Spectral Appearance of Self-gravitating Disks Powered by Stellar Objects: Universal Effective Temperature in the Optical Continuum and Application to Little Red Dots

We revisit the spectral appearance of extended self-gravitating accretion disks surrounding compact central objects such as supermassive black holes. Using dust-poor opacities, we show that all optically thick disk solutions possess a universal outer effective temperature of $T_{\rm eff}\sim 4000-4500$K, closely resembling compact, high-redshift sources known as Little Red Dots (LRDs). Assuming the extended disk is primarily heated by stellar sources, this ``disk Hayashi limit" fixes the dominant optical continuum temperature of the disk spectrum independent of accretion rate $\dot{M}$, central mass $M_\bullet$, and disk viscosity $\alpha$, and removes the parameter-tuning required in previous disk interpretations of LRDs. The formation and accretion of embedded stellar objects can both power the emission of the outer disk and hollow out the inner disk, suppressing variable UV/X-ray associated with a standard quasar. The resulting disk emission is dominated by a luminous optical continuum while a separate, non-variable UV component arises from stellar populations on the nuclear to galaxy scale. We map the optimal region of parameter space for such systems and show that LRD-like appearances naturally emerge for $\dot{M}/\alpha \gtrsim 0.1 M_\odot /{\rm yr}$, a threshold insensitive to $M_\bullet$, below which the system may transition into classical non-self-gravitating AGN disks, potentially a later evolution stage. We expect this transition to be accompanied by the enhancement of metallicity and production of dust, giving rise to far infrared emission. This picture offers a physically motivated and quantitative framework connecting LRDs with AGNs and their associated nuclear stellar population.