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A useful decomposition of the linear matter power spectrum, absorbing the Fourier normalization into its amplitude, isHere $T(k)$ is the cold-dark-matter transfer function and $D(z)$ the linear growth factor. On scales entering the horizon after matter-radiation equality, $T\simeq 1$, so $P\propto k^{n_s}$ with $n_s$ near one. Modes entering during radiation domination undergo suppressed growth relative to later entrants, producing a turnover near the equality scale and, well above it, $T\propto \ln (k/k_{\mathrm{eq}})/k^2$. Thus the linear spectrum asymptotes approximately to $k^{n_s-4}{\ln }^2(k/k_{\mathrm{eq}})$. Baryon acoustic oscillations superpose a weak oscillatory pattern. At sufficiently short scales nonlinear gravitational instability invalidates this linear extrapolation.

The plotted curve illustrates the shape rather than a precision cosmological fit; the bands are overlapping approximate scale sensitivities, not sharp experiment boundaries. Cosmic microwave background anisotropies constrain primordial amplitudes and transfer physics on very large through acoustic scales; converting them to a late-time matter spectrum requires a cosmological model. Galaxy redshift surveys constrain approximately $k\sim 0.01$–$0.2h{\mathrm{Mpc}}^{-1}$, with galaxy bias and redshift-space distortions modeled. Peculiar velocities constrain large-scale mass fluctuations; cluster abundances constrain the fluctuation normalization near cluster scales. Weak gravitational lensing measures projected matter on intermediate and smaller scales. The Lyman-alpha forest extends constraints to roughly $k\sim 0.3$–a few $h{\mathrm{Mpc}}^{-1}$ at high redshift, requiring gas, temperature and ionizing-background modeling. These are complementary measurements of the underlying spectrum rather than interchangeable direct Fourier measurements.

The matter fraction controls the equality turnover. From $a_{\mathrm{eq}}={\Omega }_{r,0}/{\Omega }_{m,0}$ and the Friedmann equation at equality,The measured background radiation fixes ${\Omega }_{r,0}{h}^2$, so the physical turnover scale measures approximately ${\Omega }_{m,0}{h}^2$. In units $h{\mathrm{Mpc}}^{-1}$ its location mainly constrains ${\Omega }_{m,0}h$; independent knowledge of $h$ is needed to extract ${\Omega }_{m,0}$. Standard radiation content gives the useful estimate $k_{\mathrm{eq}}\simeq 0.073{\Omega }_{m,0}{h}^2{\mathrm{Mpc}}^{-1}$, approximately $0.015h{\mathrm{Mpc}}^{-1}$ for ${\Omega }_{m,0}=0.3$, $h=0.7$.

The cosmological baryon density sets the inertia of the pre-decoupling photon–baryon fluid and changes its sound horizon and the oscillation pattern. The oscillation contrast and broadband suppression depend on the baryon fraction ${\Omega }_{b,0}/{\Omega }_{m,0}$; the acoustic positions depend on the sound horizon and the distance conversion. The cosmic microwave background peak-height pattern independently constrains ${\Omega }_{b,0}{h}^2$. Combining these with the equality scale and a distance or Hubble constant determination permits estimates of both requested density fractions. Spectral tilt, neutrino free streaming, galaxy bias and nonlinear evolution must be fitted rather than assigning every shape change to baryons. The acoustic signature was already observed in the 2005 galaxy correlation measurement; the first-year microwave-background parameter analysis demonstrates the complementary parameter constraints.

A survey depth $R\sim 400\mathrm{Mpc}$ makes wavelengths substantially larger than its volume poorly measurable. A characteristic inverse depth is $1/R\simeq 2.5\times 10^{-3}\mathrm{Mp}{\mathrm{c}}^{-1}$; a Fourier wavelength that fits across a dimension $L$ instead requires $k\gtrsim 2\pi /L$. For $L$ between $R$ and $2R$, this is roughly $0.008$--$0.016\mathrm{Mp}{\mathrm{c}}^{-1}$. A limited sky patch can have smaller transverse dimensions, so its survey window function raises the practical minimum in those directions. An indicative useful band for a broad survey is around $10^{-2}\mathrm{Mp}{\mathrm{c}}^{-1}$ and above, including scales near equality if its volume is large enough.

There is no upper wavenumber determined by depth alone. Mean separation of observed galaxies, galaxy power spectrum shot noise, angular/redshift resolution, and nonlinear clustering set it. Straightforward linear inference is typically restricted to roughly $k\lesssim 0.1\mathrm{Mp}{\mathrm{c}}^{-1}$ at low redshift, with smaller scales requiring modelling. The magnitude limit does not specify the galaxy comoving number density, so a more precise sampling cutoff cannot be deduced from these data.

The two measurements are complementary. The cosmic microwave background samples a large volume at an early epoch when relevant perturbations are mostly linear, and is free of galaxy bias; its matter-spectrum inference is indirect and depends on projection, cosmological parameters, foregrounds, and cosmic variance. A galaxy survey gives three-dimensional late-time information and can reach smaller scales, but observes biased, discrete tracers: galaxy bias, nonlinear gravitational growth, redshift-space distortions, selection effects, and galaxy power spectrum shot noise must be controlled. Combining both constrains the initial spectrum and its subsequent growth more strongly than either alone, while their overlapping modes test the inferred evolution.