Cortical arousals are transient events of disturbed sleep that occur spontaneously or in response to stimuli such as apneic events. The gold standard for arousal detection in human polysomnographic recordings (PSGs) is manual annotation by expert human scorers, a method with significant interscorer variability. In this study, we developed an automated method, the Multimodal Arousal Detector (MAD), to detect arousals using deep learning methods. The MAD was trained on 2,889 PSGs to detect both cortical arousals and wakefulness in 1 second intervals. Furthermore, the relationship between MAD-predicted labels on PSGs and next day mean sleep latency (MSL) on a multiple sleep latency test (MSLT), a reflection of daytime sleepiness, was analyzed in 1447 MSLT instances in 873 subjects. In a dataset of 1,026 PSGs, the MAD achieved a F1 score of 0.76 for arousal detection, while wakefulness was predicted with an accuracy of 0.95. In 60 PSGs scored by multiple human expert technicians, the MAD significantly outperformed the average human scorer for arousal detection with a difference in F1 score of 0.09. After controlling for other known covariates, a doubling of the arousal index was associated with an average decrease in MSL of 40 seconds ($\beta$ = -0.67, p = 0.0075). The MAD outperformed the average human expert and the MAD-predicted arousals were shown to be significant predictors of MSL, which demonstrate clinical validity the MAD.