Assumptions and Caveats

I have made some predictions for the microwave background temperature anisotropies which should, with sufficiently accurate measurements, distinguish between CDM and MOND dominated universes. The predictions are based on some simple assumptions, most notably that the MOND acceleration constant a₀ does not vary substantially with time, and the geometry of the universe is flat in the Robertson-Walker sense. Neither of these need hold in MOND, but plausibly may (SandersS98 1998), making this the obvious point of departure for this discussion. I have endeavored to make the most conservative assumptions in the sense that failures of these assumptions should lead to microwave background anisotropies more deviant from the standard CDM case, and hence more readily perceptible, than the cases I have discussed.

It should be noted that the signature of a purely baryonic universe is not necessarily reflected in the usual way in the power spectrum of large scale structure at z=0 [P(k) instead of $C_{\ell}$]. The calculation for the microwave background power spectrum can be made under the assumption that MOND is not yet important at the epoch of recombination. It certainly is relevant by z=0. The scale which is nonlinear now is much larger in MOND than in CDM. The rapid nonlinear growth of structure seems likely to wash out the bumps and wiggles that would otherwise be imprinted on and preserved in the power spectrum of large scale structure in the standard framework. So while one expects a definite signature of baryon domination in the microwave background, one does not necessarily expect this to be reflected in P(k).

A conventional effect which may be different in the CDM and MOND cases is reionization. I have assumed that the background radiation encounters effectively zero optical depth along the way to us. However, the optical depth can be nonzero if the universe is reionized early enough, thus perturbing the signal in the microwave background (cf. Peebles & JusziewiczPJ 1998). Structure forms faster in MOND than in CDM, so this is a greater concern. However, the degree to which it happens depends on the details of how stars and other potential ionizing sources actually form, which is not understood in either case. The main effect of a significant optical depth is to wash out the anisotropy signal. This should not much perturb the observational signatures I have discussed, which focus on the detailed structure of the peaks relative to one another. In purely baryonic models, it is conceivable that the amplitude of the second peak will be amplified by this process, which formally would invalidate the test based on the ratio of the peak-to-trough amplitudes. However, such a microwave background power spectrum would be clearly distinct from the standard CDM case.

The integrated Sachs-Wolfe effect is another matter which may be affected by the rapid growth of structure in MOND. How much depends on the unknown details of the timing. Matter domination does not occur in MOND until $z \approx 200$because of the low mass density of a baryon-only universe. Growing potentials vary rapidly, but there is not a tremendous amount of time between then and $z \approx 10$ when L^* galaxy mass objects have collapsed (SandersS98 1998). So it is not obvious how strong this effect will be, though it can potentially have a significant impact.

It seems unlikely that there are any effects which will cause CDM and MOND universes to be indistinguishable once sufficiently accurate observations of the microwave background are obtained. For the simple assumptions I have made, the distinction is surprisingly subtle, but certainly present. Any breakdown of these assumptions should lead to a greater distinction between the two. However, it remains a substantial challenge to understand some of basic effects which can impact the microwave background in the context of MOND.