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1.7.2 Lessons from Warm Dark Matter

As has been said, the chart in Fig. 1.3 only includes a few of the possibilities. But many possibilities that have been examined are not very promising. The problems with a pure Hot Dark Matter (HDM) adiabatic cosmology have already been mentioned. It will be instructive to look briefly at Warm Dark Matter, to see that some variants of CDM have less success than others in fitting cosmological observations, and also because there is renewed interest in WDM. Although CHDM and WDM are similar in the sense that both are intermediate models between CDM and HDM, CHDM and WDM are quite different in their implications. The success of some but not other modifications of the original CDM scenario shows that more is required than merely adding another parameter.

As explained above, WDM is a simple modification of HDM, obtained by changing the assumed average number density n of the particles. In the usual HDM, the dark matter particles are neutrinos, each species of which has nnu = 113 cm-3, with a corresponding mass of m(nu) = Omeganu rho0/nnu = Omeganu 92 h2 eV. In WDM, there is another parameter, m/m0, the ratio of the mass of the warm particle to the above neutrino mass; correspondingly, the number density of the warm particles is reduced by the inverse of this factor, so that their total contribution to the cosmological density is unchanged. It is true of both of the first WDM particle candidates, light gravitino and right-handed neutrino, that these particles interact much more weakly than neutrinos, decouple earlier from the Hot Big Bang, and thus have diluted number density compared to neutrinos since they do not share in the entropy released by the subsequent annihilation of species such as quarks. This is analogous to the neutrinos themselves, which have lower number density today than photons because the neutrinos decouple before e+e- annihilation (and also because they are fermions).

In order to investigate the cosmological implications of any dark matter candidate, it is necessary to work out the gravitational clustering of these particles, first in linear theory, and then after the amplitude of the fluctuations grows into the nonlinear regime. Colombi, Dodelson, & Widrow (1996) did this for WDM, and Fig. 1.4 from their paper compares the square of the linear transfer functions T (k) for WDM and CHDM. The power spectrum P (k) of fluctuations is given by the quantity plotted times the assumed primordial power spectrum Pp(k), P(k) = Pp(k) T(k)2. The usual assumption regarding the primordial power spectrum is Pp(k) = Aknp, where the ``tilt'' equals 1 - np, and the untilted, or Zel'dovich, spectrum corresponds to np = 1.

Figure 1.4a
Figure 1.4b

Figure 1.4. The square of the linear transfer function T(k) vs. wavenumber k = (2pi)/lambda (in units of h Mpc-1) for (a) Warm Dark Matter (WDM), and (b) Mixed Dark Matter (MDM - CHDM with Nnu = 1 neutrino species). (From Colombi, Dodelson, & Widrow 1996, used by permission.)

One often can study large scale structure just on the basis of such linear calculations, without the need to do computationally expensive simulations of the non-linear gravitational clustering. Such studies have shown that matching the observed cluster and galaxy correlations on scales of about 20-30 h-1 Mpc in CDM-type theories requires that the ``Excess Power'' EP approx 1.3, where

Equation 1.14 (1.14)

and as usual sigmar is the rms fluctuation amplitude in randomly placed spheres of radius r h-1 Mpc. The EP parameter was introduced by Wright et al. (1992), and Borgani et al. (1996) has shown that EP is related to the spectrum shape parameter Gamma introduced by Efstathiou, Bond, and White (1992) (cf. Bardeen et al. 1986) by Gamma approx 0.5 (EP)-3.3. For CDM and the lambdaCDM family of models, Gamma = Omegah; for CHDM and other models, the formula just quoted is a useful generalization of the spectrum shape parameter since the cluster correlations do seem to be a function of this generalized Gamma, as shown in Fig. 1.5. As this figure shows, Gamma approx 0.25 to match cluster correlation data. Peacock & Dodds (1994) have shown that Gamma approx 0.25 also is required to match large scale galaxy clustering data. This corresponds to EP approx 1.25.

Figure 1.5

Figure 1.5. The value of the J3 integral for SCDM and a number of lambdaCDM and CHDM models evaluated at R = 20h-1 Mpc is plotted against the value of the shape parameter Gamma defined in the text. As usual, J3(R) = integ0R xicc(r)r2 dr, where xicc is the cluster correlation function. The horizontal dotted line is the J3 value for the Abell/ACO sample. The squares connected by the dashed line correspond from left to right to CHDM with nnu = 1 neutrino species and Omeganu = 0.5, 0.3, 0.2, 0.1, and 0 (SCDM); the square slightly below the dashed line corresponds to CHDM with Nnu = 2 and Omeganu = 0.2; all these models have Omega = 1, h = 0.5, and no tilt. The triangles correspond (l-to-r) to lambdaCDM with (Omega0, h) = (0.3,0.7), (0.4,0.6) and (0.5,0.6). The two circles on the left correspond to CDM with h = 0.4 and (l-to-r) tilt (1 - np) = 0.1 and 0. These points and error bars are from a suite of truncated Zel'dovich approximation (TZA) simulations, checked by N-body simulations. (From Borgani et al. 1996.)

Figure 1.6

Figure 1.6. Excess power EP in the two models discussed that interpolate between CDM and HDM. Solid curve shows EP as a function of the WDM parameter m/m0; note how quickly it becomes similar to CDM. Dashed curve shows how EP for Mixed Dark Matter (MDM - CHDM with Nnu = 1 neutrino species) depends on Omeganu. The observationally preferred value is EP approx 0.25. (From Colombi, Dodelson, & Widrow 1996, used by permission.)

Since calculating sigma(r) is a simple matter of integrating the power spectrum times the top-hat window function,

Equation 1.15 (1.15)

the linear calculations shown in Fig. 1.4 immediately allow determination of EP for WDM and CHDM. The results are shown in Fig. 1.6, in which the lower horizontal axis represents the values of the WDM parameter m/m0 (with m/m0 = 1 representing the HDM limit), and the upper horizontal axis represents the values of the CHDM parameter Omeganu. This figure shows that for WDM to give the required EP, the parameter value m/m0 approx 1.5-2, while for CHDM the required value of the CHDM parameter is Omeganu approx 0.3. But one can see from Fig. 1.4 that in WDM with m/m0 gtapprox 2, the spectrum lies a lot lower than the CDM spectrum at k gtapprox 0.3 h-1 Mpc (length scales lambda ltapprox 20 h-1 Mpc). This in turn implies that formation of galaxies, corresponding to the gravitational collapse of material in a region of size ~ 1 Mpc, will be strongly suppressed compared to CDM. Thus WDM will not be able to accommodate simultaneously the distribution of clusters and galaxies. But CHDM will do much better - note how much lower T(k)2 is at k gtapprox 0.3 h Mpc-1 for WDM with m/m0 = 2 than for CHDM with Omeganu = 0.3. Actually, as we will discuss in more detail shortly, CHDM with Omeganu = 0.3 turns out, on more careful examination, to have several defects - too many intermediate-size voids, too few early protogalaxies. Lowering Omeganu to about 0.2, corresponding to a total neutrino mass of about 4.6(h / 0.5)-2 eV, in a model in which Nnu = 2 neutrino species share this mass, fits all this data (PHKC95).

Probably the only way to accommodate WDM in a viable cosmological model is as part of a mixture with hot dark matter (Malaney, Starkman, & Widrow 1995), which might even arise naturally in a supersymmetric model (Borgani, Masiero, & Yamaguchi 1996) of the sort in which the gravitino is the LSP (Dimopoulos et al. 1996). Cold plus ``volatile'' dark matter is a related possibility (Pierpaoli et al. 1996); in these models, the hot component arises from decay of a heavy unstable particle rather than decoupling of relativistic particles.

There are many more parameters needed to describe the presently available data on the distribution of galaxies and clusters and their formation history than the few parameters needed to specify a CDM-type model. Thus it should not be surprising that at most a few CDM variant theories can fit all this data. Once it began to become clear that standard CDM was likely to have problems accounting for all the data, after the discovery of large-scale flows of galaxies was announced in early 1986 (Burstein et al. 1986), Jon Holtzman in his dissertation research worked out the linear theory for a wide variety of CDM variants (Holtzman 1989; cf. also Blumenthal, Dekel, & Primack 1988) so that we could see which ones would best fit the data (Primack & Holtzman 1992, Holtzman & Primack 1993; cf. Schaefer & Shafi 1993). The clear winners were CHDM with Omeganu approx 0.3 if h approx 0.5, and lambdaCDM with Omega0 approx 0.2 if h approx 1. CHDM had first been advocated several years earlier (Bonometto & Valdarnini 1984, Dekel & Aarseth 1984, Fang et al. 1984, Shafi & Stecker 1984) but was not studied in detail until more recently (starting with Davis et al. 1992, Klypin et al. 1993).

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