The Hubble constant H_{0} is the constant of proportionality between recession speed v and distance d in the expanding Universe;
The subscripted ``0'' refers to the present epoch because in general H changes with time. The dimensions of H_{0} are inverse time, but it is usually written
where h is a dimensionless number parameterizing our ignorance. The inverse of the Hubble constant is the Hubble time t_{H}
and the speed of light c times the Hubble time is the Hubble distance D_{H}
These quantities set the scale of the Universe, and often cosmologists work in geometric units with c = t_{H} = D_{H} = 1.
The mass density of the Universe and the value of the cosmological constant are dynamical properties of the Universe, affecting the time evolution of the metric, but in these notes we will treat them as purely kinematic parameters. They can be made into dimensionless density parameters _{M} and _{} by
(Peebles 1993, pp. 310-313), where the subscripted ``0''s indicate that the quantities (which in general evolve with time) are to be evaluated at the present epoch. A third density parameter _{k} measures the ``curvature of space'' and can be defined by the relation
These parameters totally determine the geometry of the Universe if it is homogeneous, isotropic, and matter-dominated. By the way, the critical density = 1 corresponds to 7.5 x 10^{21} h^{-1} M_{} D_{H}^{-3}, where M_{} is the mass of the Sun.
Most theorists believe that it is in some sense ``unlikely'' that all three of these density parameters be of the same order, and we know that _{M} is significantly larger than zero, so many guess that (_{M}, _{}, _{k}) = (1, 0, 0), with (_{M}, 1-_{M}, 0) and (_{M}, 0, 1-_{M}) tied for second place. If _{} = 0, then the deceleration parameter q_{0} is just half _{M}, otherwise q_{0} is not such a useful parameter. When I perform cosmographic calculations and I want to cover all the bases, I use the three world models
name | _{M} | _{} |
Einstein-de Sitter | 1 | 0 |
low density | 0.05 | 0 |
high lambda | 0.2 | 0.8 |
These three models push the observational limits.