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OK, so let me start my talk with a disclaimer,
I'm a theoretical physicist
and there will be one formula in my presentation.
More specifically, I work in the field of fundamental physics,
and an ambitious goal of fundamental physics
is to address some profound questions like:
what is the Universe made of,
what are the fundamental forces of nature
and what is the nature of space-time and matter?
There are also several questions that
we cannot address even though they're very interesting
and most of these questions start with "why".
Now I want to zoom in on the first of these questions:
what is the Universe made of?
So if I would have given this talk more than a century ago
I probably would have answered: the Universe is made of atoms.
At that time atoms were considered to be the fundamental building blocks
of matter, of everything in this room, it was atoms.
But now we know atoms are not fundamental,
they consist of electrons, neutrons, protons,
and the protons themselves, some are not fundamental either,
they consist of quarks.
So now we have a somewhat simpler picture than
the full periodic table of chemical elements.
We have this small kind of periodic table
of elementary of particles.
And these particles and all their interactions
describe everything that is going on in this room for instance.
However, when you're trying to apply this
to the Universe as a whole you find something very embarrassing.
Namely, all the particles that we know,
together with all the forces that we know,
they make up less than 5% of the energy content of the Universe.
So in other words, there's more than 95% percent of the Universe
that we don't understand
with our current understanding of elementary particle physics.
And about one quarter of the Universe consists of so-called dark matter
and almost three-quarters of dark energy.
So how do we know this and what is dark matter?
Well, let me try to address this with an historic precedent,
namely, the discovery of Neptune.
So in 1821 Alexis Bouvard published tables
that he calculated of the orbit of Uranus.
And interestingly the observations of the trajectory of Uranus
deviated from this table.
So there was a gravitational anomaly
and the observations did not coincide with these theoretical predictions.
And to bridge this gap between theoretical prediction and experimental observation,
there were two different logical possibilities available.
Either you predict something like dark matter,
you predict some invisible planet
located close to Uranus that distorts its trajectory
so that it coincides with the observed one, or you change the theory.
You declare Newton's theory to be not quite right,
but you have to modify it.
And in 1845 these two gentlemen, especially Urbain Le Verrier,
predicted a new planet.
So they used this dark matter paradigm and calculated its position.
And amazingly a year later this planet was found and it was called Neptune.
So the discovery of Neptune was
the first success of the dark matter concept.
Of course after its discovery it was no longer dark.
(Laughter)
OK, interestingly there's another story that is very related,
but it has completely the opposite ending.
So in the same century it was also discovered
that Mercury has some problem with its trajectory.
So the calculated tables of the trajectory of Mercury
did not coincide with the observed ones.
So that was another instance of a gravitational anomaly.
And again the same kind of two different logical possibilities existed;
either you predict some form of dark matter,
for instance, a new planet, or you change the theory of gravity.
And the same person who theoretically predicted Neptune
also predicted another planet, namely Vulcan.
So Urbain Le Verrier predicts a new planet and calculates its position in1859.
And interestingly this planet was discovered in 1860; so one year later.
This planet is called Vulcan and it's supposed to orbit the sun
in an orbit that is closer to the sun than Mercury,
so it would have been a planet closer than Mercury.
However, this observation was never confirmed by anyone else
and now we know that Vulcan does not exist.
This story has a different ending.
But this issue was not resolved until Albert Einstein came along in 1915,
constructed general relativity,
and this new theory of gravity then fully explains the trajectory of Mercury.
So the non-discovery of Vulcan was the first failure
of the dark matter concept.
Now, in astrophysics today, we have a couple of gravitational anomalies,
so again discrepancies between theory and observation
and I'm going to show you the most prominent example.
These are the so-called galactic rotation curves.
So what you see in this curve here is the velocity
of the stars in the galaxy around the center of the galaxy
plotted as a function of the radial distance from the center.
And if you take general relativity and calculate this velocity profile,
you get this dashed curve here, so this is the theoretical prediction.
However, when you actually observe the galaxies, what you find
is this solid line here, so this is the observation.
So at large distances we have a gravitational anomaly with a discrepancy
between theory and observation.
There's also at the smallest scale a gravitational anomaly,
the Pioneer anomaly.
Pioneer spacecraft were sent out in the seventies and eighties
to explore our solar system
and since then we've been monitoring the trajectory of the spacecrafts
very precisely so we can very accurately determine the trajectory
and from this we can deduce that there's an anomalous acceleration towards the sun,
so that's another instance of a gravitational anomaly.
The trajectory that we observe does not quite coincide
with the trajectory that we calculate.
So the key question seems to be are we in a Vulcan or in a Neptune scenario?
Now most scientists including myself, consider the Vulcan scenario to be unlikely,
and the Neptune scenario to be likely.
So it really seems that there is some sort of dark matter in our Universe,
but the problem is that it has not been detected yet directly.
And there are many different acronyms and candidates for dark matter,
perhaps one of them will be discovered at the LHC at CERN in the next decade,
but right now we don't have a good clue what dark matter is.
And we shouldn't forget that the main success
of the Neptune story was not just that Neptune was predicted theoretically,
but also that it was observed.
So at the moment we're just in the stage
where dark matter's predicted but not yet observed.
And due to this some people have quasi-religious feelings about these issues
whether Vulcan or Neptune scenario is true,
so the question is, how can we make progress?
Well, the most straightforward strategy would be
to show that either the Vulcan or the Neptune scenario is correct,
but unfortunately, both strategies are currently out of reach.
So, how can we make progress on the theory side?
Well, my strategy was to actually remain agnostic about this issue
and to rephrase the question, even though this sounds a bit like cheating.
So the question that I want to address is
what is the most general theory of gravity that is possible at large distances?
So the input was to make the model as simple as possible,
because otherwise I can't do anything,
and one of the simplifying assumptions was to impose spherical symmetry.
So if you look around yourself then
this room doesn't look spherical symmetric at all
but if you move a couple of thousand kilometers upwards
then the Earth starts to look more and more like a perfect round sphere.
So at large distances, gravity behaves as if it were spherical symmetric
and the same applies to the sun, to stars and to many galaxies.
I had to use another assumption;
the absence of certain pathologies at large distances,
this is also known as the cosmic censorship conjecture.
It basically states that gravity should not become arbitrarily large
in the Universe, except within a black hole.
This is the assumption that I used.
And then I had to use some technical assumptions that I won't discuss here.
But the output of these assumptions was a model
that predicts a force law for gravity which consist of several parts.
So, now comes the formula.
The force that contains the Newtonian piece,
which is, of course, a nice consistency check,
it also contains the centrifugal force
that was already known in the times of Kepler.
It also contains the Einstein piece,
so this harmless looking term is basically the planet Vulcan,
or what replaces the planet Vulcan, that's Einstein's modification
of Newton's theory of gravity.
Interestingly you automatically get another term,
namely the cosmology constant term.
So this harmless looking Λ times r
is what contributes to 72% to the energy budget of the Universe.
This is the so-called dark energy.
But the real surprise was that's there's another term
which I called Rindler acceleration after the Austrian physicist Wolfgang Rindler,
and this consists of one constant, the Rindler acceleration a,
and then some radial dependence.
So the upshot is that, using the simplifying assumptions,
I get a force law, and this force law predicts
a new force at large distances, a Rindler force.
Now let's apply this.
So, I tested this for galaxies in a very simple time model
and again I plotted the velocity profile as a function of the real distance.
And what you can see is that the solid line,
which is the line that you get when you take into account the Rindler force,
matches the observation considerably better
than the general relativistic force law.
OK, this is one consistency check.
The other consistency check was the Pioneer trajectory
and I didn't have to do very much,
because the statement of the Pioneer anomaly
is that there is an anomalous acceleration
towards the sun and this is precisely what this Rindler force predicts.
OK, so I'm almost finished.
Let me conclude here with some scientific conclusions.
I've presented a very simple model for gravity at large distances,
which predicts a new force law
and the observational data that I've checked
are compatible with this Rindler force.
Perhaps a useful truism to take away from this talk is
if you get stuck with a question then try to rephrase it or to avoid it,
even though it might feel like cheating,
and this may shed light on the original question.
Thank you!
(Applause)