Blog Post 37, Illustris Simulation

The Illustris Simulation is a numerical cosmological simulation which includes the effects of dark energy, dark matter, baryonic matter, and the relevant underlying physics (radiation, fluid dynamics, gravity, etc).

To look at one aspect of large scale structure in this simulation, we will consider the halo mass function (HMF). The HMF is just a fancy way of counting the humber of halos within some mass bin. The halo is the extended dark matter structure which surrounds galaxies individually, and galaxy clusters.

Create a histogram of the data with bins that have a width of 0.5 in log(M). Are the low mass halos more numerous, or are the high mass halos?

We first used the Illustris Explorer tool to examine subhalos within a 400.0 kpc/h radius. We performed this search several times, with each search returning the position, halo mass, and star mass of up to 20 subhalos. We combined the output for several searches to get the following histogram

As you can see in the histogram, low mass halos are significantly more numerous than the high mass halos.

On average, what fraction of the halo mass is the stellar mass? 

On average, the stellar mass makes up around 90% of the halo mass.

Exploring Structure and Reionization in the Illustris Simulation

Next, we compared the structures between the gas density and dark matter density in the simulation. On the large scale (full box), the simulation looks like the following, with gas density on the right and dark matter density on the left:

On the small scale, the simulation looks like the following, again with dark matter density on the left, and gas density on the right:

Compare the Gas Density and the Dark Matter Density on both large (full box) and small scales (single cluster).  Describe how the Gas and Dark Matter are similar/different on each scale and speculate as to why.

As you can see in the photos above, while the gas density and dark matter density both follow similar patters, the dark matter density is much more structured, while the gas density is more diffuse, both on the large scale and on the small scale. This is particularly noticeable on the small scale, as the dark matter appears to be much more clustered than the gas density, which is more evenly spread out.


Which, the gas or the dark matter, is more confined to the filamentary structure and why?

The dark matter is more confined to the filamentary structure, as you can see on the large scale image of the dark matter density. This could possibly be because of the increased density of the dark matter; as dark matter is thought to constitute around 80% of the matter in our galaxy, this greater mass results in a larger gravitational attraction between the dark matter particles than the gas particles.

In most of medium to large galaxies, is the gas densest towards the nucleus or the disk?

In most the medium to large-scale galaxies, the gas appears to be denser towards the nucleus of the galaxy, as shown in the following snapshot:

Are the most massive galaxies in the field or in clusters?

The most massive galaxies are in the cluster. As this is where the majority of gas is concentrated, it makes sense that the galaxies that form within the clusters are larger on average than those in the field.

Next, we watched the following video, which shows the evolution of the dark matter and gas within our universe in the time since the big bang:

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

When do the first stars form in this simulation?

The first stars appear around 0.6 billion years after the Big Bang, when there is a redshift of about 8.

For what approximate redshift range is the rate at which stars are forming fastest?

The stars appear to be forming at the fastest rate when the redshift is between 1.0 and 2.0.

When, in years and redshift, does the gas temperature brighten (go from blue to having green?) This is the beginning of the "Epoch of Reionization" or the end of the "Dark Ages."

We begin to see the first ionized gases - or when the gas temperature first turns green - about 1.0 billion years after the Big Bang, with a redshift of 5.80.

In this simulation, when structures are forming, are smaller structures combining to form larger ones, or are large objects breaking up? Why do you think this is?

In the process of gas formation, we see structures form as large objects are breaking up. This makes sense, as structures gain more matter, they are likely to collapse under their own gravity in a supernovae.

Why do you think structures form along filaments?

As we saw above, dark matter forms along the filamentary structures. As dark matter makes up the majority of matter in the universe, gravitational attraction from dark matter results in the formation of structures along these filaments as well.

Blog Post 36, Worksheet 12.1, Problem 1 and 2(d)

1. Linear perturbation theory. 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 \(\bar{\rho}(t)\). Nonetheless, there are some tiny fluctuations and and not everywhere exactly the same. So let us define the density at comoving position r and time t as \(\rho (x, t)\) and the relative density contrast as \[\delta (r, t) \equiv \frac{\rho (r, t) - \bar{\rho} (t)}{\bar{\rho}(t)}.\] In this exercise we focus on the linear theory, namely, the density contrast in the problem remains small enough so we need only consider terms linear in \(\delta\). 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 \(\delta\) satisfies the following second-order differential equation \[\frac{d^2\delta}{dt^2}+\frac{2\dot{a}}{a}\frac{d\delta}{dt} = 4πG\bar{\rho}\delta.\] 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 is comoving coordinates, only their amplitude changes. Namely this means that we can factorize \[\delta(x, t) = D(t)\bar{\delta}(x),\] where \(\bar{\delta}(x)\) 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. 

We are given the equation in the form \[\frac{d^2\delta}{dt^2}+\frac{2\dot{a}}{a}\frac{d\delta}{dt} = 4πG\bar{\rho}\delta.\] We want to show that the equation for \(\delta\) can be written in the form \(\delta(x, t) = D(t)\bar{\delta}(x),\). To show this, we can use this equation as a substitution for \(\delta\) in above differential equation: \[\ddot{D}(t)\bar{\delta}(x) + \frac{2\dot{a}}{a}(\dot{D}(t)(\bar{\delta}(x)) = 4πG\bar{\rho}D(t)\bar{\delta}(x)\] This simplifies to: \[\ddot{D}(t) + \frac{2\dot{a}}{a}\dot{D}(t) = 4πG\bar{\rho}D(t),\] which is in the same form as our first equation.

(b) Now let us consider a matter dominated flat universe, so that \(\bar{\rho} = a^{-3}\rho_{c,0}\), where \(\rho_{c,0}\) is the critical density today, \(2H_0^2/8πG\) 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 \(a(t) = (3H_0t/2)^{2/3}\), which you learned in previous worksheets, and solve the differential equation for D(t). Hint: you can use the ansatz \(D(t) \propto t^q\) 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 is \(D_+(t)\); another decreasing function in t, denoting it as \(D\_(t)\). 

We are given that \(a(t) = (3H_0t/2)^{2/3}\), and thus we know that \(\dot{a}(t) = H_0 (\frac{3H_0t}{2})^{-1/3}\). We combine these to get the quantity: \[\frac{\dot{a}}{a} = \frac{H_0 (\frac{3H_0t}{2})^{-1/3}}{(3H_0t/2)^{2/3}}\] \[\frac{\dot{a}}{a} = \frac{2}{3t}\] We are also given that \(\bar{\rho} = a^{-3}\rho_{c,0}\). We can substitute in our expression for a above to get a new expression for \(\bar{\rho}\): \[\bar{\rho} = [3H_0t/2)^{2/3}]^{-3}(2H_0^2/8πG) = \frac{1}{6πGt^2}\] Now we can plug these two expressions (for \(\bar{\rho}\) and for \(\frac{\dot{a}}{a}\)) into our differential equation from part (a): \[\ddot{D}(t) + 2(\frac{2}{3t})\dot{D}(t) = \frac{4πGD(t)}{6πGt^2}\] \[\ddot{D}(t) + \frac{4}{3t}\dot{D}(t) = \frac{2}{3t}D(t)\] We can now use the substitution suggested in the hint above, \(D(t) \propto  t^q\) to make a substitution for \(D(t), \dot{D}(t)\), and \(\ddot{D}(t)\): \[q(q=1)t^{q-2} + \frac{4}{3t}qt^{q-1} - \frac{2}{33t^2}t^q = 0\] This simplifies to: \[\frac{1}{3}t^{q-2}[3q62 + q - 2] = 0\] We can use the quadratic equation to solve for our two possible answers for q, \(q = \frac{2}{3}\) and \(q = -1\). We plug these bag into the ansatz given in the question to get: \[D_+(t) \propto t^{2/3}\] \[D\_(t)\propto \frac{1}{t}\] (c) Explain why the \(D_+\) component is generically the dominant one in structure formation, and show that in the Einstein-de Sitter model, \(D_+(t) \propto a(t).\)

As t increases over time, \(D\_(t)\) will get closer and closer to 0, as \(D\_(t)\propto \frac{1}{t}\), and thus the \(D_+\) component is generically the dominant one in structure formation.

In the Einstein-de Sitter model, we know that \(a(t) = (3H_0t/2)^{2/3}\), and thus \(a(t) \propto t^{2/3}\). Thus we have: \[D_+(t) \propto t^{2/3}\] \[a(t) \propto t^{2/3}\] \[D_+(t) \propto a(t)\]
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.

(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 collapses; 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. 

Closed case

\[r = A(1 - \text{cos }\eta)\] \[t = B(\eta - \text{sin }\eta)\]

In this plot, t is on the x-axis, and r(t) is on the y-axis. (Scaled to not include the multiplication factors of A and B). As the time increases, the particle at first moves outward some distance r, and then turns around and returns to a distance of r = 0. 

Open case

\[r = A(\text{cosh }\eta - 1 \] \[t = B(\text{sinh }\eta - \eta)\]

The following two plots show the open case with with relatively large and small limits for \(\eta\), respectively, with t on the x-axis, and r(t) on the y-axis:

As time increases, the distance r from the center of the sphere continues to increase. The velocity of the particle is always positive, and thus the particle keeps expanding with some asymptotically positive velocity. 

Flat case

\[r = A\eta^2/2\] \[t = B\eta^3/6\]

In this case, as time increases, while the distance of the particle continues to increase, the velocity of the particle decreases. Thus, the particle reaches an infinite radius but with a velocity that approaches zero.