Baryon Acoustic Oscillation (BAO) Measurements

• BAO measurements are the imprints of primordial sound waves that form a standard ruler for determining cosmic distances.
• State-of-the-art analyses use Fourier and configuration space methods, redshift slicing, and reconstruction techniques to reduce non-linear damping and systematic uncertainties.
• BAO results anchor the cosmic distance ladder, breaking degeneracies and enabling precise constraints on parameters like matter density and the Hubble constant.

Baryon Acoustic Oscillation (BAO) measurements underpin a key standard ruler for mapping the expansion history of the Universe. BAO are a relic signature of sound waves propagating in the baryon-photon plasma of the early universe, imprinting a characteristic scale (the sound horizon at the drag epoch, $r_s(z_d)$) in the spatial clustering of matter. This scale manifests as a quasi-periodic modulation (the "acoustic peak" or "wiggles") in both the two-point correlation function and the power spectrum of galaxy distributions. BAO measurements provide robust constraints on cosmological distances at multiple redshifts, facilitating stringent tests of cosmic geometry, dark energy, and curvature, and enabling high-precision determinations of key cosmological parameters.

1. Methodological Framework for BAO Measurement

Contemporary BAO analyses rely on the precise measurement of galaxy clustering statistics—primarily the spherically averaged power spectrum $P(k)$ and the spatial correlation function—derived from large-volume redshift surveys such as SDSS DR7+DR9, BOSS, and 2dFGRS.

• Fourier-Space Methods:

Weighted galaxy overdensity fields are constructed and analyzed using the Feldman, Kaiser, and Peacock (FKP) scheme, with $P(k)$ measured after applying optimal weights and survey geometry corrections. The observed power spectrum is modeled as:

$$P_{\mathrm{obs}}(k) = S(k) \times P_{\mathrm{BAO}}(k)$$

where $S(k)$ is a smooth cubic spline representing the non-wiggly broadband, and $P_{\mathrm{BAO}}(k)$ is extracted from linear theory (CAMB) with the smooth part removed and oscillatory features damped by a Gaussian factor:

$$G_{\mathrm{damp}}(k) = \exp\left[-\frac{1}{2}k^2 D_{\mathrm{damp}}^2\right]$$

so that the observed BAO is:

$$\mathrm{BAO}_{\mathrm{obs}} = G_{\mathrm{damp}} \cdot \mathrm{BAO}_{\mathrm{lin}} + (1 - G_{\mathrm{damp}})$$

• Configuration-Space Methods:

The Landy–Szalay estimator is used to compute the correlation function, often modeled as:

$\xi^{\mathrm{fit}}(r) = B^2 \xi_m(\alpha r) + A(r)$

where $\xi_m$ is an acoustic template from the Fourier transform of the model $P_m(k)$, $\alpha$ is a stretch (dilation) parameter encoding potential shifts in the BAO scale, and $A(r)$ is a polynomial nuisance term that marginalizes over broadband uncertainties.

• Redshift Slicing and Reconstruction:

BAO are fit in several redshift bins to extract both "scale" (absolute distance) and "gradient" (distance ratio) information. Non-linear structure growth tends to broaden/smooth the acoustic feature; density field reconstruction techniques "undo" bulk flows (e.g., via the Zel'dovich approximation), recovering sharper peaks and reducing the impact of non-linear damping and redshift-space distortions.

• Window Function Correction:

The mapping from observed redshift-space coordinates to physical distances depends on a fiducial cosmology; a carefully computed "window function" relates observed $P(k)$ to its true cosmological equivalent, facilitating joint fits across multiple test cosmologies.

2. BAO-Derived Distance Measures and Constraints

BAO measurements constrain spherically-averaged distances through the ratio of the sound horizon to the measured distance:

$d_z \equiv \frac{r_s(z_d)}{D_V(z)}$

where the "volume-averaged" effective distance is

$$D_V(z) = \left[(1+z)^2 D_A^2(z) \frac{c z}{H(z)}\right]^{1/3}$$

with $D_A(z)$ the angular diameter distance and $H(z)$ the Hubble parameter.

Key reported constraints from SDSS DR7/2dFGRS analysis include:

• At $z=0.275$:

$r_s(z_d)/D_V(0.275) = 0.1390 \pm 0.0037$

representing a 2.7% distance measurement.

• Distance gradient over $0.2 < z < 0.35$:

$f \equiv D_V(0.35)/D_V(0.2) = 1.736 \pm 0.065$

These two quantities are nearly independent and jointly characterize both absolute and relative cosmic scale at low redshift.

Such measurements can be directly compared to the predictions of a ΛCDM cosmology, providing model-independent tests of cosmic geometry and expansion.

3. Refined Error Analysis and Likelihood Considerations

Accurate error quantification is critical for BAO-based cosmological inference:

• Non-Gaussianity in Likelihood:

Noise in the data produces a non-Gaussian shape for the likelihood surface of the BAO scale. Extensive tests with log-normal and Gaussian mock catalogs show that errors computed assuming Gaussianity underestimate the true confidence intervals.

• Empirical Error Correction:

To match the actual distribution of BAO scale fits, the band power errors are increased by 14–21% (depending on the number of slices), bringing the likelihood to near-Gaussianity and leading to more conservative, robust confidence intervals.

This rigorous error propagation, combined with mock-based covariance estimation and extended redshift slicing, results in a more reliable measurement and improved internal consistency compared to earlier BAO analyses. Methodological advances in modeling the window function and the use of increased random sampling for the survey selection function have further minimized systematic offsets.

4. Cosmological Parameter Inference and Model Robustness

BAO distances, when combined with external constraints from the CMB (e.g., WMAP5) and Type Ia supernovae, break degeneracies in the matter density–Hubble constant parameter space and allow for robust constraints on fundamental parameters:

• The combined BAO+WMAP5+Union SN sample yields:

$$\Omega_m = 0.286 \pm 0.018,\quad H_0 = 68.2 \pm 2.2~\text{km\,s}^{-1}\,\text{Mpc}^{-1}$$

• In extended cosmologies (allowing both curvature $\Omega_k$ and dark energy equation of state $w$ to vary), constraints remain tight:

$$\Omega_k = -0.006 \pm 0.008,\quad w = -0.97 \pm 0.10$$

This demonstrates that the BAO distance anchors at $z\sim0.275$ are nearly immune to assumptions regarding late-time dark energy models or spatial curvature.

The decoupling of the BAO-measured $r_s(z_d)/D_V$ from the detailed physics of dark energy at $z>0.35$ grants a high degree of model-independence, provided the CMB-calibrated physical densities are robust.

5. Impact and Role of BAO as a Standard Ruler

• Internal Consistency with ΛCDM:

The BAO-derived distance ratios align with predictions of a flat ΛCDM model. The measured value of $f$ is consistent at the 1.1$\sigma$ level, and discrepancies evident in earlier analyses (e.g., DR5) have lost significance with increased data volume and improved error handling.

• Breakdown of Degeneracies:

The combination of BAO, CMB, and supernovae establishes the "inverse distance ladder," achieving precise, absolute Hubble parameter calibration that is robust to late-time dark energy physics or geometric curvature.

• Standard Cosmological Formulas:

The analysis relies centrally on:

$$D_V(z) \equiv \left[(1+z)^2 D_A^2(z) \frac{c z}{H(z)}\right]^{1/3},\qquad d_z \equiv r_s(z_d)/D_V(z)$$

$$G_{\mathrm{damp}} = \exp\left[-\frac{1}{2}k^2 D_{\mathrm{damp}}^2\right],\quad \mathrm{BAO}_{\mathrm{obs}} = G_{\mathrm{damp}} \cdot \mathrm{BAO}_{\mathrm{lin}} + (1-G_{\mathrm{damp}})$$

These expressions fully specify the relationship between BAO observables and cosmic expansion.

6. Systematic Uncertainties and Future Directions

• The methodology includes extensive validation using multiple types of mock catalogs (log-normal, Gaussian), quantifying the effect of noise, survey geometry, and window function corrections.
• Increasing the number of redshift slices, refining the random catalog sampling, and using flexible spline-based distance models all serve to reduce systematic uncertainties seen in earlier BAO results.
• The approach is robust against physically motivated extensions to ΛCDM, e.g., spatial curvature or non-standard dark energy equations of state, with no significant degradation in parameter precision.

BAO as a standard ruler remains central to constraining cosmic expansion, offering a model-independent, percent-level anchor for cosmological parameters. Ongoing and future surveys will continue to exploit these methodological advances and rigorous statistical treatments to probe cosmic acceleration and the nature of dark energy.