# 物理代写|宇宙学代写cosmology代考|PHYC90009

## 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|Einstein equations

In the previous chapter, we have dealt with gravity only in terms of the metric, which gives us a notion of distances and straight lines (geodesics) in general spacetimes. These results were built on the principle of general covariance alone. We now turn to the second aspect of general relativity, which relates the metric to the constituents of the universe. This second part is contained in the Einstein equations, which relate the Einstein tensor describing the geometry to the energy-momentum tensor of matter. ${ }^1$ This set of equations can be summarized as the following celebrated tensor equality (Fig. 3.1):
$$G_{\mu v}+\Lambda g_{\mu v}=8 \pi G T_{\mu v}$$

Here $G_{\mu \nu}$ is the Einstein tensor defined through
$$G_{\mu v} \equiv R_{\mu v}-\frac{1}{2} g_{\mu \nu} R .$$
$R_{\mu \nu}$ is the Ricci tensor, which depends only on the metric and its derivatives; $R$, the Ricci scalar, is the contraction of the Ricci tensor $\left(R \equiv g^{\mu v} R_{\mu v}\right)$. Further, $\Lambda$ is the famous cosmological constant, $G$ is Newton’s constant, and $T_{\mu \nu}$ is the energy-momentum tensor, whose expression in the background universe we have already encountered in Sect. 2.3. Thus, the left-hand side of Eq. (3.1) is a function of the metric, the right a function of the constituents of the universe: the Einstein equations relate the two.

The simplicity of Eq. (3.1) belies the rich physics encoded in the Einstein equations. it. On small scales, Newtonian gravity is included, as we will see in Sect. 3.3, as are black holes which we will not deal with in this book. We will later encounter a different purely general-relativistic effect contained in Eq. (3.1) though: gravitational waves.

## 物理代写|宇宙学代写cosmology代考|暗能量

$\Omega_{\mathrm{m}} \simeq 0.3$的证据从1980年以来一直在积累，总密度等于临界密度的理论论点与膨胀有关，这是大约在同一时间提出的。后一种说法得到了20世纪90年代末对宇宙微波背景辐射的观测的支持(第9篇)。大约在同一时间，观察超新星的两个小组(Riess等人，1998年，Perlmutter等人，1999年)报告了加速宇宙的直接证据，最好的解释是假设暗能量的存在。证据是基于光度距离的测量。如2.2节所述，光度距离取决于宇宙在过去膨胀的速度:$d_L \propto \int d z / H(z)$。一个加速的宇宙，一个过去膨胀率较低的宇宙，因此会有更大的光度距离，因此像超新星这样的标准蜡烛会显得更暗淡

$$m-M=5 \log \left(\frac{d_L}{10 \mathrm{pc}}\right)+K$$
，其中$K$是一个校正(“k校正”)，用于由于扩展而使光谱移进或移出观测波长范围。$m-M$被称为距离模量。

## 物理代写|宇宙学代写cosmology代考|爱因斯坦方程

$$G_{\mu v}+\Lambda g_{\mu v}=8 \pi G T_{\mu v}$$

$$G_{\mu v} \equiv R_{\mu v}-\frac{1}{2} g_{\mu \nu} R .$$
$R_{\mu \nu}$定义的爱因斯坦张量，它只取决于度规及其导数;$R$，里奇标量，是里奇张量$\left(R \equiv g^{\mu v} R_{\mu v}\right)$的收缩。此外，$\Lambda$是著名的宇宙学常数，$G$是牛顿常数，$T_{\mu \nu}$是能量动量张量，其在背景宇宙中的表达式我们已经在2.3节中遇到过。因此，式(3.1)的左边是度规的函数，右边是宇宙成分的函数:爱因斯坦方程将两者联系起来

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