Dark Matter initial fluctuations (?)

[Widdekind]

BACKGROUND: Scale-independent (Harrison-Zeldovich) Initial Curvature Fluctuations

 

According to Prof. Martin Rees (New Perspectives in Astrophysical Cosmology, pp. 40-80), the most "natural" assumption, regarding the growth of density fluctuations in the early Universe, is that when any region becomes causally connected, it's density contrast ([math]\delta \rho / \rho[/math]) has become the same constant Q, for all regions across all scales at all times. (Of course, this argument is made solely in a statistical sense, and actually applies to the mean density contrast.)

 

During the matter-dominated epoch, these causally-connected "Hubble Volumes" contain, at time t (>t_eq), a mass equal to (ignoring the slight over density):

[math]M(t) = \rho(t) \left( c \; t \right)^{3}[/math]

[math]\; = \rho_{0} \left( \frac{R_{0}}{R(t)}\right)^{3} \left( c \; t \right)^{3}[/math]

[math] \; = \rho_{0} \left( \left( \frac{t_{0}}{t}\right)^{2/3} \right)^{3} \left( c \; t \right)^{3}[/math]

[math] \; = \rho_{0} (c \; t_{0})^{3} \left( \frac{t}{t_{0}}\right)[/math]

[math]\equiv M_{0} \left( \frac{t}{t_{0}}\right)[/math]

Now, in their linear regime, slight over-densities grow linearly with the Universal Scale Factor R(t) (Carroll & Ostlie. Intro. Mod. Astrophys., pp. 1295-1300). Thus, whereas at time t, mass M(t) has compacted into an over-density of Q, back at the Recombination era, that over-density was less by a factor of:

[math]\frac{R(t_{rec})}{R(t)} = \left( \frac{t_{rec}}{t}\right)^{2/3} = \left( \frac{M_{0}}{M(t)} \frac{t_{rec}}{t_{0}}\right)^{2/3} \propto M^{-2/3}[/math]

Note that, whereas, back at time t when it had attained density contrast Q, mass-scale M(t) spanned the "Hubble Distance" c t, at the present epoch t₀ that mass-scale has stretched by the factor of R(t₀)/R(t) = (t₀/t)^2/3, and now spans the size-scale:

[math]L = (c \; t) (t_{0} / t)^{2/3} = (c \; t_{0}) (t/t_{0})^{1/3} \propto t^{1/3} \propto M(t)^{1/3}[/math]

Thus, in particular, the spectrum of "initial" density fluctuations, at Recombination, which falls off as M^-2/3 in mass, falls off as L^-2 in size (see below)*.

*

M(t) scales as L

³

, as it should

(yes?)

, since this analysis ignores the (slight) over-densities associated with those volumes.

But, consider the causally-connected Hubble Volume at the transition from Matter- to Radiation-dominated epochs. Up until that equilibrium, "acoustic" effects in the "photonic-plamsa" apparently damped the gravitational growth of density contrasts. Thus, the Mass-scale [math]M_{crit} \equiv M_{0} ( t_{eq} / t_{0} )[/math] defines a threshold — above this amount of mass, density contrasts were largely unaffected by the (comparatively) brief period of Radiation-dominance and had developed to density contrasts proportional to Q x M^-2/3 at Recombination; but, below this amount of mass, density contrasts were largely damped down to the same amplitude:

Now, if this analysis were strictly & utterly true, we would see the following spectrum of Initial Curvature (Density-Contrast) Fluctians (curve overlain in green):

But, in fact, at Recombination, there were apparently slightly larger density fluctuations, at "small" scales (M < M_crit), than this simple model suggests. And, yet, those "initial" density fluctuations were not as large as indicated by standard CDM computer simulations*.

*

The non-plotted minimum density contrast, at size scales < 1 Mpc, is ~0.005

(Carroll & Ostlie,

ibid

.)

. This can be quickly seen, by noting that said density contrasts had to account for fully formed

Quasars

at z ~ 5. And, a density contrast which was just going supra-linear at z ~ 5, would (ignoring non-linearities) be about 6 today (1+z), and so about 6e-3 at

Recombination

(z ~ 1000).

 

 

 

QUESTIONS:

 

(1) Our threshold mass-scale (M_crit) seems special. For, not only does it distinguish between "Large-scale Structures" (Superclusters, Cosmic Web) and "Small-scale Structures" (Galaxies, Groups), but it is also just beginning to go non-linear at the present epoch. Thus, the Universe is "middle aged" (if you will), in that all Small-scale Structures (M < M_crit) have already virialized, while all Large-scale Structures (M > M_crit) have yet to do so. Is this some sort of special coincidence, and what could account for it ??

 

(2) Prof. Rees seems to say, that S-CDM computer simulations treat DM as "cold" and "weakly-interacting", so that (in the computer-simulated Universe) the DM can begin to gravitationally coalesce even before the Recombination era, giving DM a "head-start" on forming the gravitational potentials into which the (simulated) baryons only later begin to fall. Is this true (of the simulations), and does the above analysis imply that DM is also actually (partially) damped at small scales (M < M_crit) along with the baryons ??

 

(3) To account for high redshift compact objects (e.g. Quasars, z ~ 5), the density fluctuations, at Recombination, and on the smallest scales, must have been about 0.005 (see above). Adding this information, to the afore-cited figure, with a linear best-fit (overlain in cyan), yields the following figure:

Thus, it seems that S-CDM is accurate at large scales (M > M_crit), and at "tiny" scales, but is an over-estimate for intermediate mass scales. Does this say something subtle & significant ??

 

Thanks !!

Edited December 15, 2009 by Widdekind