## Abstract

The recent observations of cross temperature-polarization power spectra of the cosmic microwave background (CMB) made by the *Wilkinson Microwave Anisotropy Probe* (*WMAP*) satellite are in better agreement with a high value of the Thomson scattering optical depth τ≈ 0.17. This value is close to τ= 0.3, which is taken as the upper limit in the parameter extraction analysis made by the *WMAP* team. However, models with τ| 0.3 provide a good fit to current CMB data and are not significantly excluded when combined with large-scale structure data. By making use of a self-consistent reionization model, we verify the astrophysical feasibility of models with τ| 0.3. It turns out that current data on various observations related to the thermal and ionization history of the intergalactic medium are not able to rule out τ| 0.3. The possibility of a very extended reionization epoch can significantly undermine the *WMAP* constraints on crucial cosmological parameters such as the Hubble constant, the spectral index of primordial fluctuations and the amplitude of dark matter clustering.

## 1 Introduction

The recent results on cosmic microwave background (CMB) anisotropy from the *Wilkinson Microwave Anisotropy Probe* (*WMAP*) satellite (Bennett et al. 2003; Spergel et al. 2003) represent an extraordinary success for the standard cosmological model of structure formation based on cold dark matter (CDM) and adiabatic primordial perturbations. Furthermore, data releases from the Sloan Digital Sky Survey (SDSS; Tegmark et al. 2004a) are living up to expectations and combined analyses of all these data sets are placing strong constraints on most cosmological parameters.

However, even if theory and observations are in spectacular agreement, discrepancies seem to be present and have already stimulated the interest of several authors. One of most unexpected results from the first-year *WMAP* data is indeed the detection of an excess in the large angular scale cross temperature-polarization power spectrum (Kogut et al. 2003). In the standard scenario, this excess may be interpreted as due to a late reionization process which produces a second last scattering surface at lower redshifts (|10–20). The CMB radiation can therefore become polarized through anisotropic Thomson scattering on angular scales comparable to the horizon at reionization (θ > 5°).

Analysis made by the *WMAP* team of the cross temperature-polarization data constrains the best-fitting value of the Thomson optical depth of the Universe to τ≈ 0.17 (the exact best-fitting value and the errors depend on the analysis technique employed). Following this, considerable effort is spent in constructing models, both numerical (Ciardi, Ferrara & White 2003; Ricotti & Ostriker 2004a; Sokasian et al. 2003, 2004; Gnedin 2004) and semi-analytical (Hui & Haiman 2003; Wyithe & Loeb 2003; Choudhury & Ferrara 2005), analysing the feasibility of such a high τ. In this context, it was shown that an early population of metal-free stars, with reasonably low values of star-forming efficiency and escape fraction (Ciardi et al. 2003; Choudhury & Ferrara 2005), is sufficient to produce such a high τ.

In this paper we want to emphasize another aspect of this result and investigate the compatibility of cosmological and astrophysical data with a very extended reionization process, i.e. τ| 0.3. As we illustrate in the next section, the parameter extraction made by the *WMAP* team has not been considered this case, since a top-hat prior of τ < 0.3 has been included in their analyses (see Peiris et al. 2003; Spergel et al. 2003). Given the fact that the reionization process is still to be fully understood, one could wonder about the robustness of this assumption. In the next section we will indeed show that τ= 0.3 is compatible with the data and, as already pointed out in Tegmark et al. (2004b) and Fosalba & Szapudi (2004), letting τ vary freely up to τ= 0.5 more than doubles the error bars on cosmological parameters. Studying the astrophysical feasibility of a very extended reionization is therefore not only important for the theoretical understanding of the process but also for determining the cosmological parameters with highest possible accuracy and reliability.

The paper is organized as follows. In the next section, we analyse current CMB and large-scale structure data to show that high values of the optical depth are not excluded and we discuss the implications of having τ| 0.3 on cosmological parameters. In Section 3, we verify whether models with τ| 0.3 violate any observational constraints related to the ionization and thermal history of the Universe. In the final section we present our conclusions.

## 2 Cosmological Constraints on the Optical Depth

In this section we investigate the possibility of having a high optical depth τ| 0.3 by analysing current CMB and large-scale structure data. The method we adopt is based on the Markov chain Monte Carlo package cosmomc^{1} (Lewis & Bridle 2002). We use individual chains and we adopt the Raftery and Lewis convergence diagnostics.^{2} We sample the following six-dimensional set of cosmological parameters, adopting flat priors on them: the physical baryon and CDM densities, ω_{b}≡Ω_{b}*h*^{2} and ω_{dm}≡Ω_{dm}*h*^{2}; the ratio of the sound horizon to the angular diameter distance at decoupling, θ_{s}; the scalar spectral index and the overall normalization of the spectrum, *n*_{s} and *A*_{s}; and, finally, the optical depth to reionization, τ. Furthermore, we consider purely adiabatic initial conditions, we impose flatness and we do not include gravitational waves. Relaxing flatness does not affect τ in a very significant way; however, it does alter the values of the other basic parameters, and thus may further change the interpretation and astrophysical implications of the overall results. We restrict our analysis to the case of three massless neutrino families. We include the first-year temperature and polarization data (Bennett et al. 2003) with the routine for computing the likelihood supplied by the *WMAP* team (Verde et al. 2003) as well as the Cosmic Background Imager (CBI; Readhead et al. 2004) and Arcminute Cosmology Bolometer Array Receiver (ACBAR) measurements of the CMB.

In addition to the CMB data, we also consider the constraints on the real-space power spectrum of galaxies from the SDSS (Tegmark et al. 2004a). We restrict the analysis to a range of scales over which the fluctuations are assumed to be in the linear regime (*k* < 0.2 *h*^{−1} Mpc). When combining the matter power spectrum with CMB data, we marginalize over a bias *b*, constant with scale and redshift, considered as an additional nuisance parameter.

The results of our analysis are reported in Table 1 and displayed in Figs 1-4. In Fig. 1 we show the likelihood distribution function obtained by analysing the *WMAP* data only. As we can see, the data do not provide strong constraints on τ and values as large as τ| 0.5 are compatible with the data. As a further example, we plot in Fig. 2 the two-dimensional likelihood distribution on the *H*_{0}−τ plane after marginalization over the remaining nuisance parameters. As we can see, there is a clear degeneracy between the two parameters and the value of the Hubble constant is not constrained by the *WMAP* data alone. The claimed constraint in Spergel et al. (2003) of *h*= 0.72 ± 0.05 at 68 per cent c. l. is mostly due to the top prior on τ. For comparison, we report in Table 1 the results from a similar analysis but now with the inclusion of an external top-hat prior of τ < 0.3, as in the analysis made by the *WMAP* team (Spergel et al. 2003). As we can see, the inclusion of the optical depth prior more than halves the error bars and greatly improves the constraints on the Hubble parameter, the spectral index and the baryon density to values in agreement with those reported by the *WMAP* team.

It is useful to compare the *WMAP* constraints derived in the case of τ| 0.30 with independent cosmological probes. If we assume a high value of τ= 0.30 ± 0.01, then the likelihood best-fitting values are ω_{b}= 0.026, *h*= 0.78 and *n*_{s}= 1.07. A value of *h*| 0.8 is still in reasonable agreement with the *Hubble Space Telescope* (*HST*) constraint of *h*= 0.72 ± 0.07 (Freedman et al. 2001). Values of ω_{b} > 0.025 are difficult to reconcile with abundances of primordial light elements and standard big bang nucleosynthesis, which provide ω_{b}= 0.020 ± 0.001 (Burles & Tytler 1998). However, neutrino physics is still rather unknown and deviations from the standard model, like the inclusion of the neutrino chemical potential or extrarelativistic degrees of freedom, could be present and larger values of ω_{b} may be possible (see, for example, Hansen et al. 2002). We can therefore conclude that the high τ case is in agreement with the *WMAP* data and that the best-fitting cosmological model with τ= 0.3 is in reasonable agreement with several other observables.

Including the SDSS data, as we do in Fig. 3, breaks several degeneracies, mainly between the spectral index *n*_{s} and τ, and improves the constraints. However, as we can see from Fig. 4, models with τ= 0.3 are still inside the 2σ level and are not strongly disfavoured. In the *WMAP*+SDSS analysis, the best-fitting model under the assumption of τ= 0.30 ± 0.01 has ω_{b}= 0.027, *h*= 0.75, *n*_{s}= 1.08 and σ_{8}= 1.17. These results are close to those from the previous *WMAP* analysis, further confirming the reasonable agreement between the SDSS data and τ= 0.3.

Including ACBAR and CBI experiments in *WMAP*+SDSS improves the constraints on τ and we obtain an upper limit of τ≤ 0.25. While this disfavours the τ| 0.3 hypothesis, one should also note that other effects such as massive neutrinos (see Fogli et al. 2004), running spectral index and a scale-dependent bias, just to name a few, could all be present and improve the agreement between models with high τ and the data. For example, including a running spectral index in the primordial spectrum, which may be motivated by inflation and/or by a scale-dependent bias in galaxy clustering, enlarges the constraints again to τ≤ 0.32. In the case of running spectral index and τ= 0.30, the best-fitting parameters are ω_{b}= 0.024, *h*= 0.67, *n*_{s}= 0.93, d *n*/d ln *k*=−0.08 and σ_{8}= 1.11.

An accurate determination of σ_{8} may be the best way to determine τ. Unfortunately, there is a large scattering in the constraints on σ_{8} from current observations. A value of σ_{8}= 1.1–1.2 is compatible with the determinations inferred from the local cluster X-ray temperature function (see, for example, Eke et al. 1996; Borgani & Guzzo 2001; Pierpaoli, Scott & White 2001) and cosmic shear data (Van Waerbeke et al. 2001; Refregier, Rhodes & Groth 2002; Bacon et al. 2003), but incompatible with other analysis (see, for example, Seljak 2002; Viana, Nichol & Liddle 2002). Recent measurements of galaxy clustering at redshift *z*= 0.6 from the COMBO-17 survey gives σ_{8}= 1.02 ± 0.17 at zero redshift (Phleps et al. 2005). It is interesting to point out that, if the small-scale CMB anisotropy excess measured by the CBI experiment (see Readhead et al. 2004) is indeed due to the integrated Sunyaev-Zel'dovich effect, this would be compatible with a value of σ_{8}= 1.04 ± 0.12 (Komatsu & Seljak 2002), therefore in agreement with a very extended reionization scenario.

## 3 Astrophysical Constraints

A potential challenge to models with high values of τ can come from the astrophysical constraints. To address this issue, we study the implications of a high value of τ| 0.3 on the reionization history of the Universe. For this purpose, we use a semi-analytican used widely (see, for example, Cen 2003a,b; Venkatesan, Tumlinson & Shull 2003), and is in agreement with various sets of observations (Choudhury & Ferrara 2005) including the NIRB data (Magliocchetti, Salvaterra & Ferrara 2003; Salvaterra & Ferrara 2003). In such scenarios, Pop III stars with a photon production efficiency ε_{III}≡ε_{SF,III}*f*_{esc,III}≈ 0.006 (where ε_{SF,III} is the star-forming efficiency and *f*_{esc,III} is the escape fraction of ionizing photons) are able to produce a τ= 0.17 (the best-fitting value obtained by fitting the temperature-polarization cross power spectrum to ΛCDM models in which all parameters except τ assume their best-fitting values based on the temperature power spectrum; Kogut et al. 2003). One might then expect that a considerably higher value of ε_{III} must be required to produce a τ of 0.3. However, one should also keep in mind that the model with τ= 0.3 has higher values of σ_{8}= 1.18 and ω_{b}= 0.026 compared to the standard one. This means that (i) the number of collapsed haloes is higher and (ii) the baryonic mass within the collapsed haloes is larger. In fact, the value of ε_{III} required to match the high τ is ≈ 0.0035, which is similar to the value in standard scenarios. So we conclude that production of a large number of ionizing photons at high redshift does not pose any serious difficulty for models with high τ.

Various other mechanisms for early reionization have been suggested in order to produce a high τ, most notably being an early population of accreting black holes (Dijkstra, Haiman & Loeb 2003; Madau et al. 2004; Ricotti & Ostriker 2004b). The accretion of gas on to these black holes or miniquasars would produce a background of hard photons, which have a large mean free path and can ionize large regions of the IGM. However, the constraints from the present-day soft X-ray background imply that these sources can, at most, produce | three photons per hydrogen atom (Dijkstra, Haiman & Loeb 2003; Salvaterra, Haardt & Ferrara 2005), which is much less than the requirement for fully ionizing the IGM — hence the reionization from these sourced would be partial. While the nature of reionization (topology of the ionized regions, thermal state of the IGM, etc.) would be quite different if it is achieved by the black holes rather than Pop III stars, whether the photons from the miniquasars can produce a high value of τ≈ 0.3 without violating any other observational constraints needs to be verified.

The second difficulty regarding high values of τ is whether such a high number of photons at high redshift violate any observational constraints (see Fig. 5). The first set of such constraints comes from the observations related to the IGM. Since there are virtually no observational data on IGM at *z* > 6, the most severe constraints on the model come from the Lyman α Gunn-Peterson optical depth τ_{GP} at *z*≈ 6. In the standard scenario, τ_{GP} observations are very well matched by the flux from normal Pop II stars with an efficiency ε_{II}≈ 0.0005 along with a population of QSOs as required by data on optical luminosity function. It is thus necessary that the Pop III stars do not have a significant flux at *z*≈ 6. For the models studied in this paper, this provides a mild constraint on the transition redshift of Pop III stars *z*_{trans}≳ 10.5. However, there is another physical process which affects τ_{GP} and thus provides further constraints. This has to do with the mean free path λ_{H} of hydrogen-ionizing photons. At high redshifts, the high flux from Pop III stars creates huge ionized regions, thus producing high values of λ_{H}| 60–80 Mpc. Once the Pop III stars disappear at *z*_{trans}, these ionized regions start becoming neutral over a recombination time-scale, resulting in a gradual decrease of λ_{H}. However, the constraints on τ_{GP} (taking into account the uncertainty in the value of the slope of the IGM temperature-density relation γ) imply that λ_{H}≲ few Mpc at *z*≈ 6. To obtain such low values of λ_{H}, one has to make sure that the Pop III stars start disappearing around *z*_{trans}≈ 11. Once this condition is satisfied, the model is consistent with various available observations, namely, the redshift evolution of Lyman-limit systems, the temperature corresponding to the mean IGM density, the cosmic star formation rate and τ_{GP}, as shown in Fig. 5.

The second set of observational consequence is related to the NIRB, which is believed to be due to the Pop III stars. The observations of NIRB constrain the combination ε_{SF,III}(1 −*f*_{esc,III}) and *z*_{trans} (Magliocchetti et al. 2003; Salvaterra & Ferrara 2003). First, note that the value of τ is sensitive only to the combination ε_{SF,III}*f*_{esc,III}; hence, it is not difficult to match the NIRB constraints on ε_{SF,III}(1 −*f*_{esc,III}) by suitably choosing the values of ε_{SF,III} and *f*_{esc,III}. Secondly, the decline of the NIRB at wavelengths below the *J* band constrains *z*_{trans}≈ 10, whereas the τ_{GP} constraints at *z*≳ 6 require that *z*_{trans} > 11 (as discussed in the previous paragraph). However, the constraint from the NIRB is not too strong, mainly due to lack of reliable data at wavelengths shortwards of the *J* band. It is thus difficult to rule out models where the Pop III star formation rate starts decreasing gradually from *z*≈ 11 with a substantial contribution remaining until *z*≈ 9.

In conclusion, current observations, related to the IGM and NIRB, are not able to rule out models with τ| 0.3, although they are only marginally consistent in some cases.

## 4 Conclusions

In this paper we have investigated the feasibility of a very extended reionization. We can summarize our results as follows.

The first-year

*WMAP*data are in perfect agreement with a Thomson scattering optical depth parameter as high as τ= 0.5. Not considering this possibility in parameter extraction analysis may bias the results on several quantities such as the Hubble constant, the baryon density and σ_{8}.Including galaxy clustering data from the SDSS disfavours models with τ= 0.3 but not at high significance. Similarly, inclusion of the CMB data from CBI and ACBAR improves the constraints to τ≤ 0.25. On the other hand, if additional parameters such as a running spectral index or scale-dependent bias are considered, high τ models are found to be in better agreement with observations.

We have therefore studied the astrophysical feasibility of such a high value of τ, particularly with respect to the observations related to the reionization history of the Universe. Using a simple semi-analytical model (Choudhury & Ferrara 2005), we find that an early population of massive metal-free Pop III stars with photon production efficiency ε

_{III}≡ε_{SF,III}*f*_{esc,III}≈ 0.0035 is sufficient to produce such a high τ. However, in order that the photon flux does not violate the Lyman α Gunn-Peterson optical depth constraints at*z*≳ 6, the Pop III star formation rate should start decreasing around*z*_{trans}≈ 11. This value of*z*_{trans}is marginally consistent with the observations of NIRB.

We can therefore conclude that the possibility of a very extended reionization epoch is still not completely ruled out by current data and should be taken into account, especially when deriving reliable 2σ constraints from CMB anisotropy measurements. In future, more accurate CMB polarization measurements, such as those expected from the *Planck* satellite experiment, together with high accuracy measurements of σ_{8} from weak lensing and galaxy surveys, will be able to verify or rule out this interesting possibility.

### Acknowledgments

The authors would like to thank thank C. Baccigalupi and S. Leach for useful discussions. The work of AM is part of the GEMINI-SZ project funded by MURST through COFIN contract no. 2004027755.

## References

*ISO*Survey of a Dusty Universe