People who learn a second language early in life are more likely to achieve high levels of grammatical proficiency than those who learn a second language later in life. Researchers have begun to use language models in an effort to address explanations for this age of acquisition (AOA) effect. Here, we argue that this work has failed to consider an important factor that may be relevant to AOA: younger learners may experience a greater density of exposure to a second language than older learners. While earlier work neglecting this density factor in LMs failed to find an AoA effect, we demonstrate empirically and theoretically that language models that receive second language input at lower densities learn to make second language grammatical judgments more slowly and also plateau at lower levels of ability. Crucially, this remains true even in the long run when cumulative second language input is held constant. Our findings highlight the possibility that second language AOA effects in humans as well as models might reflect differences in exposure density that covary with age. This account stands in contrast to classic maturational and experiential accounts of second language age of acquisition effects, and adds an important yet missing dimension to the effort to use language models to interrogate this effect.