Cosmology Lecture 20

The Sound Speed of Baryons

Last time we showed that after decoupling, $P_{B}\propto \rho _{B}^{\gamma }$, where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \gamma ={C_{P} \over C_{V}}={5 \over 3}} for a monotonic ideal gas at constant entropy. As a result:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v_{s}^{2}={\partial P_{B} \over \partial \rho _{B}}{\big |}_{S}=\gamma {P_{B} \over \rho _{B}}={5 \over 3}{kT_{B} \over m_{H}}\,\!}

We also showed that using the third law of thermodynamics (Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T\propto V^{1-\gamma }\propto a^{3(1-\gamma )}\propto a^{-2}} , for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \gamma ={5 \over 3}} ), our expression for Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v_{s}^{2}} becomes:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v_{s}^{2}={5 \over 3}{kT_{dec} \over m_{H}}\left({a_{dec} \over a}\right)^{2}\,\!}

Where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle T_{dec},a_{dec}} are the temperature and scaling factor at the time of decoupling. Thus, Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v_{s}\propto {1 \over a}} after decoupling. Moreover, the baryonic sound speed around decoupling is given by:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v_{s}(a_{dec})={\sqrt {{5 \over 3}{kT_{dec} \over m_{H}}}}\,\!}

Using that $kT_{dec}\sim 10^{3}kT_{\gamma ,0}\sim 3000K\sim {1 \over 4}eV$, we have:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle v_{s}(a_{dec})={\sqrt {{5 \over 3}{{1 \over 4}eV \over 10^{9}eV}}}\cdot c\sim 6{km \over s}\,\!}

So at decoupling, the sound speed of baryons drops from Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {c \over {\sqrt {3}}}} to Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 6{km \over s}} . This causes the Jeans mass to drop dramatically, and structure formation can begin. We’ll investigate this in PS#9.

Statistical Properties of Perturbation Fields Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \delta _{i}}

Recall our definition of $\delta$ for “ripples about the mean”:

$\delta ({\vec {x}},t)\equiv {\rho ({\vec {x}},t) \over {\bar {\rho }}(t)}-1={\partial \rho ({\vec {x}},t) \over {\bar {\rho }}(t)}\,\!$

In Fourier space, this looks like:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\delta ({\vec {x}},t)&={1 \over (2\pi )^{3}}\int {d^{3}ke^{i{\vec {k}}{\vec {x}}}\delta ({\vec {k}},t)}\\\delta ({\vec {k}},t)&=\int {d^{3}xe^{-i{\vec {k}}{\vec {x}}}\delta ({\vec {x}},t)}\\\end{aligned}}\,\!}

We should note that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \delta ({\vec {x}})} is real, so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \delta ^{*}({\vec {k}})=\delta (-{\vec {k}})} .

Now we’ll examine the power spectrum of these perturbations as a function of $k$:

Power Spectrum

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \langle \delta ({\vec {k}})\delta ^{*}({\vec {k}}^{\prime })\rangle =P(k)(2\pi )^{3}\delta _{D}({\vec {k}}-{\vec {k}}^{\prime })\,\!}

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \delta _{D}} is the Dirac delta function. Note that in $P(k)$, we dropped the vector information about $k$. This is because the power spectrum should only depend on Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle |{\vec {k}}|} due to isotropy. The average we are taking in this equation is supposed to be an average over many universes. However, we only have one universe to work with, so we approximate that over a large amount of space, the average should be similar to that of several universes. Thus, we interpret this as a spatial average, even though that’s not exactly what was meant in the equation.

The Importance of $P(k)$:

• If $\delta$ is a Gaussian field (as predicted by many theories), then $P(k)$ completely specifies its statistical properties ($\langle \delta \rangle =0$).
• $P(k)$ quantifies the amount of gravitational clustering for each $k$ mode.
• The Fourier Transform of $P(k)$ is the two-point correlation function(see PS#9):

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \xi (r)=\langle \delta ({\vec {x}}_{1})\delta ({\vec {x}}_{2})\rangle \,\!}

where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\vec {r}}={\vec {x}}_{1}-{\vec {x}}_{2}} . A two-point correlation function describes the excess (above Poisson) probability of finding pairs of points at separation ${\vec {r}}$. The Sloan Digital Sky Survey has been finding that the galaxy-galaxy correlation Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \xi _{gg}(r)\propto \left({r \over r_{0}}\right)^{-1.8}} , which is to say, you are more likely to find galaxies close together than far apart. 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 \xi } describes clustering.

One more note 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 P(k)} : it has units 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 length^3} . We can define a dimensionless version:

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 \Delta ^2(k)\equiv {4\pi k^3P(k)\over (2\pi )^3} \,\!}

Angular Power Spectrum (with CMB as an example)

For 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 \rho _\gamma \propto T_\gamma ^4\ \Rightarrow \ {\delta T_\gamma \over T_\gamma }= {1\over 4}\delta _\gamma \,\!}

We then define the Angular Power Spectrum 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 C(\theta )} :

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 {\Delta T\over T}(\hat n_ i){\Delta T\over T}(\hat n_2)\rangle \equiv C(\theta )= {1\over 4\pi }\sum _{\ell =0}^\infty {(2\ell +1)C_\ell \cdot P_\ell (\cos \theta )} \,\!}

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 P_\ell } ’s are Legendre Polynomials, 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 C_\ell } is what everyone typically plots for galaxy surveys. In general, we can make the substitution:

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} {\Delta T\over T}(\hat n)& =\sum _{\ell =0}^\infty {\sum _{m=-\ell }^\ell { a_{\ell m}Y_{\ell m}(\theta , \phi )}} \Leftrightarrow a_{\ell m}=(-i)^\ell 4\pi \int {d^3kY_{\ell m}^*(\vec k){\Delta T\over T}(\vec k)} \\ \end{align} \,\!}

So doing out the math:

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} \langle {\Delta T\over T(\hat n_1){\Delta T\over T}(\hat n_2)}& = \sum _{\ell m}{\sum _{\ell m}{ \underbrace{\langle a_{\ell m}a_{\ell ^\prime m^\prime }\rangle }_{ C_\ell \delta _{\ell \ell ^\prime }\delta {mm^\prime }} Y_{\ell m}(\theta , \phi }Y_{\ell ^\prime m^\prime }(\theta ^\prime , \phi ^\prime )}\rangle \\ & =\sum _{\ell m}C_\ell Y_{\ell m}(\hat n_1)Y_{\ell m}(\hat n_2)\\ & ={1\over 4\pi }\sum _{\ell =0}^\infty {(2\ell +1)C_\ell P_\ell (\cos \theta )}\\ \end{align} \,\!}

Where the last step was taken using the addition theorum of spherical harmonics.

Non-Gaussian Fields

For non-Gaussian fields, the lowest order statistic is the 3-point correlation function, which is equivalent to a bi-spectrum. 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 k} space, this says 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 B(k_1, k_2, k_3)\delta _ D(k_1, k_2, k_3)\propto \langle \delta (k_1)\delta (k_2)\delta (k_3)\rangle \,\!}

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 k_1, k_2, k_3} describe a triangle 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 k} space.