This document discusses methods for probing dark matter using weak gravitational lensing and X-ray observations. It outlines the basic equations for gravitational lensing by point masses and extended mass distributions. Weak lensing of galaxies and clusters is approximated using a linearized treatment. Mass models like the singular isothermal sphere are introduced. X-ray methods assume hydrostatic equilibrium and use density models like isothermal or polytropic to estimate total mass from temperature profiles. Examples of mass reconstructions from lensing and estimates from X-rays are given.

1. Probing Dark Matter
with Weak Lensing and X-Rays

2. Lens Geometry
Deflection angle is given by,

3. Lensing by a Point Mass
Taking,
The lens equation is,
The equation becomes,

4. Lensing by a Point Mass
The rays undergo differential deflection,
Magnification for both images is
given by,

5. Weak Lensing by a Galaxy or a
Cluster
For a weak field,
If mass in galaxy is virialized,
Typical velocities in a galaxy ~ 200 km/s
Typical velocities in a galaxy cluster ~ 1000 km/s
For a weak field,
gravitational effects can be
linearized,
For a Geometrically Thin lens,
For continuous mass
distribution,

6. Weak Lensing by a Galaxy or
Cluster
Scaled Deflection is given by,
More convenient form,

7. Axially Symmetric Distribution
Deflection angle is given
by,
Scaled Deflection becomes,
Lens equation is given as,
For a point mass,
For a Singular Isothermal Sphere
(SIS),

8. Axially Symmetric Distribution
For SIS modelled Galaxy,
Putting this in the Lens equation,
For an initial estimate of mass,
For SIS modelled Galaxy Cluster,
Where,

9. Elliptical Lens

10. Mass Estimation
QSO 2237+0305 (Quasar) lensed by a foreground spiral galaxy (Einstein’s
Cross),
Mass within 0.9 arcsec (elliptical lens),

11. Mass Contours
Clowe et.al
Mass reconstruction for cluster A2204,

12. X-Ray Method (Clusters)
 Isotropic fluid (collision timescale < cooling or
heating timescale)
 Hydrostatic equllibrium with the potential (time
to cross < age)
 Spherical symmetry
On solving the hydrostatic equillibrium equation,

13. X-Ray Method
 Density Modelling (Isothermal)
 Velocity dispersion is related to core
density and radius as,
For isothermal case,
Where,
The emperical relation for
temperature,
Total mass comes out as,

14. X-Ray Method
 Density Modelling (Polytropic)
Hydrostatic equation becomes,
This gives,
Density as a function of temperature,

15. References
 Extragalactic Astronomy and Cosmology by Peter Schneider
 Clowe, D., and P. Schneider. "Wide field weak lensing observations of A1835
and A2204." Astronomy & Astrophysics 395.2 (2002): 385-397.
 Lea, Susan M., et al. "Thermal-Bremsstrahlung interpretation of cluster X-ray
sources." The Astrophysical Journal 184 (1973): L105.
 Bahcall, J. N., and C. L. Sarazin. "Parameters and predictions for the X-ray
emitting gas of Coma, Perseus, and Virgo." The Astrophysical Journal 213
(1977): L99-L103.
 https://ned.ipac.caltech.edu/level5/March02/Sarazin/