Cosmology Lecture 20

The Sound Speed of Baryons

Last time we showed that after decoupling, $P_{B}\propto \rho _{B}^{\gamma }$, where $\gamma ={C_{P} \over C_{V}}={5 \over 3}$ for a monotonic ideal gas at constant entropy. As a result:

$v_{s}^{2}={\partial P_{B} \over \partial \rho _{B}}{\big |}_{S}=\gamma {P_{B} \over \rho _{B}}={5 \over 3}{kT_{B} \over m_{H}}\,\!$

We also showed that using the third law of thermodynamics ($T\propto V^{1-\gamma }\propto a^{3(1-\gamma )}\propto a^{-2}$, for $\gamma ={5 \over 3}$), our expression for $v_{s}^{2}$ becomes:

$v_{s}^{2}={5 \over 3}{kT_{dec} \over m_{H}}\left({a_{dec} \over a}\right)^{2}\,\!$

Where $T_{dec},a_{dec}$ are the temperature and scaling factor at the time of decoupling. Thus, $v_{s}\propto {1 \over a}$ after decoupling. Moreover, the baryonic sound speed around decoupling is given by:

$v_{s}(a_{dec})={\sqrt {{5 \over 3}{kT_{dec} \over m_{H}}}}\,\!$

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

$v_{s}(a_{dec})={\sqrt {{5 \over 3}{{1 \over 4}eV \over 10^{9}eV}}}\cdot c\sim 6{km \over s}\,\!$

So at decoupling, the sound speed of baryons drops from ${c \over {\sqrt {3}}}$ to $6{km \over s}$. 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 $\delta _{i}$

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:

${\begin{aligned}\delta ({\vec {x}},t)&={1 \over (2\pi )^{3}}\int {d^{3}ke^{i{\vec {k}}{\vec {x}}}\delta ({\vec {k}},t)}\\\delta ({\vec {k}},t)&=\int {d^{3}xe^{-i{\vec {k}}{\vec {x}}}\delta ({\vec {x}},t)}\\\end{aligned}}\,\!$

We should note that $\delta ({\vec {x}})$ is real, so $\delta ^{*}({\vec {k}})=\delta (-{\vec {k}})$.

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

Power Spectrum

$\langle \delta ({\vec {k}})\delta ^{*}({\vec {k}}^{\prime })\rangle =P(k)(2\pi )^{3}\delta _{D}({\vec {k}}-{\vec {k}}^{\prime })\,\!$

where $\delta _{D}$ 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 $|{\vec {k}}|$ 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):

$\xi (r)=\langle \delta ({\vec {x}}_{1})\delta ({\vec {x}}_{2})\rangle \,\!$

where ${\vec {r}}={\vec {x}}_{1}-{\vec {x}}_{2}$. 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 $\xi _{gg}(r)\propto \left({r \over r_{0}}\right)^{-1.8}$, which is to say, you are more likely to find galaxies close together than far apart. $\xi$ describes clustering.

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

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

Angular Power Spectrum (with CMB as an example)

For photons:

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

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

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

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

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

So doing out the math:

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

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 $k$ space, this says that

$B(k_{1},k_{2},k_{3})\delta _{D}(k_{1},k_{2},k_{3})\propto \langle \delta (k_{1})\delta (k_{2})\delta (k_{3})\rangle \,\!$

where $k_{1},k_{2},k_{3}$ describe a triangle in $k$ space.