Luminosity-duration relation of fast radio bursts
  Nature of dark energy remains unknown. Especially, to constrain the time
variability of the dark-energy, a new, standardisable candle that can reach
more distant Universe has been awaited. Here we propose a new distance measure
using fast radio bursts (FRBs), which are a new emerging population of $\sim$
ms time scale radio bursts that can reach high-$z$ in quantity. We show an
empirical positive correlation between the time-integrated luminosity
(L$_{\nu}$) and rest-frame intrinsic duration ($w_{\rm int,rest}$) of FRBs. The
L$_{\nu}-w_{\rm int,rest}$ correlation is with a weak strength but
statistically very significant, i.e., Pearson coefficient is $\sim$ 0.5 with
p-value of $\sim$0.038, despite the smallness of the current sample. This
correlation can be used to measure intrinsic luminosity of FRBs from the
observed $w_{\rm int,rest}$. By comparing the luminosity with observed flux, we
measure luminosity distances to FRBs, and thereby construct the Hubble diagram.
This FRB cosmology with the L$_{\nu}-w_{\rm int,rest}$ relation has several
advantages over SNe Ia, Gamma-Ray Burst (GRB), and well-known FRB dispersion
measure (DM)-$z$ cosmology; (i) access to higher redshift Universe beyond the
SNe Ia, (ii) high event rate that is $\sim$ 3 order of magnitude more frequent
than GRBs, and (iii) it is free from the uncertainty from intergalactic
electron density models, i.e., we can remove the largest uncertainty in the
well-debated DM-$z$ cosmology of FRB. Our simulation suggests that the
L$_{\nu}-w_{\rm int,rest}$ relation provides us with useful constraints on the
time variability of the dark energy when the next generation radio telescopes
start to find FRBs in quantity.
