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 $M / L{\mathrm{B}} \sim 10^{-3} \mathrm{M}{\odot} / \mathrm{L}_{\odot, \mathrm{B}}$

• Most of the mass is in the far more numerous and fainter low-mass stars $M / L{\mathrm{B}} \sim 10^{3} \mathrm{M}{\odot} / \mathrm{L}_{\odot, \mathrm{B}}$

• 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 $M{\mathrm{dyn}}^{\mathrm{MW}} \simeq 1 \times 10^{12} \mathrm{M}{\odot}$

    • 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

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

• 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

  • $r\ll r0, \rho=\rho\rm{c}$

  • $r\gg r_0$, 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

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