Optical and X-ray GRB Fundamental Planes as Cosmological Distance Indicators

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.