J. Steinberger Nov. 2004
Review of Cold Dark Matter in Astrophysics.
Content:
Preliminaries: Einstein equation of Hubble expansion.
Big Bang Nucleo Synthesis.
Formation of CDM “Halos”
Galactic rotation curves.
Measurement of Ωm using x-ray measurements of galaxy
clusters.
Measurement of Ωm using gravitational lensing.
Why neutralinos are hopeful candidates for CDM.
1. Astrophysical preliminaries; Einstein-Friedman equation for a
homogeneous universe. Definition of Ωm .
Metric of homogeneous universe. a is the scale, x, y and z are “comoving”
coordinates, that is, they do not change with the expansion of the universe:
ds2 = dt2 + a(t)2(dx2 + dy2 + dz2)
Einstein (Friedmann) equation for homogeneous universe:
H(t) ≡ ((da/dt)/a)2 = 8πG/3*ρ(t) + Λ , where H is the hubble
expansion parameter, ρ is the energy density and Λ is the “cosmological
constant”.
For relativistic matter, that is kT >> mc2, the variation of ρ with a is as a-4,
ρrad = ρrad,0(a0/a)4. Subscript zero, here and in the following, refers to the
present universe.
For nonrelativistic matter, that is kT << mc2, the variation of ρ with a is as a3
, ρm = ρm,0(a0/a)3 .
Only in recent times, that is for a0/a < ≈ 2, has the cosmological constant
term become important; before it was negligible.
In the early universe the energy density was dominated by radiation, now it
is dominated by non relativistic matter. After cold matter domination,
H(t)2 = H02(Ωm(a0/a(t))3 + ΩΛ); Ωm+ΩΛ = 1.
Ωm is the sum of baryonic matter and cold dark matter:
Ωm = Ωb + Ωcdm.
Present best values: Ωb = .045 +/- .002, Ωm = Ωb + Ωcdm = .26 +/-.015. ΩΛ
= .74 +/- .03.
2. Big Bang Nucleo Synthesis (BBNS). t =1 sec. to 3 minutes. “The
first three minutes”. T=1 Mev – 10 KeV.
Before one second, at temperatures above 1 MeV, neutrons and protons
were in equilibrium with the photons, about 0.7 neutrons per proton, and
did not combine to higher nuclei. As the temperature and density
decreased, a) the neutrinos dropped out of equilibrium, and therefore the
neutrons decayed, with mean life ~ 900 seconds, and b) protons and
neutrons combined to form deuterons, tritium, He3, He4 and Lithium7. The
present abundances of these elements are function of the nuclear reaction
rates, the photon energy density , the barium/photon ratio, as well as
neutrino energy density, despite the fact that the neutrinos had decoupled,
since the neutrino energy densities influences the rate of the Hubble
expansion. This effect confirms that there are three neutrino families. The
present observation of the relative abundances are the first observational
verification of the Big Bang. They also yield the very important number,
the number of barions relative to the number of photons, the so-called
barion entropy:
η = nb/nγ = 6*10-10. corresponding to Ωb = .04
(NB: The observation of the early abundances is complicated by
subsequent nucleo synthesis in star chemistry).
3. Formation of CDM “haloes” under gravity.
It is assumed that since matter domination, CDM interactions other than
gravity can be neglected. Following matter domination, the primordial
inhomogeneities with scale greater than the horizon scale at radiationmatter equality. For inhomogeneities larger than the mean density of the
universe ρc of the universe, the so called “non-linear” domain,
characteristic of galaxies and other large scale structure, typically of the
order of density ~ 1000 times ρc, the calculations are numerical
simulations. The resultant evolution of the shapes and distributions of gas
clouds, galaxies and galaxy clusters in the ΛCDM model, both in
distribution and in shape, is beautifully in agreement with the observations.
The typical halo is essentially independent of scale. It is roughly spherical,
and has the shape (Navarro, Frenk and White, ApJ 490, 493 (1997)):
ρ(r) = c*ρc/((x(1+x)2),
Here ρc is the “critical density” of the universe,
ρc = H02*8πGN/3 = 1.9*10-29g/cm3, x = r/rc, rc gives a measure of the size,
and the magnitude c is typically of order 1000.
4. Galaxy Rotation Curves. These are measurements of the velocities of
galactic objects around the centres of spiral galaxies, as function of the
radius. They measure the mass M(r) included inside the radius,
v2 = M(r)GN/r. Typically they are measured using the Doppler shifts of the
21 cm hyperfine radiation of atomic hydrogen gas clouds, which represent
a non negligible fraction of the galactic barion masses. They show that the
dominant galactic gravitational mass distributions, especially at large
distances, are much in excess of the masses represented by the galaxy’s
luminous objects (stars), and is distributed out to larger radii. The rotation
curves are among the earliest and most graphic evidence for dark matter.
5. Measurements of Ωm, using x-ray emission of galaxy clusters.
These are the earliest quite accurate determinations of Ω. Galaxy clusters
are groups of typically of the order of 1000 galaxies, with radius ~ 1 Mpc
(1pc = 3light years, typical galactic radii are 50Kpc), and masses of 1015
solar masses (typical galaxies have ~ 1011 solar masses).
~85% of their baryonic content is in the form of ionised gas plasma, with
typical temperatures ~ 2 – 8 KeV, emitting in the x-ray region. X-ray
emission is due to bremsstrahlung in electron-electron collisions, and so
proportional to the square of the optical depth of the electron density at the
radius corresponding to a particular direction of view.
The baryonic plasma is, in reasonable approximation, in hydrostatic
equilibrium with the gravitational potential. On the basis of the temperature
and x-ray flux intensity, both as function of radius, the mass distribution
and gas fraction, fgas = mgas/mmatter can be determined. Typically, fgas =
0.12. Then Ωm is obtained, using the BBNS determination of Ωb:
Ωm = Ωb(BBNS)/(fgas/.85) = 0.3 +/- 0.06.
6. Ωm from CMBR.
The very nearly uniform cosmic microwave background is a Planck
spectrum of photons, last scattered when the baryonic plasma combined
into neutral atoms, at a temperature of .25 eV (30000K), at the age of the
universe 300,000 years (present age, 14 billion years). The present
temperature, after 1100 fold expansion, is measured to be 2.730K, and
provides a measure of the present photon number density, and so, given
the BBNS barion-photon ratio, the present barion density.
The CMBR inhomogeneities ΔT/T, of order ~ 10-5, reflect the primordial
inhomogeneities which, under the action of gravity, generate the present
structure in the universe, galaxies etc. The essential information of the two
dimensional angular map of ΔT/T is contained in the one dimensional
“power spectrum” of (ΔT/T)2, as function of the angular scale, l = π/θ. The
originally essentially uniform primordial power spectrum as function of
scale, at the time of photon decoupling, has evolved with the formation of
peaks and valleys, due to the plasma oscillations produced under the
action of the dark matter gravity and the photon pressure. Positions and
magnitudes of the peaks and valleys permit a precise determination of
several basic cosmological parameters, including
Ωb = 0.05 +/- .003 and Ωm = 0.30 +/- 0.05
7. Gravitational lensing of distant objects by massif intermediate
masses.
Strong lensing, that is multiple image formation and formation of parts of
Einstein rings of distant point sources, as well as “weak lensing” small
elliptical deformation of hundreds of thousands of distant galaxies, have
yielded basic confirmation of our ideas of CDM, but it is difficult to get very
precise results.
8. Why are neutralinos attractive candidates for CDM?
Assuming we know the neutralino mass, say mn = 200 GeV, we know, from
Ωm = 0.3, the neutralino density, and so, the ration of neutralino to photon
number,
nn/nγ = (Ωmρc/mn)/(1.2(T=2.730K))3/π2) = 8*10-11.
Earlier, when kT >> mn, there were equal densities for all degrees of
freedom, including the neutralinos, in equilibrium with photons and other
particles.
As the temperature drops below mn, but as long as the density is still
sufficiently large to maintain the neutralinos in equilibrium given their
interaction crosssection, their density relative to the photons drops
exponentially, as e-mn/T.
With further expansion of the universe, when the density has decreased
such that the neutralino annihilation crosssection is no longer sufficient to
cause a collision within the hubble time, as σvnnn becomes less than 1/H,
the neutralinos decouple; their number no longer decreases.
The Susy weak annihilation crosssection is about right to give the
presently observed CDM density. This gives good reason for hope that the
Neutralino is in fact the CDM particle.