**3.2. Age of the universe and cosmological constant**

From equation (24) we can also determine the current age of the universe by the integral

(34) |

Since most of the contribution to this integral comes from late times,
we can ignore the radiation term and set
_{R}
0.
When both
_{NR} and
_{} are present and
are arbitrary, the age of the universe is determined by the integral

(35) |

The integral, which cannot be expressed in terms of elementary
functions, is well approximated by the numerical fit given in the second
line. Contours of constant *H*_{0}*t*_{0}
based on the (exact) integral are shown in
figure 11.
It is obvious that, for a given
_{NR}, the
age is higher for models with
_{}
0.

Observationally, there is a consensus
[49,
50]
that *h* 0.72 ±
0.07 and
*t*_{0}
13.5 ± 1.5 Gyr
[83].
This will give *H*_{0} *t*_{0} = 0.94 ± 0.14.
Comparing this result with the fit in (35), one can immediately draw
several conclusions:

- If
_{NR}> 0.1, then_{}is non zero if*H*_{0}*t*_{0}> 0.9. A more reasonable assumption of_{NR}> 0.3 we will require non zero_{}if*H*_{0}*t*_{0}> 0.82. - If we take
_{NR}= 1,_{}= 0 and demand*t*_{0}> 12 Gyr (which is a conservative lower bound from stellar ages) will require*h*< 0.54. Thus a purely matter dominated = 1 universe would require low Hubble constant which is contradicted by most of the observations. - An open model with
_{NR}0.2,_{}= 0 will require*H*_{0}*t*_{0}0.85. This still requires ages on the lower side but values like*h*0.6,*t*_{0}13.5 Gyr are acceptable within error bars. - A straightforward interpretation of observations suggests maximum
likelihood for
*H*_{0}*t*_{0}= 0.94. This can be consistent with a = 1 model only if_{NR}0.3,_{}0.7.

If the universe is populated by dust-like matter (with *w* = 0) and
another component with an equation of state parameter
*w*_{X}, then the age
of the universe will again be given by an integral similar to the one in
equation (35) with
_{}
replaced by
_{X}(1 +
*z*)^{3(1+wX)}. This will give

(36) |

The integrand varies from 0 to
(_{NR} +
_{X})^{-1/2} in the range of
integration for *w* < 0 with the rapidity of variation decided
by *w*. As a result, *H*_{0} *t*_{0}
increases rapidly as *w* changes from 0 to -3 or so and then
saturates to a plateau. Even an absurdly negative value for *w*
like *w* = - 100 gives *H*_{0} *t*_{0} of
only about 1.48. This shows that even if some exotic dark energy is
present in the universe with a constant, negative *w*, it cannot
increase the age of the universe beyond about
*H*_{0} *t*_{0}
1.48.

The comments made above pertain to the current age of the universe.
It is also possible to obtain an expression similar to (34) for the age
of the universe at any given redshift *z*

(37) |

and use it to constrain
_{}.
For example, the existence of high redshift galaxies with evolved stellar
population, high redshift radio galaxies and age dating of high redshift
QSOs can all be used in conjunction with this equation to put constrains
on _{}
[84,
85,
86,
87,
88,
89].
Most of these observations require either
_{}
0 or
_{tot} <
1 if they have to be consistent with
*h* 0.6.
Unfortunately, the interpretation of these observations at present requires
fairly complex modeling and hence the results are not water tight.