Spectrophotometric fitting method

In the rest-frame wavelength range investigated here, the resolution of the BC03 templates is 3 ˚A and that of the M05 templates ∼ 15 ˚A. When the model spectra are redshifted at z ∼ 1, the resolution of the M05 templates becomes lower than that of the FORS2 spectra we used (∼ 12 ˚A). The resolution of our observed spectra would thus need to be degraded in order for them to be compared to models computed from M05 SSPs. For this reason, we decided to compute our composite stellar population models from BC03 SSPs instead. We did however use M05 models to compare stellar mass estimates (see Chapter 3). From 2.2.1, it becomes clear that the effect of age on the SED and spectrum (a deeper 4000 ˚A break, shallower Balmer features) can be reproduced assuming a different metal content. This results in an anticorrelation between age and metallicity, the well-known

“age-metallicity degeneracy”. This is illustrated in Fig. 2.5, where we plot the SED and spectra of a solar and suprasolar metallicity model. The model SED and spectra are very similar, despite an age difference of several Gyr between the two models. For this reason, when comparing composite stellar population models to the observed data, we as-sumed a constant metallicity of the models. In addition to solar metallicity templates, the BC03 models offer the choice of several subsolar metallicities and one suprasolar (Z=2.5Z⊙) metallicity. As the metal content of early-type galaxies appears to be slightly higher than

2.2 Fitting the spectrophotometric data 21

Figure 2.4: Relevant passbands and spectral features in the wavelength ranges used in this work. Top: model SED and spectrum of a 4 Gyr (blue circles, dark grey) and 0.5 Gyr (red squares, light grey) single stellar population model redshifted toz ∼1. Bottom: spectra of the same two models, in the rest- frame wavelength range of 3600 to 4400 ˚A. The effective wavelength of each bandpass and the spectral features are indicated by dashed lines.

Figure 2.5: Age-metallicity degeneracy: comparison of the SED and spectra of two single stellar population BC03 templates at solar (blue) and 2.5 solar (red) metallicity and with ages of 2 and 6 Gyr respectively. The spectrophotometry of the two models is remarkably similar although the suprasolar metallicity model is 4 Gyr younger than the solar metallicity one.

solar to twice the solar value (e.g. Gallazzi et al. [2006], Jimenez et al. [2008]) and because we expect the stellar library to be more complete at solar metallicity, we always considered solar metallicity models first.

The resolution of the BC03 templates redshifted toz ∼1 (6 ˚A) is still twice as high as the resolution of the FORS2 spectra used in this work. We therefore downgraded the resolution of our model spectra to that of the observed ones using a Gaussian broadening function.

The broadened model spectrum F_λ^′ is then given by F_λ^′(λ) = 1

σ√ 2π

Z ∞

0

dλ^′Fλ(λ^′)e^−(λ−λ

′)2

2σ2 (2.5)

whereFλ is the original spectrum andσ =p

Γ²_{F ORS2}−Γ²_{ST ELIB}/2.3548, with ΓF ORS2 and Γ_{ST ELIB} being the FWHM resolution of the observed and model spectra respectively. Also, to account for the broadening of lines due to the distribution of stellar velocities in a galaxy, we applied a Gaussian velocity dispersion to our composite stellar population spectra:

F_λ^′(λ) = 1 σ_v√

2π Z ∞

−∞

dvFλ

λ(1 + v c)⁻¹

e⁻

v2 2σ2

v (2.6)

where c is the speed of light and σv the stellar velocity dispersion, which we let vary be-tween 0 and 400 km/s. Finally, each model spectrum was interpolated at the wavelenghts of the observed spectrum it was compared with.

We derived stellar population parameters for our sample galaxies by comparing the grid of composite stellar population models described above with the observed SEDs and spectra.

2.2 Fitting the spectrophotometric data 23

This was done by minimizing a chi-square (χ²) estimator, defined for the photometry as χ²(T, τ, M) =X

i

(F_i,o−M ×F_i(T, τ))²

σ_i² (2.7)

whereFi,o andσi are the observed flux and flux error in the i-th band respectively, Fi(T, τ) is the flux in the i-th band of the {T, τ} model spectrum, per solar mass, and M is the stellar mass of the model. For the fit to the observed spectrum, the χ² is

χ²(T, τ, σv) =X

λ

(Fλ,o−Fλ(T, τ, σv))²

σ_λ² (2.8)

where Fλ,o and Fλ(T, τ, σv) are the flux value at λ of the observed and model spectrum respectively and σλ is the flux error at λ. This latter value was derived from the signal-to-noise ratio (S/N) of the observed spectrum. As absorption spectra of old populations, such as those of early-type galaxies, have very few true continuum regions, we estimated the S/N from the residuals of fitting the Hδ absorption feature with the combination of a Gaussian profile and a first-degree polynomial. Since we expect the true star formation history of a galaxy to be more complex than a simple delayed exponential (e.g. Marri &

White [2003], De Lucia et al. [2006]), and because the spectra of galaxies at z ∼ 1 are often very noisy, the best fit models are likely to be not sufficient to properly describe the actual star formation history of the studied galaxies. And as the photometry and spec-troscopy are likely to be affected by different systematic uncertainties, the best fits to the observed SED and spectrum might not coincide. Therefore, in order to cover most of the star formation histories associated with the observed data for a given set of models, we considered all models within both the 99.7% (hereafter, “3σ”) confidence regions, in the space of model parameters, of the fit to the SED and spectrum. We will hereafter refer to these models as “best fitting” models.

For a χ² statistic, the 3σ confidence region is defined as χ²_α =χ²₀+ ∆(ν, α) (Avni [1976]), whereχ²₀ is the minimumχ² value from the fit and ∆(ν, α) is such thatP(χ² ≤∆(ν)) =α.

Hereα= 0.997 andν is the number of degrees of freedom, i.e. the number of independent variables minus the number of free parameters, the latter being T, τ and M (for the fit to the SED) or σv (for the fit to the spectrum). In the case of the SED, the variables are the fluxes in each bandpass. For the spectrum however, the data points can not be considered as independent variables since the wavelength sampling is in this case finer than the actual resolution. As our spectra show little to no continuum in the age and wave-length ranges considered, we assume the independent variables to be the spectral features fitted by our models (some of which are shown in Fig. 2.4). To test this hypothesis, we performed a set of Monte Carlo simulations on each τ-model in a grid with T from 0.2 to 5 Gyr and τ from 0 to 2 Gyr. We perturbed the SED and spectrum of each model a thousand times, assuming bandpasses and random normal errors consistent with the data, and fitted each separately using the same grid of models. We then compared the results of the simulations with those of a fit using the same parameter grid but to unperturbed

Figure 2.6: Fraction of best fit results from the Monte-Carlo simulations within the 3σ confidence region of the fit to the spectrum (left) and SED (right) of each model in the grid, for values of T from 0.1 to 5 Gyr and τ from 0.01 to 2 Gyr, assuming wavelength ranges and errors consistent with the observed data (see following Chapters).

models. In Fig. 2.6, we show for each model in the grid the fraction of best fit models (i.e. with χ² =χ²₀) from the simulation that are within the 3σ confidence region of the fit to the unperturbed model. We found that while the confidence regions of both fits were systematically different, even if slightly, for old (T−tf in &2.5 Gyr, see Section 2.3) models the correction to ∆(ν) is small enough as to not change the overall distribution of models within the confidence region, thus validating our assumption. For models with younger ages, the confidence regions of the fit are underestimated and would need to be corrected.

We note that, in this work, the best fit τ-models to the spectrophotometric data of our early-type galaxies almost always fell within the “old models” region of parameter space defined above (the mean star formation history parameters of the models within 3σ can however fall outside this region). When a more complex star formation history was used (see Chapter 4) or a different wavelength range (see Chapter 6) we carried out the same test, but on the best fit models to the data only.

Next we estimated the constraining power of our spectrophotometric data. Because the quality of the photometry used in this work was relatively constant, we focused on the spectroscopy. We performed the fitting procedure on each of the τ-models in a grid with T varying from 0 to 5 Gyr and τ from 0 to 2 Gyr. We used the same set of filters and wavelength range as for the observed data and assumed photometric errors comparable to those of the data, but let the S/N of the model spectra vary. In Fig. 2.7, we plot as a function of T and τ the maximum S/N for which the confidence region of the fit to the simulated spectrum completely overlaps the confidence region of the fit to the simulated SED, i.e. the maximum S/N for which the spectroscopy does not add further constraints on the star formation histories than those given by the photometry. We found that on average a S/N of 6.5, and not lower than 4, is needed in order for the spectroscopy to be