  Gamma-Ray Bursts (GRBs), 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 (the Dainotti
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 $\Omega_{\text{M}}= 0.299 \pm 0.009$ assuming a flat
$\Lambda$CDM cosmology with and without correcting GRBs for selection biases
and redshift evolution. Using a 3D optical Dainotti correlation, we find this
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 simulate
additional GRBs. We determined how many GRBs are needed as standalone probes to
achieve a comparable precision on $\Omega_{\text{M}}$ to the one obtained by
SNe Ia only. We reach the same error measurements derived using SNe Ia in 2011
and 2014 with 142 and 284 simulated optical GRBs, respectively, considering the
errorbars on the variables halved. These error limits will be reached in 2038
and in 2047, respectively. Using a doubled sample (obtained by future machine
learning approaches allowing a lightcurve reconstruction and the estimates of
GRB redhifts when z is unknown) compared to the current sample, with errorbars
halved we will reach the same precision as SNe Ia in 2011 and 2014, now and in
2026, respectively. If we consider the current SNe precision, this will be
reached with 390 optical GRBs by 2054.

Cosmology and Nongalactic Astrophysics;High Energy Astrophysical Phenomena