falloff can be greater or less than 1/a . So the relationship between observed brightness and distance depends on the geometry of the universe. If A = 0, and q0 > 0, and if z is not too large, then the relationship between distance modulus and redshift is given by m - M = 25 + 5 log10(cz/H0) + 1.086z (1 - q0) (20.38)

where q0 is the current value of the deceleration parameter. In this expression, cz/H0 is in mega-parsecs, accounting for the factor of 25 in front (see Problems 20.14 and 20.15).

The geometry of the universe also determines how the apparent angular size varies with distance. For flat geometry, and an object that is not too distant by cosmological standards, the angle subtended by an object of length L at distance D is L/D (in radians) as long as L V D. For an object at a cosmologically significant distance, the angular size is

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