Energy in Radiation in the Early Universe

早期宇宙中的辐射能量

Electromagnetic radiation and the flux of neutrinos were the dominant form of energy in the early universe, becoming more dominant as one models earlier times in the big bang. It is reminiscent of the Biblical phrase "Let there be light". But of course this is not visible light but far beyond gamma rays since the average photon energy increases as one models earlier times and photon energies become high enough to accomplish pair production of all known particles. For example at a temperature of 10¹¹ K, modeled at a time of about 20 milliseconds into Weinberg's model of the big bang, the thermal energy kT = 8.6 MeV. At this temperature, the peak energy given by the Wien displacement law is 42.8 MeV. The radiation energy was sufficient not only for electron-positron pair production, but also to maintain essentially equal populations of protons and neutrons. So in that era, radiation was truly dominant.

在宇宙早期，电磁辐射和中微子通量是主要的能量形式，随着对大爆炸早期时间的建模，这种形式的能量变得更为主导。这让人联想到《圣经》中的短语“要有光”。但当然，这并不是可见光，而是远超伽马射线，因为随着对更早时期的时间建模，平均光子能量增加，光子能量变得足够高，能够实现所有已知粒子的对产生。例如，在温度为10¹¹ K时，对应于韦恩模型中大爆炸约20毫秒时，热能kT = 8.6 MeV。在这一温度下，根据维恩位移定律给出的峰值能量为42.8 MeV。此时，辐射能量不仅足以产生电子-正电子对，还能维持质子和中子的几乎相等数量。因此，在那个时代，辐射确实是主导的能量形式。

The radiation energy density in astronomy texts is usually put in the form

where a is called the radiation constant and has the value

For consistency with other particle quantities that are expressed in terms of the number of degrees of freedom, this is sometimes written in the form

where the photon "g-factor" is g_rad = 2 is the number of effective degrees of freedom associated with a photon: it's polarization can be parallel or anti-parallel to its motion. This notation facilitates its combination with neutrino energy density in describing the contribution of relativistic particles to the universe's energy density.

为与以自由度数目表达的其他粒子量一致，这种形式有时也写成：光子的‘g因子’为g_rad = 2，这是与光子相关的有效自由度数目：其极化可以与运动方向平行或反平行。这种记法便于将其与中微子能量密度结合，以描述相对论粒子对宇宙能量密度的贡献。

To assess the role of radiation in the expansion of the early universe, one must take a step beyond the simple Newtonian expansion model and include the radiation pressure. When the radiation pressure is included, the effective density represented by the radiation as a function of the scale factor R is

要评估辐射在早期宇宙膨胀中的作用，必须超越简单的牛顿膨胀模型，纳入辐射压。当纳入辐射压时，辐射所表示的有效密度作为尺度因子R的函数为

The dependence on the fourth power of R distinguishes the radiation energy density from the mass density, which depends upon the third power of R,

implying that the balance of mass and radiation energy density changes with the expansion of the universe, making a transition from a radiation dominated era to a mass dominated era.

对R的四次方的依赖性使辐射能量密度区别于质量密度，后者依赖于R的三次方，这意味着质量与辐射能量密度的平衡随宇宙的膨胀而变化，从而从辐射主导时期过渡到质量主导时期。

Matter and radiation in the early universe

早期宇宙中的物质和辐射

How old is the universe?

宇宙有多古老？

天文学文献中通常将辐射能量密度表示为一种形式，其中a被称为辐射常数，并具有值

Index

Reference
Carroll & Ostlie
Ch 29

索引参考Carroll & Ostlie第29章

HyperPhysics***** Astrophysics

HyperPhysics ***** 天体物理学

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Neutrinos in the Early Universe

早期宇宙中的中微子

Neutrinos joined Electromagnetic radiation as the dominant form of energy in the early universe, becoming more dominant as one models earlier times in the big bang. To assess the energy density in neutrinos, it would be helpful to place it in a form similar to the energy density in radiation,

but this requires a bit of analysis. For one thing, the neutrinos are fermions while the photons are bosons, so the neutrinos follow Fermi-Dirac statistics while the photons obey Bose-Einstein statistics. There are also three types of neutrinos, the electron, muon and tau neutrinos and with each is associated an antineutrino. One might expect two spin states, but one of the strange facts about neutrinos is their "chirality" - they are all "left-handed" and confined to one spin state.

Applying all the statistical factors gives a neutrino energy density expression

where g_ν = 6 for the 3 neutrinos plus their antiparticles, and the multiple 7/8 comes from the fact that they are fermions rather than bosons. Each of the 6 species has only one spin state.

应用所有统计因子后，得到一个中微子能量密度表达式，其中g ν = 6，对应3种中微子及其反粒子，而7/8的倍数则源于它们是费米子而非玻色子。每种6种粒子仅有一个自旋态。

Another difference for the neutrinos is the fact that their effective temperature at the present time is different from the 2.725K of the cosmic microwave background. Analysis of the neutrino temperature leads to the prediction that the neutrino temperature is

More detail

Relativistic Particle Density Ω_Rel

相对论粒子密度 Ω Rel

Neutrinos joined Electromagnetic radiation as the dominant form of energy in the early universe, becoming more dominant as one models earlier times in the big bang. In characterizing the effective density of the present universe, the mass of the neutrinos is generally neglected, and they are combined with photons as the effective density of the relativistic particles.

中微子与电磁辐射一起成为早期宇宙中占主导地位的能量形式，并且随着对大爆炸早期时期的建模，其主导地位进一步增强。在描述当前宇宙的有效密度时，中微子的质量通常被忽略，它们与光子一起作为相对论粒子的有效密度。

The combination of the radiation energy density

and the neutrino energy density

gives the total energy of the relativistic particles which can be written

where

We can now project a value for the total effective density of the relativistic particles as

and obtain a value for the density parameter Ω_rel for the relativistic particles by dividing by the critical density.

This shows that the amount of energy density in the relativistic particles is very small in the present era of the universe compared to the mass density = 0.27 obtained by WMAP: