4. Results and discussion
The shock-heated gas within proto-galaxies interacts with the CMB photons through the SZ effect (thermal and kinetic). These interactions generate secondary temperature anisotropies and spectral distortions. We simulate maps of the secondary anisotropies generated by a population of seeded proto-galaxies formed between redshift 5 and 10. The maps have a resolution of about 0.2 arcseconds to resolve the galaxies and contain pixels. The number of sources of mass M at redshift z is derived from the PS mass function. Their positions are drawn at random in the map. The y and profiles for, respectively, the thermal and the kinetic effects are directly derived from the integration of the gas profile along the line of sight (Eqs. 1 and 2) assuming spherical symmetry.
Similarly to the case of galaxy clusters, we assume that in the early stages of formation the gas settles into a hydrostatic equilibrium within the DM potential. A universal density profile is motivated by Navarro et al. (1996). However, the gas profile may be softer than that of the DM, and moreover the existence of a central cusp is "unobserved" (Kravtsov et al. 1998). We thus conservatively adopt the following parametrised profile for the gas distribution:
where is the central density. is left as a free parameter describing the steepness of the profile, whereas is identified with a core radius as in galaxy clusters. On cluster scales, is typically 10 to 30 times smaller than the cluster virial radius . In our model, we introduce the parameter which we vary, similarly to clusters, between 10 and 30. The central density can be derived from the gas mass of the proto-galaxy using the following equation:
where the virial radius of the structure is given by:
for a critical universe. It is fixed solely by the mass and the collapse redshift .
We will give the results for a flat model with no cosmological constant () and an open model (). Varying the cosmological parameters will vary the number of proto-galaxies along the line of sight as well as their peculiar velocities. It will also modify their physical properties, i.e. the size and velocity of the shock and thus the gas temperature. The two cosmological models represent the upper and lower bounds between which all other cosmological models involving a non-zero cosmological constant fall.
4.1. Compton distortion
The CMB photons, scattering off the electrons of the ionised hot gas, induce a spectral distortion whose amplitude is given by Eq. 1. The FIRAS experiment has measured the mean Compton parameter resulting from all the interactions undergone by the photons. The result is (Fixsen et al. 1996). This stringent observational limit incorporates the (negligible) contribution of the rather cold intergalactic medium and that of all other extragalactic signals. Among these signals, there is the contribution of the hot ionised gas in galaxy clusters. The global distortion induced by intra-cluster gas has been computed (De Luca et al. 1995; Barbosa et al. 1996), and found to be of the order of a few . In addition to galaxy clusters, one has to take into account the contribution of the proto-galaxy population in terms of the overall Compton distortion, , induced by the scattering of CMB photons on the shock-heated gas.
Based on simulated maps, we predict and we compare it to the limit set by COBE-FIRAS. Among all the parameters of the model, there are four major quantities that substantially affect the predictions of the mean Compton parameter. Two of them, and p, are related to the gas distribution (Eq. 7). The two others are the fraction, f, of BH-seeded proto-galaxies and the BH-to-spheroid mass ratio . We compare our predicted overall distortion to the COBE-FIRAS limit and look for the combinations of parameters for which our predictions fit the observations. This allows us to constrain the assumptions of our model.
For , %, and both cosmological models, we find which exceeds the observational value. In order for our prediction to be reconciled with the COBE-FIRAS limit, f must be only a few percent. This constraint on f strongly violates the actual observations (Magorrian et al. 1998; Richstone et al. 1998). is thus excluded by the limit on the global distortion whatever value we choose for .
For (i.e. an isothermal profile), an model, % and , we find whatever we adopt for p. The fraction f must be smaller than 75% for the prediction to be compatible with the observational limit. Again, this fraction is significantly smaller than the 95% advocated by Magorrian et al. (1998). Such a constraint could rule out the isothermal profile. However, up to now, was assumed to be constant and equal to . If we now use the lower limit of Magorrian et al. (1998), that is , together with % or higher there is only a marginal agreement for between the predicted and measured distortions. In the open model case (), we find approximately the same results. For to be compatible with , if , the fraction of BH seeded proto-galaxies should be smaller than 80% if or if . The predictions for agree with observations in all the cases.
For (i.e. the gas profile approximates a King profile) and for both cosmological models, we find of about to a few , a prediction compatible with . This result remains valid for all values of p between 10 and 30 including the highest boundaries of f and , respectively, 100% and .
4.2. Predicting the angular power spectrum
We choose the set of parameters associated with the isothermal profile which agrees with the COBE-FIRAS limit: , , % and . Within this context, we predict the upper limit on the contribution to secondary temperature anisotropies induced by the SZ, thermal and kinetic effects, of the proto-galactic gas. We express this contribution in terms of an angular power spectrum plotted in Fig. 1 together with the main other well-known secondary anisotropies.
At very small scales ( a few ) corresponding to galactic scales, the kinetic SZ contribution of the shock-heated gas (Fig. 1, thick solid line for and thick dashed line for ) is very large. It is interesting to note the good agreement between our results and those obtained by Peebles & Juszkiewicz (1998) for the scattering of the CMB photons by the cloudy proto-galactic plasma. The power spectrum of the kinetic SZ anisotropies for the model is significantly larger than the model. This is mainly due to the higher number of sources per unit comoving volume in open models. In all other flat cosmological models involving a non-zero cosmological constant, the power spectrum will lie between the two curves. The expected power spectrum due to the thermal SZ effect is not plotted in this figure. It is more than one order of magnitude smaller than the kinetic effect contribution. This is due to the efficiency of bremsstrahlung cooling which lowers the temperature down to a few K.
We compare the contribution of the proto-galaxies due to their SZ kinetic effect to the major sources of secondary temperature anisotropies. In each case, we choose the most extreme cases for the comparison with our upper limit prediction. The power spectra displayed in Fig. 1 are taken from the literature. The dotted line represents the upper limit of the contribution of the inhomogeneous reionisation as computed by Aghanim et al. (1996) for a quasar lifetime of yrs. The dot-dashed line represents the Rees-Sciama effect (Rees & Sciama 1968) taken from Seljak (1996) (, ). The dashed line represents the galaxy cluster contribution due to kinetic SZ effect from Aghanim et al. (1998) with a cut-off mass of . The triple-dot-dashed line represents the Vishniac-Ostriker effect (Ostriker & Vishniac 1986; Vishniac 1987) computed by Hu & White (1996) with a total reionisation occurring at . Finally, the solid thin line represents the power spectrum of the primary CMB anisotropies in a standard cold DM model computed using the CMBFAST code (Seljak & Zaldarriaga 1996). The primary CMB anisotropies dominate at all scales larger than the damping around 5 arcminutes. At intermediate scales, several effects take place among which the inhomogeneous reionisation, the Ostriker-Vishniac and the SZ effect. In Fig. 1, we do not plot the power spectra of the thermal SZ effect of galaxy clusters. It is about one order of magnitude larger than the kinetic SZ effect. At very small scales, the anisotropies are totally dominated by the proto-galactic contribution.
© European Southern Observatory (ESO) 2000
Online publication: May 3, 2000