Pick one of the 4 projects below, write a report similar to HW6 and
prepare a 10-15 min presentation to be given in class on Dec 5, Dec 10
or Dec 12 (we will decide the order of the presentations later in
class). The projects are described in some detail in the next pages.

You can work in small groups if you prefer (2-3 people) but each of you have to produce an independent report and presentation (or a section of a longer report/presentation). If you work in a group you are expected to produce a commensurately larger amount of work than if you work on your own (the grade will reflect the level of effort).

**Project 1: Your first cosmological simulation of a black hole!****Project 2: Parallelization of an N-BODY code using OPENMP****Project 3: Parallelization of an N-BODY code using CUDA****Project 4: 3D volume rendering**

(also known as secondary infall problem)

In order to solve this problem you need to generalize the integration
package you wrote for PS5 so that it can solve the -body problem in
an expanding Universe assuming spherical geometry and fixed time steps.
The following ODEs should be integrated in order to solve the
problem (see also my handwritten notes):

(1) | |||

(2) |

This equations describe the motion of spherical shells of dark matter around the black hole in comoving coordinates: is the peculiar velocity of a shell at comoving distance from the black hole and is the difference between the mass within at time and the original mass at time when the Universe was homogeneous (constant density). To convert the distances in physical units use the formula: , where the redshift depends on time as and where Gyr is the age of the Universe. The Hubble parameter in a flat matter dominated Universe is: , where km/s.

Alternatively you can integrate the equation as a function of redshift
rather than time (it may be easier). Using the
relationship you get the following equation:

(3) | |||

(4) |

Assume initial conditions as follows: start the simulation at redshift and assume that the dark matter has constant density g/cm. Remember to place a black hole of mass M at . If I failed to provide enough explanations to solve the problem feel free to ask me. Have fun!

*Hints: The dark matter should create a spherical halo around the
black hole with mass
and have a total mass
within the halo radius that increases with decreasing redshift as
.*