We study the quantum cosmology of a flat Friedmann-Lemaitre-Robertson-Walker universe filled with a (free) massless scalar field and a perfect fluid that represents radiation or a cosmological constant whose value is not fixed by the action, as in unimodular gravity. We study two versions of the quantum theory: the first is based on a time coordinate conjugate to the radiation/dark energy matter component, i.e., conformal time (for radiation) or unimodular time. As shown by Gryb and Thebault, this quantum theory achieves a type of singularity resolution; we illustrate this and other properties of this theory. The theory is then contrasted with a second type of quantisation in which the logarithm of the scale factor serves as time, which has been studied in the context of the "perfect bounce" for quantum cosmology. Unlike the first quantum theory, the second one contains semiclassical states that follow classical trajectories and evolve into the singularity without obstruction, thus showing no singularity resolution. We discuss how a complex scale factor best describes the semiclassical dynamics. This cosmological model serves as an illustration of the problem of time in quantum cosmology.

中文翻译：

---

奇点分辨率取决于时钟

我们研究了一个平坦的弗里德曼-勒梅特-罗伯逊-沃克宇宙的量子宇宙学，这个宇宙充满了一个（自由）无质量标量场和一个代表辐射的完美流体或一个其值不受作用固定的宇宙学常数，如单模引力。我们研究量子理论的两个版本：第一个版本基于与辐射/暗能量物质分量共轭的时间坐标，即共形时间（对于辐射）或单模时间。正如 Gryb 和 Thebault 所展示的，这种量子理论实现了一种奇点分辨率；我们说明了这个理论的这个特性和其他特性。然后将该理论与第二种类型的量化进行对比，在第二种类型的量化中，比例因子的对数用作时间，后者已在量子宇宙学的“完美反弹”的背景下进行了研究。与第一个量子理论不同，第二个量子理论包含遵循经典轨迹并不受阻碍地演化为奇点的半经典态，因此没有显示出奇点分辨率。我们讨论复杂的比例因子如何最好地描述半经典动力学。这个宇宙学模型可以说明量子宇宙学中的时间问题。