7 Ly$\alpha$ and He II $\lambda$1640 emission at various metallicities

To first order recombination line luminosities are proportional to the ionising photon flux and are thus simply expressed as (in units of erg s^-1)

$$L_{\rm l}(t) ~~ [{\rm erg ~ s^{-1}}] = c_{\rm l} ~~ (1 - f_{\rm esc}) ~~ Q_{\rm i}(t) ~~~ [{\rm s^{-1}}],$$ (7)

where $c_{\rm l}$ are the line emission coefficients given in Table 1, $f_{\rm esc}$ is the photon escape fraction out of the galaxy or observed aperture, and $Q_{\rm i}$ is the ionising photon flux (in units of s^-1) corresponding to the appropriate recombination line.

However, as is well known, Ly$\alpha$ constitutes a particular case due to its very large line optical depth, which implies that several effects (e.g. dust absorption, ISM geometry and velocity structure) can alter the total Ly$\alpha$ emission and lead to complex line profiles (cf. Charlot & Fall 1993; Valls-Gabaud 1993; Chen & Neufeld 1994; Kunth et al. 1998; Loeb & Rybicki 1999; Tenorio-Tagle et al. 1999). Furthermore for Ly$\alpha$ source at redshifts close to or above the redshift of reionisation the intrinsic Ly$\alpha$ emission may be further reduced or suppressed by absorption in nearby line of sight HI clouds (cf. Miralda-Escude & Rees 1997; but also Haiman 2002; Madau 2002). A proper treatment of these effects requires a complex solution of radiation transfer which depends strongly on geometrical properties of the ISM and IGM, and for which no general solution is possible. One must thus caution that depending on the application our simplifying assumptions may not apply and the predicted Ly$\alpha$ emission should thus be considered as an upper limit. Note that these difficulties do not affect other recombination lines such as H$\alpha$ and He  II $\lambda$1640, whose optical depth is strongly reduced compared to Ly$\alpha$.

Bearing the above in mind, the time evolution of the Ly$\alpha$ and He  II $\lambda$1640 line luminosities can be deduced from the evolution of $Q({\rm H})$ and $Q({\rm He^+})$ respectively given in Fig. 3.

In the case of constant star formation, at equilibrium, recombination line luminosities $L_{\rm l}$ are proportional to the star formation rate (SFR), i.e.

$$L_{\rm l} = (1-f_{\rm esc}) f_{\rm l} \left(\frac{\rm SFR}{M_{\odot} ~{\rm yr}^{-1}}\right)\cdot$$ (8)

For $L_{\rm l}$ in erg s^-1 the proportionality constants $f_{\rm l}$ for Ly$\alpha$, H$\alpha$, and He  II $\lambda$1640 are listed in Cols. 9-11 of Table 4. The variations of the ionising flux $Q({\rm H})$ already discussed above imply in particular lower star formation rates at low metallicity when identical recombination lines are used. We note that our H$\alpha$ SFR conversion factors are in good agreement with other computations at various metallicities (e.g. Kennicutt 1998; Sullivan et al. 2001) when rescaled for their $M_{\rm low}$ using $c_{\rm M}$ (Table 2).

The predicted Ly$\alpha$ and He  II $\lambda$1640 emission line equivalent widths of ageing bursts of different metallicities and all the IMF cases considered are shown in Fig. 7. Note that our revised Pop III models show smaller Ly$\alpha$ equivalent widths compared to the previous calculation in S02. This is due to an erroneous continuum definition in the earlier computations. The new models supercede those of S02. Good agreement is also found with the W(Ly$\alpha$) predictions of Tumlinson et al. (2002). The reader is also reminded that none of these recent calculations include stellar Ly$\alpha$ absorption (cf. Valls-Gabaud 1993; Charlot & Fall 1993; and Valls-Gabaud & Schaerer 2002 for new predictions).

Maximum Ly$\alpha$ equivalent widths of $\sim$240-350 Å are predicted for metallicities between solar and $4 \times 10^{-4}$. For Z down to zero (Pop III), max(W(Ly$\alpha$)) may reach up to $\sim$800-1500 Å for the various adopted IMFs (cf. S02) For comparison, the equilibrium values for SFR  =  const. are in the range of W(Ly$\alpha$$) \sim$175-275, 240-350, 500-930 Å for IMF A, B, and C respectively at $Z \le 10^{-5}$, and $\sim$70-100 Å for higher metallicities (IMF A). Note that the increased Lyman continuum output of young very metal-poor populations alone does not explain the strong increase of W(Ly$\alpha$) (cf. Table 3). In addition the reduced stellar UV continuous luminosity at $\lambda \sim 1215$ Å, due to the shift of the SED peak far into the Lyman continuum (Fig. 2), also contributes to increase W(Ly$\alpha$).

A high median Ly$\alpha$ equivalent width ($\sim$430 Å) was found in the Large Area Lyman Alpha (LALA) survey of Malhotra & Rhoads (2002) at z=4.5 and interpreted as due to AGN, starbursts with flat IMFs, or even Pop III objects. Indeed, if constant star formation is appropriate for their objects and the IMF slope is universally that of Salpeter, the observed large W(Ly$\alpha$) would require very metal-poor populations with large upper mass cut-offs and/or an increased lower cut-off (e.g. IMFs B or C). Alternatively their observations could also be explained by predominantly young bursts, with metallicities $\la$10^-5and no need for extreme IMFs (Fig. 7). This issue will be addressed in detail in a subsequent publication (Valls-Gabaud & Schaerer 2002).

As expected from the softening of the radiation field with increasing metallicity, the He  II $\lambda$1640 equivalent widths decreases strongly with Z; values W(He  II $\lambda$1640$) \ga 5$ Å are expected only at metallicities $Z \la 10^{-7}$, except if hot WR-like stars (cf. Sect. 6.2) or non-stellar sources (e.g. X-rays, AGN) provide sufficient amounts of He⁺ ionising photons.

Part of the relative weakness of W(He  II $\lambda$1640) compared to W(Ly$\alpha$)is due to a relatively strong, mostly nebular, continuum flux at 1640 Å (see S02). As Ly$\alpha$ emission may be strongly reduced due to the effects discussed earlier in objects beyond the re-ionisation redshift, and the He  II $\lambda$1640 luminosity is potentially strong enough to be detected (cf. Tumlinson et al. 2001; Oh et al. 2001; Schaerer & Pelló 2001), it is a priori not clear if both lines may be observed simultaneously and if so which of the two lines would be stronger.