The Dark Stars

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Problem 1

Estimate how much will the period of rotation of the Earth around the Sun change in one year due to gravitational capture of dark matter particles.

Problem 2

Estimate the rate of energy outcome due to the WIMP-annihilation process using the parameter values $m_{WIMP}=100GeV$ and $\langle \sigma v \rangle_{ann}=3\cdot 10^{-26} cm^3/sec$ for the annihilation cross section.

Problem 3

It is theorized that the dark matter particles' annihilation processes could be a competitive energy source in the first stars. Why did those processes play an important role only in the early Universe and why are they not important nowadays?