE> 915<BR> 916<DIV ALIGN="CENTER"><A ID="QUr"></A> 917<!-- MATH 918 \begin{eqnarray} 919\langle T_1T_2 \rangle&\myequal &\sum_l {2l+1 \over 4 \pi} 920C_{Tl} P_l(\cos \beta) \nonumber \\ 921\langle Q_{r}(1)Q_{r}(2) \rangle&\myequal &\sum_l {2l+1 \over 4 \pi} [C_{El} 922F_{1,l2}(\beta)-C_{Bl} F_{2,l2}(\beta)] \nonumber \\ 923\langle U_{r}(1)U_{r}(2) \rangle&\myequal &\sum_l {2l+1 \over 4 \pi} 924[C_{Bl} F_{1,l2}(\beta)-C_{El} F_{2,l2}(\beta) ] \nonumber \\ 925\langle T(1)Q_{r}(2) 926\rangle&\myequal & - \sum_l {2l+1 \over 4 \pi} C_{Cl} F_{1,l0}(\beta)\nonumber \\ 927\langle T(1)U_{r}(2) \rangle&\myequal &0. 928\end{eqnarray} 929 --> 930<SPAN CLASS="MATH"> 931<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 932<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img93.png" 933 ALT="$\displaystyle \langle T_1T_2 \rangle$"></TD> 934<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 935 ALT="$\textstyle \myequal $"></TD> 936<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 5.65ex; vertical-align: -2.66ex; " SRC="intro_img94.png" 937 ALT="$\displaystyle \sum_l {2l+1 \over 4 \pi} 938C_{Tl} P_l(\cos \beta)$"></TD> 939<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 940 </TD></TR> 941<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img95.png" 942 ALT="$\displaystyle \langle Q_{r}(1)Q_{r}(2) \rangle$"></TD> 943<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 944 ALT="$\textstyle \myequal $"></TD> 945<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 5.65ex; vertical-align: -2.66ex; " SRC="intro_img96.png" 946 ALT="$\displaystyle \sum_l {2l+1 \over 4 \pi} [C_{El} 947F_{1,l2}(\beta)-C_{Bl} F_{2,l2}(\beta)]$"></TD> 948<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 949 </TD></TR> 950<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img97.png" 951 ALT="$\displaystyle \langle U_{r}(1)U_{r}(2) \rangle$"></TD> 952<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 953 ALT="$\textstyle \myequal $"></TD> 954<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 5.65ex; vertical-align: -2.66ex; " SRC="intro_img98.png" 955 ALT="$\displaystyle \sum_l {2l+1 \over 4 \pi} 956[C_{Bl} F_{1,l2}(\beta)-C_{El} F_{2,l2}(\beta) ]$"></TD> 957<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 958 </TD></TR> 959<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img99.png" 960 ALT="$\displaystyle \langle T(1)Q_{r}(2) 961\rangle$"></TD> 962<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 963 ALT="$\textstyle \myequal $"></TD> 964<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 5.65ex; vertical-align: -2.66ex; " SRC="intro_img100.png" 965 ALT="$\displaystyle - \sum_l {2l+1 \over 4 \pi} C_{Cl} F_{1,l0}(\beta)$"></TD> 966<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 967 </TD></TR> 968<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img101.png" 969 ALT="$\displaystyle \langle T(1)U_{r}(2) \rangle$"></TD> 970<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 971 ALT="$\textstyle \myequal $"></TD> 972<TD ALIGN="LEFT" WIDTH="50%" NOWRAP>0.</TD> 973<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 974(<SPAN CLASS="arabic">16</SPAN>)</TD></TR> 975</TABLE> 976</SPAN></DIV> 977<BR CLEAR="ALL"><P></P><BLOCKQUOTE><SMALL CLASS="FOOTNOTESIZE"> 978The subscript <SPAN CLASS="MATH"><I>r</I></SPAN> 979here indicate that the Stokes parameters are measured in this 980particular coordinate system. 981We can use the transformation laws in equation (<A HREF="#QUtrans">8</A>) 982to write <SPAN CLASS="MATH">(<I>Q</I>,<I>U</I>)</SPAN> in terms of <SPAN CLASS="MATH">(<I>Q</I><SUB><i>r</i></SUB>,<I>U</I><SUB><i>r</i></SUB>)</SPAN>. 983</SMALL> 984</BLOCKQUOTE> 985<P> 986Using the fact that, when <!-- MATH 987 $\beta \rightarrow 0$ 988 --> 989<SPAN CLASS="MATH"><IMG STYLE="height: 2.10ex; vertical-align: -0.55ex; " SRC="intro_img102.png" 990 ALT="$\beta \rightarrow 0$"></SPAN>, <!-- MATH 991 $P_\ell(\cos\beta) \rightarrow 1$ 992 --> 993<SPAN CLASS="MATH"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img103.png" 994 ALT="$P_\ell(\cos\beta) \rightarrow 1$"></SPAN> and <!-- MATH 995 $P_\ell^2(\cos 996\beta) \rightarrow \sin^2 \beta \frac{(\ell+2)!}{8 (\ell-2)!}$ 997 --> 998<SPAN CLASS="MATH"><IMG STYLE="height: 3.50ex; vertical-align: -1.28ex; " SRC="intro_img104.png" 999 ALT="$P_\ell^2(\cos 1000\beta) \rightarrow \sin^2 \beta \frac{(\ell+2)!}{8 (\ell-2)!}$"></SPAN>, 1001the definitions above imply that the variances of the temperature and 1002polarization are related to the power spectra by 1003<BR> 1004<DIV ALIGN="CENTER"><A ID="var"></A> 1005<!-- MATH 1006 \begin{eqnarray} 1007\langle TT \rangle&\myequal &\sum_\ell {2\ell+1 \over 4 \pi} 1008C_{T\ell} \nonumber \\ 1009\langle QQ \rangle + \langle UU\rangle &\myequal &\sum_l {2\ell+1 \over 4 \pi} \left(C_{E\ell} 1010+C_{B\ell}\right) \nonumber \\ 1011\langle TQ\rangle = \langle TU\rangle&\myequal & 0. 1012\end{eqnarray} 1013 --> 1014<SPAN CLASS="MATH"> 1015<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 1016<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img105.png" 1017 ALT="$\displaystyle \langle TT \rangle$"></TD> 1018<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1019 ALT="$\textstyle \myequal $"></TD> 1020<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 5.65ex; vertical-align: -2.66ex; " SRC="intro_img106.png" 1021 ALT="$\displaystyle \sum_\ell {2\ell+1 \over 4 \pi} 1022C_{T\ell}$"></TD> 1023<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1024 </TD></TR> 1025<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img107.png" 1026 ALT="$\displaystyle \langle QQ \rangle + \langle UU\rangle$"></TD> 1027<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1028 ALT="$\textstyle \myequal $"></TD> 1029<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 5.65ex; vertical-align: -2.66ex; " SRC="intro_img108.png" 1030 ALT="$\displaystyle \sum_l {2\ell+1 \over 4 \pi} \left(C_{E\ell} 1031+C_{B\ell}\right)$"></TD> 1032<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1033 </TD></TR> 1034<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img109.png" 1035 ALT="$\displaystyle \langle TQ\rangle = \langle TU\rangle$"></TD> 1036<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1037 ALT="$\textstyle \myequal $"></TD> 1038<TD ALIGN="LEFT" WIDTH="50%" NOWRAP>0.</TD> 1039<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1040(<SPAN CLASS="arabic">17</SPAN>)</TD></TR> 1041</TABLE> 1042</SPAN></DIV> 1043<BR CLEAR="ALL"><P></P> 1044 1045<P> 1046It is also worth noting that with these conventions, the cross power <SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img110.png" 1047 ALT="$C_{C\ell}$"></SPAN> 1048for scalar perturbations 1049must be positive at low <SPAN CLASS="MATH"><IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="intro_img36.png" 1050 ALT="$\ell$"></SPAN>, in order to produce <EM>at large scales</EM> a radial pattern of 1051polarization around cold temperature spots (and a tangential pattern around hot 1052spots) as it is expected from scalar perturbations (<A 1053 HREF="intro_Bibliography.htm#crco">Crittenden et al., 1995</A>). 1054 1055<P> 1056Note that Eq. (<A HREF="#Pexpansion2">12</A>) implies that, if the Stokes parameters are 1057rotated <EM>everywhere</EM> via 1058<BR> 1059<DIV ALIGN="CENTER"><A ID="eq:rotateQU"></A> 1060<!-- MATH 1061 \begin{eqnarray} 1062\left(\begin{array}{c} 1063 Q'\\U' 1064\end{array}\right) = 1065\left(\begin{array}{c c} 1066 \cos2\psi & \sin2\psi \\ 1067 -\sin2\psi & \cos2\psi 1068\end{array} \right) 1069\left(\begin{array}{c} 1070 Q\\U 1071\end{array} \right),%\nonumber 1072\end{eqnarray} 1073 --> 1074<SPAN CLASS="MATH"> 1075<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 1076<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="intro_img111.png" 1077 ALT="$\displaystyle \left(\begin{array}{c} 1078Q'\\ U' 1079\end{array}\right) = 1080\left(\be... 1081...array} \right) 1082\left(\begin{array}{c} 1083Q\\ U 1084\end{array} \right),%\nonumber 1085$"></TD> 1086<TD> </TD> 1087<TD> </TD> 1088<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1089(<SPAN CLASS="arabic">18</SPAN>)</TD></TR> 1090</TABLE> 1091</SPAN></DIV> 1092<BR CLEAR="ALL"><P></P> 1093then the polarized <SPAN CLASS="MATH"><IMG STYLE="height: 1.46ex; vertical-align: -0.45ex; " SRC="intro_img26.png" 1094 ALT="$a_{\ell m}$"></SPAN> coefficients are submittted to the same rotation 1095<BR> 1096<DIV ALIGN="CENTER"><A ID="eq:rotateEB"></A> 1097<!-- MATH 1098 \begin{eqnarray} 1099\left(\begin{array}{c} 1100 a_{E,\ell m}'\\a_{B,\ell m}' 1101\end{array}\right) = 1102\left(\begin{array}{c c} 1103 \cos2\psi & \sin2\psi \\ 1104 -\sin2\psi & \cos2\psi 1105\end{array} \right) 1106\left(\begin{array}{c} 1107 a_{E,\ell m}\\a_{B,\ell m} 1108\end{array} \right).%\nonumber 1109\end{eqnarray} 1110 --> 1111<SPAN CLASS="MATH"> 1112<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 1113<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="intro_img112.png" 1114 ALT="$\displaystyle \left(\begin{array}{c} 1115a_{E,\ell m}'\\ a_{B,\ell m}' 1116\end{arra... 1117...\begin{array}{c} 1118a_{E,\ell m}\\ a_{B,\ell m} 1119\end{array} \right).%\nonumber 1120$"></TD> 1121<TD> </TD> 1122<TD> </TD> 1123<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1124(<SPAN CLASS="arabic">19</SPAN>)</TD></TR> 1125</TABLE> 1126</SPAN></DIV> 1127<BR CLEAR="ALL"><P></P> 1128 1129<P> 1130Finally, with these conventions, a polarization with (<SPAN CLASS="MATH"><I>Q</I>>0,<I>U</I>=0</SPAN>) will be along the 1131North-South axis, and (<SPAN CLASS="MATH"><I>Q</I>=0,<I>U</I>>0</SPAN>) will be along a North-West to South-East axis 1132(see Fig. <A HREF="#fig:reftqu">5</A>) 1133<P> 1134 1135<H3><A ID="SECTION632"></A> 1136<A ID="subsub:relatoldversion"></A> 1137<BR> 1138Relation to previous releases 1139</H3>Even though it was stated otherwise in the documention, <b>HEALPix</b> used a different 1140convention for the polarization in its previous releases. The tensor harmonics approach 1141(<A 1142 HREF="intro_Bibliography.htm#kks">Kamionkowski et al. (1997)</A>, hereafter KKS) was used, instead of 1143the current spin weighted spherical harmonics. These two approaches differ by 1144the normalisation and sign of the basis functions used, which in turns change 1145the normalisation of the power spectra. 1146Table 1 summarize the relations between the CMB power spectra in the different 1147releases. 1148See § <A HREF="#subsec:cmbfast">A.2.1</A> about the interface between <b>HEALPix</b> and CMBFAST. 1149 1150<P> 1151<TABLE STYLE="width:100%;"> 1152<TR><TD> 1153<SMALL CLASS="SMALL">Table 1: Relation between CMB power spectra conventions used in HEALPix, CMBFAST and 1154KKS. The power spectra on the same row are equal.</SMALL> 1155<TABLE CELLPADDING=3 BORDER="1"> 1156<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=94>Component</TD> 1157<TD ALIGN="CENTER"><b>HEALPix</b> <SPAN CLASS="MATH"><IMG STYLE="height: 1.87ex; vertical-align: -0.41ex; " SRC="intro_img113.png" 1158 ALT="$\ge$"></SPAN> 1.2<A ID="tex2html17" 1159 HREF="#footmp620"><SUP><SPAN CLASS="arabic">1</SPAN></SUP></A></TD> 1160<TD ALIGN="CENTER">CMBFAST</TD> 1161<TD ALIGN="CENTER">KKS</TD> 1162<TD ALIGN="CENTER"><b>HEALPix</b> <SPAN CLASS="MATH"><IMG STYLE="height: 1.87ex; vertical-align: -0.41ex; " SRC="intro_img114.png" 1163 ALT="$\le$"></SPAN> 1.1<A ID="tex2html18" 1164 HREF="#footmp621"><SUP><SPAN CLASS="arabic">2</SPAN></SUP></A></TD> 1165</TR> 1166<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=94>Temperature</TD> 1167<TD ALIGN="CENTER"><!-- MATH 1168 $C_{\ell}^{\mathrm{TEMP}}$ 1169 --> 1170<SPAN CLASS="MATH"><IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="intro_img115.png" 1171 ALT="$C_{\ell}^{\mathrm{TEMP}} $"></SPAN></TD> 1172<TD ALIGN="CENTER"><!-- MATH 1173 $C_{\mathrm{T},\ell}$ 1174 --> 1175<SPAN CLASS="MATH"><IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="intro_img116.png" 1176 ALT="$C_{\mathrm{T},\ell} $"></SPAN></TD> 1177<TD ALIGN="CENTER"><!-- MATH 1178 $C_{\ell}^{\mathrm{T}}$ 1179 --> 1180<SPAN CLASS="MATH"><IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="intro_img117.png" 1181 ALT="$C_{\ell}^{\mathrm{T}} $"></SPAN></TD> 1182<TD ALIGN="CENTER"><!-- MATH 1183 $C_{\ell}^{\mathrm{TEMP}}$ 1184 --> 1185<SPAN CLASS="MATH"><IMG STYLE="height: 2.51ex; vertical-align: -0.67ex; " SRC="intro_img115.png" 1186 ALT="$C_{\ell}^{\mathrm{TEMP}} $"></SPAN></TD> 1187</TR> 1188<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=94>Electric or Gradient</TD> 1189<TD ALIGN="CENTER"><!-- MATH 1190 $C_{\ell}^{\mathrm{GRAD}}$ 1191 --> 1192<SPAN CLASS="MATH"><IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="intro_img118.png" 1193 ALT="$C_{\ell}^{\mathrm{GRAD}} $"></SPAN></TD> 1194<TD ALIGN="CENTER"><!-- MATH 1195 $C_{\mathrm{E},\ell}$ 1196 --> 1197<SPAN CLASS="MATH"><IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="intro_img119.png" 1198 ALT="$C_{\mathrm{E},\ell} $"></SPAN></TD> 1199<TD ALIGN="CENTER"><!-- MATH 1200 $2C_{\ell}^{\mathrm{G}}$ 1201 --> 1202<SPAN CLASS="MATH"><IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="intro_img120.png" 1203 ALT="$2C_{\ell}^{\mathrm{G}} $"></SPAN></TD> 1204<TD ALIGN="CENTER"><!-- MATH 1205 $2C_{\ell}^{\mathrm{GRAD}}$ 1206 --> 1207<SPAN CLASS="MATH"><IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="intro_img121.png" 1208 ALT="$2C_{\ell}^{\mathrm{GRAD}} $"></SPAN></TD> 1209</TR> 1210<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=94>Magnetic or Curl</TD> 1211<TD ALIGN="CENTER"><!-- MATH 1212 $C_{\ell}^{\mathrm{CURL}}$ 1213 --> 1214<SPAN CLASS="MATH"><IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="intro_img122.png" 1215 ALT="$C_{\ell}^{\mathrm{CURL}} $"></SPAN></TD> 1216<TD ALIGN="CENTER"><!-- MATH 1217 $C_{\mathrm{B},\ell}$ 1218 --> 1219<SPAN CLASS="MATH"><IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="intro_img123.png" 1220 ALT="$C_{\mathrm{B},\ell} $"></SPAN></TD> 1221<TD ALIGN="CENTER"><!-- MATH 1222 $2C_{\ell}^{\mathrm{C}}$ 1223 --> 1224<SPAN CLASS="MATH"><IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="intro_img124.png" 1225 ALT="$2C_{\ell}^{\mathrm{C}} $"></SPAN></TD> 1226<TD ALIGN="CENTER"><!-- MATH 1227 $2C_{\ell}^{\mathrm{CURL}}$ 1228 --> 1229<SPAN CLASS="MATH"><IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="intro_img125.png" 1230 ALT="$2C_{\ell}^{\mathrm{CURL}} $"></SPAN></TD> 1231</TR> 1232<TR><TD ALIGN="LEFT" VALIGN="TOP" WIDTH=94>Temp.-Electric cross correlation</TD> 1233<TD ALIGN="CENTER"><!-- MATH 1234 $C_{\ell}^{\mathrm{T-GRAD}}\rule[.3cm]{0cm}{.2cm}$ 1235 --> 1236<SPAN CLASS="MATH"><IMG STYLE="height: 2.68ex; vertical-align: -0.68ex; " SRC="intro_img126.png" 1237 ALT="$C_{\ell}^{\mathrm{T-GRAD}}\rule[.3cm]{0cm}{.2cm}$"></SPAN></TD> 1238<TD ALIGN="CENTER"><!-- MATH 1239 $C_{\mathrm{C},\ell}$ 1240 --> 1241<SPAN CLASS="MATH"><IMG STYLE="height: 2.27ex; vertical-align: -0.75ex; " SRC="intro_img127.png" 1242 ALT="$C_{\mathrm{C},\ell} $"></SPAN></TD> 1243<TD ALIGN="CENTER"><SPAN CLASS="MATH"><IMG STYLE="height: 2.39ex; vertical-align: -0.43ex; " SRC="intro_img128.png" 1244 ALT="$-\sqrt{2}$"></SPAN> <!-- MATH 1245 $C_{\ell}^{\mathrm{TG}}$ 1246 --> 1247<SPAN CLASS="MATH"><IMG STYLE="height: 2.56ex; vertical-align: -0.67ex; " SRC="intro_img129.png" 1248 ALT="$C_{\ell}^{\mathrm{TG}} $"></SPAN></TD> 1249<TD ALIGN="CENTER"><!-- MATH 1250 $\sqrt{2}C_{\ell}^{\mathrm{T-GRAD}}$ 1251 --> 1252<SPAN CLASS="MATH"><IMG STYLE="height: 2.68ex; vertical-align: -0.68ex; " SRC="intro_img130.png" 1253 ALT="$\sqrt{2}C_{\ell}^{\mathrm{T-GRAD}} $"></SPAN></TD> 1254</TR> 1255</TABLE></TD></TR> 1256<TR CLASS="LEFT"> 1257<TD><DL> 1258<DD><A ID="footmp620"><SUP><SPAN CLASS="arabic">1</SPAN></SUP></A> Version 1.2 (Feb 2003) or more recent of <b>HEALPix</b> package 1259</DD> 1260 1261<DD><A ID="footmp621"><SUP><SPAN CLASS="arabic">2</SPAN></SUP></A> Version 1.1 or older of <b>HEALPix</b> package 1262</DD> 1263</DL></TD></TR> 1264</TABLE> 1265 1266<P> 1267Introducing the matrices 1268<BR> 1269<DIV ALIGN="CENTER"> 1270 1271<!-- MATH 1272 \begin{eqnarray} 1273M_{\ell m} &\myequal & \left( 1274 \begin{array}{cc} X_{1,\ell m} & i X_{2,\ell m} \\ 1275 -i X_{2,\ell m} & X_{1,\ell m} 1276 \end{array} 1277 \right) 1278 1279\end{eqnarray} 1280 --> 1281<SPAN CLASS="MATH"> 1282<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 1283<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img131.png" 1284 ALT="$\displaystyle M_{\ell m}$"></TD> 1285<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1286 ALT="$\textstyle \myequal $"></TD> 1287<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="intro_img132.png" 1288 ALT="$\displaystyle \left( 1289\begin{array}{cc} X_{1,\ell m} & i X_{2,\ell m} \\ 1290-i X_{2,\ell m} & X_{1,\ell m} 1291\end{array} \right)$"></TD> 1292<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1293(<SPAN CLASS="arabic">20</SPAN>)</TD></TR> 1294</TABLE> 1295</SPAN></DIV> 1296<BR CLEAR="ALL"><P></P> 1297where the basis functions <SPAN CLASS="MATH"><I>X</I><SUB><i>1</i></SUB></SPAN> and <SPAN CLASS="MATH"><I>X</I><SUB><i>2</i></SUB></SPAN> have been defined in 1298Eqs. (<A HREF="#def:basis">13</A>) and above, 1299the decomposition in spherical harmonics coefficients (<A HREF="#Pexpansion2">12</A>) of a 1300given map of the Stokes parameter 1301<SPAN CLASS="MATH"><I>Q</I></SPAN> and <SPAN CLASS="MATH"><I>U</I></SPAN> can be written in the case of <b>HEALPix</b> 1.2 as 1302<BR> 1303<DIV ALIGN="CENTER"><A ID="QU:12"></A> 1304<!-- MATH 1305 \begin{eqnarray} 1306\phantom{1.2}{ 1307\left( 1308\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\\U \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array} 1309\right) 1310} &\myequal & \sum_{\ell m} M_{\ell m} { 1311\left( 1312\begin{array}{c} -a_{\ell m}^{\mathrm{GRAD}} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\\-a_{\ell m}^{\mathrm{CURL}} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array} 1313\right) 1314}. 1315\end{eqnarray} 1316 --> 1317<SPAN CLASS="MATH"> 1318<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 1319<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 8.22ex; vertical-align: -3.92ex; " SRC="intro_img133.png" 1320 ALT="$\displaystyle \phantom{1.2}{ 1321\left( 1322\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\... 1323...}{.2cm}\\ U \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right) 1324}$"></TD> 1325<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1326 ALT="$\textstyle \myequal $"></TD> 1327<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 8.22ex; vertical-align: -3.92ex; " SRC="intro_img134.png" 1328 ALT="$\displaystyle \sum_{\ell m} M_{\ell m} { 1329\left( 1330\begin{array}{c} -a_{\ell m}^{... 1331...thrm{CURL}} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right) 1332}.$"></TD> 1333<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1334(<SPAN CLASS="arabic">21</SPAN>)</TD></TR> 1335</TABLE> 1336</SPAN></DIV> 1337<BR CLEAR="ALL"><P></P> 1338 1339<P> 1340For KKS, with the same definition of <SPAN CLASS="MATH"><I>M</I></SPAN>, the decomposition reads 1341<BR> 1342<DIV ALIGN="CENTER"><A ID="QU:KKS"></A> 1343<!-- MATH 1344 \begin{eqnarray} 1345{ 1346\left( 1347\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\\-U \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array} 1348\right) 1349} &\myequal & \sum_{\ell m} M_{\ell m} { 1350\left( 1351\begin{array}{c} \sqrt{2}a_{\mathrm{E},\ell m} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\\\sqrt{2}a_{\mathrm{B},\ell m} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array} 1352\right) 1353}, 1354 1355\end{eqnarray} 1356 --> 1357<SPAN CLASS="MATH"> 1358<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 1359<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 8.22ex; vertical-align: -3.92ex; " SRC="intro_img135.png" 1360 ALT="$\displaystyle { 1361\left( 1362\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\\ -U \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right) 1363}$"></TD> 1364<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1365 ALT="$\textstyle \myequal $"></TD> 1366<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 8.22ex; vertical-align: -3.92ex; " SRC="intro_img136.png" 1367 ALT="$\displaystyle \sum_{\ell m} M_{\ell m} { 1368\left( 1369\begin{array}{c} \sqrt{2}a_{\m... 1370...{B},\ell m} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right) 1371},$"></TD> 1372<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1373(<SPAN CLASS="arabic">22</SPAN>)</TD></TR> 1374</TABLE> 1375</SPAN></DIV> 1376<BR CLEAR="ALL"><P></P> 1377whereas in <b>HEALPix</b> 1.1 it was 1378<BR> 1379<DIV ALIGN="CENTER"><A ID="QU:11"></A> 1380<!-- MATH 1381 \begin{eqnarray} 1382\phantom{1.1}{ 1383\left( 1384\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\\U \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array} 1385\right) 1386}&\myequal &\sum_{\ell m} M_{\ell m} { 1387\left( 1388\begin{array}{c} -\sqrt{2}a_{\ell m}^{\mathrm{GRAD}} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\\\sqrt{2}a_{\ell m}^{\mathrm{CURL}} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array} 1389\right) 1390}. 1391 1392\end{eqnarray} 1393 --> 1394<SPAN CLASS="MATH"> 1395<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 1396<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 8.22ex; vertical-align: -3.92ex; " SRC="intro_img137.png" 1397 ALT="$\displaystyle \phantom{1.1}{ 1398\left( 1399\begin{array}{c} Q \rule[.3cm]{0cm}{.2cm}\... 1400...}{.2cm}\\ U \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right) 1401}$"></TD> 1402<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1403 ALT="$\textstyle \myequal $"></TD> 1404<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 8.22ex; vertical-align: -3.92ex; " SRC="intro_img138.png" 1405 ALT="$\displaystyle \sum_{\ell m} M_{\ell m} { 1406\left( 1407\begin{array}{c} -\sqrt{2}a_{\... 1408...thrm{CURL}} \rule[.3cm]{0cm}{.2cm}\rule[-.3cm]{0cm}{.2cm}\end{array}\right) 1409}.$"></TD> 1410<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1411(<SPAN CLASS="arabic">23</SPAN>)</TD></TR> 1412</TABLE> 1413</SPAN></DIV> 1414<BR CLEAR="ALL"><P></P> 1415The difference between KKS and 1.1 was due to an error of sign on one the basis functions. 1416 1417<P> 1418 1419<H3><A ID="SECTION633"> 1420Relation with IAU convention</A> 1421</H3> 1422In a cartesian referential with axes <SPAN CLASS="MATH"><I>x</I></SPAN> and <SPAN CLASS="MATH"><I>y</I></SPAN>, the Stokes parameters for 1423linear polarisation are defined such that <SPAN CLASS="MATH">+<I>Q</I></SPAN> is aligned with <SPAN CLASS="MATH">+<I>x</I></SPAN>, <SPAN CLASS="MATH">-<I>Q</I></SPAN> with <SPAN CLASS="MATH">+<I>y</I></SPAN> and <SPAN CLASS="MATH">+<I>U</I></SPAN> with the 1424bisectrix of <SPAN CLASS="MATH">+<I>x</I></SPAN> and <SPAN CLASS="MATH">+<I>y</I></SPAN>. Although this definition is universally accepted, 1425some confusion may still arise from the relation of 1426this local cartesian system to the global spherical one, as described below 1427(<A 1428 HREF="intro_Bibliography.htm#hamakerleahy">Hamaker & Leahy, 2003</A>), and as illustrated in Fig. <A HREF="#fig:reftqu">5</A>. 1429<P> 1430 1431<DIV class="CENTER"><A ID="fig:reftqu"></A><A ID="intro:fig:reftqu"></A><A ID="1072"></A> 1432<TABLE> 1433<CAPTION class="BOTTOM"><STRONG>Figure 5:</STRONG> 1434Coordinate conventions for <b>HEALPix</b> (<EM>lhs</EM> panels) and IAU (<EM>rhs</EM> panels). The 1435 upper panels illustrate how the spherical coordinates are measured, and the 1436 lower panel how the <SPAN CLASS="MATH"><I>Q</I></SPAN> and <SPAN CLASS="MATH"><I>U</I></SPAN> Stokes parameters are identified in the 1437 tangential plan. 1438</CAPTION> 1439<TR><TD> 1440<DIV CLASS="centerline" ID="par4474" ALIGN="CENTER"> 1441<IMG STYLE="" 1442 SRC="./merge_reftqu.png" 1443 ALT="Image merge_reftqu"></DIV> 1444 1445<P></TD></TR> 1446</TABLE> 1447</DIV> 1448 1449<P> 1450The polarization conventions defined by the International Astronomical Union 1451(<A 1452 HREF="intro_Bibliography.htm#iau74">IAU, 1974</A>) are summarized in <A 1453 HREF="intro_Bibliography.htm#hambreg">Hamaker & Bregman (1996)</A>. They define at each point on the 1454celestial sphere a cartesian referential with the <SPAN CLASS="MATH"><I>x</I></SPAN> and <SPAN CLASS="MATH"><I>y</I></SPAN> axes pointing 1455respectively toward the North and East, and the <SPAN CLASS="MATH"><I>z</I></SPAN> 1456axis along the line of sight pointing toward the observer (ie, inwards) for a 1457right-handed system. 1458 1459<P> 1460On the other hand, following the mathematical and CMB litterature tradition, 1461<b>HEALPix</b> defines a cartesian referential with the <SPAN CLASS="MATH"><I>x</I></SPAN> and <SPAN CLASS="MATH"><I>y</I></SPAN> axes pointing 1462respectively toward the <EM>South</EM> and East, and the <SPAN CLASS="MATH"><I>z</I></SPAN> axis along the line of sight 1463pointing away from the observer (ie, <EM>outwards</EM>) for a right-handed 1464system. The <EM>Planck</EM> CMB mission follows the same convention (<A 1465 HREF="intro_Bibliography.htm#ansari">Ansari et al., 2003</A>). 1466 1467<P> 1468The consequence of this definition discrepency is a change of sign of <SPAN CLASS="MATH"><I>U</I></SPAN>, 1469which, if not accounted for, jeopardizes the calculation of the Electric and Magnetic CMB 1470polarisation power spectra. 1471 1472<P> 1473 1474<H3><A ID="SECTION634"></A> 1475<A ID="intro:polcconv"></A> 1476<BR> 1477How <b>HEALPix</b> deals with these discrepancies: <SPAN CLASS="texttt">POLCCONV</SPAN> keyword 1478</H3> 1479The FITS keyword <SPAN CLASS="texttt">POLCCONV</SPAN> has been introduced in <b>HEALPix</b> 2.0 to describe the 1480polarisation coordinate convention applied to the data contained in the file. 1481Its value is either <SPAN CLASS="texttt">'COSMO'</SPAN> for files following the HEALPix/CMB/<EM>Planck</EM> convention 1482(default for sky map synthetized with HEALPix routine <A HREF="./fac_synfast.htm#fac:synfast"><SPAN CLASS="texttt">synfast</SPAN></A>) 1483or <SPAN CLASS="texttt">'IAU'</SPAN> for those 1484following the IAU convention, as defined above. Absence of this keyword is 1485interpreted as meaning <SPAN CLASS="texttt">'COSMO'</SPAN> (as it is the case for WMAP maps). 1486 1487<P> 1488Starting with <b>HEALPix</b> 3.40, when dealing with a polarized (full-sky or cut-sky) signal map, 1489<BR>- the F90 subroutine <A HREF="./sub_input_map.htm#sub:input_map"><SPAN CLASS="texttt">input_map</SPAN></A> in its default mode, 1490<BR>- the F90 facilities calling it and dealing with the <SPAN CLASS="MATH"><I>I</I></SPAN>, <SPAN CLASS="MATH"><I>Q</I></SPAN> and <SPAN CLASS="MATH"><I>U</I></SPAN> Stokes parameters as a whole, <EM>ie</EM> 1491<A HREF="./fac_anafast.htm#fac:anafast"><SPAN CLASS="texttt">anafast</SPAN></A> and 1492<A HREF="./fac_smoothing.htm#fac:smoothing"><SPAN CLASS="texttt">smoothing</SPAN></A>, 1493<BR>- as well as their IDL wrappers 1494<A HREF="./idl_ianafast.htm#idl:ianafast"><SPAN CLASS="texttt">ianafast</SPAN></A> and 1495<A HREF="./idl_ismoothing.htm#idl:ismoothing"><SPAN CLASS="texttt">ismoothing</SPAN></A>, 1496<BR>- the IDL visualisation routines 1497<A HREF="./idl_mollview.htm#idl:mollview"><SPAN CLASS="texttt">azeqview</SPAN>, <SPAN CLASS="texttt">cartview</SPAN>, <SPAN CLASS="texttt">gnomview</SPAN>, <SPAN CLASS="texttt">mollview</SPAN> and <SPAN CLASS="texttt">orthview</SPAN></A> 1498called with 1499<A HREF="./idl_mollview.htm#idl:mollview:polarization"><SPAN CLASS="texttt">Polarization=2</SPAN> or <SPAN CLASS="texttt">3</SPAN></A>, 1500<BR>- and all C++ facilities (and the input routine <SPAN CLASS="texttt">read_Healpix_map_from_fits</SPAN>) 1501<BR> 1502will all 1503<BR>- issue an error message and 1504crash if <SPAN CLASS="texttt">POLCCONV</SPAN> is explicitely set to a value different from <SPAN CLASS="texttt">'COSMO'</SPAN> and <SPAN CLASS="texttt">'IAU'</SPAN>, 1505<BR>- issue a warning (except in C++), and <EM>swap the sign of the <SPAN CLASS="MATH"><I>U</I></SPAN> polarisation</EM> stored into memory if the FITS file being read contains <SPAN CLASS="texttt">POLCCONV='IAU'</SPAN>, 1506<BR>- issue a warning (except in C++) if the keyword <SPAN CLASS="texttt">POLCCONV</SPAN> is totally absent, and then carry on with the original data, 1507<BR>- or work silently with the original data if <SPAN CLASS="texttt">POLCCONV='COSMO'</SPAN>. 1508<BR> 1509On the other hand, and as in previous releases, routines treating or showing 1510each of <SPAN CLASS="MATH"><I>I</I></SPAN>, <SPAN CLASS="MATH"><I>Q</I></SPAN> and <SPAN CLASS="MATH"><I>U</I></SPAN> fields separately, 1511such as the F90 facilities 1512<A HREF="./fac_median_filter.htm#fac:median_filter"><SPAN CLASS="texttt">median_filter</SPAN></A>, 1513<A HREF="./fac_ud_grade.htm#fac:ud_grade"><SPAN CLASS="texttt">ud_grade</SPAN></A>, or 1514<A HREF="./fac_map2gif.htm#fac:map2gif"><SPAN CLASS="texttt">map2gif</SPAN></A> 1515as well as their IDL counterparts 1516<A HREF="./idl_median_filter.htm#idl:median_filter"><SPAN CLASS="texttt">median_filter</SPAN></A>, 1517<A HREF="./idl_ud_grade.htm#idl:ud_grade"><SPAN CLASS="texttt">ud_grade</SPAN></A>, or 1518<A HREF="./idl_mollview.htm#idl:mollview"><SPAN CLASS="texttt">mollview</SPAN> et al</A> run with 1519<A HREF="./idl_mollview.htm#idl:mollview:polarization"><SPAN CLASS="texttt">Polarization=0</SPAN> or <SPAN CLASS="texttt">1</SPAN></A> will 1520ignore the value of <SPAN CLASS="texttt">POLCCONV</SPAN> (copying it unchanged into their output files, when applicable) 1521and preserve the sign of <SPAN CLASS="MATH"><I>U</I></SPAN>. 1522 1523<P> 1524Finally, 1525the IDL subroutine <A HREF="./idl_change_polcconv.htm#idl:change_polcconv"><SPAN CLASS="texttt">change_polcconv.pro</SPAN></A> 1526and the Python facility <SPAN CLASS="texttt">change_polcconv.py</SPAN> are 1527provided to add the <SPAN CLASS="texttt">POLCCONV</SPAN> keyword or 1528change/update its value and swap the sign of the <SPAN CLASS="MATH"><I>U</I></SPAN> Stokes parameter, when applicable, in 1529an existing FITS file. 1530 1531<P> 1532 1533<H2><A ID="SECTION640"></A> 1534<A ID="sphericalstuff"></A> 1535<BR> 1536Spherical harmonic conventions 1537</H2> 1538 1539<P> 1540The Spherical Harmonics are defined as 1541<BR> 1542<DIV ALIGN="CENTER"><A ID="eq:ylm_def"></A> 1543<!-- MATH 1544 \begin{eqnarray} 1545Y_{\ell m}(\theta,\phi) &\myequal & \lambda_{\ell m}(\cos\theta) e^{{i} 1546 m\phi} 1547\end{eqnarray} 1548 --> 1549<SPAN CLASS="MATH"> 1550<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 1551<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img139.png" 1552 ALT="$\displaystyle Y_{\ell m}(\theta,\phi)$"></TD> 1553<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1554 ALT="$\textstyle \myequal $"></TD> 1555<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 2.62ex; vertical-align: -0.68ex; " SRC="intro_img140.png" 1556 ALT="$\displaystyle \lambda_{\ell m}(\cos\theta) e^{{i} 1557m\phi}$"></TD> 1558<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1559(<SPAN CLASS="arabic">24</SPAN>)</TD></TR> 1560</TABLE> 1561</SPAN></DIV> 1562<BR CLEAR="ALL"><P></P> 1563where 1564<BR> 1565<DIV ALIGN="CENTER"><A ID="eq:lam_def"></A> 1566<!-- MATH 1567 \begin{eqnarray} 1568\lambda_{\ell m}(x) &\myequal & \sqrt{ \frac{2\ell+1}{4\pi} 1569 \frac{(\ell-m)!}{(\ell+m)!} } P_{\ell m}(x), \quad\textrm{for~} 1570 m\ge 0\\ 1571\lambda_{\ell m} &\myequal & (-1)^m \lambda_{\ell |m|}, \quad\textrm{for~} 1572 m < 0, \nonumber \\ 1573\lambda_{\ell m} &\myequal & 0, \quad\textrm{for}\, |m| > \ell.\nonumber 1574\end{eqnarray} 1575 --> 1576<SPAN CLASS="MATH"> 1577<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 1578<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img141.png" 1579 ALT="$\displaystyle \lambda_{\ell m}(x)$"></TD> 1580<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1581 ALT="$\textstyle \myequal $"></TD> 1582<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 7.05ex; vertical-align: -2.81ex; " SRC="intro_img142.png" 1583 ALT="$\displaystyle \sqrt{ \frac{2\ell+1}{4\pi} 1584\frac{(\ell-m)!}{(\ell+m)!} } P_{\ell m}(x), \quad\textrm{for~} 1585m\ge 0$"></TD> 1586<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1587(<SPAN CLASS="arabic">25</SPAN>)</TD></TR> 1588<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img143.png" 1589 ALT="$\displaystyle \lambda_{\ell m}$"></TD> 1590<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1591 ALT="$\textstyle \myequal $"></TD> 1592<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 2.56ex; vertical-align: -0.85ex; " SRC="intro_img144.png" 1593 ALT="$\displaystyle (-1)^m \lambda_{\ell \vert m\vert}, \quad\textrm{for~} 1594m < 0,$"></TD> 1595<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1596 </TD></TR> 1597<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img143.png" 1598 ALT="$\displaystyle \lambda_{\ell m}$"></TD> 1599<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1600 ALT="$\textstyle \myequal $"></TD> 1601<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img145.png" 1602 ALT="$\displaystyle 0, \quad\textrm{for}\, \vert m\vert > \ell.$"></TD> 1603<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1604 </TD></TR> 1605</TABLE> 1606</SPAN></DIV> 1607<BR CLEAR="ALL"><P></P> 1608 1609<P> 1610Introducing <!-- MATH 1611 $x\equiv\cos\theta$ 1612 --> 1613<SPAN CLASS="MATH"><IMG STYLE="height: 1.69ex; vertical-align: -0.10ex; " SRC="intro_img146.png" 1614 ALT="$x\equiv\cos\theta$"></SPAN>, the associated Legendre Polynomials <SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img147.png" 1615 ALT="$P_{\ell m}$"></SPAN> 1616solve the differential equation 1617<BR> 1618<DIV ALIGN="CENTER"><A ID="eq:diff_eq"></A> 1619<!-- MATH 1620 \begin{eqnarray} 1621(1-x^2)\frac{d^2}{dx^2}P_{\ell m} - 2x \frac{d}{dx}P_{\ell m} 1622 + \left(\ell(\ell+1) - \frac{m^2}{1-x^2}\right) P_{\ell m} &\myequal & 0. 1623\end{eqnarray} 1624 --> 1625<SPAN CLASS="MATH"> 1626<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 1627<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 5.83ex; vertical-align: -2.42ex; " SRC="intro_img148.png" 1628 ALT="$\displaystyle (1-x^2)\frac{d^2}{dx^2}P_{\ell m} - 2x \frac{d}{dx}P_{\ell m} 1629+ \left(\ell(\ell+1) - \frac{m^2}{1-x^2}\right) P_{\ell m}$"></TD> 1630<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1631 ALT="$\textstyle \myequal $"></TD> 1632<TD ALIGN="LEFT" WIDTH="50%" NOWRAP>0.</TD> 1633<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1634(<SPAN CLASS="arabic">26</SPAN>)</TD></TR> 1635</TABLE> 1636</SPAN></DIV> 1637<BR CLEAR="ALL"><P></P> 1638They are related to the ordinary Legendre Polynomials <SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img149.png" 1639 ALT="$P_\ell$"></SPAN> by 1640<BR> 1641<DIV ALIGN="CENTER"><A ID="eq:legendreass"></A> 1642<!-- MATH 1643 \begin{eqnarray} 1644P_{\ell m} &\myequal & (-1)^m (1-x^2)^{m/2} \frac{d^m}{dx^m} P_{\ell}(x), 1645\end{eqnarray} 1646 --> 1647<SPAN CLASS="MATH"> 1648<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 1649<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img150.png" 1650 ALT="$\displaystyle P_{\ell m}$"></TD> 1651<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1652 ALT="$\textstyle \myequal $"></TD> 1653<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 4.84ex; vertical-align: -1.69ex; " SRC="intro_img151.png" 1654 ALT="$\displaystyle (-1)^m (1-x^2)^{m/2} \frac{d^m}{dx^m} P_{\ell}(x),$"></TD> 1655<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1656(<SPAN CLASS="arabic">27</SPAN>)</TD></TR> 1657</TABLE> 1658</SPAN></DIV> 1659<BR CLEAR="ALL"><P></P> 1660which are given by the Rodrigues formula 1661<BR> 1662<DIV ALIGN="CENTER"><A ID="eq:rodrigues"></A> 1663<!-- MATH 1664 \begin{eqnarray} 1665P_{\ell}(x) &\myequal & \frac{1}{2^\ell \ell!}\frac{d^\ell}{dx^\ell} (x^2-1)^\ell. 1666\end{eqnarray} 1667 --> 1668<SPAN CLASS="MATH"> 1669<TABLE CELLPADDING="0" ALIGN="CENTER" WIDTH="100%"> 1670<TR VALIGN="MIDDLE"><TD NOWRAP WIDTH="50%" ALIGN="RIGHT"><IMG STYLE="height: 2.33ex; vertical-align: -0.68ex; " SRC="intro_img152.png" 1671 ALT="$\displaystyle P_{\ell}(x)$"></TD> 1672<TD ALIGN="CENTER" NOWRAP><IMG STYLE="height: 0.58ex; vertical-align: -0.10ex; " SRC="intro_img13.png" 1673 ALT="$\textstyle \myequal $"></TD> 1674<TD ALIGN="LEFT" WIDTH="50%" NOWRAP><IMG STYLE="height: 5.13ex; vertical-align: -1.69ex; " SRC="intro_img153.png" 1675 ALT="$\displaystyle \frac{1}{2^\ell \ell!}\frac{d^\ell}{dx^\ell} (x^2-1)^\ell.$"></TD> 1676<TD CLASS="eqno" WIDTH=10 ALIGN="RIGHT"> 1677(<SPAN CLASS="arabic">28</SPAN>)</TD></TR> 1678</TABLE> 1679</SPAN></DIV> 1680<BR CLEAR="ALL"><P></P> 1681 1682<P> 1683Note that our <SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img11.png" 1684 ALT="$Y_{\ell m}$"></SPAN> are identical to those of <A 1685 HREF="intro_Bibliography.htm#edmonds">Edmonds (1957)</A>, 1686even though our definition of the <SPAN CLASS="MATH"><IMG STYLE="height: 2.04ex; vertical-align: -0.45ex; " SRC="intro_img147.png" 1687 ALT="$P_{\ell m}$"></SPAN> differ from his by a factor 1688<SPAN CLASS="MATH">(-1)<SUP><i>m</i></SUP></SPAN> (<I>a.k.a.</I> Condon-Shortley phase). 1689 1690<P> 1691 1692<DIV CLASS="navigation"><HR> 1693<!--Navigation Panel--> 1694<A 1695 HREF="intro_HEALPix_Software_Package.htm"> 1696<IMG WIDTH="63" HEIGHT="24" ALT="previous" SRC="prev.png"></A> 1697<A 1698 HREF="intro_Introduction_HEALPix.htm"> 1699<IMG WIDTH="26" HEIGHT="24" ALT="up" SRC="up.png"></A> 1700<A 1701 HREF="intro_Pixel_window_functions.htm"> 1702<IMG WIDTH="37" HEIGHT="24" ALT="next" SRC="next.png"></A> 1703<A ID="tex2html97" 1704 HREF="intro_TABLE_CONTENTS.htm"> 1705<IMG WIDTH="65" HEIGHT="24" ALT="contents" SRC="contents.png"></A> 1706<BR> 1707<B> Previous:</B> <A 1708 HREF="intro_HEALPix_Software_Package.htm">The HEALPix Software Package</A> 1709 1710<B>Up:</B> <A 1711 HREF="intro_Introduction_HEALPix.htm">Introduction to HEALPix</A> 1712 1713<B> Next:</B> <A 1714 HREF="intro_Pixel_window_functions.htm">Pixel window functions</A> 1715<B> Top:</B> <a href="main.htm">Main Page</a></DIV> 1716<!--End of Navigation Panel--> 1717<ADDRESS> 1718Version 3.50, 2018-12-10 1719</ADDRESS> 1720</BODY> 1721</HTML> 1722
