Cosmology Lecture 02

The Friedmann Equations, continued

Recall we had the following equations:

$\left({{\dot {a}} \over a}\right)^{2}={8\pi \over 3}G\rho -{k \over a^{2}}\,\!$ ${\dot {\rho }}=-3{{\dot {a}} \over a}(\rho +P)\,\!$ ${{\ddot {a}} \over a}=-{4\pi \over 3}G(\rho +3P)\,\!$

To close the equations, we need to relate $P$ and $\rho$ with an equation of state:

${P=w\rho (c^{2})}\,\!$

Note that we will generally set $c=1$ in this class. Combined with (2), this gives us:

${\begin{aligned}{\dot {\rho }}&=-3{{\dot {a}} \over a}(1+w)\rho \\{{\dot {\rho }} \over \rho }&=-3(1+w){{\dot {a}} \over a}\\\rho &\propto a^{-3(1+w)}\\\end{aligned}}\,\!$

Note that we’ve assumed ${\dot {w}}=0$, which is okay most of the time. Some special cases of interest are:

• Pressure-less “dust” $P=0$, $w=0\Rightarrow \rho \propto a^{-3}$ because volume goes as $V\propto {1 \over a^{3}}$.
• Relativistic particles (photons, bosons): $w={1 \over 3}$, $P={\rho \over 3}\Rightarrow \rho \propto a^{-4}$ because $VV\propto {1 \over a^{3}}$, and energy is given by $E\propto {1 \over a}$.
• ($\Lambda$)/Dark Energy: $w=-1$, $P=-\rho \Rightarrow \rho =$ constant in time.

To get density ($\rho$) as a function of time, want to solve for $w$.

Critical Density, $\rho _{crit}$

We define $\rho _{crit}$ to be the critical density at which $k=0$ (and $E=0$):

${\rho _{crit}}={3H^{2} \over 8\pi G}\,\!$

Today, we measure: $\rho _{crit,0}={3H_{0}^{2} \over 8\pi G}={3 \over 8\pi }{\left(100h{km \over sMpc}\right)^{2} \over G}=2.78\cdot 10^{11}h^{2}{M_{\odot } \over Mpc^{3}}=1.88\cdot 10^{-29}h^{2}{g \over cm^{3}}$. Note that the mass of the sun is $M_{\odot }=2\cdot 10^{33}g$, and the mass of the proton is $M_{p}=1.67\cdot 10^{-24}g$.

Density Parameter, $\Omega (t)$

$\Omega$ measures the ratio of the density of the universe to the critical density:

$\Omega (t)\equiv {\rho (t) \over \rho _{c}(t)}={8\pi G\rho (t) \over 3H^{2}(t)}\,\!$ $\Omega {\begin{cases}\leq 1&\Rightarrow open\ (k\leq 0)\\=1&\Rightarrow flat\ (k=0)\\\geq 1&\Rightarrow closed\ (k\geq 0)\end{cases}}\,\!$

In general, $\Omega$ can consist of multiple components: $\Omega =\sum _{i}{\Omega _{i}}$ e.g.

$\Omega {\begin{cases}r&=radiation\\m&=matter\ (dark\ and\ luminous)\\b&=baryons\ (dark\ and\ luminous)\\\nu &=neutrinos\\\Lambda &=dark\ energy\end{cases}}\,\!$

$\Omega =1$ is an unstable equilibrium; any perturbation from $\Omega =1$ in the early universe ensures $\Omega$ is far from 1 today. That we measure $\Omega _{m}\approx 0.3$ today implies that the early universe must have been extremely finely tuned.

Evolution of $H(t)$

Today (at $a=1$): $k={8\pi \over 3}G\rho _{0}-H_{0}^{2}=H_{0}^{2}(\Omega _{0}-1)$. Using the $1^{st}$ Friedmann equation (1), we have:

${H^{2}=H_{0}^{2}({\Omega _{0} \over a^{3+3w}}+{1-\Omega _{0} \over a^{2}})}\,\!$

(THE Friedmann Equation) where $H_{0}\equiv {\sqrt {8\pi G\rho \over 3{\rho _{crit}}}}$, and it is understood that $\Omega _{0}=\sum _{i}{\Omega _{i,0}}$. Each Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Omega _{i}} has it’s own $w_{i}$, so really Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\Omega _{0} \over a^{3+3w}}=\sum _{i}{\Omega _{i,0} \over a^{3+3w_{i}}}} .

Evolution of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Omega (a)}

We’ll show in PS#1 that for any single component:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {1-\Omega (a) \over \Omega (a)}={1-\Omega _{0} \over \Omega _{0}}a^{1+3w}\,\!}

Plotting Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Omega (a)} , we will find that for early $a$, $\Omega$ is extremely close to 1.

Evolution of $a(t)$: Solving the Friedmann Equation

The evolution of the expansion of the universe is governed by:

$\left({{\dot {a}} \over a}\right)^{2}={8\pi \over 3}G\rho -{k \over a^{2}}\,\!$

We can apply this to several models of the universe:

• The Einstein-deSitter (flat) Model: $k=0$, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega (a)=\Omega _0=1} . Using that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho \propto a^{-3(1+w)}} , we have:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \left({\dot a \over a}\right)^2& \propto a^{-3(1+w)}\\ a^{-1}a^{{3\over 2}(1+w)}da& \propto dt\\ a^{-{3 \over 2}(1+w)}& \propto t\\ \end{align} \,\!} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {a(t)\propto t^{{2\over 3(1+w)}}} \,\!}

Thus, the rate of expansion of the universe depends on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle w} : (a) The matter-dominated era: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega \approx \Omega _ m\Rightarrow w=0, P=0, a\propto t^{2 \over 3}} . (b) The radiation-dominated era: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega \approx \Omega _ r\Rightarrow w={1\over 3} \Rightarrow a(t)\propto t^\frac12} . (c) The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Lambda } -dominated era: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega \approx \Omega _\Lambda \Rightarrow w=-1, P=-\rho , \rho } constant in time Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow a(t)\propto e^{Ht}} , where H is now actually a constant. This is exponential inflation. We used to think that this only happened early on (like Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-34}} seconds), but now we think that this has also been happening recently. Next time, we will do the harder two cases: open and closed.