The Possible Initial States of The Universe

Most people when talking about cosmology typically talk about the universe in one context, that is, as a particular solution to the Einstein field equations. Part of my research in mathematical cosmology is to try to determine whether the present-day universe which we observe to be very close to spatially flat and homogeneous, and very close to isotropic could have emerged from a more general geometric state.

What is often not discussed adequately is the fact that not only has our universe emerged from special initial conditions, but the fact that these special initial conditions also must include the geometry of the early universe, and the type of matter in the early universe. Below, I have attached a simulation that shows how the early universe can evolve to different possible states depending on the type of physical matter parametrized by an equation of state parameter \gamma . In particular, some examples are:

  • \gamma = 0: Vacuum energy
  • \gamma = 4/3: Radiation
  • \gamma = 2: Stiff Fluid

Note: Click the image below to access the simulation!

In these simulations, we present phase plots of solutions to the Einstein field equations for spatially homogeneous and isotropic flat, hyperbolic, and closed universe geometries. The different points are:

  1. dS: de Sitter universe – Inflationary epoch
  2. M: Milne universe
  3. F: spatially flat FLRW universe – our present-day universe
  4. E: Einstein static universe

Note how by changing the value of \gamma , the dynamics lead to different possible future states. Dynamical systems people will recognize the problem at hand requires one to determine for which values of \gamma is F a saddle or stable node.

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Attempts at a General Einstein Equation for an Arbitrary FLRW Cosmology

I tried to derive a general Einstein field equation for an arbitrary FLRW cosmology. That is, one that can handle any of the possible spatial curvatures: hyperbolic, spherical, or flat. Deriving the equation was easy, solving it was not! It ends up being a nonlinear, second-order ODE, with singularities at a=0, which turns out to be the Big Bang singularity, which obviously is of physical significance. Anyways, here’s a log of my notebook, showing the attempts. More to follow! 

A Really Quick Derivation of The Cauchy-Riemann Equations

Here is a really quick derivation of the Cauchy-Riemann equations of complex analysis.

Consider a function of a complex variable, z, where z = x + iy, such that:

f(z) = u(z) + i v(z) = u(x+ iy) + i v(x+iy),

where u and v are real-valued functions.

An analytic function is one that is expressible as a power series in z.
That is,

f(z) = \sum_{n=0}^{\infty} a_{n} z^{n}, \quad a_{n} \in \mathbb{C}.

Then,

u(x+iy) + i v(x+iy) = \sum_{n=0}^{\infty} a_{n} (x+iy)^{n}.

We formally differentiate this equation as follows. First, differentiating with respect to x, we obtain

u_{x} + i v_{x} = \sum_{n=1}^{\infty} n a_{n} \left(x+iy\right)^{n-1}.

Differentiating with respect to y, we obtain

u_{y} + i v_{y} = i \sum_{n=1}^{\infty} n a_{n} \left(x + i y\right)^{n-1}.

Multiplying the latter equation by -i and equating to the first result, we obtain

-iu_{y} + v_{y} = \sum_{n=1}^{\infty} na_{n} \left(x+iy\right)^{n-1} = u_{x} + i v_{x}.

Comparing imaginary and real parts of these equations, we obtain

\boxed{u_{x} = v_{y}, \quad u_{y} = -v_{x}},

which are the famous Cauchy-Riemann equations.

 

The Mathematics of “Filling the Triangle”

I’ve been fascinated by the triangle offense for a long time. I think it is a beautiful way to play basketball, and the right way to play basketball, in the half-court, a “system-based” way to play. For those of you that are interested, I highly recommend Tex Winter’s classic book on the topic.

There is this brief video as well where Tex Winter explains how the triangle offense and a basketball are grounded in geometric principles:

 

I don’t think people recognize though how deep of a geometry problem this is actually. Looking at when the triangle is filled, as in the video above, we have the following situation:

trianglesetup
The 3 triangles that form when one triangle is filled involving all 5 players. The letters a,b,c,d,e,f,g,h,i denote the angles within the triangles. We are assuming NBA court dimensions where the 1/2 court is 47′ long and the team bench area which roughly corresponds to the top of the three-point line is 28′ from the baseline.

The problem I wanted to study was given 5 players’ random positions on the court, could a series of equations be solved yielding (x,y) coordinates that would yield where players should “go” to fill the triangle? 

Using simple geometry, from the diagram above, we see that each player’s position in the triangle offense is governed by the following system of nonlinear equations:

\left(x_4-x_2\right) \left(x_4-x_5\right)+\left(y_4-y_2\right) \left(y_4-y_5\right)=\cos (a) \sqrt{\left(x_2-x_4\right){}^2+\left(y_2-y_4\right){}^2} \sqrt{\left(x_4-x_5\right){}^2+\left(y_4-y_5\right){}^2}

\left(x_4-x_2\right) \left(x_2-x_5\right)+\left(y_4-y_2\right) \left(y_2-y_5\right)=\cos (b) \sqrt{\left(x_2-x_4\right){}^2+\left(y_2-y_4\right){}^2} \sqrt{\left(x_2-x_5\right){}^2+\left(y_2-y_5\right){}^2}

\left(x_2-x_5\right) \left(x_4-x_5\right)+\left(y_2-y_5\right) \left(y_4-y_5\right)=\cos (c) \sqrt{\left(x_2-x_5\right){}^2+\left(y_2-y_5\right){}^2} \sqrt{\left(x_4-x_5\right){}^2+\left(y_4-y_5\right){}^2}

\left(x_2-x_1\right) \left(x_2-x_5\right)+\left(y_2-y_1\right) \left(y_2-y_5\right)=\cos (d) \sqrt{\left(x_1-x_2\right){}^2+\left(y_1-y_2\right){}^2} \sqrt{\left(x_2-x_5\right){}^2+\left(y_2-y_5\right){}^2}

\left(x_2-x_1\right) \left(x_1-x_5\right)+\left(y_2-y_1\right) \left(y_1-y_5\right)=\cos (e) \sqrt{\left(x_1-x_2\right){}^2+\left(y_1-y_2\right){}^2} \sqrt{\left(x_1-x_5\right){}^2+\left(y_1-y_5\right){}^2}

\left(x_1-x_5\right) \left(x_2-x_5\right)+\left(y_1-y_5\right) \left(y_2-y_5\right)=\cos (f) \sqrt{\left(x_1-x_5\right){}^2+\left(y_1-y_5\right){}^2} \sqrt{\left(x_2-x_5\right){}^2+\left(y_2-y_5\right){}^2}

\left(x_1-x_3\right) \left(x_1-x_5\right)+\left(y_1-y_3\right) \left(y_1-y_5\right)=\cos (h) \sqrt{\left(x_1-x_3\right){}^2+\left(y_1-y_3\right){}^2} \sqrt{\left(x_1-x_5\right){}^2+\left(y_1-y_5\right){}^2}

\left(x_1-x_3\right) \left(x_3-x_5\right)+\left(y_1-y_3\right) \left(y_3-y_5\right)=\cos (i) \sqrt{\left(x_1-x_3\right){}^2+\left(y_1-y_3\right){}^2} \sqrt{\left(x_3-x_5\right){}^2+\left(y_3-y_5\right){}^2}

\left(x_1-x_5\right) \left(x_3-x_5\right)+\left(y_1-y_5\right) \left(y_3-y_5\right)=\cos (g) \sqrt{\left(x_1-x_5\right){}^2+\left(y_1-y_5\right){}^2} \sqrt{\left(x_3-x_5\right){}^2+\left(y_3-y_5\right){}^2}

Further, the angles obviously must satisfy the following constraints:

a + b + c = \pi, \quad d + e + f = \pi, \quad g + h + i = \pi

Finally, we require that each player be about 15-20 feet apart in the triangle offense (because the offense is predicated on spacing), and thus have some additional constraints:

15\leq \sqrt{\left(x_2-x_4\right){}^2+\left(y_2-y_4\right){}^2}\leq 20

15\leq \sqrt{\left(x_4-x_5\right){}^2+\left(y_4-y_5\right){}^2}\leq 20

15\leq \sqrt{\left(x_2-x_5\right){}^2+\left(y_2-y_5\right){}^2}\leq 20

15\leq \sqrt{\left(x_1-x_2\right){}^2+\left(y_1-y_2\right){}^2}\leq 20

15\leq \sqrt{\left(x_1-x_5\right){}^2+\left(y_1-y_5\right){}^2}\leq 20

15\leq \sqrt{\left(x_1-x_3\right){}^2+\left(y_1-y_3\right){}^2}\leq 20

15\leq \sqrt{\left(x_3-x_5\right){}^2+\left(y_3-y_5\right){}^2}\leq 20

Solving this highly nonlinear system of equations with constraints is not a trivial problem! It fact, because of the high degree of nonlinearity and dimension of the problem, it is safe to assume that no closed-form solution exists, and therefore, must be solved numerically.

For this task, we used MATLAB, and experimented with the lsqnonlin() and fsolve() commands. The only issue is that (as with all such numerical algorithms) convergence depends very highly on the choice of initial conditions. It is very difficult to choose a priori this many initial conditions, so I wrote a script that randomized initial conditions. I then ran several numerical experiments and obtained the following results:

In the plot above, I have labeled the plots that converged to the triangle formation with the title “this one”. In addition, the five black circles denote the initial positions of the players on the court before they fill the triangles in the offense. One sees just by the diagram above, how difficult such a problem is to solve mathematically, even through a numerical approach. Running more trials would perhaps yield better results, but, it works! I am truly fascinated by this. In the coming days, I will work on optimizing the numerical algorithm, and post my updates as they come.

Here is an animation of one of the scenarios above when the algorithm converges correctly:

In this animation above, the black dots represent the positions of the players on the court. They begin at initial (random) positions and attempt to fill the triangles as described above.

Thanks for reading!

New paper published on cosmological singularities

My new paper has now been published in Annalen der Physik, which is a great honour, because 100 years ago, Einstein’s General Theory of Relativity was also published in the same journal.

This paper describes a method by which one is able to determine whether a given spatially flat cosmological model produces finite-time singularities, and also gives some examples of interesting cosmological model configurations. 

The paper can be accessed by clicking the image below: 


The preprint can be accessed here on the arXiv. 

Making a Cosmological Model

What goes into making a cosmological model? Here is a presentation (that was part of my Ph.D. dissertation) that I have reproduced and embedded here to describe what actually goes into the making of a cosmological model. After describing some general properties, I describe specifically a early-universe model that contains a viscous fluid and a magnetic field. 

The background mathematics can be found in this old presentation of mine here: