Spherical collapse and the limits of a halo model

A solvable sphere isolates one part of gravitational collapse

An overdense region can expand initially while its self-gravity slows that expansion. In a sufficiently bound ideal construction, its physical radius reaches a maximum and then contracts. This turnaround is different from the density simply becoming larger than the background.

Choose a uniform pressureless sphere of fixed mass M, with a flat matter-only Einstein–de Sitter exterior. Neglect pressure, angular momentum, tides, radiation and vacuum energy. Follow the uniform interior only before shell crossing or the singular endpoint.

Choose the common bang-time branch, which approaches the early growing perturbation rather than adding a separate initial bang-time offset. The construction supplies one restrictive nonlinear example, rather than the general evolution of an arbitrary irregular region.

The exterior background has physical mass density ρ̄_m = 1/(6πGt²). The sphere has ρ_m = 3M/(4πR³), where R is its physical radius. Their ratio compares densities at the same cosmic time t.

A parameter labels both radius and time

Write R = A₀(1 − cos θ) and t = B₀(θ − sin θ), with positive constants satisfying A₀³ = GMB₀². The parameter θ is dimensionless; it is not cosmic time or an angle describing where a particle sits around the sphere.

The specific mechanical energy of a shell at the outer boundary is −GM/(2A₀). Radius reaches its maximum R_ta = 2A₀ at θ = π, when t_ta = πB₀. Subscript ta denotes turnaround.

On the expanding branch before π, R increases; on the contracting branch after π, R decreases. The derivative dt/dθ = B₀(1 − cos θ) stays positive on the displayed interval, so the parameter still orders cosmic time.

Display only 0.5 ≤ θ ≤ 3π/2. This interval includes expansion, turnaround and regular contraction while stopping before the formal dust singularity at 2π. No post-singularity trajectory is supplied.

Actual density and a linear comparison answer different questions

Substituting the parametric radius and time into the two physical densities gives 1 + δ_NL = (9/2)(θ − sin θ)²/(1 − cos θ)³. The actual nonlinear density contrast is δ_NL, so the density ratio is one larger than that contrast.

A separate quantity, δ_L = (3/5)[(3/4)(θ − sin θ)]²ᐟ³, is the early growing linear amplitude extrapolated to the same time. It is a comparison label calculated by continuing a linear solution, not the actual nonlinear density.

At turnaround, the actual density ratio is 9π²/16 ≈ 5.551652. Thus δ_NL ≈ 4.551652, whereas δ_L ≈ 1.062410. Those three numbers refer to a density ratio, a true contrast and an extrapolated contrast respectively.

After turnaround, the exterior continues to dilute while the sphere contracts. Both changes raise the interior-to-background density ratio. A large ratio therefore cannot be inferred from radius alone without knowing the comparison time and background.

A singular endpoint is a limit of the model

At θ = 3π/2 the radius has returned to half its turnaround value, but the actual density ratio is about 146.84. At θ = π/2 the same radius ratio occurred earlier, with density ratio only about 1.466. The expanding background makes these different states.

The formal θ → 2π dust endpoint sends the radius to zero and the interior density to infinity. A corresponding linearly extrapolated label is about 1.68647. This endpoint lies outside the displayed regular trajectory.

The finite linear label does not remove the diverging physical density. It also does not by itself establish a black hole, an equilibrated halo or a universal threshold applicable to every background and halo definition.

Assigning a later virialised size requires additional assumptions about energy, relaxation and external influences. The parametric dust equations alone supply no velocity distribution, equilibrium density profile or prescription for surviving the singularity.

Real haloes require trajectories and a definition

In collisionless structure formation, matter trajectories can cross and create several velocity streams at one position. A single-stream pressureless fluid description then loses information even when particles continue to follow regular trajectories.

Numerical gravitational calculations can follow such particle motion, mergers and nonspherical forces within stated assumptions and resolution. Connecting the resulting dark-matter structures to luminous galaxies additionally requires gas, cooling, star formation and feedback models.

A halo mass also belongs to an operational boundary convention, often involving a specified enclosed overdensity relative to a chosen reference density. It is not supplied uniquely by the word collapsed or by the value of δ_L alone.

The sphere remains useful because it separates turnaround, genuine nonlinear density and linear extrapolation in a transparent example. A realistic halo abundance, mass profile or galaxy population requires further dynamics, definitions and validation.

Worked reasoning

Try the reasoning

1. Is θ itself the cosmic time?

   Show the answer and reasoning

   No.

   Cosmic time is B₀(θ − sin θ); θ is a dimensionless trajectory parameter.

2. Why do θ = π/2 and 3π/2 have different interior-to-background density ratios despite equal R/R_ta?

   Show the answer and reasoning

   They occur at different times while the exterior density continues to fall.

   Fixed mass and equal physical radius give the same interior density, but the later background is more dilute.

3. Does the dust singularity or the linear label near 1.68647 specify a realistic equilibrium halo profile?

   Show the answer and reasoning

   No.

   The model ends at a singularity; relaxation, multistreaming, external forces and a halo definition require additional treatment.

What to carry forward

• A finite spherical dust model can show turnaround and regular contraction under explicit restrictive assumptions.
• Actual nonlinear density, density ratio and an extrapolated linear amplitude are distinct quantities.
• An ideal collapse endpoint does not determine a realistic halo profile, universal threshold or luminous galaxy.