Linear growth in an expanding background

Expansion changes the conditions for gravitational growth

A structure-growth calculation needs a background expansion as well as a description of the fluctuations. Here choose a spatially flat pressureless-matter-only general-relativistic background, often called the Einstein–de Sitter model.

Let A = a/a_i, with H = H_i A⁻³ᐟ² for arbitrary positive H_i. Its physical background mass density is ρ̄_m = 3H_i²A⁻³/(8πG). Density here is mass per volume, so no extra conversion from energy density is left implicit.

Consider a weak nonrelativistic matter-density perturbation on scales much smaller than the instantaneous Hubble length c/H. Neglect pressure and velocity dispersion, and use the pressureless single-stream fluid description in its Newtonian weak-perturbation regime.

These conditions define the example. Radiation, vacuum energy, significant pressure, an all-scale relativistic perturbation treatment and later multistreaming are not included. Density contrast is not being asserted to be gauge independent on every scale.

A second-order equation needs two initial conditions

Within that regime, the linear density-mode amplitude obeys δ̈ + 2Hδ̇ − 4πGρ̄_mδ = 0. Dots denote cosmic-time derivatives. The H-dependent term and gravitational source belong to the same specified background.

Define N = ln A, so d/dt = H d/dN. The matter-only relation Ḣ = −3H²/2 and 4πGρ̄_m = 3H²/2 reduce the equation to δ_NN + (1/2)δ_N − (3/2)δ = 0.

The two independent solutions are A and A⁻³ᐟ². The general amplitude is therefore δ = C_g A + C_d A⁻³ᐟ². The constants are determined by both initial contrast and initial derivative, rather than by contrast alone.

The initial derivative is tied to initial peculiar-velocity divergence by linear continuity. Converging or diverging matter motion can therefore change the later amplitude even when the starting density perturbation is the same.

Equal starting contrasts can have different futures

Choose δ = 0.01 at A = 1 in all three cases. A pure growing contribution has (C_g,C_d) = (0.01,0) and initial δ_N = 0.01. A pure decaying contribution has (0,0.01) and initial δ_N = −0.015.

For a third case, choose zero initial δ_N. The two equations C_g + C_d = 0.01 and C_g − (3/2)C_d = 0 give C_g = 0.006 and C_d = 0.004.

Zero initial derivative is thus a mixture of growing and decaying contributions, not a pure growing mode and not a static perturbation for all later times. Its subsequent change follows from the supplied gravitational equation.

These are three alternative initial states in the same background. They are not three measurements of one source, and no observational likelihood has been used to choose their coefficients.

The distinction remains visible over the displayed interval

At A = 4, the pure growing amplitude is 0.04 and the pure decaying amplitude is 0.00125. The zero-initial-derivative mixture gives 0.0245.

At A = 8, their amplitudes are 0.08, approximately 0.000441942 and approximately 0.0481768. The background H/H_i has meanwhile fallen to about 0.0441942. Expansion and perturbation amplitude are different quantities in the table.

The largest displayed contrast is the chosen value 0.08. It illustrates the small-amplitude calculation, but its size alone does not guarantee a specified accuracy for neglected nonlinear terms or for an application outside the stated physical regime.

The table ends at A = 8. Extending the growing power law formally to order-unity contrast would not supply the true nonlinear density, a collapsed object or a halo mass profile.

Initial conditions and validity must travel with the growth law

A pure growing mode is a specific initial-condition choice. A density snapshot alone does not prove that its decaying contribution is absent; initial motion or earlier physical evolution must supply that information.

Different backgrounds and matter properties give different growth equations. The powers A and A⁻³ᐟ² here are consequences of the stated matter-only pressureless system, not universal growth laws for every cosmic epoch or component.

As fluctuations become nonlinear or particle trajectories produce multistreaming, a linear single-stream fluid solution no longer supplies the full dynamics. More complete perturbation methods and numerical descriptions have their own physical and resolution limits.

The next lesson gives a finite spherical-collapse construction to contrast actual nonlinear density with an artificially extrapolated linear amplitude. Keeping those quantities separate prevents a convenient growth label from being read as the physical density of a halo.

Worked reasoning

Try the reasoning

1. Does the single initial contrast δ = 0.01 uniquely select a pure growing solution?

   Show the answer and reasoning

   No.

   The second-order equation also needs an initial derivative, related to initial peculiar motion.

2. At A = 4, what are the supplied pure growing and pure decaying amplitudes?

   Show the answer and reasoning

   0.04 and 0.00125.

   They follow from 0.01A and 0.01A⁻³ᐟ² under the same matter-only assumptions.

3. Does the displayed δ = 0.08 establish a universal error bound for linear theory?

   Show the answer and reasoning

   No.

   An accuracy claim requires checking omitted terms and the physical regime; the chosen amplitude alone is insufficient.

What to carry forward

• A growth law belongs to a specified background, matter description and scale regime.
• Initial density and initial motion determine the growing and decaying contributions.
• A formal linear extrapolation is distinct from the actual nonlinear density of a collapsed structure.