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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础
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The epoch at which the energy density in matter equals that in radiation is called matterradiation equality. It has a special significance for the generation of large-scale structure and for the development of CMB anisotropies, because perturbations grow at different rates in the two different eras (note that for large-scale structure, there is a third era: that of dark energy domination today; see Exercise 2.14). It is therefore a useful exercise to calculate the epoch of matter-radiation equality. To do this, we need to compute the energy density of both matter and radiation, and then find the value of the scale factor at which they were equal.

Using Eq. (2.76) and Eq. (2.82), we see that, as long as $T_{v}$ is much larger than all neutrino masses, the total energy density in radiation is
$$\frac{\rho_{\mathrm{r}}}{\rho_{\mathrm{cr}}}=\frac{4.15 \times 10^{-5}}{h^{2} a^{4}} \equiv \frac{\Omega_{\mathrm{r}}}{a^{4}} .$$
To calculate the epoch of matter-radiation equality, we equate Eqs. (2.85) and (2.72) to find
$$a_{\mathrm{eq}}=\frac{4.15 \times 10^{-5}}{\Omega_{\mathrm{m}} h^{2}} .$$
A different way to express this epoch is in terms of redshift $z$; the redshift of equality is
$$1+z_{\text {eq }}=2.38 \times 10^{4} \Omega_{\mathrm{m}} h^{2} .$$
Note that, as the amount of matter in the universe today, $\Omega_{\mathrm{m}} h^{2}$, goes up, the redshift of equality also goes up.

## 物理代写|宇宙学代写cosmology代考|Dark energy

We now know that there is an additional ingredient in the universe’s energy budget, dark energy, a substance whose equation of state $w$ is neither 0 (as it would be if the substance was nonrelativistic) or $1 / 3$ (ultra-relativistic), but rather close to $-1$. A multitude of independent pieces of evidence has accumulated for the existence of dark energy, a substance that has this negative equation of state and does not participate in gravitational collapse. For one, we have strong evidence that the universe is Euclidean, with total density parameter close to 1 . Since $\Omega_{\mathrm{m}}=0.3$ is very far from 1 (and radiation is totally negligible today), something that does not clump as does matter has to make up this budgetary shortfall. Second, the expansion of the universe is accelerating, as measured by standard candles and rulers. As we will see in Ch. 3, accelerated expansion $(\ddot{a}>0)$ occurs only if the dominant constituent in the universe has a negative equation of state, i.e. negative pressure.

Evidence that $\Omega_{\mathrm{m}} \simeq 0.3$ has been accumulating since about 1980 , and theoretical arguments that the total density is equal to the critical density are tied to inflation, which was proposed around the same time. The latter claims were bolstered by observations of the CMB in the late 1990s (Ch. 9). Around the same time, two groups (Riess et al., 1998, Perlmutter et al., 1999) observing supernovae reported direct evidence for an accelerating universe, one that is best explained by postulating the existence of dark energy. The evidence is based on measurements of the luminosity distance. As discussed in Sect. 2.2, the luminosity distance depends on the how rapidly the universe expanded in the past: $d_{L} \propto \int d z / H(z)$. An accelerating universe, one in which the expansion rate was lower in the past, would therefore have larger luminosity distances, and therefore standard candles like supernovae would appear fainter.

More concretely, the luminosity distance of Eq. (2.43) can be used to find the apparent magnitude $m$ of a source with absolute magnitude $M$. Magnitudes are related to fluxes and luminosities via $m=-(5 / 2) \log (F)+$ constant and $M=-(5 / 2) \log (L)+$ constant. Since the flux scales as $d_{L}^{-2}$, the apparent magnitude $m=M+5 \log \left(d_{L}\right)+$ constant. The convention is that
$$m-M=5 \log \left(\frac{d_{L}}{10 \mathrm{pc}}\right)+K$$
where $K$ is a correction (” $K$-correction”) for the shifting of the spectrum into or out of the observed wavelength range due to expansion. $m-M$ is referred to as distance modulus.

## 物理代写|宇宙学代写cosmology代考|The fundamental equations of cosmology

Cosmology is, essentially, an application of general relativity coupled with statistical mechanics. The only relevant long-range force is gravity, which also provides the background spacetime within which matter moves, as we have seen in the last chapter. Since cosmology deals with the evolution of the entire universe, we are not interested in the fate of individual particles. Instead, we care about the collective, average behavior of matter, which is described by statistical mechanics. This is why essentially all results in cosmology can be derived from the combination of two equations: the Einstein equations on the gravity side, and the Boltzmann equations of statistical mechanics for matter and radiation.
These are formidable equations, and their application can quickly get technical. In this chapter, we will present the general form of the Einstein and Boltzmann equations, and describe their physical content. We will then apply them to the homogeneous universe, which, for the Einstein equations, allows us to derive the Friedmann equation (1.3). These results will also allow us to compute the expansion history and thermal history of the universe in this chapter and the next. Further, with the experience we gain in this chapter, there will be nothing particularly difficult about the subsequent chapters which deal with perturbations in the universe. So, becoming familiar with the framework laid out in this chapter will pay off greatly when going through the rest of the book.

## 宇宙学代考

$$\frac{\rho_{\mathrm{r}}}{\rho_{\mathrm{cr}}}=\frac{4.15 \times 10^{-5}}{h^{2} a^{4}} \equiv \frac{\Omega_{\mathrm{r}}}{a^{4}} .$$

$$a_{\text {eq }}=\frac{4.15 \times 10^{-5}}{\Omega_{\mathrm{m}} h^{2}}$$

$$1+z_{\text {eq }}=2.38 \times 10^{4} \Omega_{\mathrm{m}} h^{2} .$$

## 物理代写|宇宙学代写cosmology代考|Dark energy

$$m-M=5 \log \left(\frac{d_{L}}{10 \mathrm{pc}}\right)+K$$

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