Introduction

Central to cosmology is the resolution of the mass discrepancy problem. In the current standard picture, the discrepancy between observed luminous mass and inferred dynamical mass in extragalactic systems is attributed to the presence of nonbaryonic cold dark matter (CDM). However, the predictions of CDM (e.g., Navarro, Frenk, & WhiteNFW 1997) fail the precision tests afforded by the rotation curves of low surface brightness galaxies (McGaugh & de Blok 1998aMBa; Moore et al.MQGL 1999). In contrast, the modified Newtonian dynamics (MOND) introduced by MilgromM83 (1983) as an alternative to dark matter accurately predicted the behavior of these systems well in advance of the observations (McGaugh & de Blok 1998bMBb; de Blok & McGaugh 1998BM; see also Begeman, Broeils, & Sanders 1991BBS; SandersS96 1996; Sanders & VerheijenSV 1998).

In conventional cosmology, CDM is required for two fundamental reasons. One is that the dynamically inferred mass density of the universe greatly exceeds that appropriate for baryons as determined from primordial nucleosynthesis ($\Omega_m \gg \Omega_b$; e.g., Copi, Schramm, & Turner 1995CST). The other is that gravitational formation of large scale structure proceeds slowly (as t^2/3) in an expanding universe. It is only possible to reach the rich amount of structure observed at z=0 from the smooth ($\Delta T/T \sim 10^{-5}$) microwave background at $z \approx 1400$ if there is a nonbaryonic component whose density fluctuations can grow unimpeded by radiation pressure.

It does appear possible to explain these points with MOND. MOND is an alteration of the force law at very small acceleration scales, $a < a_0 = 1.2 \times 10^{-8}\;{\rm cm}\,{\rm s}^{-2}$. The low acceleration scale applies in most disk galaxies and the universe as a whole. In the context of MOND, conventional measures of the dynamical mass density of the universe are overestimated by a factor which depends on the typical acceleration. Accounting for this leads to a very low density universe with $\Omega_m \approx \Omega_b$(McGaugh & de Blok 1998bMBb; SandersS98 1998).

Perhaps the simplest possible MOND universe one can consider is one in which a₀ remains constant in time (Felten 1984F84; SandersS98 1998). In this case, the universe does not enter the MOND regime of very low acceleration until $1+z \approx 2.33(\Omega_m h/0.02)^{-1/3}$, or $z \approx 1.6$. Everything is normal at higher redshift, so conventional results like primordial nucleosynthesis and recombination are retained. However, small regions can enter the MOND regime at early times as the phase transition begins (SandersS98 1998). Once radiation releases its hold on the baryons ($z \approx 200$ in a low density universe), these regions will behave as if they possess a large quantity of dark matter. Consequently, structure forms very rapidly. Indeed, early structure formation is another promising way to distinguish between CDM and MOND (SandersS98 1998). For the present purpose, it suffices to realize that if the MOND force law is operative, structure forms much more rapidly than the Newtonian t^2/3. It is not necessary to have CDM for a rich amount of large scale structure to grow from an initially smooth cosmic microwave background.

In this paper I begin to explore how anisotropies in the microwave background might help to distinguish between CDM and MOND. As a starting point, for MOND I make two basic assumptions: that a₀ is constant, and the background metric is flat in the usual Robertson-Walker sense. This is not the only possibility for a MOND universe (MilgromM89 1989). The acceleration constant may vary with time, and the nature of the background geometry is unclear. Still, this seems like the most obvious point of departure. In essence, I am examining the microwave background anisotropy properties of a conventional cosmology in which all the matter is baryonic in the amount specified by primordial nucleosynthesis, but the amplitude of the anisotropies is not constrained to be large by the slow growth of structure. Failure of the assumptions should result in a more pronounced effect on the microwave background than what I discuss below, making it easier to distinguish between CDM and MOND.