After the discovery of the Higgs boson
and gravitational waves two of the most
fundamental questions of physics still
remain unanswered the origin of dark
matter and the origin of ordinary matter
in the universe to address the origin of
matter we have developed a technique to
model first order of phase transitions in
the early universe which are thought to
be the cause amazingly these first order
phase transitions are very similar to
the phase transitions we experience
daily take boiling water for example
just as bubbles of steam form as we boil
water which is the new phase bubbles of
a new phase of space can form when the
temperature of the universe drops from a
higher temperature to a lower
temperature these bubbles form in the
middle of the old phase of space in
random spots and they grow and propagate
outwards as these bubbles collide they
produce detectable gravitational waves
which can shed light on the origin of
matter
our approach reformulates the bubbles
equations of motion which are described
by quantum fields in a simpler path
trajectory problem using a coordinate
shift called intrinsic coordinates the
problem can now be thought of as a
particle rolling down an inverted
potential where the question that you
need to ask is where should you place a
particle such that it can roll down one
hill and come to rest on top of another
this problem becomes increasingly more
difficult to generalise in higher
dimensions as the path it tastes becomes
a lot more convoluted and in addition to
this the system's friction reduces over
time making the system more and more
slippery our method looks at creating a
initial guess trajectory between the
first hill and the second hill then
iteratively pushing the path to the
path of least resistance at the same
time another another algorithm is used
to find the correct starting position
using an overshoot or undershoot method
just to find the exact amount of energy
so that the particle will fall down on
top of the other hill metastably in
creating this model we are one step
closer in creating a larger simulation which will allow us to find the
result of colliding bubbles and find the
gravitational waves which will allow us
to find the fundamental question the
origin of matter hi everyone thank you
for watching that video now we delve
into a little bit more of the of a
technical summary of modeling phase
transitions in the early universe so
first of all we'll start with the objectives
of what my model aims to do and then the
reformulated bubble equations of motion
the 1d algorithm and then the 2d
algorithm and we'll finish off with a
short clip of the results so the
objectives of this project was to
develop a 1d solution and a 2d solution
to the this path projectory problem that
we have or the the bubble equations of
motion and then from the 2d solution we
tested and and improve the stability of
the solution as well as its convergence
speed so first we reformulate the bubble
equations of motion in a tangential
equation and a normal equation so the
tangential equation here up at the top
describes the position of the particle
at any given time so this X variable
here is a trajectory that the user
initially gives the program so you guess a
trajectory from the highest hill or the
true vacuum as we call it and the lower
hill or the false vacuum and this
tangential equation will result in the
position of that particle on this
trajectory with respect to time the
normal equation on the other hand gives
you the normal force that the particle
experiences on the trajectory so say we
are at some point here the particle will
experience some normal force which will
attempt to
push it off the path so this normal
force is essentially the normal force
you need to push on to the particle to
keep it on that trajectory for this for
our code to work properly we require
that the normal force is zero so there's
no force trying to push the particle off
the trajectory that it takes so let's
first start with the simple 1d algorithm
the 1-day algorithm is just in its crux
a overshoot undershoot method so let's
start by drawing two hills here in one
dimension and so here is the high hill
or as we'll call it the true vacuum hill
and then this is the lower hill or the
false vacuum hill and so the problem
requires us to place a particle
somewhere along this hill such that it
rolls down and remains at the
false vacuum metastably meaning that it
remains here for a very long period of
time so what our solution does is that
it looks at positions between the
highest point of the true vacuum hill so
the peak and the peak of the false
vacuum hill here and it starts off at
the mid point and so it tests this as
the starting position of the particle and
sees if the particle overshoots the
false vacuum hill or undershoots it and
so depending on this it'll move the
position of the particle higher or lower
to give it more potential energy or less
potential energy as required this is
done iteratively until it gets a stable
enough solution and then it decides yes
in the convergence block the convergence
decision block and go straight to the
action calculation this actually action
calculation for all intensive purposes
is a validation technique that we can
use to check if we're reaching a
solution similar to others in literature
and now we'll have a look at the 2d
algorithm so here the to 2d algorithm is
a little bit complex first thing you'll
see here however is that the 1d
algorithm is incorporated in our 2d
algorithm and this is because the
tangential equation is basically the
look of the equations of motion in one
dimension so if we go back you
wouldn't expect to see a normal force on
this path so it's just essentially the
tangential equation ok so first we
start the algorithm and we go straight
into the maximum finder and potentially
rescaling slash normalising block what
these blocks do is that for any given
hills
or true vacuum and false vacuum hill it
pushes those into known positions so we
make the true vacuum hill
always at (1,1) and the false vacuum hill
is always rescaled to (0,0) this is
just for a simplification for all our
functions and it makes it easier to
compute some of the properties and then
we go straight to the spline
interpolation block so this is the block
which draws a trajectory between the
true and false vacuum so say we have the
false vacuum here and the true vacuum
here this spline perturbation I mean
spline interpolation block given a
couple of control points will give a
path towards the false vacuum this
path then parameterised by the
variable X and then we do normal and
tangential vector calculations and then
we run the run the 1d algorithm so the
1d algorithm will calculate the position
of the particle along the path that we
give it with respect to time and then it
will push a part of that solution to the
normal force calculation to calculate
the normal force on the whole trajectory
so then we check if the normal force is
zero if it's not we uses path
deformation algorithm and so this path
deformation algorithm will push the path
to the path of least normal force so if
say we have a force pushing the particle
at some point to this point here then
the next path should look something a
little bit more like say this and then
we iterate this algorithm to push the
trajectory until there's zero normal
force and then we move to the
action calculation block so that we
can validate with other techniques in
literature that brings us to the end of
the technical summary here's a short
clip of the 2d algorithm running as you
can see at every iteration the spine is
deformed the normal force is reduced and
the initial starting position is changed
in the future this program will be
required to handle an arbitrary number
of dimensions so the problem will become quite a bit more complex and in
addition to this it will be subject to
potentials which have numerous hills
this path trajectory formulation however
does provide a very promising way to
handle these more complex problems so we hope that as we make this more and more
complex we will not need to change our
reformulation lastly I'd like to thank
all my supervisors Prof. Csaba Balazs, Dr. Andrew Fowlie and
Prof. Malin Premaratne for making this
whole project possible
I hope you enjoyed this video and thank
you for watching if you have any
questions please ask me
