**3.1.3 Cosmic time**

Time as one coordinate of the four-dimensional world was well
established when
Friedmann (1922)
introduced the concept of the
*time-dependence* of cosmic parameters, represented by the first and
second time derivatives of the scale factors *R (t)*, and thus the
possibility of measuring *cosmic time*.

Well aware of the fact that the cosmological constant cannot be determined independently:

``It should be remarked that the `cosmological' quantity remains indetermined in our formulae'',

Friedmann considered solutions for *t* in a world of positive curvature
for different values of . For
this he introduced the constant *A*,
which is directly proportional to the mass *M* of the universe, and
integrated his time-dependent form of Einstein's equations.

Friedmann's ``monotonous world of the first kind'' is described as follows:

``The time of growth of

Rfrom 0 toR_{0}we will callthe time since the creation of the world^{(5)}, this timet'is given by

The time since the creation of the (monotonous) world (of the first kind) considered as function of

R_{0},A, has the following properties:

1. it grows with growingR_{0}; it declines with increasinga, e.g., the mass in space decreases; 3. it decreases with increasing .''

Friedman's monotonous world of the second kind, for which it can also
be shown that *R* is an *increasing function of time*, starts from

The ``*periodic world*'' requires that lies within the limits (-, 0),
with those values excluded which lead to more than infinite period.

``Our knowledge is completely insufficient to permit number computation and to decide which world is ours . . . . With = 0 and

M= 5 x 10^{21}solar masses, we find that the world period is of the order 10 billion years. But these numbers can, of course, only serve as an illustration for our computations.''

The *expansion age* of the universe is given by the inverse Hubble
constant (defined below). The *evolutionary age* can be determined from
the evolutionary times of its oldest members plus the time from the
beginning of the universe to their formation. Two direct methods are
currently used:

- age of the oldest solid matter from our vicinity (meteorites, earth, moon) from radioactive dating

- age of the oldest members of the galaxy (globular clusters).

The difference between the observed *expansion age* and the observed
*physical age* of the universe has plagued cosmologists almost from the
beginning.