Dynamical Tests

The idea for the dynamical tests is straightforward. We, in a sense, try to determine whether the initial kick given to the Universe was large enough to cause the Universe to exceed its escape velocity.

How do we go about this exercise?

Well, the procedure is:

• determine the escape speed of the Universe.
  • the escape speed is determined by how strongly gravity retards the expansion and so, we can get a measure of the escape speed by measuring the mass of the Universe
• determine how fast the Universe is expanding.
  • Hubble's law, v = H(now) x D ===> H(now) and the Hubble constant tells how fast the Universe is currently expanding.
  • Compare the expansion rate to the escape speed.

Seems simple. Let's do it.

Start with:

• -k = 1/2 [dR(t)/dt]**2 -(4 pi/3) G rho(t) R(t)**2 -(CC/3) R(t)**2
  • Define the critical density for a universe with CC = 0 ===> 0 = 1/2 [H(now) R(t)]**2 - (4 pi G/3) rho(t) R(t)**2 - 0

    ===> rho(critical) = (3 H(now)**2 / [8 pi G] )

  • So, what if CC is not 0? Well, then in a closed Universe we must have

    ===> rho(t) > rho(critical) - (CC/[4 pi G] = rho(critical) [ 1 - 2 CC/(3 H(now)**2) ], or

    ===> rho(t) + CC/(4 pi G) > rho(critical)

  I write the relation in the last way to show explicitly how the Cosmological constant enters. If CC > 0, then the mass density, etc. can be less than the critical density and the Univere can still be closed (or flat).

Weighing the Universe

There are obvious contributions to rho(t), e.g., planets, galaxies, ... . However, there are other not so obvious contributions. We need

• rho(t) = (rho(baryons) + rho(neutrinos) + rho(photons) + ... ) + rho(Cosmological constant)

I lumped in the CC/(4 pi G) term into the density in the above. Using the above, define

• Omega = Omega(baryons) + Omega(neutrinos) + Omega(photons) + ... + Omega(CC)

So the issue becomes how do we measure all of these contributions to Omega?

But before we go on, photons are in there, but why? Aren't photons massless? And what if neutrinos turn out to be massless? Why are they there?

• According to Einstein, E = mc**2, and mass and energy are equivalent. They are just different forms of a single entity

  ===> mass = Energy / c**2

• So, for the CMBR ===> rho(photons) = U / c**2 = a T**4 / c**2 ===> rho(photons) = 4.7 x 10**(-34) grams per cubic centimeter << than any reasonable rho(critical)

• The neutrinos are more problematic.

  For the Cosmic Neutrino Background Radiation (CNBR)

  ====> rho(neutrinos) = U / c**2 = a T**4 / c**2 for massless neutrinos

  ====> rho(neutrinos) = 0.3 rho(photons) since T(neutrinos) ~ 1.9 Kelvin as compared to T(photons) ~ 2.74 Kelvin. If neutrinos are massless then the CNBR does not make a large contribution to the matter density of the Universe.

• What if neutrinos are not massless?
  • number of neutrinos per c.c. is N(neutrinos) ~ aT**4/kT ~ 375 per c.c.
  • ===> rho(neutrinos) ~ 375 m(neutrino) grams per c.c.
  • To get an idea about when neutrinos are important, let's use H(now) ~ 25 [km/s] per Million light years ===> rho(critical) ~ 1.2 x 10**(-29) grams per cubic centimeter
  • ===> rho(neutrinos) > rho(critical) if m(neutrino) > 3.2 x 10**(-32) grams ~ 0.000035 m(electron) or 18 eV (or so).

    Neutrinos can either make a significant contribution to the mass of the Universe or not, depending upon whether they have a significant rest mass.

Galactic Masses (Individuals and Groups)

In this section, I will really be discussing using galaxies as probes of the mass of the Universe. In some cases, there will be direct counting of galaxies, in other cases, the galaxies will be used as test particles. Define

• Omega(gal) = Omega(baryon) + Omega(other?)

We will measure Omega(gal) using light and dynamical techniques

Light Method

This is seemingly straight-forward, we simply count up galaxies. The complication is that we observe primarily the galaxies which are easy to see. However, there are also very many galaxies which are hard to see. These hard to see galaxies are the ones with low surface brightness (which means that they are very hard to see against the pervasive background of the night sky. The number of these low surface brightness galaxies is not yet well-established, however, the current density distribution of galaxies is interesting. But anyway, let's press onward under the assumption that we may be making a factor of 2 error or so by only considering the bright galaxies in our discussion.

Galaxies are divided up into different types (morphological classification scheme):

The different types of galaxies have different Mass-to-Light (M/L) ratios. To get a feel for what M/L tells us, consider:

• A galaxy made of 10**11 stars like the Sun has M/L = 10**11 M(Sun) / 10**11 L(Sun) ===> M/L = 1 M(Sun)/L(Sun). If we measure M/L in units of solar masses to solar luminosities, then this galaxy has M/L = 1.
• Suppose that this same galaxy also had 10**11 black holes of mass 1 M(Sun) ===> M/L = 2 x 10**11 M(Sun) / 10**11 L(Sun) ===> M/L = 2
• Suppose that we had a galaxy composed of 10**11 stars of mass 10 M(Sun) and 10**4 L(Sun) ===> M/L = 10**11 x 10 M(Sun) / [10**11 x 10**4 L(Sun)] ===> M/L = 0.001

  The Mass-to-Light depends on the types of stars that populate galaxies and how much unseen matter exists in the galaxies.

The typical values for the visible parts of galaxies are:

• Ellipticals: (M/L) ~ 7 - 20
• Spirals: (M/L) ~ 3 - 5

===> Omega(gal,visible) ~ 0.007 (factor of 2 uncertainty) << 1

Dynamical Methods

The dynamical methods are indirect in the sense that you do not count-up things. You rely on Newton's laws of physics and the nature of gravity. The dynamical methods are applied on all scales -- from individual galaxies to pairs of galaxies to clusters of galaxies to clusters of clusters of galaxies to ... .

GALAXY ROTATION CURVES (1970's)

Consider individual disk galaxies. Recall the discussion of the Tully-Fisher method. We imagine the stars and the gas in the disks of spiral galaxies are in orbit about the centers of the disk. In this case, we have that
• mv**2/R - GM(galaxy)m/R**2 = 0 ===> M(galaxy) = Rv**2/G

So, if we say, pick out a star and then measure how far it is from the center of the galaxy and how fast it is moving ===> mass contained within the orbit of the star (if the mass of the galaxy is distributed spherically). This is a powerful method, for example,

• Measure rotation curves
• Milky Way rotation curve

  ===> R(Sun) ~ 25,000 light years ~ 2.5 x 10 **22 cm and V(Sun) ~ 220 km/s = 2.2 x 10**7 cm.s

  ===> M(Milky Way) ~ 1.8 x 10**44 grams ~ 10**11 M(Sun) !

This is a nice way to measure the masses of galaxies. However, let's look at the rotation curves of some galaxies. The curves are flat at large radii. What does this mean?

Based upon rotation curves, one concludes that the dark matter in the halos contributes at least 3 - 10 times the mass density of the luminous parts of the galaxies ===> Omega > 0.02 --> 0.07

BINARY GALAXIES

Consider two galaxies in orbit. Then use Kepler's third law of planetary motion (as stated by Newton):
• P**2 = [4 pi**2/(G{M+m})] a**3
Studies ===> M/L ~ 90 with an error of 40. Results aren't too good, but we note that M/L is still >> the M/L for the luminous parts of galaxies.

CLUSTERS OF GALAXIES (1930's)

So, for example, for one galaxy we have (roughly speaking)

• F(motion) + F(gravity) = 0

  where

  • the motion force is like a centrigual force and so is ~ Mv**2/R, R is the distance to the galaxies and M is the mass of the galaxy
  • the gravity force is the sum of the forces due to all of the other galaxies. We can represent this sum as roughly GM(total)M/{average R}**2 in the direction of the center of the cluster.

  Result: - GM(total)/{average R}**2 + v**2/{average R} = 0 ===> M(total) ~ v**2 {average R} / G

  Using galaxies, (M/L) for the cluster ~ 10 (M/L) for the luminous part of the cluster. ===> Omega(cluster) ~ 0.1 - 0.3 < 1

Gravitational Lenses

Due to the curvature of space caused by concentrations of mass, light-rays bend as they pass by stars, galaxies, cluster of galaxies, ... . The bending makes masses act like lenses. A recent release from the HST of a lens produced by Abell 2218 (see HST press release. The amount of bend is determined by the mass of the lensing object.

Recent work using faint blue objects (young galaxies at redshifts of z = 1 - 3 ?) is quite suggestive. In deep exposures of relatively distant clusters (z ~ 0.2 - 0.5), the background faint blue objects are lensed. Although, these studies are in their early stages, preliminary results give results consistent with the other more traditional ways of inferring the mass of clusters of galaxies.