Annu. Rev. Astron. Astrophys. 1991. 29: 325-362 Copyright © 1991 by Annual Reviews. All rights reserved

5. INFLATION AND DARK MATTER

Observations indicate that our universe may contain a large amount of nonluminous, ``dark'' matter. Inflation adds an interesting new dimension to this issue.

In the noninflationary cosmologies, there is no preferred value for the density parameter . This quantity was treated by most astronomers as an input from observations into the theory. Inflationary models, however, make a very definite prediction about : In the usual scenarios, = 1 + O(10^-4) [the correction of O(10^-4) arising because of the fluctuations in the curvature induced by density inhomogeneities]. This prediction can be tested in two ways: First, this constraint, added to the fact that the universe is matter-dominated today, implies that H₀t₀ = (2/3). If, based on stellar and galactic ages, we take t₀ = (12-20) Gyr, we have to rule out values of h greater than 0.65. An independent determination of H₀ leading to, say, h 0.65 will rule out inflation.

More directly, inflationary models can be ruled out if observations conclusively point out a value of < 1. Since luminous matter in the universe contributes far less than unity, one may say that the inflationary idea can be correct only if there exists some form of dark matter in sufficient quantities.

The observational status of the value of is not very certain. The following claims have been made in the literature (for a review, see 13, Ch. 10, 100, 115):

1. The mean density in the solar neighborhood gives about 0.003h^-1.

2. Studies based on the Magellanic stream and timing arguments in local groups give a higher value of about 0.06h^-1.

3. The mass density in groups of galaxies contributes about 0.16 and that in large clusters give about 0.25.

4. The Virgo centric fall also suggests a mass density of 0.25.

5. The constraints from primordial nucleosynthesis imply a constraint on the baryonic contribution to mass density: _B = (0.014-0.026)h^-2; or if we take 0.5 < h < 1, we get 0.014 < _B <0.104. (Authors differ somewhat on the upperbound on _B and the cited values are in the range 0.1 to 0.2; 69.)

Two features stand out in the above estimates if they are all correct. First, there seems to be a tendency for to increase with the scale over which it is measured. (This conclusion is somewhat tentative and quite controversial.) If we use gravitational effects occurring in a system of size L to measure , we will miss out on matter distributed smoothly over sizes significantly larger than L; thus, if the universe has a significant fraction of mass distributed smoothly over scales larger than, say 50 Mpc, we can still reconcile the above observations with = 1. Second, the observations are just marginally consistent with a fully baryonic universe with _B 0.2 if (and only if) h = 0.4. Thus, these observations alone probably do not rule out a completely baryonic universe (yet!).

Inflation makes definite claims regarding the above situation. First, inflation is not compatible with a purely baryonic universe (since < 0.2). Thus not only does inflation demand dark matter, it demands nonbaryonic dark matter. Second, inflation requires most of the dark matter (a fraction of about 0.75) to be distributed smoothly at scales larger than about 20 Mpc or so, to escape small-scale bounds on .

Non-baryonic dark matter may be needed for a completely different reason. As discussed above, perturbations in the baryonic matter are constrained to be less than about 10^-4 at decoupling. This does not give sufficient time for these perturbations to grow into the structures we see today. The constraints on non-baryonic matter, which does not couple directly to radiation, are often less severe. Dark matter perturbations can grow by a factor of at least (T_eq / T_dec) from the epoch of matter domination till the time of decoupling. Baryonic matter can be allowed to ``catch up'' with dark matter perturbations after the decoupling. This can reduce the (T / T) value by a factor of about 50-100 or so in many models (e.g. 109). A model for galaxy formation incorporating these constraints is not very easy to build, however. If we assume that the dark matter is made of weakly interacting massive particles (WIMPs), we can broadly distinguish between two scenarios for structure formation depending on the mass m_x of the WIMP.

The growth of structures in a universe dominated by a non-baryonic dark matter particle of mass m_x proceeds as follows: These particles will become nonrelativistic at a temperature T_NR = m_x corresponding to the time t_NR g^-1/2 (m_p³ / T²_NR) 1.2 x 10⁷ s (m_x / 1 KeV)^-2 r², where the r is the ratio of the g-factors for the x-particle and photons. This implies that these particles were moving with relativistic velocities from the time t_D they decoupled from other matter until t_NR. Since these particles can cover a distance of c (t_NR - t_D) ct_NR (``free streaming'') in this time, all perturbations at scales less than ct_NR would have been wiped out (14, 15, 99). This corresponds (today) to the wavelength _FS (ct_NR) (T_NR / T₀) r 1 Mpc (m_x / KeV)^-1 r, containing a mass of M_FS 1.5 x 10¹¹ M ( h²) (m_x / 1 KeV)^-3 r³. This drastically reduces the power in the spectrum k³ |_k|² for M < M_FS. Thus the spectrum has significant power only in the range M_FS < M < M_eq; within this range, the spectrum is also reasonably flat.

Candidates for dark matter with M_FS M_eq (i.e. for m_x 100 eV) are called hot dark matter; their power spectrum will be peaked around M_FS M_eq 10¹⁴-10¹⁵ M. Candidates with m_x 10 KeV are called cold dark matter; they will have a relatively flat, gently declining, power spectrum from about 10⁸ M to 10¹⁵ M.

For m_x 100 eV (hot dark matter), the first structures that form contain masses of about 10¹⁵ M; these structures collapse, forming ``pancakes'' and ``filaments'' around which baryons cluster (132, 133). Numerical simulations suggest that in order to reproduce the observed galaxy-galaxy correlation function, the collapse should have occurred rather recently (z 1-2), a requirement that is difficult to reconcile with the existence of quasars of higher redshift (23, 126, 127). Many astronomers, therefore, view the hot dark matter scenarios with disfavor.

If WIMPs have considerably higher mass [say m_x O(1 GeV) or so], it is possible to form galactic size objects first and evolve larger structures by gravitational clustering. Numerical simulations with these scenarios produce results that are in better agreement with observation, provided that h 0.2 or so (29, 30, 31, 32, 33). This, of course, is a priori inconsistent with inflation. In contrast, HDM can easily accommodate = 1.

Several attempts have been made to reconcile cold dark matter (CDM) scenarios with inflation. This can be done (for example) by invoking a very smooth contribution _smooth = 1 - _CDM 0.8 due to some relativistic particle (85, 95, 117), or, more brazenly, by postulating a cosmological constant of the required magnitude. All alternative would be to produce a scenario in which galaxy formation is ``biased'' - i.e. only very overdense regions (3 peaks) lead to the galaxies we see (9, 65). The more numerous, 1-peaks will be distributed ill a relatively smoother fashion. No final consensus has emerged regarding these ideas and none of them appears to have a compelling simplicity or naturalness.

In conclusion, it is worth pointing out one loophole in the conclusion = 1 in an inflationary scenario. If there is a residual -term, then =1 need not correspond to k = 0. Should it turn out that the FRW models with k = 0, = 1 are inconsistent with data, the inflationary scenario can still survive with a nonzero , although such a formula for survival would have been purchased dearly at the cost of simplicity and Occam's razor.

We next discuss the role of the -term in the context of inflation.