Recognize when a density expansion or a numerical method has failed

Write density as a sum of terms ordered in the initial fluctuation amplitude. Nonlinear continuity and Euler terms couple Fourier modes, so the power receives contributions beyond the square of the linear field. For Gaussian initial conditions the odd initial contractions vanish, leaving familiar leading terms P11,P22 and twice P13 under the convention that P13 is one cross contraction. A primordial non-Gaussian field can change that counting and must not silently inherit the Gaussian truncation.

Small loop corrections over a tested scale range support an approximation; computing one more term is not an automatic convergence proof. Long-wavelength displacements, baryonic effects, ultraviolet sensitivity, and redshift-space mapping can alter the relevant organization. Effective descriptions can encode unresolved short-scale stress with fitted coefficients and power counting, but need a domain of validity and uncertainty estimate. Their success does not mean that a dust fluid remains microscopically exact inside every halo.

Consider a one-dimensional prescribed displacement map of Zel’dovich form, x=q−λ sin(kq)/k, where q labels the initial comoving position and λ is a growing dimensionless displacement amplitude. Before crossing, local mass conservation gives one plus density contrast equal to the inverse Jacobian. At q=0 the Jacobian is 1−λ. Expanding its inverse produces a geometric density series whose higher terms grow as λ approaches one, despite the simple linear displacement.

When the Jacobian vanishes, initially distinct trajectories can reach the same position. A single-valued pressureless velocity cannot describe the resulting multiple streams; velocity dispersion and higher phase-space moments become relevant. The divergence of the cold idealized density at a caustic is not a claim that the spacetime metric becomes singular. This mapping is an illustrative pre-crossing model, not a general three-dimensional collapse solution, and continuing its one-stream density formula beyond crossing would be physically misleading.

The linear matter variance per logarithmic wave-number interval is k³Plin/(2π²). Values approaching unity often warn that a simple density expansion is losing control, but are not a universal rigorous error bound. Smoothing and spectral shape matter, and a particular observable can need resummation or additional effective terms earlier. Because a sub-Hubble potential is suppressed relative to density by expansion-to-wave-number factors, weak-field gravity can remain useful well into a nonlinear density regime.

The optimization lab illustrates numerical instability separately. Reset and set step size 0.5,initial position 2, then Step twice; predict position 0.5 and objective 0.125. Reset and set step size 2.5,initial position 2, then Step twice; predict position 4.5 and objective 10.125, outside the plot with unstable behavior. This exact gradient-descent sequence has no gravitational collapse or shell crossing. Its divergence comes from an unstable update factor; it must not be interpreted as evidence that a cosmological density singularity occurred.