Transcript Section9x - University of Chicago

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 overdensity 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 Criterion
The 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 overdensity) 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
8πGρmH
1/2
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
GρmH
1/2
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 overdensity 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
GρmH
1/2
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 Stars
The 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 Stars
There 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 Light
Locating 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
Finding Distant Quasars
Quasars can be identified as radio or X-ray sources. Many of
discrete sources we see at
the dicrete
these wavelengths are
Radio
AGN, and some are very
distant.
X-ray
(and, of course, some quasars
are very bright!)
Finding Distant Quasars
Also, 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 Forest
Observations 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 Galaxies
Early 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 Galaxies
A 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
Finding Distant Galaxies
Using lensing to find distant galaxies is now the
leading technique. For example:
Finding Distant Galaxies
THE 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
Reionization
In 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!
quasar redshift
Ly-A line in quasar
dark gap
image from Xiaohui Fan
Reionization
The 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
Reionization
Simulations 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 Structure
The 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
from Loh et al. 2008
The Formation of Structure
The 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 both
ΩM and ΩΛ
two simulations at
z=0 using the
same
start
conditions
with
different
cosmologies
ΩM = 0.3
ΩM =1.0
*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