# The magnificent ACT of flavor-specific neutrino self-interaction

Anirban Das  and Subhajit Ghosh

###### Abstract

We revisit the cosmology of neutrino self-interaction and use the latest
cosmic microwave background data from the Atacama Cosmology Telescope (ACT)
and the Planck experiment to constrain the interaction strength. In both
flavor-universal and nonuniversal coupling scenarios, we find that the ACT
data prefers strong neutrino self-interaction that delays neutrino free
streaming until just before the matter-radiation equality. When combined with
the Planck 2018 data, the preference for strong interaction decreases due to
the Planck polarization data. For the combined dataset, the flavor-specific
interaction still provides a better fit to the CMB data than $\Lambda$CDM.
This trend persists even when neutrino mass is taken into account and extra
radiation is added. We also study the prospect of constraining such strong
interaction by future terrestrial and space telescopes, and find that the
upcoming CMB-S4 experiment will improve the upper limit on neutrino self-
interaction by about a factor of three.

##  1 Introduction

Observational cosmology has made great achievements over the last three
decades, especially in the frontier of cosmic microwave background (CMB)
observation. Starting from the COBE-FIRAS experiment in the early 1990s,
continuing with the WMAP satellite in the 2000s, and finally, the Planck
experiment in the 2010s have continued the legacy measurement of the CMB
anisotropy power spectra with ever-increasing precision between generations.
Just within the past 30 years, the temperature measurement sensitivity of the
instruments has improved by about 100 times which has resulted in a $~{}\sim
0.1\%$ precision on anisotropy measurement. Today, even though Planck has
released its final data in 2018, other experiments are continuing to observe
the microwave sky and more missions are being designed for the future.

Currently, multiple terrestrial CMB experiments are ongoing, such as, Atacama
Cosmology Telescope (ACT) [1] in Chile, South Pole Telescope-3G (SPT-3G) [2],
and BICEP3 [3] at the South Pole. While Planck has been able to measure the
CMB power spectrum with $\mathcal{O}(50\,\,\mu\mathrm{K\text{-}arcmin})$ level
sensitivity and few arcmin angular resolution, its data goes only upto
$\ell_{\mathrm{max}}=2500$ for temperature and 2000 for polarization spectrum
[4, 5]. Measuring the polarization power spectra with better precision and at
smaller scales is one of the key goals of ongoing and future missions. For
example, ACT collaboration has published temperature and polarization power
spectra with the angular resolution a few arcmins (corresponding to
$\ell_{\mathrm{max}}\simeq 4000$), which is a major improvement over the
Planck data [1]. The SPT-3G has measured the spectra at the level of
$\ell_{\mathrm{max}}=3000$ [2]. In combination with the Planck data, these
experiments have helped us improve the measurement of the parameters of the
$\Lambda$CDM model of cosmology and constrain its various extensions.

Looking into the future, there are multiple telescopes that are being built or
planned. CMB-S4 is a terrestrial experiment that has been approved and is in
its final phase of technical designing. One of the key goals of CMB-S4 is to
achieve sub-arcmin angular resolution and unprecedented sensitivity to measure
the small-scale power spectrum [6]. It is likely to use both the Chilean and
the South Pole sites to perform ultra-deep and wide-field surveys. A space-
based mission CORE has been proposed in the European Space Agency’s M5 call
for mid-sized missions in 2016 [7]. According to their plan, CORE is going to
focus on small-scale polarization data. PICO has been proposed with an
ambitious goal of surveying the whole CMB sky in a broad range of frequencies
(21-799 GHz) with a sensitivity about 100 times better than the Planck
experiment [8]. Another space-based mission LiteBIRD is being set to launch
later in this decade to measure the B-mode polarization of the CMB [9]. The
BICEP array is being built to measure the B-mode polarization signal with
greater precision and will replace the Keck array [3]. More futuristic
experiments like CMB-HD, if materialized according to its current plan, would
provide a remarkably valuable tool to look for new physics in CMB, thanks to
its significantly better angular resolution and improved sensitivity, and help
us detect or constrain beyond-$\Lambda$CDM models [10].

Many extensions of the $\Lambda$CDM have been considered as alternative models
of the Universe. Among them, models with extra relativistic degrees of freedom
in the form of neutrinos and/or other light dark sector particles with secret
self-interaction are particularly interesting as they often arise in many
particle physics models. Moreover, neutrinos hold a guaranteed key to the
discovery of a new beyond Standard Model physics with its mass not explained
by the current theory. The terrestrial neutrino oscillation experiments have
improved the measurements of the neutrino mixing parameters over the last few
decades [11]. The existence of neutrino flavor oscillation proves that they
have nonzero mass. However, the absolute mass scale of the neutrinos and any
knowledge about self-interaction other than the Standard Model weak force
remains unknown to us. Future CMB measurements might hold the key to success
in this area.

Neutrino self-interaction has been a topic of active research since the early
days of CMB observation [12, 13, 14]. The existence of free-streaming
relativistic species in the early Universe affects the observed CMB anisotropy
power spectra by suppressing its amplitude and shifting it toward larger
angular scales. This happens due to the large relativistic speed of the
neutrinos, the effect of which is transferred to other species via gravity.
However, if there is self-interaction among the particles, then it impedes
their free-streaming and lessens these effects on the power spectra. This in
turn cancels the phase shift and boosts the power spectra at the small angular
scale compared to the free streaming scenario. The new effects of neutrino
self-interaction have been a topic of great interest over the last several
years in various contexts including the so-called Hubble tension [15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35] For
more details on self-interacting neutrino, a recent comprehensive review can
be found in Ref. [36]. However, for an alternative interpretation of the
Hubble tension, see Ref. [37, 38] for example.

The effects of self-interacting neutrino (SINU) are present at scales that
entered the horizon long before the onset of their free streaming which
typically happens before recombination in the models of our interest.
Therefore, very precise measurements of the CMB power spectra at smaller
angular scales are crucial. The recent result from the ACT, which extends up
to multipole $\ell\simeq 4000$ with significantly improved precision than
Planck, is a huge progress in this direction. In this work, we use this latest
ACT data to update the constraint on both flavor-universal and nonuniversal
self-interaction using massless neutrinos. We also combine ACT with the 2018
Planck data to investigate their combined constraining power. In concordance
with earlier works, we find a strongly interacting (SI) and a moderately
interacting (MI) mode in the posterior distribution of the coupling strength.
We find that ACT prefers a slightly stronger self-interaction for the SI mode
compared to Planck. It also increases the overall significance of the SI mode
relative to a Planck-only analysis. These observations could be attributed to
a specific feature in the ACT data. The results for the strong coupling mode
remain unchanged qualitatively when a nonzero neutrino mass is included with
expected changes in a few other cosmological parameters showing the validity
of our baseline analysis with massless neutrinos. We also study an extension
of the $\Lambda$CDM model of cosmology by varying the number of extra
relativistic degrees of freedom. The extra radiation does not affect the
neutrino self-coupling strength in any significant manner except changing some
of the background parameters.

Finally, we perform a Fisher forecast analysis of SINU cosmology for a few
future CMB experiments, such as CMB-S4 and CORE. We point out the multipole
range that contains the most information about SINU, and find that CMB-S4 will
improve the constraint on the coupling strength approximately by a factor of
three. This future forecast about neutrino self-interaction for terrestrial
and space-based experiments would be useful in planning future experiments.

The outline of the paper is as follows. In section 2, we give a brief
introduction to the cosmology of self-interacting neutrino. We show our data
analysis and the results in Sec. 3 followed by the future prospect analysis in
Sec. 4. Finally, we discuss the results in this paper and conclude in Sec. 5.

##  2 Cosmology of self-interacting neutrino

Following our previous work in Ref. [22], we consider a phenomenological model
of self-interacting neutrino (SINU) where the interaction is mediated by a
heavy mediator. This means the mediator mass is greater than the typical
momentum transfer in the scattering during all relevant cosmological epochs
for CMB. In this case, the interaction term in the Lagrangian can be written
as a four-Fermi effective operator,

$\mathcal{L}\supset
G^{(ijkl)}_{\mathrm{eff}}\bar{\nu}_{i}\nu_{j}\bar{\nu}_{k}\nu_{l}\,,$ (2.1)

where $i,j,k,l$ are neutrino flavor indices, and $G^{(ijkl)}_{\mathrm{eff}}$
is the effective coupling strength for such states. This coupling is
proportional to the inverse square of the mediator mass. In this work, we will
consider only flavor-diagonal coupling matrices,

$G_{\mathrm{eff}}=\mathrm{diag}\\{G_{\mathrm{eff}}^{(1)},G_{\mathrm{eff}}^{(2)},\ldots
G_{\mathrm{eff}}^{(N)}\\}\,.$ (2.2)

These diagonal coupling strengths will be free parameters. We will also always
take the self-interacting neutrinos to be massless. As a result, there is no
mixing between different flavors in this scenario. Hence, we are free to
define flavor-universal or nonuniversal interactions in this basis without
worrying about the basis transformation of the coupling matrix.

With this effective coupling, neutrino scattering cross section is
$\langle\sigma v\rangle\propto(G_{\mathrm{eff}})^{2}T^{2}$, and the scattering
rate is $\Gamma\propto(G_{\mathrm{eff}})^{2}T_{\nu}^{5}$ with $T_{\nu}$ as the
neutrino bath temperature. Therefore, the covariant neutrino opacity
$\dot{\tau}_{\nu}$ due to self-scattering can be defined as

$\dot{\tau}_{\nu}\equiv-a(G_{\mathrm{eff}})^{2}T_{\nu}^{5}\,.$ (2.3)

Here, $a$ is the scale factor of the Universe. In the presence of self-
coupling, the neutrino perturbation equations for the anisotropic stress and
higher multipoles in the Boltzmann hierarchy are augmented with a collision
term proportional to $\dot{\tau}_{\nu}$. This collision term has a damping
action on the neutrino propagation before their decoupling and results in a
late onset of neutrino free streaming compared to $\Lambda$CDM. In the absence
of self-interaction, the neutrinos travel with a supersonic speed after
decoupling, dragging the photon-baryon plasma with it through gravitational
interaction. However, in the presence of strong self-interaction, the late
onset of free streaming changes this and affects the growth of fluctuations in
the photons and baryons. The SINU induced changes in the CMB and matter power
spectra are discussed in detail in Ref. [39, 20, 22]. To summarize, SINU
causes those perturbations, that entered the horizon before neutrino
decoupling, to grow more than in $\Lambda$CDM which enhances the CMB
temperature and polarization anisotropy power spectra. Additionally, it causes
a phase shift of the oscillations in the power spectra toward smaller angular
scales by changing the distance to the last scattering surface from us.

Both the effects on power amplitude and phase shift are proportional to the
fraction of the free streaming radiation energy density which is quantified by
$R_{\nu}$ [40],

$R_{\nu}=\dfrac{\rho_{\nu}^{\mathrm{free}}}{\rho_{\gamma}+\rho_{\nu}}\,.$
(2.4)

Here, $\rho_{\gamma}$ and $\rho_{\nu}$ are the total photon and neutrino
energy densities, respectively, and $\rho_{\nu}^{\mathrm{free}}$ is the total
free streaming neutrino density. Clearly, this ratio is maximum in
$\Lambda$CDM when all neutrinos are free streaming. Whereas in
$\mathbf{3c+0f}$, these changes are almost absent as all three flavors are
strongly interacting. The effects are intermediate in $\mathbf{2c+1f}$ and
$\mathbf{1c+2f}$ where the neutrinos are partially free streaming. These will
be demonstrated in the results in the next section. Finally, we emphasize that
even though we consider only neutrinos as the free-streaming species in this
paper, the same physics applies to any radiation-like species.

##  3 Results from ACT and Planck + ACT

We use the CMB data of ACT from four observing seasons to constrain both the
flavor universal and non-universal self-interaction scenarios. The flavor
universal case has been previously studied in Ref. [29]. However, our analysis
of flavor universal self-interaction differs from the previous study in a few
key aspects. In majority of the analyses in Ref. [29], the authors have varied
both the effective number of degrees of freedom in neutrinos
$N_{\mathrm{eff}}$ and neutrinos mass while deriving constraints of flavor
universal self-interaction. Ref. [29] did consider a one-parameter
$(\log_{10}G_{\mathrm{eff}})$ extension which is similar to our case, however,
the neutrino mass was set to a non-zero value. In our baseline models, we have
treated the neutrinos to be massless and fixed the $N_{\mathrm{eff}}=3.046$
according to the Big Bang nucleosynthesis calculation assuming the Standard
Model. Therefore, our analysis is the minimal extension of the $\Lambda$CDM
scenario where we have added only one new parameter which is the neutrino
self-interaction strength. We also study cases where we vary both the
$N_{\mathrm{eff}}$ and included the effects of neutrino mass. Whereas, for the
flavor non-universal case, we derive the first constraints using the ACT data.

###  3.1 Datasets & Methodology

In our analysis, we have used the ACT DR4 which includes small-scale
measurements up to $\ell=4126$. ACT is a ground-based telescope whose dataset
does not extend to the largest observable scale. Therefore, to get the most
stringent constraints, we combined the ACT dataset with the Planck measurement
following the prescription in Ref [1]. Below, we detail all the datasets used
in the analysis.

  * •

ACT: We use the actpollite_dr4 likelihood code from the ACT collaboration
which includes the DR4 data release [1]. The likelihood contains TT
measurement $(600<\ell<4126)$ and TE and EE measurements $(350<\ell<4126)$.
Because ACT lacks the large-scale data, it cannot constrain the reionization
optical depth $\tau_{\mathrm{reio}}$. Hence, we use a Gaussian prior centered
at $\tau_{\mathrm{reio}}=0.06\pm 0.01$ while analyzing the ACT data alone
[41].111This value differs slightly from the value used in Ref. [1, 29]:
$\tau_{\mathrm{reio}}=0.065\pm 0.015$

  * •

Planck: We use the Planck 2018 high-$\ell$ plik_lite222Following earlier
analyses, the nuisance parameter marginalized ‘lite’ likelihood is used to
reduce the dimensionality of the parameter space for MCMC runs using
Multinest. likelihood containing TT measurements $(30<\ell<2508)$ and TE and
EE measurements $(30<\ell<1996)$ [42]. We also include the low-$\ell$ TT
likelihood which includes TT measurements below $\ell<29$ and the lensing
likelihood.

  * •

Planck + ACT: When combining ACT with Planck data we removed ACT TT data below
$\ell<1800$ following Ref. [1].

We used a modified version of CLASS [43, 22], MontePython [44] and Multinest
[45, 46, 47, 48] for the MCMC runs, and GetDist [49] to analyze the chains and
plot the posterior distributions.

###  3.2 Baseline analysis: Massless neutrinos

In our baseline analysis, we have considered the neutrinos to be massless and
the $N_{\mathrm{eff}}$ is set to $3.046$. We analyzed the three scenarios
$\mathbf{3c+0f}$, $\mathbf{2c+1f}$, and $\mathbf{1c+2f}$ where three, two, and
one neutrino species are self-interacting, respectively, among the total three
species. Similar to the previous analysis we have used the following priors in
Table 1 for baseline analysis.

Parameters |  Prior
---|---
$10^{2}\omega_{b}$ |  $[1.00,4.00]$
$\omega_{\rm cdm}$ |  $[0.08,0.16]$
$100\theta{}_{s}$ |  $[1.03,1.05]$
$\ln 10^{10}A_{s}$ |  $[2.0,4.0]$
$n_{s}$ |  $[0.9,1.02]$
$\tau_{\rm reio}$ |  $[0.004,0.25]$
$\log_{10}[{\rm G_{eff}}/{\rm MeV}^{-2}]$ |  $[-5.0,-0.8]$
Table 1: Prior ranges for primary cosmological parameters for the baseline
analysis. Although ACT data extends to higher multipole,
$\log_{10}G_{\mathrm{eff}}=-5.0$ still corresponds to the mode for which
neutrinos decouple before the horizon entry of the smallest scale probed by
ACT (see Fig.2 of Ref. [22]). For $\log_{10}G_{\mathrm{eff}}\simeq-5$
neutrinos would decouple at a redshift much higher than $10^{5}$, which
corresponds to multipoles $\ell>10^{4}$ in the CMB spectrum well beyond the
scales probed by ACT. Thus, $\log_{10}G_{\mathrm{eff}}=-5.0$ safely represents
the $\Lambda$CDM limit even for the ACT dataset.

Figure 1: 1D posteriors of $\log_{10}G_{\mathrm{eff}}$ in three coupling
scenarios using the ACT (left) & ACT+Planck (right) data. The ACT data shows a
preference for the SI over MI mode irrespective of the number of coupled
neutrino species, which stands in stark contrast with our earlier analysis
using Planck 2018 data in Ref. [22]. When combined with Planck, the SI mode
significance is reduced compared to the MI mode for the flavor-universal case.
For flavor-specific cases, SI mode yields a better fit to the data.
Additionally, the SI mode in the ACT-only analysis favors a slightly larger
value of $\log_{10}G_{\mathrm{eff}}$.

Fig. 1 summarizes the main results of this work for the massless neutrino
self-interaction. As in earlier works, we find two separate modes in the
$\log_{10}G_{\mathrm{eff}}$ posterior distribution. The strongly interacting
(SI) mode peaks at a higher value of $\log_{10}G_{\mathrm{eff}}$, and the
moderately interesting (MI) mode shows up at a comparatively weaker value
closer to the $\Lambda$CDM limit. The ACT data alone prefers the SI mode over
the MI mode for all the scenarios. This preference for strong interaction has
also been observed in the flavor-universal case (similar to the
$\mathbf{3c+0f}$ scenario) studied in Ref. [29]. This can be attributed to the
better fit to the ACT EE data by the SI mode in the $700\lesssim\ell\lesssim
1000$ range of multipoles [29].

In the flavor non-universal scenarios, namely the $\mathbf{2c+1f}$ and
$\mathbf{1c+2f}$ cases, the relative significance of the MI mode relative to
the SI mode is higher. In the flavor non-universal cases, the changes in the
CMB spectrum corresponding to the MI and SI points are smaller compared to the
flavor-universal case because a significant portion of the total radiation is
simply free-streaming in non-universal case [22]. This results in a smaller
difference in $\chi^{2}$ between the best points of the two modes and, thus,
results in a smaller peak height difference in the posterior plot
corresponding to the two modes.

Moreover, the SI mode peak is slightly shifted to higher
$\log_{10}G_{\mathrm{eff}}$ compared to the Planck-only result (see Fig. 1 in
Ref. [22]). Thus ACT data prefer a delayed neutrino decoupling compared to the
Planck data. In the right panel of Fig. 1 we show the results of the combined
analysis with the ACT and Planck data. The addition of ACT data with Planck
increases the significance of the SI mode for all cases. Particularly for the
flavor non-universal cases, the SI mode provides a better fit to the combined
data. Since Planck prefers a smaller value of $\log_{10}G_{\mathrm{eff}}$ for
the SI mode, in the combined analyses, the SI mode peak positions are
stretched toward a slightly smaller value compared to the ACT result.

In Fig. 2 and 3, we show the triangle plots for some relevant cosmological
parameters using ACT and ACT+Planck data, respectively. For both datasets,
there is an expected shift of the parameters $A_{s}e^{-2\tau_{\mathrm{reio}}}$
and $n_{s}$ toward larger values for the SI mode when the number of
interacting neutrino species is increased, which can be clearly seen from the
2D plots. As explained in Ref. [22], this is because the scale-dependent power
enhancement from large $G_{\mathrm{eff}}$ depends on the total number of
interacting neutrinos. This shift is apparent from Fig. 2 as for the ACT
analysis the posterior is entirely dominated by the SI mode. However, in the
Planck + ACT analysis in Fig. 3, the MI mode becomes dominant in
$\mathbf{3c+0f}$. This leads to the bigger peak to the right in the
$A_{s}e^{-2\tau_{\mathrm{reio}}}$ and $n_{s}$ posteriors which should not be
confused with the dominant SI mode peaks for $\mathbf{2c+1f}$ and
$\mathbf{1c+2f}$. In fact, it is easier to identify the individual modes
(SI/MI) for any parameter by comparing with the 2D posterior with
$\log_{10}G_{\mathrm{eff}}$.

To study the individual modes in detail, we performed a separate analysis
where we fixed the prior range on $\log_{10}G_{\mathrm{eff}}$ to separate out
the MI and SI mode according to Table 2 [22].

Mode | Prior range on $\log_{10}G_{\mathrm{eff}}$
---|---
MI | $[-5.0,-2.7]$
SI | $[-2.7,-0.8]$
Table 2: Prior ranges for $\log_{10}G_{\mathrm{eff}}$ for MI and SI mode
separation. The ranges for other cosmological parameters remain the same as in
Table 1.

The parameter values and limits of the cosmological parameters for the ACT and
Planck + ACT datasets are quoted in Table 3 and 4, respectively. In the
tables, we also quote the $\chi^{2}$ difference for each of the models from
the corresponding 6-parameter $\Lambda$CDM model. We also quote the Akaike
Information Criterium (AIC) criterion defined as [50],

$\Delta{\rm AIC}=\chi^{2}_{{\rm min},\mathcal{M}}-\chi^{2}_{{\rm
min},\Lambda{\rm CDM}}+2(N_{\mathcal{M}}-N_{\Lambda{\rm CDM}}).$ (3.1)

Here $\chi^{2}_{{\rm min},i}$ and $N_{i}$ are the minimum $\chi^{2}$ and
number of parameters for $i$ model, respectively. Negative values $\Delta{\rm
AIC}$ signify the preference of the models over $\Lambda$CDM. Additionally,
for the baseline model, we also compute the Bayes factor $(\mathcal{B}_{\rm
SI})$ which is the ratio of Bayesian evidence of the SI over the MI mode.

$\mathcal{B}_{\rm SI}={\mathcal{Z}_{\rm SI}\over\mathcal{Z}_{\rm MI}}$ (3.2)

The Bayesian evidence which is defined below is computed using MultiNest
algorithm.

$\mathcal{Z}_{M}\equiv\mathrm{Pr}(\mathrm{d}|M)=\int\mathrm{Pr}(\mathrm{d}|\theta,M)\mathrm{Pr}(\theta|M)\mathrm{d}\theta\,,$
(3.3)

where $\mathrm{d}$ is the data and $\theta$ are the cosmological parameters
for model $M$. $\mathcal{B}_{\mathrm{SI}}>1$ signifies that the SI mode is
more significant than the MI mode. In Table 5, we show the Bayes for all the
self-interaction scenarios for the baseline model.

For the ACT dataset, both $\Delta$AIC and $\mathcal{B}_{\rm SI}$ show that
more interacting flavors provide a better fit to the data compared to
$\Lambda$CDM. Note that, the $\mathbf{3c+0f}$ model has the most negative
$\Delta$AIC, however, $\mathbf{2c+1f}$ has the largest Bayes factor333This is
not surprising because these two methods measure different quantities for
model comparison. The Bayes factor is related to the ratio of areas under the
SI and MI modes in the 1D posterior, respectively. We can see from Fig. 1 that
visibly the area under the SI mode of $\mathbf{2c+1f}$ is greater than
$\mathbf{3c+0f}$ which can offset the slight increase in area under the MI
mode.. Since ACT data lacks large-scale information, the changes induced by
the strong neutrino self-interaction can be relatively easily compensated by
modifying other parameters, most notably, $A_{s}e^{-2\tau_{\mathrm{reio}}}$
and $n_{s}$ as can be seen from the 2D panels in Fig. 2. While analyzing in
combination with the Planck data, this freedom is taken away and it has been
established by previous studies that $\mathbf{3c+0f}$ scenario does not
provide a good fit to the Planck data [22]. Therefore, for Planck + ACT
results, a lesser amount of self-interacting neutrino is favored. This shows
that Planck has more statistical power (more precision) than the ACT data as
the combined analysis prefers the MI mode in the $\mathbf{3c+0f}$ case. While
for $\mathbf{2c+1f}$ and $\mathbf{1c+2f}$ both Planck and ACT have a stronger
preference for the SI mode compared to the flavor universal case which results
in a greater significance of the SI mode over the MI one compared to the
$\mathbf{3c+0f}$ scenario.

Figure 2: The posterior distributions of
$\log_{10}G_{\mathrm{eff}},H_{0},A_{s}e^{-2\tau_{\mathrm{reio}}}$, and $n_{s}$
using the ACT data alongwith the $\tau$-prior. Note the large boost in the
significance of the SI mode in the $\log_{10}G_{\mathrm{eff}}$ posterior. See
Fig. 14 in Appendix. C for an extended parameter plot.

Massless $\nu$ | $\mathbf{3c+0f}$ | $\mathbf{2c+1f}$ | $\mathbf{1c+2f}$
---|---|---|---
Parameters |  MI |  SI |  MI |  SI |  MI |  SI
$10^{2}\omega_{\rm b}$ |  $2.151\pm 0.030$ |  $2.153\pm 0.031$ |  $2.154\pm 0.030$ |  $2.153\pm 0.030$ |  $2.155\pm 0.029$ |  $2.155\pm 0.030$
$\omega_{\rm cdm}$ |  $0.1190\pm 0.0037$ |  $0.1215\pm 0.0039$ |  $0.1192\pm 0.0037$ |  $0.1205\pm 0.0039$ |  $0.1194\pm 0.0037$ |  $0.1199\pm 0.0038$
$100\theta{}_{s}$ |  $1.04358\pm 0.00076$ |  $1.04872^{+0.00084}_{-0.00058}$ |  $1.04345\pm 0.00073$ |  $1.04684\pm 0.00079$ |  $1.04335\pm 0.00070$ |  $1.04495\pm 0.00076$
$\ln 10^{10}A_{s}$ |  $3.038\pm 0.024$ |  $2.974\pm 0.024$ |  $3.042\pm 0.023$ |  $2.998\pm 0.024$ |  $3.045\pm 0.023$ |  $3.022\pm 0.023$
$n_{s}$ |  $0.991^{+0.019}_{-0.013}$ |  $0.960^{+0.016}_{-0.018}$ |  $0.996^{+0.017}_{-0.011}$ |  $0.975\pm 0.016$ |  $0.999^{+0.016}_{-0.0094}$ |  $0.990\pm 0.015$
$\tau_{\rm reio}$ |  $0.0596\pm 0.0097$ |  $0.0588\pm 0.0098$ |  $0.0600\pm 0.0098$ |  $0.0594\pm 0.0097$ |  $0.0600\pm 0.0099$ |  $0.0598\pm 0.0097$
$\log_{10}[{\rm G_{eff}}/{\rm MeV}^{-2}]$ |  $-3.72^{+0.79}_{-0.41}$ |  $-1.62^{+0.14}_{-0.10}$ |  $-3.77\pm 0.60$ |  $-1.62^{+0.17}_{-0.13}$ |  $-3.79^{+0.87}_{-0.75}$ |  $-1.65^{+0.29}_{-0.18}$
$H_{0}({\rm km/s/Mpc})$ |  $68.1\pm 1.5$ |  $68.9\pm 1.5$ |  $68.0\pm 1.5$ |  $68.6^{+1.4}_{-1.7}$ |  $67.9\pm 1.5$ |  $68.3\pm 1.5$
$\sigma_{8}$ |  $0.840\pm 0.015$ |  $0.840\pm 0.016$ |  $0.840^{+0.015}_{-0.014}$ |  $0.837\pm 0.016$ |  $0.840\pm 0.015$ |  $0.838\pm 0.016$
$\chi^{2}-\chi^{2}_{\Lambda{\rm CDM}}$ |  $-2.08$ |  $-8.26$ |  $-1.34$ |  $-6.9$ |  $-0.69$ |  $-4.07$
$\Delta{\rm AIC}$ |  $-0.08$ |  $-6.26$ |  $0.66$ |  $-4.9$ |  $1.31$ |  $-2.07$

Table 3: Mean values and 68% confidence limits for ACT dataset for the
baseline massless $\nu$ scenario. We also show the $\chi^{2}$ difference from
the bestfit 6-parameter $\Lambda$CDM model and the corresponding $\Delta$AIC.
Two important derived parameters $H_{0}$ and $\sigma_{8}$ are also shown. In
all tables, the varied and derived parameters are separated by a horizontal
line unless mentioned otherwise.

Another interesting feature of the analysis with the ACT data is the values of
$H_{0}$ for the various scenarios. Historically, flavor-universal neutrino
self-interaction was proposed as a solution to the Hubble tension, which is
the discrepancy between the CMB and local measurement of $H_{0}$ using the
distance ladder, currently standing at $\sim\,4.4\sigma$ [51]. However, with
the release of Planck 2018 polarization data, the overall Planck data
predominantly favors the MI mode (which is the $\Lambda$CDM limit) over the SI
mode [23, 22]. In the flavor non-universal case, although the significance of
the SI mode peak increases dramatically, the corresponding $H_{0}$ value does
not increase appreciably to resolve the tension [22]. The changes in the CMB
spectra compared to $\Lambda$CDM are milder in flavor-specific neutrino self-
interaction compared to the flavor universal case as only a smaller fraction
of the total radiation is self-interacting. Therefore, the shifts in other
cosmological parameters to compensate for these changes (to arrive at a good
fit to the data) are also smaller in flavor-specific cases. This is the reason
for a smaller upward shift of the $H_{0}$ for flavor non-universal cases. This
is a robust prediction that also holds true when the effective number of
neutrinos and neutrino mass are varied.

The increase in $H_{0}$ in our baseline scenario results from the modification
of the acoustic phase shift. Free streaming neutrinos induce a scale-dependent
phase shift on photon-baryon acoustic oscillation which moves the CMB peaks
toward smaller multipole. Due to the self-interaction, neutrinos stop free-
streaming and the acoustic phase shift is modified. The shift in the CMB peak
position due to the reduction of the phase shift can be compensated by the
increase of the $H_{0}$ [52]. In Fig. 4, we show the shifts of the CMB peaks
for the TT and EE spectra for flavor dependent neutrino self-interaction. This
is the primary mechanism for the increase of $H_{0}$ when $N_{\rm eff}$ is not
allowed to vary. The amount of neutrino-induced phase follows the scaling
relation:

$\phi_{\nu}\propto
R_{\nu}=R_{\nu}^{\text{$\Lambda$CDM{}}}\times\begin{cases}0,~{}~{}\,\qquad{\rm
for~{}}\text{$\mathbf{3c+0f}$}\\\\[6.45831pt]
{\Large\nicefrac{{1}}{{3}}},\qquad{\rm
for~{}}\text{$\mathbf{2c+1f}$}\\\\[6.45831pt]
{\Large\nicefrac{{2}}{{3}}},\qquad{\rm
for~{}}\text{$\mathbf{1c+2f}$}\end{cases}$ (3.4)

Here $R_{\nu}$ is the free-streaming neutrino energy fraction defined in
Eq.(2.4) and $\phi_{\nu}$ is the acoustic phase induced by neutrinos. The
proportionality of the phase shift with the number of interacting neutrino
flavors can be clearly seen in Fig. 4. A tale-tell signature of this mechanism
is a shift in the parameter $\theta_{s}$ which is the angular size of the
sound horizon at recombination [52, 22]:

$\frac{\Delta\theta_{s}}{\theta_{s}}\propto\Delta\phi\;,$ (3.5)

where $\Delta\phi$ is the change of the neutrino-induced phase compared to
$\Lambda$CDM444Since the MI mode is the $\Lambda$CDM limit for neutrino self
interaction,
$\Delta\theta_{s}=\theta_{s}^{\rm(SI)}-\theta_{s}^{(\Lambda\rm{CDM})}\approx\theta_{s}^{\rm(SI)}-\theta_{s}^{(\rm
MI)}$ .. Thus the following scaling relation is a robust prediction for the
flavor-specific scenario,

$\Delta\theta_{s}|_{\text{$\mathbf{1c+2f}$}}\approx\frac{1}{2}\Delta\theta_{s}|_{\text{$\mathbf{2c+1f}$}}\approx\frac{1}{3}\Delta\theta_{s}|_{\text{$\mathbf{3c+0f}$}}.$
(3.6)

One can check the validity of Eq.(3.6) from Table 3 and 4555While comparing
with numbers from the tables $\Delta\theta_{s}$ is the difference between the
MI and SI mode _mean_ values.. Later in the paper, we will present results for
varying $N_{\mathrm{eff}}$ of both coupled and free-streaming nature. The
above equations also hold true in those cases by the appropriate modification
of $R_{\nu}$.

However, a sizable positive change in $\Delta\theta_{s}$ does not always
translate to a proportionately higher value of the $H_{0}$. The exact relation
between $\theta_{s}$ and $H_{0}$ involves other cosmological parameters such
as baryon and dark matter energy densities. As the ACT data lack large-scale
information, these energy densities are less constrained, resulting in a
slightly weaker increase in $H_{0}$ in the ACT case. As the modification of
$\theta_{s}$ is the primary driver for $H_{0}$ increase for our baseline
model, a similar scaling in terms of $H_{0}$ also holds true,

$\Delta H_{0}|_{\text{$\mathbf{1c+2f}$}}\approx\frac{1}{2}\Delta
H_{0}|_{\text{$\mathbf{2c+1f}$}}\approx\frac{1}{3}\Delta
H_{0}|_{\text{$\mathbf{3c+0f}$}}.$ (3.7)

Figure 3: The posterior distributions of
$\log_{10}G_{\mathrm{eff}},H_{0},A_{s}e^{-2\tau_{\mathrm{reio}}}$, and $n_{s}$
using the combined ACT and Planck data. See Fig. 15 in Appendix. C for an
extended parameter plot.

Figure 4: Shift in the CMB acoustic peak positions compared to $\Lambda$CDM
for the TT (Left) and EE (Right) spectra for $\log_{10}G_{\mathrm{eff}}=-1.0$.
The origin of the peak shift is the change of the neutrino-induced phase-shift
in the CMB acoustic peaks due to neutrino-self interaction [40, 52, 39]. The
phase shift, hence the peak shift, is proportional to the amount of coupled
neutrino flavor. The phase shift is the primary driver for the increase in
$H_{0}$ in the neutrino-self interaction scenarios, specifically when $N_{\rm
eff}$ is kept fixed.

Massless $\nu$ | $\mathbf{3c+0f}$ | $\mathbf{2c+1f}$ | $\mathbf{1c+2f}$
---|---|---|---
Parameters |  MI |  SI |  MI |  SI |  MI |  SI
$10^{2}\omega_{\rm b}$ |  $2.232\pm 0.013$ |  $2.245^{+0.012}_{-0.014}$ |  $2.233\pm 0.012$ |  $2.239\pm 0.013$ |  $2.235\pm 0.012$ |  $2.236\pm 0.013$
$\omega_{\rm cdm}$ |  $0.1197\pm 0.0011$ |  $0.1213\pm 0.0011$ |  $0.1197\pm 0.0011$ |  $0.1208\pm 0.0011$ |  $0.1197\pm 0.0011$ |  $0.1203\pm 0.0011$
$100\theta{}_{s}$ |  $1.04210\pm 0.00029$ |  $1.04738^{+0.00044}_{-0.00038}$ |  $1.04211\pm 0.00029$ |  $1.04560^{+0.00040}_{-0.00034}$ |  $1.04209^{+0.00026}_{-0.00029}$ |  $1.04382^{+0.00038}_{-0.00031}$
$\ln 10^{10}A_{s}$ |  $3.048\pm 0.012$ |  $2.981\pm 0.012$ |  $3.049\pm 0.012$ |  $3.004\pm 0.012$ |  $3.051\pm 0.012$ |  $3.029\pm 0.012$
$n_{s}$ |  $0.9639^{+0.0061}_{-0.0044}$ |  $0.9373\pm 0.0042$ |  $0.9646^{+0.0057}_{-0.0041}$ |  $0.9469\pm 0.0043$ |  $0.9661\pm 0.0042$ |  $0.9569\pm 0.0041$
$\tau_{\rm reio}$ |  $0.0549\pm 0.0057$ |  $0.0539\pm 0.0058$ |  $0.0551\pm 0.0059$ |  $0.0542\pm 0.0059$ |  $0.0554^{+0.0052}_{-0.0061}$ |  $0.0548\pm 0.0058$
$\log_{10}[{\rm G_{eff}}/{\rm MeV}^{-2}]$ |  $-4.05^{+0.62}_{-0.40}$ |  $-1.737^{+0.086}_{-0.064}$ |  $-3.94\pm 0.53$ |  $-1.69^{+0.11}_{-0.090}$ |  $-3.87^{+0.83}_{-0.69}$ |  $-1.65^{+0.18}_{-0.12}$
$H_{0}({\rm km/s/Mpc})$ |  $68.01\pm 0.49$ |  $69.38\pm 0.49$ |  $68.04\pm 0.48$ |  $68.87\pm 0.49$ |  $68.05\pm 0.48$ |  $68.44\pm 0.48$
$\sigma_{8}$ |  $0.8291\pm 0.0054$ |  $0.8335\pm 0.0062$ |  $0.8289\pm 0.0053$ |  $0.8293\pm 0.0062$ |  $0.8286\pm 0.0053$ |  $0.8270\pm 0.0060$
$\chi^{2}-\chi^{2}_{\Lambda{\rm CDM}}$ |  $-1.89$ |  $-1.21$ |  $-1.41$ |  $-5.25$ |  $-0.79$ |  $-5.17$
$\Delta{\rm AIC}$ |  $0.11$ |  $0.79$ |  $0.59$ |  $-3.25$ |  $1.21$ |  $-3.17$

Table 4: Mean values and 68% confidence limits for the Planck + ACT dataset.
We also show the $\chi^{2}$ difference from the bestfit 6-parameter
$\Lambda$CDM model and the corresponding $\Delta$AIC.

Massless $\nu$ |  $\mathbf{3c+0f}$ |  $\mathbf{2c+1f}$ |  $\mathbf{1c+2f}$
---|---|---|---
ACTDR4+tauprior |  $6.576$ |  $6.706$ |  $2.519$
Planck-TTTEEE+lowE+lensing |  $0.024$ |  $0.179$ |  $0.529$
Planck-TTTEEE+lowE+lensing+ACTDR4 |  $0.161$ |  $1.291$ |  $2.227$

Table 5: Bayes factors for SINU for all datasets.

In Fig. 5 and 6 we show the residuals of the bestfit SINU models w.r.t. the
corresponding $\Lambda$CDM models for both ACT (left) and Planck (right) for
the TT and EE modes, respectively. To analyze the consistency between the two
datasets, we plot the residual with respect to both these datasets in each
plot. For better visualization of the residuals, we plot the effective
$\chi^{2}$ for each dataset bin ignoring the correlation between adjacent
bins,

$\tilde{\chi}^{2}_{\ell,{\rm Data}}=\frac{(C_{\ell,{\rm Data}}-C_{\ell,{\rm
bf}})^{2}}{(\Delta C_{\ell,{\rm
Data}})^{2}}\qquad\text{for}~{}\ell\in\ell_{\rm Data(binned)}.$ (3.8)

Figure 5: (Top) TT residuals of the bestfit models relative to the
corresponding $\Lambda$CDM analyses for the ACT (left) and Planck (right)
datasets. (Bottom) The multipole distribution of the $\chi^{2}_{\ell}$ for the
two bestfit models.

Figure 6: Same as Fig. 5 for EE spectrum.

We use the binning provided in the official data release by the respective
collaborations.

For the ACT TT residual plot (Fig. 5, left) we see that SINU provides a better
fit to the data than the $\Lambda$CDM. This can be verified from the plot of
$\tilde{\chi}^{2}_{\ell,{\rm ACT}}$ where the $\mathbf{3c+0f}$ overall gives
the least $\tilde{\chi}^{2}$ across all the bins. However, we can see from the
$\tilde{\chi}^{2}_{\ell,{\rm Planck}}$ panel that Planck data highly disfavor
this fit. This discrepancy is particularly very large for $\ell\lesssim 500$
since ACT does not have any data in that multipole range. For the Planck TT
residual plot (Fig. 5, right) we see that $\mathbf{1c+2f}$ provides an overall
better fit to the among SINU scenarios and is very close to the $\Lambda$CDM
fit. This can be clearly seen from the $\tilde{\chi}^{2}_{\ell,{\rm Planck}}$
panel. Interestingly, from both the $\tilde{\chi}^{2}_{\ell,{\rm ACT}}$ panels
in the figure, we see that in general SINU scenarios give smaller
$\tilde{\chi}^{2}$ than $\Lambda$CDM for ACT. The overall increase in
$\tilde{\chi}^{2}$ is visibly quite smaller going from ACT bestfit to Planck
bestfit for $\tilde{\chi}^{2}_{\ell,{\rm ACT}}$. Thus ACT TT data prefers SINU
over $\Lambda$CDM for the bestfit with ACT runs and also for the bestfit with
Planck runs. Thus for Planck+ACT results, the significance of the SI mode
increases owing to the addition of ACT data. In the EE residual plots (Fig.
5), we see the overall same features as in the TT residual plots.

We can also comment on the compatibility between ACT and Planck data from the
same residual plots. TT residual in Fig. 5 shows that compared to ACT bestfit
(left panel) $\tilde{\chi}^{2}_{\ell,{\rm ACT}}$ increases substantially in
the multipole range $1600<\ell<2000$ for the Planck bestfit (right panel).
Whereas, for EE residuals (Fig. 5) $\tilde{\chi}^{2}_{\ell,{\rm ACT}}$
increases significantly in the range $600<\ell<1000$. Therefore, TT and EE
data from ACT and Planck have a mild disagreement in those corresponding
multipole ranges mentioned above. This perhaps points to some unknown
systematics in the intermediate-scale data. Future data from ACT, SPT, and
other experiments will be crucial in confirming this.

Note that for SI interaction strength neutrinos decouple around redshift
$z\sim 4000$ which corresponds to multipole $\ell\sim 300$ [22]. Neutrino
self-interaction models fit the ACT EE data better in the $600<\ell<1000$
range which is relevant for SI mode interaction strength. Thus, it can be
inferred from this analysis that ACT polarization data is the likely driver
for the enhancement of the SI mode, which was pointed out in Ref. [29]. For
the ACT TT data, the preferred SI mode interaction strength, which yields a
better fit for the multipole range $1600<\ell<2000$, corresponds to a smaller
value of $\log_{10}G_{\mathrm{eff}}$ (see Fig. 2 in Ref. [29]). The high
multipole ACT data does not have large constraining power because of the large
error bars as can be seen from the $\tilde{\chi}^{2}_{\ell,{\rm ACT}}$ plots
(inset) in Fig. 5 and 6. The increase in the significance of the SI mode is
driven by the intermediate scales, rather than at smaller scales ($\ell>2500$)
that were unexplored by Planck.

Figure 7: Changes in the TT and EE CMB power spectra
$(D_{\ell}\equiv\ell(\ell+1)C_{\ell}/2\pi)$ due to neutrino self-interaction.
The coupling strength is fixed at $\log_{10}G_{\mathrm{eff}}=-1$. Note the
drop in $\Delta D_{\ell}$ above $\ell\gtrsim 2000$.

At smaller scales, the CMB spectrum drops significantly due to diffusion
damping. In Fig. 7, we show the changes in the CMB spectra from $\Lambda$CDM
$(\Delta D_{\ell})$ for flavor-specific scenarios. After $\ell\gtrsim 2000$,
$\Delta D_{\ell}$ drops rapidly because the overall signal is damped. For
example, the changes in the CMB spectrum due to SINU effects are $\Delta
D_{\ell}\sim 10^{-2}$ at $\ell\sim 6000$. Probing such small changes at high
multipole is going to be highly challenging. Future experiments will require
high sensitivities and ultra-low angular resolution. In sec. 4, we will study
the reach of a few future CMB measurements. Furthermore, for SI mode
interaction strength, the neutrino decoupling takes place at a much smaller
multipole. Therefore, more precise CMB measurement in the intermediate region
$600<\ell<1000$ will be crucial to determine the fate of the SI mode.

The constraints derived in this section for the flavor-universal case are
similar (approximately consistent within the error bar) to the ones derived in
Ref. [29] for the one-parameter extension. The slight discrepancies may be
attributed to the different choice of prior ranges for
$\log_{10}G_{\mathrm{eff}}$, the assumption on neutrino mass, the choice of
$\tau_{reio}$-prior (specifically for the ACT-only analysis), and slight
differences in dataset combination. For example, the ACT+Planck analysis in
Ref. [29] does not include the Planck Lensing data, whereas this work
does666In Ref. [29] Planck Lensing data was included in their analysis with
the BAO data..

###  3.3 Effects of massive neutrino

Figure 8: Comparison between the posterior distributions from the analyses
with massless (dashed) and massive (solid) neutrino in the $\mathbf{2c+1f}$
(blue) and $\mathbf{1c+2f}$ (green) cases. In massive case, for each of the
models, one of the _free streaming_ neutrinos is modeled as massive with mass
$0.06~{}{\rm eV}$. Adding neutrino mass does not change the
$\log_{10}G_{\mathrm{eff}}$ posteriors by any discernible amount. See Fig. 16
in Appendix. C for an extended parameter plot for Planck + ACT analysis.

In this section, we show the results of a separate analysis including massive
neutrino. For this purpose, we added the massive neutrino as a free-streaming
neutrino with a fixed total mass, but not as coupled neutrinos. Hence, we
study the cosmology of $\mathbf{1c+2f}$ and $\mathbf{2c+1f}$ where one
neutrino species among the respective $\mathbf{2f}$ and $\mathbf{1f}$ species
is massive with a mass of $m_{\nu}=0.06\,{\rm eV}$. We want to emphasize that
such separation of the massive neutrinos and interacting neutrinos is not
expected to give rise to any unusual results given the negligible correlation
between $\log_{10}G_{\mathrm{eff}}$ and $\sum m_{\nu}$ [23, 29].

Fig. 8 shows the 1D posterior of relevant parameters for $\mathbf{1c+2f}$ and
$\mathbf{2c+1f}$ scenarios for all the datasets. From the
$\log_{10}G_{\mathrm{eff}}$ posteriors, we see that adding neutrino mass does
not change the favored interaction strength in SI and MI modes. However, as
expected with massive neutrinos, they increase the total matter density in the
Universe and change the time of matter-radiation equality. This affects the
background density evolution and results in slightly smaller values of $H_{0}$
and $\sigma_{8}$ relative to the massless case. The change in the matter-
radiation equality also changes the size of the sound horizon at recombination
and the distance to the last scattering surface. Together, these shift the CMB
power spectra toward smaller $\ell$. The damping of the matter power spectrum
at small scale additionally damps the CMB spectra at smaller angular scales
[53]. Note that the shifts in $H_{0}$ and $\sigma_{8}$ due to neutrino mass
are greater for the Planck dataset compared to ACT because of Planck’s large-
scale data. In appendix B, we provide the constraints on all the cosmological
parameters for the massive SINU scenario.

###  3.4 Effects of varying $N_{\mathrm{eff}}$

A well-motivated extension of the $\Lambda$CDM cosmology is the addition of
extra relativistic degrees of freedom, denoted by $\Delta N_{\mathrm{eff}}$.
The baseline model $\Lambda$CDM has three active neutrinos which together
yield $N_{\mathrm{eff}}=3.046$.

Because we have flavor-nonuniversal self-interaction, the additional neutrino
states may or may not be self-interacting. In this section, we will study the
effects of varying both the free streaming and coupled neutrino fraction in
SINU scenarios. The effects of adding more radiation species are qualitatively
opposite to that of neutrino self-interaction [54, 20, 22]. Larger
$N_{\mathrm{eff}}$ increases the net radiation density which results in a
greater expansion rate of the Universe during radiation domination and a delay
of the matter-radiation equality. This shifts the CMB power spectrum peaks
toward low $\ell$ and damps the tail of the spectrum.

####  3.4.1 Additional free streaming neutrinos

Figure 9: Comparison between the posterior distributions from the analyses
with fixed $N_{\mathrm{eff}}=3.046$ and varying $N_{\mathrm{eff}}$. The extra
$\Delta N_{\mathrm{eff}}$ component is treated as noninteracting. See Fig. 17
in Appendix. C for an extended parameter plot for the varying
$N_{\mathrm{eff}}$ analysis with Planck + ACT data.

Parameters
---
$10^{2}\omega_{\rm b}$
$\omega_{\rm cdm}$
$100\theta{}_{s}$
$\ln 10^{10}A_{s}$
$n_{s}$
$\tau_{\rm reio}$
$\log_{10}[{\rm G_{eff}}/{\rm MeV}^{-2}]$
$\Delta N_{\rm eff}$
$H_{0}({\rm km/s/Mpc})$
$\sigma_{8}$
$\chi^{2}-\chi^{2}_{\overline{\Lambda{\rm CDM}}}$
$\Delta{\rm AIC}$
ACTDR4+tauprior
---
$3c+(0+\Delta N_{\rm eff})f$ |  $2c+(1+\Delta N_{\rm eff})f$ |  $1c+(2+\Delta N_{\rm eff})f$
$2.172\pm 0.034$ |  $2.169\pm 0.033$ |  $2.168\pm 0.031$
$0.1244^{+0.0044}_{-0.0052}$ |  $0.1231^{+0.0043}_{-0.0048}$ |  $0.1218^{+0.0037}_{-0.0045}$
$1.0477^{+0.0016}_{-0.00051}$ |  $1.0459^{+0.0015}_{-0.00058}$ |  $1.0442\pm 0.0011$
$2.989^{+0.024}_{-0.033}$ |  $3.012^{+0.027}_{-0.030}$ |  $3.035\pm 0.025$
$0.979^{+0.025}_{-0.022}$ |  $0.989^{+0.022}_{-0.015}$ |  $>0.994$
$0.060\pm 0.010$ |  $0.0599\pm 0.0098$ |  $0.0601\pm 0.0096$
$-1.79^{+0.32}_{+0.052}$ |  $-1.89^{+0.47}_{+0.12}$ |  $-2.3^{+1.1}_{-1.2}$
$<0.315$ |  $<0.243$ |  $<0.172$
$70.7^{+1.9}_{-2.3}$ |  $69.9^{+1.7}_{-2.1}$ |  $69.1^{+1.5}_{-1.8}$
$0.849\pm 0.018$ |  $0.846\pm 0.018$ |  $0.845\pm 0.016$
$-4.61$ |  $-3.54$ |  $-0.49$
$-2.61$ |  $-1.54$ |  $1.51$
Planck-TTTEEE+lowE+lensing
---
$3c+(0+\Delta N_{\rm eff})f$ |  $2c+(1+\Delta N_{\rm eff})f$ |  $1c+(2+\Delta N_{\rm eff})f$
$2.244^{+0.015}_{-0.018}$ |  $2.246^{+0.015}_{-0.019}$ |  $2.246^{+0.015}_{-0.018}$
$0.1215^{+0.0014}_{-0.0020}$ |  $0.1215^{+0.0013}_{-0.0021}$ |  $0.1215^{+0.0014}_{-0.0020}$
$1.04174^{+0.00029}_{-0.00041}$ |  $1.042001^{+0.000099}_{-0.00074}$ |  $1.0422^{+0.0012}_{-0.0010}$
$3.045\pm 0.017$ |  $3.042^{+0.021}_{-0.014}$ |  $3.040\pm 0.018$
$0.9664\pm 0.0066$ |  $0.9652^{+0.0083}_{-0.0054}$ |  $0.9647^{+0.0070}_{-0.0064}$
$0.0544\pm 0.0074$ |  $0.0544\pm 0.0074$ |  $0.0545^{+0.0068}_{-0.0076}$
$<-4.06$ |  $<-3.76$ |  $<-2.63$
$<0.133$ |  $<0.134$ |  $<0.130$
$68.68^{+0.64}_{-0.97}$ |  $68.78^{+0.63}_{-1.1}$ |  $68.80^{+0.64}_{-0.99}$
$0.8285^{+0.0065}_{-0.0078}$ |  $0.8287^{+0.0068}_{-0.0077}$ |  $0.8277^{+0.0067}_{-0.0078}$
$1.09$ |  $1.17$ |  $1.36$
$3.09$ |  $3.17$ |  $3.36$

Parameters
---
$10^{2}\omega_{\rm b}$
$\omega_{\rm cdm}$
$100\theta{}_{s}$
$\ln 10^{10}A_{s}$
$n_{s}$
$\tau_{\rm reio}$
$\log_{10}[{\rm G_{eff}}/{\rm MeV}^{-2}]$
$\Delta N_{\rm eff}$
$H_{0}({\rm km/s/Mpc})$
$\sigma_{8}$
$\chi^{2}-\chi^{2}_{\overline{\Lambda{\rm CDM}}}$
$\Delta{\rm AIC}$
Planck-TTTEEE+lowE+lensing+ACTDR4
---
$3c+(0+\Delta N_{\rm eff})f$ |  $2c+(1+\Delta N_{\rm eff})f$ |  $1c+(2+\Delta N_{\rm eff})f$
$2.240^{+0.014}_{-0.016}$ |  $2.243\pm 0.014$ |  $2.241\pm 0.014$
$0.1210^{+0.0012}_{-0.0018}$ |  $0.1215^{+0.0015}_{-0.0017}$ |  $0.1210^{+0.0012}_{-0.0015}$
$1.04290^{-0.00049}_{-0.0013}$ |  $1.0441^{+0.0018}_{-0.0024}$ |  $1.04318^{+0.00099}_{-0.0014}$
$3.040^{+0.029}_{-0.0058}$ |  $3.025\pm 0.025$ |  $3.038^{+0.015}_{-0.018}$
$0.962^{+0.012}_{-0.0035}$ |  $0.956^{+0.014}_{-0.012}$ |  $0.9622^{+0.0054}_{-0.0072}$
$0.0552\pm 0.0058$ |  $0.0549\pm 0.0058$ |  $0.0552\pm 0.0058$
$-3.58^{+2.0}_{-0.76}$ |  $-2.5^{+1.0}_{-1.3}$ |  $-2.29^{+0.99}_{-1.3}$
$<0.0807$ |  $<0.0894$ |  $<0.0770$
$68.74^{+0.50}_{-1.1}$ |  $69.09^{+0.74}_{-0.87}$ |  $68.79^{+0.56}_{-0.74}$
$0.8328^{+0.0055}_{-0.0070}$ |  $0.8318\pm 0.0064$ |  $0.8299\pm 0.0062$
$3.71$ |  $0.56$ |  $0.4$
$5.71$ |  $2.56$ |  $2.4$

Table 6: Mean values and 68% confidence limits for SINU with additional free
streaming neutrinos. Note that the values and limits are for the full sample
where we have not separated out the MI and SI modes. We also show the
$\chi^{2}$ difference from the bestfit 7-parameter $\Lambda$CDM model where
$N_{\rm eff}$ is also varied (denoted as $\overline{\Lambda\rm CDM}$). The
corresponding $\Delta$AIC is also shown.

In this section, we added extra free-streaming radiation $(\Delta N_{\rm
eff})$ in all SINU scenarios and dubbed them as: $\mathbf{3c+(0+\Delta N_{\rm
eff})f}$, $\mathbf{2c+(1+\Delta N_{\rm eff})f}$ and $\mathbf{1c+(2+\Delta
N_{\rm eff})f}\,$777We have added a flat prior on $\Delta N_{\rm eff}$ for all
SINU scenarios in the range $[0,1]$ . The results of adding extra free
streaming are fairly simple to understand as it just increases
$N_{\mathrm{eff}}$ without changing the amount of self-interaction. In Fig. 9,
we show the 1D posteriors of all parameters using the three dataset
combinations. The most significant changes happen in $n_{s}$ and $H_{0}$. The
extra free streaming $\Delta N_{\mathrm{eff}}$ increases their values, which
is expected from the change in the matter-radiation equality and the sound
horizon during recombination. Even though the ACT data prefer the SI mode and
$\Delta N_{\mathrm{eff}}$ cancels the effects of $G_{\mathrm{eff}}$, we do not
see any increase in the significance or the position of the SI mode in
$G_{\mathrm{eff}}$ in Fig. 9. This suggests that the effects of $\Delta N_{\rm
eff}$ are compensated mostly by $n_{s}$. In the $\mathbf{3c+0f}$ case, there
is a preference for larger $\Delta N_{\mathrm{eff}}$ than the other cases and,
as a result, larger $H_{0}$ too.

When only Planck data is used, the difference between the three SINU cases
goes away almost completely for all parameters except for
$\log_{10}G_{\mathrm{eff}}$. This is because the dominant modes, the MI mode
of $\mathbf{3c+0f}$ and the SI mode of $\mathbf{1c+2f}$, are close to each
other for other parameters. Finally, in none of the cases shown in Fig. 9, we
find any preference for a nonzero value of $\Delta N_{\mathrm{eff}}$.

####  3.4.2 Additional self-interacting neutrinos

As in our setup, we always use an integer number of self-interacting neutrinos
to assign their individual coupling strengths, we cannot directly use the
$N_{\mathrm{eff}}$ parameter to vary their number. However, as far as the
total relativistic energy density of free streaming species is concerned, it
can be varied either by changing the number of states or the temperatures. So,
we use the relative temperature $\xi_{i}\equiv T_{i}/T_{\gamma}$ to vary the
energy density of the _coupled_ neutrinos888The (flat) prior range for the
temperature of the coupled was chosen as $\xi_{\nu,c}=[0.0,1.0]$. Only for
this subsection, we varied $H_{0}$ instead of $100\theta_{s}$ with a flat
prior between $[58,78]$.. This is possible only for a thermal species. In
fact, from a particle physics perspective, varying $\xi_{i}$ is more physical
as the number of flavor eigenstates cannot be non-integer.

However, when a radiation species is self-interacting with a temperature-
dependent interaction rate, varying $\xi_{i}$ is not straightforward. For free
streaming neutrinos, there is a complete degeneracy between temperature and
the number of states since the relevant cosmological quantities like energy
density, pressure density depends on the combination $N_{\nu}T_{\nu}^{4}$,
where $N_{\nu}$ is the number of free-streaming neutrino states. On the other
hand, for self-interacting neutrinos the interaction rate has a different
dependence of $T_{\nu}$ as shown in Eq.(2.3):

$\dot{\tau}_{\nu}=-a(G_{\mathrm{eff}})^{2}T_{\nu}^{5}\;.$ (3.9)

Therefore, when $T_{\nu}$ (and hence $\xi_{\nu}$) is varied, to achieve the
same interaction rate, $G_{\mathrm{eff}}$ also needs to be scaled according to
Eq.(2.3). In other words, since $\dot{\tau}_{\nu}$ is the physical quantity,
the cosmological signatures depend on the combination
$(G_{\mathrm{eff}})^{2}T_{\nu}^{5}$. So, to compare with the fixed temperature
baseline model, we define a rescaled coupling strength

$\tilde{G}_{\mathrm{eff}}\equiv
G_{\mathrm{eff}}\left(\frac{T_{\nu}}{T_{\nu,{\rm SM}}}\right)^{5/2}\;,$ (3.10)

Figure 10: The posterior distributions when the $\Delta N_{\mathrm{eff}}$ for
coupled neutrinos is varied. For comparison, we also show the respective
distributions for fixed $N_{\mathrm{eff}}=3.046$. See Fig. 18 in Appendix. C
for an extended parameter plot for the varying $N_{\mathrm{eff}}$ analysis
with Planck + ACT data.

where $T_{\nu,{\rm SM}}$ is the SM neutrino temperature. Thus, for the
baseline base, $\tilde{G}_{\mathrm{eff}}=G_{\mathrm{eff}}$. In Appendix. A, we
explicitly demonstrate via MCMC analysis the degeneracy between
$G_{\mathrm{eff}}$ and $T_{\nu}$. As an example, we studied the
$\mathbf{3c+0f}$ scenario with varying temperatures via two parametrizations:
$N_{c}=1$ and $N_{c}=3$ coupled neutrinos with their corresponding varying
temperature. We show that after proper rescaling (as shown in Eq.(3.10)), both
models give the exact same constraints on _all_ cosmological parameters (Fig.
13).

Parameters
---
$10^{2}\omega_{\rm b}$
$\omega_{\rm cdm}$
$100\theta{}_{s}$
$\ln 10^{10}A_{s}$
$n_{s}$
$\tau_{\rm reio}$
$\log_{10}[{\rm G_{eff}}/{\rm MeV}^{-2}]$
$\frac{T_{{\rm c},1\nu}}{T_{{\rm f},1\nu}}$
$\log_{10}[{\rm\tilde{G}_{eff}}/{\rm MeV}^{-2}]$
$H_{0}({\rm km/s/Mpc})$
$\sigma_{8}$
$\Delta N_{\rm c}$
$\chi^{2}-\chi^{2}_{\overline{\Lambda{\rm CDM}}}$
$\Delta{\rm AIC}$
ACTDR4+tauprior
---
$(3+\Delta N_{\rm c})c+0f$ |  $(2+\Delta N_{\rm c})c+1f$ |  $(1+\Delta N_{\rm c})c+2f$
$2.123\pm 0.051$ |  $2.109^{+0.043}_{-0.051}$ |  $2.089^{+0.030}_{-0.039}$
$0.1157^{+0.0079}_{-0.0091}$ |  $0.1122^{+0.0062}_{-0.0083}$ |  $0.1084^{+0.0032}_{-0.0052}$
$1.0472\pm 0.0021$ |  $1.0462^{+0.0017}_{-0.0011}$ |  $1.04516^{+0.00086}_{-0.00070}$
$2.987^{+0.030}_{-0.038}$ |  $2.999^{+0.027}_{-0.030}$ |  $3.012\pm 0.024$
$0.951^{+0.025}_{-0.034}$ |  $0.951^{+0.025}_{-0.034}$ |  $0.948^{+0.014}_{-0.025}$
$0.0583^{+0.010}_{-0.0093}$ |  $0.0582\pm 0.0097$ |  $0.0578\pm 0.0096$
$-2.5^{+1.2}_{-1.7}$ |  $-2.5^{+1.4}_{-2.1}$ |  $-2.8\pm 1.2$
$0.966^{+0.049}_{-0.045}$ |  $0.913\pm 0.070$ |  $<0.658$
$-2.5^{+1.2}_{-1.4}$ |  $-2.6^{+1.4}_{-1.8}$ |  $-4.0^{+2.5}_{-1.4}$
$65.8^{+3.8}_{-5.0}$ |  $64.1^{+2.4}_{-4.7}$ |  $61.9^{+1.1}_{-2.7}$
$0.825\pm 0.026$ |  $0.814^{+0.022}_{-0.026}$ |  $0.803^{+0.015}_{-0.019}$
$-0.36\pm 0.50$ |  $-0.57^{+0.31}_{-0.57}$ |  $-0.830^{+0.032}_{-0.20}$
$-4.73$ |  $-3.76$ |  $-1.07$
$-2.73$ |  $-1.76$ |  $0.93$
Planck-TTTEEE+lowE+lensing
---
$(3+\Delta N_{\rm c})c+0f$ |  $(2+\Delta N_{\rm c})c+1f$ |  $(1+\Delta N_{\rm c})c+2f$
$2.223\pm 0.022$ |  $2.223\pm 0.022$ |  $2.220\pm 0.022$
$0.1177\pm 0.0029$ |  $0.1175\pm 0.0029$ |  $0.1172\pm 0.0029$
$1.04246^{+0.00028}_{-0.00081}$ |  $1.04275^{+0.00019}_{-0.0012}$ |  $1.04266^{+0.00071}_{-0.00090}$
$3.030^{+0.021}_{-0.015}$ |  $3.026^{+0.025}_{-0.018}$ |  $3.027\pm 0.018$
$0.956^{+0.011}_{-0.0076}$ |  $0.954^{+0.012}_{-0.0091}$ |  $0.9542^{+0.0081}_{-0.0091}$
$0.0526\pm 0.0074$ |  $0.0526\pm 0.0074$ |  $0.0525\pm 0.0073$
$-4.25^{+0.20}_{-0.73}$ |  $-3.8^{+2.0}_{-1.1}$ |  $-3.4^{+1.5}_{-1.2}$
$0.987^{+0.017}_{-0.015}$ |  $0.977^{+0.027}_{-0.023}$ |  $0.944^{+0.063}_{-0.044}$
$-4.26^{+0.18}_{-0.73}$ |  $-3.9^{+2.0}_{-1.2}$ |  $-3.4^{+1.8}_{-1.6}$
$66.9\pm 1.4$ |  $66.9\pm 1.4$ |  $66.7\pm 1.4$
$0.8171^{+0.0093}_{-0.011}$ |  $0.8168^{+0.0093}_{-0.011}$ |  $0.815\pm 0.010$
$-0.16\pm 0.19$ |  $-0.18\pm 0.19$ |  $-0.19\pm 0.19$
$0.25$ |  $0.19$ |  $0.15$
$2.25$ |  $2.19$ |  $2.15$

Parameters
---
$10^{2}\omega_{\rm b}$
$\omega_{\rm cdm}$
$100\theta{}_{s}$
$\ln 10^{10}A_{s}$
$n_{s}$
$\tau_{\rm reio}$
$\log_{10}[{\rm G_{eff}}/{\rm MeV}^{-2}]$
$\frac{T_{{\rm c},1\nu}}{T_{{\rm f},1\nu}}$
$\log_{10}[{\rm\tilde{G}_{eff}}/{\rm MeV}^{-2}]$
$H_{0}({\rm km/s/Mpc})$
$\sigma_{8}$
$\Delta N_{\rm c}$
$\chi^{2}-\chi^{2}_{\overline{\Lambda{\rm CDM}}}$
$\Delta{\rm AIC}$
Planck-TTTEEE+lowE+lensing+ACTDR4
---
$(3+\Delta N_{\rm c})c+0f$ |  $(2+\Delta N_{\rm c})c+1f$ |  $(1+\Delta N_{\rm c})c+2f$
$2.202\pm 0.020$ |  $2.204\pm 0.021$ |  $2.200\pm 0.020$
$0.1150\pm 0.0026$ |  $0.1152^{+0.0025}_{-0.0029}$ |  $0.1145\pm 0.0026$
$1.04326^{+0.00010}_{-0.00096}$ |  $1.0439^{+0.0021}_{-0.0016}$ |  $1.04338^{+0.00074}_{-0.00063}$
$3.026^{+0.022}_{-0.011}$ |  $3.019^{+0.026}_{-0.022}$ |  $3.025^{+0.014}_{-0.016}$
$0.949^{+0.011}_{-0.0066}$ |  $0.946\pm 0.010$ |  $0.9483^{+0.0070}_{-0.0079}$
$0.0533\pm 0.0058$ |  $0.0532\pm 0.0057$ |  $0.0531\pm 0.0057$
$-4.06^{+0.15}_{-0.89}$ |  $-3.3\pm 1.2$ |  $-2.9^{+1.6}_{-2.0}$
$0.970\pm 0.015$ |  $0.954\pm 0.024$ |  $0.882^{+0.070}_{-0.048}$
$-4.09^{+0.15}_{-0.90}$ |  $-3.3\pm 1.2$ |  $-3.1^{+1.6}_{-2.0}$
$65.7^{+1.2}_{-1.4}$ |  $65.9\pm 1.4$ |  $65.4\pm 1.3$
$0.8139\pm 0.0094$ |  $0.8144\pm 0.0095$ |  $0.8113\pm 0.0093$
$-0.35\pm 0.17$ |  $-0.34\pm 0.17$ |  $-0.38^{+0.16}_{-0.18}$
$-0.06$ |  $-2.08$ |  $-2.26$
$1.94$ |  $-0.08$ |  $-0.26$

Table 7: Mean values and 68% confidence limits for SINU with additional self-
interacting neutrinos. Note that the values and limits are for the full sample
where we have not separated out the MI and SI modes. In addition to the six
$\Lambda$CDM parameters here we have varied $\log_{10}G_{\mathrm{eff}}$ and
$\xi_{\nu}$ for the coupled neutrino flavor. Here $\Delta N_{c}\equiv N_{\rm
eff}-3.046=\Delta N_{\rm eff}$. We also show the $\chi^{2}$ difference from
the bestfit 7-parameter $\Lambda$CDM model where $N_{\rm eff}$ is also varied
(denoted as $\overline{\Lambda\rm CDM}$). The corresponding $\Delta$AIC is
also shown. ${T_{{\rm c},1\nu}}/{T_{{\rm f},1\nu}}$ in the table is the ratio
of coupled to the free streaming temperature (SM neutrino temperature) and is
proportional to $\xi_{\nu}$ for the coupled flavor. For all datasets ${T_{{\rm
c},1\nu}}/{T_{{\rm f},1\nu}}<1$ (equivalently $\Delta N_{c}<0$) and thus the
mean values of $\log_{10}[{\rm\tilde{G}_{eff}}/{\rm MeV}^{-2}]$ are always
lower than the corresponding $\log_{10}[{\rm G_{eff}}/{\rm MeV}^{-2}]$
following Eq. (3.10).

This discussion is also relevant for comparison with existing results in the
literature for the varying $N_{\mathrm{eff}}$ (coupled) scenarios for flavor
universal coupling. Since those analyses usually do not vary the neutrino
temperature, but rather the number of states directly, the $G_{\mathrm{eff}}$
does not scale in those cases. Therefore, $G_{\mathrm{eff}}$ in those analyses
(where $N_{\mathrm{eff}}$ is varied directly), should be compared with
$\tilde{G}_{\mathrm{eff}}$ in our analysis. Note that, for a _physical_ model
where $N_{\mathrm{eff}}$ is varied through the temperature of the species, the
scaling in Eq.(3.10) should be properly taken into account. For the analysis
in this section, we have studied the temperature-varying coupled neutrinos as
$\mathbf{(3+\Delta N_{\rm c})c+0f}$, $\mathbf{(2+\Delta N_{\rm c})c+1f}$ and
$\mathbf{(1+\Delta N_{\rm c})c+2f}$ cases. In each of the scenarios, only the
temperatures of the corresponding _coupled_ species were varied.

In Fig. 10 we show the constraints on some relevant cosmological parameters
along with $\log_{10}\tilde{G}_{\mathrm{eff}}$. It has been shown that the ACT
data prefers a smaller value of $N_{\rm eff}$ in general [1]. This also can be
seen from the $N_{\mathrm{eff}}$ plots. This happens due to the strong
degeneracy with $n_{s}$ which arise from the lack of large-scale structure
data. In general, Planck and ACT both prefer a smaller value of
$N_{\mathrm{eff}}$ compared to the standard value of 3.046. Therefore, due to
the less amount of silk damping the preferred value of $n_{s}$ are smaller
compared to the baseline models. The decrease in $N_{\mathrm{eff}}$ also
results in a smaller value of the $H_{0}$ worsening the $H_{0}$ tension in
this case. The central values for SI mode in $G_{\mathrm{eff}}$ moved slightly
to the smaller values and increased in significance for all datasets. The
shift in SI mode $G_{\mathrm{eff}}$ can be understood through the additional
degeneracy with $N_{\mathrm{eff}}$. In this scenario, the changes induced by
$n_{s}$ are compensated by both $N_{\mathrm{eff}}$ and $G_{\mathrm{eff}}$ (or
$\tilde{G}_{\mathrm{eff}}$). Thus, the additional freedom of varying
$N_{\mathrm{eff}}$ limits the values allowed for $\tilde{G}_{\mathrm{eff}}$
and pushes the peak to a slightly smaller value. Note that this freedom is
also present in the free-streaming $\Delta N_{\mathrm{eff}}$ result in the
earlier section. However, in that case, because only positive $\Delta
N_{\mathrm{eff}}$ is allowed, the role of $\Delta N_{\mathrm{eff}}$ to
compensate for the changes in the CMB spectrum is quite limited. Therefore,
the SI mode values of $G_{\mathrm{eff}}$ are virtually unmodified from the
baseline model in that case. This constrained fit is also reflected in the
$\chi^{2}$ difference from $\Lambda$CDM in Table 6 and 7 where we see that
coupled $N_{\mathrm{eff}}$ gives a better fit of the data compared to free-
streaming $N_{\mathrm{eff}}$ for each of the respective scenarios.

##  4 Prospects of future CMB experiments

As demonstrated in the previous section, the main constraining power of
current CMB experiments on the neutrino self-interaction comes from the
intermediate $500\lesssim\ell\lesssim 1500$ range of the multipoles. Improving
the constraint on $G_{\mathrm{eff}}$ would require more precise measurements
of the CMB spectra, especially the EE spectrum, to determine the amplitude and
the position of the peaks more accurately. Even though neutrino self-
interaction is not a part of the primary science goals of any of the upcoming
CMB experiments, instrument requirements, and survey strategies for other
goals such as primordial gravitational wave and light relics could benefit the
search for signatures of SINU. CMB-S4 plans to constrain $\Delta
N_{\mathrm{eff}}<0.06$ at 95% confidence level by conducting a high-resolution
survey, and will also perform an ultra-low noise survey aimed for primordial
gravitational wave [6]. Both CMB-S4 and CORE are aiming for
$\sim\mathcal{O}$(few) $\,\mu\mathrm{K\text{-}arcmin}$ sensitivity in the mid-
frequency range. Such strategies, if realized, will certainly improve the
constraint on $G_{\mathrm{eff}}$. In this section, we use the planned design
specifications of CMB-S4 and CORE to forecast future constraints on
$G_{\mathrm{eff}}$. We also show the limit for a cosmic-variance limited
experiment.

Figure 11: The distribution of Fisher information for
$\log_{10}G_{\mathrm{eff}}$ over the multipoles $\ell$ for CMB-S4, CORE, and
cosmic variance limited experiments. The instrumental noise dominates over
cosmic variance for $\ell\gtrsim 2000$ which shows that a large portion of the
information from the high $\ell$ tail of the CMB anisotropy power spectra
would still be unavailable in upcoming experiments.

###  4.1 Fisher forecast

We use the Fisher formalism to forecast limits using the same set of seven
parameters used in the previous section:
$\\{H_{0},\omega_{b},\omega_{c},A_{s},n_{s},\tau_{\mathrm{reio}},\log_{10}G_{\mathrm{eff}}\\}$.
The Fisher matrix can be written as [55, 56]

$F_{ij}=\sum_{\ell}\dfrac{2\ell+1}{2}~{}f_{\mathrm{sky}}~{}\mathrm{Tr}\left(\mathbf{C}_{\ell}^{-1}\dfrac{\partial\mathbf{C}_{\ell}}{\partial\lambda_{i}}\mathbf{C}_{\ell}^{-1}\dfrac{\partial\mathbf{C}_{\ell}}{\partial\lambda_{j}}\right)\,,$
(4.1)

where $i,j$ are the parameter indices, $\lambda_{i}$ are the parameters,
$f_{\mathrm{sky}}$ is the fraction of the sky observed by the experiment. The
covariance matrix $\mathbf{C}_{\ell}$ is defined as

$\mathbf{C}_{\ell}=\begin{bmatrix}C_{\ell}^{TT}&C_{\ell}^{TE}\\\
C_{\ell}^{TE}&C_{\ell}^{EE}\end{bmatrix}+\mathbf{N}_{\ell}\,,$ (4.2)

with a diagonal noise covariance matrix
$\mathbf{N}_{\ell}=\mathrm{diag}(N_{\ell}^{TT},N_{\ell}^{EE})$. We do not
include the lensing data. The noise can be taken to be [55]

$N_{\ell}^{TT}=\Delta_{T}^{2}\exp\left(\ell(\ell+1)\dfrac{\theta_{\mathrm{FWHM}}^{2}}{8\ln
2}\right)\,,$ (4.3)

where $\Delta_{T}$ is the instrumental noise in $\mu$K-radian and
$\theta_{\mathrm{FWHM}}$ is the beam width. The polarization noise is
$\Delta_{P}=\sqrt{2}\Delta_{T}$. CMB experiments observe in multiple frequency
channels with the corresponding beam size and sensitivity depending on the
frequency. We use the publicly available design parameters for CMB-S4 and CORE
[57, 7].

Figure 12: Fisher forecast for a few cosmological parameters for CMB-S4
(green), CORE (red), and cosmic variance limited (orange) experiments. Planck
2018 TTTEEE+lensing posteriors are shown in gray for comparison.

Finally the diagonal elements of the inverse Fisher matrix $F^{-1}_{ii}$ give
the marginalized variance of the parameters. Therefore, the parameter
uncertainty is given by

$\sigma_{i}=\sqrt{F^{-1}_{ii}}\,.$ (4.4)

We choose the fiducial model as the bestfit cosmology from Ref. [22] for the
SI mode in $\mathbf{3c+0f}$ case with Planck 2018 TTTEEE+lowE data. Here we
note that the SI mode posterior in SINU cosmology is not a perfect Gaussian.
It is contiguous with the MI mode at smaller $G_{\mathrm{eff}}$. However, for
the purpose of forecasting the upper limit, we expect the Fisher formalism to
be applicable as long as we compute the derivatives within a small enough
region around the fiducial point where the posterior can be taken to be
approximately Gaussian. This is a reasonable assumption given the shapes of
the SI mode in Figs. 2 and 3. We ignore the left side of the
$\log_{10}G_{\mathrm{eff}}$ posterior. While computing the derivatives in
Eq.(4.1), we checked the convergence with decreasing value of the step size.
We found taking a step size of $1\%$ of the fiducial value is sufficient for
all parameters. We use $\ell_{\mathrm{max}}=4000$ in all forecasts.

First, we investigate the constraining power of different multipoles. The
diagonal elements of the terms on the right-hand side of Eq.(4.1) contain this
information. In Fig. 11, we plot the diagonal term corresponding to
$\log_{10}G_{\mathrm{eff}}$ as a function of $\ell$ for CMB-S4, CORE, and
cosmic variance limited experiments. Looking at the last case, when the
instrumental noise is negligible, most of the constraining power comes from
the high-$\ell$ tail in the range $2000\lesssim\ell\lesssim 4000$ with a
substantial part coming from the intermediate multipoles $\ell\lesssim 2000$.
However, when instrumental noise is taken into account, the contribution from
the high-$\ell$ region goes down significantly as $\mathbf{N}_{\ell}$
dominates at large $\ell$. Due to its better angular resolution, CMB-S4 can
still access the multipoles above 2000 up to $\ell\lesssim 2800$. In the case
of CORE, the range is limited to $\ell\lesssim 1900$.

When we look at the data from the current experiments (cf. Fig. 5 and 6), the
decisive multipole range for SINU is $500\lesssim\ell\lesssim 1200$. It was
also pointed out in Ref.[58]. Such is the case because the current experiments
do not have enough precision at small angular scales. However, this should be
contrasted with the forecast for future CMB experiments discussed here which
suggests that more information from the CMB spectra can be acquired from
higher $\ell\gtrsim 2000$ multipoles if small enough instrumental noise is
achieved.

Fig. 12 shows the 1$\sigma$ error ellipses for a few parameters. From the
panel of $\log_{10}G_{\mathrm{eff}}$, we see that CMB-S4 is expected to yield
about three times better $1\sigma$ (68% confidence level) upper limit compared
to the latest and the final data from the Planck experiment. We quote the
future sensitivities of $\log_{10}G_{\mathrm{eff}}$ in Table 8. We note that
our results for CMB-S4 matches with the previous work [59]. The $1\sigma$
errorbar for CORE is understandably bigger because of its lesser sensitivity.
Finally, a cosmic variance-limited experiment with negligible instrumental
noise would perform even better as expected. The effectiveness of the future
experiments can also be measured by defining a meaningful Figure-of-merit
(FoM). It has been defined in multiple ways in the literature. However, we
follow the simple definition of FoM as the inverse of the volume of the
7-dimensional hyperellipsoid defined by the inverse Fisher matrix, FoM
$=\sqrt{\mathrm{det}(F)}$ [60, 61]. These FoMs are given in Table 8.

Experiment |  Sensitivity for $\log_{10}[G_{\mathrm{eff}}/\mathrm{MeV^{-2}}]$ |  Figure-of-merit (FoM) |  FoM relative to CV-limited
---|---|---|---
CMB-S4 |  $<-1.87$ |  $1.1\times 10^{28}$ |  $1.8\times 10^{-3}$
CORE |  $<-1.84$ |  $3.9\times 10^{26}$ |  $6.4\times 10^{-5}$
CV-limited |  $<-1.9$ |  $6.1\times 10^{30}$ |  1
Table 8: The future sensitivity of $\log_{10}G_{\mathrm{eff}}$ and overall
figure-of-merit of the upcoming CMB experiments, and comparison with a cosmic
variance-limited experiment. Note that the sensitivities are based on the SI
mode bestfit point of Planck 2018 TTTEEE+lowE data.

##  5 Discussion & Outlook

We revisit the study of the cosmology of self-interacting neutrinos using the
new CMB data from the Atacama Cosmology Telescope (ACT) experiment in
combination with the latest 2018 data from the Planck experiment. Prior
cosmological data was known to yield two distinct modes in the neutrino
interaction strength ($G_{\mathrm{eff}}$) – the moderately interacting (MI)
mode with a weaker coupling that merges with the $\Lambda$CDM cosmology, and a
strongly interacting (SI) mode, in which case, neutrinos decouples just prior
to the matter-radiation equality epoch. However, the 2018 Planck polarization
data strongly disfavored the SI mode [22].

In this work, on the other hand, we find that the new ACT data favors SI over
the MI mode irrespective of the number of coupled neutrino species. The reason
behind this can be assigned to a certain feature in the ACT polarization data
in the $500\lesssim\ell\lesssim 1200$ multipole range that is not present in
the Planck 2018 data. This conclusion corroborates with the findings of
another work in Ref. [29] in the flavor-universal case. Such a mismatch
between the two datasets could be resolved with upcoming experiments that will
measure the polarization data with more precision. We do not find a
significant change in the value of the Hubble parameter. However, the combined
ACT and Planck data prefer the SI mode over $\Lambda$CDM for flavor-specific
interaction.

We also studied the effect of adding neutrino mass to this analysis. We found
slightly smaller values of the Hubble parameter $H_{0}$ and the dark energy
density $\Omega_{\Lambda}$ as massive neutrinos change the total matter
density of the Universe for all experiments. These changes are greater for
Planck due to its large-scale data. There is no other qualitative difference
between the massless and massive neutrino results proving the robustness of
our baseline massless neutrino model discussed in Sec. 3.2.

Adding more free streaming radiation in the early Universe has the opposite
effect of neutrino self-coupling on the CMB power spectra. It increases
bestfit values of $H_{0}$ and $n_{s}$ but does not change the position of the
SI mode in $\log_{10}G_{\mathrm{eff}}$. We note that none of these extended
models yield a significantly large value of $H_{0}$ or change the position of
the SI mode in $G_{\mathrm{eff}}$. On the other hand, when we vary the $\Delta
N_{\mathrm{eff}}$ for the coupled neutrinos, we find that $H_{0}$ and $n_{s}$
get slightly smaller relative to the fixed $N_{\mathrm{eff}}$ case. These
changes are explained by the net decrease in $N_{\rm eff}$.

Finally, we did a Fisher analysis of the future prospect of a few
upcoming/planned CMB experiments. We chose CMB-S4 and CORE as two benchmark
terrestrial and space-based experiments, and use their publicly available data
and technical design parameters. The arcminute level angular resolution of
CMB-S4 will significantly improve the sensitivity at smaller angular scales.
It will help constrain or detect signatures of neutrino self-interaction in
the large multipole ($\ell\gtrsim 2000$) tail of the CMB spectrum. The
forecast sensitivity of CMB-S4 is roughly twice as good as CORE for
$\ell_{\mathrm{max}}=4000$. This is due to the relatively lesser sensitivity
of CORE at the small scale. However, we note that although CMB-S4 is an
approved experiment and it is currently in the design phase, CORE is yet to be
approved and its instrument requirements are subject to change in the future.
Nonetheless, the complementarity of the science scopes of ground and space-
based future experiments is expected to be crucial and will be key to
continuing the legacy of CMB observation in the coming decades [62].

## Acknowledgment

The work of A.D. was supported by the U.S. Department of Energy under contract
number DE-AC02-76SF00515 and Grant Korea NRF-2019R1C1C1010050. S.G. is
supported by the U.S. National Science Foundation grant PHY2112540. The
computational part of the work is supported by the Fermi National Accelerator
Laboratory, managed and operated by Fermi Research Alliance, LLC under
Contract No. DE-AC02-07CH11359 with the U.S. Department of Energy, Office of
Science, Office of High Energy Physics. A.D. gratefully acknowledges the
hospitality at APCTP during the completion of this work.

Numerical computation |
---|---
Total core hour [$\mathrm{h}$] | 359,201
Thermal Design Power per core [$\mathrm{W}$] [63] | 11.88
Total energy consumption for computations [$\mathrm{kWh}$] | 4266
Average emission of CO2 in Illinois, US [$\mathrm{kg/kWh}$] [64] | 0.3
Total CO2-emission for numerical computations [$\mathrm{kg}$] | 1280
Were the emissions offset? | No
Travel |
Total CO2-emission for travel [$\mathrm{kg}$] | 0
Total CO2-emission [$\mathrm{kg}$] | 1280
Table 9: Approximate estimate of the total CO2 emission during the course of
this work.

## Carbon footprint estimate

As this work involved intensive numerical computation, in Table 9, we give an
estimate of the breakdown and total CO2 emission from the activities related
to it [65]. The total emission is approximately 1280 kg, equivalent to about 5
hours of air travel.

##  Appendix A Temperature scaling of $G_{\mathrm{eff}}$

Figure 13: 1D plots of cosmological parameters with self-interacting neutrino
species with $N_{c}=3$ and $N_{c}=1$ respectively where the temperature is
allowed to vary. The constraints on $\log_{10}\tilde{G}_{\mathrm{eff}}$ and
other $\Lambda$CDM cosmological parameters are identical in both cases.

In this section, we perform mcmc analyses of $\mathbf{3c+0f}$ scenarios with
varying temperatures for the Planck dataset. We choose two different methods
which outline the dependence of $\log_{10}G_{\mathrm{eff}}$ on temperature. In
the first case, we choose the number of coupled flavors $N_{c}=3$, and in the
latter scenario, we choose $N_{c}=1$. Fig. 13 shows the 1D posterior
distributions in these cases. The $N_{c}=1$ expectedly prefers a higher
neutrino temperature $\xi_{\nu}$. As expected the constraint on
$\log_{10}G_{\mathrm{eff}}$ is weaker in this scenario which follows from the
arguments in sec. 3. However, the scaled coupling constant
$\log_{10}\tilde{G}_{\mathrm{eff}}$, as defined in Eq.(3.10), is identical in
both cases. Moreover, the other $\Lambda$CDM cosmological parameter values are
also identical in both cases. This shows that varying the temperature of
neutrinos to account for varying $N_{\mathrm{eff}}$ produces identical
constraints on $\Lambda$CDM parameters as varying $N_{\mathrm{eff}}$ directly.
However, the constraints on $\log_{10}G_{\mathrm{eff}}$ scale according to the
temperature. We also show constraints on a derived parameter ${T_{{\rm
c},1\nu}}/{T_{{\rm f},1\nu}}$ which is the ratio of coupled vs free streaming
_one-neutrino_ temperature. $T_{{\rm c},1\nu}$ is defined as $T_{{\rm
c},1\nu}\equiv T_{{\rm c}}/N_{c}^{1/4}$ and ${T_{{\rm f},1\nu}}$ is the free
streaming neutrino temperature in SM.

##  Appendix B Parameter value for massive SINU

$\mathbf{2c+1f}$ | ACTDR4+tauprior | Planck-TTTEEE+lowE+lensing |  Planck-TTTEEE+lowE+lensing +ACTDR4
---|---|---|---
Parameters |  Massless $\nu$ |  Massive $\nu$ |  Massless $\nu$ |  Massive $\nu$ |  Massless $\nu$ |  Massive $\nu$
$10^{2}\omega_{\rm b}$ |  $2.154\pm 0.031$ |  $2.153\pm 0.030$ |  $2.237^{+0.014}_{-0.016}$ |  $2.234\pm 0.015$ |  $2.236\pm 0.013$ |  $2.234\pm 0.013$
$\omega_{\rm cdm}$ |  $0.1202\pm 0.0038$ |  $0.1206\pm 0.0039$ |  $0.1200\pm 0.0012$ |  $0.1204\pm 0.0012$ |  $0.1203\pm 0.0012$ |  $0.1207\pm 0.0012$
$100\theta{}_{s}$ |  $1.0462^{+0.0018}_{-0.00039}$ |  $1.0461^{+0.0019}_{-0.00039}$ |  $1.042230^{+0.000020}_{-0.00070}$ |  $1.042197^{+0.000036}_{-0.00068}$ |  $1.0440^{+0.0020}_{-0.0022}$ |  $1.0441^{+0.0017}_{-0.0021}$
$\ln 10^{10}A_{s}$ |  $3.007^{+0.025}_{-0.032}$ |  $3.009^{+0.025}_{-0.033}$ |  $3.037^{+0.020}_{-0.013}$ |  $3.040^{+0.020}_{-0.013}$ |  $3.025\pm 0.025$ |  $3.026\pm 0.025$
$n_{s}$ |  $0.979\pm 0.018$ |  $0.978\pm 0.018$ |  $0.9611^{+0.0078}_{-0.0034}$ |  $0.9602^{+0.0077}_{-0.0035}$ |  $0.955\pm 0.010$ |  $0.953^{+0.015}_{-0.013}$
$\tau_{\rm reio}$ |  $0.0595\pm 0.0098$ |  $0.0597\pm 0.0097$ |  $0.0536\pm 0.0073$ |  $0.0544\pm 0.0073$ |  $0.0546\pm 0.0058$ |  $0.0551\pm 0.0059$
$\log_{10}[{\rm G_{eff}}/{\rm MeV}^{-2}]$ |  $-2.04^{+0.66}_{+0.22}$ |  $-2.07^{+0.69}_{+0.23}$ |  $<-3.82$ |  $<-3.84$ |  $-2.7\pm 1.2$ |  $-2.6^{+1.1}_{-1.3}$
$H_{0}({\rm km/s/Mpc})$ |  $68.5\pm 1.5$ |  $67.8\pm 1.5$ |  $68.02^{+0.55}_{-0.65}$ |  $67.33^{+0.52}_{-0.69}$ |  $68.50\pm 0.64$ |  $67.83\pm 0.64$
$\sigma_{8}$ |  $0.838\pm 0.016$ |  $0.826\pm 0.016$ |  $0.8240^{+0.0057}_{-0.0067}$ |  $0.8129\pm 0.0062$ |  $0.8290\pm 0.0059$ |  $0.8179\pm 0.0058$
$\chi^{2}-\chi^{2}_{\Lambda{\rm CDM}}$ |  $-6.92$ |  $-6.99$ |  $0.17$ |  $1.26$ |  $-5.31$ |  $-4.9$
$\Delta{\rm AIC}$ |  $-4.92$ |  $-4.99$ |  $2.17$ |  $3.26$ |  $-3.31$ |  $-2.9$

$\mathbf{1c+2f}$ | ACTDR4+tauprior | Planck-TTTEEE+lowE+lensing |  Planck-TTTEEE+lowE+lensing +ACTDR4
---|---|---|---
Parameters |  Massless $\nu$ |  Massive $\nu$ |  Massless $\nu$ |  Massive $\nu$ |  Massless $\nu$ |  Massive $\nu$
$10^{2}\omega_{\rm b}$ |  $2.155\pm 0.030$ |  $2.154\pm 0.030$ |  $2.237\pm 0.014$ |  $2.234\pm 0.015$ |  $2.235\pm 0.013$ |  $2.233\pm 0.013$
$\omega_{\rm cdm}$ |  $0.1197\pm 0.0037$ |  $0.1200\pm 0.0037$ |  $0.1200\pm 0.0012$ |  $0.1204\pm 0.0012$ |  $0.1200\pm 0.0011$ |  $0.1205\pm 0.0011$
$100\theta{}_{s}$ |  $1.0443\pm 0.0011$ |  $1.0443\pm 0.0011$ |  $1.0423^{+0.0012}_{-0.00091}$ |  $1.04228^{+0.00015}_{-0.00091}$ |  $1.0432^{+0.0010}_{-0.0013}$ |  $1.0432^{+0.0010}_{-0.0013}$
$\ln 10^{10}A_{s}$ |  $3.031\pm 0.025$ |  $3.032^{+0.023}_{-0.026}$ |  $3.036^{+0.017}_{-0.016}$ |  $3.039^{+0.017}_{-0.015}$ |  $3.036^{+0.014}_{-0.019}$ |  $3.038^{+0.013}_{-0.019}$
$n_{s}$ |  $0.994^{+0.017}_{-0.013}$ |  $0.993^{+0.016}_{-0.013}$ |  $0.9610^{+0.0065}_{-0.0051}$ |  $0.9602^{+0.0066}_{-0.0048}$ |  $0.9601^{+0.0052}_{-0.0072}$ |  $0.9590^{+0.0056}_{-0.0072}$
$\tau_{\rm reio}$ |  $0.0598\pm 0.0097$ |  $0.0601\pm 0.0096$ |  $0.0536\pm 0.0073$ |  $0.0545\pm 0.0073$ |  $0.0551\pm 0.0058$ |  $0.0555\pm 0.0058$
$\log_{10}[{\rm G_{eff}}/{\rm MeV}^{-2}]$ |  $-2.5\pm 1.1$ |  $-2.4\pm 1.1$ |  $<-2.84$ |  $<-2.92$ |  $-2.4^{+1.1}_{-1.3}$ |  $-2.4^{+1.1}_{-1.3}$
$H_{0}({\rm km/s/Mpc})$ |  $68.1\pm 1.5$ |  $67.5\pm 1.5$ |  $68.04\pm 0.58$ |  $67.35\pm 0.57$ |  $68.32^{+0.55}_{-0.49}$ |  $67.63\pm 0.52$
$\sigma_{8}$ |  $0.838\pm 0.015$ |  $0.826^{+0.014}_{-0.016}$ |  $0.8235\pm 0.0062$ |  $0.8124\pm 0.0061$ |  $0.8275\pm 0.0059$ |  $0.8162\pm 0.0057$
$\chi^{2}-\chi^{2}_{\Lambda{\rm CDM}}$ |  $-4.08$ |  $-4.04$ |  $0.13$ |  $1.22$ |  $-5.16$ |  $-4.79$
$\Delta{\rm AIC}$ |  $-2.08$ |  $-2.04$ |  $2.13$ |  $3.22$ |  $-3.16$ |  $-2.79$

Table 10: Mean values and 68% confidence limits for massless-$\nu$ and
massive-$\nu$ cases for $\mathbf{2c+1f}$ and $\mathbf{1c+2f}$. Note that the
values and limits are for the full sample where we have not separated out the
MI and SI modes. We also show the $\chi^{2}$ difference from the bestfit
6-parameter $\Lambda$CDM model and the corresponding $\Delta$AIC

##  Appendix C Triangle plots for ACT and Planck+ACT analysis

Figure 14: Triangle plot for an extended set of cosmological parameters for
the baseline model analysis with ACT dataset. Figure 15: Triangle plot for an
extended set of cosmological parameters for the baseline model analysis with
Planck + ACT dataset. Figure 16: Triangle plot for an extended set of
cosmological parameters for the analysis with massive neutrino for Planck +
ACT dataset. Figure 17: Triangle plot for an extended set of cosmological
parameters for additional free-streaming $\Delta N_{\rm eff}$ analysis with
Planck + ACT dataset. Figure 18: Triangle plot for an extended set of
cosmological parameters for additional coupled $\Delta N_{\rm eff}$ analysis
with Planck + ACT dataset.

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