# Chapter 9 Dark Matter

## Introduction

* A **mismatch** between
  * The amount of matter known (or at least suspected) to be present
  * The amount of matter we see ina given environment or volume
* Baryonic dark matter
  * Stars & remnants (mostly WDs) account for < 10% of the baryons (deduced from the relative abundances of light elements created in BBN + fluctuations in CMB)
  * In the form of intergalactic gas (Ly $\alpha$ forests)
  * High temperature - WHIM (Warm-Hot Intergalactic Medium), FUV+X ray radiation
* Non-Baryonic dark matter

## Mass-to-Light Ratio

* In **B-band** ($\sim3800 \overset\circ{\text{A}}-4800 \overset\circ{\text{A}}$), measured in solar units $\mathrm{M}*{\odot} / \mathrm{L}*{\odot, \mathrm{B}}$

### The Milky Way

$$
L\_{\mathrm{MW}, \mathrm{B}} \simeq2.3 \times 10^{10}\ \mathrm{L}\_{\odot, \mathrm{B}}
$$

* Dependence of stellar luminosity on mass (quite steep) $L \propto M^{\sim 3.5}$
* Most of the luminosity of a stellar population is contributed by the most massive stars $M / &#x4C;*{\mathrm{B}} \sim 10^{-3} \mathrm{M}*{\odot} / \mathrm{L}\_{\odot, \mathrm{B}}$
* Most of the mass is in the far more numerous and fainter low-mass stars $M / &#x4C;*{\mathrm{B}} \sim 10^{3} \mathrm{M}*{\odot} / \mathrm{L}\_{\odot, \mathrm{B}}$
* Within 1 kpc from the Sun

  $$
  \left\langle M / L\_{\mathrm{B}}\right\rangle \approx 4\ \mathrm{M}*{\odot} / \mathrm{L}*{\odot, \mathrm{B}}
  $$

### Larger scale

* Integrating the luminosity function (LF, the number of galaxies per unit volume with luminosity in the range $\[L, L+d L]$) of galaxies within hundreds of Mpc from our location

  * Assumption 1 - the slope in the LF applies for unobserved lower massive galaxies
  * Total stellar luminosity

    $$
    j\_{\text {stars,} \mathrm{B}}=1.1 \times 10^{8} \mathrm{L}\_{\odot, \mathrm{B}}\ \mathrm{Mpc}^{-3}
    $$
  * If there are too many low mass galaxies, the integration might diverge
  * Assumption 2 - the mass-to-light ratio in the neighborhood of the solar system also applies to most of the galaxies

  Under these assumptions, the mass density is

  $$
  \rho\_{\text {stars,} \mathrm{B}}=1.1 \times 10^{8} \mathrm{L}*{\odot, \mathrm{B}} \cdot 4 \mathrm{M}*{\odot} / \mathrm{L}*{\odot, \mathrm{B}} \approx 4.4 \times 10^{8} \mathrm{M}*{\odot} \mathrm{Mpc}^{-3}
  $$

  $$
  \Omega\_{\text{stars}}\approx3\times10^{-3}
  $$

## Galaxy Rotation Curves

* Spiral galaxy - thin disk rotating about the center
* Galaxy rotation curve

  $$
  v(R)=\frac{v\_{\mathrm{r}}(R)-v\_{\mathrm{gal}}}{\sin i}=\frac{v\_{\mathrm{r}}(R)-v\_{\mathrm{gal}}}{\sqrt{1-b^{2} / a^{2}}}
  $$

  * $v\_{\mathrm{r}}(R)$ - velocity deduced from the Doppler shift of spectral lines from **stars/gas in the disk**
  * $v\_{\mathrm{gal}}$ - systemic velocity of the galaxy indicated by the Doppler shift of **the nucleus**
  * $i$ - inclination deduced from semi-major axis $a$ and semi-minor axis $b$
    * $i=0$ - face on
    * $i=\pi/2$ - edge on
* Newtonian gravity

  $$
  v(R)=\sqrt{\frac{G M(R)}{R}}
  $$

  * Inside the bulge, $M(R)\propto R^3$, $v\propto R$
  * Outside the bulge, $M(R)=M$, $v\propto R^{-1/2}$
  * For a distribution of matter in hydrostatic equilibrium, $\rho\sim R^{-2}$, $v=Constant$
* Evidence of non-baryonic dark matter
  * For most of the spiral galaxies, the rotational velocity first increases with galactic distance $R$ and then **flattens**
  * The integrated stellar light of the disks of spiral galaxies generally falls exponentially with $R$

    $$
    I(R)=I(0) \exp \left(-\frac{R}{R\_{\mathrm{s}}}\right)
    $$

    where $I$ is the surface brightness and $R\_\text{s}$ is a characteristic scale-length

    * Beyond $R>3R\_\text{s}$, only 3% of the light remains - mass of stars inside $R$ becomes essentially constant
    * The gravitational mass continues to increase
  * Dynamical mass of the Milky Way $&#x4D;*{\mathrm{dyn}}^{\mathrm{MW}} \simeq 1 \times 10^{12} \mathrm{M}*{\odot}$
    * Based on the motions od satellites orbiting our galaxy
    * High values of $M / L\_{\mathrm{B}}$ implied

      $$
      \left\langle M / L\_{\mathrm{B}}\right\rangle \simeq 50 \mathrm{M}*{\odot} / \mathrm{L}*{\odot, \mathrm{B}}\left(\frac{R\_{\mathrm{halo}}}{100\ \mathrm{kpc}}\right)
      $$

      which reflect the fact that the Milky Way disk lies at the centre of a spherical halo of non-baryonic dark matter
  * The inner regions (bulges) of spirals and ellipticals are dominated by baryons
  * Dwarf galaxies are dominated by dark matter - has led to searches for gamma-rays produced by the annihilation of particle-antiparticle pairs of some non-baryonic dark matter candidates
  * No evidence for dark matter in globular clusters

## Dark Matter in Galaxy Clusters

* Greatest concentration of matter in the Universe
* Doppler shifts
  * Systemic redshift
  * Velocity dispersion

### Virial Mass

* Relaxed cluster - no longer expanding nor contracting
  * Velocity dispersion - depth of the gravitational potential well within which dark matter, galaxies and intracluster gas move
* Virial theorem

  $$
  -2\langle K\rangle=\langle U\rangle
  $$

  where the brackets denote time averages

  * Kinetic energy

    $$
    K=\frac{1}{2} M\langle v^{2}\rangle
    $$

    where

    $$
    \langle v^{2}\rangle \equiv \frac{1}{M} \sum\_{i} m\_{i}\left|\dot{\mathbf{x}}\_{i}\right|^{2}
    $$

    is a 3-D velocity!
  * Potential energy

    $$
    U=-\frac{1}{2} G \sum\_{i, j \atop j \neq i} \frac{m\_{i} m\_{j}}{\left|\mathbf{x}*{j}-\mathbf{x}*{i}\right|}=-\alpha \frac{G M^{2}}{r\_{\mathrm{h}}}
    $$

    * $\alpha\sim0.4$ - obtained through simulations
    * $r\_\text{h}$ - half-mass radius, the radius of a sphere centered on the center of mass of the cluster and within which half of the cluster mass is contained
  * The dynamical mass of a cluster

    $$
    M=\frac{\langle v\rangle^{2} r\_{\mathrm{h}}}{\alpha G}\sim 1.7 \times 10^{15} \mathrm{M}*{\odot}\left(\frac{\sigma*{\mathrm{r}}}{1000 \mathrm{km} \mathrm{s}^{-1}}\right)^{2}\left(\frac{r\_{\mathrm{h}}}{1 \mathrm{Mpc}}\right)
    $$

    where $\sigma\_\text{r}$ is the one-dimensional velocity dispersion
  * The mass-to-light ratio of Coma (which has quenched)

    $$
    \left\langle\frac{M}{L\_{\mathrm{B}}}\right\rangle \approx \frac{2 \times 10^{15} \mathrm{M}*{\odot}}{8 \times 10^{12} \mathrm{L}*{\odot, \mathrm{B}}} \approx \frac{250 \mathrm{M}*{\odot}}{\mathrm{L}*{\odot, \mathrm{B}}}\sim5\left\langle\frac{M}{L\_\text{B}}\right\rangle\_\text{MW}
    $$

    Actually, the Milky Way $\sim10^{12}M\_\odot$ has very high star formation efficiency

![](https://1509032923-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MPMxe8Bu9WDT3p-DA_8%2Fsync%2Fc590a1744a48af4b2eea0f68025204cc4518e9c4.png?generation=1608873186548003\&alt=media)

### Hydrostatic Equilibrium

* Use temperature, density and chemical composition to determine the mass of clusters
* Equation of hydrostatic equilibrium

  $$
  \frac{d P}{d r}=-G \frac{M(r) \rho(r)}{r^{2}}
  $$

  $M$ is the total (DM + baryons) mass inside $r$

  For ideal gas

  $$
  P=\frac{\rho k T}{\mu m\_{\mathrm{p}}}
  $$

  where $\mu$ is the mean molecular weight (average mass per particle divided by $m\_\ce{H}$, $\sim0.6$ for a fully ionized plasma of solar composition)

  $$
  M(r)=\frac{k T(r) r}{G \mu m\_{\mathrm{p}}}\left\[-\frac{d \ln \rho}{d \ln r}-\frac{d \ln T}{d \ln r}\right]
  $$
* By tracking $T$, $\rho$ and the composition from the core to the outskirts

  $$
  M\_{\mathrm{hydro}}^{\mathrm{Coma}}=(1-2) \times 10^{15} \mathrm{M}*{\odot}\sim M*{\mathrm{vir}}^{\mathrm{Coma}} \approx 2 \times 10^{15} \mathrm{M}\_{\odot}
  $$
* SZ effect of CMB - from $n\_e$ to the abundance of baryons of the intracluster gas

## Gravitational Lensing

* A light ray traversing a region where the gravitational field has a gradient, for example near a point mass, will bend towards the mass
* Allow us to probe the distribution of matter in galaxies and in clusters independently of the nature of the matter

### Classification

* Depending on the projected distance between the light source and the lens

#### Strong lensing

* Occurs at **small angular separations** between source and lens
* Einstein rings, multiple images, highly distorted images, arcs

#### Weak lensing

* Occurs when the alignment between observer, lens and source is not close
* Slightly distorted single images of background galaxies
* Cosmic shear
  * Large-scale distribution of galaxies in the Universe acts as weak lens
  * Analysed by statistical means, averaging over many distorted galaxy images

#### Microlensing

* Occurs when two stars (at the appropriate distances from Earth) become closely aligned as seen from Earth due to their relative transverse velocities
* Usually images cannot be resolved (Subo NB)
* Give rise to a characteristic light curve - the background source increases in brightness on a timescale of $\sim1$ month
* Achromatic - no chromatic effects

### Basics

* Assumptions
  * No cosmic shear
    * Assume that the lensing action is dominated by a single matter inhomogeneity at some location between source and observer
  * Thin lens approximation
    * All the action of light deflection takes place at a single distance
    * lens
      * Halo of galaxies $\sim100$ kpc
      * Cluster of galaxies $\sim$ a few Mpc
    * Source-lens and lens-observer distances $\sim$ Gpc
  * Weak field approximation
    * Impact parameter is much greater than the Schwarzchild radius

      $$
      \xi \gg \frac{2 G M}{c^{2}}
      $$

#### The Lens Equation

$$
\tilde{\alpha}(\xi)=\frac{4 G M(\xi)}{c^{2}} \frac{1}{\xi}
$$

where $\tilde{\alpha}(\xi)$ is the deflection angle

![](https://1509032923-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MPMxe8Bu9WDT3p-DA_8%2Fsync%2F8b1811b2a4dbaffe62721f1ce680f15b44eaf271.png?generation=1608873186761009\&alt=media)

* Reduced deflection angle

  $$
  \alpha(\theta)=\frac{\mathrm{D}*{\mathrm{LS}}}{\mathrm{D}*{\mathrm{S}}} \tilde{\alpha}(\theta)\Rightarrow\beta=\theta-\alpha(\theta)
  $$

#### Einstein Radius

* Under small angle approximation $\xi=\mathrm{D}\_{\mathrm{L}} \theta$

  $$
  \beta(\theta)=\theta-\frac{\mathrm{D}*{\mathrm{LS}}}{\mathrm{D}*{\mathrm{L}} \mathrm{D}\_{\mathrm{S}}} \frac{4 G M}{c^{2} \theta}
  $$

  when $\beta=0$

  $$
  \theta\_{\mathrm{E}}=\sqrt{\frac{4 G M}{c^{2}} \frac{\mathrm{D}*{\mathrm{LS}}}{\mathrm{D}*{\mathrm{L}} \mathrm{D}\_{\mathrm{S}}}}
  $$

  A ring like image is formed
* $\beta\lesssim\theta\_\rm{E}$ - strong magnification
* $\beta\gg\theta\_\rm{E}$ - little magnification
* Strong lensing

  $$
  \frac{\theta\_{\mathrm{E}}}{\operatorname{arcsec}}=\left(\frac{M}{10^{11.09} \mathrm{M}*{\odot}}\right)^{1 / 2}\left(\frac{\mathrm{D}*{\mathrm{L}} \mathrm{D}*{\mathrm{S}} / \mathrm{D}*{\mathrm{LS}}}{\mathrm{Gpc}}\right)^{-1 / 2}
  $$
* Microlensing of stars in the bulge by a solar-mass disk star, $\mathrm{D}*{\mathrm{LS}} / \mathrm{D}*{\mathrm{S}} \approx 1 / 2$

  $$
  \theta\_{\mathrm{E}}=0.64 \times 10^{-3} \operatorname{arcsec}\left(\frac{M}{\mathrm{M}*{\odot}}\right)^{1 / 2}\left(\frac{\mathrm{D}*{\mathrm{L}}}{10\ \mathrm{kpc}}\right)^{-1 / 2}
  $$

#### Image positions and Magnifications

* Positions

  $$
  \beta=\theta-\frac{\theta\_{\mathrm{E}}^{2}}{\theta}\Rightarrow \theta\_{1,2}=\frac{1}{2}\left(\beta \pm \sqrt{\beta^{2}+4 \theta\_{\mathrm{E}}^{2}}\right)
  $$
* Magnification on fluxes - conserves surface brightness - the ratio between **the solid angles sustended by the image and the source**

  $$
  \mu=\frac{\theta}{\beta} \frac{d \theta}{d \beta}\Rightarrow \mu\_{1,2}=\left(1-\left\[\frac{\theta\_{\mathrm{E}}}{\theta\_{1,2}}\right]^{4}\right)^{-1}=\frac{u^{2}+2}{2 u \sqrt{u^{2}+4}} \pm \frac{1}{2}
  $$

  where $\mu\equiv \beta/\theta\_{\mathrm{E}}$

  * $\beta\to0$ - diverge (in the limit of geometrical optics)
  * $\theta<\theta\_\rm{E}$ - negative magnification - mirror inverted
  * Total magnification $\mu=|\mu\_1|+|\mu\_2|>1$

#### Singular Isothermal Sphere

* SIS
  * 1-D velocity dispersion of gas and stars is only weakly dependent on distance $r$ fromt the center
  * Treating the galaxy as 'gas' of stars with $p=\rho k T / m$
  * Isothermal

    $$
    m\sigma^2=kT
    $$
* 3-D density distribution

  $$
  \rho(r)=\frac{\sigma\_{\mathrm{v}}^{2}}{2 \pi G} \frac{1}{r^{2}}\Rightarrow\rho=\frac{\rho\_{\mathrm{c}}}{1+\left(\frac{r}{r\_{0}}\right)^{2}}
  $$

  The latter formula is obtained to avoid the singularity in the center, $r\_0$ is the core radius

  * $r\ll r*0, \rho=\rho*\rm{c}$
  * $r\gg r\_0$, the SIS behavior is recovered
* Total mass enclosed within $\xi$

  $$
  M(\xi)=\frac{\pi \sigma\_{\mathrm{v}}^{2}}{G} \xi
  $$
* The deflection angle

  $$
  \tilde{\alpha}(\xi)=\frac{4 \pi}{c^{2}} \sigma\_{\mathrm{v}}^{2}=1.4^{\prime \prime}\left(\frac{\sigma\_{\mathrm{v}}}{220\ \mathrm{km}\ \mathrm{s}^{-1}}\right)^{2}
  $$

  For cored model

  $$
  \tilde{\alpha}(\xi)=\frac{4 \pi}{c^{2}} \sigma\_{\mathrm{v}}^{2} \frac{\xi}{\left(\xi\_{\mathrm{c}}^{2}+\xi^{2}\right)^{1 / 2}}
  $$