# Stellar Population Models

*Note*: This page describes algorithms applied to the DR12 BOSS data and included in DR13 via symlinks. They were not re-applied to the latest e/BOSS spectral pipeline data in DR13.

## Spectro-Photometric Model Fitting

The stellar population models of Maraston (2005) and Maraston *et al.* (2009) are used to perform a best-fit to the observed `ugriz` magnitudes of BOSS galaxies with the spectroscopic redshift determined by the BOSS pipeline, using an adaptation of the publicly available Hyper-Z code of Bolzonella, Miralles & Pelló (2000). The fit is carried out on extinction corrected model magnitudes that are scaled to the `i`-band c-model magnitude, *i.e.*:

mag_x = modelmag_x - extinction_x + (cmodelmag_i - modelmag_i),

where x denotes the photometric band (`ugriz`).

Two sets of template spectra are used (see Maraston *et al.* 2012 for details):

- a passively evolving galaxy with a two-component metallicity of same age and no ongoing star formation or reddening, as in Maraston
*et al.*(2009), - an ensemble of canonical star formation modes, including exponentially-declining, constant with truncation, and constant, star formation, for various timescales and various metallicities, as in Maraston
*et al.*(2006). In order to minimize the event of low-age, high-dust fake solutions, reddening is not included (Pforr, Maraston & Tonini 2012 [submitted to MNRAS]). Both template sets are available for Salpeter and Kroupa initial mass functions.

The output of the fit for each galaxy includes: age, star formation mode, metallicity, k-corrected absolute magnitudes in `ugriz`, plus the reduced χ^{2}. Additionally, the best fit model spectrum as well as the probability distribution function (PDF) for the stellar mass are provided. Stellar masses and star formation rates are computed from the best-fit SED as in Maraston *et al.* (2006). Furthermore the stellar mass at the median PDF and 68% confidence levels are provided. Mass loss due to stellar evolution and the account of mass in stellar remnants is included.

The algorithm assumes a ΛCDM cosmology with `H`_{0} = 71.9, Ω_{m} = 0.258, and Ω_{Λ} = 0.742.

## Emission Line Kinematics

An adaptation of the publicly available Gas AND Absorption Line Fitting (GANDALF, Sarzi *et al.* 2006) and penalised PiXel Fitting (pPXF, Cappellari & Emsellem 2004) codes are used to fit the stellar population synthesis models of Maraston & Strömbäck (2011) and Thomas, Maraston & Johansson (2011) to observed BOSS galaxy spectra. We chose these codes as GANDALF simultaneously fits stellar templates and Gaussian emission-line models.

The stellar population synthesis models used were all at fixed solar metallicity to limit computing time, utilised the MILES stellar library and had age ranges from 6.5 Myr to 11 Gyr. Since we were not using this fit to extract stellar population metallicities and ages, considering only solar metallicity did not impact on our results and age-metallicity degeneracy effects were unimportant. We adopted the MILES resolution calculated in Beifiori *et al.* (2011).

Outputs from this fitting process that we are providing include the emission-line fluxes (both observed and de-reddened) and equivalent widths, the gas kinematics, the stellar kinematics, an `E(B-V)` value and derived BPT classifications. Our reddening values can be obtained by plugging the `E(B-V)` value for each object into the dust attenuation equation of Calzetti *et al.* (2000).

Outputs from this fitting process that we are providing include the emission-line fluxes (both observed and de-reddened) and equivalent widths, the gas kinematics, the stellar kinematics, two `E(B-V)` values (explained below), absorption line indices and derived BPT classifications.

A reduced χ^{2} value is also provided for the fit, as well as Amplitude-over-Noise (AoN) values for each of the emission lines.

## Principal Component Analysis (PCA) Method

Chen *et al.* (2012) model physical galaxy parameters based on a library of model spectra for which principal components (PCs) have been identified. The method is applied in the following steps.

### Create library of model spectra

The library of model spectra is based on Bruzual & Charlot (2003) stellar population synthesis models. The model is parameterized by the following characteristics.

#### Star formation histories (SFHs)

Each SFH consists of three parts: an underlying continuous model + a series of super-imposed stochastic bursts + a random probability for star formation to stop exponentially (*i.e.* truncation). Figure 1 shows three examples of SFHs. The top panel is a continuous model, middle panel shows a continuous model with two random bursts, a truncation can be found in the bottom panel.

#### Metallicity

95% of the model galaxies in the library are distributed uniformly in metallicity from 0.2 - 2.5 `Z`_{☉}; 5% of the model galaxies are distributed uniformly between 0.02 and 0.2 `Z`_{☉}.

#### Dust extinction

Dust extinction is modelled using the two-component model described in Charlot & Fall (2000). The `V`-band optical depth has a Gaussian distribution over the range 0 < τ_{V} < 6. with a peak at 1.2 and 68% of the total probability distribution distributed over the range 0~2. This prior distribution of τ_{V} values is motivated by the observed distribution of Balmer decrements in SDSS spectra (Brinchmann *et al.* 2004). The fraction of the optical depth that affects stellar populations older than 0.01 Gyr is parametrized as μ, which is again modeled as a Gaussian with a peak at μ = 0.3, and a 68 percentile range of 0.1~1.

#### Velocity dispersion

Each of the model spectra is convolved to a velocity that is uniformly distributed over the range of values from 75 to 400 km/s.

### Principal components (PCs) are identified from the model library

The regions around nebular emission lines are masked in the model spectra. We mask 500 km/s around the [OII]3726.03, [OII]3728.82, Hζ3889.05, [NeIII]3869.06, Hδ4101.73, Hγ4340.46, Hβ4861.33, [OIII]4959.91, and [OIII]5007.84 Å lines. Each spectrum in the masked library is normalized to its mean flux between 3700-5500 Å (this is the range we use for analysis). The mean spectrum of the masked library is calculated and subtracted from each of the model spectra.

PCA code is run on the "residual" spectra. Figure 2 presents the mean spectrum and the top seven PCs for the input model library.

### Project BOSS data and models onto PCs

### Estimate galaxy physical parameters for BOSS

`z`, we select only models that have an age smaller than the age of the universe at that redshift. We step through the models one at a time, calculating the χ

^{2}as follows:

where `C`_{α} (α = 1–7) represents the projection coefficients. The superscript "m" and "d" refer to model and data. `i` represents the `i`-th model. `P`_{α,α′} is the inverse of the covariance matrix of `C`_{α}. The covariance matrix of `C`_{α} is calculated in the projection process.

## Full spectral fitting

### firefly

In DR14 is introduced a new stellar population model fitting code: firefly described here