Explore the matter power spectrum with BAO wiggles—adjust baryon fraction to see acoustic oscillations
The matter power spectrum P(k) describes how matter density fluctuations vary with scale in the universe. It's measured from galaxy surveys and gravitational lensing, encoding the entire history of structure formation—from quantum fluctuations during inflation, through baryon acoustic oscillations before recombination, to gravitational growth in the dark matter era. The characteristic "wiggles" at k ~ 0.05-0.3 h/Mpc are the imprint of sound waves in the primordial plasma, providing a "standard ruler" for measuring cosmic distances.
This log-log plot shows P(k) vs wavenumber k (in h/Mpc). Large scales (small k) are on the left; small scales (large k) are on the right. The overall declining shape comes from the primordial spectrum P(k) ∝ kn with n ≈ 0.96 (slightly red-tilted) multiplied by the transfer function T(k), which encodes how modes entered the horizon and were processed by radiation, baryons, and dark matter. The peak around k ~ 0.02 h/Mpc marks the transition from the matter-radiation equality scale to the free-streaming regime.
Before recombination at z ≈ 1100, photons and baryons were tightly coupled by Thomson scattering, forming a relativistic fluid that supported sound waves with speed c_s = c/√3. Dark matter overdensities seeded these oscillations, creating acoustic peaks that propagated outward. At recombination, the photons decoupled and the sound speed dropped to zero, "freezing" the oscillations at a characteristic scale—the sound horizon r_s ≈ 150 Mpc. This scale appears as periodic wiggles in P(k) with wavelength Δk ~ 2π/r_s. Measuring these wiggles provides a standard ruler for cosmology, analogous to standard candles but in Fourier space.
The transfer function T(k) = δ_k(a_late) / δ_k(a_early) describes how initial density fluctuations evolve. This toy uses the Eisenstein-Hu (1998) fitting formula, which accurately captures the suppression at small scales (large k) from Silk damping and the BAO oscillations from baryon-photon coupling. The "no-wiggle" fit removes the oscillatory component, showing the smooth underlying shape. The difference between wiggle and no-wiggle is pure BAO signal, used in surveys like BOSS and DESI to constrain dark energy.
The sound horizon r_s = ∫ c_s dt is the comoving distance sound waves traveled from the Big Bang to recombination. It depends on the baryon-to-photon ratio η = Ω_b h² and the matter density. For Planck parameters, r_s ≈ 147 Mpc. This physical scale is imprinted in both the CMB (as the angular scale of the first acoustic peak) and the matter power spectrum (as the BAO wavelength). By measuring r_s at different redshifts using galaxy clustering, we can reconstruct the expansion history H(z) and constrain dark energy.
The baryon fraction f_b = Ω_b / Ω_m controls the BAO amplitude. More baryons mean stronger coupling to the photon fluid, creating larger pressure forces that enhance the oscillations. Dark matter, which doesn't interact with photons, stays put. The competition between baryonic pressure (pushing out) and dark matter gravity (pulling in) creates the oscillatory pattern. Increasing Ω_b boosts the odd peaks (compression) relative to even peaks (rarefaction), a signature measurable in galaxy surveys.
Planck Cosmology: Use the Planck 2018 preset (Ω_m = 0.315, Ω_b = 0.049, h = 0.674, σ_8 = 0.81). This is our best-fit universe from CMB data. Notice the clear BAO wiggles at k ~ 0.05-0.3 h/Mpc. The first peak around k ≈ 0.015 h/Mpc marks the transition from superhorizon to subhorizon modes at matter-radiation equality.
Vary Baryon Density: Increase Ω_b from 0.045 to 0.08 and watch the BAO wiggles grow stronger. The oscillation amplitude scales roughly as f_b². High baryon content enhances the acoustic signal, making it easier to detect in galaxy surveys. Lower Ω_b (0.02) nearly erases the wiggles—baryons barely affect the dark matter dynamics.
Show BAO Oscillations Only: Enable "Show BAO Oscillations Only" to isolate the pure wiggle component: δP/P = (P_wiggle - P_no-wiggle) / P_no-wiggle. This oscillatory signal has roughly constant period in k-space and decays at small scales due to Silk damping (photon diffusion). The BAO feature is typically a ~10% effect in P(k).
Sound Horizon Scale: The main BAO wavelength Δk ≈ 0.06 h/Mpc corresponds to r_s ≈ 150 Mpc. Adjusting h changes the conversion between physical and comoving scales: larger h shifts the wiggles to higher k. This is why BAO measurements are sensitive to the Hubble constant.
Normalization σ_8: The parameter σ_8 is the RMS mass fluctuation in 8 h⁻¹ Mpc spheres today. Changing it rescales P(k) vertically (in log space it shifts up/down). Current tension exists between CMB measurements (σ_8 ≈ 0.81) and weak lensing surveys (σ_8 ≈ 0.75), potentially signaling new physics or systematic errors.
Shape vs Wiggles: The no-wiggle fit shows the smooth "shape" of P(k), which depends on Ω_m h² (matter density per Hubble²) and the matter-radiation equality scale. The wiggles are added on top. Galaxy surveys measure both: the shape constrains Ω_m, while the wiggles provide the standard ruler for H(z) and D_A(z).
The matter power spectrum is the rosetta stone of large-scale structure cosmology:
The matter power spectrum connects the largest and smallest scales in cosmology—from quantum fluctuations in the inflaton field to the sound of the primordial plasma to the present-day distribution of galaxies. Current surveys like DESI, Euclid, and the Vera Rubin Observatory are measuring P(k) to exquisite precision, using BAO and other features to map the expansion history, weigh neutrinos, constrain inflation, and test General Relativity on cosmic scales. Any deviation from the Eisenstein-Hu prediction would signal new physics in the dark sector.