Automatic detection of fast oscillations of dark matter scalar field and updated cosmological constraints on QCDM

The paper introduces a new technique for automatically detecting the onset of rapid oscillations in scalar field dark matter and implements it in the Boltzmann solver CLASS. The authors apply this method to the QCDM model, a field-theoretic model with interacting dark matter and dark energy, and update cosmological constraints using recent data.

Problem: Scalar field dark matter with a quadratic potential undergoes fast oscillations when the mass scale in the Klein-Gordon equation becomes much smaller than the Hubble parameter, making numerical solutions intractable. Existing methods either switch from solving the Klein-Gordon equation to fluid equations at the onset of oscillations or introduce new variables to absorb the oscillations. However, these techniques rely on an estimate of when oscillations start, which can be cumbersome for large parameter space scans and have been used mainly for non-interacting models.

New method: The authors introduce an averaging technique with automatic detection of the onset of oscillations. The algorithm tracks the ratio r = y/θ, where y and θ are variables in a reformulated version of the Klein-Gordon equation. A sudden drop in this ratio signals the start of oscillations. The algorithm also independently confirms oscillations by checking for sign changes in the dark matter equation of state wχ. The transition phase θ∗ is then set, and averaging is triggered. The method is implemented in CLASS and tested on the QCDM model.

Key results:

• The automatic detection method accurately identifies the onset of oscillations and triggers averaging, as shown in Fig. 2.

• The technique correctly handles the presence of interactions, ensuring that the averaged equation of state of dark matter receives no spurious contribution from the interaction term.

• The QCDM model predicts a suppression of the matter power spectrum relative to ΛCDM by roughly 30%–35%, with the largest dip around k ∼ 10−3 − 10−2 h Mpc−1.

• The CMB power spectrum of QCDM matches ΛCDM closely, with only small oscillatory features at high multipoles.

Updated constraints: The authors perform MCMC analysis using Cobaya with data from DESI DR2 BAO, Pantheon+ supernovae (with and without SH0ES calibration), Planck 2018, ACT DR6, and SPT-3G. Key results from Table 3:

• For CMB+DESI+PPS: H0 = 68.37+0.30−0.26 km/s/Mpc, S8 = 0.8119 ± 0.0079, omegam = 0.3010 ± 0.0033.

• The interaction parameter λ remains weakly constrained (log(λ/Mpc−2) < 3.99 for the full dataset).

• The Bayes factor ln Bij = −2.189 for CMB+DESI+PPS, indicating a preference for ΛCDM over QCDM.

Conclusion: The automatic detection technique is efficient and accurate. While QCDM provides a viable fit to the data and produces modest shifts in cosmological parameters, the additional model freedom is not sufficiently favored by the data to overcome the Bayesian complexity penalty. Both QCDM and ΛCDM can explain the data equally well.

Improvements to AI systems based on this paper:

1. Adaptive numerical solver with automatic regime-switching detection

• Implement a general-purpose module that monitors dimensionless ratios (e.g., r = y/θ) of dynamical variables in real-time during ODE integration. When a sudden drop or threshold crossing occurs, the solver automatically switches from stiff (e.g., Klein-Gordon) to averaged/fluid equations without user-specified transition times.

• The improved AI system can autonomously handle multi-scale physics (fast oscillations vs. slow cosmological expansion) across arbitrary parameter scans, eliminating manual tuning and reducing numerical instability.

2. Self-validating oscillation detection via multiple independent observables

• Build a cross-checking mechanism where the AI verifies the onset of a new dynamical regime using two or more physical indicators (e.g., ratio drop AND sign changes in equation-of-state parameter wχ). If indicators disagree, the system flags uncertainty or retries with finer time-stepping.

• This enables robust, trustworthy automation for complex interacting field theories where single-metric detection may fail due to coupling terms.

3. Interaction-aware averaging for effective equations of state

• Extend the averaging technique to automatically separate background interaction terms from intrinsic field oscillations, ensuring no spurious contributions leak into averaged quantities (e.g., effective wχ).

• The AI can now correctly model interacting dark sectors (like QCDM) without manual correction, enabling fast and accurate predictions for coupled scalar-field models.

4. Bayesian model-comparison with built-in complexity penalty

• Integrate the automatic detection method into MCMC samplers (e.g., Cobaya) to compute Bayes factors (ln B) directly during parameter exploration, rather than post-hoc.

• The improved system can autonomously decide whether added model complexity (e.g., interaction parameter λ) is justified by data, providing real-time guidance on model selection and preventing overfitting.

5. High-precision power spectrum prediction with oscillation-averaged initial conditions

• Use the detected transition phase θ∗ to set initial conditions for Boltzmann solvers (CLASS) that smoothly match averaged fluid equations, avoiding artificial discontinuities in matter power spectrum (P(k)) or CMB spectra.

• This yields accurate predictions for suppression features (e.g., 30–35% dip at k 10−3–10−2 h Mpc−1) and high-multipole oscillatory CMB signatures, enabling direct comparison with next-generation surveys (DESI, Euclid, CMB-S4).

6. Automated robustness checks for dataset combinations

• Implement a pipeline that automatically reruns the detection and averaging for each new dataset combination (e.g., CMB+DESI vs. CMB+Pantheon+SH0ES), flagging if the oscillation onset or averaged dynamics change significantly.

• This allows the AI to quantify systematic uncertainties from numerical choices and data selection, improving reliability of reported H0 and S8 constraints.

It is well known that a scalar field dark matter with a quadratic potential undergoes fast oscillations when the time period represented by the mass scale in the Klein-Gordon equation becomes much smaller than that set by the Hubble parameter. This makes a solution to the equation numerically intractable. Many works in the literature have addressed the problem by either switching between solving the Klein-Gordon equation in the well-behaved regime to solving the fluid equations at the onset of oscillations, or by introducing a new set of variables that can absorb these oscillations. Despite being successful, these techniques rely on an estimate of when the oscillations start. For large scale scans of a model's parameter space, this can become cumbersome. Furthermore, the techniques have been used mainly for non-interacting dark matter models. In this work, we introduce an averaging technique with an automatic detection of the onset of oscillations, capable of capturing the non-interacting as well as the interacting dark matter scenarios. The technique, implemented in CLASS, is tested on the QCDM model and shows excellent detection and averaging abilities. We also update the cosmological constraints on the model using the most recent public data.