Introduction to Cosmology

Structure Formation 233

Problems

1. The mean free path퓁of photons in homogeneous interstellar dust can be found
   from Equation (1.4) assuming that the radius of dust grains is 10−^7 m. Extinction
   observations indicate that퓁≈1kpc at the position of the Solar System in the
   Galaxy. What is the number density of dust grains [4]?


2. To derive Jeans wavelength휆Jand Jeans mass푀J[see Equation (10.22)], let
   us argue as follows. A mass푀J=휌휆^3 Jcomposed of a classical perfect gas will
   collapse gravitationally if its internal pressure푃=휌kT∕푚cannot withstand the
   weight of a column of material of unit area and height휆J.Here푚is the mean
   mass of the particles forming the gas. If we set the weight of the latter greater
   than or equal to푃,
   퐺푀J
   휆^2 J

휌휆J≳

휌kT

푚

,

we will obtain a constraint on the sizes of fragment which will separate gravi-

tationally out of the general medium. Show that this leads to Equation (10.22)

[5].

3. The universal luminosity density radiated in the blue waveband by galaxies is
   퐿U=( 2 ± 0. 2 )× 108 ℎ퐿⊙Mpc−^3.

Show that the Coma value훶=푀∕퐿=300 in solar units then gives훺m= 0 .30.

4. Assuming that galaxies form as soon as there is space for them, and that
   their mean radius is 30ℎ−^1 kpc and their present mean number density is
   0. 03 ℎ^3 Mpc−^3 , estimate the redshift at the time of their formation [2].

References

[1] Planck Collaboration: Ade, P. A. R.et al. 2014 Astron. Astrophysand preprint arXiv:
1303.5076 [astro-ph.CO].
[2] Hawkins, E.et al. 2003 Mon. Not. R. Astron. Soc. 346 , 78.
[3] Peebles, P. J. E. 1993Principles of physical cosmology. Princeton University Press,
Princeton, NJ.
[4] Shu, F. H. 1982The physical universe. University Science Books, Mill Valley, CA.
[5] York,D.G.et al. 2000 Astr. J. 120 , 1579.