Dark Acoustic Oscillations: Dark Sector Signatures

• Dark Acoustic Oscillations are distinctive imprints in the matter power spectrum produced by pressure-supported sound waves in a coupled dark matter–radiation plasma.
• They are modeled using analytical transfer functions that capture collisional damping and nonlinear evolution, revealing key microphysical parameters like coupling strength and decoupling redshift.
• Observational signatures appear in the CMB, galaxy clustering, halo mass functions, and 21-cm cosmology, providing stringent constraints on diverse dark sector interactions.

Dark acoustic oscillations (DAOs) are a distinct set of imprints in the cosmological matter power spectrum arising from interactions between dark matter (DM) and a relativistic species—either within the dark sector (e.g., dark radiation such as a dark photon) or with a Standard Model particle (notably, cosmic neutrinos). Analogous to baryon acoustic oscillations (BAOs), DAOs are a direct consequence of pressure-supported sound waves sourced by these interactions before kinetic decoupling in the early Universe. Their presence, morphology, and damping are highly sensitive to the microphysical parameters of the coupled species, the coupling strength, and the fraction of interacting DM. DAOs provide a potent probe for exploring non-minimal dark sector physics, offering distinctive signatures in both the linear and nonlinear cosmic structure and serving as stringent targets for current and next-generation astrophysical and cosmological observations (Buckley et al., 2014, Cyr-Racine et al., 2013, Akita et al., 2023, Muñoz et al., 2020).

1. Microphysical Origin and Theoretical Description

DAOs originate from the period when DM is sufficiently coupled to a relativistic particle bath (dark radiation, DR, or neutrinos) to form a tightly coupled “dark plasma.” This coupled fluid supports acoustic waves with a characteristic sound speed: $$c_D^2 \approx \frac{1}{3(1 + R_D^{-1})},\qquad R_D = \frac{4\rho_{\mathrm{DR}}}{3\rho_{\mathrm{DM}}}$$ where $\rho_{\mathrm{DR}}$ is the DR density, and $\rho_{\mathrm{DM}}$ the DM density (Cyr-Racine et al., 2013, Buckley et al., 2014). The Boltzmann hierarchy for perturbations includes collision terms reflecting the DM–DR momentum transfer rate, typically parameterized via a cross section $\sigma_T$ for Thomson or analogous scattering.

Coupling persists until a redshift $z_{\mathrm{dec}}$ when the Hubble rate $H(z)$ exceeds the momentum exchange rate and DM “kinetically decouples.” Perturbation modes that enter the horizon before this decoupling undergo acoustic oscillations; those entering afterwards evolve collisionlessly. Collisional (Silk-like) damping erases small-scale fluctuations below a characteristic damping scale: $$r_{\mathrm{SD}} \sim \Bigl( \frac{4 a_d^3 m_\chi}{81 H_0 \sqrt{\Omega_r \Omega_{\mathrm{DM}}}} \Bigr)^{1/2}$$ where $a_d$ is the scale factor at decoupling and $m_\chi$ the DM mass (Buckley et al., 2014).

The central analytic scale, the DAO sound horizon, is given by: $$r_{\mathrm{DAO}} = \int_{z_{\mathrm{dec}}}^{\infty} \frac{c_D(z)}{H(z)} dz$$ and controls the oscillation frequency in the resulting matter power spectrum.

2. Linear Power Spectrum and Transfer Function

The effect of DAOs on the linear matter power spectrum $P(k)$ is a series of damped oscillations—a “ringing” superposed on a global suppression of power at wavenumbers exceeding $k_{\mathrm{DAO}} \sim \pi/r_{\mathrm{DAO}}$: $$T_{\mathrm{DAO}}(k) = T_{\Lambda\mathrm{CDM}}(k) \left[ 1 + A \exp\left(-\left(\frac{k}{k_{\text{damp}}}\right)^p\right) \cos(kr_{\mathrm{DAO}} + \varphi) \right]$$ Here, $k_{\text{damp}} \sim 1/r_{\mathrm{SD}}$, and $A$, $p$, $\varphi$ are determined by the specific microphysics (Cyr-Racine et al., 2013, Buckley et al., 2014). Oscillations are separated by $\Delta k \sim \pi/r_{\mathrm{DAO}}$ with an envelope damped by collisional diffusion.

In the ETHOS framework, $P(k)$ is well captured by a two-parameter transfer function: $$T_L(k) = \begin{cases} 1, & k \ll k_{\text{peak}} \ 1 - (k/k_{\text{peak}})^\alpha e^{-(k/k_{\text{peak}})^\beta} [1 - h_{\text{peak}} \sin(\omega k/k_{\text{peak}})], & k \gtrsim k_{\text{peak}} \end{cases}$$ where $k_{\text{peak}}$ locates the first DAO peak, and $h_{\text{peak}}$ its amplitude; $T_L^2(k)$ multiplies the CDM spectrum to yield $P(k)$ (Muñoz et al., 2020).

3. Nonlinear Evolution and Halo Mass Function

Nonlinear gravitational evolution smooths high-frequency DAOs in the power spectrum through mode coupling, yet it does not entirely erase substructure imprints (Schaeffer et al., 2021, Buckley et al., 2014). Notably, DAO-induced features persist robustly in the halo mass function (HMF) even at $z = 0$:

• The HMF inherits oscillatory “bumps” and “dips” corresponding to DAO and Silk scales, in contrast to the monotonic suppression seen for warm dark matter (WDM) (Buckley et al., 2014).
• This nontrivial structure in $dn/dM$ survives to low redshifts, making the satellite and subhalo populations sensitive to DAOs, especially for strongly coupled scenarios (Akita et al., 2023).

Semi-analytical predictions employ extended Press-Schechter (EPS) formalisms modified to incorporate DAO-altered $P(k)$, supplemented with realistic treatments of subhalo tidal evolution (Akita et al., 2023).

4. Observational Signatures and Constraints

The multi-scale impact of DAOs gives rise to several classes of observable signatures:

• Cosmic Microwave Background: DAOs induce nontrivial modifications to the temperature and polarization angular spectra (TT, EE, TE) at $\ell \gtrsim 600$, producing out-of-phase oscillatory residuals, amplitude modulations, and nonmonotonic peak shifts. These cannot be mimicked by vanilla $\Lambda$CDM plus simple $\Delta N_{\text{eff}}$ extensions (Cyr-Racine et al., 2013).
• Large-Scale Structure: BAO-scale analogues in the galaxy power spectrum are highly constrained; CMB + BAO + galaxy P(k) + lensing data require that the DAO scale be $r_{\mathrm{DAO}} \lesssim 10\,h^{-1}$Mpc or that the interacting DM fraction $f_{\mathrm{int}} \lesssim 5\%$ for “baryon-strength” coupling (Cyr-Racine et al., 2013).
• Halo Mass Function and Satellites: The predicted number and mass function of subhalos are sensitive to DAOs. Milky Way satellite counts from DES and Pan-STARRS1 limit the DM–neutrino cross section for various energy dependencies:

$$\begin{aligned} &\sigma_{\rm DM-\nu,0} \lesssim 4\times10^{-34}\,{\rm cm}^2(m_{\rm DM}/{\rm GeV})\ &\sigma_{\rm DM-\nu,2}^0 \lesssim 10^{-46}\,{\rm cm}^2(m_{\rm DM}/{\rm GeV})(E_\nu/E_\nu^0)^2\ &\sigma_{\rm DM-\nu,4}^0 \lesssim 7\times10^{-59}\,{\rm cm}^2(m_{\rm DM}/{\rm GeV})(E_\nu/E_\nu^0)^4 \end{aligned}$$

(Akita et al., 2023).

• 21-cm Cosmology: DAOs generate distinctive delays and nonmonotonic features in the cosmic dawn 21-cm brightness-temperature global history and power spectrum. Current and planned interferometers (e.g., HERA) will be sensitive to DAO models with cutoffs at $k_{\text{peak}} \lesssim 250-300\,h/{\rm Mpc}$ and can distinguish DAO from WDM suppression and CDM+feedback at high statistical significance (Muñoz et al., 2020).

5. Models, Parameter Dependencies, and Phenomenological Variety

DAOs arise in a range of dark sector constructs:

• DM–Neutrino Scattering: Described by a cross section $\sigma_{\rm DM-\nu,n} \propto E_\nu^n$, with decoupling and damping scales dependent on the interaction’s temperature scaling (Akita et al., 2023).
• Self-Interacting DM with Light Mediator: The “dark atom” scenario, where DM is charged under an unbroken U(1) and couples via a Yukawa (massive vector) potential. The coupling strength $\alpha_D$, DM mass $m_D$, mediator mass $m_\phi$, and relic DR-to-photon temperature ratio $\xi$ set $r_{\mathrm{DAO}}$ and $r_{\mathrm{SD}}$ (Buckley et al., 2014, Cyr-Racine et al., 2013).
• ETHOS Formalism: Provides a mapping from DM microphysics to transfer function parameters ($k_{\text{peak}}$, $h_{\text{peak}}$), enabling systematic paper of DM–DR acoustic regimes (Muñoz et al., 2020).

Observable phenomenology interpolates between CDM, WDM, and more complex nonmonotonic features (e.g., multiple HMF “bumps,” modifications to high-z reionization, and galaxy scaling relations).

6. Nonlinear Persistence and Challenges for Detection

Nonlinear mode coupling degrades the contrast of DAOs in the density power spectrum at late times, particularly toward low redshift (Schaeffer et al., 2021). However, in the HMF and the subhalo/cluster mass functions, DAOs persist as broad, low-amplitude features:

• DAO residuals are detectable in the HMF and subhalo mass statistics, provided observations can resolve mass scales below the cutoffs and distinguish oscillatory imprints from stochastic scatter (Schaeffer et al., 2021, Buckley et al., 2014).
• The amplitude of these features is typically low and diffuse, especially after nonlinear evolution, posing a challenge for detection with current survey precision.
• Observational windows offering greatest sensitivity are the high-redshift (e.g., cosmic dawn, $z \sim 10-30$) 21-cm power spectrum, the abundance of faint dwarf galaxies, small-scale strong gravitational lensing anomalies, and future surveys with increased dynamic range and completeness at low masses (Muñoz et al., 2020, Akita et al., 2023).

7. Constraints, Implications, and Future Prospects

Current cosmological data (CMB, galaxy clustering, Lyman-α forest, and Milky Way satellites) place stringent limits on the parameter space accessible to DAOs:

• Any scenario producing strong DAOs at large scales is excluded unless the interacting DM fraction is $\lesssim 5\%$ or decoupling occurs sufficiently early ($r_{\mathrm{DAO}} \lesssim 10\,h^{-1}$Mpc) (Cyr-Racine et al., 2013).
• Models tuned to suppress small-scale structure enough to resolve “missing satellites” or “too big to fail” problems via DAOs are disfavored by current upper bounds, especially in elastic DM–neutrino scenarios (Akita et al., 2023).
• Next-generation probes—including high-precision CMB polarization/lensing, 21-cm tomography, and dense galaxy group catalogs—are forecast to further tighten constraints and may provide an avenue for detecting suppressed but nonzero DAO signatures (Muñoz et al., 2020, Buckley et al., 2014).

DAOs thus remain a critical, actively constrained theoretical possibility for extending the ΛCDM paradigm and probing the detailed microphysics of the dark sector via astrophysical observations and cosmological structure formation.