Cosmology Lecture 18

Still More on Metric Perturbations

Recall that we have our Boltzmann equation, which describes metric perturbations. To gain physical intuition into this equation, started taking velocity moments of this equation (in the non-relativistic limit).

$0^{th}$ moment:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \int {d^{3}pf}=n={p \over m}\,\!}

$1^{st}$ moment:

${\int {d^{3}p{\vec {v}}f} \over \int {d^{3}pf}}=\langle {\vec {v}}\rangle \,\!$

$2^{nd}$ moment:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle v_{i}v_{j}\rangle ={1 \over n}\int {d^{3}pv_{i}v_{j}f}\,\!}

and we defined Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\sigma _{ij}}^{2}=\langle v_{i}v_{j}\rangle -\langle v_{i}\rangle \langle v_{j}\rangle } . We can iterate to calculate any Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle n^{th}} moment. We then used these moments inside the Boltzmann equation.

$0^{th}$ moment of Boltzmann (the continuity equation):

${\partial \over \partial t}n+\nabla \cdot (n\langle {\vec {v}}\rangle )=0\,\!$

$1^{st}$ moment of Boltzmann:

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 {\partial \over \partial t}\langle v_ j\rangle +\langle v_ i\rangle \partial _ i\langle v_ j\rangle =-\partial _ j\phi -{1\over n} \partial _ i(\overbrace{n{\sigma _{ij}}^2}^{anisotropic\atop pressure}) \,\!}

In the Newtonian limit (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 \dot p_ i=-\partial _ i\phi \cdot m} ), dropping 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 \langle \rangle } ’s, and using 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 {\sigma _{ij}}^2\sim P{\delta _{ij}}+\tau _{ij}} (where 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 _{ij}} is the non-diagonal piece), 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 {\partial {\vec v}\over \partial t}+({\vec v}\cdot \nabla ){\vec v}=-\nabla \phi -{1\over \rho }\nabla P \,\!}

Which is just Euler’s Equation for a perfect fluid.

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 2^{nd}} moment of Boltzmann. T his is messy, so we’ll use that for a perfect gas, 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 {\sigma _{ij}}^2\Leftrightarrow {P\over \rho }\propto T} , which gives us:

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 diffusion\ term={3\over 2}{dT\over dk}+(\nabla \cdot {\vec v})T+{1\over \rho }\nabla \cdot \vec q \,\!}

where 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 q} is the heat flux.

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^{th}} moment: Recall the linearized continuity and Euler’s equations (in comoving coordinates):

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{cases} \dot\delta _ k+{\vec k+{\vec v}_ k\over a}=0\\ {\partial \over \partial t}(a{\vec v}_ k)+iv_ s^2\vec k\delta _ k+i\vec k\phi _ k=0 \end{cases} \,\!}

Defining 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 \theta =\nabla \cdot {\vec v}} (and 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 \theta _ k=i\vec k{\vec v}_ k} ):

${\begin{cases}{\dot {\delta }}_{k}+{1 \over a}\theta _{k}=0\\{\dot {\theta }}_{k}+{{\dot {a}} \over a}\theta _{k}-{v_{s}^{2}k^{2} \over a}\delta _{k}-{k^{2} \over a}\phi _{k}=0\end{cases}}\,\!$

So compare the Boltzmann equation:

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 {\partial f\over \partial t}+\dot x_ i{\partial f\over \partial x_ i}+\dot p_ i{\partial f\over \partial p_ i} =\left({\partial f\over \partial t}\right)_ c \,\!}

with the full theory (the coupled Boltzmann and Einstein equations in a conformal Newtonian gauge):

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 ds^2=c^2dt^2(1+2\phi )-a^2(t)(1-2\Psi )dx^ idx_ i \,\!}

This gives us the Geodesic equation (which is the GR equation of motion):

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 {\dot p=p\dot\Psi -\epsilon n_ i\partial _ i\phi } \,\!}

where 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 \dot\Psi } term is specific to GR, 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 \epsilon =\sqrt{p^2+m^2}} , and 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_ i=pn_ i} . In GR, the Boltzmann equation is written using the energy-momentum tensor:

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 T_{\mu \nu }=\int {d^3p(-g)^{-\frac12}{P_\mu P_\nu \over p^0}f} \,\!}

where 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 f} is the phase space distribution. The first-order part of the energy-momentum conservation equation (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 T^{\mu \nu }_{i\mu }=0} ) gives us the following moments:

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} & \dot\delta _ k+{(1+w)\over a}\theta _ k-3(1+w)\dot\Psi _ k=-3{\dot a \over a}(v_ s^2-w)\delta _ k\\ & \dot\theta _ k+(1-3w){\dot a \over a}\theta _ k+{\dot w\over 1+w}\theta _ k-{v_ s^2\over 1+w}{k^2\over a}\delta _ k+{k^2\over a}\sigma _ k-{k^2\over a}\phi _ k=0\\ \end{align} \,\!}

where 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 \sigma _ k\equiv (k_ ik_ j-{1\over 3}{\delta _{ij}}){\sigma _{ij}}_ k} . There are, of course, higher moments as well. Compare the above result to the result we got using just the linearized fluid equations. That simplistic result was almost right, except for the factor of 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 {1\over 3}} . We can look at different components of this energy-momentum conservation equation:

• CDM: here we are pressure-less, 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 v_ s=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 w=0} , so 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} \dot\delta _ c+{1\over a}\theta _ c-3\dot\Psi & =0\\ \dot\theta _ c+{\dot a \over a}\theta _ c-{k^2\over a}\phi & =0\\ \end{align} \,\!}

• Baryons: we have a non-relativistic fluid, 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 v_ s^2\ll c^2} , and 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 {\sigma _{ij}}\approx 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 \begin{align} \dot\delta _ b+{1\over a}\theta _ b-3\dot\Psi & =0\\ \dot\theta _ b+{\dot a \over a}\theta _ b-{v_ s^2k^2\over a}\delta _ b-{k^2\over a}\phi - \underbrace{{4\over 3}{\bar\rho _\gamma \over \bar\rho _ b}n_ e\sigma _ T(\theta _\gamma -\theta _ b)}_{{momentum\ transfer\atop between}\atop photons\ and\ baryons}& =0\\ \end{align} \,\!}

• Photons: 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={1\over 3}} , 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 v_ s^2={c^2\over 3}} :

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} \dot\delta _\gamma +{4\over 3a}\theta _\gamma -4\dot\Psi & =0\\ \dot\theta _\gamma \dots \sigma _\gamma \dots -n_ e\sigma _ T(\theta _ b-\theta _\gamma )& =0\\ \end{align} \,\!}

• Neutrinos: no Thomson scattering, collision-less, etc.

Coupled Baryon-Photon Fluid

Before photon decoupling, baryons and photons are tightly coupled via Thomson scattering (it behaves like a single fluid). Recall that Thomson scattering is the low-energy limit of Compton scattering, which is the general mechanism for photon-electron scattering. We are in this low-energy regime because 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 kT_\gamma \ll m_ ec^2\ll m_ pc^2} after about 1 second. The cross-section for Thomson scattering is 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 \sigma _ T=.665\cdot 10^{-24}cm^2} , and

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 {d\sigma \over d\Omega }={3\over 15\pi }\sigma _ T(1+\cos ^2\theta ) \,\!}

The optical depth to Thomson scattering is given by:

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 =\int {n_ e\sigma _ Tds} \,\!}

where 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_ e} is the electron density, and 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 ds} is distance along the line of sight. 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 \gg 1} is optically thick (opaque), and 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 \ll 1} is optically thin (transparent).

We can calculate when photons decouple (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 \tau } from Thomson scattering $\sim 1$):

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 =\int {n_ e\sigma _ Tc{dt\over dz}dz} \,\!}

where integral over distance is written in terms of redshift. We can use the Friedmann equation to replace 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 dt\over dz} , and we’ll use that the free electron density is given by:

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_ e(z)=x_ e(z)n_ H(z)=x_ e(z)(1+z)^3n_{H,0} \,\!}

where 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_ H} is the # density of hydrogen and 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 x_ e} is the ionization fraction. We can use the Y parameter to calculate 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_ H,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 Y={4 n_{He}\over n_ H+4n_{He}}={2n_{neutron}\over n_ B}={n_ B-n_ H\over n_ B} \,\!}

which tells us 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 n_ H=(1-Y)n_ B} . 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 n_ e(z)=x_ e(z)(1+z)^3(1-Y)\underbrace{\Omega _{B,0}\over m_\rho }_{n_{B,0}\over \rho _{crit,0}}\underbrace{3 H_0^2\over 8\pi G}_{\rho _{crit,0}} \,\!}

We’ll actually do this out to get 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 } as a function of 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 z} and cosmological parameters on PS#8. Next time we’ll see if we can calculate the sound speed 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 v_ s} .