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

LMW,B≃2.3×1010 L⊙,BL_{\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

    ⟨M/LB⟩≈4 M⊙/L⊙,B\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

      jstars,B=1.1×108L⊙,B Mpc−3j_{\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

    ρstars,B=1.1×108L⊙,B⋅4M⊙/L⊙,B≈4.4×108M⊙Mpc−3\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}
    Ωstars≈3×10−3\Omega_{\text{stars}}\approx3\times10^{-3}

Galaxy Rotation Curves

  • Spiral galaxy - thin disk rotating about the center

  • Galaxy rotation curve

    v(R)=vr(R)−vgalsin⁡i=vr(R)−vgal1−b2/a2v(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)=GM(R)Rv(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⁡(−RRs)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

        ⟨M/LB⟩≃50M⊙/L⊙,B(Rhalo100 kpc)\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⟨K⟩=⟨U⟩-2\langle K\rangle=\langle U\rangle

    where the brackets denote time averages

    • Kinetic energy

      K=12M⟨v2⟩K=\frac{1}{2} M\langle v^{2}\rangle

      where

      ⟨v2⟩≡1M∑imi∣x˙i∣2\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=−12G∑i,jj≠imimj∣xj−xi∣=−αGM2rhU=-\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=⟨v⟩2rhαG∼1.7×1015M⊙(σr1000kms−1)2(rh1Mpc)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)

      ⟨MLB⟩≈2×1015M⊙8×1012L⊙,B≈250M⊙L⊙,B∼5⟨MLB⟩MW\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

    dPdr=−GM(r)ρ(r)r2\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=ρkTμmpP=\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)=kT(r)rGμmp[−dln⁡ρdln⁡r−dln⁡Tdln⁡r]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

    MhydroComa=(1−2)×1015M⊙∼MvirComa≈2×1015M⊙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

        ξ≫2GMc2\xi \gg \frac{2 G M}{c^{2}}

The Lens Equation

α~(ξ)=4GM(ξ)c21ξ\tilde{\alpha}(\xi)=\frac{4 G M(\xi)}{c^{2}} \frac{1}{\xi}

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

  • Reduced deflection angle

    α(θ)=DLSDSα~(θ)⇒β=θ−α(θ)\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 $\xi=\mathrm{D}_{\mathrm{L}} \theta$

    β(θ)=θ−DLSDLDS4GMc2θ\beta(\theta)=\theta-\frac{\mathrm{D}_{\mathrm{LS}}}{\mathrm{D}_{\mathrm{L}} \mathrm{D}_{\mathrm{S}}} \frac{4 G M}{c^{2} \theta}

    when $\beta=0$

    θE=4GMc2DLSDLDS\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

    θEarcsec⁡=(M1011.09M⊙)1/2(DLDS/DLSGpc)−1/2\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, $\mathrm{D}{\mathrm{LS}} / \mathrm{D}{\mathrm{S}} \approx 1 / 2$

    θE=0.64×10−3arcsec⁡(MM⊙)1/2(DL10 kpc)−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

    β=θ−θE2θ⇒θ1,2=12(β±β2+4θE2)\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

    μ=θβdθdβ⇒μ1,2=(1−[θEθ1,2]4)−1=u2+22uu2+4±12\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

      mσ2=kTm\sigma^2=kT

  • 3-D density distribution

    ρ(r)=σv22πG1r2⇒ρ=ρc1+(rr0)2\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(ξ)=πσv2GξM(\xi)=\frac{\pi \sigma_{\mathrm{v}}^{2}}{G} \xi

  • The deflection angle

    α~(ξ)=4πc2σv2=1.4′′(σv220 km s−1)2\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

    α~(ξ)=4πc2σv2ξ(ξc2+ξ2)1/2\tilde{\alpha}(\xi)=\frac{4 \pi}{c^{2}} \sigma_{\mathrm{v}}^{2} \frac{\xi}{\left(\xi_{\mathrm{c}}^{2}+\xi^{2}\right)^{1 / 2}}
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