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In [[Big Bang]] [[cosmology]], '''neutrino decoupling''' refers to the epoch at which neutrinos ceased interacting with [[baryonic matter]], and thereby ceased influencing the dynamics of the universe at early times.<ref>Longair (2006), p. 290</ref> Prior to decoupling, [[neutrino]]s were in [[thermal equilibrium]] with [[proton]]s, [[neutron]]s, and [[electron]]s, which was maintained through the [[weak interaction]]. Decoupling occurred approximately at the time when the rate of those weak interactions was slower than the rate of [[expansion of the universe]]. Alternatively, it was the time when the time scale for weak interactions became greater than the age of the [[universe]] at that time. Neutrino decoupling took place approximately one second after the Big Bang, when the [[temperature]] of the universe was approximately 10 [[1000000000 (number)|billion]] [[kelvin]]s, or 1 [[MeV]].<ref>Longair (2006), p. 291</ref> | |||
==Derivation of decoupling time== | |||
Neutrinos are kept from [[free streaming]] by their interactions with [[electron]]s and [[positron]]s, such as the reaction | |||
<math>e^- + e^+ \longleftrightarrow \nu_e + \bar{\nu}_e</math>. | |||
The approximate rate of these interactions is set by the [[number density]] of electrons and positrons, the averaged product of the [[Cross section (physics)|cross section]] for interaction and the [[velocity]] of the particles. The number density <math>n</math> of the [[Theory of relativity|relativistic]] electrons and positrons depends on the cube of the temperature <math>T</math>, so that <math>n \propto T^3</math>. The product of the cross section and velocity for weak interactions for temperatures (energies) below W/Z boson masses (~100 GeV) is given approximately by <math>\langle \sigma v \rangle \sim G_F^2 T^2</math>, where <math>G_F</math> is [[Fermi's constant]] (as is standard in [[particle physics]] calculations, factors of the [[speed of light]] <math>c</math> are set equal to 1). Putting it all together, the rate of weak interactions <math> \Gamma</math> is | |||
<math>\Gamma = n \langle \sigma v \rangle \sim G_F^2 T^5</math>. | |||
This can be compared to the expansion rate which is given by the [[Hubble parameter]] <math>H</math>, with | |||
<math>H = \sqrt{\frac{8\pi}{3}G \rho}</math>, | |||
where <math>G</math> is the [[gravitational constant]] and <math>\rho</math> is the [[energy density]] of the universe. At this point in cosmic history, the energy density is dominated by radiation, so that <math>\rho \propto T^4</math>. As the rate of weak interaction depends more strongly on temperature, it will fall more quickly as the universe cools. Thus when the two rates are approximately equal (dropping terms of order [[unity (mathematics)|unity]], including an effective [[degenerate energy level|degeneracy]] term which counts the number of states of particles which are interacting) gives the approximate temperature at which neutrinos decouple: | |||
<math> G_F^2 T^5 \sim \sqrt{G T^4}</math>. | |||
Solving for temperature gives | |||
<math>T \sim \left( \frac{\sqrt{G}}{G_F^2} \right) ^{1/3} \sim 1~\textrm{MeV}</math>.<ref name="Bernstein">Bernstein (1989), p. 27.</ref> | |||
While this is a very rough derivation, it illustrates the important physical phenomena which determined when neutrinos decoupled. | |||
==Observational evidence== | |||
While neutrino decoupling can not be observed directly, it is expected to have left behind a [[cosmic neutrino background]], analogous to the [[cosmic microwave background radiation]] of visible light which was emitted at a much later epoch. "The detection of the neutrino background is far beyond the capabilities of the present generation of neutrino detectors."<ref>Longair (2006), p. 302.</ref> There is data, however, which indirectly indicates the presence of a neutrino background. One piece of evidence is damping of the angular [[power spectrum]] of the CMB, which results from anisotropies in the neutrino background.<ref>Trotta (2005), p. 1.</ref> | |||
Another indirect measurement of neutrino decoupling is allowed by the role that neutrino decoupling plays in setting the ratio of [[neutron]]s to [[proton]]s. Before decoupling, the number of neutrons and protons are maintained in their equilibrium abundances by weak interactions, specifically [[beta decay]] and [[electron capture]] (or inverse beta decay) according to | |||
<math>n \leftrightarrow p + e^- + \bar{\nu}_e</math> | |||
and | |||
<math>p + e^- \leftrightarrow \nu_e + n</math>. | |||
Once the rate of weak interactions is slower than the characteristic rate of the expansion of the universe, this equilibrium cannot be maintained, and the abundance of neutrons to protons "freezes in," at a value | |||
<math>\left[\frac{n}{n + p}\right] = 0.21</math>.<ref>Longair (2006), p. 291–292.</ref> | |||
This value is simply found by evaluating the [[Boltzmann factor]] for neutrons and protons at decoupling time, according to | |||
<math>\frac{n_n(T)}{n_p(T)} = \exp\left(\frac{-\Delta m}{T}\right)</math>, | |||
where <math>\Delta m</math> is the mass difference between neutrons and protons and <math>T</math> is the temperature at decoupling.<ref name="Bernstein" /> This ratio is critical to the synthesis of [[atom]]s during [[Big Bang nucleosynthesis]], the process which formed the majority of [[helium]] atoms in the universe, as it "is the dominant factor in determining the amount of helium produced."<ref>Grupen (2005), p. 218.</ref> As helium atoms are stable, the neutrons are locked in, and [[beta decay]] of neutrons into protons, electrons, and neutrinos can no longer occur. Thus the abundance of neutrons remained essentially fixed to the present day. This abundance can be measured by [[astronomer]]s, and, as it was determined by the ratio of neutrons to protons at neutrino decoupling, the helium abundance indirectly measures the temperature at which neutrino decoupling took place, and is in agreement with the figure derived above.<ref>Longair (2006), p. 293.</ref> | |||
==See also== | |||
* [[Timeline of the Big Bang]] | |||
==References== | |||
{{reflist}} | |||
==Bibliography== | |||
* {{cite journal | author = Bernstein, J., Brown, L.S., and Feinberg, G. | title = Cosmological helium production simplified | journal = Reviews of Modern Physics | volume = 61 | issue = 1 | year = 1989 | pages = 25–39 | bibcode = 1989RvMP...61...25B | doi = 10.1103/RevModPhys.61.25}} | |||
* {{cite book | author = Grupen, C., Cowan, G., Eidelman, S., and Stroh, T. | title = Astroparticle Physics | publisher = Springer | year = 2005 | isbn = 978-3-540-25312-9 | url = http://books.google.com/?id=wlVGJ4X1BKkC}} | |||
* {{cite book | last = Longair | first = Malcolm | title = Galaxy Formation | publisher = Springer | location = Berlin | year = 2006 | isbn = 978-3-540-73477-2 | url = http://books.google.com/?id=blzugw-83nAC}} | |||
* {{cite journal | author = Trotta, R., Melchiorri, A. | title = Indication for Primordial Anisotropies in the Neutrino Background from the Wilkinson Microwave Anisotropy Probe and the Sloan Digital Sky Survey | journal = Physical Review Letters | volume = 95 | issue = 1 | year = 2005 | bibcode = 2005PhRvL..95a1305T | doi = 10.1103/PhysRevLett.95.011305 | pages = 011305 | pmid = 16090604|arxiv = astro-ph/0412066 }} | |||
==External links== | |||
* [http://setiathome.ssl.berkeley.edu/~aparsons/lecture_notes/Ay228_Cosmology.pdf Student notes from a UC Berkeley cosmology course] (see page 16) | |||
* [http://zebu.uoregon.edu/~rayfrey/321/lecture19.pdf Professor notes from a University of Oregon astrophysics course] | |||
[[Category:Physical cosmology]] | |||
Revision as of 16:05, 25 January 2014
In Big Bang cosmology, neutrino decoupling refers to the epoch at which neutrinos ceased interacting with baryonic matter, and thereby ceased influencing the dynamics of the universe at early times.[1] Prior to decoupling, neutrinos were in thermal equilibrium with protons, neutrons, and electrons, which was maintained through the weak interaction. Decoupling occurred approximately at the time when the rate of those weak interactions was slower than the rate of expansion of the universe. Alternatively, it was the time when the time scale for weak interactions became greater than the age of the universe at that time. Neutrino decoupling took place approximately one second after the Big Bang, when the temperature of the universe was approximately 10 billion kelvins, or 1 MeV.[2]
Derivation of decoupling time
Neutrinos are kept from free streaming by their interactions with electrons and positrons, such as the reaction
.
The approximate rate of these interactions is set by the number density of electrons and positrons, the averaged product of the cross section for interaction and the velocity of the particles. The number density of the relativistic electrons and positrons depends on the cube of the temperature , so that . The product of the cross section and velocity for weak interactions for temperatures (energies) below W/Z boson masses (~100 GeV) is given approximately by , where is Fermi's constant (as is standard in particle physics calculations, factors of the speed of light are set equal to 1). Putting it all together, the rate of weak interactions is
.
This can be compared to the expansion rate which is given by the Hubble parameter , with
,
where is the gravitational constant and is the energy density of the universe. At this point in cosmic history, the energy density is dominated by radiation, so that . As the rate of weak interaction depends more strongly on temperature, it will fall more quickly as the universe cools. Thus when the two rates are approximately equal (dropping terms of order unity, including an effective degeneracy term which counts the number of states of particles which are interacting) gives the approximate temperature at which neutrinos decouple:
.
Solving for temperature gives
.[3]
While this is a very rough derivation, it illustrates the important physical phenomena which determined when neutrinos decoupled.
Observational evidence
While neutrino decoupling can not be observed directly, it is expected to have left behind a cosmic neutrino background, analogous to the cosmic microwave background radiation of visible light which was emitted at a much later epoch. "The detection of the neutrino background is far beyond the capabilities of the present generation of neutrino detectors."[4] There is data, however, which indirectly indicates the presence of a neutrino background. One piece of evidence is damping of the angular power spectrum of the CMB, which results from anisotropies in the neutrino background.[5]
Another indirect measurement of neutrino decoupling is allowed by the role that neutrino decoupling plays in setting the ratio of neutrons to protons. Before decoupling, the number of neutrons and protons are maintained in their equilibrium abundances by weak interactions, specifically beta decay and electron capture (or inverse beta decay) according to
and
.
Once the rate of weak interactions is slower than the characteristic rate of the expansion of the universe, this equilibrium cannot be maintained, and the abundance of neutrons to protons "freezes in," at a value
.[6]
![{\displaystyle \left[{\frac {n}{n+p}}\right]=0.21}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfa7cd7cbfe5ce23a758958134198e6300713986)
This value is simply found by evaluating the Boltzmann factor for neutrons and protons at decoupling time, according to
,
where is the mass difference between neutrons and protons and is the temperature at decoupling.[3] This ratio is critical to the synthesis of atoms during Big Bang nucleosynthesis, the process which formed the majority of helium atoms in the universe, as it "is the dominant factor in determining the amount of helium produced."[7] As helium atoms are stable, the neutrons are locked in, and beta decay of neutrons into protons, electrons, and neutrinos can no longer occur. Thus the abundance of neutrons remained essentially fixed to the present day. This abundance can be measured by astronomers, and, as it was determined by the ratio of neutrons to protons at neutrino decoupling, the helium abundance indirectly measures the temperature at which neutrino decoupling took place, and is in agreement with the figure derived above.[8]
See also
References
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Bibliography
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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