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Baryon acoustic oscillations

Baryon acoustic oscillations (BAOs) are periodic fluctuations in the distribution of galaxies and matter on large scales, arising from acoustic waves that propagated through the hot, dense baryon-photon plasma in the early universe until the epoch of recombination about 380,000 years after the Big Bang.^[1] These waves, driven by gravitational instabilities and radiation pressure, created regions of enhanced and reduced density with a characteristic comoving scale of approximately 150 megaparsecs (or roughly 500 million light-years), which froze into the cosmic structure when the plasma cooled and neutral atoms formed, decoupling photons from baryons.^[2] This scale, known as the sound horizon, represents the distance sound waves could travel before recombination and serves as a cosmic fossil imprinted in both the cosmic microwave background (CMB) and the large-scale structure (LSS) of the universe.^[1] The BAO signature appears as a series of oscillations in the matter power spectrum P(k), where k is the wavenumber, with the fundamental mode corresponding to the sound horizon scale.^[2] Theoretically, these features emerge from the tight coupling between baryons and photons before recombination, leading to baryonic density perturbations that evolve into the observed clustering of galaxies today.^[2] The first observational detection of the BAO peak occurred in 2005 using over 46,000 luminous red galaxies from the Sloan Digital Sky Survey (SDSS) at redshifts $0.16 < z < 0.47, revealing a correlation function peak at $105 \pm 6 h^{-1} Mpc that matched predictions from CMB data and validated the standard \LambdaCDM model.^[3] Subsequent analyses confirmed this feature across multiple tracers, including emission-line galaxies and quasars, with the baryon fraction \Omega_b h^2 influencing the amplitude and position of the oscillations.^[3] As a standard ruler, BAOs provide a fixed physical length that can be measured in both transverse (angular) and radial (redshift) directions, enabling precise determinations of the angular diameter distance D_A(z) and the Hubble parameter H(z) as functions of redshift without relying on assumptions about galaxy bias or non-linear evolution.^[4] This geometric probe has been instrumental in constraining cosmological parameters, such as the matter density \Omega_m and the dark energy equation-of-state parameter w, with early SDSS results yielding \Omega_m = 0.273 \pm 0.025 (for a flat universe with a cosmological constant).^[3] Major surveys like the Baryon Oscillation Spectroscopic Survey (BOSS) and its extension eBOSS achieved sub-percent precision on the BAO scale across redshifts up to z \approx 2, while the Dark Energy Spectroscopic Instrument (DESI) has further tightened constraints, measuring the expansion history with 0.3% precision in its DR2 release (March 2025) and providing evidence for evolving dark energy at 4.2\sigma significance when combined with CMB observations.^[5]^[6] BAOs complement other cosmological probes, such as supernova distances and CMB anisotropies, by offering a model-independent test of cosmic acceleration and the universe's flatness, with \Omega_k \approx 0.000 \pm 0.002 from recent joint CMB + BAO analyses (as of 2025).^[7] Ongoing and future missions, including NASA's Nancy Grace Roman Space Telescope, aim to extend BAO measurements to higher redshifts (z > 2) and fainter tracers, potentially resolving tensions in H_0 and probing the nature of dark energy with unprecedented detail.^[1]

Origins in the Early Universe

Plasma Epoch and Baryon-Photon Coupling

In the early universe, the plasma epoch extended from shortly after the Big Bang until approximately 380,000 years later, during which the universe was dominated by radiation and matter existing in a hot, ionized plasma state.^[8] This plasma primarily consisted of photons, free electrons, protons, and other baryons, with the universe's temperature around 3000 K at the epoch's end, maintaining full ionization. The expansion was governed by the interplay of radiation pressure and gravitational forces acting on primordial density perturbations. Baryons and photons were tightly coupled during this period through Thomson scattering, the primary interaction process where photons scatter elastically off free electrons. Since electrons were electrostatically bound to protons, this scattering effectively dragged baryons along with the photon fluid, preventing baryons from collapsing under gravity and instead responding collectively to photon pressure gradients. The strength of this coupling decreased as the universe expanded and the baryon-to-photon ratio evolved, but remained sufficiently strong to treat the baryon-photon mixture as a cohesive fluid until recombination. The sound speed in this baryon-photon plasma, which characterizes wave propagation, is expressed as

c_s = \frac{c}{\sqrt{3(1 + R)}}

where c is the speed of light and R = \frac{3\rho_b}{4\rho_\gamma} represents the ratio of baryon energy density \rho_b to photon energy density \rho_\gamma. This formula arises from the relativistic fluid equations, accounting for the inertial contribution of baryons to the otherwise radiation-dominated pressure. As R increases with cosmic expansion due to the dilution of photon density, c_s decreases from near c/\sqrt{3} in the radiation-dominated regime. Initial density perturbations in the early universe originated from quantum fluctuations amplified during cosmic inflation, injecting a characteristic scale into the primordial power spectrum. These perturbations, once entering the horizon during the plasma epoch, were amplified by the tight baryon-photon coupling, driving oscillations in the fluid density that set the stage for baryon acoustic oscillations.

Recombination and Acoustic Damping

The recombination epoch, occurring at a redshift of approximately z \sim 1100, marks the transition in the early universe from a hot, ionized plasma to a neutral gas, as free electrons combined with protons to form neutral hydrogen atoms. This process decoupled photons from baryons, allowing the cosmic microwave background (CMB) radiation to propagate freely and imprinting the state of density perturbations at that time. The timeline of recombination was rapid, spanning a narrow range in redshift due to the sensitivity of the ionization fraction to temperature, with the universe cooling to about 3000 K. During this epoch, Silk damping emerged as a key dissipative mechanism, arising from the random diffusion of photons through the baryon-photon fluid, which smeared out small-scale density perturbations. This diffusion damping, first quantified by Silk in 1968, suppresses acoustic oscillations on scales smaller than the photon mean free path at recombination, effectively erasing power in the perturbation spectrum below roughly 10 comoving Mpc. The damping scale is set by the interplay of photon scattering and the expanding universe, leading to an exponential suppression of baryon density fluctuations on sub-horizon scales.^[9] At the end of recombination, the acoustic oscillations in the baryon-photon plasma froze as the coupling between photons and baryons weakened dramatically, imprinting the prevailing density fluctuations onto the baryonic matter distribution. This freezing preserved the characteristic scales of the oscillations, which later served as seeds for large-scale structure formation after matter domination. Just prior to recombination, the baryon loading parameter R = \frac{3 \rho_b}{4 \rho_\gamma}, which measures the relative inertia of baryons to photons, reached its peak value of around 0.6, enhancing the amplitude of compression-phase waves while suppressing rarefaction phases in the oscillations.

Physics of Acoustic Oscillations

Sound Waves in Cosmic Plasma

In the early universe, following the inflationary epoch, primordial density perturbations provided the initial seeds for structure formation. These quantum fluctuations, amplified during inflation, evolved into classical density contrasts in the hot cosmic plasma dominated by photons, baryons, electrons, and dark matter. Gravitational instabilities arising from these over- and under-densities drove the excitation of pressure waves in the tightly coupled baryon-photon fluid, where photons scattered off charged baryons and electrons, maintaining tight coupling until recombination around redshift z ≈ 1100.^[2] The baryon-photon plasma behaved as a relativistic fluid with photons supplying the dominant pressure, enabling the propagation of sound waves analogous to those in a neutral gas. These acoustic waves featured alternating phases of compression, where gravitational attraction pulled baryons and photons into denser regions, increasing local pressure, and rarefaction, where the restoring photon pressure dispersed the fluid, creating under-densities. The sound speed in this fluid was c_s \approx c / \sqrt{3(1 + R_b)}, with R_b = 3 \rho_b / 4 \rho_\gamma accounting for the baryon loading effect that reduced c_s from the photon-dominated value of c / \sqrt{3}. The dynamics of these perturbations followed a wave equation in the expanding universe, approximated by the simplified Jeans equation:

\ddot{\delta} + \frac{\dot{a}}{a} \dot{\delta} + c_s^2 k^2 \delta = 4\pi G \rho \delta,

where \delta is the density contrast, a is the scale factor, the dot denotes derivative with respect to cosmic time, k is the comoving wavenumber, c_s is the sound speed, G is Newton's constant, and \rho is the total background density. This equation captures the competition between gravitational collapse (right-hand side), pressure support ( c_s^2 k^2 \delta term), and Hubble friction from cosmic expansion (damping term).^[2]^[10] Dark matter played a crucial passive role in this process, as its collisionless nature prevented participation in the acoustic oscillations. Instead, cold dark matter clustered solely under gravity, forming deeper potential wells that amplified the initial primordial perturbations and sourced the gravitational driving force for the baryon-photon waves without itself oscillating. This separation allowed the acoustic signatures to imprint distinctly on the baryonic component while dark matter provided the stable gravitational framework. The waves propagated until recombination, when baryons decoupled from photons, freezing the oscillations and setting the scale for later structure formation.^[2]

Formation of the Sound Horizon

The sound horizon represents the characteristic comoving scale over which acoustic waves in the early universe's baryon-photon plasma propagate before recombination, marking the maximum distance sound can travel from initial density perturbations. This scale is defined as the integral r_s = \int_0^{t_{\rm rec}} \frac{c_s \, dt}{a(t)}, equivalently expressed in conformal time as r_s = \int_0^{\eta_{\rm rec}} c_s \, d\eta, where c_s is the adiabatic sound speed in the plasma, t_{\rm rec} (or \eta_{\rm rec}) denotes the time (or conformal time) at recombination, and a(t) is the scale factor.^[11]^[12] The value of r_s depends sensitively on key cosmological parameters that shape the early universe's expansion and plasma properties. Specifically, it is influenced by the Hubble expansion rate through the overall matter and radiation densities, which determine the conformal time to recombination; higher densities lead to faster expansion and a smaller horizon. The baryon density \Omega_b h^2 affects the sound speed c_s via the baryon-to-photon ratio, reducing c_s and thus r_s for higher baryon fractions, while the radiation content sets the early equality epoch and initial conditions for wave propagation.^[12] In the standard \LambdaCDM cosmology with parameters consistent with cosmic microwave background constraints (\Omega_m h^2 \approx 0.14, \Omega_b h^2 \approx 0.022), theoretical calculations yield r_s \approx 150 Mpc at the present epoch, though the physical scale at recombination (redshift z \approx 1090) is much smaller due to the contracted scale factor. This sound horizon scale imprints the fundamental wavelength of baryon acoustic oscillations in the matter power spectrum, with the primary peak in the galaxy correlation function occurring at separations near r_s, corresponding to wavenumbers k \approx \pi / r_s for the dominant mode.^[12] The acoustic damping beyond this scale further sharpens the feature, making r_s the defining length for the oscillatory signature observed in large-scale structure.

BAO as a Cosmological Probe

Standard Ruler Concept

Baryon acoustic oscillations manifest as a characteristic peak in the two-point correlation function of galaxies, occurring at a comoving separation of approximately 100 h^{-1} Mpc. This peak arises from the frozen sound horizon scale imprinted on the matter distribution during the early universe. The fixed physical size of this scale, denoted as the comoving sound horizon r_s, serves as a "standard ruler" in cosmology, allowing measurements of cosmic distances without relying on intermediate calibrations. By observing the apparent size of this ruler in galaxy surveys at redshift z, cosmologists can infer key expansion history parameters. In the transverse direction, the angular size of the BAO feature provides a measure of the angular diameter distance D_A(z), while the redshift-space distortion along the line of sight yields the Hubble parameter H(z). These inferences arise from comparing the observed separation of the peak to the theoretically predicted r_s, which is precisely determined from cosmic microwave background data.^[13] The standard ruler's utility is encapsulated in the relation for the observed transverse BAO scale:

\theta_{\rm BAO} = \frac{r_s}{D_M(z)},

where D_M(z) = (1 + z) D_A(z) is the comoving angular diameter distance.^[14] For isotropic measurements combining transverse and radial directions, the dilation scale D_V(z) is used, defined as D_V(z) = \left[ (1+z)^2 D_A^2(z) \, c z / H(z) \right]^{1/3}. This equation links the early-universe scale r_s directly to late-time observables, enabling robust distance measurements.^[13] Crucially, the BAO standard ruler remains independent of late-time cosmic evolution because r_s is set during the recombination era at z \approx 1100, prior to the dominance of dark energy or other late-universe phenomena. This early origin ensures that the ruler's length is unaffected by subsequent modifications to the expansion history, providing a clean probe of cosmology across redshifts.^[15]

Imprint on Large-Scale Structure

The baryon acoustic oscillations (BAO) from the early universe leave a distinct signature in the large-scale distribution of matter, manifesting as subtle oscillations in the matter power spectrum P(k). These features, often called baryon wiggles, arise because the acoustic waves imprint density perturbations at scales tied to the sound horizon r_s at the epoch of recombination, with oscillations appearing at wavenumbers k \sim \pi / r_s and harmonics thereof.^[12] The wiggles reflect the frozen-in phase of the plasma oscillations, modulating the power spectrum with a characteristic periodicity that persists into the late universe through linear evolution of structure.^[2] The amplitude of these BAO wiggles in the matter power spectrum is typically a 5-10% modulation relative to the smooth underlying spectrum, providing a faint but detectable statistical feature in the clustering of dark matter and galaxies.^[4] This modulation is damped on small scales due to Silk damping and nonlinear effects, but remains prominent on large scales relevant to cosmological probes. Additionally, the presence of baryons alters the overall shape of the transfer function, suppressing power at small scales (k \gtrsim 0.1 \, h \, \mathrm{Mpc}^{-1}) compared to a pure cold dark matter scenario; this baryon loading effect reduces the amplitude of fluctuations by factors depending on the baryon density parameter \Omega_b h^2, as the baryons add pressure support that delays gravitational collapse during the early phases of structure formation.^[12] In configuration space, the BAO imprint appears in the two-point correlation function \xi(r) as an excess probability of finding pairs of matter particles or galaxies separated by approximately the sound horizon scale, typically r \approx 100 \, h^{-1} \, \mathrm{Mpc}. This peak in \xi(r) quantifies the enhanced clustering at the BAO separation, arising from the Fourier transform of the oscillatory P(k), and serves as a direct tracer of the frozen acoustic scale in the observed galaxy distribution. The feature's persistence in \xi(r) underscores its role as a standard ruler for measuring cosmic expansion, distinct from the broader clustering induced by gravitational instability alone.^[12]

Observational Evidence

Detection in the Sloan Digital Sky Survey

The Baryon Oscillation Spectroscopic Survey (BOSS), conducted as part of the third phase of the Sloan Digital Sky Survey (SDSS-III) from 2009 to 2014, targeted the measurement of baryon acoustic oscillations (BAO) by obtaining spectra and redshifts for over 1.5 million galaxies and quasars across a wide sky area.^[16] This effort built on earlier SDSS phases by focusing on luminous massive galaxies at intermediate redshifts, enabling precise mapping of large-scale structure to detect the BAO signature.^[16] The first robust detection of the BAO feature in galaxy clustering came from the SDSS-II phase, using a sample of 46,748 luminous red galaxies (LRGs) at redshifts 0.16 < z < 0.47.^[3] In this analysis, Eisenstein et al. identified a clear peak in the large-scale two-point correlation function at a separation of approximately 105 h^{-1} Mpc, corresponding to the sound horizon scale imprinted during recombination.^[3] This detection, with a significance exceeding 4σ, provided the initial empirical confirmation of the BAO as a standard ruler in the galaxy distribution.^[3] To enhance the BAO signal degraded by nonlinear evolution and redshift-space distortions, BOSS analyses employed density field reconstruction techniques, which shift observed galaxy positions based on the estimated linear displacement field to partially recover the primordial acoustic signature.^[17] Additionally, the Alcock-Paczynski test was applied to assess cosmological distortions in the observed clustering, comparing radial and transverse scales to constrain the isotropic dilation parameter α, defined as α = D_V(z) / r_s, where D_V(z) is the volume-averaged distance and r_s is the comoving sound horizon at the drag epoch.^[3] These methods sharpened the BAO peak and mitigated systematic effects in the redshift surveys.^[17] Key results from SDSS, particularly at an effective redshift of z ≈ 0.35 using LRG samples, achieved a precision of approximately 3% on α, translating to a relative distance measurement D_V(z=0.35) / r_s = 10.23 ± 0.17 before reconstruction improvements in later BOSS data releases refined this to sub-2% levels.^[3]^[18] This precision established BAO as a reliable probe for cosmological parameters, with the detected scale aligning closely with theoretical predictions from cosmic microwave background data.^[3]

Findings from Other Galaxy Surveys

The 6dF Galaxy Survey provided the first detection of baryon acoustic oscillations (BAO) at low redshift, analyzing the large-scale correlation function of approximately 75,000 galaxies at an effective redshift z_{\rm eff} = 0.106. This measurement yielded a spherically averaged distance scale of D_V(z_{\rm eff}) = 457 \pm 27 Mpc (at a fixed dark matter fraction \Omega_m = 0.3), confirming the BAO feature at a scale of $105 \, h^{-1} Mpc with a significance of about 2σ.^[19] The WiggleZ Dark Energy Survey extended BAO detections to higher redshifts, measuring the acoustic signature in the clustering of over 200,000 emission-line galaxies spanning $0.2 < z < 1.0. At effective redshifts z = 0.44, 0.60, and $0.73, the survey achieved BAO constraints on the distance-redshift relation with precisions of 7–10%, providing the highest-redshift galaxy-based BAO measurement at the time and enabling tests of dark energy evolution.^[20] The extended Baryon Oscillation Spectroscopic Survey (eBOSS), part of SDSS-IV, detected BAO using the Lyman-α forest in quasar spectra, probing neutral hydrogen absorption at an effective redshift z = 2.33 with over 500,000 quasars. This yielded a precise measurement of the BAO scale with 2.5% accuracy in the transverse direction and 4% in the line-of-sight direction, extending the BAO probe into the early universe where galaxy clustering is sparse.^[21] Results from the Dark Energy Spectroscopic Instrument (DESI), as of its second data release (DR2) in March 2025 using the first three years of observations, have further improved BAO precision with data on more than 14 million galaxies, quasars, and Lyman-α forests up to z \approx 3.5. These measurements achieve sub-percent precision on the sound horizon scale, with aggregate distance measurements at 0.28% precision (improved from 0.5% in DR1) and individual isotropic scales down to ~1.5%, surpassing previous surveys in volume, accuracy, and providing stronger evidence for dynamical dark energy when combined with other data.^[5] When combined across surveys like 6dF, WiggleZ, eBOSS, and DESI, BAO data integrated with cosmic microwave background (CMB) observations from Planck constrain the Hubble constant to H_0 \approx 68 km/s/Mpc, highlighting the ongoing tension with local measurements around 73 km/s/Mpc but providing robust tests of cosmological consistency.

Applications to Dark Energy

BAO in Cosmological Distance Measurements

Baryon acoustic oscillations (BAO) serve as a cosmic standard ruler, enabling precise measurements of cosmological distances by comparing the observed separation of the acoustic peak in galaxy clustering to the comoving sound horizon scale r_s at the drag epoch. This scale, imprinted in the large-scale structure, allows calibration of the expansion history across cosmic time. In the isotropic approximation, where clustering is averaged over all directions, BAO measurements constrain the volume-averaged comoving distance, defined as

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

where D_A(z) is the angular diameter distance, H(z) is the Hubble parameter at redshift z, and c is the speed of light. This parameterization combines radial and transverse distance information into a single observable, first utilized in analyses of the Sloan Digital Sky Survey (SDSS) luminous red galaxy sample to yield a 4% precise measurement of D_V(z=0.35)/r_s. By anchoring r_s with independent determinations from the cosmic microwave background (CMB), such as the Planck satellite's value of r_s = 147.09 \pm 0.26 Mpc, absolute distances are obtained, providing a robust, low-redshift extension to the cosmic distance ladder. To disentangle transverse and radial expansions, analyses exploit the anisotropy in BAO clustering, where the Alcock–Paczyński effect distorts the observed scales due to the conversion of observed angles and redshifts into comoving distances under an assumed cosmology. This distortion parameter quantifies the mismatch between the true geometry and the fiducial model, enabling separate constraints on D_A(z) from transverse modes and H(z) from radial modes, or specific combinations like D_A(z) / H(z)^{-1/2} that reflect the underlying expansion. Seminal anisotropic BAO measurements from the Baryon Oscillation Spectroscopic Survey (BOSS) demonstrated this approach, achieving percent-level precision on these quantities at intermediate redshifts by fitting scaling parameters \alpha_\perp and \alpha_\parallel to the two-point correlation function or power spectrum. These measurements are particularly powerful because they directly probe the product D_M(z) H(z), where D_M(z) = (1 + z) D_A(z), offering insights into the Hubble expansion without reliance on intermediate distance indicators like supernovae. The model-independent nature of BAO arises from its reliance on linear perturbation theory and the well-understood physics of the early universe, allowing tests of cosmic flatness and the expansion history without presupposing a dark energy equation of state. When integrated with CMB data, BAO constraints on r_s / D_V(z) at low redshifts complement high-redshift CMB measurements of the angular scale of the sound horizon, enabling detection of spatial curvature deviations at the percent level; for instance, combined analyses consistently favor a flat universe within uncertainties. This synergy provides a clean probe of late-time acceleration, with BAO offering geometric constraints that can be used to infer dark energy properties in subsequent modeling.

Constraints on Dark Energy Parameters

Baryon acoustic oscillations (BAO) measurements provide robust constraints on the dark energy density parameter \Omega_\Lambda and equation of state w within the Lambda-CDM model, favoring \Omega_\Lambda \approx 0.70 and w = -1. In the flat Lambda-CDM framework, combining DESI DR2 BAO data (as of 2025) with Planck cosmic microwave background (CMB) observations yields \Omega_m = 0.3027 \pm 0.0036, implying \Omega_\Lambda = 1 - \Omega_m \approx 0.697 \pm 0.004, consistent with a cosmological constant-dominated universe at high precision.^[5] These results align BAO-inferred expansion history with early-universe CMB constraints, reinforcing the standard model's viability without significant deviations.^[5] To probe potential time evolution in dark energy, analyses often employ the Chevallier-Polarski-Linder (CPL) parametrization, where the equation of state is modeled as w(a) = w_0 + w_a (1 - a) with a as the scale factor. BAO data, particularly from DESI DR2, tightly constrain these parameters when combined with CMB and supernova datasets, revealing tensions with Lambda-CDM. For instance, DESI DR2 BAO + Planck CMB + Pantheon+ supernovae yield w_0 = -0.838 \pm 0.055 and w_a = -0.62^{+0.22}_{-0.19}, indicating a 2.8\sigma preference for dynamical dark energy with w_0 > -1 and w_a < 0.^[5] Similar combinations with other supernova compilations, such as Union3 or DESY5, strengthen this deviation to 3.8\sigma–4.2\sigma, highlighting BAO's role in bounding evolving dark energy models.^[5] Key recent results from DESI DR2 BAO integrated with Planck further refine constant-w constraints, reporting w = -1.00 \pm 0.03 (approximate, aligned with combinations), which rules out w > -0.8 at high confidence and supports behavior near w = -1.^[5] This precision underscores BAO's sensitivity to late-time acceleration, tightening bounds on dark energy dynamics. Additionally, BAO-derived Hubble constant values, such as H_0 = 68.51 \pm 0.58 km/s/Mpc from DESI DR2 + big bang nucleosynthesis + sound horizon priors, align with CMB estimates around 68 km/s/Mpc but exhibit a 4.5\sigma tension with local measurements like SH0ES (H_0 = 73.04 \pm 1.04 km/s/Mpc), suggesting potential inconsistencies in the expansion rate.^[5]

Theoretical Extensions

Integration with General Relativity

Baryon acoustic oscillations (BAO) are analyzed within the framework of general relativity (GR) through cosmological perturbation theory, where relativistic corrections account for effects beyond the standard non-relativistic approximation. One key correction arises from redshift-space distortions (RSD), caused by the peculiar velocities of galaxies relative to the Hubble flow, which distort the observed clustering along the line of sight. These velocities, induced by gravitational interactions, elongate structures in redshift space, particularly on large scales. The linear regime of this effect, known as the Kaiser effect, boosts the power spectrum amplitude in the parallel direction by a factor involving the growth rate f and the angle to the line of sight, as derived in the plane-parallel approximation.^[22] In the context of BAO, RSD modifies the observed scale of the acoustic peak in the correlation function or power spectrum, requiring careful modeling to extract the underlying real-space BAO feature. The Kaiser formula provides the linear anisotropic power spectrum as P^s(k, \mu) = P^r(k) (1 + \beta \mu^2)^2, where \beta = f / b (with b the galaxy bias), \mu is the cosine of the angle between the wavevector and line of sight, and P^r(k) is the real-space power spectrum containing the BAO signature. Relativistic corrections, such as gravitational redshift and lensing contributions, become relevant on very large scales (k \lesssim 0.01 h/Mpc) but are subdominant for typical BAO analyses, which focus on scales around the sound horizon. These effects are incorporated via the full relativistic Boltzmann code or approximations in N-body simulations tuned to GR.^[23] A primary observable from RSD is the combination f \sigma_8(z), where f(z) is the logarithmic growth rate of structure, \sigma_8(z) is the rms density fluctuation on 8 h^{-1} Mpc scales smoothed at redshift z, and in GR, f(z) \approx \Omega_m(z)^{0.55} for matter domination transitioning to dark energy influence. This quantity is extracted from the anisotropic clustering signal via the power spectrum multipoles or wedge, often modeled with perturbation theory extensions like the Taruya-Nishimichi-Saito formula to include non-linear damping. When combined with BAO measurements of the angular diameter distance D_A(z) and Hubble parameter H(z), f \sigma_8(z) breaks degeneracies, enabling tight constraints on the matter density parameter \Omega_m. For instance, joint analyses yield \Omega_m \approx 0.30 with percent-level precision, assuming flat \LambdaCDM.^[24] To test GR, BAO and RSD data are used to constrain parametrized deviations in the perturbed Friedmann-Lemaître-Robertson-Walker metric, such as modifications to the Poisson equation via the parameter \mu(a,k), which alters the relation between the gravitational potential and matter overdensity (the lapse function modification in the conformal Newtonian gauge). In GR, \mu = 1; deviations parametrize theories like f(R) gravity or scalar-tensor models. The complementary \Sigma(a,k) parameter modifies lensing and growth. Full-shape analyses of galaxy power spectra from surveys like BOSS and DESI, incorporating BAO and RSD, fit these parameters while marginalizing over bias and other systematics. Seminal frameworks for this approach include the effective field theory of dark energy, enabling scale- and time-dependent tests.^[25] As of 2025, analyses using the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2) data from the first three years of observations, combining full-shape clustering (including BAO and RSD) with cosmic microwave background and supernova observations, continue to show no significant deviations from GR. Constraints on the deviation parameters, such as \mu_0 = 0.05 \pm 0.22 in a time-dependent parametrization (where \mu(a) = 1 + \mu_0 g(a) and g(a) is a growth suppression function), remain consistent with GR values within 1\sigma, corresponding to modifications limited to below 5% in effective strength for most models (updated from DR1 analyses with tighter uncertainties). Similar results hold for binned redshift and scale dependencies, reinforcing GR's validity on cosmological scales up to z \approx 1.^[25]^[5]

Current and Future Observational Prospects

As of November 2025, the Dark Energy Spectroscopic Instrument (DESI) has delivered significant advancements in BAO measurements through its Data Release 2 (DR2) results, released in March 2025 and incorporating data from over 14 million galaxies and quasars across redshifts up to z \approx 2.3, which have further tightened constraints on the dark energy equation-of-state parameter w compared to previous releases and pre-2020 surveys like those from the Baryon Oscillation Spectroscopic Survey (BOSS), with the aggregate BAO distance measurement precision improving to 0.28% from 0.5% in DR1.^[5] These improvements stem from enhanced statistical power and refined modeling of the BAO feature, enabling more precise mapping of the acoustic scale as a standard ruler for cosmic distances and providing stronger evidence (up to 3.1\sigma) for evolving dark energy when combined with other probes.^[5] The Euclid space mission, launched in July 2023 and commencing routine observations in February 2024, is providing space-based BAO measurements with reduced atmospheric distortions, targeting galaxies up to redshift z \approx 2 through its spectroscopic survey of at least 30 million objects.^[26] Its Quick Data Release 1 (Q1) in March 2025 included initial spectroscopic data from deep fields, contributing to early BAO analyses and joint studies with DESI that enhance constraints on the expansion history. Forecasts indicate that Euclid will achieve approximately 1% precision on the angular diameter distance D_A(z) by leveraging the BAO scale in combination with weak lensing, significantly enhancing constraints on the expansion history over ground-based efforts. Looking ahead, the Nancy Grace Roman Space Telescope, scheduled for launch in 2027, will extend BAO observations via wide-field imaging and spectroscopy of emission-line galaxies, such as those selected by H\alpha emission, across vast sky areas up to z \approx 2.5 in its High Latitude Spectroscopic Survey. This approach promises to map billions of galaxies with high density, yielding BAO precisions competitive with Euclid while focusing on underrepresented high-redshift regimes to cross-validate distance measurements.^[27] Key challenges in these observational prospects include foreground contamination from interstellar medium effects and nonlinear gravitational evolution that damps the BAO signal at low redshifts (z < 0.5), potentially biasing distance inferences. Mitigation strategies involve advanced emulation techniques, such as perturbation theory-based models trained on N-body simulations, to reconstruct the linear BAO regime accurately and achieve the targeted precisions without excessive computational overhead.

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