An acoustic imprint in the statistics of structure

A preferred separation is a statistical imprint

Before photons and baryons cease to exchange momentum efficiently, pressure and gravity support acoustic motion in their coupled medium. The resulting perturbations can leave a characteristic comoving length in later matter statistics.

The relevant sound-travel integral has the form ∫c_s dt/a over a specified early interval. Photon last scattering and baryon drag describe related but distinct endpoints; a sound horizon must state which endpoint and physical history it uses.

Later structure contains a superposition of responses to many initial perturbations. An acoustic feature therefore concerns a small statistical preference in pair separations. It does not require every galaxy to sit on one spherical wall surrounding a central object.

Bulk motions, nonlinear growth and tracer properties affect the measured feature. Here a supplied analytic spectrum and a supplied smoothing operation isolate some statistical ideas without fitting an observed cosmology or calculating its early sound horizon.

Define one hypothetical isotropic field

Let L be an arbitrary positive comoving length. Choose P₀ = 0.1L³, α = 0.5 and r_s = 5L. These are teaching parameters; r_s is a chosen length label, not an evaluated physical sound horizon or a value in megaparsecs.

Define j₀(z) = sin z/z with j₀(0) = 1, and choose P(k) = P₀ exp(−σ²k²/2)[1 + αj₀(kr_s)]. The wavenumber k has units of inverse length, P has units of length cubed and σ is a comoving smoothing length.

Use the Fourier convention ξ(r) = ∫₀∞ k²P(k)j₀(kr) dk/(2π²). The two-point correlation ξ is dimensionless. All analytic integrals run over the full positive wavenumber range, even though a drawing may show only a finite window.

Because |j₀| ≤ 1 and α = 0.5, this spectrum is nonnegative. It specifies a hypothetical statistically homogeneous and isotropic perturbation field. Its small variance alone does not guarantee positive physical density in every possible random realisation.

The transform separates a central term from an acoustic excess

For this spectrum, ξ(r) = P₀[G_σ(r) + αK_σ(r,r_s)], where G_σ(r) = (2π)⁻³ᐟ²σ⁻³ exp[−r²/(2σ²)]. The Gaussian central term and the acoustic excess are explicitly separate.

For positive r, K_σ = (2π)⁻³ᐟ²{exp[−(r−r_s)²/(2σ²)] − exp[−(r+r_s)²/(2σ²)]}/(2σrr_s). At r = 0 use its regular limit K_σ(0,r_s) = G_σ(r_s), rather than dividing by zero.

Equivalently, K_σ is the uniform angular average of G_σ(|r − r_s n|) over unit vectors n. This makes the three-dimensional geometry explicit. The maximum of the acoustic excess need not fall exactly at r_s.

The table shows E(r) = P₀αK_σ, rather than the full ξ. For σ = L, the full variance is ξ(0) ≈ 0.0063493754; for σ = 2L it is about 0.0008111062. A finite radial plot from 0 to 10L does not redefine those full-transform results.

Filtering density amplitude squares the filter in power

Now prescribe a density-mode amplitude filter W(k) = exp(−Σ²k²/2), with Σ = √(3/2)L. Multiplying each density Fourier amplitude by W multiplies its power by W² = exp(−Σ²k²).

Starting from σ = L, the filtered spectrum has the same analytic form with σ_eff² = σ² + 2Σ² = 4L². Thus σ_eff = 2L. Using W only once in the power would give the wrong smoothing length.

At r = 5L, the acoustic excess decreases from about 0.0000634936 to 0.0000317467, while excess at some more distant separations increases as the feature broadens. The central Gaussian contribution also changes, so these excess values are not the full correlation.

For example, at kL = 1 the initial power is about 0.0548369L³ and the filtered power about 0.0122358L³. Their ratio is exp(−1.5), the square of the chosen amplitude filter at that wavenumber.

A clean filter is only one part of real evolution

The supplied filter illustrates suppression of small-scale power and broadening of a chosen correlation feature. It is not a derivation of gravitational displacements, a calibrated nonlinear evolution law or a source of all new power produced by mode coupling.

A measured galaxy correlation additionally depends on sample selection, tracer bias, redshift-space effects, distance-coordinate choices and measurement covariance. Extracting a distance requires a physical ruler model and a likelihood, rather than reading r_s directly from this toy.

The preferred length in the early transfer process is also distinct from the exact maximum of a later, smoothed three-dimensional correlation function. Geometry, the broad spectrum and subsequent evolution matter when translating one into the other.

An acoustic imprint can link early plasma physics with later statistical structure. Establishing that link in data requires the full transfer calculation, observational controls and model comparison; the analytic example supplies no current detection, precision claim or uniquely confirmed origin of perturbations.

Worked reasoning

Try the reasoning

1. Does a preferred pair separation require every galaxy to lie on one physical shell?

   Show the answer and reasoning

   No.

   The correlation describes statistical pair relationships in superposed perturbations, rather than one universal material wall.

2. At r = 5L, is the table entry E the complete correlation ξ?

   Show the answer and reasoning

   No.

   The central contribution P₀G_σ must also be added; the table deliberately isolates the acoustic excess.

3. Does this Gaussian amplitude filter predict all nonlinear gravitational power or an observed acoustic distance?

   Show the answer and reasoning

   No.

   The operation is supplied for teaching. Actual evolution, tracer effects, physical ruler calibration and a data likelihood require additional work.

What to carry forward

• An early acoustic process can leave a preferred statistical separation without forming one universal galaxy shell.
• A Fourier convention, units and the full transform distinguish power from the correlation feature.
• A density-amplitude filter enters power squared; a prescribed smoothing model does not replace full nonlinear evolution or observation.