[4097] Physics 316: Extragalactic Astronomy and Cosmology (Summary of Lectures)
Lecture 35

Review of Quiz Results

[L.S.Structure] The Jeans Length & Mass

The Jeans length (lambj) is the critical length scale defined by the boundary between
  • whether the crossing time (lambj/cs) for a pressure wave traveling at the sound speed
    • is
  • shorter or longer than the timescale for gravitational collapse (G rhom)-1/2)
    • Thus it is the largest length scale that pressure fluctuations, moving at cs, have been able to travel and hence resist gravitational collapse.

    • fluctuations on scales less than lambj will not have been able to grow - such growth will be resisted by the internal pressure
    • fluctuations on scales greater than lambj will be able to grow - the internal pressure cannot travel over these scales fast enough.

    Clearly
    • lambj = cs(pi/G rhom)1/2

    One can then define the The Jeans mass as the mass contained within a sphere of diameter lambj

    • Mj = (pi/6) rhom (lambj)3 = (pi/6) rhom (pi (cs)2/G rhom)3/2
    Note how
    • the effect of gravity is reflected in the appearance of the density rhom (in the denominator)
    • the effect of pressure is reflected in the the appearance of the sound speed cs (in the numerator)
    again reflecting the competition between internal pressure and self-gravity.

    [L.S.Structure] Instability & Expansion

    Main points from Bothun [end Sect5.1.3 & Sect5.1.4]

    Clearly the growth rate of perturbations will be slower than the static case - the expansion of the universe will retard the growth

    • The expressions for the Jeans length & Jeans Mass are still correct - it is just that there is a time-dependence associated with cs and rhom

    Furthermore, since we are aiming to make structures with density enhancements drho/rho > 1 we are forced to use non-linear perturbation theory. This is not easy, and requires N-body simulations

    • Since they have a smaller "overdensity" these simulations are likely to be more successful for the largest-scale structures (superclusters drho/rho ~20) than for small-scale structures (eg. galaxies drho/rho > 104) where non-linear effects will be most extreme.

    For Omegatot=1 (Bothun [Sect5.1.5]) it can be shown that after recombination the time dependence of a density fluctuation (delta = drho/rho) grows as a powerlaw with time

    • ie delta propto (t)alpha
    where alpha = 2/3 (as opposed to the exponential growth for a static universe).
    • This is the same time-dependence as for the universal scale factor in the Matter-Dominated Era
      • so delta propto R(t) propto 1/(1+z)

    [L.S.Structure] Jeans Mass Prior to Recombination

    As we just discussed, the Jeans length is proportional to the the sound speed:
    • lambj = cs(pi/G rhom)1/2
    Prior to recombination the effect of the photons is a greatly enhaced sound speed
    • cs = (c/SQRT(3)) / ( (3/4) rhom/rhor + 1 )1/2
    where rhor is the radiation density. Thus prior to recombination rhom < < rhor and cs ~ constant. Thus the Jeans length is large

    Furthermore the photons interact with the matter

    • This provides an additional pressure that strongly resists the growth of gravitational collapse
      • and in some cases "damp" any density pertubations This leads to a characteristic scale (~11 Mpc) below which even the "seeds" of structure will be washed away

    In the Radiation Dominated Era the Jeans mass increases as the temperature falls
    • Mj propto T-3
    		This can be seen by noting the different dependences of of Mj on rhom and rhotot
    ie Mj propto rhom (rhotot)-3/2
    Now in the RDE,
    rhotot ~ rhor = a T4/c2
    rhom propto R-3(t) propto T3
    Thus
    Mj propto T3 (T4)-3/2 propto T-3

    Most significantly, prior to recombination the Jeans length exceeds the Horizon size (ct). There insufficient matter within the horizon for an adiabatic density fluctuation to undergo gravitational collapse.
    • Note that density fluctuations on scales in excess of the photon damping length can exist (these are the acoustic waves we discussed earlier), they just cannot grow.

    [L.S.Structure] Jeans Mass Post Recombination

    At the time of recombination, the temperature has fallen sufficiently that electrons can remain bound to the nucleons. The photons stop interacting significantly with the matter (the relic of this radiation is now seen as the CMB of course).

    • The radiation therefore no longer contributes to the pressure resisting gravitational collapse.
    • There is a large reduction in the sound speed
      • "sound" is now carried primarily by the baryons, rather than by photons
        • cs = (5kT/3 mp)1/2
        where mp is the mass of a proton
    Thus there is a large reduction in the Jeans length and hence Jeans Mass at the time of recombination
    • Mj = (pi/6) rhom (pi kT /G mp rhom)3/2
    for an ideal gas (see Bothun [Sect5.1.7]).
    Thus Mj decreases as T decreases (and hence as the universe ages).

    The photon damping mass is generally thought to be few x1012Msun at recombination

    It should be noted that we have primarily been discussing baryons. The situation is slightly more complex for the case on the non-baryonic DM candidates ...

    		[Image Credit: Longair 1989]

    pre-recombination post-recombination
    Jeans Length
    (lambj)     		121 Mpc 15 kpc
    Horizon size
    (at recombination)
    90 Mpc     		greater than lambj smaller than lambj
    Jeans Mass 1017 Msun 2x105 Msun
    Photon Damping Mass few x1012 Msun ---

    End of lecture

    On to Next Lecture Back to Summary of Lectures