Diffusion damping
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In modern
cosmological Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
theory, diffusion damping, also called photon diffusion damping, is a physical process which reduced density inequalities (
anisotropies Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
) in the early
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the universe. ...
, making the universe itself and the
cosmic microwave background radiation In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all space ...
(CMB) more uniform. Around 300,000 years after the Big Bang, during the epoch of '' recombination'', 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 In cosmology, baryon acoustic oscillations (BAO) are fluctuations in the density of the visible baryonic matter (normal matter) of the universe, caused by acoustic density waves in the primordial plasma of the early universe. In the same way ...
, the Doppler effect, and the effects of gravity on electromagnetic radiation, for the eventual formation of galaxies and
galaxy clusters A galaxy cluster, or a cluster of galaxies, is a structure that consists of anywhere from hundreds to thousands of galaxies that are bound together by gravity, with typical masses ranging from 1014 to 1015 solar masses. They are the second-l ...
, these being the dominant large scale structures which are observed in the universe. It is a damping ''by'' diffusion, not ''of'' diffusion.Hu, Sugiyama & Silk (1996-04-28), p. 2 The strength of diffusion damping is calculated by a mathematical expression for the ''
damping factor Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inc ...
'', which figures into the
Boltzmann equation The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
, an equation which describes the amplitude of perturbations in the CMB. The strength of the diffusion damping is chiefly governed by the distance photons travel before being scattered (diffusion length). The primary effects on the diffusion length are from 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. 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.


Introduction

Diffusion damping took place about 13.8 billion years ago, during the stage of the early universe called '' recombination'' or matter-radiation ''decoupling''. This period occurred about 320,000 years after the Big Bang. This is equivalent to a redshift of around ''z'' = 1090. Recombination was the stage during which simple atoms, e.g.
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
and
helium Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
, began to form in the cooling, but still very hot, soup of protons,
electrons The electron ( or ) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no ...
and photons that composed the universe. Prior to the recombination epoch, this ''soup'', a plasma, was largely
opaque Opacity or opaque may refer to: * Impediments to (especially, visible) light: ** Opacities, absorption coefficients ** Opacity (optics), property or degree of blocking the transmission of light * Metaphors derived from literal optics: ** In lingu ...
to the
electromagnetic radiation In physics, electromagnetic radiation (EMR) consists of waves of the electromagnetic (EM) field, which propagate through space and carry momentum and electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) li ...
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.Hu (1995-08-26), p. 6 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.Liddle & Lyth (2000-04-13), p. 63, 120 There was no significant diffusion damping for electrons, which could not diffuse nearly as far as photons could in similar circumstances. Thus all damping by electron diffusion is negligible when compared to photon diffusion damping.Padmanabhan (1993-06-25), p. 171–2 Acoustic perturbations of initial density fluctuations in the universe made some regions of space hotter and denser than others.Harrison (1970-05-15) These differences in
temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measurement, measured with a thermometer. Thermometers are calibrated in various Conversion of units of temperature, temp ...
and
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
are called ''
anisotropies Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physic ...
''. 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 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.Madsen (1996-05-15), p. 99–100Longair (2008-01-08), p. 355Jetzer & Pretzl (2002-07-31), p. 6Rich (2001-06-15), p. 256 Photon diffusion was first described in Joseph Silk's 1968 paper entitled "Cosmic Black-Body Radiation and Galaxy Formation",Silk (1968-02-01) which was published in ''
The Astrophysical Journal ''The Astrophysical Journal'', often abbreviated ''ApJ'' (pronounced "ap jay") in references and speech, is a peer-reviewed scientific journal of astrophysics and astronomy, established in 1895 by American astronomers George Ellery Hale and ...
''. As such, diffusion damping is sometimes also called Silk damping, though this term may apply only to one possible damping scenario.Partridge (1995-09-29), p. 302Bonometto, Gorini & Moschella (2001-12-15), p. 55 Silk damping was thus named after its discoverer.Madsen (1996-05-15), p. 99–101Hu (1994-06-28), p. 15


Magnitude

The magnitude of diffusion damping is calculated as a ''damping factor'' or ''suppression factor'', represented by the symbol \mathcal , which figures into the
Boltzmann equation The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
, an equation which describes the amplitude of perturbations in the CMB. 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. : \mathcal(\mathit) = \int_^ \dote^\; d\eta. Where: * \eta is the conformal time. * \dot is the "differential optical depth for Thomson scattering".
Thomson scattering Thomson scattering is the elastic scattering of electromagnetic radiation by a free charged particle, as described by classical electromagnetism. It is the low-energy limit of Compton scattering: the particle's kinetic energy and photon frequen ...
is the scattering of electromagnetic radiation (light) by charged particles such as electrons. * \mathit is the
wave number In the physical sciences, the wavenumber (also wave number or repetency) is the ''spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to temp ...
of the wave being suppressed.Longair (2008-01-08), p. 450 * (\dote^) is the visibility function. * (\eta) = /\lambda_\mathit The damping factor \mathcal , when factored into the
Boltzmann equation The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
for the cosmic microwave background radiation (CMB), reduces the amplitude of perturbations: : Theta_0+\Psi\eta_\ast) = hat_0+\Psi\eta_\ast)\mathcal(\mathit). Where:Jungman, Kamionkowski, Kosowsky & Spergel (1995-12-20), p. 2–4Hu (1995-08-26), p. 146 *\mathit_\ast is the conformal time at decoupling. * \Theta_0 is the "monopole erturbationof the photon distribution function" * \Psi is a "gravitational-potential erturbationin the Newtonian gauge". The Newtonian gauge is a quantity with importance in the
General Theory of Relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the differential geometry, geometric scientific theory, theory of gravitation published by Albert Einstein in 1915 and is the current descr ...
. * Theta_0+\Psi\eta) is the effective temperature. Mathematical calculations of the damping factor depend on \mathit_\mathit , or the ''effective diffusion scale'', which in turn depends on a crucial value, ''the diffusion length'', \lambda_\mathit .Hu, Sugiyama & Silk (1996-04-28), p. 5 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 In physics, mean free path is the average distance over which a moving particle (such as an atom, a molecule, or a photon) travels before substantially changing its direction or energy (or, in a specific context, other properties), typically as a ...
'', and is denoted by \lambda_\mathit. As the direction of these steps are randomly taken, \lambda_\mathit is approximately equal to \sqrt \lambda_\mathit , where \mathit is the number of steps the photon takes before the conformal time at decoupling (\mathit_\ast).Hu (1995-08-26), p. 12–13 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'', \mathit_\mathit , decreases as ionised
hydrogen Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic ...
and
helium Helium (from el, ἥλιος, helios, lit=sun) is a chemical element with the symbol He and atomic number 2. It is a colorless, odorless, tasteless, non-toxic, inert, monatomic gas and the first in the noble gas group in the periodic table. ...
bind with the free, charged electrons. As this occurs, the mean free path increases proportionally: \lambda_\mathit \varpropto ^ . That is, the mean free path of the photons is
inversely proportional In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constan ...
to the electron ionisation fraction and the baryon number density (\mathit_\mathit). 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. Small changes to these values before or during recombination can augment the damping effect considerably. 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 history of ionisation. The effect of diffusion damping is greatly augmented by the finite width of the
surface of last scattering In Big Bang cosmology the cosmic microwave background (CMB, CMBR) is electromagnetic radiation that is a remnant from an early stage of the universe, also known as "relic radiation". The CMB is faint cosmic background radiation filling all space ...
(SLS).(1995-08-26), p. 137 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.Durrer (2001-09-17), p. 5 It also means that during recombination, the diffusion length changed dramatically, as the ionisation fraction shifted.Hu (1995-08-26), pp. 156–7


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.Hu (1995-08-26), p. 136–8Hu & White (1997-04-20), p. 568–9 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 Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not a ...
, and diffuses with them. 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 corresponding increases in photon density, and the lower the photon density, the less diffusion there would be: the less diffusion, the less damping. Photon diffusion is not dependent on the causes of the initial fluctuations in the density of the universe.


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 Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving ...
.


The Silk scale and the Silk mass

Diffusion damping exponentially decreases anisotropies in the CMB on a scale (the Silk scale) much smaller than a degree, or smaller than approximately 3 megaparsecs.Bonometto, Gorini & Moschella (2001-12-15), p. 227–8 This angular scale corresponds to a
multipole moment A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly ...
\mathit \gtrsim 800.Papantonopoulos (2005-03-24), p. 63 The mass contained within the Silk scale is the ''silk mass''. Numerical evaluations of the Silk mass yield results on the order of 10^ solar masses at recombinationJedamzik, Katalinić & Olinto (1996-06-13), p. 1–2 and on the order of the mass of a present-day galaxy or
galaxy cluster A galaxy cluster, or a cluster of galaxies, is a structure that consists of anywhere from hundreds to thousands of galaxies that are bound together by gravity, with typical masses ranging from 1014 to 1015 solar masses. They are the second-lar ...
in the current era. : \mathit_\mathit \approx \frac. Scientists say diffusion damping affects ''small'' angles and corresponding anisotropies. Other effects operate on a scale called ''intermediate'' 10 \lesssim \mathit \lesssim 100 or ''large'' \mathit \lessapprox 10 . 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.Kaiser & Silk (1986-12-11), p. 533


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.Hu & Sugiyama (1994-07-28), p. 2Sunyaev & Zel'dovich (Sept. 1980), p. 1 Diffusion may also have had a significant effect on the evolution of primordial cosmic magnetic fields, 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 A magnetosonic wave, also called a magnetoacoustic wave, is a linear magnetohydrodynamic (MHD) wave that is driven by thermal pressure, magnetic pressure, and magnetic tension. There are two types of magnetosonic waves, the ''fast'' magnetosonic w ...
s (waves of ions travelling through a magnetised plasma). This process began before the era of
neutrino decoupling In Big Bang cosmology, neutrino decoupling was the epoch at which neutrinos ceased interacting with other types of matter, and thereby ceased influencing the dynamics of the universe at early times. Prior to decoupling, neutrinos were in thermal eq ...
and ended at the time of recombination.Brandenburg, Enqvist & Olesen (January 1997), p. 2


See also

*
Chronology of the universe The chronology of the universe describes the history and future of the universe according to Big Bang cosmology. Research published in 2015 estimates the earliest stages of the universe's existence as taking place 13.8 billion years ago, wit ...
* Joseph Silk *
Photon diffusion Photon diffusion is a situation where photons travel through a material without being absorbed, but rather undergoing repeated scattering events which change the direction of their path. The path of any given photon is then effectively a random wa ...
*
Timeline of cosmological theories This timeline of cosmological theories and discoveries is a chronological record of the development of humanity's understanding of the cosmos over the last two-plus millennia. Modern cosmological ideas follow the development of the scientific di ...


Notes


References


Bibliography

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External links


Diffusion damping explained in a "1997 Travelguide to CMB physics" by Wayne Hu
{{good article Physical cosmology Plasma physics