We develop a spherical self-similar model for the formation of a galaxy through gas collapsing in an isolated self-gravitating dark matter halo. As is well known, the self-similarity assumption makes the problem eminently tractable by reducing it to a system of ordinary differential equations. We improve upon the existing literature on self-similar collapse in two ways. First, we include the effects of radiative cooling and the formation of a pseudo-disk at the center of collapse, in a parametrised manner. More importantly, we solve for the evolution of gas and dark matter simultaneously and self-consistently using a novel iterative approach. As a result, our model produces shell trajectories of both gas and dark matter that qualitatively agree with the results of full hydrodynamical simulations of self-gravitating systems. We discuss the impact of various ingredients such as the accretion rate, gas equation of state, disk radius and cooling rate amplitude on the evolution of the gas shells, although we leave the inclusion of stellar and black hole activity to future work. The self-consistent evolution of gas and dark matter allows us to study the response (or `quasi-adiabatic relaxation') of the dark matter trajectories to the presence of collapsing gas, an effect that has gained increasing importance recently in the context of precision estimates of small-scale statistics like the matter power spectrum. Our default configuration produces a relaxation relation in qualitative agreement with that seen in cosmological hydrodynamical simulations, and further allows us to easily study the impact of the model ingredients mentioned above. As an initial application, we vary one ingredient at a time and find that the accretion rate and gas equation of state have the largest impact on the relaxation relation, while the cooling amplitude plays only a minor role. Our model thus provides a convenient framework to rapidly explore the coupled nonlinear impact of multiple astrophysical processes on the mass and velocity profiles of dark matter in galactic halos, and consequently on observables such as rotation curves and gravitational lensing signals.

中文翻译：

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星系形成和暗晕弛豫的自相似模型


我们开发了一个球形自相似模型，用于通过气体在孤立的自引力暗物质晕中塌缩形成星系。众所周知，自相似假设通过将问题简化为常微分方程组，使问题变得非常容易处理。我们通过两种方式改进了有关自相似崩溃的现有文献。首先，我们以参数化的方式包括辐射冷却的影响和在塌陷中心形成伪盘的影响。更重要的是，我们使用一种新颖的迭代方法同时且自洽地解决了气体和暗物质的演化。因此，我们的模型产生了气体和暗物质的壳轨迹，其质量与自引力系统的完整流体动力学模拟的结果一致。我们讨论了吸积率、气体状态方程、圆盘半径和冷却速率幅度等各种成分对气体壳层演化的影响，尽管我们将恒星和黑洞活动的纳入留待未来的工作。气体和暗物质的自洽演化使我们能够研究暗物质轨迹对塌缩气体存在的响应（或“准绝热弛豫”），这种效应最近在精度方面变得越来越重要小规模统计数据的估计，例如物质功率谱。我们的默认配置产生了与宇宙流体动力学模拟中所见的定性一致的松弛关系，并且进一步使我们能够轻松地研究上述模型成分的影响。 作为最初的应用，我们一次改变一种成分，发现吸积率和气体状态方程对弛豫关系的影响最大，而冷却幅度仅起次要作用。因此，我们的模型提供了一个方便的框架，可以快速探索多个天体物理过程对银河晕中暗物质的质量和速度分布的耦合非线性影响，从而对旋转曲线和引力透镜信号等可观测值产生影响。