objects moving in straight paths through
curved space. This can be likened to a
basketball sitting on a rubber sheet, it causes
a well in the sheet so a tennis ball that tries to
travel straight will end up moving in a line that
curves towards the basketball (Figure 1).
ince the dawn of history, humans
have been wondering about their
fate in the cosmos, and have over
time have developed new tools to
answer questions about the world and the
universe. One natural question to ask is “Will
we ever explore the whole universe?” or the
more basic question, “Can we explore the
whole universe?” With Einstein’s discovery of
general relativity, this question could be
answered quantitatively and was studied
extensively.
Background
General relativity is a theory describing how
objects move through space and time (or
space-time), and how space-time curves due
to matter. Its main revelation is that gravity
isn’t a conventional force, it’s just the
consequence of
Figure 1: A 3D representation of general relativity
This theory doesn’t only work with objects
though, it can be used to calculate the
behaviour of the universe. The problem with
general relativity is that the equations used
1
are notoriously hard to solve but in the 1920s,
Friedman, Robertson and Walker came up
with a solution for a universe that is both
homogenous and isotropic, meaning it is the
same and looks the same at every point (see
The FRW Model). On the large scale, this is
similar to our observations of our own
universe so this model turned out to be very
useful for calculating the how the universe
would expand, giving a simple term, called the
“scale factor” which determines the size of
the universe over time.
density, the universe would slow down in its
expansion till it almost stops, but it would
keep on going. Lastly, if there was too much
matter in the universe, it would expand but
then slow down and begin to contract, ending
in a “Big Crunch” (Figure 2). It was observed
that the amount of matter was very close to
the critical density.
The FRW Model
The FRW model is based on the following
derived equation for the scale factor “a”,
governing the size of the universe.
𝑎̇ 2 Ω𝑟 Ω𝑚
=
+
+ a2 Ω𝑑 + Ω𝑘
𝑎
𝐻02 𝑎2
The different Ω terms relate to the different
types of energy in the universe, the subscripts
are r=radiation, m=matter, d=dark energy (see
main article) and k=curvature. H0 is Hubble’s
constant, which is a measurement of the scale
factor at the present time.
Figure 2: A plot of the size of the universe vs. time for a
critical density universe and an over dense universe.
This equation was derived by Friedman
Robertson and Walker from the first Einstein
equation which gives the metric for a given
distribution of energy and mass (which are
equivalent in relativity). The “metric” is a
quantity which describes the structure of space
time and its change over time, used to arrive at
the scale factor “a”. The FRW equation was
solved in MATLAB, an interpreter program, using
its differential equation solving function ode45,
and then calculating and plotting the results.
Using this model, they found that there were
three possible scenarios for the universe,
depending on the amount of matter in it. If
the amount of matter was below a critical
density, the universe would just expand
forever, the gravitational attraction not
enough to hold it together. At the critical
2
For all these fates, the answer to the main
question, “can the whole universe be
explored?” was a promising everything, that
is, the whole universe could be explored. In
two of the three cases, an observer would
have infinite time so they could be as leisurely
as they liked, however, in the over dense (too
much matter) case, an observer would have
to travel at close to light speeds to explore the
universe, due to the lack of time.
Dark Energy
Some time later, in 1998, distant supernovae
were observed to have extremely high red
shifts, suggesting they are moving away from
us at speeds too fast to be explained by the
model discussed above. The only way to
explain this phenomenon was to introduce a
repulsive kind of energy, called dark energy
that is thought to be causing the expansion of
the universe to accelerate. Dark energy is
believed to make up around 72% of the
energy of the universe, with matter making up
almost all of the rest (Figure 3).
Dark energy plays an important role in
determining the fate of the universe, and
luckily was able to be introduced quite easily
into the FRW model of the universe. This is
due to Einstein’s addition of a cosmological
constant to his equations, which was intended
to prevent the
equations suggesting
that the universe would
collapse in on itself. It is
of the form of a
repulsive pressure, very
similar to dark energy,
and so dark energy can
be modeled in the FRW
equation by changing
the value of this
constant.
It must be mentioned
that there is at present
no good (or even bad)
explanation for dark
energy, the title “dark”
implying it can’t be seen
directly and is in many
senses unknown. One
attempt to model dark
energy, which is at least
consistent with theory,
treats it as a fluid with a
certain equation of
state. Its equation of
state is just a relation
between its density and
Our Universe
Using the observed energy densities for our
universe, that is, dark energy makes up 72%,
matter 28%, we can then use the FRW model
to calculate how our
universe will expand,
and hence determine
its fate and our fate
within it. Using
numerical integration
techniques, a computer
can give a plot of the
scale factor “a” over
time for our universe
(Figure 4).
Figure 3
pressure, and dark
Figure 4: A plot of the scale factor “a” vs. time
energy is observed to
have a negative equation of state, that is, it
has a negative pressure or energy density.
This makes sense, as it seems to be acting as a
repulsive force, which would happen with
either negative pressure, or negative energy
density.
3
The time is set to 0
today and the scale
factor is set to 1 at this
time, as an initial
condition. The red line
is the expansion of our
universe with time and
when “a” is at 0, this
corresponds to the time
of the big bang. See
“Models with Different
Parameters” for the
meaning of the other
lines.
Models with Different Parameters
It is mentioned that the energy percentages, or
densities, play a role in determining the
expansion of the universe. We can therefore see
what kinds of universes arise from assuming
different energy densities other than the ones
we observe. These different examples are drawn
in different colours in Figure 4.




The set of densities (Ωm Ω r Ωd) for our
universe are (0.3, 0, 0.7) in red.
For a universe with only matter, the
model used before the discovery of dark
energy, the parameters are (1, 0, 0) in
green.
A universe with only radiation is similar
to that with matter, (0, 1, 0) in blue.
Lastly a universe that contains only dark
energy is (0, 0, 1) in purple.
An interesting feature of such a universe is that
it has no big bang, in fact it has no beginning.
Any universe that contains even the smallest
amount of dark energy will end up accelerating
like our own, as dark energy always comes to
dominate over matter in the end.
Figure 4 (as above)
4
So what does this show us? Firstly, we can
read off how long ago the universe began,
around 13 billion years ago. We can also see
that in the future, the universe will begin to
expand at an accelerating rate, and is just
beginning to do so now. This suggests that
there is no possible way for the universe to
collapse back in on itself, a possibility in
models with no dark energy. Instead, the
universe will expand forever at faster and
faster speeds of expansion, with all matter
drifting apart. This means that the universe
will go on forever, so if we have infinite time,
can we explore the whole universe? At the
moment it seems like the answer is yes.
Light rays
When we want to examine what we can and
can’t observe, and what we can and can’t
explore, relativity tells us that we must use
light rays. That’s because if nothing can travel
faster than light, we can tell if we can observe
an event by testing whether light rays given
off by it will reach us. Similarly, we can test
where we can go or what we can influence by
how far light rays given off by us can go. If
light from us cannot reach a certain object,
then we have no way of reaching it because if
light can’t get there, nothing can. Using this
idea, we can plot different light paths in the
universe to gather information about what
can be influenced, explored or observed by
what.
Figure 5: A plot of the distance between two objects vs. time
Figure 6: A plot of comoving distance vs. time
Different important light rays or “light cones”
are plotted in the universe to show their
separation with time (Figure 5). The green
light rays are light rays given off at the big
bang and are known as the particle horizon of
the universe. They represent the farthest
anything can have travelled since the big
bang, and provide the way of quantifying the
size of our universe. The blue rays are our
“past and future light cones”. The past light
cone shows what can have influenced us and
what we can observe today, as anything inside
the cone could have emitted light that
reached as today. Our future light cone shows
what we can influence and explore, by the
same token, anything inside the cone can be
reached by light emitted by us today. Lastly,
the red light cone is known as the “event
horizon” of the universe and it shows
everything that could have possibly influenced
an observer who is at the end of the universe,
in this case, an observer an infinite time in the
future. This will be discussed more in the next
section.
curved, which is defies common sense, light
normally travels in straight lines. The rays are
curved because they are both moving at the
speed of light and the universe is expanding
around them at a changing rate, causing them
to end up travelling with a curved path. To
make things easier to read and interpret we
can change the distance coordinate so that it
increases with the expansion of the universe,
instead of being an absolute distance
measure, making it “comoving distance”.
The light cones shown in Figure 6 are the
same as the ones in Figure 5. Again, it is
obvious that the light rays still aren’t travelling
in straight lines. This is because the time goes
to infinity and so the light rays become
stretched and so have curved paths through
space time. We therefore want to change the
time coordinate so that it causes light to
travel straight. This could mean that the new
time coordinate becomes finite, allowing us to
see what happens at the end of the universe
and find the answers to the questions we are
asking, including what we can explore and
what we can observe. To do this, the time
coordinate is transformed into conformal
time, for the explanation see “Conformal
time”.
Coordinate Transformations
Figure 5 is a plot of distance versus time in the
universe for some important light rays. What
we can immediately see is that they are
5
cone only takes up a small amount of the
universe, meaning that although we have
infinite time, we still cannot explore the
whole universe. Also, the red light cone or
event horizon does not cover the whole
universe (represented by the green cone)
which means that there are events that we
will never be able to observe.
Conformal time
Coordinate transformations are simple to
understand, a well known example being the
transformation of polar coordinates to
Cartesian coordinates. To make the
transformation to coordinate time we use the
following equation:
𝒅𝜼 =
𝒅𝒕
𝒂(𝒕)
It may seem strange that in an infinite amount
of time we can’t go everywhere, but the
problem is that some parts of space end up
expanding away from us faster than light,
which means we will never be able to catch
up to those regions, provided they keep on
moving faster and faster away. Although
objects may not travel faster than light, the
space in between them is allowed to in
general relativity. So it looks like we will never
be able to explore the whole universe, since
it’s physically impossible. But what if this
model isn’t exactly correct …
Where “t” is normal time, “a” is the scale factor
and “n” is conformal time. This means that a
small change in conformal time equals a small
change in time, divided by the scale factor at
each time. An interesting feature of conformal
time is that even though time may extend to
infinity, conformal time may only go up to a
certain value, it may be finite or infinite.
Whether conformal time is infinite or bounded
gives important information about what can
influence what in the universe. Generally, if
conformal time is bounded/finite, this means
that there will be only a fixed portion of the
universe which we explore, and a different
portion which we can observe.
Different Models
There are several limitations to the model
described above. Firstly, it is assumed that our
universe is homogenous (has the same energy
density everywhere) and is isotropic (is the
same in all directions). When we observe the
universe, we can tell that these assumptions
are blatantly untrue, one spot in the night sky
will look different from another, however, on
the large scale of the universe they work well
A plot of conformal time versus comoving
distance now produces light rays that travel in
straight lines which we are accustomed to.
The light rays shown in Figure 7 are the same
as above. There are several implications of
this graph, the first being that our future light
Figure 7: A plot of comoving distance vs. conformal time. Light rays travel in 45o lines.
6
as approximations. Another limitation is that
all the energy densities (matter, dark energy
etc) are assumed to stay constant throughout
the evolution of the universe, meaning there
is no interaction between matter and dark
energy, for example. This second limitation
can be fixed in some senses.
As mentioned earlier, dark energy can be
treated as a fluid with negative pressure,
having an equation of state “w”, a
relationship between its density and pressure.
For matter, w=0 and for dark energy w seems
to be equal to -1, so we can model the decay
of dark energy into matter by a changing w.
Doing this produces interesting results, it
turns out that in such a model the conformal
time of the universe is unbounded. This
means that it is possible to explore and
observe the entirety of the universe in this
case. It seems reasonable that dark energy
should interact with matter in some way and
so it is certainly possible that this model
applies to our universe. Another, less certain
model is when “w” is less than -1, known as
models with phantom energy, which provides
an alternate ending for our universe (See “The
Big Rip”).
The Big Rip
When “w” is set to be less than -1, it
produces a universe that expands to an
infinite size in a finite amount of time. This
means that at a certain time in the future,
the universe effectively gets ripped apart as
each particle recedes from the others at
faster than light speeds. This causes all
matter to be torn apart almost instantly and
the universe splits apart.
Conclusions
So it appears that if our universe does
conform to our current FRW model, there are
only parts of the universe which we can
observe, explore and even influence, even
though it is likely that our universe will last
forever. This might be disheartening to some,
but there is plenty of hope, as other viable
models show that we might, in fact, be able to
explore the entire universe. Unfortunately, or
perhaps fortunately, there are no definitive
answers at present due to the lack of
understanding of dark energy, and to a lesser
extent dark matter. Perhaps if the nature of
dark energy is discovered, we will know the
exact fate of the universe and how much we
will ever be able to explore.
An interesting feature of these models is
that they begin in a very similar fashion to
our model for our own universe. In fact,
some are quite indistinguishable, which
could mean that our own universe will rip
apart in around 20 billion years, and we
wouldn’t know it. Luckily, there appears to
be no physical reason for this to occur, but
dark energy is currently so mysterious that
anything is possible.
Nic Marks is a first year physics student at
Sydney University.
7
Acknowledgements
References/Further Reading
Thanks a lot to Associate Professor Geraint
Lewis who agreed to supervise a great project
for us, despite not originally planning to, at
the cost of his own valuable research time.
Thanks also to Richard Hunstead, TSP
coordinator for physics, who organised all
these projects. Thanks also to Hao, my project
partner, who is equally responsible for all this
work.
8

Hartle, James (2003), An Introduction
to Einstein’s General Relativity (San
Francisco: Pearson Education Inc.)

Friedmann, A (1922). "On the
Curvature of Space". General
Relativity and Gravitation 31: 1991–
2000. doi:10.1023/A:1026751225741

Caldwell, Robert R.; Kamionkowski,
Marc and Weinberg, Nevin N. (2003).
“Phantom Energy and Cosmic
Doomsday". arΧiv:astro-ph/0302506.

Lemaître, Georges (1931), "Expansion
of the universe, A homogeneous
universe of constant mass and
increasing radius accounting for the
radial velocity of extra-galactic
nebulæ", Monthly Notices of the
Royal Astronomical Society 91: 483–
490,
http://adsabs.harvard.edu/abs/1931
MNRAS..91..483L