Given a population of objects with comoving number density n (z)
(number per unit volume) and cross section 
(z) (area), what is
the incremental probability dP that a line of sight will intersect
one of the objects in redshift interval dz at redshift z?
Questions of this form are asked frequently in the study of QSO
absorption lines or pencil-beam redshift surveys. The answer is
(Peebles 1993,
pp. 323-325). The dimensionless differential
intersection probability is plotted in Figure 7.
Figure 7. The dimensionless differential
intersection probability dP / dz; dimensionless in the sense of
n (z)
Roger Blandford, Ed Farhi, Jim Peebles and Wal Sargent all contributed
generously to my understanding of this material and Kurt Adelberger,
Lee Armus, Deepto Chakrabarty, Alex Filippenko, Andrew Hamilton,
Phillip Helbig, Wayne Hu, John Huchra, Daniel Mortlock, Tom Murphy,
Gerry Neugebauer, Adam Riess, Paul Schechter and Douglas Scott caught
errors, suggested additional material, or helped me with wording,
conventions or terminology. I thank the NSF and NASA for financial support.
(z)
DH = 1.
The three curves are for
the three world models, (
M, 
) =
(1, 0), solid; (0.05, 0), dotted; and (0.2, 0.8), dashed.
Acknowledgments