Editing Cosmic microwave background (section)

==Multipole analysis==
[[File:WMAP 2008 TT spectra.png|thumb|Example Multipole Power Spectrum. WMAP Data are represented as points, curves correspond to the best-fit LCDM model<ref name="WMAP9Cosmo">{{Cite journal |last1=Hinshaw |first1=G. |last2=Larson |first2=D. |last3=Komatsu |first3=E. |last4=Spergel |first4=D. N. |last5=Bennett |first5=C. L. |last6=Dunkley |first6=J. |last7=Nolta |first7=M. R. |last8=Halpern |first8=M. |last9=Hill |first9=R. S. |last10=Odegard |first10=N. |last11=Page |first11=L. |last12=Smith |first12=K. M. |last13=Weiland |first13=J. L. |last14=Gold |first14=B. |last15=Jarosik |first15=N. |date=2013-09-20 |title=NINE-YEAR ''WILKINSON MICROWAVE ANISOTROPY PROBE'' ( ''WMAP'' ) OBSERVATIONS: COSMOLOGICAL PARAMETER RESULTS |url=https://iopscience.iop.org/article/10.1088/0067-0049/208/2/19 |journal=The Astrophysical Journal Supplement Series |volume=208 |issue=2 |pages=19 |doi=10.1088/0067-0049/208/2/19 |issn=0067-0049|arxiv=1212.5226 |bibcode=2013ApJS..208...19H }}</ref>]]
The CMB angular anisotropies are usually presented in terms of power per multipole.<ref name="cmbreview">{{cite journal |author1=P.A. Zyla et al. (Particle Data Group) |title=Review of Particle Physics |journal=Progress of Theoretical and Experimental Physics |date=2020 |volume=2020 |issue=8 |page=083C01 |doi=10.1093/ptep/ptaa104 |url=https://pdg.lbl.gov/2020/reviews/rpp2020-rev-cosmic-microwave-background.pdf|doi-access=free }} Cosmic Microwave Background review by Scott and Smoot.</ref>
The map of temperature across the sky, <math>T(\theta,\varphi),</math> is written as coefficients of [[spherical harmonics]],
<math display="block">T(\theta,\varphi) = \sum_{\ell m} a_{\ell m} Y_{\ell m}(\theta,\varphi)</math>
where the <math>a_{\ell m}</math> term measures the strength of the angular oscillation in <math>Y_{\ell m}(\theta,\varphi)</math>, and ''ℓ'' is the multipole number while ''m'' is the azimuthal number.
The azimuthal variation is not significant and is removed by applying the [[angular correlation function]], giving power spectrum term <math>C_{\ell}\equiv \langle |a_{\ell m}|^2 \rangle.</math> Increasing values of ''ℓ'' correspond to higher multipole moments of CMB, meaning more rapid variation with angle.

===CMBR monopole term (''ℓ'' = 0)===
The monopole term, {{nowrap|1=''ℓ'' = 0}}, is the constant isotropic mean temperature of the CMB, {{nowrap|1=''T''<sub>''γ''</sub> = {{val|2.7255|0.0006|u=K}}}}<ref name="cmbreview"/> with one standard deviation confidence. This term must be measured with absolute temperature devices, such as the FIRAS instrument on the [[Cosmic Background Explorer|COBE]] satellite.<ref name="cmbreview"/>{{rp|499}}

===CMBR dipole anisotropy (''ℓ'' = 1)===
CMB dipole represents the largest anisotropy, which is in the first spherical harmonic ({{nowrap|1=''ℓ'' = 1}}), a cosine function. The amplitude of CMB dipole is around {{val|3.3621|0.0010|u=mK}}.<ref name="cmbreview"/> The CMB dipole moment is interpreted as the peculiar motion of the Earth relative to the CMB. Its amplitude depends on the time due to the Earth's orbit about the barycenter of the solar system. This enables us to add a time-dependent term to the dipole expression. The modulation of this term is 1 year,<ref name="cmbreview"/><ref name="cobe">{{Cite web|title=COBE Differential Microwave Radiometers: Calibration Techniques.| url=http://articles.adsabs.harvard.edu/pdf/1992ApJ...391..466B| last= Bennett| first=C.}}</ref> which fits the observation done by COBE FIRAS.<ref name="cobe"/><ref>{{Cite journal|title=Dipole Modulation of Cosmic Microwave Background Temperature and Polarization.| last=Shosh| first=S.| journal=Journal of Cosmology and Astroparticle Physics| year=2016| volume =2016 |issue=1|page=046| doi=10.1088/1475-7516/2016/01/046 | arxiv=1507.04078 | bibcode=2016JCAP...01..046G | s2cid=118553819}}</ref> The dipole moment does not encode any primordial information.

From the CMB data, it is seen that the Sun appears to be moving at {{val|369.82|0.11|u=km/s}} relative to the reference frame of the CMB (also called the CMB rest frame, or the frame of reference in which there is no motion through the CMB). The [[Local Group]] — the galaxy group that includes our own Milky Way galaxy — appears to be moving at {{val|620|15|u=km/s}} in the direction of [[galactic longitude]] {{nowrap|1=''ℓ'' = {{val|271.9|2|u=°}}}}, {{nowrap|1=''b'' = {{val|30|3|u=°}}}}.<ref name="cmbreview"/> The dipole is now used to calibrate mapping studies.

===Multipole (''ℓ'' ≥ 2)===
The temperature variation in the CMB temperature maps at higher multipoles, or {{nowrap|''ℓ'' ≥ 2}}, is considered to be the result of perturbations of the density in the early Universe, before the recombination epoch at a redshift of around&nbsp;{{nowrap|1=''z'' ⋍ 1100}}. Before recombination, the Universe consisted of a hot, dense plasma of electrons and baryons. In such a hot dense environment, electrons and protons could not form any neutral atoms. The baryons in such early Universe remained highly ionized and so were tightly coupled with photons through the effect of Thompson scattering. These phenomena caused the pressure and gravitational effects to act against each other, and triggered fluctuations in the photon-baryon plasma. Quickly after the recombination epoch, the rapid expansion of the universe caused the plasma to cool down and these fluctuations are "frozen into" the CMB maps we observe today.<ref name="cmbreview"/>

===Data analysis challenges===
Raw CMBR data, even from space vehicles such as WMAP or Planck, contain foreground effects that completely obscure the fine-scale structure of the cosmic microwave background. The fine-scale structure is superimposed on the raw CMBR data but is too small to be seen at the scale of the raw data. The most prominent of the foreground effects is the dipole anisotropy caused by the Sun's motion relative to the CMBR background. The dipole anisotropy and others due to Earth's annual motion relative to the Sun and numerous microwave sources in the galactic plane and elsewhere must be subtracted out to reveal the extremely tiny variations characterizing the fine-scale structure of the CMBR background.
The detailed analysis of CMBR data to produce maps, an angular power spectrum, and ultimately cosmological parameters is a complicated, computationally difficult problem.

In practice it is hard to take the effects of noise and foreground sources into account. In particular, these foregrounds are dominated by galactic emissions such as [[bremsstrahlung]], [[Synchrotron radiation#Synchrotron radiation in astronomy|synchrotron]], and [[dust#Dust in other contexts|dust]] that emit in the microwave band; in practice, the galaxy has to be removed, resulting in a CMB map that is not a full-sky map. In addition, point sources like galaxies and clusters represent foreground sources which must be removed so as not to distort the short scale structure of the CMB power spectrum.

Constraints on many cosmological parameters can be obtained from their effects on the power spectrum, and results are often calculated using [[Markov chain Monte Carlo]] sampling techniques.

===Anomalies===
{{See also|Cosmological principle|Axis of evil (cosmology)|CMB cold spot}}

With the increasingly precise data provided by WMAP, there have been a number of claims that the CMB exhibits anomalies, such as very large scale anisotropies, anomalous alignments, and non-Gaussian distributions.<ref name="arXiv:0905.2854v2">{{cite journal| last1=Rossmanith |first1=G. |year=2009 |title=Non-Gaussian Signatures in the five-year WMAP data as identified with isotropic scaling indices |doi=10.1111/j.1365-2966.2009.15421.x| journal=Monthly Notices of the Royal Astronomical Society|volume=399|issue=4| pages=1921–1933|arxiv=0905.2854|bibcode = 2009MNRAS.399.1921R |last2=Räth |first2=C. |last3=Banday|first3=A. J.|last4=Morfill| first4=G. |doi-access=free |s2cid=11586058 }}</ref><ref name="arXiv:astro-ph/0511666">{{cite journal| last1=Bernui| first1=A. |year=2007 |title=Mapping the large-scale anisotropy in the WMAP data| doi=10.1051/0004-6361:20065585 |journal=Astronomy and Astrophysics| volume=464 |issue=2 |pages=479–485 |arxiv=astro-ph/0511666 |bibcode = 2007A&A...464..479B |last2=Mota |first2=B. |last3=Rebouças|first3=M. J.| last4=Tavakol|first4=R.| s2cid=16138962 }}</ref><ref name="arXiv:astro-ph/0503213">{{cite journal|last1=Jaffe|first1=T.R.| year=2005|title=Evidence of vorticity and shear at large angular scales in the WMAP data: a violation of cosmological isotropy?| doi=10.1086/444454|journal=The Astrophysical Journal|volume=629 | issue=1| pages=L1–L4|arxiv=astro-ph/0503213|bibcode = 2005ApJ...629L...1J |last2=Banday |first2=A. J. | last3=Eriksen|first3=H. K. | last4=Górski|first4=K. M.| last5=Hansen|first5=F. K.|s2cid=15521559}}</ref> The most longstanding of these is the low-''ℓ'' multipole controversy. Even in the COBE map, it was observed that the [[quadrupole]] ({{nowrap|1=''ℓ'' = 2}}, spherical harmonic) has a low amplitude compared to the predictions of the Big Bang. In particular, the quadrupole and octupole ({{nowrap|1=''ℓ'' = 3}}) modes appear to have an unexplained alignment with each other and with both the [[plane of the ecliptic|ecliptic plane]] and [[equinox]]es.<ref>{{cite journal |last1=de Oliveira-Costa |first1=A. |year=2004 |title=The significance of the largest scale CMB fluctuations in WMAP|journal=[[Physical Review D]]|volume=69 |pages=063516 | doi=10.1103/PhysRevD.69.063516 |arxiv=astro-ph/0307282 |bibcode = 2004PhRvD..69f3516D |issue=6 |last2=Tegmark |first2=Max |last3=Zaldarriaga |first3=Matias |last4=Hamilton |first4=Andrew| s2cid=119463060 |url=https://cds.cern.ch/record/628847 |type=Submitted manuscript }}</ref><ref>{{cite journal|last1=Schwarz|first1=D. J.|date=2004|title=Is the low-''ℓ'' microwave background cosmic?| journal=[[Physical Review Letters]]| volume=93| pages=221301| doi=10.1103/PhysRevLett.93.221301| pmid=15601079| arxiv=astro-ph/0403353| bibcode=2004PhRvL..93v1301S |issue=22 | last2=Starkman |first2=Glenn D. |last3=Huterer|first3=Dragan|last4=Copi|first4=Craig|s2cid=12554281|display-authors=2|url=https://cds.cern.ch/record/725179|type=Submitted manuscript}}</ref><ref>{{cite journal|last1=Bielewicz| first1=P.|last2=Gorski|first2=K. M.|last3=Banday|first3=A. J.|date=2004 |title=Low-order multipole maps of CMB anisotropy derived from WMAP|journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=355|pages=1283–1302 |doi=10.1111/j.1365-2966.2004.08405.x | arxiv=astro-ph/0405007 |bibcode=2004MNRAS.355.1283B |issue=4 | doi-access=free|s2cid=5564564}}</ref> A number of groups have suggested that this could be the signature of quantum corrections or new physics at the greatest observable scales; other groups suspect systematic errors in the data.<ref>{{Cite journal |last=Cao |first=F. J. |last2=de Vega |first2=H. J. |last3=Sánchez |first3=N. G. |date=2004-10-22 |title=Quantum inflaton, primordial perturbations, and CMB fluctuations |url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.70.083528 |journal=Physical Review D |volume=70 |issue=8 |pages=083528 |doi=10.1103/PhysRevD.70.083528}}</ref><ref name="arXiv:0907.2731v3">{{cite arXiv |last1=Liu|first1=Hao|last2=Li|first2=Ti-Pei|date=2009|title=Improved CMB Map from WMAP Data|class=astro-ph |eprint=0907.2731v3}}</ref><ref name="arXiv:1006.1270v1">{{cite arXiv|last1=Sawangwit|first1=Utane|last2=Shanks|first2=Tom|date=2010|title=Lambda-CDM and the WMAP Power Spectrum Beam Profile Sensitivity|class=astro-ph |eprint=1006.1270v1}}</ref><ref name="arXiv:1009.2701v1">{{cite journal |last=Liu|first=Hao|date=2010|title=Diagnosing Timing Error in WMAP Data |journal=Monthly Notices of the Royal Astronomical Society: Letters |volume=413 |issue=1|pages=L96–L100|arxiv=1009.2701v1 |display-authors=etal |bibcode=2011MNRAS.413L..96L| doi=10.1111/j.1745-3933.2011.01041.x |doi-access=free |s2cid=118739762}}</ref>

Ultimately, due to the foregrounds and the [[cosmic variance]] problem, the greatest modes will never be as well measured as the small angular scale modes. The analyses were performed on two maps that have had the foregrounds removed as far as possible: the "internal linear combination" map of the WMAP collaboration and a similar map prepared by [[Max Tegmark]] and others.<ref name="hinshaw07">{{cite journal|last=Hinshaw|first=G.|author2= (WMAP collaboration)|year=2007|title=Three-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: temperature analysis|journal=[[Astrophysical Journal Supplement Series]]|volume=170|issue=2|pages=288–334|doi=10.1086/513698|arxiv=astro-ph/0603451|bibcode=2007ApJS..170..288H|last3=Bennett|first3=C. L.|last4=Bean|first4=R.|author-link4=Rachel Bean|last5=Doré|first5=O.|last6=Greason|first6=M. R.|last7=Halpern|first7=M.|last8=Hill|first8=R. S.| last9=Jarosik| first9=N.| last10=Kogut| first10=A.| last11=Komatsu|first11=E.|last12=Limon|first12=M.|last13=Odegard|first13=N.|last14=Meyer|first14=S. S.| last15=Page |first15=L. |last16=Peiris |first16=H. V.|last17=Spergel|first17=D. N.|last18=Tucker|first18=G. S.| last19=Verde| first19=L.| last20 =Weiland|first20=J. L.|last21=Wollack|first21=E.|last22=Wright|first22=E. L.|display-authors=etal| citeseerx=10.1.1.471.7186|s2cid=15554608}}</ref><ref name="FirstWMAP">
{{cite journal|last=Bennett|first=C. L.|author2= (WMAP collaboration)|year=2003|title=First-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: preliminary maps and basic results|journal=[[Astrophysical Journal Supplement Series]]|volume=148|issue=1|pages=1–27|doi=10.1086/377253|arxiv=astro-ph/0302207| bibcode=2003ApJS..148....1B| last3=Hinshaw|first3=G.| last4=Jarosik| first4=N.|last5=Kogut| first5=A.| last6=Limon|first6=M.|last7=Meyer|first7=S. S.|last8=Page|first8=L.|last9=Spergel|first9=D. N.|last10=Tucker|first10=G. S.| last11=Wollack |first11=E.| last12=Wright|first12=E. L.|last13=Barnes|first13=C.|last14=Greason|first14=M. R.|last15=Hill|first15=R. S.| last16=Komatsu |first16=E.| last17=Nolta| first17=M. R.|last18=Odegard|first18=N.|last19=Peiris|first19=H. V.|last20=Verde|first20=L.|last21=Weiland|first21=J. L.|s2cid=115601|display-authors=etal}} This paper warns that "the statistics of this internal linear combination map are complex and inappropriate for most CMB analyses."</ref><ref>
{{cite journal|last1=Tegmark|first1=M.|last2=de Oliveira-Costa|first2=A.|last3=Hamilton|first3=A.|year=2003|title=A high resolution foreground cleaned CMB map from WMAP|journal=[[Physical Review D]]|volume=68|pages=123523|doi=10.1103/PhysRevD.68.123523|arxiv=astro-ph/0302496|bibcode = 2003PhRvD..68l3523T|issue=12 |s2cid=17981329}} This paper states, "Not surprisingly, the two most contaminated multipoles are [the quadrupole and octupole], which most closely trace the galactic plane morphology."</ref> Later analyses have pointed out that these are the modes most susceptible to foreground contamination from synchrotron, dust, and bremsstrahlung emission, and from experimental uncertainty in the monopole and dipole.

A full [[Bayesian analysis]] of the WMAP power spectrum demonstrates that the quadrupole prediction of [[Lambda-CDM model|Lambda-CDM cosmology]] is consistent with the data at the 10% level and that the observed octupole is not remarkable.<ref>{{cite journal|last1=O'Dwyer|first1=I.|date=2004|title=Bayesian Power Spectrum Analysis of the First-Year Wilkinson Microwave Anisotropy Probe Data|journal=[[Astrophysical Journal Letters]]|volume=617|pages=L99–L102|doi=10.1086/427386|arxiv=astro-ph/0407027|bibcode=2004ApJ...617L..99O|issue=2|last2=Eriksen |first2=H. K. |last3=Wandelt|first3=B. D.|last4=Jewell|first4=J. B.|last5=Larson|first5=D. L.|last6=Górski|first6=K. M.|last7=Banday|first7=A. J.|last8=Levin|first8=S.|last9=Lilje|first9=P. B. |s2cid=118150531 }}</ref> Carefully accounting for the procedure used to remove the foregrounds from the full sky map further reduces the significance of the alignment by ~5%.<ref>{{cite journal|last1=Slosar|first1=A.|last2=Seljak|first2=U.|date=2004|title=Assessing the effects of foregrounds and sky removal in WMAP|journal=[[Physical Review D]]|volume=70|pages=083002|doi=10.1103/PhysRevD.70.083002|arxiv=astro-ph/0404567|bibcode = 2004PhRvD..70h3002S|issue=8|s2cid=119443655|url=https://cds.cern.ch/record/732816|type=Submitted manuscript}}</ref><ref>{{cite journal|last1=Bielewicz|first1=P.|year=2005|title=Multipole vector anomalies in the first-year WMAP data: a cut-sky analysis|journal=[[Astrophysical Journal]]|volume=635|pages=750–60|doi=10.1086/497263|arxiv=astro-ph/0507186|bibcode=2005ApJ...635..750B|issue=2|last2=Eriksen |first2=H. K. |last3=Banday|first3=A. J.|last4=Górski|first4=K. M.|last5=Lilje|first5=P. B. |s2cid=1103733}}</ref><ref>{{cite journal|last1=Copi |first1=C.J. |year=2006|title=On the large-angle anomalies of the microwave sky|journal=[[Monthly Notices of the Royal Astronomical Society]]|volume=367|issue=1 |pages=79–102|doi=10.1111/j.1365-2966.2005.09980.x|arxiv=astro-ph/0508047 |bibcode=2006MNRAS.367...79C|last2=Huterer |first2=Dragan |last3=Schwarz |first3=D. J. |last4=Starkman |first4=G. D. |doi-access=free |citeseerx=10.1.1.490.6391 |s2cid=6184966 }}</ref><ref>{{cite journal|last1=de Oliveira-Costa|first1=A.| last2=Tegmark| first2=M.|year=2006|title=CMB multipole measurements in the presence of foregrounds|journal=[[Physical Review D]]|volume=74| pages=023005|doi=10.1103/PhysRevD.74.023005|arxiv=astro-ph/0603369|bibcode = 2006PhRvD..74b3005D| issue=2|s2cid=5238226|url=https://cds.cern.ch/record/934594|type=Submitted manuscript}}</ref>
Recent observations with the [[Planck (spacecraft)|Planck telescope]], which is very much more sensitive than WMAP and has a larger angular resolution, record the same anomaly, and so instrumental error (but not foreground contamination) appears to be ruled out.<ref>{{cite web| url = https://www.newscientist.com/article/dn23301-planck-shows-almost-perfect-cosmos--plus-axis-of-evil.html| title = Planck shows almost perfect cosmos – plus axis of evil}}</ref> Coincidence is a possible explanation, chief scientist from [[WMAP]], [[Charles L. Bennett]] suggested coincidence and human psychology were involved, "I do think there is a bit of a psychological effect; people want to find unusual things."<ref>{{cite web| url = https://www.newscientist.com/article/dn18489-found-hawkings-initials-written-into-the-universe.html| title = Found: Hawking's initials written into the universe}}</ref>

Measurements of the density of quasars based on [[Wide-field Infrared Survey Explorer]] data finds a dipole significantly different from the one extracted from the CMB anisotropy.<ref name="SuburSarkar">{{cite journal |title=A Test of the Cosmological Principle with Quasars|journal=The Astrophysical Journal Letters|year=2021|doi=10.3847/2041-8213/abdd40|last1=Secrest|first1=Nathan J.|last2=Hausegger|first2=Sebastian von|last3=Rameez|first3=Mohamed|last4=Mohayaee|first4=Roya|last5=Sarkar|first5=Subir|last6=Colin|first6=Jacques|volume=908|issue=2|pages=L51|arxiv=2009.14826|bibcode=2021ApJ...908L..51S|s2cid=222066749 |doi-access=free }}</ref> This difference is conflict with the [[cosmological principle]].<ref>{{Cite journal |last1=Perivolaropoulos |first1=L. |last2=Skara |first2=F. |date=2022-12-01 |title=Challenges for ΛCDM: An update |url=https://www.sciencedirect.com/science/article/pii/S1387647322000185 |journal=New Astronomy Reviews |volume=95 |pages=101659 |doi=10.1016/j.newar.2022.101659 |issn=1387-6473|arxiv=2105.05208 |bibcode=2022NewAR..9501659P }}</ref>