Difference between revisions of "SZ Effect"

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[[File:CarlstromFig4.png|thumb|500px|center| Difference between a blackbody spectrum and the post-SZ effect CMB spectrum. The number of photons with frequencies less than 218 GHz is decreased, while the number with energies greater than 218 GHz is increased. The kSZ effect is shown in the dotted line.  Data from the cluster Abell 2163. Adapted from Figure 4 of Carlstrom et al., Annual Reviews of Astronomy & Astrophysics vol 40, pg 643, 2002.]]
 
[[File:CarlstromFig4.png|thumb|500px|center| Difference between a blackbody spectrum and the post-SZ effect CMB spectrum. The number of photons with frequencies less than 218 GHz is decreased, while the number with energies greater than 218 GHz is increased. The kSZ effect is shown in the dotted line.  Data from the cluster Abell 2163. Adapted from Figure 4 of Carlstrom et al., Annual Reviews of Astronomy & Astrophysics vol 40, pg 643, 2002.]]
  
\section{Uses of SZ}
+
What is the use of the SZ effect in astrophysics? Before we answer that, let's remind ourselves of a few things and do a quick calculation. First of all, galaxy clusters are filled with hot gas. The gas comes from material accreted by the cluster, from the intergalactic medium (IGM). This matter from the IGM (mostly just pure hydrogen) falls towards the potential wells of the galaxy clusters, which represent significant "overdensities" relative to most of the universe. As the gas falls into the potential well of the cluster (from infinity), it converts gravitational potential energy to kinetic energy, reaching a speed:
 +
\begin{equation}
 +
v \sim \sqrt{GM_{\rm cluster}/{R_{\rm cluster}}}.
 +
\end{equation}
 +
The gas shocks as it approaches the edge of the cluster converting its kinetic energy to thermal energy, providing the pressure support necessary to keep the gas there. It reaches a temperature:
 +
\begin{equation}
 +
T \sim m_H v^2/k_B \sim \frac{m_H}{k_B}\frac{GM_{\rm cluster}}{R_{\rm cluster}}
 +
\end{equation}
 +
As we can see, the temperature of the gas scales with the mass of the cluster! By measuring the SZ effect on CMB photons, we can constrain the mass of the cluster. Although in reality the problem is somewhat messier than what we've presented here, studies consistently find masses (for the most massive clusters) of $\sim 10^{14} M_{\odot}$. And yet, adding up the total stellar mass of the galaxies in the clusters only gives $\sim 10^{12} M_{\odot}$. This is reason number 94389244 that we believe dark matter exists -- most of the gravitating mass of galaxy clusters isn't visible!
  
  
 
+
Note that the SZ method isn't perfect. First of all, it's useless for all but the most massive dark matter halos in the universe. This is simply because the gas filling lower mass DM halos simply isn't hot enough to significantly Compton up-scatter CMB photons. Even in massive halos, there are some astrophysical issues which can get in the way. For example, we've assumed that gravity is the only factor at play in setting the temperature of the gas. However, in reality this simply isn't true. Supernovae and AGN can drive large-scale outflows of gas from the galaxy and into the halo.  
 
 
 
 
% everything below here is old.
 
 
 
\iffalse
 
Consider a non-relativistic thermal bath of electrons at temperature $T_e$.
 
The average kinetic energy of these electrons is ${3\over2}kT_e$. Now
 
suppose that there is also a bath of photons which all have
 
energy $E=h\nu\ll kT_e$.  Putting this cold bath of photons in with
 
the hot (but non-relativistic) electrons, we find that they get up-scattered.
 
After first scattering, the mean shift in energy is:
 
$$\begin{aligned}\Delta\_E&={P_{K,single\ e^-}\over n_{photons}\sigma_Tc}=
 
{{4\over3}U_{ph}\gamma^2\beta^2\sigma_Tc\over n_{photons}\sigma_Tc}=
 
{4\over3}\gamma^2\beta^2E\\
 
{\Delta\_E\over E}&={4\over3}\gamma^2\beta^2={4\over3}{v^2\over c^2}=
 
{4\over3}{3kT_e\over m_ec^2}={4kT_e\over m_ec^2}\\ \end{aligned}$$
 
which is much the mean fractional change in photon energy after 1 scattering.
 
 
 
Last time, we'd written down the mean energy of the upscattered photons:
 
$$\Delta\_E={P_{Compton,single\ e^-}\over n_{photons}\sigma_Tc}$$
 
The bottom term $n_{photons}\sigma_Tc$ is just the collision rate for
 
photons. We used this to calculate:
 
$${\Delta\_E\over E}={4kT_e\over m_ec^2}\ll1$$
 
Thus, after a single scattering:
 
$$\_E_i=E\left(1+{4kT_e\over m_ec^2}\right)$$
 
So after $N$ scatterings:
 
$$\_E_N=E\left(1+{4kT_e\over m_ec^2}\right)^N$$
 
In the limit of $N\to\infty$:
 
$$E_N=Ee^{4kT_eN\over m_ec^2}\equiv Ee^{4y}$$
 
This is the definition of the Compton $y$ parameter: $y\equiv {kT_e\over m_e
 
c^2}N$.\par
 
Suppose we have a cloud of electrons that has radius $l$, and the [[Optical
 
Depth]] out of the cloud $\tau_e\gg1$.  The mean number of collisions a photon
 
will undergo in getting out of the cloud is $N\sim\tau_e^2$. This is because
 
each time step between collisions is ${\lambda_{mfp}\over c}$, and the number
 
of steps to get out is:
 
$${Time\ to\ get\ out\over Each\ time\ step}\sim{{l^2\over c\lambda_{mfp}}
 
\over {\lambda_{mfp}\over c}}\sim\left({l\over\lambda_{mfp}}\right)^2
 
\sim\tau_e^2$$
 
\def\ppt#1{{\partial #1\over\partial t}}
 
The reason why the time to get out is $l^2\over c\lambda_{mfp}$, is because
 
the diffusion equation gives us $\ppt{Q}=D\nabla^2Q$, which by dimensional
 
analysis, says:
 
$$\inv{T}\sim{D\over L^2}$$
 
$D$ is the diffusivity (viscosity) of the medium.\par
 
If $\tau_e\ll1$, $N_s\sim\tau$ because the intensity goes as $e^{-\tau}\approx
 
1-\tau$.  Generally:
 
$$\begin{aligned}N_s&\sim max(\tau_e,\tau_e^2)\\
 
&\sim\tau_e(1+\tau_e)\\ \end{aligned}$$
 
Thus, the Compton $y$ parameter is:
 
$$y\approx {kT_e\over m_ec^2}\tau_e(1+\tau_e)$$
 
\begin{itemize}
 
\item We'll apply this to CMB photons traveling through an intra-cluster
 
gas. We'll say: $n_e\sim3\e{-3}cm^{-3}$, $\sigma_T\sim10^{-24}cm^2$, and
 
$l\sim10^6\cdot3\e{18}cm$.  This gives us $\tau_e\sim10^{-2}$.  Now
 
$kT_e$ should be a few $keV$, so $${kT_e\over m_ec^2}\sim10^{-2}$$
 
The us the energy post-traversal of the cluster is related to the initial
 
energy of the photons by:
 
$$E_{post}=E_{init}e^{4y}=E_{init}(1+4\e{-4})$$
 
Thus, we expect to see a shift in the frequency of peak flux when looking
 
through a cluster vs. around one.  We can calculate the change in brightness
 
temperature this causes by using:
 
$$\begin{aligned}F_{\nu,before}&={2kT_{CMB}\over\lambda^2}\\
 
F_{\nu,after}\left(\lambda={\lambda_0\over1+4y}\right)
 
&={2kT_{CMB}\over\lambda^2}\\
 
&={2kT_{CMB}^\prime\over\lambda_0^2}\\ \end{aligned}$$
 
Solving for $T_{CMB}^\prime$:
 
$$\begin{aligned}T_{CMB}^\prime&=T_{CMB}\left({\lambda\over\lambda_0}\right)^2\\
 
&=T_{CMB}\left({1\over1+4y}\right)^2\approx T_{CMB}(1-8y)\\ \end{aligned}$$
 
So the change in brightness temperature is $\Delta T_{CMB}\sim8yT_{CMB}\sim
 
10mK$. This is hard to detect from ground-based telescopes, but this is
 
a much greater effect than the inherent anisotropies of the CMB (which we've
 
detected), so space-based telescopes like WMAP, if they have the angular
 
resolution, should have already detected this.  Yet they haven't.\par
 
Now we've been a little sloppy.  We've said that every photon from the CMB
 
underwent a small shift in energy, but for an optically thin cloud like the
 
one we've been discussing, only a few photons are ever scattered.  We'll examine
 
this on the Rayleigh-Jeans tail.\par
 
For any $F_\nu$ incident upon our cloud, the $F_{\nu,after}$ which results
 
from unscattered photons should be less (there are fewer photons).  Then the
 
flux from the upscattered photons gets added back in.  We'll choose some
 
frequency $\nu_0$ which scatters into $\tn=\nu_0\left(1+{4kT_e\over m_ec^2}
 
\right)$, so that:
 
$$\begin{aligned}F_{\nu,after}(\tn)
 
&={2kT_{CMB}\over c^2}\tn^2-{2kT_{CMB}\over c^2}\tn^2\tau_e
 
+{2kT_{CMB}\over c^2}\nu_0^2\tau_e\\
 
&={2kT_{CMB}\over c^2}
 
{\tn^2\over\left(1+{4kT_e\tau_e\over m_ec^2}\right)^2}\\ \end{aligned}$$
 
This matches what we estimated just by shifting, so we're okay.\par
 
 
 
\item We'll examine another situation where we can use the SZ effect
 
to measure $H_0$ in an x-ray cluster.  We'll say that $\theta_c$ is the
 
observed angular size of the cluster, and $F_x$ is the observed x-ray flux.
 
We'll say there's some observed redshift (using optical data), which is
 
related to the recession velocity $v$.  The x-ray luminosity $L_x=
 
F_x\cdot4\pi d^2$, and the recessional velocity is $v=Hd$.  Thus:
 
$$L_x\propto F_xv^2H^{-2}$$
 
We also know, from bremsstrahlung that $L_x\propto n_e^2r_c^3F(T_e)$, where
 
$F(T_e)$ is some function of the temperature of the electrons, and $r_c$ is
 
the radius of the cluster.  Now we can express $r_c$ as:
 
$$r_c\sim\theta_cd\sim\theta_c{v\over H}$$
 
Setting our two expressions for the luminosity equal:
 
$$\begin{aligned}F_xv^2H^{-2}&\propto n_e^2r_c^3F(T_e)\\
 
&\propto n_e^2\theta_c^3{v^3\over H^3}F(T_e)\\ \end{aligned}$$
 
which gives us that $n_e\propto H^\hf$.  Now SZ tells us that $y\propto T_en_e
 
r_c\propto n_er_c\propto H^\hf\theta_c\cdot d{v\over H}$. Thus:
 
$$y\propto \inv{H^\hf}$$
 
\end{itemize}
 
\fi
 
  
 
\end{document}
 
\end{document}
 
<\latex>
 
<\latex>

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<latex> \documentclass[11pt]{article} \def\inv#1Template:1 \over \def\ddtTemplate:D \over dt \def\mean#1{\left\langle {#1}\right\rangle} \def\sigot{\sigma_{12}} \def\sigto{\sigma_{21}} \def\eval#1{\big|_{#1}} \def\tr{\nabla} \def\dce{\vec\tr\times\vec E} \def\dcb{\vec\tr\times\vec B} \def\wz{\omega_0} \def\ef{\vec E} \def\ato{{A_{21}}} \def\bto{{B_{21}}} \def\bot{{B_{12}}} \def\bfieldTemplate:\vec B \def\apTemplate:A^\prime \def\xp{{x^{\prime}}} \def\yp{{y^{\prime}}} \def\zp{{z^{\prime}}} \def\tp{{t^{\prime}}} \def\upxTemplate:U x^\prime \def\upyTemplate:U y^\prime \def\e#1{\cdot10^{#1}} \def\hf{\frac12} \def\^{\hat } \def\.{\dot } \def\tnTemplate:\tilde\nu

\usepackage{fullpage} \usepackage{amsmath} \usepackage{eufrak} \begin{document}


\section*{ The Sunyaev-Zeldovich Effect: A case-study of Inverse-Compton Scattering}

A classic astrophysical example of Inverse Compton Scattering is the Sunyaev-Zel'dovich (SZ) effect.

Massive galaxy clusters throughout the Universe are filled with fast-moving free electrons. Most CMB photons have been free-streaming through the Universe since the Epoch of Recombination. However, a fraction of CMB photons pass through massive clusters before reaching our detector, and some of these will interact with high energy electrons through inverse Compton scattering. This changes the energies of the CMB photons and produces an observable deviation from the Planck function in the CMB spectrum.

To see quantitatively how interactions with energetic photons change the CMB spectrum, let's consider what happens to a single photon passing through a cluster, which for our purposes can be approximated as a cloud of hot electrons.

We'll call the kinetic temperature of electrons $T_{e}$. As derived in the lecture on Inverse Compton Scattering, the power radiated by a single electron passing through a photon field is $$P=\frac{4}{3}\sigma_{T}U_{\gamma}c\beta^{2}\gamma^{2},$$ where $\sigma_{T}$ is the Thomson cross section, $U_{\gamma}$ is the energy density of the radiation field, $\beta=v/c$, and $\gamma = 1/\sqrt{1-v^2/c^2}$.

We can calculate $\Delta E_{\gamma}$, the typical change in the energy of a photon for a single interaction with an electron, as $\Delta E_{\gamma}=P/R_Template:\rm coll$, where $R_Template:\rm coll=N_Template:\rm coll/\Delta t$ is is the number of collisions with photons per unit time for a single electron.

The rate of collisions can be written as $R=n\sigma v$, and $v=c$ in the case of electrons scattering off photons.

Thus, $$R_Template:\rm coll=n_{\gamma}\sigma_{T}c$$

Here $n_{\gamma}$ is the number density of photons, $\sigma_{T}$ is the Thomson cross section for electron-photon interactions, and $v$ is the typical electron velocity. We can then write $$\Delta E_{\gamma}=\frac{P}{R_Template:\rm coll}=\frac{\frac{4}{3}\sigma_{T}U_{\gamma}c\beta^{2}\gamma^{2}}{n_{\gamma}\sigma_{T}v}=\frac{4}{3}\gamma^{2}\beta^{2}E_{\gamma}$$ 

Where we have introduce $E_{\gamma}=U_{\gamma}/n_{\gamma}$, the average photon energy.

We'd like to write $\Delta E_{\gamma}$ in terms of temperature. From energy equipartition, we have (for a single electron): $$\frac{1}{2}m_{e}v^{2}=\frac{3}{2}kT_{e}\Rightarrow v^{2}=\frac{3kT_{e}}{m_{e}}.$$

We're going to assume that $v\ll c$, so that $\gamma\to1$.

Then we can write $$\Delta E_{\gamma}\approx\frac{4}{3}\beta^{2}E_{\gamma}=\frac{4v^{2}}{3c^{2}}E_{\gamma}=\frac{4kT_{e}}{m_{e}c^{2}}E_{\gamma}$$

This expression gives the change in the energy of a photon after a single interaction with an electron. If a 2nd interaction occurs, the electron's energy will change again. In the limit of many interactions $N$, we can change the discrete $\Delta E$ into an infinitesimal differential, and we have a differential equation: $$\frac{{\rm d}E_{\gamma}}{{\rm d}N}=\left(4\frac{kT_{e}}{m_{e}c^{2}}N\right)E_{\gamma}=4yE_{\gamma}$$

where we introduced the dimensionless Compton $y$ parameter, $y\equiv kT_{e}N/\left(m_{e}c^{2}\right)$.

This is a homogenous differential equation for $E_{\gamma}$; it's solution is $$E_{\gamma}=E_{\gamma,0}e^{4y}$$

For a typical cluster, $n_{e}\approx3\times10^{-3}\,{\rm cm}^{-3}$ and $\ell=3\times10^{24}\,{\rm cm}.$ Thus, the Optical Depth is $$\tau_{e}=n_{e}\sigma_{T}\ell\approx10^{-2}$$

I.e., clusters are optically thin. Thus, we can write $N\approx\tau$. For a cluster with electron temperature $T_{e}=10^{8}{\rm K}$ (corresponding to a mass of a few $\times10^{14}\,M_{\odot}$), we have $$\frac{kT_{e}}{m_{e}c^{2}}\approx10^{-2};\qquad y=10^{-2}\times10^{-2}=10^{-4}$$

The fact that $\tau_{e}\ll1$ means that most CMB photons passing through a cluster will not interact at all. Of the approximately $1\%$ that do, almost all will only interact once, and their energies will typically increase by a factor of $\sim e^{4\times10^{-4}}\sim1+4\times10^{-4}$.

\section*{The Kinematic Sunyaev-Zeldovich Effect: A case study of Relativistic Beaming}

The kinematic Sunyaev-Zel'dovich (kSZ) effect is a result of relativistic beaming and scattering, rather than Compton scattering. When a hot cloud of electrons (such as those found in galaxy clusters) has a net velocity relative to the background light (usually the CMB), it scatters the photons isotropically in cluster rest frame. When this is transformed back to the CMB rest frame, the scattered photons are beamed, increasing the intensity in the direction of motion, and decreasing it behind the cluster. The fully general derivation of this effect is quite complicated, but for non-relativistic cluster velocities ($\beta << 1$), the kSZ derivation becomes tractable. In this limit, 3 important things happen: \begin{enumerate}

 \item The kSZ and SZ effects can be treated separately.
 \item The effects of the Doppler shift are negligible (because the CMB spectrum has no sharp features).
 \item The relativistic beaming equation can be approximated to first order in $\beta$.

\end{enumerate}

Without further ado, let's start with the relativistic beaming effect. As a reminder, this equation tells us how a small section of solid angle in the cluster rest frame ($d\Omega^\prime$) transforms into the lab frame ($d\Omega$).

\begin{equation}

 \frac{d\Omega^\prime}{d\Omega} = \frac{1 - \beta^2}{\left(1 - \beta \cos\left(\theta\right)\right)^2}

\end{equation} To first order in $\beta$, this is \begin{equation}

 \frac{d\Omega^\prime}{d\Omega} \approx 1 + 2 \cos\left(\theta\right) \beta

\end{equation}

This gives us an approximate expression for the intensity of scattered photons an observer at an angle $\theta$ with respect to the cluster's velocity sees (in the optically thin limit): \begin{equation}

 I_{\nu,scatt}\left(\theta\right) \approx \left(1 + 2 \cos\left(\theta\right) \beta\right) \tau I_{\nu,CMB}

\end{equation} where $I_{\nu,CMB}$ is the unperturbed CMB blackbody spectrum, and $I_{\nu,scatt}$ is the intensity of the scattered photons. The full expression for the specific intensity is the sum of the scattered photons and the unscattered photons \begin{equation}

 I_{\nu,CMB} \left(1 - \tau\right) + I_{\nu,scatt} = I_{\nu,CMB}\left(1 + 2 \cos\left(\theta\right) \beta\right)

\end{equation}

This can be written more succinctly as \begin{equation}

 \frac{\Delta I_{\nu,CMB}}{I_{\nu,CMB}} = \frac{v_r}{c} \tau

\end{equation} where I $v_r = v \cos(\theta)$ is the cluster's line of sight velocity, and I have dropped the factor of 2 since we are only working to order of magnitude (and to match the kSZ literature).

\section*{Effect on CMB spectrum}

How does the SZ Effect change the CMB spectrum in practice? Figure 1 compares the predicted CMB spectrum after the SZ Effect (solid line) to the blackbody spectrum predicted without the SZ effect (dashed line). Since the energies of CMB photons are much lower than cluster electrons, all CMB photons which interact with electrons gain energy. However, because the total number of photons in conserved, the SZ effect decreases the number of photons at lower energies, moving the to higher energy regions of the spectrum.

Effect of SZ effect on the CMB spectrum. The dashed line shows the expected spectrum for a blackbody of with the temperature of the CMB. The solid line shows the spectrum after the SZ effect: photons are removed from the lower-energy part of the spectrum and upscattered to higher energies. Adapted from Figure 1 of Carlstrom et al., Annual Reviews of Astronomy & Astrophysics vol 40, pg 643, 2002.

This can be seen more clearly in Figure 2, below, which plots the difference between the CMB spectrum after the SZ effect and the pure blackbody spectrum. The number of photons with frequencies less than 218 GHz is decreased, while the number with energies greater than 218 GHz is increased. This plot also shows the kSZ effect in the dotted line.

Difference between a blackbody spectrum and the post-SZ effect CMB spectrum. The number of photons with frequencies less than 218 GHz is decreased, while the number with energies greater than 218 GHz is increased. The kSZ effect is shown in the dotted line. Data from the cluster Abell 2163. Adapted from Figure 4 of Carlstrom et al., Annual Reviews of Astronomy & Astrophysics vol 40, pg 643, 2002.

What is the use of the SZ effect in astrophysics? Before we answer that, let's remind ourselves of a few things and do a quick calculation. First of all, galaxy clusters are filled with hot gas. The gas comes from material accreted by the cluster, from the intergalactic medium (IGM). This matter from the IGM (mostly just pure hydrogen) falls towards the potential wells of the galaxy clusters, which represent significant "overdensities" relative to most of the universe. As the gas falls into the potential well of the cluster (from infinity), it converts gravitational potential energy to kinetic energy, reaching a speed: \begin{equation} v \sim \sqrt{GM_{\rm cluster}/{R_{\rm cluster}}}. \end{equation} The gas shocks as it approaches the edge of the cluster converting its kinetic energy to thermal energy, providing the pressure support necessary to keep the gas there. It reaches a temperature: \begin{equation} T \sim m_H v^2/k_B \sim \frac{m_H}{k_B}\frac{GM_{\rm cluster}}{R_{\rm cluster}} \end{equation} As we can see, the temperature of the gas scales with the mass of the cluster! By measuring the SZ effect on CMB photons, we can constrain the mass of the cluster. Although in reality the problem is somewhat messier than what we've presented here, studies consistently find masses (for the most massive clusters) of $\sim 10^{14} M_{\odot}$. And yet, adding up the total stellar mass of the galaxies in the clusters only gives $\sim 10^{12} M_{\odot}$. This is reason number 94389244 that we believe dark matter exists -- most of the gravitating mass of galaxy clusters isn't visible!


Note that the SZ method isn't perfect. First of all, it's useless for all but the most massive dark matter halos in the universe. This is simply because the gas filling lower mass DM halos simply isn't hot enough to significantly Compton up-scatter CMB photons. Even in massive halos, there are some astrophysical issues which can get in the way. For example, we've assumed that gravity is the only factor at play in setting the temperature of the gas. However, in reality this simply isn't true. Supernovae and AGN can drive large-scale outflows of gas from the galaxy and into the halo.

\end{document} <\latex>