Euclid Quick Data Release (Q1). Hunting for luminous z > 6 galaxies in the Euclid Deep Fields -- Forecasts and the first bright detections

The evolution of the rest-frame ultraviolet luminosity function (UV LF) is a powerful probe of early star formation and galaxy stellar mass build-up. At z>6, its bright end (M_ ̊m UV < -21) remains poorly constrained due to small survey volumes of existing near-infrared (NIR) space-based imaging surveys. The Euclid Deep Fields (EDFs) will cover 53,deg2$ with NIR coverage down to a magnitude of 26.5,AB, providing a factor of 100 increase in area compared to previous space-based surveys. They thus offer an unprecedented opportunity to select bright z>6 Lyman break galaxies (LBGs) and definitively constrain the bright end of the UV LF. With its NIR coverage extending to sim2, , Euclid has the power to detect galaxies out to z Observatory will distinguish genuine Lyman breaks from contaminant features, while < -23.5) candidates selected from the Q1 EDF-N dataset (Euclid Quick Data Release). If their redshifts are reliable, their magnitudes suggest a DPL UV LF model at $z>9. This highlights the power of Euclid in constraining the bright end of the UV LF in the early Universe and in identifying the most luminous sources that are valuable for further follow-up observations. Here, we present a forecast for the number densities of z>6 galaxies that Euclid is expected to observe in the final EDF dataset. Using synthetic photometry from spectral energy distribution (SED) templates of z=5--15 galaxies, z=1--4 interlopers, and Milky Way M-, L-, and T-type dwarfs, we investigate optimal selection methodologies for high-z LBGs in the EDF datasets. We find that a combination of signal-to-noise ratio (S/N) cuts with SED fitting (over optical to MIR bands) yields the highest fidelity sample, recovering more than 76% of the input synthetic z>6 LBGs, while limiting low-z contamination to less than 10%. This contamination does not include effects from instrumental artefacts, which will impact the first Euclid data releases. Auxiliary data prove critical: optical coverage from the Hyper Suprime Camera and Vera C. Rubin data are vital for recovering z>10 sources. Based on empirical double power-law LF models, we expect more than 100,000 LBGs at z=6--12 and more than 100 sources as far back as z>12 in the final Euclid data release. In contrast, the steeper Schechter LF models predict no detections of z>12 LBGs. In this work, we also present two ultra-luminous (M_ ̊m UV