Present Age of the Universe:
Determination from Matter and Radiation Dominated
Models
Written and Researched By:
Jon Zizka &
Motoharu Kawazu
Introduction
In the beginning of time, the universe was smaller, hotter, and denser than it is now. The
expansion is successfully explained through the big bang theory. However, the fate of
the universe is uncertain. With matter and radiation, the universe’s expansion is subject
to a deceleration, but is the deceleration enough to stop expansion or is the universe going
to expand forever? How large a deceleration is based on the density of radiation,
luminous matter, and dark matter as they compare to the critical density? If the total
density of the universe exceeds the critical value, the universe is closed and at some time
in the future, it will collapse into the big crunch. If the total density is less than the
critical value, the universe is open and it will expand forever with a low deceleration. A
flat universe will have a density equal to the critical value resulting in an expansion with
critical deceleration. Today it is unclear how much an impact deceleration has on the
universe because the precise value is uncertain. All three conditions are under
investigation through ongoing research. Therefore, it is still unknown which theory if
any will prevail as the consensus of the universes’ fate.
Early theories of the universe started with scientists like Newton and Olbers. Newton
initially figured that since the universe is made of a finite number of stars, the objects in
the universe, under gravitational forces, should eventually come together to form a single
blob of matter. This has not occurred, so Newton concluded that the universe must be
static and possess infinite stars.
If there are galaxies in every direction, then our sky should be completely covered by
them. In fact, if the universe has no edge, then in every direction we look, our line of
sight should eventually encounter a star. So the sky should be bright everywhere, but we
know the sky is dark at night. This is known as Olbers’ Paradox.
The first way the paradox can be avoided is if there are no stars beyond some point.
However, we see no evidence of an edge to the universe.
There is an edge in another dimension – an edge in time. Even if the universe is infinite
in extent, but has a finite age, then we would see stars whose light has had time to reach
us within the limit set by the age of the universe. Under this condition, the maximum
distance that light can travel over the age of the universe is known as the cosmic horizon,
and the space inside of the cosmic horizon is the visible universe. As the universe ages,
the visible universe becomes bigger.
When we observe to large enough distances towards the cosmic horizon, we are looking
back in time when the universe was denser and hotter. However, the radiation has been
diluted and red shifted to long wavelengths by the expansion of the universe. Therefore,
when we observe the sky at night, we are looking at a bright sky, but in the radio region
of the electromagnetic spectrum.
The red shifting by the expansion of the universe is known as the cosmological red shift.
This is very similar to the Doppler Effect, but there is one main difference. Doppler shift
occurs from an object moving through space. Wavelengths are either increased or
shortened depending on where the object is moving towards or away from the observer.
The cosmological red shift also causes the wavelengths of electromagnetic radiation to
shift redward, but it is not caused by the motions of objects. It is the expansion of space
between objects that makes it seem as though the objects are moving. Relative to an
observer, the object has moved, but relative to space surrounding it, the object has
remained stationary. With the Doppler Effect, the object moves relative to the observer
and the space around it. Measurements of the cosmological red shifts can be used to
measure the way the universe is expanding (under a deceleration) and the age of the
visible universe.
Hubble used red shift to measure distances and velocities of many objects. He plotted the
velocity versus the distance for each object and found the following correlation:
v = Hod
The slope of this linear relationship, H0, is known as the Hubble constant. The age of the
universe is given by 1/H0, if the universe is flat, i.e. with critical deceleration. To obtain
a more precise value for the age of the universe we must include factors for deceleration.
With this in mind, A. Freidman derived a conditional relationship, which is known as the
Friedman equation. Modified for densities of radiation, luminous matter, and dark
matter, the equation takes the form,
2
kc 2
$ 1 % $ d! % 8" G
(# r + # m + # d ) & 2 2
' 2 ('
( =
3
Ro !
) ! * ) dt *
The quantity G is the gravitational constant, c is the speed of light, and ! r , ! m , and
! d are the densities of radiation, matter, and dark matter respectively. The value of ! is
the ratio
!=
R(t )
Ro
where Ro is the distance between two objects today, and R(t) is the distance at some other
time t.
The densities at time t can be expressed in terms of present values through ! . They are
!
! r (t ) = r4
"
!
! m (t ) = m3
"
! d (t ) = ! d
The quantity k is a parameter which can equal 1, 0, or -1. This is a measurement of the
curvature of the universe where 1 corresponds to convergence (high deceleration) or a
closed space, 0 means a steady flat universe, and -1 indicates divergence (low
deceleration) or open space. We can define normalized densities, the ratios of each
density to the critical density, according to the following relations:
!r
!c
!
"m = m
!c
!
"d = d
!c
! = !r + !m + !d
"r =
Solutions to the Friedman equation yield a host of theoretical models that can be used to
interpret the fate of the universe. This project solves the equation for the expected age of
the universe for two models, a matter dominated universe and a radiation dominated
universe, in terms of the normalized densities. An interpretation of the results will be
presented and comparisons to dating techniques through radioactivity, globular clusters,
and white dwarfs.
First, the age of the universe will be derived for the matter dominated universe. Here, the
dark matter and radiation densities are set equal to zero:
!d = !r = 0
Using the density relations in terms of a ! with ! =
R(t )
, we get
Ro
3
! #R $
! m = o3 = % o & !c
"
' R(
Combining the conditions of density and this relation with the Friedman equation yields
2
3
1 # dR $ 8! G # Ro $ kc 2
"o & ' % 2
&
' =
R 2 ( dt )
3
( R) R
This expression can be simplified by multiplying through by R 2 and dividing through
by Ro 2 , resulting in
2
1 # dR $ 8! G # Ro $ kc 2
"o & ' % 2
&
' =
Ro 2 ( dt )
3
( R ) Ro
(1)
Since the Hubble constant can be expressed in terms of the expansion parameter as
1 d!
H=
, equation (1) can be rewritten as
! dt
H2 =
Replacing ! by
8! G "
kc 2
$ 2 2
3
Ro #
R
and dividing through by H 2 gives
Ro
1=
8! G "
kc 2
#
3H 2
R2 H 2
Finally using the normalized density and that fact that !c =
kc 2
= H 2 (! "1)
2
R
Combining equation (1), (2), and
3H 2
8" G
(2)
R
= 1 + z (Kolb 52), the integral for the age of the
R0
universe can be expressed as
t=#
R (t )
0
(1+ z )!1
dR
dx
= H !1 #
0
dR / dt
(1 + " + "x !1 )1/ 2
According to Kolb, this integral when evaluated with z = 0 gives the present age of the
universe according to
H o to =
" !1
$o
2
!1
1/2 #
cos
(2
$
!
1)
!
(
$
!1)
%
& for !o >1
o
o
2($o ! 1)3/ 2 '
$o
(
(3.1)
H o to =
$o
2(1 ! $o )3/ 2
" 2
#
(1! $o )1/2 ! cosh !1 (2$o !1 ! 1) & for !o <1
%
' $o
(
(3.2)
H oto = 2 / 3 for !o =1
(3.3)
The matter dominated model has a discontinuity at !o =1. Therefore we find three
conditional equations for the solution. All values of ! , t, and H become !o , to , and H o ,
because these are the values that correspond to z = 0. Plotting H oto versus !o will show
the relationship between the age of the universe for the matter dominated model with
differing normalized densities.
The derivation for the radiation dominated model is very similar to that of the matter
dominated. Here, the matter and dark matter densities are set equal to zero.
!d = !m = 0
4
! #R $
Also, ! m = o4 = % o & !c and when this equation is combined with the Friedman
"
' R(
equation and simplified, we get
2
2
1 # dR $ 8! G # Ro $ kc 2
"o & ' % 2
&
' =
Ro 2 ( dt )
3
( R ) Ro
Using equation (4), (2), and
(4)
R
= 1 + z (Kolb 52), the following integral can be solved for
R0
the age of the universe:
t=#
R (t )
0
(1+ z )!1
dR
dx
= H !1 #
0
dR / dt
(1 + " + "x !2 )1/ 2
Again according to Kolb, this integral when evaluated with z = 0 gives the present age of
universe in terms of the normalized density parameter. This relationship is
H o to =
!o " 1
!o "1
(5)
Plotting equation (5) will show the relationship between the age of the universe for the
radiation dominated model with differing normalized densities.
Numerical Method
The programming used to generate the graphs is not complicated. Thus a simple “for
loop” can be used to produce this graph. For the matter dominated model, the program
had to be built around the discontinuity at !o =1. A loop routine was used to generate
values of the dependent variable H oto and then plotted. The simplification of the
program is because we are working with closed form relations as solutions of the
Friedman equation for the present age of the universe. The integration is done
analytically in order to eliminate rounding errors of numerical methods.
The programming code is presented in Appendix A. The code for the graph of the
radiation dominated universe just included a “for loop” for one simple equation.
However, for the matter dominated universe, there are three equations that need to be
considered due to the singularity when !o = 1. This can be done by writing two separate
loops. The first loop will include the values where !o is less than one. The second will
include those values where !o is greater than one. There is only one value when !o is
equal to one so no for loop is needed for that. Each of these “for loops” has a separate
“print” function for the values. They must be programmed in consecutive order. First
the “for loop” is made for !o < 1, then for !o = 1, and finally for !o > 1. If this
programming is done in any other order, the graph will not look as it should. These
values can all be plotted on one graph so it illustrates a continuous smooth function that is
actually a function defined through piecewise functions.
In addition, there is no need to convert the equations into dimensionless form. The
normalized density value, !o , is already a dimensionless number since it is a ratio of two
densities. Also, H oto , is dimensionless because to has units of time, and H o has units of
1/time. Thus when they are multiplied, the unit of time cancels out leaving the quantity
dimensionless.
Results
A new arbitrary variable, y, will be introduced here. Let y equal the following values.
For a matter-dominated universe,
" !1
#
$o
2
cos (2$o !1 ! 1) !
($o !1)1/2 & for !o >1
3/ 2 %
2($o ! 1) '
$o
(
" 2
#
$o
y=
(1! $o )1/2 ! cosh !1 (2$o !1 ! 1) & for !o <1
3/ 2 %
2(1 ! $o ) ' $o
(
y=
y = 2/3 for !o =1
For a radiation dominated universe,
y=
!o " 1
!o "1
This is done to condense these equations into one equation that can characterize both
graphs at all points. In general, H oto = y . This can be rearranged in the form:
to =
y
Ho
Since H o , the Hubble constant, is indeed a constant value, then for increasing values of y,
the age of the universe will also become older. The following are the graphs for a
radiation and matter dominated universe.
Figure 1.1
!o
Figure 1.2
!o
For both graphs, as the value of !o increases, the value of y and H oto decrease which
means the age of the universe decreases as well. The graphs have the same curves.
However, the matter dominated universe has a curve that is shifted up on the graph
towards higher values of H oto . The validity of these graphs compare nicely to Kolb’s
graphs in The Early Universe (54), demonstrating that the codes are working properly.
Discussion
These graphs show many interesting results. If Ωo is zero,
t0 H o = 1
leading to
t0 =
1
,
H0
which is the expected age of the universe without any decelerating effects due to the lack
of matter and radiation. In this case, to=14 billions years if the Hubble constant equals 71
km/s/Mpc.
According to both models, the age of the universe drops with increasing density. At
Ωo=1 (flat universe), the age of the universe is 0.67(1/H0) for the matter dominated
model and 0.5(1/H0) for the radiation dominated model or 9.33 billion years and 7
billions years respectively. A flat universe (Ωo = 1) would be younger than an open
universe (Ωo<1) and older than a closed universe (Ωo>1).
Current estimates show that the universe is nearly flat and therefore Ωo is about unity
because the fantastic stretching of space during the inflationary period rendered the
observable universe geometrically flat, even if the whole universe were highly curved.
According to Physics at FSU, the inflationary theory shows that around 10-34 second after
formation, the universe experienced hyper-expansion. At about 10-24 second, the hyperexpansion stopped leaving the universe about 100 million light-years in size. Afterwards,
big bang evolution took over. The bit of universe we see today would have at that time
been only a few centimeters across.
There are at least three ways that the age of the universe can be measured. These are the
age of the chemical elements; the age of the oldest globular clusters; and the age of the
oldest white dwarf stars.
The age of the Milky Way, which would give a lower limit to the age of the universe, is
assessed through radioactive dating the Rhenium-187 and Osmium-187 ratio. Rhenium187 decays into Osmium-187 with a half-life of 40 billion years. Current research shows
that about 15% of the original Re-187 has decayed, which leads to an age of 8-11 billion
years (Wright).
Other estimates of the universe’s age through radioactive dating have been done by
comparing the abundance of Uranium-238 in the Earth to Supernovae and using the
production rate of the Uranium-238 to Thorium-232 ratio with the ratio observed in very
old, metal poor stars (Wright). Both approaches lead to the universe being 14.5 billion
years old.
The age of a globular cluster can be inferred from the turnoff point in its HertsprungRussell diagram. The turnoff point is the location where stars have reached their terminal
age on the main sequence and are about to move towards the giant region as they change
their source of energy from hydrogen fusion to helium fusion. Since all stars in the
cluster were formed at the same time, the time inferred from the turnoff is the age of the
cluster. The oldest globular clusters are between 12 and 14 billion years old, which sets a
lower limit to the age of the universe (Wright).
A white dwarf is the core of a red giant star that is revealed after the envelope is expelled
late in the stars life. Afterwards, the white dwarf cools since the temperature is not
maintained through nuclear processes. The oldest white dwarfs will be the coolest and
thus the faintest. By searching for faint white dwarfs, one can estimate the length of time
the oldest white dwarfs have been cooling. Cool white dwarfs in the globular cluster M4
have been reported to be 12.7 billion years old (Wright).
It appears that the universe is older than the ages given by the matter and radiation
models at Ωo=1. There can be a number of reasons for the inconsistency. The Hubble
constant has been reported to range from 50 km/s/MPC to 100 km/s/Mpc. If the Hubble
constant is closer to 50 km/s/Mpc, the models would be more consistent with dating
techniques. The models assume that 100% of the density is in the form of luminous
matter or radiation. Current research shows that 27% of the matter is luminous and 73%
is dark matter. Therefore, the solution to the Friedman equation should include the dark
matter density. In the beginning of the introduction, it was assumed that ! d = 0. If
instead, this value is assumed non zero, the Freidman equation used in this derivation to
map the age of the universe would be modified to resemble the following:
2
2
1 % dR & 8! G #
% Ro & $ kc
"d + "o ( ) + ' 2
(
) =
Ro 2 , dt 3 *.
, R - / Ro
The values of the density of matter are higher than those of radiation, and the age of the
universe if too young according to a matter dominated universe. Thus this eliminates the
possibility of a radiation dominated universe since its age would be even lower than a
universe which is matter dominated. Adding another density as a factor in this equation
and derivation will increase values of Ωo. Since matter has a higher density than
radiation, the matter dominating universe will have an older age. This was proved from
the results shown in the graphs (Figures 1.1 and 1.2). The matter dominated universe is
shifted up on the graph relative to the radiation dominated. Thus if a substance is added
to the equation with a higher density than luminous matter, the curve will be shifted even
higher than the matter dominating curve resulting in an older universe. The exact age
that the universe would be with dark matter can not be calculated here. However, the age
of the universe would be greater with dark matter which would correspond more with
observational values of the age of the universe.
Conclusion
We used the Friedman equation to determine the present age of the universe for matter
dominated and radiation dominated models. Graphs of the results generated with
computer code (See Appendix A) show that the age decreases with increasing density.
At no density, the relationships give an age equivalent to 1/H0, a result that is obtained
with no deceleration effects. At a normalized density of 1 (close to what is measured),
the flat universe, the present age is 9.33 billion years for the matter dominated model and
7 billion years for the radiation model for a Hubble Constant of 71 km/s/Mpc. Both
values are low compared to dating through radioactivity studies and to the ages of the
oldest globular clusters and white dwarfs.
If the Hubble constant is closer to 50 km/s/Mpc, the matter dominated model would have
an age of 13 billion years and would be in better agreement with dating studies. At this
Hubble constant value, the radiation dominated model would still be inconsistent.
If the influence of dark matter were to be included in the matter dominated model,
estimates show that the curve on the graph for the matter dominated model would shift
upwards (Kolb), resulting in a larger present age at a normalized density equal to one.
Obtaining a precise present age of the universe is challenging since there are many
factors that lead to uncertainties in the Hubble constant, the normalized density
parameters for radiation, luminous matter, and dark matter. Moreover, many of the
dating techniques give lower limits to the age for comparison. The primary result of this
project was not to find precise results, but to show the behavior of the present age of the
universe for two restricted models and by doing so, the results suggest the existence of
dark matter.
Appendix A: Codes
1. Radiation Dominated
2. Matter Dominated
Appendix B: References
Bennett, Jeffrey, et al. The Cosmic Perspective. San Francisco: Pearson, 2007.
Kaufmann III, William J., and Roger A. Freeman. Universe: Fifth Edition. New York:
Freeman, 1998.
Kolb, Edward W., and Michael S. Turner. The Early Universe. Redwood City: Addison,
1990.
Seeds, Michael A. Foundations of Astronomy. Austrailia: Thomson, 2005.
Wright, Edward L. “The Age of the Universe.” UCLA Division of Astronomy and
Astrophysics. 2 Jul 2005 < http://www.astro.ucla.edu/~wright/age.html>
Zeilik, Michael, and Stephen A. Gregory. Introduction Astronomy and Astrophysics.
Austrailia: Thomson, 1998.
“The Age of the Universe.” Physics at FSU. Spring 1998.
<http://www.physics.fsu.edu/Courses/Spring98/AST3033/Universe/age.htm>
“How Old is the Universe.” WMAP Cosmology 101: Age of the Universe. 15 Dec 2005
< http://map.gsfc.nasa.gov/m_uni/uni_101age.html>