M4 Formation of Structures

Topic

The central element of this model is the density perturbation that grows to become a structure. This perturbation grows due to gravitational instability: while it remains small, its growth proceeds in a linear regime—during the matter-dominated era—where the perturbation increases in direct proportion to the universe's scale factor.

However, once the amplitude of that perturbation becomes comparable to the surrounding average density, it ceases to follow that smooth linear growth and collapses non-linearly to form a virialized halo; this threshold causes an abrupt shift in the regime, transitioning from gradual growth to violent collapse.

Even before this growth begins, the transfer function captures how the sound horizon and the mean free path of the dark matter particles themselves distort the initial perturbation spectrum; this early distortion ensures that the structure's final form depends on the physics predating its actual growth.

The amplitude of these fluctuations, measured at a fixed reference scale, currently represents one of the most significant points of tension between the value predicted by the cosmic microwave background and the value directly measured via weak gravitational lensing; this discrepancy turns the fluctuation amplitude itself into further evidence of the tension facing the Standard Model. Thus, from the density perturbation that initiates this discussion to the tension regarding fluctuation amplitude that concludes it—encompassing linear growth, non-linear collapse into a virialized halo, and the early distortion imposed by the transfer function—the density perturbation transforms a minute difference in the early universe into the entire structure observed today, as well as into one of the enduring open questions concerning that very structure.

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