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We have already mentioned that the Omegam = 1 CHDM model with Omeganu = 0.2 was found to be the best fit to nearby galaxy data of all cosmological models [3]. But this didn't take into account the new high-z supernova data and analyses [96] leading to the conclusion that OmegaLambda - Omegamatter approx 0.2, nor the new high-redshift galaxy data. Concerning the latter, Somerville, Primack, and Faber [97] found that none of the Omegam = 1 models with a realistic power spectrum (e.g., CHDM, tilted CDM, or tauCDM) makes anywhere near enough bright z ~ 3 galaxies. But we found that LambdaCDM with Omegam approx 0.4 makes about as many high-redshift galaxies as are observed [97]. This Omegam value is also implied if clusters have the same baryon fraction as the universe as a whole: Omegam approx Omegab / fb approx 0.4, using for the cosmological density of ordinary matter Omegab = 0.019 h-2 [98] and for the cluster baryon fraction fb = 0.06 h-3/2 [99] from X-ray data or fb = 0.077 h-1 from Sunyaev-Zel'dovich data [100]. An analysis of the cluster abundance as a function of redshift based on X-ray temperature data also implies that Omegam approx 0.44 ± 0.12 [101, 102]. Thus most probably Omegam is ~ 0.4 and there is a cosmological constant OmegaLambda ~ 0.6. In the 1984 paper that helped launch CDM [14], we actually considered two models in parallel, CDM with Omegam = 1 and LambdaCDM with Omegam = 0.2 and OmegaLambda = 0.8, which we thought would bracket the possibilities. It looks like an LambdaCDM intermediate between these extremes may turn out to be the right mix.

The success of Omegam = 1 CHDM in fitting the CMB and galaxy distribution data suggests that flat low-Omegam cosmologies with a little hot dark matter be investigated in more detail. We have used CMBFAST [64] to examine LambdaCHDM models with various h, Omegam, and Omeganu, assuming Omegab = 0.019 h-2. Figure 1 shows the power spectrum P(k) for LambdaCDM and a sequence of LambdaCHDM models with increasing amounts of hot dark matter, compared to the power spectrum from APM [108]. Here we have fixed Omegam = 0.4 and Hubble parameter h = 0.65. All of these models have no tilt and the same bias parameter, to make it easier to compare them with each other. As expected, the large-scale power spectrum is the same for all these models, but the amount of small-scale power decreases as the amount of hot dark matter increases.

Figure 1

Figure 1. Nonlinear dark matter power spectrum vs. wavenumber for LambdaCDM and LambdaCHDM models with Omeganu / Omegam = 0.05, 0.1, 0.2, 0.3. Here Omegam = 0.4, the Hubble parameter h = 0.65, there is no tilt (i.e., n = 1), and the bias b = 0.85. Note that in this and the next Figure we ``nonlinearized'' all the model power spectra [103], to allow them all to be compared to the APM data (the small ``wiggles'' in the high-Omeganu power spectra are an artifact of the nonlinearization procedure).

In Figures 2 and 3 we consider a sequence of twelve LambdaCDM and LambdaCHDM models with h = 0.65, Omegam = 0.3, 0.4, 0.5, 0.6, and Omeganu / Omegam = 0, 0.1, and 0.2. We have adjusted the amplitude and tilt n of the primordial power spectrum for each model in order to match the 4-year COBE amplitude and the ENACS differential mass function of clusters [109] (cf. [110]). (We checked the CMBFAST calculation of LambdaCHDM models against Holtzman's code used in our earlier investigation of LambdaCHDM models [1]. Our results are also compatible with those of recent studies [111, 112] in which n = 1 models were considered. But we find that some LambdaCDM and LambdaCHDM models require n > 1, called ``anti-tilt'', and it is easy to create cosmic inflation models that give n > 1 - cf. [113].) In all the LambdaCHDM models the neutrino mass is shared between Nnu = 2 equal-mass species - as explained above, this is required by the atmospheric neutrino oscillation data if neutrinos are massive enough to be cosmologically significant hot dark matter. (This results in slightly more small-scale power compared to Nnu = 1 massive species, as explained above, but the Nnu = 1 curves are very similar to those shown.) In Ref. [114] we have shown similar results for Hubble parameter h = 0.6, and also plotted the best CHDM and LambdaCDM models from [3]. Note that all these Figures are easier to read in color; see the version of this paper on the Los Alamos archive [115].

Figure 2

Figure 2. Nonlinear dark matter power spectrum vs. wavenumber for 12 LambdaCHDM models with Nnu = 2 massive neutrino species and Hubble parameter h = 0.65, with tilt and sigma8 determined by COBE + ENACS cluster abundance. The bias chosen for these models is that which minimizes chi2 over the entire range of available APM data.

Figure 3

Figure 3. CMB anisotropy power spectrum vs. angular wave number for the same models as in Figure 2. The data plotted are from COBE and three recent small-angle experiments [104, 105, 106, 107].

Of the LambdaCHDM models shown, for Omegam = 0.4 - 0.6 the best simultaneous fits to the small-angle CMB and the APM galaxy power spectrum data [108] are obtained for the model with Omeganu / Omegam = 0.1, and correspondingly m(nuµ) approx m(nutau) approx 0.8 - 1.2 eV for h = 0.65. For Omegam < 0.4, smaller or vanishing neutrino mass appears to be favored. Note that the anti-tilt permits some of the LambdaCHDM models to give a reasonably good fit to the COBE plus small-angle CMB data. Thus, adding a little hot dark matter to the moderate-Omegam LambdaCDM models may perhaps improve somewhat their simultaneous fit to the CMB and galaxy data, but the improvement is not nearly as dramatic as was the case for Omegam = 1.

It is apparent that the LambdaCDM models with Omegam = 0.4, 0.5 have too much power at small scales (k gtapprox 1 h-1 Mpc), as is well known [117, 118] - although recent work [119] suggests that the distribution of dark matter halos in the Omegam = 0.3, h = 0.7 LambdaCDM model may agree well with the APM data. On the other hand, the LambdaCHDM models may have too little power on small scales - high-resolution LambdaCHDM simulations and semi-analytic models of early galaxy formation may be able to clarify this. Such simulations should also be compared to data from the massive new galaxy redshift surveys 2dF and SDSS using shape statistics, which have been shown to be able to discriminate between CDM and CHDM models [120].

Note that all the LambdaCDM and LambdaCHDM models that are normalized to COBE and have tilt compatible with the cluster abundance are a poor fit to the APM power spectrum near the peak. The LambdaCHDM models all have the peak in their linear power spectrum P(k) higher and at lower k than the currently available data (e.g., from APM). Thus the viability of LambdaCDM or LambdaCHDM models with a power-law primordial fluctuation spectrum (i.e., just tilt n) depends on this data/analysis being wrong. In fact, it has recently been argued [21] that because of correlations, the errorbars underestimate the true errors in P(k) for small k by at least a factor of 2. The new large-scale surveys 2dF and SDSS will be crucial in giving the first really reliable data on this, perhaps as early as next year.

The best published constraint on Omeganu in LambdaCHDM models is [116]. Figure 4 shows the result of their analysis, which uses the COBE and cluster data much as we did above, the P(k) data only for 0.025 (h / Mpc) < k < 0.25 (h / Mpc), the constraint that the age of the universe is at least 13.2 ± 2.9 Gyr (95% C.L.) from globular clusters [122], and also the power spectrum at high redshift z ~ 2.5 determined from Lyman-alpha forest data. The conclusion is that the total neutrino mass mnu is less than about 5.5 eV for all values of Omegam, and mnu ltapprox 2.4(Omegam / 0.17-1) eV for the observationally favored range 0.2 leq Omegam leq 0.5 (both at 95% C.L.). Analysis of additional Lyman-alpha forest data can allow detection of the signature of massive neutrinos even if mnu is only a fraction of an eV. Useful constraints on Omeganu will also come from large-scale weak gravitational lensing data [123] combined with cosmic microwave background anisotropy data.

Figure 4

Figure 4. Constraints on the neutrino mass assuming (a) Nnu = 1 massive neutrino species and (b) Nnu = 2 equal-mass neutrino species. The heavier weight curves show the effect of including the Lyman-alpha forest constraint. (From [116]; used by permission.)

JRP acknowledges support from NASA and NSF grants at UCSC.

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