Big Bang Energy and Time Example

In Weinberg's First Three Minutes, the big bang expansion is modeled in segments which are characterized by time, temperature and characteristic particle energy. The first segment in which the process is so characterized, the temperature is 10¹¹ Kelvin. Given that temperature, how do you calculate the associated time and particle energy?

在Weinberg的《第一分钟》中，大爆炸的膨胀被分为若干段，这些段由时间、温度和特征粒子能量所表征。在第一个被这样表征的阶段中，温度为10¹¹开尔文。已知温度，如何计算相应的时间和粒子能量？

The mean energy associated with a particle in thermal equilibrium at temperature T is kT, where k is the Boltzmann constant. So the characteristic particle energy at this temperature is

在温度为T的热平衡状态下，与一个粒子相关的平均能量是kT，其中k是玻尔兹曼常数。因此，该温度下的特征粒子能量是

One important piece of information that we have is the energy density in the 3K background radiation, which is about 0.25 MeV/m³ at the present time. From the Stefan-Boltzmann law, we know that the energy density is proportional to T⁴, so we could scale the energy up to the temperature 10¹¹ K. But this includes only the energy of the photons, and there are neutrinos, electrons and positrons also in thermal equilibrium at that time. The thermal energy at 8.6 MeV is so high compared to the pair-production threshold of about 1 MeV for the production of electron-positron pairs that it must be assumed that they are being continuously created. The collection of electrons and positrons is "radiation like", being in thermal equilibrium with the photons and neutrinos.

我们已知的重要信息之一是3K背景辐射中的能量密度，目前约为0.25 MeV/m³。根据斯特凡-玻尔兹曼定律，我们知道能量密度与温度的四次方成正比，因此我们可以将能量尺度扩展到10¹¹ K的温度。但这一结果仅包括光子的能量，当时还有中微子、电子和正电子也处于热平衡状态。在8.6 MeV的热能水平下，远高于电子-正电子对产生阈值约1 MeV，因此必须假设它们持续被产生。电子和正电子的集合具有“辐射-like”的特性，与光子和中微子处于热平衡状态。

The known energy density of the photon background radiation can be scaled up to the target temperature and adjusted to include the neutrinos, electrons and positrons if we know the relative populations of each in the expanding big bang. At these high temperatures, those populations can be treated statistically, and the effective number of species of the particles established. Those populations relative to the photons are:

光子背景辐射的已知能量密度可以被扩展到目标温度，并在已知每种粒子在宇宙大爆炸扩展过程中的相对数量时，调整以包括中微子、电子和正电子。在这些高温下，这些粒子的分布可以进行统计处理，从而确定粒子种类的有效数量。这些粒子相对于光子的相对分布为：

在温度为10¹¹ K的膨胀宇宙中，可以计算出能量密度：

将此代入膨胀时间的 Friedman 方程中得到

The next step in Weinberg's scenario has time 0.11 sec and temperature 3 x 10¹⁰K. If you just plug into the above relationship for that temperature, you get expansion time 0.24 seconds. I'm not clear on how you get 0.11 seconds.

温伯格情景的下一步中，时间是0.11秒，温度是3×10¹⁰ K。如果你将这个温度代入上述关系式，得到的膨胀时间是0.24秒。我不清楚是如何得到0.11秒的。