In the past few episodes, we've
delved deep into cosmology,
building up the tools we
need to really understand
dark energy and its
influence on the universe.
Pretty soon you'll
understand as much as anyone,
with the possible exception
of people who are actually
paid to understand dark energy.
But before we get there,
it's time for some homework.
Today I have a challenge
question for you.
First, if you haven't
watched these three episodes,
you'll want to do that.
We live in unusual times.
The balance between
dark energy and matter
in the modern universe is
on the side of dark energy.
The expansion of
the universe has
been accelerating for
some billions of years
now, ever since it became
large enough for regular matter
to dilute away enough so
that dark energy dominates.
But we're still very
near that tipping point.
So close, in fact, that it feels
like a huge cosmic coincidence.
This genuinely puzzles
many cosmologists.
If you add up the energy in
any large volume of space,
about 70% of it is dark
energy and the remaining 30%
is in the form of
regular matter.
And by regular, I also
mean dark matter, which
is actually most of that 30%.
Now, these percentages
aren't some abstract quantity
or influence.
I'm talking about the
relative amount of energy.
So the number of joules,
ergs, British thermal units,
energon cubes-- of this
stuff in a large volume.
Say a cubic million
light years, around 70%
dark energy, 30% matter.
So there's more dark energy
than the other stuff.
But there's something
really weird here.
These numbers are extremely
close to each other.
I hear you ask, how are 70%
and 30% close to each other?
I mean, 70% cacao
dark chocolate is
very different-- and
much tastier-- than 30%
milk chocolate, right?
Thanks for asking.
And I agree about
dark chocolate.
But if our universe
were chocolate--
which it isn't,
sadly-- then it's
only been deliciously dark
for a relatively short period.
Let me explain.
In the Einstein field equations
and in the Friedmann equations
that are derived from
those, dark energy
is described by a positive
cosmological constant.
If that constant truly
stays constant over time,
then it represents an
energy of the vacuum.
Its density stays
constant and so the amount
of dark energy increases
at the same rate
as the volume of the universe.
When the universe
has doubled in size,
there'll be twice as much
dark energy, joule for joule.
And when it was half its current
volume, there was half as much.
At the same time,
the total amount
of regular matter in
any expanding region
remains constant.
That means the density
of matter decreases
as the universe expands.
At some point in the past,
there was a perfect balance
between dark energy and matter.
How long ago?
Very roughly, it was when
the universe was half
its current volume
because that would
mean half the current
amount of dark energy.
A given giant box
in the universe
currently has around 30 parts
matter and around 70 parts
dark energy.
Halve its volume,
and it still has
those 30 parts matter, but only
around 35 parts dark energy.
Close enough to 50-50
for an astronomer.
How long ago was that?
Well, the volume of that giant
box is equal to its linear size
to the power of 3.
We can think of that
linear size as the scale
factor of the universe.
So the scale factor, the
size of the universe,
only needed to be around 20%
smaller than it is currently
for the volume of any patch of
space to be smaller by half.
Assuming a constant
expansion rate,
a constant increase in the
scale factor, that means it
was only two to three
billion years ago.
Blink of an eye, really.
No, really.
Dark energy has dominated the
universe only during the tenure
of life on Earth, although
its effect has been felt
for a bit longer than that.
Think about it this way.
Since the end of the
inflationary epoch,
when the universe was
about 10 to the power
of minus 32 seconds
old and around the size
of a grain of sand,
it has doubled
in size approximately 100 times
to get to its current size.
That's 100 doublings
of its scale factor--
so its linear dimension,
not its volume.
It'll go on to double in size
approximately infinity times
in the future.
OK, so let's think about
time not in billions of years
but in the number of doublings.
For the vast majority of
those past 100 doublings,
dark energy has had an extremely
tiny energy contribution.
Seriously.
Practically zero.
And for the vast majority
of future doublings,
regular matter will
have diluted away
and be an
infinitesimal influence
compared to dark energy.
It's only right now that both
have a measurable effect.
Here's your challenge question.
For how many of those
past 100 doublings
has dark energy had any
significant effect--
let's say at least 10%
of the energy density?
And for how many
of those infinite
future doublings will
regular matter and energy
have any significant
effect-- again, at least 10%
of the energy density?
And the extra credit question.
Answer the question I just
asked, but also tell me
how many billion
years ago dark energy
started to have a
significant effect
and how many billion
years in the future will
it take for matter
to cease to matter.
For the extra
credit part, you can
make some crude approximations
to get a rough idea
with only simple algebra.
Or you could go ahead and
solve the Friedmann equations.
Up to you.
I'll choose three
correct answers
at random for both the main
and the extra credit questions
to receive Space Time t-shirts.
To be eligible, you need
to explain your reasoning
and show your work.
Feel free to propose
any solutions
you come up with for this
apparent cosmological
coincidence.
You also need to
not give answers
in the comments section.
That will disqualify you from
this and future challenge
questions.
Email your answers to
pbsspacetime@gmail.com
with the subject line
Dark Energy Challenge
within two weeks.
We filter by subject
line, so make sure
you use exactly this
phrase to be considered.
See you next week for a fresh
new episode of Space Time.
