Cosmology Lecture 09

• "Frame 5:" $T=10^{9}K$, $t=3min$, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle kT=0.086MeV} . As was started in Frame 4, the major player is $\gamma$, with only small contributions from (p,n,$e^{-}$). Therefore:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g_{*}=2\,\!}

At this point, neutron decay becomes important (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {n_{n} \over n_{p}}\sim 0.16} ):

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p+n\to D+\overbrace {\gamma } ^{2.2MeV}\,\!}

There isn’t much deuterium in the universe because Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n_{p},n_{n}} are small and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n_{\gamma }} is large, so deuterium gets broken up as soon as it’s made.

Digression: The primary channel for energy generation in the center of stars is the pp chain, whereby H is fused into He. The pp chain generates about 85% of the energy in the sun. This chain goes as follows:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}&(1)p+p\to D+e^{+}+\nu _{e}\\&(2)p+D\to ^{3}He+\gamma \\&(3)^{3}He+^{3}He\to ^{4}He+2p\\&(net)4p\to ^{4}He+2e^{+}+2\nu _{e}+energy\\\end{aligned}}\,\!}

This generates a temperature of about Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 1.5\cdot 10^{7}K} in the core of the sun. The early universe, this temperature occurred about 9 days in. Back then, the dominant energy-releasing reaction was:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle p+n\to D+\gamma \,\!}

Why are the primary reactions different for the center of a star and the early universe? Well, in the center of a star, there aren’t so many free neutrons floating around, so the early universe reaction doesn’t work there. Also, the density of the core of a star (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sim 100{g \over cm^{3}}} ) is much greater than the density of the early universe (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sim 10^{-8}{g \over cm^{3}}} ). Notice that one of the reactions for the sun generates a neutrino, so these reactions involve weak interactions. We already discussed that the universe’s density was out of the realm of weak interactions after about 1 sec. Now back to our regularly scheduled program.

• "Frame 5.5:" $t=3.75min$. Here T finally gets low enough for deuterium to persist. Nucleosynthesis begins. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {n_{n} \over n_{p}}\sim 0.15} .
• "Frame 6:" Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T=10^{8.5}K} , $t\sim 30min$, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle kT=.027MeV} . The major players are still $\gamma$, with a little more p, $e^{-}$. Free neutrons are now mostly in Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle He} .

After Frame 6, the next important epoch is matter-radiation equality. After that, it’s the last scattering of $e^{-}$ off $\gamma$. The CMB epoch is when recombination occurs.

Frame 4 Revisited

This is the epoch of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle e^{-},e^{+}} annihilation. Energy conservation dictates that $S$ (the entropy density) be conserved. Thus, the entropy is transferred into $\gamma$ as Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle e^{-},e^{+}} annihilate. Since Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle S=constant\cdot g_{*}T^{3}} , we can calculate how much the photon gas was heated up in this epoch:

$\overbrace {e^{+}+e^{-}} ^{g_{*}={11 \over 2}}\to \overbrace {2\gamma } ^{g_{*}=2}\,\!$ Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{matrix}{11 \over 2}T_{before}^{3}=2T_{after}^{3}&\Rightarrow &{T_{after} \over T_{before}}=\left({11 \over 4}\right)^{1 \over 3}\end{matrix}}\,\!}

Since the neutrinos have already decoupled, they are unaffected by this heating. Therefore:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {T_{\nu }=\left({4 \over 11}\right)^{1 \over 3}T_{\gamma }}\,\!}

This is true even today.

Frame 5.5 Revisited

Recall that this was the epoch of nucleosynthesis. In Frame 5, there was a bottleneck where elements heavier that H could not be made because D kept being destroyed by radiation. In 5.5, that bottleneck has been lifted, and heavier elements are synthesized:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}D+D&\to ^{3}He+n\\D+D&\to ^{4}He+\gamma \\n+D&\to ^{3}H+\gamma \\D+^{3}H&\to ^{4}He+n\\^{4}He+^{3}H&\to ^{7}Li+\gamma \\^{4}He+^{3}He&\to ^{7}Be+\gamma \\^{7}Be+e^{-}&\to ^{7}Li+\nu _{e}\\\end{aligned}}\,\!}

This tree continues ad infinitum. The most abundant element in the universe is $H$, then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle He} . The abundance of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle ^{4}He} can be calculated:

$Y\equiv {mass\ in\ ^{4}He \over mass\ of\ all\ baryons}={4m_{p}\cdot n_{^{4}He} \over m_{p}(n_{p}+n_{n})}\,\!$

We can estimate that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n_{^{4}He}\approx {n_{n} \over 2}} , since all other n-containing elements are rare. Thus:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {Y={2{n_{n} \over n_{p}} \over 1+{n_{n} \over n_{p}}}}\,\!}

We can use an estimate of Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {n_{n} \over n_{p}}\sim {1 \over 7}} .

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

• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle Y} depends on the baryon-to-photon ratio=Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \eta \propto \Omega _{b}h^{2}} . As $\eta$ increases, 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 p+n\to D+\gamma } is favored over photo-dissociation, so D is more stable and the bottleneck breaks earlier, so 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 n_ n\over n_ p} is higher when 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 He} forms. Thus 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 Y} would be larger.
• 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 Y} depends on the lifetime of neutrons: 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 \tau _ n} .
• 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 Y} depends on the # of species of neutrinos, 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 N_\nu } . This parameter affects the energy density of the universe. A higher 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 N_\nu } causes the expansion of the universe to increase, since:

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 H\propto \sqrt{\rho }\propto \sqrt{g_*}T^2 \,\!}

This affects how many neutrons have decayed. We can establish a fitting formula for Y:

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 Y=0.23+0.025\log _{10}({\eta \over 10^{10}})+ 0.0075(g_*-10.75)+0.014(\tau -10.6min) \,\!}

This shows, for example, that an increase in 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 N_\nu } of 1 would change 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 g_*} :

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{matrix} \Delta g_*=2\cdot {7\over 8}={7\over 4} & \Rightarrow & \Delta Y\approx 0.013 \end{matrix} \,\!}