Bias in the tensor-to-scalar ratio from self-interacting dark radiation

TL;DR

The paper investigates whether self-interacting dark radiation (DR) can bias the inference of the tensor-to-scalar ratio $r$ from primordial $B$-mode measurements. It develops a minimal DR model as an effectively massless axion-like particle with quartic self-interaction $\lambda\phi^4/4!$ and implements the resulting collisional dynamics in a modified CLASS via a relaxation-time approximation, with DR abundance set by $\Delta N_{ m eff}$. The authors show that efficient DR self-interactions suppress anisotropic stress, reducing gravitational-wave damping and enhancing the $B$-mode spectrum in a way that depends on $\Delta N_{ m eff}$ and the recoupling redshift $z_{\rm int}$ (or $\lambda$). Using mock data and MCMC analyses, they demonstrate that neglecting these interactions can bias $r$ by amounts comparable to or larger than the expected sensitivities of upcoming experiments like the Simons Observatory, LiteBIRD, and PICO, highlighting the need to model DR interactions in precision searches for primordial gravitational waves.

Abstract

We investigate the cosmological imprint of self-interacting dark radiation (DR) on the primordial $B$-mode angular power spectrum and its impact on the estimation of the tensor-to-scalar ratio $r$. We consider a minimal model in which DR is described as an effectively massless axion-like particle with quartic self-interactions. These interactions are incorporated into the Einstein-Boltzmann equations using the relaxation time approximation and implemented in the $\texttt{CLASS}$ code. We show that increasing the strength of DR self-interactions suppresses anisotropic stress, thereby reducing the damping of gravitational waves and leading to an enhancement of the primordial $B$-mode signal relative to the free-streaming case. Using mock CMB data and Markov Chain Monte Carlo analyses, we show that neglecting DR self-interactions may bias the inferred value of $r$ by an amount comparable to the uncertainty expected in forthcoming CMB polarization experiments, such as the ground-based $\textit{Simons Observatory}$ and the satellite missions $\textit{LiteBIRD}$ and PICO. Our results emphasize the importance of properly modeling DR interactions in future precision searches for primordial $B$-modes in order to obtain unbiased constraints on inflationary gravitational waves.

Figures (5)

• Figure 1: Ratio of the DR self-interaction rate to the Hubble parameter as a function of redshift. Two distinct regimes are evident: the collisionless regime ($\alpha_2 \langle\Gamma\rangle/H < 1$), where DR free-streams and generates anisotropies, and the collision-dominated regime ($\alpha_2 \langle\Gamma\rangle/H > 1$), where self-interactions are efficient, driving the distribution function toward equilibrium and suppressing anisotropies. Since equilibrium is reached at late times, these interactions are referred to as recoupling. Solid curves show different self-interaction strengths, characterized by the recoupling redshift $z_{\rm int}$, or equivalently the self-coupling constant $\lambda$ (see Eqs. (\ref{['gamma_lambda']}) and (\ref{['gammavsz']})). The vertical gray dashed line indicates the recombination redshift $z_{\rm CMB}\sim1100$. For $z_{\rm int} > z_{\rm CMB}$, DR recouples before recombination, potentially leaving a noticeable imprint on the CMB, while for $z_{\rm int} < z_{\rm CMB}$, no significant effect is expected.
• Figure 2: Relative difference in the primordial B-mode polarization angular power spectrum induced by self-interacting DR, compared to the standard free-streaming case, as a function of multipole moment $\ell$. Each curve corresponds to a different recoupling redshift $z_{\rm int}$, with the associated self-coupling $\lambda$ and interaction timescale $\tau$ indicated. Earlier interactions (i.e., larger $z_{\rm int}$, larger $\lambda$, or shorter $\tau$) result in stronger suppression of anisotropies due to efficient self-interactions. This erases the anisotropic stress responsible for GW damping, thereby enhancing the amplitude of primordial GW (see nahuelbaymloverde2022) and thus increasing the power in the B-mode spectrum. For sufficiently early interactions, $z_{\rm int}\gtrsim 10^7$, the effect saturates and no further enhancement is observed. Conversely, for late recoupling, $z_{\rm int}\lesssim z_{\rm CMB}\sim 10^3$, the impact is negligible since the self-interactions become effective only after recombination, when they can no longer influence the primary CMB anisotropies. For this plot, we assume $\Delta N_{\rm eff}=0.5$ as a representative example resulting in a roughly 5% difference for larger $\lambda$. Decreasing the abundances to $\Delta N_{\rm eff}=0.1$ gives an effect of about 1%.
• Figure 3: Posterior distributions of the reconstructed relative bias of the tensor-to-scalar ratio, $\Delta r/r = (r_{\rm best-fit} - r_{\rm fid})/r_{\rm fid}$, obtained over 500 noise realizations for different fiducial values of $r$. Three types of simulated analysis described in Table \ref{['tab:typeofsim']} are shown: validation of the standard pipeline (Check STD-STD) in blue, validation of the interacting DR pipeline (Check INT-INT) in orange, and the main case of interest where self-interacting DR is analyzed assuming the standard model (INT-STD) in green. Colored dots and error bars represent the mean and the standard deviation, respectively. While the check cases exhibit negligible bias across the $r$ range, the INT-STD case shows a systematic overestimation of $r$ illustrating the bias introduced when self-interacting DR is incorrectly modeled as standard free-streaming.
• Figure 4: Bias in the reconstructed tensor-to-scalar ratio $\Delta r=r_{\rm best-fit}-r_{\rm fid}$, as a function of the fiducial value of $r$, for different values of $\Delta N_{\rm eff}$, $z_{\rm int} \simeq 10^7$ ($\lambda\simeq10^{-10}$), and two experimental noise configurations: Noise 1 (dark blue) and Noise 2 (yellow) representative of next-generation CMB experiments. Solid, dashed, and dotted lines correspond to $\Delta N_{\rm eff} = 0.1$, $0.3$, and $0.5$, respectively. The shaded regions indicate the standard deviation related to the posterior distribution in Fig. \ref{['fig:violin']} for Noise 1 (dark blue), Noise 2 (yellow), and a cosmic variance–limited scenario (only CV in dark gray). The results show that the bias can be comparable to or even exceed the statistical uncertainty, underscoring the need to accurately model DR interactions in future precision measurements of primordial B-modes. In this analysis, we isolate the impact of self-interacting DR on the estimation of $r$ by considering only this specific systematic effect, excluding others.
• Figure 5: Bias in the recovered tensor-to-scalar ratio, $\Delta r$, as a function of the fiducial $r$. Different line styles indicate DR self-interaction strengths, parametrized by the recoupling redshift $z_{\rm int}$ (or equivalently, the self-coupling $\lambda$). Colors correspond to DR abundance: $\Delta N_{\rm eff}=0.5$ (dark blue), $\Delta N_{\rm eff}=0.3$ (yellow), and $\Delta N_{\rm eff}=0.1$ (green). Three representative interaction regimes are shown: early ($z_{\rm int} \simeq 10^7$), intermediate ($z_{\rm int} \simeq 10^{4.6}$), and late ($z_{\rm int} \simeq 10^{0.9}$) recoupling. Shaded regions represent the $1\sigma$ statistical uncertainty on $r$ for the experimental configuration Noise 1 (light gray) and the cosmic variance–limited case (only CV in dark gray). As expected from the behavior shown in Fig. \ref{['fig:relativediff']}, the bias saturates for early interactions ($z_{\rm int}\gg z_{\rm CMB}\sim 10^3$, i.e., large $\lambda$) and it is comparable to the $1\sigma$ level, while it becomes negligible for late interactions ($z_{\rm int}\lesssim z_{\rm CMB}$, i.e. small $\lambda$), where DR interactions become efficient too late to affect the primordial B-mode signal.