## 物理代写|宇宙学代写cosmology代考|OLET1640

2022年12月28日

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## 物理代写|宇宙学代写cosmology代考|Relation between scale factor and temperature

The primordial plasma is in adiabatic expansion and its entropy is preserved. We consider an elementary volume expanding with space. Such a volume is called a covolume because its comoving coordinates $r, \theta$ and $\phi$ do not vary with time. For simplicity, a volume equal to $a^3$ is taken where $a(t)$ is the scale factor. The entropy of the primordial plasma contained in this volume is written as:
$$s=\mathcal{S}(T) a^3$$
and does not vary with time. Equation [1.43] allow the entropy $s$ to be directly expressed as a function of the temperature $T$ and the scaling factor $a$ :
$$s \equiv \frac{4 \pi^2}{45} T^3 h_{\mathrm{eff}}(T) a^3$$
We thus obtain a relation between the logarithmic derivatives of $a$ and $T$ in the form:
$$H \equiv \frac{\dot{a}}{a}=-\frac{\dot{T}}{T}\left{1+\frac{1}{3} \frac{\mathrm{d} \ln h_{\mathrm{eff}}}{\mathrm{d} \ln T}\right}$$
which now makes it possible to solve equation [1.59] relating to the expansion.
As a first approximation, we can neglect the variations of the coefficient $h_{\text {eff }}$ with temperature. Figure $1.3$ shows that $h_{\text {eff }}$ varies only by a factor of 8 when the temperature decreases by three orders of magnitude, from $10 \mathrm{GeV}$ to $10 \mathrm{MeV}$. Thereby, we deduce a value of $0.1$ for the term $\mathrm{d} \ln h_{\mathrm{eff}} / 3 \mathrm{~d} \ln T$ between the previous braces $s_4$ a small value compared to $1_{\text {. The conservation of entropy s thus implies, up }}$ to the variations of $h_{\mathrm{eff}}$, that the product $T \times a$ of the temperature by the scale factor does not vary with time. From this relation, we shall be able to derive analytically the age of the universe based on the temperature of the ylem with a quite acceptable accuracy.

One should notice however that if $h_{\text {eff }}$ varies globally only very little with time. this is not the case during the quarks/hadrons phase transition during which the QGP transforms into a plasma of pions, with traces of protons and neutrons. We have assumed this transition to be of first order, hence a sharp decrease in $h_{\text {eff }}$ from $31.03$ to 8.48. During this phase transition, the temperature remains locked at $200 \mathrm{MeV}$, so that it is the product $h_{\text {eff }} \times a^3$ which is now constant. The scale factor has increased by a factor $(31.03 / 8.48)^{1 / 3} \simeq 1.54$ when the transition ends.

## 物理代写|宇宙学代写cosmology代考|Relation between cosmic time and temperature

We can now integrate equation [1.59] by taking into account relation [1.64] that we just established between the scale factor $a$ and the temperature $T$. It expresses the fact that the product $h_{\text {eff }} T^3 a^3$ is constant during the expansion. During the cosmic time $d t$, the temperature of ylem varies by $d T$, such that:
$$d t=-\sqrt{\frac{45}{8 \pi^3}} M_{\mathrm{Pl}}\left{\frac{1+\left(\mathrm{d} \ln h_{\mathrm{eff}} / 3 \mathrm{~d} \ln T\right)}{\sqrt{g_{\mathrm{eff}}}}\right} \frac{d T}{T^3}$$
The integration of this differential equation is in principle easy, using, for example, the classical fourth-order Runge-Kutta method (Press et al. 2007). The only technical difficulty is the divergence of the derivative of $\ln h_{\text {eff }}$ with respect to $\ln T$ at the time of the quarks/hadrons phase transition.

Forgetting this problem for the moment and neglecting the variation of $h_{\mathrm{eff}}$ with temperature, it follows that:
$$d t=-\sqrt{\frac{45}{8 \pi^3}} \frac{M_{\mathrm{Pl}}}{\sqrt{g_{\mathrm{eff}}}} \frac{d T}{T^3}$$
The age of the universe $t$ is given by the integral on the temperature $T^{\prime}$ varying from infinity, a value corresponding to the Big Bang, down to $T$ :
$$t=\int_0^t d t^{\prime}=\sqrt{\frac{45}{32 \pi^3}} \int_{\infty}^T \frac{M_{\mathrm{Pl}}}{\sqrt{g_{\mathrm{eff}}\left(T^{\prime}\right)}} d\left(T^{\prime-2}\right)$$
The temperature values near $T$ provide the dominant contribution to this integral. The coefficient $g_{\text {eff }}$ can be evaluated at temperature $T$ and treated as a constant. The cosmic time is then simplified to:
$$t \simeq \sqrt{\frac{45}{32 \pi^3}} \frac{M_{\mathrm{Pl}}}{\sqrt{g_{\mathrm{eff}}(T)}} \int_{\infty}^T d\left(T^{\prime-2}\right) \equiv \sqrt{\frac{45}{32 \pi^3}} \frac{1}{\sqrt{g_{\mathrm{eff}}(T)}} \frac{M_{\mathrm{P} 1}}{T^2}$$
The previous expression is evaluated in the system of units where $c=k_{\mathrm{B}}=$ $\hbar=1$. The Planck mass $M_{\mathrm{Pl}}$ and the temperature $T$ can be expressed in $\mathrm{MeV}$, so that the ratio $M_{\mathrm{Pl}} / T^2$ is evaluated in $\mathrm{MeV}^{-1}$. To obtain a cosmic time $t$ expressed in seconds, one mainly has to multiply the result by the reduced Planck constant $\hbar=6.582 \times 10^{-22} \mathrm{MeV} \mathrm{s}$. Our final result is then written in the form:
$$t \simeq \frac{1.71 \mathrm{~s}}{\sqrt{g_{\mathrm{eff}}(T)}}\left{\frac{1 \mathrm{MeV}}{T}\right}^2$$

## 物理代写|宇宙学代写cosmology代考|Relation between scale factor and temperature

$$s=\mathcal{S}(T) a^3$$

$$s \equiv \frac{4 \pi^2}{45} T^3 h_{\mathrm{eff}}(T) a^3$$

H lequiv Ifrac ${\backslash d o t{a}}{}=-$ Ifrac ${\backslash \operatorname{dot}{T}}{T} \backslash \operatorname{lef}{1+\mid$ frac

1. The conservation of entropy s thus implies, up 的变化 $h_{\text {eff }}$ ，即产品 $T \times a$ 比例因子的温度不随时间变化。从这个 关系中，我们将能够以相当可接受的精度，根据 ylem 的温度分析推导出宇宙的年龄。
然而，人们应该注意到，如果 $h_{\mathrm{eff}}$ 在全球范围内随时间 变化很小。在夸克/强子相变期间情况并非如此，在此期 间 QGP 转变成介子等离子体，带有质子和中子的痕迹。 我们假设这种转变是一阶的，因此急剧下降 $h_{\mathrm{eff}}$ 从 31.03到 8.48。在此相变期间，温度保持锁定在 $200 \mathrm{MeV}$ ，所以它是产品 $h_{\text {eff }} \times a^3$ 现在是不变的。比 例因子增加了一个因子 $(31.03 / 8.48)^{1 / 3} \simeq 1.54$ 当过 渡结束时。

## 物理代写|宇宙学代写cosmology代考|Relation between cosmic time and temperature

$\mathrm{d} \mathrm{t}=-$ Isqrt $\left.\left{\backslash \operatorname{frac}{45}\left{8 \backslash \mathrm{pi}^{\wedge} 3\right}\right} M_{-}{\backslash \mathrm{mathrm}{\mathrm{P}}}\right} \backslash \mathrm{eft}{\backslash \operatorname{frac}{1+$

$$d t=-\sqrt{\frac{45}{8 \pi^3}} \frac{M_{\mathrm{Pl}}}{\sqrt{g_{\mathrm{eff}}}} \frac{d T}{T^3}$$

$$t=\int_0^t d t^{\prime}=\sqrt{\frac{45}{32 \pi^3}} \int_{\infty}^T \frac{M_{\mathrm{Pl}}}{\sqrt{g_{\mathrm{eff}}\left(T^{\prime}\right)}} d\left(T^{\prime-2}\right)$$

$$t \simeq \sqrt{\frac{45}{32 \pi^3}} \frac{M_{\mathrm{Pl}}}{\sqrt{g_{\mathrm{eff}}(T)}} \int_{\infty}^T d\left(T^{\prime-2}\right) \equiv \sqrt{\frac{45}{32 \pi^3}}$$

$\hbar=6.582 \times 10^{-22} \mathrm{MeVs}$. 然后我们的最终结果写成 以下形式:

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## MATLAB代写

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