# Microlensing and Weak Lensing

## Contents

### Problem 1: microlensing

The microlensing occurs when a lens passes in front of a distant source. Consider the conditions of existence of such an effect and its main features.

First of all, microlensing occurs when angular distance between the images can not be resolved. This puts constraint on hte lens mass. Considering the typical case of lensing of distant star on the star from the Milky Way or local group. In this case the distance are $\sim 10\mbox{kpc}$ and masses are $\sim M_{\odot}$. The angular width of the Einstein ring can be estimated as $$\vartheta_E=\left({4GM\over c^2}{D_{DS}\over D_D D_S}\right)^{1/2}=4\cdot 10^{-7}\left({M\over M_{\odot}}\right)^{1/2}\left({10\mbox{kpc}\over D}\right)^{1/2}\sim 10^{-3}{}''$$ Such angular distances can not be resolved, so the only opportunity do detect lensing is through variation of visible brightness of the star. The duration of the effect depends on relative velocity of star and lens or, more precisely, on period of time needed for image to cross the Einstein ring. This period can be estimated as $$t_0={D_D\vartheta_E\over v}=0.214\mbox{yrs} \left(M\over M_{\odot} \right)^{1/2}\left(D_D \over 10\mbox{kpc} \right)^{1/2}\left(D_{DS}\over D_S\right)^{1/2}\left(200\mbox{km/s}\over v\right)$$ Assuming, for example, that lens is located in our galaxy and source is located in Large Magellanic Clouds, we can take $D_{DS}/D_S\sim 1$. Observing the brightness curves on time scales from one imnute to one year, one can detect lenses with masses $\sim 10^{-6}M_{\odot} - 10^2 M_{\odot}$. Note, however, that expression for microlensing duration only relates lens mass$M$, datsnaces and relative velocity. Furthermore, brightness curves must be symmetric and similar for all spectral bands, since lensing doesn't change wavelength.

### Problem 2: optical depth

In order to describe the probability of microlensing the notion of optical depth is used. Consider its physical sense. How many sources should one observe in order to register a few events per year?

Optical depth is a probability for a given star to be inside Einstein cone of the lens at an arbitrary point in time. Thus, optical depth is an integral of number density of lenses multiplied by the area of Einstein ring for each lens: $$\tau = {1\over \delta \omega}\int dV n(D_D)\pi \vartheta_E^2$$ Substituting the expression for Einstein cone angle: $$\tau = \int_{0}^{D_S}{4\pi G\rho\over c^2}{D_DD_{DS}\over D_S}dD_D={4\pi G\over c^2}D_S^2\int_{0}^{1}\rho(x)x(1-x)dx$$ where $x=D_D D_S^{-1}$. This quantity depends only on number density of lenses and not on their mass. Optical depth can be estimated as $\tau\sim 5\cdot10^{-7}$. Thus, millions of sources must be observed during the year to register dozens of lensing events.

### Problem 3: microlensing and dark matter

How can microlensing help to detect and characterize dark matter in a galactic halo?

Assume that dark matter in the halo of our galaxy is concentrated in compact objects with the masses similar to star's masses. Such objects are called MACHO (MAssive Compact Halo Object). In that case, MACHO's are gravitational lenses. When observing the stars in LMC, one can register a microlensing effect and, thus, put constraints on the masses of dark matter objects, forming the halo. The original idea of this method is developed by Pachinsky. Such investigation was performed in MACHO project. It was determined, that optical depth in the direction of LMC is $$ \tau_{LMC} = 1.2^{+0.4}_{-0.3}\cdot 10^{-7}. $$ Data from MACHO project suggest, that 20\% of halactic halo consist of compact objects with masses $0.15M_{\odot}\div 0.9M_{\odot}$.

### Problem 4: general relation for deflection angle

Obtain the general relation for reflection angle in the lens' field, if the surface density of the lens is given.

In general, given the surface density of the lens, one have $$\Sigma (\vec{\xi})=\int\rho(\vec{\xi},z)dz$$ where $z$ is a coordinate along the line of sight. Deflection angle is an integral of deflections, caused by infinitesimal mass elements: $$\vec{\hat{\alpha}}(\vec{\xi})={4G\over c^2}\int{(\vec{\xi}-\vec{\xi}')\Sigma(\vec{\xi}')\over |\vec{\xi}-\vec{\xi}'|^2}d^2\xi'$$ With critical density $$\Sigma_{cr}={c^2D_S\over 4\pi GD_DD_{DS}}$$ and convergence $$k(\vec{\xi})={\Sigma(\vec{\xi})\over\Sigma_{cr}}$$ one can rewrite the expression for deflection angle in the form $$\vec{\hat{\alpha}}(\vec{\xi})={1\over \pi}\int{(\vec{\xi}-\vec{\xi}')k(\vec{\xi})\over |\vec{\xi}-\vec{\xi}'|^2}d^2\xi'$$ This expression allows to easily introduce the scalar potential (previously rewriting the expression to observation angle terms): $$\psi(\vec{\vartheta})={1\over \pi}\int\ln(\vec{\vartheta}-\vec{\vartheta}') k(\vec{\vartheta}')d^2\vartheta'$$ and obtain the deflection angle in the form: $$\vec{\hat{\alpha}}=\nabla\psi(\vec{\vartheta})$$ $$k={1\over 2}\nabla^2\psi(\vec{\vartheta}).$$

### Problem 5: deflection Jacobian

Introduce the Jacobian of the transformation from the coordinates without lensing to the coordinates after lensing. What is its physical sense?

Jacobian for transformation from original coordinates to coordinates аfter lensing describe the local properties of the lensing. It has the form: $$ A\equiv {\partial \vec{\beta}\over \partial \vec{\vartheta}} = \delta_{ij} - {\partial^2\psi(\vec{\vartheta})\over \partial \vartheta_i \partial \vartheta_j} $$ Using the expression for $\Delta \psi$: $$ \Delta \psi = 2k $$ and introducing the quantity $$ \gamma \equiv \gamma_1+i\gamma _2 = |\gamma |e^{2i\phi} $$ we obtain $$ A=\left( \begin{array}{cc} 1-k-\gamma_1 & -\gamma_2\\ -\gamma_2 & 1-k+\gamma_1 \end{array} \right) $$ or, in more clear form: $$ A = (1-k)\left( \begin{array}{cc} 1& 0\\ 0& 1 \end{array} \right) - |\gamma|\left( \begin{array}{cc} \cos 2\phi & \sin 2 \phi \\ \sin 2\phi & -\cos 2\phi \end{array} \right) $$ Thus, one can conclude, that $k$ describes the amplification of the source without variation of its form, $|\gamma|$ describes the amplitude of the deformation, and $\phi$ its orientation.

### Problem 6: mass profile reconstruction

Propose a method to reconstruct the mass distribution basing on the measured surface brightness of the images.

Suppose, the surface brightness without lensing is known and equals $I_0(\vec\beta)$. The visible brightness after lensing is $I(\vec\vartheta) = I_0 (\vec\beta(\vec{\vartheta}))$. In weak lensing limit: $$ I(\vec\vartheta)\simeq I_0(\vec\beta_0+A(\vec\vartheta_0)(\vec\vartheta-\vec\vartheta_0)) $$ Mass distribution must be derived from measured $I(\vec\vartheta)$. Since the surface density is in general unknown, one must make reasonable assumptions regarding the distribution of source galaxies. First, their form can be assumed to be elliptic. Second, the angular distribution of galaxies we assume to be uniform, i.e. galaxies are directed in random. Under these assumptions and averaging, we can derive the observable quantity: $$ g(\vec\vartheta) = {\gamma(\vec\vartheta)\over 1-k(\vec\vartheta)}, $$ which can be used to restore surface distribution of mass. Due to its ability to restore the mass profile, weal lensing plays an important role in determining the constraints on the density of dark matter in the Universe. One of the most representative example of weak lensing was the detection of dark matter in 1Е0657-556 object, which is called the Bullet Cluster. It was formed as a result of collision of two star clusters. During the collision the three main components of the clusters, namely, visible star matter, hot interstellat gas and dark matter, separated from each other. It was reliably shown, that centers of mass, derived from the observations in visible light (star matter) and in X-ray range (hot gas), do not coincide with center of masss derived from gravitational lesing measurements.