Notes
  • Notes
  • 恒星结构与演化
    • Chapter 7. Equation of State
    • Chapter 3. Virial Theorem
    • Chapter 11. Main Sequence
    • Chapter 4. Energy Conservation
    • Chapter 12. Post-Main Sequence
    • Chapter 2. Hydrostatic Equilibrium
    • Chapter 6. Convection
    • Chapter 9. Nuclear Reactions
    • Chapter 10 Polytrope
    • Chapter 8. Opacity
    • Chapter 14. Protostar
    • Chapter 13. Star Formation
    • Chapter 5. Energy Transport
  • 天体光谱学
    • Chapter 6 气体星云光谱
    • Chapter 5 磁场中的光谱
    • Chapter 7 X-射线光谱
    • Chapter 3 碱金属原子
    • Chapter 1 光谱基础知识
    • Chapter 9 分子光谱
    • Chapter 4 复杂原子
    • Chapter 2 氢原子光谱
  • 物理宇宙学基础
    • Chapter 2 Newtonian Cosmology
    • Chapter 1 Introduction
    • Chapter 5* Monochromatic Flux, K-correction
    • Chapter 9 Dark Matter
    • Chapter 10 Recombination and CMB
    • Chapter 8 Primordial Nucleosynthesis
    • Chapter 7 Thermal History of the Universe
    • Chapter 6 Supernova cosmology
    • Chapter 5 Redshifts and Distances
    • Chapter 4 World Models
    • Chapter 3 Relativistic Cosmology
  • 数理统计
    • Chapter 6. Confidence Sets (Intervals) 置信区间
    • Chapter 1. Data Reduction 数据压缩
    • Chapter 7. Two Sample Comparisons 两个样本的比较
    • Chapter 3. Decision Theory 统计决策
    • Chapter 4. Asymptotic Theory 渐近理论
    • Chapter 5. Hypothesis Testing 假设检验
    • Chapter 9. Linear Models 线性模型
    • Chapter 10 Model Selection 模型选择
    • Chapter 2. Estimation 估计
    • Chapter 11 Mathematical Foundation in Causal Inference 因果推断中的数理基础
    • Chapter 8. Analysis of Variance 方差分析
  • 天体物理动力学
    • Week8: Orbits
    • Week7: Orbits
    • Week6: Orbits
    • Week5: Orbits
    • Week4: Orbits
    • Week3: Potential Theory
    • Week2
    • Week1
  • 天体物理吸积过程
    • Chapter 4. Spherically Symmetric Flow
    • Chapter 2. Fluid Dynamics
    • Chapter 5. Accretion Disk Theory
    • Chapter 3. Compressible Fluid
  • 天文技术与方法
    • Chapter1-7
  • 理论天体物理
    • Chapter 6 生长曲线的理论和应用
    • Chapter 5 线吸收系数
    • Chapter 4 吸收线内的辐射转移
    • Chapter 3 恒星大气模型和恒星连续光谱
    • Chapter 2 恒星大气的连续不透明度
    • Chapter 1 恒星大气辐射理论基础
  • 常微分方程
    • 线性微分方程组
    • 高阶微分方程
    • 奇解
    • 存在和唯一性定理
    • 初等积分法
    • 基本概念
  • 天体物理观测实验
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  • Robertson-Walker Metric
  • Friedmann Equation
  • Cosmological constant
  1. 物理宇宙学基础

Chapter 3 Relativistic Cosmology

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Last updated 4 years ago

Robertson-Walker Metric

  • Flat (Euclid) space

    (dl)2=(dx)2+(dy)2+(dz)2⇒Δl=∫12(dx)2+(dy)2+(dz)2(\text{d}l)^2=(\text{d}x)^2+(\text{d}y)^2+(\text{d}z)^2\\ \Rightarrow \Delta l=\int_1^2\sqrt{(\text{d}x)^2+(\text{d}y)^2+(\text{d}z)^2}(dl)2=(dx)2+(dy)2+(dz)2⇒Δl=∫12​(dx)2+(dy)2+(dz)2​
  • Flat (Minkowski) spacetime

    (dl)2=(cdt)2−(dx)2−(dy)2−(dz)2⇒Δl=∫12(cdt)2−(dx)2−(dy)2−(dz)2(\text{d}l)^2=(c\text{d}t)^2-(\text{d}x)^2-(\text{d}y)^2-(\text{d}z)^2\\ \Rightarrow \Delta l=\int_1^2\sqrt{(c\text{d}t)^2-(\text{d}x)^2-(\text{d}y)^2-(\text{d}z)^2}(dl)2=(cdt)2−(dx)2−(dy)2−(dz)2⇒Δl=∫12​(cdt)2−(dx)2−(dy)2−(dz)2​
  • Proper distance

    distance at $t_A=t_B$ (hypothesis on the existence of a well-defined universal time)

  • Curvature

    K=3πlim⁡D→02πD−CD3K=\frac{3}{\pi}\lim_{D\to 0}\frac{2\pi D-C}{D^3}K=π3​D→0lim​D32πD−C​

    where $D​$ is the radius, $C​$ is the perimeter

    Some 2-D​ examples

    • Plain - zero curvature

    • Sphere - positive curvature

    • Hyperbolic paraboloid (马鞍面;双曲抛物面) - negative curvature

  • 2-D curvature

    Spherical polar coordinates $(r,\theta,\phi)$

    (dl)2=(dD)2+(rdϕ)2=(Rdθ)2+(rdϕ)2=(drcos⁡θ)2+(rdϕ)2=(dr1−r2/R2)2+(rdϕ)2\begin{aligned} (\text{d}l)^2&=(\text{d}D)^2+(r\text{d}\phi)^2\\ &=(R\text{d}\theta)^2+(r\text{d}\phi)^2\\ &=\left(\frac{\text{d}r}{\cos\theta}\right)^2+(r\text{d}\phi)^2\\ &=\left(\frac{\text{d}r}{\sqrt{1-r^2/R^2}}\right)^2+(r\text{d}\phi)^2 \end{aligned}(dl)2​=(dD)2+(rdϕ)2=(Rdθ)2+(rdϕ)2=(cosθdr​)2+(rdϕ)2=(1−r2/R2​dr​)2+(rdϕ)2​

    In general, 2-D curvature

    (dl)2=(dr1−kr2)2+(rdϕ)2(\text{d}l)^2=\left(\frac{\text{d}r}{\sqrt{1-kr^2}}\right)^2+(r\text{d}\phi)^2(dl)2=(1−kr2​dr​)2+(rdϕ)2
  • 3-D curvature

    (dl)2=(dr1−kr2)2+(rdθ)2+(rsin⁡θdϕ)2(\text{d}l)^2=\left(\frac{\text{d}r}{\sqrt{1-kr^2}}\right)^2+(r\text{d}\theta)^2+(r\sin\theta\text{d}\phi)^2(dl)2=(1−kr2​dr​)2+(rdθ)2+(rsinθdϕ)2
  • R-W Metric

    (ds)2=(cdt)2−(dr1−kr2)2−(rdθ)2−(rsin⁡θdϕ)2(\text ds)^2=(c\text dt)^2-\left(\frac{\text{d}r}{\sqrt{1-kr^2}}\right)^2-(r\text{d}\theta)^2-(r\sin\theta\text{d}\phi)^2(ds)2=(cdt)2−(1−kr2​dr​)2−(rdθ)2−(rsinθdϕ)2
    • Proper distance

      ΔL=(−Δs)2, dt=0\Delta L=\sqrt{(-\Delta s)^2},\ \text dt=0ΔL=(−Δs)2​, dt=0
    • Comoving distance

      K(t)=ka2(t), r(t)=a(t)x⇒(ds)2=(cdt)2−a2(t)[(dx1−kx2)2−(xdθ)2−(xsin⁡θdϕ)2]K(t)=\frac{k}{a^2(t)},\ r(t)=a(t)x\\ \Rightarrow (\text ds)^2=(c\text dt)^2-a^2(t)\left[\left(\frac{\text{d}x}{\sqrt{1-kx^2}}\right)^2-(x\text{d}\theta)^2-(x\sin\theta\text{d}\phi)^2\right]K(t)=a2(t)k​, r(t)=a(t)x⇒(ds)2=(cdt)2−a2(t)[(1−kx2​dx​)2−(xdθ)2−(xsinθdϕ)2]

      Traditionally, we use $r$ to represent comoving radial distance

      ⇒(ds)2=(cdt)2−a2(t)[dr21−kr2−r2(dθ2+sin⁡2θdϕ2)]\Rightarrow (\text ds)^2=(c\text dt)^2-a^2(t)\left[\frac{\text{d}r^2}{1-kr^2}-r^2(\text{d}\theta^2+\sin^2\theta\text{d}\phi^2)\right]⇒(ds)2=(cdt)2−a2(t)[1−kr2dr2​−r2(dθ2+sin2θdϕ2)]
  • SR model - - empty universe

    • Time dilution

      τ=γ−1t(t,r)⇒(τ,l)\tau=\gamma^{-1}t\\ (t,r)\Rightarrow(\tau,l)τ=γ−1t(t,r)⇒(τ,l)

Friedmann Equation

  • Einstein's field equation

    Gαβ=8πGc2TαβG_{\alpha\beta}=\frac{8\pi G}{c^2}T_{\alpha\beta}Gαβ​=c28πG​Tαβ​
    • Einstein tensor

      Gαβ=Rαβ−12gαβRG_{\alpha\beta}=R_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}RGαβ​=Rαβ​−21​gαβ​R
    • Metric

      ds2=gαβdxαdxβ\mathrm ds^2=g_{\alpha\beta}\mathrm dx^\alpha\mathrm dx^\betads2=gαβ​dxαdxβ
  • Friedmann equation

    (a˙a)2+kc2a2=8πG3ρa¨a=−4πG3(ρ+3Pc2)\left(\frac{\dot{a}}{a}\right)^2+\frac{kc^2}{a^2}=\frac{8\pi G}{3}\rho\\ \frac{\ddot a}{a}=-\frac{4\pi G}{3}\left(\rho+\frac{3P}{c^2}\right)(aa˙​)2+a2kc2​=38πG​ρaa¨​=−34πG​(ρ+c23P​)

    the second is known as acceleration equation

  • At present time

    Ω0=ρ0ρc, ρc=3H028πG\Omega_0=\frac{\rho_0}{\rho_c},\ \rho_c=\frac{3H_0^2}{8\pi G}Ω0​=ρc​ρ0​​, ρc​=8πG3H02​​

    rewrite Friedmann equation

    a˙02=8π3Ga02ρ0−kc2⇒H02a02=H02a02Ω0−kc2⇒kc2=H02a02(Ω0−1), k=+1,0,−1\dot a_0^2=\frac{8\pi}{3}Ga_0^2\rho_0-kc^2\\ \Rightarrow H_0^2a_0^2=H_0^2a_0^2\Omega_0-kc^2\\ \Rightarrow kc^2=H_0^2a_0^2(\Omega_0-1),\ k=+1,0,-1a˙02​=38π​Ga02​ρ0​−kc2⇒H02​a02​=H02​a02​Ω0​−kc2⇒kc2=H02​a02​(Ω0​−1), k=+1,0,−1
    • $\Omega_0>1$, overcritical density, $k=+1$

    • $\Omega=1$, critical density, $k=0$

    • $0<\Omega_0<1$, undercritical density, $k=-1$

  • Difference between Newtonion cosmology

    • Newtonion

      kc2=−2Umc2kc^2=-\frac{2U}{mc^2}kc2=−mc22U​
    • GR, $k$ stands for curvature

Cosmological constant

  • Static universe

    • $a$ is a constant, $\dot a=\ddot a=0$

    • $H=0$

    • Age of the universe is infinite

    • Friedmann equation

      kc2a2=8πG3ρ=−8πGc2P0\frac{kc^2}{a^2}=\frac{8\pi G}{3}\rho=-\frac{8\pi G}{c^2}P_0a2kc2​=38πG​ρ=−c28πG​P0​
    • To make sure that $\rho>0$, $P_0<0$ , Einstein introduced a constant, Lorentz-invariant term $\Lambda$

      Gαβ−Λgαβ=8πGc4TαβG_{\alpha\beta}-\Lambda{g_{\alpha\beta}}=\frac{8\pi G}{c^4}T_{\alpha\beta}Gαβ​−Λgαβ​=c48πG​Tαβ​

      rewrite Friedmann equation

      (a˙a)2+kc2a2=8πG3ρ+Λc232a¨a+(a˙a)2+kc2a2=−8πGc2P+Λc2\left(\frac{\dot{a}}{a}\right)^2+\frac{kc^2}{a^2}=\frac{8\pi G}{3}\rho+\frac{\Lambda c^2}{3}\\ 2\frac{\ddot a}{a}+\left(\frac{\dot{a}}{a}\right)^2+\frac{kc^2}{a^2}=-\frac{8\pi G}{c^2}P+\Lambda c^2(aa˙​)2+a2kc2​=38πG​ρ+3Λc2​2aa¨​+(aa˙​)2+a2kc2​=−c28πG​P+Λc2
    • Define vacuum energy density

      ρvac=Λc28πG\rho_{vac}=\frac{\Lambda c^2}{8\pi G}ρvac​=8πGΛc2​

      we have

      2a¨a+(a˙a)2+kc2a2=−8πGc2(P−ρvac)2\frac{\ddot a}{a}+\left(\frac{\dot{a}}{a}\right)^2+\frac{kc^2}{a^2}=-\frac{8\pi G}{c^2}(P-\rho_{vac})2aa¨​+(aa˙​)2+a2kc2​=−c28πG​(P−ρvac​)
    • For Newtonion cosmology, we have to introduce an additional potential energy term

      VΛ≡−16Λmc2r2V_\Lambda\equiv-\frac{1}{6}\Lambda mc^2r^2VΛ​≡−61​Λmc2r2
      ⇒U=T+V+VΛ=12mr˙2−4π3Gρr2m−16Λmc2r2\begin{aligned} \Rightarrow U&=T+V+V_\Lambda\\ &=\frac{1}{2}m\dot{r}^2-\frac{4\pi}{3}G\rho r^2m-\frac{1}{6}\Lambda mc^2r^2 \end{aligned}⇒U​=T+V+VΛ​=21​mr˙2−34π​Gρr2m−61​Λmc2r2​
      ⇒F⃗Λ=−∇VΛ\Rightarrow \vec F_\Lambda=-\nabla V_\Lambda⇒FΛ​=−∇VΛ​
the Milne Model