UNIVERSITY OF GLASGOW
Department of Physics and Astronomy
Astronomy A3/A4 Laboratory Projects
Friedmann Model Universes
Summary of Goals
1. To write a computer program capable of solving the Friedmann Equations for the evolution
of Friedmann Cosmological Models. Students are free to choose any suitable programming
language and platform (clearly acquiring programming skills in the chosen language is
another important goal of the project), although advice will be given on those choices which
make best use of Astronomy teaching resources and learning opportunities.
2. To apply this program to investigate the cosmological significance of the Friedmann
solutions – e.g. through exploring their theoretical dependence on cosmological parameters,
or through directly estimating those parameters by comparison of the models with real data.
Overview (See also A1Y Introduction to Cosmology and/or Honours Cosmology Notes)
The Hot Big Bang model has evolved over the past 50 years or so to now represent the standard
theoretical framework within which cosmologists understand the origin and evolution of the
Universe. Central to the Hot Big Bang model is the idea that the Universe originated in a primeval
fireball – a state of almost unimaginably high density and temperature – about 15 billion years ago,
and has been expanding and cooling ever since. Key evidence in support of the Hot Big Bang
model includes:
 The expansion of the Universe, first detected by Hubble in the 1920s
 The evolution of the Universe, as seen by the fact that the properties of very distant galaxies
(i.e. observed in the very distant past) are significantly different from galaxies we see today
 The abundances of light elements, believed to be ‘cooked’ during the first few minutes after
the Big Bang and showing excellent agreement between theory and observations
 The Cosmic Microwave Background Radiation (CMBR), which is the relic radiation
emitted at the time – approximately 300,000 years after the Big Bang – when the Universe
had cooled sufficiently to allow neutral hydrogen to form
Another key idea underpinning the Hot Big Bang Model is the Cosmological Principle – the
notion that the Universe is homogeneous and isotropic on large scales, however lumpy it may
appear in our immediate neighbourhood. The smoothness of the CMBR and the regularity of the
largest galaxy redshift surveys provide excellent observational evidence in support of the
Cosmological Principle. An immediate theoretical consequence of its validity is that we can
describe mathematically the evolution of the Universe1 in terms of a single function: the Scale
Factor, R (t ) , where t denotes the time since the Big Bang. More specifically the time evolution of
the Scale Factor can be determined by solving the Friedmann Equations. The principal aim of
this Laboratory Project is to write a computer program to solve numerically the Friedmann
Equations and investigate their solution as we vary the parameters of our cosmological model.
By Universe here we mean the homogeneous and isotropic background universe, which is the ‘backdrop’ on which
the structure which we observe in the Universe today evolved – through the influence of gravity. The properties of that
structure are of no concern to us here; the Friedmann model describes only the homogenous and isotropic background.
1
Possible Directions
To some extent this is an open-ended project: solution of the Friedmann Equations is the basic goal,
but there are many different directions in which the project may then proceed. These include:




Investigation of how the age of the Universe depends upon the cosmological parameters.
Investigation of the various types of solution for the evolution of the Scale Factor – e.g.
‘open’, ‘closed’ and ‘flat’ Universes; indefinite expansion, eventually re-collapse,
‘oscillatory’ Universes – depends on the cosmological parameters.
Investigation of how measurable properties of galaxies – e.g. apparent brightness, angular
diameter, number density – evolve as a function of redshift, for different values of the
cosmological parameters. (These quantities are used in the ‘classical tests’ of cosmological
models).
Comparison of recent cosmological data, from e.g. distant supernovae, with Friedmann
models – applying the classical tests to constrain the values of cosmological parameters.
Any or all of the above possibilities would (hopefully) make a stimulating and educational project
topic. Once the basic framework (i.e. a working code to solve the Friedmann Equations) is in
place, groups can discuss with staff and demonstrators options for its development and application.
(Groups are also welcome to consider their own ideas for specific directions in which the project
may proceed, and these ideas needn’t be set in stone at the beginning of the lab project).
Theoretical Background
A rigorous derivation of the Friedmann Equations follows from Einstein’s General Relativity, and
MSci students will encounter the basic ideas necessary to derive them in the A3/A4 Relativity and
Gravitation course. Here we simply state the equations, under the approximation that the Universe
is matter dominated and its matter content can be treated as a perfect fluid. They are:
2
 R 
8G  kc 2
  
  2
3
3 R
R
(1)

R
4G 
3P  

  2  
R
3 
c  3
(2)
Here  and P denote the mean matter density and mean pressure of the Universe respectively, c is
the speed of light, G is the gravitational constant,  is known as the cosmological constant (and
mathematically can be thought of as an integration constant in Einstein’s field equations of General
Relativity) and k is known as the curvature constant which defines the geometry of the Universe:
k  1 implies an open geometry; k  1 implies a closed geometry and k  0 implies a flat
geometry.

We define the Hubble parameter H (t )  R R , which measures the expansion rate of the Universe
(with present day value H 0 ). We assume first that   0 . From eq. (1) this means that
 8G 
k 0  
2
 3H 
1
  crit
(3)
 crit is the critical density: the density required to make the Universe flat. We also define
m  
 crit

8G

kc2





;
;

k
3H 2
3H 2
R2H 2
(4)
It follows immediately from eq. (1) that, at any time, t:
m    k  1
(5)
If we assume that the Universe is pressureless (i.e. P  0 , which is reasonable for the matterdominated regime) then we require only eq. (1) to solve for the evolution of the Scale Factor. We
also assume mass conservation, which implies:
3
R 3   0 R03  constant     0 R0
R3
(6)
Eq. (1) then can be re-written as:
8G 0 R03 R 2
R 2 

 kc2
3R
3
(7)
To make further progress we introduce new variables defined as follows:
YR
R0
; X  H 0 (t  t 0 )
(8)
Thus Y expresses the Scale Factor of the Universe in units of its present day value (note that we
also have the relation Y  (1  z ) 1 , where z is the redshift) and X measures the age of the Universe
in units of the present value of the Hubble time (the characteristic timescale for the expansion of
the Universe). Making the substitution
dR d
dY
dY dX
dY
  R 0 Y   R0
 R0
 R0 H 0
dt dt
dt
dX dt
dX
(9)
eq. (7) becomes, after some algebra:
8G 0 1
 2
kc2
 dY 

Y


 
3H 02 Y 3H 02
R02 H 02
 dX 
(10)

 dY 
2

  m 0    0Y  1   m 0    0 
Y
 dX 
(11)
2
or, using eqs. (4) and (5):
2
Here  m 0 is the present-day value of the dimensionless matter density and   0 is the present-day
value of the so-called dimensionless vacuum energy density. Note that the generic prediction of
inflationary models is that the dimensionless curvature density  k 0  0 (this is because the
rapid expansion during the inflationary phase of the very early Universe drives the geometry of the
Universe towards being flat – think of inflating the curved surface of a football, say, to the size of
the Earth: locally the geometry of the football would now appear to be flat, just as locally the Earth
appears flat). In inflationary models, therefore, the final term of eq. (11) vanishes. Note also that,
if   0    0  0 then eq. (11) has simple analytic solution:
Y  AX 2 / 3  C
(12)
where A and C are constants. Requiring Y  1 when X  0 , and then substituting for Y and X in
terms of R and t, it is easy to show that eq. (12) reduces (exactly as it should) to the Einstein de
Sitter solution of the Friedmann Equations, for a spatially flat critical density Universe, which was
derived in Astronomy A1Y.
Numerical Solution of the Friedmann Equations
In general, eq. (11) cannot be solved analytically. The object of this computational project is, in
essence therefore, to solve eq. (11) numerically for specified values of  m 0 and   0 , and explore
the nature of this solution as these parameters are varied. The solution is rendered simpler by the
adoption of the transformed variables Y and X , but it is of course straightforward to relate these to
the (more familiar) variables R and t.
For example, by integrating eq. (11) numerically for negative values of X , starting from X  0 , it
can be determined what value of X yields Y  0 . This, then, tells us the Age of the Universe (for
the chosen combination of cosmological parameters) in units of the Hubble time.
Some (reasonably simple) questions to think about:










How should we implement the numerical integration of eq. (11)? How should we choose,
e.g., the step size in X to ensure we obtain an accurate solution in a reasonable time?
How do we know whether to take the positive or negative square root of eq. (11) ?
Should we always take the same square root? If not, why not?
Why do we integrate eq. (11) for negative values of X to obtain the age of the Universe?
What would be the physical significance of integrating for positive values of X ?
Can you understand why, for an Einstein de Sitter Universe, Y  0 for X   2 3 ? (Consult
your A1Y notes for some inspiration!)
Strictly speaking, the age of the Universe derived by solving eq. (11) is incorrect, since the
Universe is not matter dominated all the way back to the Big Bang. Comment on the size of
error this will introduce into your determination of the age of the Universe?
Can you turn your solution into a relation between redshift and age of the Universe, or
redshift and lookback time?
Current observational evidence seems to suggest that the Universe is currently accelerating.
How would this acceleration show up in eq. (11) ?
How might it be best to present results showing e.g. the age or geometry of the Universe as
a function of the cosmological parameters?
Friedmann Model Universes: What next?
As discussed briefly above, once a code has been successfully implemented to solve eq. (11),
numerous directions are open for developing the project. In particular, several avenues involve the
comparison of Friedmann model predictions with observations of distant galaxies or supernovae.
This type of analysis will require the use of the Friedmann Equations to define theoretical Hubble
diagrams for e.g. high-redshift supernovae – these diagrams consist of curves of predicted
apparent magnitude (assuming SNIa are standard candles) versus redshift. Since the shape of
such a curve depends on the cosmological parameters, comparing the predictions with real
observations allows the parameters to be measured. This approach has been pioneered since 1995
in a worldwide effort led by two different research groups. In 1998 both groups reached the
(startling) conclusion that the Universe is accelerating, since its dynamics are now dominated by a
positive cosmological constant. In principle, students should be able to confirm this result for
themselves, using real data. If this direction is chosen for the project, further tutorial notes on the
construction of theoretical Hubble diagrams – and the definition of e.g. luminosity distance – will
be provided in due course.
Dr Martin Hendry
October 2005