Lecture 25: Large Scale Structure

Studying the distribution of matter on all scales may give clues as to what the distribution of matter was like at earlier times -- especially on the largest scales as the Universe isn't old enough for the largest scale distributions to have changed much.

• this is a relatively new area in astronomy (such issues didn't arise when galaxies were still thought to be part of the Milky Way!).
• Concept of hierarchical structures meaning small objects group together to form larger entities which group together to form yet larger entities and so on has been one theme for studying structures in the Universe.

Galaxies => Groups => Clusters => Superclusters

Milky Way is a member of the Local Group which lies on the outskirts of the Virgo Supercluster. The Virgo Cluster is the nearest cluster. The Local Group consists of two large spirals (Milky Way and M31), a small spiral (M33), and a number of irregulars and small ellipticals for a total of about 20 members.

Note that large empty regions between clusters and superclusters have also been found -- these are called voids.

Cosmological Principle: We do not live in a special location in the Universe. In other words, "The Universe is the same everywhere". Clearly the Universe is not the same everywhere on small scales -- galaxies exist and there are spaces between them. Likewise clusters exist with spaces between them. On what scale is the Universe uniform?

What does "uniform" mean? -- It means that the Universe is not only isotropic (meaning the same when viewed in any direction) but homogenous, meaning the same in very suitably sized volume element.

Hubble's Study of the Distribution of Galaxies

Hubble could not take spectra of large numbers of galaxies to determine their distance (exposures were measured in hours for one galaxy in those days) so he devised the following scheme to study the distribution of galaxies throughout space:

R=distance to a shell surrounding the observer

dR=thickness of the shell corresponding to a change in brightness of 1 magnitude.

           

Compute the number of galaxies in this "one magnitude" thick shell:

This result says that if Hubble counted galaxies to a magnitude limit and then counted to a limit of one magnitude fainter, he would have 10^0.6=4 times as many galaxies in the fainter bin. This result assumes that galaxies are distributed uniformly in space and that space has a Euclidean geometry. To the levels that Hubble could count, he saw that the log of the counts increased by 0.6 for each magnitude fainter that he counted. Modern counting tests show that the Universe is not Euclidean. An additional complication is that galaxies are not all the same and appear systematically brighter in the past when more massive stars were still on the main sequence in all galaxies.

Modern Studies of Large Scale Structure

Much progress has been made in our ability to take many spectra of galaxies at once. We can now produce various 3-dimensional views of the sky:

The Local Supercluster

Clearly visible are clusters and superclusters as well as voids. From an observational standpoint, the Universe appears to have a structure like foam or bubbles or filaments when viewed on scales of ~100s of Mpc.

Two very different schemes for modeling how this structure has developed in the Universe have been proposed:

Top-Down Formation:               Bottom-up Formation:

    Early Universe                           Early Universe

Formation of Superclusters     gas clouds, stars, galaxies

Formation of clusters                 galaxies form clusters

Formation of galaxies         clusters form superclusters

Stars form in galaxies

The bottom-up model suffers from gravity not being able to act quickly enough over large distances to cause the formation of superclusters. The spherical appearance of some clusters also suggests that they have been gravitationally bound for a long enough period of time that gravity has acted to equilibrate their shapes which is also a potential strike against this model.

However, HST observations of distant galaxies suggests that formation of galaxies from smaller units may be common which would support at least part of this picture.

The top-down model has suffered from the need to have relatively large fluctuations in mass present shortly after the Big Bang. These fluctuations may have been observed by the COBE (Cosmic Background Explorer) satellite which would lend support to this model.

Output from a computer simulation of the structure of the Universe:

Friedmann Models, Evolution of Galaxies

To study how galaxies change over time, we can take advantage of how long it takes light to travel to us from a distant galaxy. For example, a galaxy lying at a distance of 1 Gpc is seen as it was over 3 billion years ago (recall that 1 parsec = 3.26 light-years).

Some Preliminaries about the Universe as a whole:

Hubble's Law implies directly that the Universe is expanding:

V = H_oxd => Rate at which an object is moving away from us is proportional to its distance.

This property of velocity away from us being linearly proportional to the object's distance is a simple consequence of the fact that the Universe is expanding.

Consider a one-dimensional example. A rubber ruler at time t=0 has marks on it at points A,B,C as listed. The ruler is then stretched so that it becomes twice as long and the separation of the three points is noted at a later time t=t'. The observer is located at D=0.

A three-dimensional analog to this expanding rubber ruler is a loaf of raisin bread -- the dough (to be equated with space) contains raisins (to be equated with galaxies) that move further apart as the dough expands.

Note a key concept here: it is space itself which is expanding. The sizes of objects like hydrogen atoms are not changing nor are the sizes of objects held together by gravitational attraction such as clusters of galaxies.

Significance of Hubble's Constant

Hubble's Constant is the rate at which the Universe is expanding divided by the current size of the Universe denoted as R_o:

If the Universe has been expanding at a uniform velocity since t=0, we can equate the reciprocal of Hubble Constant to the current age of the Universe:

Compute what density is required for a galaxy's velocity to just equal the escape speed:

r = distance to the galaxy

M=total mass contained within the sphere with radius r

Toting up all of the known baryonic matter gives a density of less than .04.

If we observe a very distant galaxy, its motion will refer to a time much earlier than the present and so the expansion of the universe will not have slowed as much as now. Consequently Hubble's Constant will have a slightly different value. The rate of change of the expansion rate of the Universe is called the deceleration parameter with the current value usually denoted as q_o. Formally, q_o is written as

This somewhat peculiar form for the scaling of this second derivative arises from casting the derivative into a form that relates the second derivative of the size of the Universe to observable properties of the Universe.

The amount of matter in the Universe and hence are related to q_o since it is the matter that is doing the deceleration:

Type	Discoverers	Spatial Curvature	Volume	Density	q0	Age
Closed	Friedmann-Lemaitre	Positive	Finite		>0.5	<2/3Ho-1
Flat	Einstein-deSitter	Zero	Infinite		0.5	2/3Ho-1
Open	Friedmann-Lemaitre	Negative	Infinite		<0.5	2/3Ho-1<to <Ho-1

 Spatial Curvature: recall that in Einstein's General Theory of Relativity, the presence of a mass causes space to be curved; here we see this concept extended to the Universe as a whole. Essentially this means that light cannot travel along a straight line; it must follow a curved path.

Closed (also called bound): Universe expands and re-collapses cyclically

Flat: Expansion stops at infinite time

Open (also called unbound): Universe expands forever

If you are curious where this all comes from:

Using General Relativity, an expression for the radius R of the Universe can be derived:

k is the spatial curvature and is called the Cosmological Constant and has no physical interpretation known now but some data on distance supernovae imply that it may not equal 0!

Returning to computing the "lookback" time to a distant galaxy, the lookback time would be easy to compute if the Universe were expanding at a uniform rate and if light traveled along a straight line path. But neither of these conditions appears to be correct, and the differential equations above must be solved to compute the lookback time. Luckily, the solution is available but rather messy. The time dependence of quantities is explicitly shown and t₁=time when light was emitted by our distant galaxy and t_o=now (note that these are in parametric form, is called the development angle and is a computational convenience):

(and recall the version good for high velocities given in Lecture 24)

For q_o > 0.5,

For q_o=0.5,

For q_o<0.5,

Beware! You must use a form of H_o with units of inverse time in the above relations!

Example:

Compute the lookback time to a galaxy with z=.87 (which would have been measured from the wavelength shift of its spectral lines). Use q_o=.52 and H_o=75km/sec/Mpc=1/(1.3x10¹⁰yrs) in this example.

First compute and :

Now compute the value of t_o-t₁:

So it has taken 5.3 billion yrs for the light from this galaxy to reach us -- the Sun had not been formed when the light we are observing now left this galaxy!

Studying Galaxy Evolution

The preceding example illustrates that if we observe a particular type of galaxy at a variety of redshifts, we can hope to see how it has changed over time. Using our knowledge of stellar evolution will also be useful since any galaxy is comprised of a population of stars whose properties will change over time.

Caveats:

1) We need to know H_o and q_o.

2) We need to be able to identify the same type of galaxy at a great distance as we observe nearby.

Item 1 has been partially addressed using the Hubble Space Telescope to determine H_o; q_o can be derived from some data completely outside the current expansion rate of the Universe but its value may be ~0.1 but is still controversial and may be 0.5 if the recent supernovae data on the Cosmological Constant are correct.

Item 2 also relies heavily on HST but since HST has instruments that cannot be adjusted to observe at sufficiently long wavelengths, one is forced to observe the shapes of distant galaxies at what were ultraviolet wavelengths in the rest frame. Because small amounts of star formation can appear very bright at ultraviolet wavelengths, this regime is not the best for checking whether a distant galaxy is the same or different in shape from a nearby galaxy. A new mission, Next Generation Space Telescope (NGST) is being designed to help solve this problem.