Jeans Instability Criterion

The structures we see around us today and even in the distant universe (galaxies, clusters of galaxies, etc.) must form gravitationally from the growth of initial density fluctuations in the early universe. To consider how this happens let’s establish the basic criterion for collapse of an over-density:

Consider a uniform spherical region with the following basic properties: temperature, T, uniform density, ρ, and size (radius), R. The total mass, M, follows as

M=4πR3ρ/3

For a spherical region, the gravitational force on that particle due to the over-density acts as though the mass were concentrated at the center (this is exactly analogous to the situation of a body falling at the surface of the Earth, in which the net gravitational effect is computed by placing the Earth's mass at a point at the Earth's center).

Jeans Instability CriterionThe escape velocity for that particle is

Vesc=(2GM/R)1/2

The motions of particles in a gas is fundamentally related to the temperature (hotter = faster); this is specified by relating the typical kinetic energy of a particle to the typical thermodynamic energy of a particle in a gas, i.e.

3kT/2=mHv2/2

Note here that the mass is the particle mass (NOT the mass of the over-density) and that we use the mass of a hydrogen atom here since hydrogen is by far the dominant constituent of the universe. From this we can readily deduce the typical particle velocity as

v=(3kt/mH)1/2

If the typical particle velocity is larger than the escape velocity then atoms at the edge of the cloud are unbound and will move away. Over time, the cloud will dissipate.

Jeans Instability Criterion

The over-density will only collapse under the force of gravity if the typical particle velocity is less than escape velocity. We can find this threshold criterion (which is a relationship between the cloud's temperature, radius, and density) by equating the two velocities. So, to start we have

(2GM/R)1/2=(3kT/mH)1/2

and thus

2GM/R=3kT/mH

Recall M=4πR3ρ/3 and therefore

8πR2Gρ/3=3kT/mH

Jeans Instability Criterion

Now, rearrange to solve for R, and we find that for a given value of ρ and T, a region just on the brink between collapse and dispersal has a radius of

R= 9kT 1/2

8πGρmH

This radius is called the Jeans radius, RJ, after James Jeans who derived this condition in 1920.

The simplified spherical treatment we have used here is approximate, but gives the same approximate result as more complex theoretical treatments. The textbook gives the Jeans length as

L= πkT 1/2

GρmH

which is very similar, noting that the text’s value nominally refers to size (i.e. diameter, i.e. L=2R)

Jeans Instability Criteria

The importance of this stability criterion is as follows:

Given an over-density of temperature T and particle density n, at some radius R it will be balanced between collapse and expansion

If the length is less than the Jeans length then the over-density does NOT have enough mass to keep particles from escaping. The over-density will evaporate.

However, if the scale of the over-density is larger than the Jean's length then particles at the edge of the over-density are gravitationally bound, and the ove-rdensity will collapse under its self-gravity.

Jeans Instability Criteria

What was the temperature and density at recombination? The current temperature of CMD is 2.7 K, and the redshift of recombination was ~1080. Therefore at recombination the temperature of photons (and hence mater, because they were close-couples by photon-electron scatter prior to that!) is

Trec=2.7*(1+1080) i,e, Trec is approximately 3000K

Remember also that we can compute the critical density for the universe currently

Jeans Instability Criteria

Also, recall that the matter density is only about a quarter of the critical value, so that

from which we get that currently

The matter density simply goes as the scale factor (i.e. 1+z) cubed, so the matter density at recombination is simply

ρm=2.4x10-27 (1+1080)3 or at recombination we have a density of

approximately ρm=3x10-18 kg/m3

Jeans Instability Criterion

The textbook Jeans length is

L= πkT 1/2

GρmH

so after substituting in the values of temperature and density of matter at recombination we get

L=6.2x1017 m

or about 66 light years!

At the typical density, a region this size has a total mass of about 5x105 Msun

This is about the size and mass of a globular cluster! Its much more than a star, and much less than galaxy...anything large than this size can begin to collapse.

The First ‘Galaxies’

The gravitationally driven collapse of objects after recombination is actually more complex than the simple Jeans calculation would suggest. Remember this picture?

The First ‘Galaxies’

At recombination, dark matter over-densities already exist! It is these over-densities that seed the collapse of structure – indeed without them simulations of early structure formation basically fail to assemble baryonic objects which become galaxies.

The First StarsThe Jeans length at recombination is about the size of

a globular cluster - much much larger than a star! As the universe expands, the matter density drops as (1+z)3 and the temperature as1+z and so the Jeans length grows as the universe expands. That is, from this simple criterion it becomes even harder and harder for star-sized objects to collapse!

So, although galaxy-sized and larger objects can collapse, the formation of the first stars requires some additional complexity. The basic solution is to invoke an additional cooling mechanism and form star-sized clumps down inside of the collapsing globular cluster sized objects. Such stars have never been seen – yet. We call this expected population of stars “Population III”

The First StarsThere is ongoing

debate as to exactly how the first stars form ; the debate centers on how the gas cools, and how angular momentum gets transported. Much of the research is driven by complex simulations:

First Stars and Galaxies• Population III stars -> very massive

>30M☉, no “metals”, and not seen (yet!) since short lifetime

• Population II stars -> low “metal” abundance, in globular clusters and galactic halo

• Population I stars -> high “metal” abundance, seen in galaxy disk and nucleus. Are born from gas which has been polluted by lots of prior supernovae

The Quest for First LightLocating and characterizing the first stars and galaxies to form is referred to as the search for ‘first light’. Since the discovery of the first quasars in the late 1960s, the record-holding ‘most distant object’ has gotten more and more distant. The current record holders are galaxies and quasars at z~8 or so! The most distant objects are extraordinarily faint and discovering them requires big telescopes and sophisticated methods.

Finding Distant QuasarsQuasars can be identified as radio or X-ray sources. Many of the dicrete

Radio

X-ray

discrete sources we see at these wavelengths are AGN, and some are very distant.

(and, of course, some quasars are very bright!)

Finding Distant QuasarsAlso, because quasars have unique spectra (they have a very blue spectral continuum, and have very strong and wide emission lines) they can be reliably isolated from stars either using multi-color images, or spectral surveys. In particular, quasars show a UV excess relative to stars, even if both appear as point sources in telescopes.

Typical quasar

Sun-like star

The Lyman-Alpha ForestObservations of distant quasars reveal absorption features due to intervening materiel. Any neutral hydrogen between us and the quasar will produce absorption lines of hydrogen – particularly the so-called Lyman-alpha line (electron jumping from n=1 to n=2 energy level). The aggregate of these absorption lines is called the ‘Lyman alpha forest’

low z high z

The Lyman-Alpha Forest

Every one of these absorption lines is due to an intervening cloud of hydrogen!

Finding Distant GalaxiesEarly discoveries of distant galaxies were also made by observing radio sources. In this case the galaxies found are the hosts of active radio emitting jets – distant analogs of objects like Centaurus A Cygnus A and M87 in the nearby universe. Centaurus A

Finding Distant Galaxies

In the mid 1990s it was recognized that the aggregate effect of the Lyman-alpha forest could be used to find very distant galaxies and quasars; since then, almost all of the most distant objects have been found at optical and near-IR wavelengths sensitive to this feature. Exploiting this technique requires very deep images – usually only achievable with large telescopes or space-base observatories

The Hubble Ultra-Deep Field

Finding Distant GalaxiesA powerful method to find distant objects is to use the magnifying power of gravitational lenses to aid in the search. For example, in 1997 the then most distant galaxy known was this one:

Lensing makes the very faint distant galaxies much brighter, allowing their discovery with smaller telescopes or more modest observations.

Finding Distant GalaxiesUsing lensing to find distant galaxies is now the leading technique. For example:

Finding Distant GalaxiesTHE most distant galaxies known as of right now are at z~9, and there is alot of debate about how many of these ‘candidate’ distant galaxies are actually valid discoveries. Here’s a recent paper (14 May 2010):

Reionization

As the first massive stars and quasars form they will be emitting lots of UV photons. These will tend to to re-ionize the hydrogen in the Universe (prior to this, hydrogen was last ionized at the surface of last scattering.) In fact, the Universe we see around us today has neutral hydrogen only in dense regions (the disks of spiral galaxies for example) that are self-shielded from these ionizing photons. The outskirts of galaxies and the gas in clusters and groups (the intra-cluster medium) is all ionized! This transition from smoothly distributed neutral hydrogen in the Universe after recombination to clumpy neutral hydrogen surrounded by ionized hydrogen marks the end of the so-called ‘epoch of reionization’

Reionization and the Lyman-alpha Forest

The ionization state of the diffuse gas in the outskirts of galaxies (and even the diffuse gas in small dark matter clumps that perhaps have never even formed stars) can be deduced from the spectra of quasars. The aggregate properties of the Lyman-alpha forest encodes this information.

low z high z

ReionizationIn the nearby universe there are not too many absorbers

But in the distant universe there are! Small clumps of neutral hydrogen are more common at high redshift.

Reionization

But even at z=4 lots of quasar light gets through – showing that neutral hydrogen is very clumpy even then (though there are lots of clumps!) However, recent observations of quasars at z=6 and beyond show something else – a complete absence of light at intervening Ly-A wavelengths!

Ly-A line in quasarquasar redshift

dark gap

image from Xiaohui Fan

ReionizationThe onset of this apparent opacity at Ly-A wavelengths is due to an increase in the amount of diffuse neutral hydrogen – in other words, we have seen the tail-end of the epoch of reionization in the spectra of very distant quasars!

Fan et al. 2006

ReionizationSimulations of reionization suggest that it occurs in a clumpy fashion – with individual zones of ionized hydrogen first forming around the first stars and quasars, and then growing and connecting until most of the universe is reionized – by about z=6 as quasar data suggest.

Reionization

One final question: if a fully ionized universe at z=1080 (when the CMB forms) is opaque to photons at all wavelengths, why doesn’t the universe at z=6 turn opaque again as it re-ionizes?

Answer: density

At z=6, the matter in the universe is on average

(1080/6)3 = 5.8 million times less dense!

The Formation of Structure

Once the first stars and galaxies form we are basically in territory for which there is good observational data. We now have observed many galaxies to about z~4 and can trace how those objects evolve down to lower redshifts.

In fact the evolution of galaxy populations is itself another piece of evidence for a Universe of finite age (i.e. Big Bang cosmology). We can directly observe that galaxies at high redshifts have younger stellar populations!

The Formation of StructureThe first piece of direct evidence for a change in galaxy populations came in the 1980s from observations of something now known as the Butcher-Oemler effect. Basically this is the observation that distant clusters of galaxies have proportionately more blue (actively star forming) galaxies in them than nearby clusters.

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The Formation of StructureThe details of how large scale structure* evolves over cosmic time are sensitive to cosmology. In particular the number of massive galaxy clusters and also the size and amount of under-density of voids yields a measure of bothΩM and ΩΛ

*large scale structure is a grab-bag name that describes the arrangement of both visible and dark matter on scales larger than galaxies and up to 100’s of Mpc

ΩM = 0.3 ΩM =1.0

two simulations at z=0 using the same start conditions with different cosmologies