Consideraciones necesarias para la implementación de un repetidor

In this section we describe the quantities that will be used to evaluate the results of the training tests.

3.5.1

Quantities Derived From Training

To evaluate the success of the training procedure, we begin by examining the training products themselves. The training process and products have been outlined in Figure 3.4 and described in Sec §3.2.

Final M₀ and M₁ components, color laws, and color dispersions will be compared directly with those of the input model.

3.5.2

Quantities Derived From Light Curve Fitting

Other quantities, such as HD bias, can be obtained from the light curve fit results.

Recall that each realization of a particular training test is subsequently used to fit a test set of SNe light curves generated from the same input model (see Figure 3.5). For each test set SN, the best-fitting scale (x₀), stretch (x₁), and color (c) parameters are determined by minimizing a χ2 based on the difference between the SN photometry and synthetic photometry of the model flux. For each SN, the fitted distance modulus is given by

µf it=mB−MB+αx1−βc (3.8)

where the effectiveB-band magnitude m_B is defined as m_B =−2.5 log₁₀(x₀) +10.635 and the global parametersα,β, andMBare determined by a fit of the entire test set using the “SALT2mu” program described in Marriner et al. (2011).

For a given training test, it is interesting to compare the mean recovered α and β parameters with their input values. Various quantities may also be constructed from the fitted distance moduliµf it. These are described in the following subsections.

Hubble scatter

The simplest quantity to calculate from the fitted distance moduli is the Hubble scatter. This quantity is defined as the dispersion on∆µ, the difference between the fitted distance modulus and the distance modulus calculated from the best-fit cosmological parameters.

Like K12, we simplify this quantity slightly by computing the dispersion of

∆µ ≡µf it−µcalc(z,ΩM,ΩΛ,w) (3.9)

where the fitted distance modulus µf it comes from (3.8) and the calculated distance modulusµ_calc is obtained by assuming aΛCDM cosmology with ΩM=0.3, ΩΛ=0.7,

andw=−1.

Hubble bias

For a single test SN i, we begin by calculating the average fitted distance modulus over all training realizations N:

<µf it,i>≡( N

∑

We can then define the Hubble bias∆<µi>as the difference between the average fitted distance modulus (3.10) and the actual distance modulusµsim:

∆<µi>≡<µf it,i>−µsim,i. (3.11) In figures, this quantity has been binned as a function of redshift and the mean bias has been plotted against the mean redshift in each bin. Therefore, as plotted, the Hub- ble bias tells us on average how correct a training’s measured distance modulus is for a supernova in a particular redshift bin. We expect that an ideal training should have bias measurements consistent with zero in all redshift bins.

Training model scatter

For a single test SN i, we define the training model scatter δ µi as the dispersion of the fitted distance moduli µ_{(f it,i),j} (where j runs over training realizations 1 to N) about the mean fitted distance modulus<µi>:

δ µi≡ v u u t N

∑

This quantity tells us how reproducible an individual SNe’s measured distance modu- lus is from realization to realization for a particular training test. In figures, this quantity has been binned as a function of redshift and the mean model scatter has been plotted against the mean redshift in each bin. Therefore, this plot tells us how stable a training’s average distance modulus is for a supernova in a particular redshift bin. In the limit of a perfect training, the model uncertainty should go to zero in all redshift bins.

3.5.3

Best-Fit Cosmologies

Finally, for each training test we can determine the ensemble of best-fit cosmology pa- rameters recovered from the test set.

We’d like to know whether our trained models are able to accurately recover the input cosmology, or whether the training procedures result in a systematic bias. To measure these biases, we fit the simulated Hubble diagram in a manner similar to theFwCDMfits described in Kessler et al. (2009a).

To obtain best-fit cosmology parameters, SN distance moduli are combined with Baryon Acoustic Oscillation (BAO) and Cosmic Microwave Background (CMB) con- straints.

For the BAO constraint, we use the quantityAdefined by Eisenstein et al. (2005),

A(z₁;w,ΩM,ΩDE) = √ ΩM E(z₁)1/3 × " |Ωk|1/2 z₁ Sk |Ωk|1/2 Z z₁ 0 dz0 E(z0) #2/3 , (3.13)

and for the CMB we use the shift parameterR R(z_CMB;w,ΩM,ΩDE) = p ΩM Z z_CMB 0 dz0 E(z0). (3.14)

Rather than taking the best fitAandRvalues from data, we calculate them from the SN simulation cosmology parameters (H =70,ΩM =0.3,ΩDE =0.7,w=−1.0,Ωk=0.0) and the experimentally determined redshiftsz₁=0.35 andz_CMB=1090 (Eisenstein et al.

2005; Komatsu et al. 2009). However, we keep the published uncertainties, yielding the constraints

χ_BAO2 = [(A(0.35;w,ΩM,ΩDE)−0.487)/0.017]2 (3.15) and

χ_CMB2 = [(A(1090;w,ΩM,ΩDE)−1.750)/0.019]2. (3.16) In the absence of input SN data, these constraints yield the best-fit cosmology param- etersΩM=0.299±0.052 andw=−1.010±0.3.