Spherical Top-Hat Formalism

• Spherical Top-Hat Formalism is a model using a uniform-density sphere to analytically capture nonlinear gravitational collapse and predict structure formation.
• It extends classical collapse models by incorporating inhomogeneities through the LTB metric and accounting for dark energy, exotic fluids, and viscosity.
• The formalism underpins practical applications such as protocluster identification and benchmarking N-body simulations against theoretical predictions in various cosmological models.

The spherical top-hat formalism is a theoretical framework for modeling the nonlinear gravitational collapse of an overdense region in the universe, idealized as a sphere with uniform (top-hat) density. It provides a tractable but powerful analytic approach to understanding structure formation, the formation criteria for bound objects such as clusters and black holes, and the impact of various cosmological models—including those with dark energy, exotic fluids, viscosity, modified gravity, and coupled dark sectors.

1. Foundations of the Spherical Top-Hat Formalism

The classical spherical top-hat collapse model considers a spherically symmetric region with uniform overdensity embedded in an expanding background. The internal dynamics of the sphere are often described by a closed Friedmann–Robertson–Walker (FRW) metric, while the background is modeled as a flat or open FRW universe. The equation governing the evolution of the sphere’s radius $R(t)$ is

$\frac{d^2R}{dt^2} = -\frac{GM}{R^2}$

for pressureless matter (dust), with $M$ the total mass enclosed. The solution is a cycloidal trajectory parametrized by

$$t = \frac{t_{\mathrm{ta}}}{\pi}(\theta - \sin\theta), \qquad R = \frac{R_{\mathrm{ta}}}{2}(1 - \cos\theta)$$

where $R_{\mathrm{ta}}$ and $t_{\mathrm{ta}}$ are the turnaround radius and time, respectively. The turnaround is the epoch where expansion halts and contraction begins.

The non-linear spherical collapse threshold is usually defined in terms of the density contrast $\delta$ between the sphere and the background. For an Einstein–de Sitter (EdS) universe, the critical overdensity for collapse at $z=0$ is $$\delta_c^{\mathrm{sc}} = (9\pi^2/16)-1 \approx 4.55$$. In $\Lambda$CDM or other backgrounds, this critical threshold must be solved numerically from the collapse evolution equation.

2. Extensions: Inhomogeneous and Relativistic Spherical Collapse

The classical top-hat formalism assumes homogeneity inside the sphere: each region at fixed radius collapses simultaneously. Generalizations abandon this assumption using the Lemaître–Tolman–Bondi (LTB) metric, accommodating initial radial inhomogeneities (Pereira et al., 2010). The spacetime is described by

$$ds^2 = dt^2 - \frac{R'(r,t)^2}{1+f(r)} dr^2 - R(r,t)^2 d\Omega^2$$

with $f(r)$ a curvature function and $R(r,t)$ the areal radius for each comoving shell. This yields a local density contrast $\delta(r,t)$ that explicitly depends on $r$ and is governed by the nonlinear equation

$$\ddot{\delta} + 2H\dot{\delta} - 4\pi G\bar{\rho} \delta (1+\delta) - \frac{4}{3}\frac{\dot{\delta}^2}{1+\delta} = \frac{\Lambda}{2}(1+\delta) + \frac{2}{3}\left[\frac{\partial}{\partial t} \ln \frac{R}{R'}\right]^2 (1+\delta)$$

Specializing to a homogeneous profile eliminates the gradient terms, recovering the top-hat result, but the general LTB solution supports realistic, shell-dependent collapse times. With a cosmological constant, outer shells may fail to collapse entirely, leading to dividing shells with distinct dynamics.

3. Top-Hat Collapse in Exotic and Unified Dark Sector Models

The top-hat formalism has been extensively adapted to test the nonlinear evolution of cosmologies where the dark sector is non-trivial:

• Chaplygin Gas Models: Both generalized and modified Chaplygin gas (gCg, MCG), characterized by equations of state such as $p = -C/\rho^{1+\alpha}$ or $p = A\rho - B/\rho^\alpha$, interpolate between dark matter and dark energy behaviors. The effective sound speed, governing the pressure response, is crucial in these models. In the nonlinear regime, the effective sound speed drops rapidly as the collapse proceeds (e.g., $c_{\text{eff}}^2 \simeq -\alpha w$ for small $\delta$, and more generally derived exactly), allowing otherwise DE-like fluids to mimic dust in the collapsed region (Fernandes et al., 2011, Karbasi et al., 2015). Enhanced values of $\alpha$ or $A$ often accelerate collapse by suppressing pressure support in the overdense region.
• Viscous Unified Dark Fluids (VUDF, VMCG, VGCCG): The inclusion of bulk viscosity modifies the effective equation of state, leading to terms such as $p = \alpha\rho - \zeta_0\rho - A$ or bulk viscous additive contributions. Viscosity damps density perturbations; collapse delays for increased viscosity, but if the viscosity coefficient $\zeta_0$ is small ($\lesssim 10^{-3}$), the model closely tracks $\Lambda$CDM predictions (Li et al., 2014, Debnath et al., 2015, Jawad et al., 2016). In loop quantum cosmology (LQC), the Friedmann equation is modified by quantum corrections, which further alter the collapse threshold and growth rate.
• Scalar-Field Dark Sector Models with Non-minimal Coupling: When the dark sector contains oscillating scalar fields (mimicking dark matter) coupled non-minimally to quintessence or phantom dark energy fields (e.g. via $q\psi^2\phi^2$ terms), the collapse outcome depends on coupling strength. Weak coupling preserves dust-like collapse (unclustered DE); strong coupling induces effective pressure in the dark matter and may cluster dark energy locally, modifying the virialization process (Saha et al., 28 Oct 2024).

4. Spherical Top-Hat in Observational and Simulation Contexts

The analytic framework provides direct means to connect the theory to observable quantities and simulations:

• Protocluster Identification: The critical nonlinear overdensity from the top-hat model can be used to select regions in high-redshift surveys or simulations that will evolve into massive clusters by $z=0$. The physical extent is set by the turnaround radius (the zero-velocity surface), and candidate regions are flagged if the mass within exceeds a cluster threshold (e.g., $10^{14} M_\odot$) (Lee et al., 2023). Such definitions correlate tightly with final cluster masses in simulations, and the identification is robust against redshift-space distortions.
• Comparison to N-body Simulations: While TSC dynamics well approximate simulated halos before turnaround, systematic deviations appear post-turnaround: real halos have larger ($\sim$16-20%) virial radii and delayed collapse, attributed to non-uniform density profiles (e.g., NFW-like), inside-out collapse, and nontrivial internal velocity dispersions (shell-crossing), all absent from the analytical top-hat model (Suto et al., 2015). Correcting for these by incorporating velocity dispersion via the Jeans equation improves agreement.
• Primordial Black Hole Formation: Full general-relativistic, spherically symmetric collapse frameworks (e.g., Misner–Sharp, Hernandez–Misner) generalize and encompass the top-hat picture. They are necessary for precise threshold criteria (e.g., $2m/R \geq 1$), initial and boundary condition handling, and accurate modeling of profiles and stability, especially when pressure or gradients become important (Bloomfield et al., 2015).

5. Nonlinear Physics, Energy Conservation, and Dark Energy Clustering

Extending the spherical top-hat model to scenarios where dark energy can cluster or exchange energy with matter requires careful reformulation:

• The radius evolution within the overdense region includes pressure and dark energy terms, generalizing to

$$\frac{\ddot{r}}{r} = -\frac{4\pi G}{3}(\rho_m + \rho_{\text{de}} + 3p_{\text{de}})$$

with clustering described by explicit dependence on local $\delta_{\text{de}}$ and, importantly, the clustering parameter $\gamma$ (not to be confused with the potential parameter) (Herrera et al., 2019). $\gamma=0$ means fully clustered DE (local energy conservation), $\gamma=1$ corresponds to homogeneity (no local clustering, non-conservation). $\delta_c$, the linear threshold for collapse, becomes sensitive to DE perturbations: overdense DE suppresses collapse (raising $\delta_c$), underdense (phantom) accelerates it (lowering $\delta_c$).

• In models with non-minimal coupling or where the dark sector is composed of coupled fields, the energy–momentum is further nontrivially split between components, and the collapse path depends explicitly on the coupling strength, background field amplitudes, and whether the interior region maintains unclustered or clustered DE (Saha et al., 28 Oct 2024).

6. Mathematical Formalism and Key Equations

The core equations in spherical top-hat collapse and its extensions include:

Quantity / Concept	Equation / Expression	Model Context
Spherical collapse dynamics	$\frac{d^2R}{dt^2} = -\frac{GM}{R^2}$, $t=\frac{t_{\mathrm{ta}}}{\pi}(\theta - \sin\theta)$	Classical Top-Hat
Density contrast evolution	$$\ddot{\delta} + 2H\dot{\delta} - 4\pi G\bar{\rho}\delta(1+\delta) - \frac{4}{3}\frac{\dot{\delta}^2}{1+\delta}= \text{(extra terms)}$$	LTB/Generalized
Critical density for collapse	$\delta_c = [\delta_{\text{tot}}]^{\text{lin}}(z_c)$	With clustering dark energy
Effective sound speed (gCg)	$$c_{\text{eff}}^2 = w\frac{(1+\delta)^{-\alpha} - 1}{\delta}$$	Generalized Chaplygin Gas
Bulk viscous pressure	$p = \alpha\rho - \zeta_0\rho - A$, $c_s^2 = \alpha - \zeta_0$	Viscous unified dark fluid
Dynamical radius evolution	$y'' + [y'+(1/a_i)] (H'/H + 1/a) = \dots$	Clustering DE, nonlinear

These mathematical relations encode the nonlinear, dynamical, and stability properties of the top-hat collapse and its generalizations to more complex cosmological models.

7. Applications and Theoretical Implications

The spherical top-hat formalism remains central for:

• Determining thresholds for nonlinear structure formation and predictions of mass functions (e.g., Press–Schechter, halo model).
• Formulating semi-analytic galaxy and protocluster identification algorithms consistent with simulations and survey data (Lee et al., 2023).
• Testing the impact of exotic components (e.g., unified dark sector, scalar fields with non-minimal coupling) and new physics (viscosity, quantum gravity corrections) on structure formation (Li et al., 2014, Debnath et al., 2015, Jawad et al., 2016, Saha et al., 28 Oct 2024).
• Benchmarking analytic predictions against full N-body simulations and hydrodynamic codes, revealing the limitations imposed by simplified assumptions and informing refinements (Suto et al., 2015).

The model’s adaptability to various physical scenarios and direct linkage to analytic, numerical, and observational results ensure its continued relevance in cosmological structure formation studies. Extensions to non-homogeneous, multi-component, or non-minimally coupled systems reveal the nuanced role of nonlinear physics in the growth of cosmic structures, the clustering of dark energy, and the interpretation of future high-precision observations.