Dipole of the Luminosity Distance: A Direct Measure of $H(z)$

We show that the dipole of the luminosity distance is a useful observational tool which allows us to determine the Hubble parameter as a function of redshift $H(z)$. We determine the number of supernovae needed to achieve a given precision for $H(z)$ and to distinguish between different models for dark energy. We analyze a sample of nearby supernovae and find a dipole consistent with the cosmic microwave background at a significance of more than $2\sigma$.

Figure 1

Direction of the luminosity dipole from a low-redshift supernovae sample, in a celestial coordinate system centered on the CMB dipole (1 and $2\sigma$ contours). The two directions agree well.

Figure 2

We show the relative error in $H(z)$ for one supernova, as a function of the redshift, in a flat $\Lambda \mathrm{CDM}$ universe with ${\Omega }_{\Lambda }=0.7$ and for two different values of the intrinsic error $\Delta m=0.1$ and $\Delta m=0.15$. This represents as well the relative error in the dipole $d_L^{(1)}(z)$.

Figure 3

We show the number of supernovae needed to measure $H(z)$ with an accuracy of 30%, in a flat $\Lambda \mathrm{CDM}$ universe with ${\Omega }_{\Lambda }=0.7$, as a function of the redshift and for two different values of the intrinsic error $\Delta m=0.1$ and $\Delta m=0.15$.

Figure 4

We show the number of supernovae needed to differentiate the measured Hubble parameter $H(z)$ from the theoretical one in a flat $\Lambda \mathrm{CDM}$ universe $H_{\Lambda \mathrm{CDM}}(z)$, as a function of the relative difference $\frac{|H(z)-H_{\Lambda \mathrm{CDM}}(z)|}{H_{\Lambda \mathrm{CDM}}(z)}$, for redshifts $z=0.1$, $z=0.2$, and $z=0.3$ and intrinsic error $\Delta m=0.1$.