lens}}{d\sigma dz}=4\pi \frac{c}{H_{0}}\frac{(1+z)^{2}D_{\rm A}^{2}}{E(z)}\phi(\sigma),
\label{number_lens}$$ where $D_{\rm A}$ is the angular diameter distance, and $E(z)=H(z)/H_{\rm 0}$ is the dimensionless Hubble parameter. We note that the simulated redshift distribution of lens galaxies is independent of the Hubble constant adopted in our work and the only role of the assumed cosmological model is to extrapolate the redshift distribution from the redshifts of the main galaxy sample in the SDSS to higher redshifts.

Supernovae
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The total number of observed supernovae per unit time as a function of redshift is given by $$\frac{dN_{\rm SN}}{dz}=4\pi \frac{c}{H_{0}}\frac{(1+z)^{2}D_{\rm A}^{2}}{E(z)}\frac{1}{1+z}n_{\rm SN}(z),$$ where $n_{\rm SN}(z)$ is the volumetric supernova rate in a local rest frame. Compared to eq. (\[number\_lens\]), the additional factor $1/(1+z)$ accounts for a conversion from a local to the observer rest frame.

![Comoving volumetric rate of type Ia and core-collapse supernovae assumed in our study. For core-collapse type, the rate is given by the star formation history from @Mad2014; for type Ia, the rate is the convolution of a delay time distribution given by eq. (\[dtd\]) with the adopted model of star formation history. Data points show observational measurements of the rate for type Ia, as compiled by @Gra2014. The various distinct symbols highlight selected 