Formula used in the simulation

SN Ia flux and distance moduli

The flux of a SN Ia in a band b at the obs-frame phase p is simulate by sncosmo with the following formula :

\[F_b(p) = \frac{1}{1+z}\frac{1}{4\pi d_L^2}\int_0^{+\infty} \phi_b\left(\frac{\lambda}{1+z}, \frac{p}{1+z}\right)T_b\left(\lambda\right)\frac{\lambda}{hc} d\lambda\]

where \(\mathbf{\phi_b}\) is the restframe flux density and \(\mathbf{\lambda}\) the obs-frame wavelength.

The flux is re-scaled in ADU units by applying the following factor:

\[F_b^{ADU} = 10^{-0.4 m_B} 10^{\left(ZP_{obs} - ZP_{AB}\right)}\]

The observed magnitude is given by the absolute magnitude \(\mathbf{M_B}\) and the distance moduli \(\mathbf{\mu}\) :

\[m_B = M_B + \mu\]

In the simulation :math:`mu` is computed as:

\[\mu = 5 \log\left((1+z_{vp})^2 (1+z_{2cmb}) (1+z_{cos})r(z_{cos})\right) + 25\]

with :

  • \(\mathbf{z_{cos}}\) the cosmological redshift

  • \(\mathbf{z_{vp}}\) the redshift due to the peculiar velocity of the SN / CMB

  • \(\mathbf{z_{2cmb}}\) the redshift due to our peculiar motion / CMB

  • \(\mathbf{r(z)}\) the comoving distance

Noise formula

The flux error is computed as :

\[\sigma^2_F = \frac{F}{G} + \sigma_{skynoise}^2 + \left(\sigma_\mathrm{CCD}^2\right) \times 4\pi\sigma_{PSF}^2+ \left(\frac{\ln(10)}{2.5}F\sigma_{zp}\right)^2\]

The first term is the Poisson noise with G the gain in $ e^- $ / ADU.

The second term is the noise from sky flux.

If you use limiting magnitude at 5σ, sky-noise is computed as :

\[\sigma_{skynoise} = \frac{1}{5}10^{0.4\left(ZP - m_{5\sigma}\right)}\]

The last term is the propagation of the zero point incertitude.