Neutrinos remain an elusive component of the Standard Models of particle physics and cosmology. While cosmological measurements have placed some of the strongest constraints on the sum of neutrino masses (see e.g., Refs. [1,2]), we do not yet know the value. The precise mechanism for the generation of such neutrino masses is also still uncertain. Further, the presence of anomalies in terrestrial neutrino experiments [3][4][5][6][7][8][9] may indicate, if confirmed, that yet unknown physics exists in the neutrino sector, hence providing a window into physics beyond the Standard Model.
In particular, new physics altering the free-streaming nature of neutrinos in the early Universe has received renewed interest in recent years (see e.g., Refs. ). In the Standard Model, neutrinos decouple from the primordial plasma and begin to free stream when the universe has cooled to a temperature of ∼1.5 MeV. While freely streaming, neutrinos still interact gravitationally with the rest of the Universe, tugging on any particles in their paths while they pass by. Observationally, this damps the amplitude of photon fluctuations and shifts them to slightly larger scales [49][50][51], impacting the amplitude and phase of the observed cosmic microwave background (CMB) temperature and polarization power spectra.
Introducing new physics in the neutrino sector by allowing them to self-interact can, however, significantly delay the time at which neutrinos begin to free-stream. Such a delay abates how long neutrinos gravitationally tug on the photons, leaving a measurable imprint on the CMB [52,53]. The delay in free-streaming also impacts the evolution of the two Newtonian gravitational potentials ϕ and ψ, leading to scale-dependent effects on the growth of matter fluctuations. See Ref. [54] for a thorough discussion of these effects. Through the combination of effects, neutrino self-interactions can be constrained with CMB measurements, baryon acoustic oscillation (BAO) measurements, and other large-scale structure (LSS) measurements (see e.g., Refs. [13,33,34,44,52,[54][55][56][57][58]).
In its simplest implementation, self-interactions can be described by an effective four-fermion interaction parametrized by a dimensionful Fermi-like constant G eff [59]. This effective coupling constant determines the neutrino self-interaction rate, Γ ν ∝ G 2 eff T 5 ν where T ν is the homogeneous temperature of the neutrino bath. Since this interaction only happens between neutrinos, it does not alter the physics and timing of neutrinos decoupling from the rest of the primordial plasma, as discussed in Ref. [60].
We note however that any ultraviolet completion of the effective four-fermion interaction is subject to several strong constraints, including from supernovae [61][62][63][64][65][66][67][68][69][70][71][72][73][74], big bang nucleosynthesis [75][76][77], neutrino observations with the IceCube experiment [59,78,79], particle colliders [80][81][82][83][84], and those arising from meson, leptons, tritium, and gauge-boson decay kinematics [83][84][85][86][87][88]. Taken literally, these bounds exclude values of G eff large enough to affect cosmological observables. Therefore, the G eff parametrization used in this work should not be interpreted as an actual particle model of neutrino self-interaction, but rather as a proxy controlling the onset of neutrino free-streaming in our Universe.
With this in mind, previous works [13,[33][34][35]44,52,[54][55][56][57][58] have interestingly found that an effective neutrino selfinteraction strength orders of magnitude larger than the standard electroweak interaction can be compatible with CMB and BAO data. Unlike other popular cosmological extensions, the G eff posterior probability distribution is characterized by two distinct islands in parameter space; a strongly interacting mode, SIν, with G eff ∼ 10 -1.5 MeV -2 , and a moderately interacting mode, MIν, with G eff ∼ 10 -4 MeV -2 that is nearly indistinguishable from ΛCDM. This bimodality stems from a multiparameter degeneracy with G eff [56] involving the angular size of the baryon-photon sound horizon at last scattering θ Ã , the amplitude of scalar fluctuations A s , and the scalar spectral index n s . The strong neutrino self-interactions of the SIν mode shift the phase and boost the amplitude of the multipoles entering the causal horizon before the onset of neutrino free-streaming. To reconcile these effects with cosmological data, larger θ Ã and lower n s values are preferred when G eff is large, which in turns result in a lower value of A s to ensure consistency with low CMB multipoles. On the other hand, the MIν mode is characterized by values of θ Ã , A s , and n s approximately equal to their ΛCDM values.
The recent special interest for this model arises from the fact that the SIν mode belongs to the family of scenarios that bring the CMB and Cepheid-calibrated SNIa measurements of the Hubble constant, H 0 , closer by introducing a new species relevant in the early universe to reduce the sound horizon at recombination (see e.g., Refs. [89,90]). Preferring a higher value of N eff , the SIν mode is coincident with larger values of H 0 and lower values of σ 8 , the amplitude of linear density fluctuations at 8 h -1 Mpc, offering a potential simultaneous resolution to both the σ 8 tension and discrepancies in H 0 .
Nevertheless, as discussed in Refs. [33][34][35]44,54], the inclusion of the Planck CMB polarization data [1] disfavors the SIν mode compared to the MIν, casting serious doubt on the viability of this mechanism to resolve the current tensions (see Ref. [91] for a more detailed explanation of this limitation). Despite being statistically suppressed, the SIν mode is not entirely ruled out by these analyses. This then asks the question of what kind of cosmological data could eliminate the viability of a late onset of neutrino free streaming.
The low value of the spectral index n s associated with the SIν mode provides an important clue: cosmological data probing a broad range of scales can provide an important lever arm to constrain the spectral tilt and detect any deviation from its ΛCDM value. High-resolution observations of the CMB temperature and polarization spectra probing small angular scales inaccessible to the Planck satellite are a promising candidate for such an observational constraint. Changes to neutrino free-streaming also leave a distinct fingerprint via a phase shift in the angular peak position θ s . Thus, discriminating polarization measurements can offer another window into constraining neutrino interactions [92]. In this work, we use high-resolution CMB data from four observing seasons of the Atacama Cosmology Telescope (ACT) [93] to probe neutrino selfinteraction in the early universe, which is a step towards higher sensitivity measurements from the complete ACT dataset, the Simons Observatory [94] and CMB-S4 [95].
Previous works have shown the compatibility of ACT measurements with other models increasing the sound horizon at recombination, such as early dark energy (EDE) and pseudoscalar sterile neutrino self-interactions [96,97]. Though the physics of how they increase the inferred H 0 from the CMB is similar, the underlying physics and the subsequent perturbation theory and phenomenology can be vastly different, thus motivating further exploration of these class of models with a wide variety of future datasets.
We show below that a delayed onset of neutrino free streaming brought on by significant neutrino self interactions still appears compatible with CMB observations at small angular scales from ACT. This is the case for both ACT alone and in combination with data from the Wilkinson Microwave Anisotropy Probe (WMAP) [98,99].
The paper is organized as follows. In Sec. II, we present the cosmological models, data, parameter choices, and statistical tools used in our analyses. Our main results are presented in Sec. III. In Sec. IV we highlight the importance of the ACT E-mode polarization for our results. We consider the impact of BAO measurements on our results in Sec. V and briefly discuss our systematic tests in Sec. VI. We conclude in Sec. VII.
A. Models, data, and parameter choices Our baseline cosmological model includes three massive neutrinos with degenerate masses that can scatter among themselves with an interaction rate Γ ν ∝ G 2 eff T 5 ν . This parallels the analysis done by Planck [1]. 1 The details of the Boltzmann equations involving such massive selfinteracting neutrinos are provided in Ref. [54] (see also Ref. [100]). Within this baseline model, the parameters N eff (which is used to adjust the neutrino temperature T ν ) and the sum of neutrino masses P m ν are also allowed to vary freely from their standard values of 3.046 eV and 0.06 eV respectively. Our baseline model is thus described by a total of nine parameters once the six standard ΛCDM parameters 2 are included. This model is denoted as
We also consider a simpler extension of ΛCDM in which only the neutrino interaction strength G eff is added, with N eff and P m ν fixed at their standard values. In this case, we adopt the standard practice of considering a single neutrino mass eigenstate carrying all the mass. This model is simply referred to as "G eff ".
Throughout our analysis, we used modified versions of the codes CAMBfoot_0 [101] and CosmoMCþMultinest [102,103], equivalent to those used in Ref. [54]. This analysis framework assumes a linear evolution of perturbations. Although negligible at the scales probed by Planck, at smaller scales like those measured by ACT, nonlinear gravitational lensing effects impact Σm ν and N eff estimates and therefore it is likely that they will also alter estimates of G eff . However, we will show later that, despite entering the nonlinear regime for lensing, multipoles above 2500 contribute negligible information to the constraint and therefore a linear analysis here is valid. For future analyses with even stronger influence from l > 2500, analyzing neutrino selfinteractions in the non-linear regime may become necessary.
We use uniform priors on all parameters, except for the Planck absolute map-level calibration parameter by the square which all Planck spectra are divided which has the Gaussian prior y cal ¼ 1.0000 AE 0.0025, and the optical depth to reionization which has the Gaussian prior τ ¼ 0.065 AE 0.015 (see Ref. [93]) and replaces low-l polarization data. As in previous work, we place a uniform prior on the logarithm of the coupling constant G eff , but we here extend the lower range of the prior to -8.0 to include smaller coupling values and allow for any shift in the MIν mode location. (All prior ranges are summarized in Table I.) We also utilize nested sampling [104] to thoroughly sample the multimodal posteriors. For this we use 2000 live points, set the target sampling efficiency to 0.3, set the accuracy threshold on the log Bayesian evidence to 20% to ensure the accuracy of credible intervals. We refer to "modes" as disjoint regions of parameter space that 1 Our analysis here differs from that presented in Ref. [54], which used a single massive neutrinos containing all the mass, in addition to massless neutrinos. isolate the islands of the multimodal posterior distribution. We utilize the mode separation feature in the Multinest algorithm [103] to isolate each mode and compute the parameter posterior distributions for that mode.
In this work we present results on neutrino selfinteractions in the presence of the ACT DR4 data. We further combine ACT with WMAP and Planck CMB data, as well as a selection of low-redshift datasets. We denote the data as follows:
( [110]. This is only included when also adding BAO to combinations with Planck. Thus, we do not make note of the lensing measurement when labeling the data combination. When combining ACT with Planck or WMAP, we follow the procedure described in Ref. [93], i.e., we remove ACT TT data below l < 1800 in analyses with Planck while we use the full ACT dataset in combination with WMAP.
We use a variety of techniques to assess the statistical significance of the models we consider. We assess the relative statistical significance between two modes of the posterior by computing their maximum likelihood ratio,
where θ SIν and θ MIν are the parameters describing the SIν and MIν modes' best-fit parameters, respectively.
To assess how well the different models fit the data, we also compute Δχ 2 values. We then translate these values into a significance in terms of Gaussian standard deviations (i.e., σs). We assume that Δχ 2 is distributed according to the χ 2 -distribution. When comparing fits across different numbers of free parameters, we assume the χ 2 -distribution with k degrees of freedom where k is the difference in parameter numbers. This effectively penalizes the additional parameters. Then we find the σ (Gaussian significance) whose C.L. matches the cumulative probability function at the given Δχ 2 , following the same method as in [96]. We note, however, that the multidimensional posteriors may not be sufficiently Gaussian for either mode (especially MIν), so Gaussian significances should be interpreted with caution.
We finally compute the Akaike information criterion (AIC) [111] to penalize the extra parameters added to the interacting neutrino model. The ΔAIC between two models is computed as
A lower AIC value provides a better fit, so a negative ΔAIC value indicates preference for interacting neutrinos.
To compare different dataset combinations' preference for either mode, we compute the Bayes factor
which compares the modes' Bayesian evidence Z, defined as the parameter-averaged likelihood of the data,
Pr ðdjθ; M j Þ Pr ðθjM j Þdθ; ð4Þ
The lower bound employed in previous works was -6. Here we take -8 as the lower bound in case higher ls are sensitive to extremely small changes in G eff .
where d is the data, θ are the parameters describing model M, M j denotes the jth mode of the posterior distribution on the space of θ, and Ω θ is the entire parameter space. Note that we are using the Bayes factor to compare the modes' statistical significances, instead of model comparison. [56] We note that Z is weakly sensitive to the prior volume when the likelihood is uninformative near the prior boundary (which will be the case for the MIν in our analyses). Thus, the exact values of B SIν we report here have a mild (linear) dependence on our choice of prior distribution. However, even significantly extending our prior distribution on G eff to include the Standard Model Fermi constant value (G eff ∼ 10 -11 MeV -2 ) could at most change B SIν by a factor of order unity, hence not qualitatively changing our conclusions. Moreover, given a choice of prior, the relative values of B SIν for different dataset combinations are an informative indicator of the corresponding preference for one mode versus the other among these data.
The parameter constraints on our baseline
and Planck data are shown in Fig. 1. ACT data, both alone and when combined with WMAP, show some preference for strong neutrino self-interactions and the resulting delayed onset of neutrino free streaming. The distribution of G eff is bimodal for all dataset combinations presented, but the strongly interacting mode is relatively preferred with the ACT data alone as well as the ACT data combined with WMAP data, whereas the weakly interacting mode, compatible with ΛCDM, is preferred by the Planck data. Adding ACT data to Planck increases the probability of the SIν mode compared to Planck alone, but keeps an overall preference for the MIν mode. It is interesting to note that the ACT þ WMAP combination, which together probe a broad range of angular scales, comparatively to Planck favors a late onset of neutrino free streaming, although ΛCDM still provides a good fit to the data. The improvement in overall χ 2 for this interacting model, compared to ΛCDM, is 13.7 for three extra parameters for ACT þ WMAP, which we translate to a 2.9σ preference. For ACT þ Planck the improvement in χ 2 is only 1.7 for the SIν mode, with ΛCDM preferred. The physical reasons driving these preferences and differences will be discussed in the next few sections.
In Table II we quantify the relative significance of the SIν and MIν modes within our baseline model. For ACT-only, we find a Bayes factor of 1.5 AE 0.2. 4 With WMAP added, the Bayes factor grows to 2.8 AE 0.6, corresponding to an increased preference for SIν [see Table 4 in Ref. [112] where the inverse corresponds to our definition of the Bayes factor]. Planck alone and ACT þ Planck show values below 1, showing a greater preference for MIν than the other dataset combinations. Similar conclusions can be drawn from the maximum likelihood ratios.
Credible intervals for the different cosmological parameters within our baseline model are given in Appendix A in Table IV for the SIν mode and in Table V for the MIν mode.
We note that our ACT results do not yet have the sensitivity to weigh on the discussions around tensions between the CMB and the local Universe estimates of both
We limit the range of the horizontal axis displayed here (despite the prior extending farther in either direction) to better show the details of the two posterior modes. We minimally smooth the posteriors in order to maintain the bimodal features. As such, small values of G eff can appear unconverged, as indicated by the wiggly lines for low G eff . We note that Multinest was not able to automatically separate the modes with ACT-only. To compare evidence, we ran two separate chains with a prior on log 10 ðG eff MeV 2 Þ of ½-4.0; -2.0 and ½-2.0; 0.0, respectively, and with equivalent priors for all other parameters. We chose these G eff priors in order to maintain the same prior volume between the two modes, and we safely are able to cut at -4 instead of -8 since the ACT-only 1D G eff posterior drops off, as in Fig. 1.
the Hubble constant, H 0 , and the amplitude of matter clustering, σ 8 (see also Appendix E). 5
To explore how much of the slight preference for the SIν mode we see when ACT Why does G eff have such an impact on the fit to ACT þ WMAP data? As a single parameter, it influences the CMB power spectra in three ways; amplitude, phase, and tilt as described in Ref. [54]. This can be seen by looking at the correlation between G eff and the parameters A s e -2τ , θ Ã , and n s (shown in Fig. 5 in Appendix C). Changing the value of G eff between the two modes of the distribution is, thus, able to accomplish the equivalent of adjusting three parameters. We note, however, that G eff 's impacts on the power spectra are coupled; large G eff causes a boost in amplitude which coincides with a shift towards small scales and a blue tilt (all compared to ΛCDM). With three parameters that can either increase or decrease in values, there are a total of eight different combinations of changes to the power spectra. Remarkably, G eff is able to capture the one combination 6 that is actually favored by the data (discussed more in Sec. IV).
The three effects generated by G eff are also able to compensate for other fluctuations seen in the ACT fits.
As discussed in Ref. [93], ACT data alone prefer a 2.3 -2.7σ low fluctuation of Ω b h 2 and high fluctuation of n s within ΛCDM compared to WMAP and Planck. The low Ω b h 2 value damps the first acoustic peak and small angular scales, and also produces a slight phase shift towards larger scales (due to the reduced baryon loading of the primeval plasma). This small-scale damping is then compensated by the high n s value, which tilts the spectra up. This movement occurs along a strong degeneracy line and is alleviated when large angular scales are added to ACT. Allowing also N eff to vary for ACT data alone yields similarly low Ω b h 2 values, but with an n s value more similar to Planck accompanied by a low N eff centered at ∼2.3, disfavoring the larger relativistic degrees of freedom N eff ¼ 3.5 at 4σ, as discussed in Ref. [93]. N eff and n s are highly degenerate due to large N eff causing small-scale damping [50] and higher n s being able to undo this damping. While most of the Ω b h 2 -n s ACT fit can be shifted with WMAP data, the low fluctuation in N eff remains (ACT þ WMAP prefers N eff values ∼2.3σ lower than the standard value) and leaves room for the neutrino strong mode.
Strong neutrino self-interactions, i.e., large values of G eff , are able to undo the damping and phase shift from the low preferred Ω b h 2 by boosting the spectra and shifting it towards small scales. This is done by exploiting the multiparameter degeneracy between G eff , A s and n s described in Ref. [56], which results in lower values of both A s and n s as compared to ΛCDM. At the same time, strong neutrino self-interactions allow N eff to be closer to its standard value due to absence of free-streaming phase and amplitude shift [49] on the CMB in this case (see Table IV in Appendix A). Appendix G explores the additional degeneracy with the primordial helium abundance.
ACT's preference for a delayed onset of neutrino free streaming is predominantly driven by its E-mode polarization spectrum. We show in Table III the Δχ 2 between SIν (nine parameters) and ΛCDM (six parameters) for different CMB data combinations. The largest Δχ 2 in favor of the SIν mode occurs for ACT's E-mode polarization data for ACT alone, with Δχ 2 ACT∶EE ¼ -7.3, and ACT þ WMAP, with Δχ 2 ACT∶EE ¼ -6.0. In Fig. 2 we show the G eff posterior distribution for the ACT combined dataset and separate constraints from its TT, TE, and EE data. 7 While the TT and TE data show some preference for the SIν mode, ACT's EE data dominates the preference for a delayed onset of neutrino free streaming as the MIν mode nearly disappears in this case.
The addition of Planck data neutralizes ACT's E-mode polarization preference. Planck polarization data is signal dominated in windows below l ≈ 700 (see Fig. 17 in Ref. [113]), and can therefore statistically overpower ACT's polarization preferences in this multipole range, which, as we will see below, plays a significant role in these constraints. Given that WMAP does not include EE data, the inclusion of WMAP does not suppress ACT EE data's preference for SIν. Further, this combination also sees a preference for SIν coming from the ACT TE data, with Δχ 2 ACT∶TE ¼ -7.0.
Why does ACT's E-mode polarization prefer a delayed onset of neutrino free streaming while Planck's does not? To answer this question, we show the -χ 2 per degree of freedom (-χ 2 =d:o:f:) for the SIν mode, MIν mode, and ΛCDM as a function of the maximum l used for the TT, TE, EE, and combined ACT data in Fig. 3. We use the ACT þ WMAP best-fit models and we indicate with a dashed line a reference value of -χ 2 =d:o:f: ¼ -1.
Overall, both the SIν and MIν modes provide a better fit than ΛCDM to the ACT E-mode polarization data (see Appendix B for tables comparing the MIν mode to ΛCDM). Both modes have larger -χ 2 =d:o:f: than ΛCDM in the bottom right panel of Fig. 3, which focuses on the E-mode polarization. For the range 700 ≲ l ≲ 1000 in this panel, as l max increases the SIν mode rapidly provides a better fit to the data than either the MIν mode or ΛCDM. The MIν mode fits the TE cross-correlation data better than the SIν mode in the same range, but the margin between the modes is twice as large with the EE autocorrelation. Ultimately the effect in the 700 ≲ l ≲ 1000 range of the E-mode polarization drives the total likelihood in favor of the SIν mode. Beyond l ∼ 1000, there is not significant information added by smaller scales to further differentiate the two interacting neutrino modes. Finally, angular scales corresponding to l ≳ 1700 do not incorporate additional information to further improve the fit of ΛCDM over the 2 modes. More specifically, we find that removing ACT data at l > 2500 does not significantly change the posterior distribution on G eff . By contrast, keeping only data at l > 1000 entirely removes the modest preference for G eff ∼ 10 -1.6 MeV -2 and leaves a nearly flat posterior, reinforcing the fact that the preference for the SIν mode is driven by larger angular scales.
Our findings mirror those in Refs. [96,97] showing that the most pertinent feature for the respective extensions to ΛCDM is in the l ≲ 1000 range of the EE spectrum of ACT data. This is also in agreement with the theoretical expectation provided in Ref. [114], which concludes that CMB experiments are most sensitive to neutrino interactions at l ≲ 1000.
Having identified this important range of angular scales in the E-mode polarization spectrum, we now study the residuals between the best-fit Planck ΛCDM cosmological model and the three leading datasets of ACT, Planck, and SPT-3G [115] E-mode polarization data in this range. As shown in Fig. 4, residuals for all experiments show a slight upward fluctuation in the range 700 ≲ l ≲ 1000, with ACT showing the largest shift. A delayed onset of neutrino free streaming can better capture this upward fluctuation, driving the preference for the SIν mode in the ACT data. Indeed, as described in Ref. [56], Fourier modes corresponding to this particular l range enter the causal horizon close to the onset of neutrino free streaming for the SIν mode, allowing them to be significantly influenced by the modified evolution of the gravitational potentials. The figure also clarifies why the preference is reduced when including Planck data: while consistent with both ACT and SPT-3G in this multipole range, Planck does not possess such a strong E-mode upward fluctuation. FIG. 2. ACT 1D posteriors for G eff in the baseline model for the full TT þ TE þ EE data and for fits of separate spectra. We note again that we minimally smooth the posteriors, so low values of G eff can appear unconverged. FIG. 4. Residuals of E-mode polarization data for ACT (orange), Planck (blue), and SPT-3G (green) data relative to Planck's best-fit ΛCDM model. We overlay the best-fit for SIν for just ACT E-mode polarization in gray (best-fit further described in Sec. IV), and the gray band highlights the multipoles that drive ACT's preference for SIν. Each bin spans 50 multipoles. Note the upwards deviation from the best fit in the 700 ≲ l ≲ 1000 range of the ACT and SPT-3G data. The residuals of TT, TE, and EE power spectrum for the full l range are presented in Appendix D. FIG. 3. -χ 2 =N bins as a function of l max where the former is the negative reduced χ 2 and the latter the maximum l used in the likelihood, shown for SIν (in red), MIν (in blue), and ΛCDM (in black) best fits to ACT þ WMAP data. This is shown for the combined ACT data in top left, TT only in top right, TE only in bottom left, and EE only in bottom right. Overall, it is clear that for most choices of l max , SIν has the largest -χ 2 =N bins across all sectors of ACT data. The top left figure shows the greatest difference between SIν and MIν in the 700 ≲ l ≲ 1000 range, which is echoed only by the bottom right, indicating the importance of E-mode polarization in that l range in promoting the favorability of SIν.
We now expand our analysis beyond CMB-only results and as previously mentioned incorporate BAO and Planck CMB lensing data. Overall we find that adding BAO measurements increases the significance of the SIν compared to the MIν mode. 8 Such preference is not surprising for the ACT þ WMAP CMB dataset; the values of our baseline model's parameters most relevant to BAO (i.e. H 0 , Ω m , and N eff ) for ACT þ WMAP within the SIν mode are closer to their concordance ΛCDM values than those within the MIν mode and therefore BAO further consolidates this preference. We find that the SIν mode has a Δχ 2 ¼ -13.2 compared to ΛCDM for ACT þ WMAP once BAO is included (Table X in Appendix B), roughly corresponding to a 2.9σ preference for a delayed onset of neutrino free streaming. ΛCDM continues to be preferred when adding BAO to ACT þ Planck.
We test the robustness of our results-and in particular the contribution to them from ACT-by examining different partitions of the ACT data independently, and revisiting some of the analysis assumptions:
(i) Figure 2 shows that different spectra (TT, TE, EE) in ACT give very consistent results. (ii) ACT data are further composed of "deep" maps, spanning 20-340 deg 2 , and "wide" maps, spanning 210-1400 deg 2 (see Refs. [93,105]). Both sets contain TT, TE, and EE power spectra spanning the full-l ranges detailed in Sec. II. After constraining the interaction with each set separately, we conclude that the two patches give consistent results but the wide data prefer the SIν mode more than the deep data does. This weaker significance in the deep data is not unexpected, however, as the wide data has smaller errors at large scales, and therefore stronger constraining power, than the deep data at scales that are relevant for this model (See Appendix F for further discussion). (iii) The difference in mode preference between ACT and Planck could also point to a different behavior at large and small scales since the two experiments' constraining power peaks in different regimes [93].
Even if ACT on its own does not yet allow for a high signal-to-noise comparison of the two regimes we run a few tests to check any effect on the results.
We find that the G eff 1D posteriors when we vary the full nine parameter space with all of ACT, or ACT small scales (l > 2500) removed, or ACT large scales (l < 1000) removed are all consistent. As mentioned before, for constraints of neutrino selfinteraction with ACT data, the critical information is in the large scales and small scales hardly contribute to the demonstrated preference for SIν. This also supports the framework of a linear analysis. (iv) Finally, we note that multiplying or dividing the ACT TE power spectra by an artificial calibration factor of 1.05 to account for some unknown amplitude offset as in Ref. [93] does not substantially change the results.
We have shown that ACT DR4 data, both alone and when combined with WMAP, display a slight preference for a delayed onset of neutrino free streaming due to neutrino self interactions at the 2-3σ level. Like in previous analyses using CMB data, we find the parameter posteriors to be bimodal. When using ACT the strongly interacting mode representing the delayed onset of neutrino free streaming has a higher likelihood than the ΛCDM analogous, weakly interacting mode. The preference reverts towards ΛCDM when Planck data is added to ACT, with the stronglyinteracting mode suppressed and the bimodality moving more in favor of the ΛCDM analog model as seen in previous works. We determined that ACT data's preference is primarily driven by angular scale corresponding to 700 ≲ l ≲ 1000 in the ACT E-mode polarization data. The impact of Planck in the joint fit is mostly due to a wider range of cosmological information provided by Planck and in particular in this EE range.
In summary, despite the high-resolution observations of the CMB temperature and polarization spectra we find the posterior to be persistently bimodal and the preference between the modes to be dependent on choice of model (G eff only or G eff þ N eff þ Σm ν ) and dataset combination. This indicates that we need more data to make any decisive judgements on this model, and motivates searches for unaccounted-for systematic effects in the 500 ≲ l ≲ 1000 region in polarization that we highlighted (see also Refs. [96,97]). Alternative new-physics models could also be examined, including those where only a fraction of the neutrinos can self-interact (see e.g., Refs. [33][34][35]). Our results also call for detailed scrutiny of upcoming ACT polarization data, especially in the l ≲ 1000 region.
In this appendix, we list the credible intervals for the different model parameters used in our analyses. In Table IV, we give the mean parameter values and their 1σ error bars for the SIν mode for four data combinations. The corresponding MIν values are given in Table V. The parameter values within our simpler G eff oneparameter extension are given in Table VI. We note that for the MIν mode, the log 10 ðG eff MeV 2 Þ posterior is very non-Gaussian. As such, the 1σ error bar on G eff (especially the lower bound), are only indicative.
Table VII shows the Δχ 2 between ΛCDM and its extension with Σm ν þ N eff and (the strong mode of) only G eff . For both models the SIν mode prefers slightly higher values of H 0 and σ 8 than the MIν mode given all the listed dataset combinations. Overall the estimated values of H 0 are lower than in ΛCDM due to its anticorrelation with neutrino mass, which is held fixed in ΛCDM.
In this appendix we supplement the comparison of the χ 2 values of SIν, MIν, and ΛCDM best-fit models with respect to different datasets and their components. The G eff model does not change the posteriors of H 0 and σ 8 significantly from the ΛCDM result but the other extensions do. The similarity of the N eff þ P m ν contours and the baseline model contours indicate that the shift to lower H 0 is largely due to the variation in N eff and P m ν . This is in contrast to the results of Ref. [96] where fitting the EDE model to ACT data resulted in a shift to higher values of H 0 . Though the new physics introduced here and in Ref. [96] both appear compatible with ACT data and capture similar features, their relationships with the data are not identical.
We present CMB power spectra from data and from theory predictions as residuals to the ΛCDM best fit predictions. Figure 6 shows the TT plots, Fig. 7 APPENDIX F: ACT WIDE/DEEP Fig. 10 shows the 1D posterior of log 10 ðG eff MeV 2 Þ for ACT "deep," ACT "wide," and the full ACT dataset. We find that both the posteriors produced with "deep" and "wide" have a preference for log 10 ðG eff MeV 2 Þ ∼ -1.3 around which their tallest peaks and best-fit models are located. However, the preference for SIν over MIν is much stronger with "wide" than with "deep," resulting in the combined ACT posterior which is between the two other posteriors. We show Δχ 2 values comparing SIν to ΛCDM for the isolated "deep" and "wide" components of ACT in Table XII. The "wide" E-mode polarization is the only component with a strong enough preference for SIν to bring ΔAIC < 0 compared to ΛCDM.
We show the "wide" and "deep" E-mode polarization residuals in Fig. 11. While the "wide" data show a stronger fluctuation in 700 ≲ l ≲ 1000, the large error bars of the "deep" data are still consistent with this fluctuation. Therefore, the discrepancies between the preferences of "wide" and "deep" are within the margins of error of the two components of ACT. This is again similar to the findings in Ref. [96] that the feature which the EDE model fits better than ΛCDM is more pronounced in the "wide" component (specifically "wide" EE) than the "deep," further suggesting that the two different physical models maybe describing the same feature in ACT data. FIG. 11. Difference for the E-mode polarization data for ACT wide, deep, and total compared to the best-fit ΛCDM model from Planck. In the 700 ≲ l ≲ 1000 range of interest, the three datasets are not significantly far apart. FIG. 10. 1D posterior for G eff , obtained by varying the 6 ΛCDM parameters, Σm ν , N eff , and G eff with all of ACT (in green), "deep" ACT (in blue), and "wide" ACT (in red). The "wide" only posterior shows the strongest preference for SIν but the blue posterior also does slightly prefer SIν.
https://github.com/ckreisch/IntNuCAMB.
From Fig.5of Sec. C we also infer that fitting SIν to ACT þ WMAP does not result in a simultaneous higher H 0 and lower σ 8 , as was the case with SIν and Planck in previous works. Introducing G eff and N eff does not change the values of H 0 and σ 8 away from the ΛCDM values much. Introducing Σm ν actually pushes both H 0 and σ 8 lower.
It captures only one combination rather than two because when G eff is small, the physics is equivalent to ΛCDM.
These were obtained by running separate nested sampling runs with ACT TT data alone, TE data alone, and EE data alone.
A slight preference for the SIν mode when including BAO and distance ladder H 0 measurements had been noted in Ref.[54], but their analysis only includes Planck temperature data. We see that ACT þ WMAP data when combined with BAO measurements, as well as when combined with local H 0 measurements from SH 0 ES, have some preference for late neutrino free streaming. See Appendix E for an updated discussion on neutrino free streaming and local H 0 measurements in light of ACT data.