Monday, December 7, 2015

Blog Post 37, ILLUSTRIS Simulation

The Illustis Simulation is a computer simulation of an entire universe using numerical calculations.  This project simulates most if not all relevant physics to high accuracy and results in a statistically accurate model of the universe in which one can study the distributions of gas, heat, dark matter, strs, etc.

First we will examine a random overdense region for halo data.  We will then plot this data in a histogram and interpret it.
This plot shows log(M) of the galactic halos on the x axis, and how many of these occurred in our sample location.  The histogram shows a clear trend in halo size, in that low mass halos are much more common than high mass halos.  Remember, we are using a log scale, so a halo in the 14-14.5 bin is 10,000 times more massive than one in the 10-10.5 bin.
Quickly analyzing the full dataset for our selected halos, we find that on average, 15% of the halo mass is stellar mass.  That implies that the rest is primarily dark matter.  Cool!

Using the simulation, we can make observations about the structure of a universe like our own.

Gas and Dark Matter Densities:

(Gas density (left) and Dark Matter density (right))

On a large scale, Gas and Dark Matter densities seem to correlate with one another, with the gas density being more spread out than the dark matter.  On a small scale (see below) this same trend is visible, but with the gas being very poorly defined compared to the stark definition of the dark matter filamentary structure.  This is likely due to baryonic interaction of the gas matter with itself, causing a counter-force to simply gravitation.  This greater degree of randomness could serve as a viable explanation for the greater dispersion of normal matter.

(Gas Density and Dark Matter density of a single cluster)

In a single galaxy (see below), the Gas is highly concentrated at the nucleus, while the dark matter is more dispersed in the halo.
(Gas density and dark matter density above the stellar material of the galaxy)

The most massive galaxies tend to be found in clusters, not in the background field.

Gas Temperature Evolution:
The following observations are based upon a video derived from the simulation.  This video can be found at: http://www.illustris-project.org/movies/illustris_movie_cube_sub_frame.mp4

Stars first begin to form win the early universe along the dark matter filaments as shown by the increase in gas temperature.  This starts slowly, then accelerates, reaching maximum star formation rate around redshift 1.5 to 1.0, where massive amounts of stars form.  The first stars begin to form at about 1 billion years after the Big Bang (Redshift 5.75), although they do not widely populate the universe until about 2.5 billion years after the Big Bang (Redshift: 2.75).  This marks the end fate "Dark Ages."  In the simulation, structure formation usually occurs through parts of very large structures collapsing due to gravity, breaking the very large structures into smaller more compact ones.  However, over time, these smaller structures tend to consume their neighbors, forming larger high density structures throughout the universe.  These structures form along filaments because gravity from dark matter is strongest there.  This gravity will pull both dark matter and gas together to form structures.

References:
http://www.illustris-project.org/explorer/
http://www.illustris-project.org/movies/illustris_movie_cube_sub_frame.mp4

Sunday, December 6, 2015

Blog Post 36, WS 12.1, Problem 1, 2(d): Large Scale Structures

Linear perturbation theory. In this and the next exercise we study how small fluctuations in the initial condition of the universe evolve with time, using some basic fluid dynamics.
In the early universe, the matter/radiation distribution of the universe is very homogeneous and isotropic. At any given time, let us denote the average density of the universe as ¯ρ(t). Nonetheless, there are some tiny fluctuations and not everywhere exactly the same. So let us define the density at comoving position r and time t as ρ(x,t)  and the relative density contrast as
δ(r,t)=ρ(r,t)¯ρ(t)¯ρ(t)

In this exercise we focus on the linear theory, namely, the density contrast in the problem remains small enough so we only need consider terms linear in δ. We assume that cold dark matter, which behaves like dust (that is, it is pressureless) dominates the content of the universe at the early epoch. The absence of pressure simplifies the fluid dynamics equations used to characterize the problem.

(a) In the linear theory, it turns out that the fluid equations simplify such that the density contrast δ satisfies the following second-order differential equation
d2δdt2+2˙aadδdt=4πG¯ρδ

where a(t) is the scale factor of the universe. Notice that remarkably in the linear theory this equation does not contain spatial derivatives. Show that this means that the spatial shape of the density fluctuations is frozen in comoving coordinates, only their amplitude changes. Namely this means that we can factorize
δ(x,t)=D(t)˜δ(x)
where δ ̃(x) is arbitrary and independent of time, and D(t) is a function of time and valid for all x. D(t) is not arbitrary and must satisfy a differential equation. Derive this differential equation.

(b) Now let us consider a matter dominated flat universe, so that ¯ρ(t)=a3ρc,0 where ρc,0 is the critical density today, 3H20/8πG as in Worksheet 11.1 (aside: such a universe sometimes is called the Einstein-de Sitter model). Recall that the behaviour of the scale factor of this universe can be written a(t)=(3H0t/2)2/3, which you learned in previous worksheets, and solve the differential equation for D(t). Hint: you can use the ansatz D(t)tq and plug it into the equation that you derived above; and you will end up with a quadratic equation for q. There are two solutions for q, and the general solution for D is a linear combination of two components: One gives you a growing function in t, denoting it as D+(t); another decreasing function in t, denoting it as D(t)

(c) Explain why the D+ component is generically the dominant one in structure formation, and show that in the Einstein-de Sitter model, D+(t)a(t).

2: Spherical collapse. Gravitational instability makes initial small density contrasts grow in time. When the density perturbation grows large enough, the linear theory, such as the one presented in the above exercise, breaks down. Generically speaking, non-linear and non-perturbative evolution of the density contrast have to be dealt with in numerical calculations. We will look at some amazingly numerical results later in this worksheet. However, in some very special situations, analytical treatment is possible and provide some insights to some important natures of gravitational collapse. In this exercise we study such an example.

(d) Plot r as a function of t for all three cases (i.e. use y-axis for r and x-axis for t), and show that in the closed case, the particle turns around and collapse; in the open case, the particle keeps expanding with some asymptotically positive velocity; and in the flat case, the particle reaches an infinite radius but with a velocity that approaches zero.

Let's begin.

(a) In the linear theory, it turns out that the fluid equations simplify such that the density contrast δ satisfies the following second-order differential equation
d2δdt2+2˙aadδdt=4πG¯ρδ

where a(t) is the scale factor of the universe. Notice that remarkably in the linear theory this equation does not contain spatial derivatives. Show that this means that the spatial shape of the density fluctuations is frozen in comoving coordinates, only their amplitude changes. Namely this means that we can factorize
δ(x,t)=D(t)˜δ(x)
where δ ̃(x) is arbitrary and independent of time, and D(t) is a function of time and valid for all x. D(t) is not arbitrary and must satisfy a differential equation. Derive this differential equation. 

Let's start with our equation:
d2δdt2+2˙aadδdt=4πG¯ρδ
Now, let's use the second equate to substitute in for delta.
d2D(t)˜δ(x)dt2+2˙aadD(t)˜δ(x)dt=4πG¯ρD(t)˜δ(x)
Now ˜δ cancels.
d2D(t)dt2+2˙aadD(t)dt=4πG¯ρD(t)
This resultant equation is entirely dependent upon t, as x does not appear anywhere in it, thus it is time independent, and is our differential equation of interest.

(b) Now let us consider a matter dominated flat universe, so that ¯ρ(t)=a3ρc,0 where ρc,0 is the critical density today, 3H20/8πG as in Worksheet 11.1 (aside: such a universe sometimes is called the Einstein-de Sitter model). Recall that the behaviour of the scale factor of this universe can be written a(t)=(3H0t/2)2/3, which you learned in previous worksheets, and solve the differential equation for D(t). Hint: you can use the ansatz D(t)tq and plug it into the equation that you derived above; and you will end up with a quadratic equation for q. There are two solutions for q, and the general solution for D is a linear combination of two components: One gives you a growing function in t, denoting it as D+(t); another decreasing function in t, denoting it as D(t)

We will start with the equation from before.
d2D(t)dt2+2˙aadD(t)dt=4πG¯ρD(t)
We now substitute in for D(t) and ¯ρ.
d2tqdt2+2˙aadtqdt=4πGρc,0a3tq
Now, let's focus on the ˙aa term.
We are told that a(t)=(3H0t/2)2/3, so:
˙aa=23t
Plugging this in, we get:
d2tqdt2+43tdtqdt=16πGρc,09H20t2tq
Taking our derivatives, we get:
q(q1)tq2+4qtq23=16πGρc,09H20tq2
tq2 cancels, leaving:
q2+q3=16πGρc,09H20
We now substitute in for ρc,0.
q2+q3=16πG3H209π8GH20
This simplifies to:
q2+q3=23
Solving for q we get:
q=1,2/3
Finalizing this, we get:
D+(t)t2/3 D(t)t1

(c) Explain why the D+ component is generically the dominant one in structure formation, and show that in the Einstein-de Sitter model, D+(t)a(t).

The D+ component will always be dominant, because as t increases, the D- term will go to 0 since 1=0.
Time for just a little more math:
D+(t)t2/3
a(t)=(3H0t/2)2/3t2/3
Thus:
D+(t)t2/3a(t)
D+(t)a(t)

d) Plot r as a function of t for all three cases (i.e. use y-axis for r and x-axis for t), and show that in the closed case, the particle turns around and collapse; in the open case, the particle keeps expanding with some asymptotically positive velocity; and in the flat case, the particle reaches an infinite radius but with a velocity that approaches zero.

The three cases are an Open Universe (Blue), a Flat Universe (Red), and an a Closed Universe (Black).


I worked with B. Brzycki and N. James on this (last) problem.