Optical and X-ray GRB Fundamental Planes as Cosmological Distance
  Indicators
  Gamma-Ray Bursts (GRBs), observed up to $z=9.4$, can be employed as
standardized candles, extending the distance ladder beyond Supernovae Type Ia
(SNe Ia, $z=2.26$). We standardize GRBs using the 3D fundamental plane relation
among the rest-frame end time of the X-ray plateau emission, its corresponding
luminosity, and the peak prompt luminosity. Combining SNe Ia and GRBs, we
constrain the matter content $\Omega_{\text{M}}= 0.299 \pm 0.009$ assuming a
flat $\Lambda$CDM cosmology with and without correcting for selection biases
and redshift evolution. Using a similar 3D correlation in optical, we find that
our optical GRB sample is as efficacious in the determination of
$\Omega_{\text{M}}$ as the X-ray sample. We trimmed our GRB samples to achieve
tighter planes to be used to simulate additional GRBs. We determined how many
GRBs are needed to be used as standalone probes to achieve a comparable
precision on $\Omega_{\text{M}}$ to the one obtained by SNe Ia only. {\bf We
achieve the error measurements derived using SNe Ia in (Conley et al. 2011)
with 142 GRBs in optical considering the errorbars on the variables halved and
in Betoule et al. (2014) with 284 optical GRBs, by considering that the error
bars on the variables halved. Using a doubled sample (obtained by future
machine learning approaches that allow both a lightcurve reconstruction and the
estimates of GRBs for which the GRB is unknown) compared to our current one,
with errorbars halved we will reach the same precision of (Scolnic et al. 2018)
with 390 GRBs which will be uses as efficient standalone probes as SNe Ia by
2054.
