Main Page: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
mNo edit summary
No edit summary
 
(245 intermediate revisions by more than 100 users not shown)
Line 1: Line 1:
{{cosmology}}
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.
[[File:WMAP TT power spectrum.png|thumb|right|375px|The power spectrum of the cosmic microwave background radiation temperature anisotropy in terms of the angular scale (or [[multipole moment]]). Diffusion damping can be easily seen in the suppression of power peaks when ''l''&nbsp;≳&nbsp;1000.<ref name=bonometto227228 />]]


In modern [[cosmology|cosmological]] theory, '''diffusion damping''', also called '''photon diffusion damping''',<!--'''collisionless damping''',CAN'T FIND A SOURCE TO SUPPORT THIS--> is a physical process which reduced density inequalities ([[anisotropies]]) in the early [[universe]], making the universe itself and the [[cosmic microwave background radiation]] (CMB) more uniform. Around 300,000 years after the [[Big Bang]], during the epoch of ''[[recombination (cosmology)|recombination]]'', [[Photon diffusion|diffusing photons]] travelled from hot regions of space to cold ones, equalising the temperatures of these regions. This effect is responsible, along with [[baryon acoustic oscillations]], the [[Doppler effect]], and [[gravitational redshift|the effects of gravity on electromagnetic radiation]], for the eventual formation of [[galaxies]] and [[galaxy clusters]], these being the dominant large scale structures which are observed in the universe. It is a [[damping]] ''by'' diffusion, not ''of'' diffusion.<ref name=hu_physics2>Hu, Sugiyama & Silk (1996-04-28), p. 2</ref>
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
<!--ANOTHER (newer) STORY? Photon diffusion is one of two effects which damp small scale ''temperature'' anisotropies. The other may be termed [[statistical damping]]; it occurs when the width of the fluctuation is less than the width of the [[last scattering surface]].<ref name=longair450 />
* Only registered users will be able to execute this rendering mode.
-->
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


The strength of diffusion damping is calculated by a mathematical expression for the ''damping factor'', which figures into the [[Boltzmann equation]], an equation which describes the amplitude of perturbations in the CMB.<ref name=jungman24 /> The strength of the diffusion damping is chiefly governed by the distance photons travel before being scattered (diffusion length). What affect the diffusion length are primarily the properties of the plasma in question: different sorts of plasma may experience different sorts of diffusion damping. The evolution of a plasma may also affect the damping process.<ref name=hu1213 /> The scale on which diffusion damping works is called the '''Silk scale''' and its value corresponds to the size of galaxies of the present day. The mass contained within the Silk scale is called the '''Silk mass''' and it corresponds to the mass of the galaxies.<ref name=madsen99101 />
Registered users will be able to choose between the following three rendering modes:  


==Introduction==
'''MathML'''
Diffusion damping took place about 13.7 billion years ago, during the stage of the early universe called ''[[recombination (cosmology)|recombination]]'' or  matter-radiation ''decoupling''. This period occurred between 240,000 and 310,000 years after the [[Big Bang]].<ref name=ryden>Ryden (2003), p. 158</ref><ref name=hu6>Hu (1995-08-26), p. 6</ref><ref group="nb">This is equivalent to a [[redshift]] of ''z''&nbsp;=&nbsp;1000.</ref> Recombination was the stage during which simple [[atoms]], e.g. [[hydrogen]] and [[helium]], began to form in the cooling, but still very hot, soup of [[proton]]s, [[electrons]] and [[photons]] that composed the universe. Prior to the recombination epoch, this ''soup'', a [[plasma (physics)|plasma]], was largely [[Opacity (optics)|opaque]] to the [[electromagnetic radiation]] of photons. This meant that the permanently excited photons were scattered by the protons and electrons too often to travel very far in straight lines.<ref name="hu6" /> During the recombination epoch, the universe cooled rapidly as free electrons were captured by atomic nuclei; atoms formed from their constituent parts and the universe became transparent: the amount of photon scattering decreased dramatically. Scattering less, photons could diffuse (travel) much greater distances.<ref name=hu_physics2 /><ref name=liddle63120>Liddle & Lyth (2000-04-13), p. 63, 120</ref> There is no significant diffusion damping for electrons, which cannot diffuse nearly as far as photons can in similar circumstances. Thus all damping by electron diffusion is negligible when compared to photon diffusion damping.<ref name=padmanabhan171172>Padmanabhan (1993-06-25), p. 171&ndash;2</ref>
:<math forcemathmode="mathml">E=mc^2</math>


Acoustic perturbations of initial density fluctuations in the universe made some regions of space hotter and denser than others.<ref name=harrison1>Harrison (1970-05-15)</ref> These differences in [[temperature]] and [[density]] are called ''[[anisotropies]]''. Photons diffused from the hot, overdense regions of plasma to the cold, underdense ones: they dragged along the protons and electrons: the photons pushed electrons along, and these, in turn, pulled on protons by the [[Coulomb force]]. This caused the temperatures and densities of the hot and cold regions to be averaged and the universe became less [[anisotropies|anisotropic]] (characteristically various)  and more ''isotropic'' (characteristically uniform). This reduction in anisotropy is the ''damping'' of diffusion damping. Diffusion damping thus damps temperature and density anisotropies in the early universe. With baryonic matter (protons and electrons) escaping the dense areas along with the photons; the temperature and density inequalities were ''[[adiabatically]]'' damped. That is to say the ratios of photons to baryons remained constant during the damping process.<ref name=hu1213 /><ref name=madsen99101>Madsen (1996-05-15), p. 99&ndash;100</ref><ref name=longair355>Longair (2008-01-08), p. 355</ref><ref name=jetzer6>Jetzer & Pretzl (2002-07-31), p. 6</ref><ref name=rich256>Rich (2001-06-15), p. 256</ref>
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


Photon diffusion was first described in [[Joseph Silk]]'s 1968 paper entitled "Cosmic Black-Body Radiation and Galaxy Formation",<ref name=silk>Silk (1968-02-01)</ref> which was published in ''[[Astrophysical Journal|The Astrophysical Journal]]''. As such, diffusion damping is sometimes also called '''Silk damping''',<ref name=bonometto227228 /> though this term may apply only to one possible damping scenario.<ref name=padmanabhan171172 /><ref name=partridge302>Partridge (1995-09-29), p. 302</ref><ref name=bonometto55>Bonometto, Gorini & Moschella (2001-12-15), p. 55</ref> Silk damping was thus named after its discoverer.<ref name=madsen99101>Madsen (1996-05-15), p. 99&ndash;101</ref><ref name="bonometto55"/><ref name=hu_nature15>Hu (1994-06-28), p. 15</ref>
'''source'''
<!--Silk damping vs. diffusion damping: need in-depth section-->
:<math forcemathmode="source">E=mc^2</math> -->


==Magnitude==
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].
The magnitude of diffusion damping is calculated as a ''damping factor'' or ''suppression factor'', represented by the symbol <math> \mathcal{D} </math>, which figures into the [[Boltzmann equation]], an equation which describes the amplitude of perturbations in the CMB.<ref name=jungman24 /> The strength of the diffusion damping is chiefly governed by the distance photons travel before being scattered (diffusion length). What affect the diffusion length are primarily the properties of the plasma in question: different sorts of plasma may experience different sorts of diffusion damping. The evolution of a plasma may also affect the damping process.<ref name=hu1213 />


:<math> \mathcal{D}(\mathit{k}) = \int_{0}^{\eta_0} \dot{\tau}e^{-[\mathit{k}/{\mathit{k}_\mathit{D}(\eta)}]^2}\; d\eta. </math><ref name=jungman24 />
==Demos==


Where:
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
*<math> \eta </math> is the [[conformal time]].
*<math> \dot{\tau} </math> is the "differential optical depth for Thomson scattering". [[Thomson scattering]] is the scattering of electromagnetic radiation (light) by charged particles such as electrons.<ref name=jungman24 />
*<math> \mathit{k} </math> is the [[wave number]] of the wave being suppressed.<ref name=longair450>Longair (2008-01-08), p. 450</ref>
*<math> (\dot{\tau}e^{-[\mathit{k}/{\mathit{k}_\mathit{D}(\eta)}]^2})</math> is the [[visibility function]].<ref name=jungman24 />
*<math> {\mathit{k}_\mathit{D}}(\eta) = {2\pi}/\lambda_\mathit{D}</math>


The damping factor <math> \mathcal{D} </math>, when factored into the [[Boltzmann equation]] for the cosmic microwave background radiation (CMB), reduces the amplitude of perturbations:


:<math> [\Theta_0+\Psi](\eta_\ast) = [\hat{\Theta}_0+\Psi](\eta_\ast)\mathcal{D}(\mathit{k}). </math><ref name=jungman24 />
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


Where:<ref name=jungman24>Jungman, Kamionkowski, Kosowsky & Spergel (1995-12-20), p. 2&ndash;4</ref><ref name=hu146>Hu (1995-08-26), p. 146</ref>
==Test pages ==
*<math>\mathit{\eta}_\ast</math> is the conformal time at decoupling.
*<math> \Theta_0 </math> is the "monopole [perturbation] of the photon distribution function"<ref name=jungman24 />
*<math> \Psi </math> is a "gravitational-potential [perturbation] in the Newtonian gauge". The [[Newtonian gauge]] is a quantity with importance in the [[General Theory of Relativity]].<ref name=jungman24 />
*<math> [\Theta_0+\Psi](\eta) </math> is the effective temperature.
<!--Another (more complicated) definition of lambda_D:
<math>
1/6 \int \frac{\mathit{R}^2+\frac{4(1+\mathit{R})}{5}}{\dot{\tau}{(1+\mathit{R})}^2} d\eta
</math> DEFINE R<ref name=hu_nature15 /> -->


[[File:Walk3d 0.png|thumb|320px|Three random walks in three dimensions. In diffusion damping, photons from hot regions diffuse to cold regions by random walk, so after <math>\mathit{N}</math> steps, the photons have travelled a distance <math>\lambda_D=\sqrt{\mathit{N}}\lambda_C</math>.]]
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


Mathematical calculations of the damping factor depend on <math> \mathit{k}_\mathit{D} </math>, or the ''effective diffusion scale'', which in turn depends on a crucial value, ''the diffusion length'', <math> \lambda_\mathit{D} </math>.<ref name=hu_physics_5>Hu, Sugiyama & Silk (1996-04-28), p. 5</ref> The diffusion length relates how far photons travel during diffusion, and comprises a finite number of short steps in random directions. The average of these steps is the ''Compton [[mean free path]]'', and is denoted by <math>\lambda_\mathit{C}</math>. As the direction of these steps are randomly taken, <math> \lambda_\mathit{D} </math> is approximately equal to <math> \sqrt{\mathit{N}} \lambda_\mathit{C} </math>, where <math>\mathit{N}</math> is the number of steps the photon takes before the [[conformal time]] at decoupling (<math>\mathit{\eta}_\ast</math>).<ref name=hu1213>Hu (1995-08-26), p. 12&ndash;13</ref>
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
The diffusion length increases at recombination because the mean free path does, with less photon scattering occurring; this increases the amount of diffusion and damping. The mean free path increases because the ''electron ionisation fraction'', <math> \mathit{x}_\mathit{e} </math>, decreases as ionised [[hydrogen]] and [[helium]] bind with the free, charged electrons. As this occurs, the mean free path increases proportionally: <math> \lambda_\mathit{C} \varpropto {(\mathit{x}_\mathit{e} \mathit{n}_\mathit{b})}^{-1} </math>. That is, the mean free path of the photons is [[proportionality|inversely proportional]] to the electron ionisation fraction and the baryon number density (<math>\mathit{n}_\mathit{b}</math>). That means that the more baryons there were, and the more they were ionised, the shorter the average photon could travel before encountering one and being scattered.<ref name=hu1213 /> Small changes to these values before or during recombination can augment the damping effect considerably.<ref name=hu1213 /> This dependence on the baryon density by photon diffusion allows scientists to use analysis of the latter to investigate the former, in addition to the <!--enigmatic?--> history of ionisation.<ref name=hu_physics_5 />
==Bug reporting==
 
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
The effect of diffusion damping is greatly augmented by the finite width of the [[surface of last scattering]] (SLS).<!--need great graph in Hu's thesis--><ref name=hu137>(1995-08-26), p. 137</ref> The finite width of the SLS means the CMB photons we see were not all emitted at the same time, and the fluctuations we see are not all in phase.<ref name=durrer5>Durrer (2001-09-17), p. 5</ref> It also means that during recombination, the diffusion length changed dramatically, as the ionisation fraction shifted.<ref name=hu156>Hu (1995-08-26), pp. 156–7</ref>
 
=== Model dependence ===
In general, diffusion damping produces its effects independent of the cosmological model being studied, thereby masking the effects of other, model-''dependent'' phenomena. This means that without an accurate model of diffusion damping, scientists cannot judge the relative merits of cosmological models, whose theoretical predictions cannot be compared with observational data, this data being obscured by damping effects. For example, the peaks in the power spectrum due to acoustic oscillations are decreased in amplitude by diffusion damping. This deamplification of the power spectrum hides features of the curve, features that would otherwise be more visible.<ref name=hu136138>Hu (1995-08-26), p. 136&ndash;8</ref><ref name=hu_tail568>Hu & White (1997-04-20), p. 568&ndash;9</ref>
 
Though general diffusion damping can damp perturbations in collisionless dark matter simply due to photon dispersion, the term ''Silk damping'' applies only to damping of adiabatic models of baryonic matter, which is coupled to the diffusing photons, not [[dark matter]],<ref name=padmanabhan171172 /> and diffuses with them.<ref name=partridge302 /><ref name=bonometto55 /> Silk damping is not as significant in models of cosmological development which posit early isocurvature fluctuations (i.e. fluctuations which do not require a constant ratio of baryons and photons). In this case, increases in baryon density do not require a corresponding increases in photon density, and the lower the photon density, the less diffusion there would be: the less diffusion, the less damping.<ref name=rich256 /> Photon diffusion is not dependent on the causes of the initial fluctuations in the density of the universe.<ref name=hu_physics_5 />
 
== Effects ==
=== Speed ===
Damping occurs at two different scales, with the process working more quickly over short ranges than over longer distances. Here, a short length is one that is lower than the mean free path of the photons. A long distance is one that is greater than the mean free path, if still less than the diffusion length. On the smaller scale, perturbations are damped almost instantaneously. On the larger scale, anisotropies are decreased more slowly, with significant degradation happening within one unit of [[Hubble time]].<ref name=padmanabhan171172 />
 
=== The Silk scale and the Silk mass ===
Diffusion damping exponentially decreases anisotropies in the CMB on a scale (the '''Silk scale''')<ref name=madsen99101 /> much smaller than a [[degree (angle)|degree]], or smaller than approximately 3 [[parsec|megaparsecs]].<ref name=bonometto227228>Bonometto, Gorini & Moschella (2001-12-15), p. 227&ndash;8</ref> This angular scale corresponds to a [[multipole moment]] <math>\mathit{l} \gtrsim 800</math>.<ref name=jetzer6 /><ref name=papantonopoulos63>Papantonopoulos (2005-03-24), p. 63</ref> The mass contained within the Silk scale is the ''silk mass''.  Numerical evaluations of the Silk mass yield results on the order of <math>10^{13} </math> solar masses at recombination<ref name=jedamzik12>Jedamzik, Katalinić & Olinto (1996-06-13), p. 1&ndash;2</ref>  and on the order of the mass of a present-day [[galaxy]] or [[galaxy cluster]] in the current era.<ref name=madsen99101 /><ref name=padmanabhan171172 />
 
:<math> \mathit{M}_\mathit{s} \approx \frac{\mathit{m}_\mathit{p} {\mathit{t}_\mathit{rec}}^{3/2}}{\sqrt{\mathit{n}_\mathit{rec} \sigma^3}}.</math>
<!--for alternate calculation, see Longair (2008-01-08), p. 356-->
 
As diffusion damping works at this level, scientists say it affects ''small'' angles and corresponding anisotropies. This is to be contrasted with other effects which operate on a scale called ''intermediate'' <math>10 \lesssim \mathit{l} \lesssim 100</math> or ''large'' <math>\mathit{l} \lessapprox 10 </math>. Searches for anisotropies on a small scale are not as difficult as those on larger scales, partly because they may employ ground-based telescopes and their results can be more easily predicted by current theoretical models.<ref name=kaiser533>Kaiser & Silk (1986-12-11), p. 533</ref>
 
=== Galaxy formation ===
Scientists study photon diffusion damping (and CMB anisotropies in general) because of the insight the subject provides into the question, "How did the universe come to be?". Specifically, primordial anisotropies in the temperature and density of the universe are supposed to be the causes of later large-scale structure formation. Thus it was the amplification of small perturbations in the pre-recombination universe that grew into the galaxies and galaxy clusters of the present era. Diffusion damping made the universe isotropic within distances on the order of the Silk Scale. That this scale corresponds to the size of observed galaxies (when the passage of time is taken into account) implies that diffusion damping is responsible for limiting the size of these galaxies. The theory is that clumps of matter in the early universe became the galaxies that we see today, and the size of these galaxies is related to the temperature and density of the clumps.<ref name=hu_analytic2>Hu & Sugiyama (1994-07-28), p. 2</ref><ref name=sunyaev1>Sunyaev & Zel'dovich (Sept. 1980), p. 1</ref>
 
Diffusion may also have had a significant effect on the evolution of primordial [[cosmic magnetic field]]s, fields which may have been amplified over time to become galactic magnetic fields. However, these cosmic magnetic fields may have been damped by radiative diffusion: just as acoustic oscillations in the plasma were damped by the diffusion of photons, so were [[magnetosonic wave]]s (waves of ions travelling through a magnetised plasma). This process began before the era of [[neutrino decoupling]] and ended at the time of recombination.<ref name=jedamzik12 /><ref name=brandenburg2>Brandenburg, Enqvist & Olesen (January 1997), p. 2</ref>
 
== See also ==
*[[Timeline of the Big Bang]]
*[[Timeline of cosmology]]
*[[Joseph Silk]]
*[[Photon diffusion]]
 
== Notes ==
{{Reflist|group=nb|2}}
 
== References ==
{{reflist|2}}
 
== Bibliography ==
*{{Cite journal
| doi = 10.1016/S0370-2693(96)01566-3
| volume = 392
| issue = 3-4
| pages = 395–402
| last = Brandenburg
| first = Axel
| coauthors = Kari Enqvist, Poul Olesen
| title = The effect of Silk damping on primordial magnetic fields
| journal = Physics Letters B
| accessdate = 2008-08-16
| date = 1997-01
| url = http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVN-3SPCNP6-7G&_user=10&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_version=1&_urlVersion=0&_userid=10&md5=662906675e5ddd5290cd0131289fc062
|arxiv = hep-ph/9608422 |bibcode = 1997PhLB..392..395B }}
* {{Cite book
| edition = 1st
| publisher = Taylor & Francis
| isbn = 0-7503-0810-9
| pages = 416
| last = Bonometto
| first = S.
| coauthors = V. Gorini, U. Moschella
| title = Modern Cosmology
| date = 2001-12-15
}}
*{{Cite arxiv
| last = Durrer
| first = Ruth
| title = Physics of Cosmic Microwave Background anisotropies and primordial fluctuations
| date = 2001-09-17
| eprint = astro-ph/0109274
| class = astro-ph
}}
*{{Cite journal
| doi = 10.1103/PhysRevD.1.2726
| volume = 1
| issue = 10
| pages = 2726
| last = Harrison
| first = E. R.
| title = Fluctuations at the Threshold of Classical Cosmology
| journal = Physical Review D
| accessdate = 2009-03-14
| date = 1970-05-15
| url = http://link.aps.org/abstract/PRD/v1/p2726
|bibcode = 1970PhRvD...1.2726H }}
*{{Cite arxiv
| last = Hu
| first = Wayne
| title = The Nature versus Nurture of Anisotropies
| date = 1994-06-28
| eprint = astro-ph/9406071
| class = astro-ph
}}
*{{Cite arxiv
| last = Hu
| first = Wayne
| title = Wandering in the Background: A CMB Explorer
| date = 1995-08-26
| eprint = astro-ph/9508126
| class = astro-ph
}}<!-- GREAT PICTURES: MUST GET THEM -->
*{{Cite arxiv
| last = Hu
| first = Wayne
| coauthors = Naoshi Sugiyama
| title = Anisotropies in the Cosmic Microwave Background: An Analytic Approach
| date = 1994-07-28
| eprint = astro-ph/9407093
| class = astro-ph
}}<!-- a good photo-->
*{{Cite arxiv
| last = Hu
| first = Wayne
| coauthors = Naoshi Sugiyama, Joseph Silk
| title = The Physics of Microwave Background Anisotropies
| date = 1996-04-28
| eprint = astro-ph/9604166
| class = astro-ph
}}<!-- Get the picture with the dotted and solid diagonal lines "damping length"?: it's the increase in damping during recombination -->
*{{Cite journal
| volume = 479
| issue = 2
| pages = 568–579
| last = Hu
| first = Wayne
| coauthors = Martin White
| title = The Damping Tail of Cosmic Microwave Background Anisotropies
| journal = The Astrophysical Journal
| date = 1997-04-20
| doi=10.1086/303928
| bibcode=1997ApJ...479..568H
|arxiv = astro-ph/9609079 }}
*{{Cite arxiv
| last = Jedamzik
| first = K.
| coauthors = V. Katalinić, A. Olinto
| title = Damping of Cosmic Magnetic Fields
| date = 1996-06-13
| eprint = astro-ph/9606080
| class = astro-ph
}}
*{{Cite book
| publisher = Springer
| isbn = 1-4020-0666-7
| pages = 328
| last = Jetzer
| first = Ph.
| coauthors = K. Pretzl
| others = Rudolf von Steiger (ed.)
| title = Matter in the Universe
| series = Space Sciences Series of ISSI
| date = 2002-07-31
}}
*{{Cite arxiv
| last = Jungman
| first = Gerard
| coauthors = Marc Kamionkowski, Arthur Kosowsky, David N Spergel
| title = Cosmological-Parameter Determination with Microwave Background Maps
| date = 1995-12-20
| eprint  = astro-ph/9512139
| class = astro-ph
}}
*{{Cite journal
| doi = 10.1038/324529a0
| volume = 324
| issue = 6097
| pages = 529–537
| last = Kaiser
| first = Nick
| coauthors = Joseph Silk
| title = Cosmic microwave background anisotropy
| journal = Nature
| date = 1986-12-11
|bibcode = 1986Natur.324..529K }}
*{{Cite book
| publisher = Cambridge University Press
| isbn = 0-521-57598-2
| pages = 400
| last = Liddle
| first = Andrew R.
| coauthors = David Hilary Lyth
| title = Cosmological Inflation and Large-Scale Structure
| date = 2000-04-13
}}
*{{Cite book
| edition = 2nd
| publisher = Springer
| isbn = 3-540-73477-5
| pages = 738
| last = Longair
| first = Malcolm S.
| title = Galaxy Formation
| date = 2008-01-08
}}
*{{Cite book
| edition = 1
| publisher = Chapman & Hall/CRC
| isbn = 0-412-62300-5
| pages = 144
| last = Madsen
| first = Mark S.
| title = Dynamic Cosmos
| date = 1996-05-15
}}
*{{Cite book
| publisher = Cambridge University Press
| isbn = 0-521-35254-1
| pages = 393
| last = Partridge
| first = R. B.
| title = 3K: The Cosmic Microwave Background Radiation
| date = 1995-09-29
}}
*{{Cite book
| publisher = Cambridge University Press
| isbn = 0-521-42486-0
| pages = 499
| last = Padmanabhan
| first = T.
| title = Structure Formation in the Universe
| date = 1993-06-25
}}
*{{Cite book
| edition = 1
| publisher = Springer
| isbn = 3-540-41350-2
| pages = 302
| last = Rich
| first = James
| title = Fundamentals of Cosmology
| date = 2001-06-15
}}
*{{Cite book
| publisher = Addison Wesley
| isbn = 0-8053-8912-1
| pages = 300
| last = Ryden
| first = Barbara
| title = Introduction to Cosmology
| date = 2002-11-12
}}
*{{Cite journal
| volume = 151
| pages = 459
| last = Silk
| first = Joseph
| title = Cosmic Black-Body Radiation and Galaxy Formation
| journal = Astrophysical Journal
| date = 1968-02-01
| bibcode = 1968ApJ...151..459S
| doi = 10.1086/149449
}}<!--paper which originally introduced photon diffusion-->
*{{Cite book
| edition = 1
| publisher = Springer
| isbn = 3-540-22712-1
| pages = 300
| last = Papantonopoulos
| first = E.
| title = The Physics of the Early Universe
| date = 2005-03-24
}}
*{{Cite journal
| volume = 18
| issue = 1
| pages = 537–560
| last = Sunyaev
| first = R. A.
| coauthors = Y. B. Zel'dovich
| title = Microwave background radiation as a probe of the contemporary structure and history of the universe
| journal = Annual review of astronomy and astrophysics
| date = Sept 1980
| doi = 10.1146/annurev.aa.18.090180.002541
| bibcode=1980ARA&A..18..537S
}}
 
== External links ==
*[http://background.uchicago.edu/~whu/physics/damping.html  Diffusion damping explained in a "1997 Travelguide to CMB physics" by Wayne Hu]
 
{{CMB_experiments}}
{{good article}}
 
[[Category:Physical cosmology]]
[[Category:Plasma physics]]
 
[[de:Silk-Dämpfung]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

Retrieved from "https://en.formulasearchengine.com/index.php?title=Main_Page&oldid=329754"