[4097] Physics 316: Extragalactic Astronomy and Cosmology (Summary of Lectures)
Lecture 36

Recent Results: HST observations detect white dwarfs with ages 12-13 Gyr in the Milky Way Globular cluster M4, placing the age of the universe at 13-14 Gyr [Press Release]

[L.S.Structure] Recap+ of last Time

Small density perturbations are parameterized by delta, the ratio of the density within the pertubation to the mean density of the universe at that time. These are thought to arise as the result of statistical (quantum) fluctuations during/after Inflation.

The question then becomes which (if any) of these perturbations can

  1. survive (not get damped) as a function of time
  2. grow (get enhanced - meaning delta increases) given the expansion of the universe
Of course what we're really interested in are those perturbations that can
  • survive & grow to the point where they undergo a gravitational instability as parameterized by the Jeans Mass
Such density perturbations are presumeably the "seeds" of the structures we see in the universe today.
Right is a slightly different plot of Jeans Mass vs Universal Scale factor
  • (dont worry about the slightly different behaviour to the one I showed last time in class between radaition-matter equality and recombination)
As we discussed, in the
  • RDE
    • Mj propto R3(t)
    • R(t) propto t1/2
  • MDE
    • Mj propto R-3/2(t)
    • R(t) propto t2/3
(see also General scaling summary) 		[Image Credit: E. Battaner & E. Florido]
The structures we see in the universe today are due to baryons. However the effects of dark matter can be detected (amonst other ways) via
  • the flat rotation curves of Spiral galaxies
    • the velocity of the baryonic matter (stars) as a function of radius is inconsistent with the observed distribution of baryonic matter as a function of radius
      • however if we add "extra" (unseen [dark]) matter distributed with the "natural" function of radius, then the flat rotation curves can be reconciled with the "revised" gravitational potential as a function of radius
  • the X-ray emitting "halos" of elliptical galaxies & clusters of galaxies
    • the X-ray emitting gas has thermal velocities in excess of the escape velocities the gravitational potential implied by observed distribution of baryonic matter
      • however if we add "extra" (unseen [dark]) matter, then we can decrease the escape velocities such that the gas is indeed gravitationally bound

[L.S.Structure] Evolution of a Density Perturbation

So how might a density perturbation"grow" ?
  • During the RDE the amplitude of these perturbations (ie. delta) is thought to grow linearly with time.
    • although a full General Relativistic treatment is necessary
  • During the "Acoustic Era" (if it really exists) the surviving (on scales such that they are not damped) perturbations (delta) are thought to be constant with time
    • those with scales above the photon damping scale will survive, but will not grow
[Image Credit: E. Battaner & E. Florido]
After recombination there is a huge reduction in Jeans Mass). All surviving density perturbations are suddenly above the Jeans Mass and hence subject to gravitational instability.

  • delta initially grows as t2/3 but thereafter quickly becomes non-linear
    • Thus requiring N-body simulations rather than direct (simple) analytical analsis.

[L.S.Structure] The "Acoustic" Period & CDM

One of the major uncertainties associated with the behaviour of the baryons is whether the Acoustic Era (for baryons) is actually important (or even exists) at all
  • For example, in CDM scenarios in which the CDM particles no longer interact with the photons once they become "cold" (ie.non-relativistic), then these particles might "clumping" (due to gravity) much earlier in the history of the universe. Thus prooducing the gravitational potential "seeds" into which baryons start to fall once they are "decoupled" from photons at recombination.
Until will can identify the CDM particle (or particles !) and its rest mass (their rest masses)
  • we have to "guess" when it (they) become non-relativistic and first start to clump
    • hence the maximium amplitude of delta it/they can give rise to prior to recombination
      • hence the depth of the gravitational "seed" potentials into which the baryons can fall into immediately after recombination
All of this impacts whether the period of "acoustic oscillation" of the baryons is actually of any consequence to structure formation scenarios (& indeed whether it exists at all).

Despite all the uncertainties, the Jeans Masses that "pop" out of the standard paradigm are in the range thet we observe. This is couraging !

[L.S.Structure] Top-down or Bottom-up ?

So there are various model-dependent possibilities as to what density pertubations (amplitude & size/mass scale)
  • survive until recombination
and (for those that do) whether & how quickly they can/do grow into structures of the mass/sacle-size we see today.

To put it another way:

  • What comes first - a sub-galaxy-sized potential, galaxy-sized potential or a supercluster-sized potential ?

Adding non-baryonic DM to mix complicates the situation. Once these particles become non-relativistic they can start to clump via the gravitational instability (assuming these particles not coupled to the photons). This can be long before recombination (perhaps even soon after the Inflationary period if the WIMPS are sufficiently massive). After recombination the baryons are able to "accrete" onto these "seeds".

  • The HDM candidates (neutrinos) remain free-streaming and damp fluctuations on scales up to the horizon size at the epoch they become non-relativistic.
    For a neutrino mass of 30 eV, this corresponds to For a neutrino mass of 1 eV, this corresponds to
    Damping Length 41 Mpc 1.2 Gpc
    Damping Mass 3x1015 Msun 3x1018 Msun

    Masses on these scales would be first to form (ie. Superclusters or larger). After recombination baryons "collect" on these scales. Later fragmentation then leads to smaller-scale structure. Thus HDM predict a top-down scenario for structure formation.
  • The more massive CDM candidates become non-relativistic and start to clump due to the gravitational instability earlier (long before recombination). Since the timescale for collapse is longer for larger regions (ie more massive regions), a Bottom-up scenario for structure formation therefore applies to CDM models. After recombination the baryons are able to "accrete" onto these "seeds", eventually making globular cluster & galaxy-sized objects by the gravitational coalescence of these subunits, then clusters etc etc. The greatest strength is the natural prediction that the baryonic (visible) matter should be at the center (deepest part of the DM potential) and hence that the observed small-scale structures be surrounded by DM halos.
    • clusters at high z are not predicted by this this scenario
Links to some of the Movies shown in class
CDM [Image Credit: GC3] MDM [Image Credit: GC3]

[L.S.Structure] The Statistics of the Large-scale Structure

Clearly there is a need to quantify both the observed and simulated distributions. There are two (related) techniques commonly employed to measure the observed structure as a function of scale-size
  • The Two-point correlation function
  • The Power Spectrum
Both are applied to large samples of objects. The shape of these functions can then be compared to the shape predicted by various structure formation scenarios.

Due to rather limited sample sizes, many early analyses were only able to accurately constrain the angular correlation function (basically a 2-D version of two-point correlation function). However since sufficiently large samples are now becoming available, we will not discuss the galaxy-galaxy or cluster-cluster angular correlation functions

  • of course we have already in connection with the fluctuations analysis on the CMB

[L.S.Structure] The Two-Point Correlation Function

(See also Bothun [Sect3.5.5] although I use a different notation).

Given a possionian distribution of objects, the probability (dP) of finding an object in a 3-D cell of volume dV is simply
dP = nm dV
where nm is the mean number density. Thus the probability of finding an object in both of two cells separated by a distance s is
dP = (nm)2 dV1 dV2

Thus one can parameterize any excess probability seen on the scale s as Xi(s) via
dP = (nm)2 dV1 dV2 [1+Xi(s)]
For a sample of objecs (eg from our redshift survey), we can then calculate
Xi(s) = < dn1 dn2 (nm)-2 >
where dn1 & dn2 are the excess densities (over nm) in our two cells separated by s, and the < > denote we take the ensemble average

There are a number of potential problems associated with this technique.
  • These are all related to the fact that the location of the objects in 3-D space requires knowledge of the redshift. As we discussed in class a couple of lectures ago, the peculiar velocities can lead to distortions in the observed function Xi(s)

[L.S.Structure] The Two-Point Correlation Function for Galaxies

For galaxies this two-point correlation function Xigg(s) is well represented by a power law
Xigg(s) = (s0/ s )g
over the range 200 kpc < hs < 20Mpc (where h=0.65 for H0=65 km s-1 Mpc-1) with
g = 1.71+/-0.05
and a "clustering length "
s0 = 5.1+/-0.2 h-1 Mpc
[Image Credit: Luigi Guzzo]

[L.S.Structure] The Two-Point Correlation Function for Clusters

For clusters of galaxies the two-point correlation function Xicc(s) is also well represented by a power law over the range 2 Mpc < hs < 200 Mpc (where h=0.65 for H0=65 km s-1 Mpc-1) with
g = 1.9
and a "clustering length "
s0 = 17 h-1 Mpc
[Image Credit: Luigi Guzzo]

The previous two two-point correlation functions were constructed using data from optical surveys.

For clusters of galaxies we can also use X-ray surveys.

Since the X-ray luminosity is more closely related to the (total) mass than the optical light, this will serve as a good check.

Indeed there is good agreement.

[Image Credit: Luigi Guzzo]

End of lecture

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