Search for High-energy Neutrinos from Ultraluminous Infrared Galaxies with IceCube

The ULIRGs for this analysis are obtained from three different catalogs, primarily based on data from the Infrared Astronomical Satellite (IRAS; Neugebauer et al. 1984), listed below. The infrared luminosity L_IR listed in the catalogs is determined using the IRAS flux measurements at 12, 25, 60, and 100 μm (see e.g., Sanders & Mirabel 1996). The three catalogs are:

• 1.  
  The IRAS Revised Bright Galaxy Sample (RBGS; Sanders et al. 2003). This catalog contains the brightest extragalactic sources observed by IRAS. These RBGS objects are sources with a 60 μm infrared flux f₆₀ > 5.24 Jy and a Galactic latitude $\left|b\right|\gt 5^\circ$ to exclude the Galactic Plane. The RBGS provides the total infrared luminosity between 8 and 1000 μm for all objects in the sample, containing 21 ULIRGs.
• 2.  
  The IRAS 1 Jy Survey of ULIRGs (Kim & Sanders 1998) selected from the IRAS Faint Source Catalog (FSC; Moshir et al. 1992) contains sources with f₆₀ > 1 Jy. The survey required the ULIRGs to have a Galactic latitude $\left|b\right|\gt 30^\circ$ to avoid strong contamination from the Galactic Plane. Furthermore, in order to have accessible redshift information from observatories located at Maunakea, Hawaii, this survey is restricted to declinations δ > − 40°. The resulting selection is a set of 118 ULIRGs.
• 3.  
  The ULIRG sample used by Nardini et al. (2010). Their selection is primarily based on the redshift survey (Saunders et al. 2000) of the IRAS Point Source Catalog (PSC; Beichman et al. 1988). This catalog contains objects with f₆₀ ≳ 0.6 Jy and covers 84% of the sky. In addition, the authors require the ULIRGs to be observed by the Infrared Spectograph (IRS; Houck et al. 2004) on the Spitzer Space Telescope (Werner et al. 2004). As such, they obtain a sample of 164 ULIRGs.

There exists some overlap between the ULIRGs of the above three catalogs. Therefore, the NASA/IPAC Extragalactic Database ⁶⁴ (NED) is used to cross-identify these sources. This cross-identification was done by obtaining the NED source list of each catalog, and then cross-identifying sources with the same NED information in each list. The result is a selection of 189 unique ULIRGs. For uniformity, NED is also used to obtain the equatorial coordinates and redshift for each object. Since L_IR values depend on IRAS flux measurements, which are optimized separately for the different IRAS surveys, they are taken in the following order:

• 1.  
  From Catalog 1 if available;
• 2.  
  From Catalog 2 if not available in Catalog 1;
• 3.  
  From Catalog 3 if not available in Catalog 1 or 2.

Note that Catalog 3 is the only catalog that provides uncertainties on these values. We therefore include all objects from this catalog that are consistent with L_IR = 10¹² L_⊙ within one standard deviation. ⁶⁵

To obtain a representative sample of the local ULIRG population, which we define as a sample of ULIRGs that is complete up to a given redshift, we place a cut on the redshifts in our initial ULIRG selection. We find that keeping only those objects with z ≤ 0.13 allows the least luminous ULIRGs (i.e., L_IR = 10¹² L_⊙) to be observed, given a conservative IRAS sensitivity f₆₀ = 1 Jy (see Appendix A, where we also take into account the effect of the limited sky coverage of Catalog 2). The result is a representative sample of 75 ULIRGs with z ≤ 0.13, shown in Figure 1. A listing of these 75 selected ULIRGs is provided in Appendix A.

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The IceCube Neutrino Observatory is a 1 km³ optical Cerenkov detector located at the geographic South Pole (Aartsen et al. 2017b). The detector is buried in the ice at depths between 1450 and 2450 m below the surface. It consists of 5160 digital optical modules (DOMs), which are distributed over 86 vertical strings. Each DOM contains a photomultiplier tube and on-board read-out electronics (Abbasi et al. 2009, 2010). When a high-energy neutrino interacts with the ice or the bedrock in the vicinity of the detector, secondary relativistic particles are produced that emit Cerenkov radiation, which can be registered by the DOMs. The number of photons collected by the DOMs gives a measure of the energy deposited by these secondary charged particles. This feature combined with the arrival time of the light at the DOMs is used to reconstruct the particle trajectories, which are subsequently used to determine the arrival direction of the original neutrino.

IceCube is sensitive to all neutrino flavors, although neutrinos and antineutrinos can generally not be distinguished. Neutral-current interactions of the three flavors, and charged-current interactions of electron neutrinos (${\nu }_{e}+{\bar{\nu }}_{e}$) and tau neutrinos (${\nu }_{\tau }+{\bar{\nu }}_{\tau }$) are observed by means of their particle cascades. These cascades lead to a rather spherical pattern of recorded photons inside the detector, with a relatively poor angular resolution (≳8°; see Aartsen et al. 2019c). However, charged-current interactions of muon neutrinos (${\nu }_{\mu }+{\bar{\nu }}_{\mu }$) produce muons, which can leave extended track-like signatures in the detector. ⁶⁶ The typical angular resolution of these tracks is ≲1° for muons with energies ≳ 1 TeV (Aartsen et al. 2017c, 2020a).

For this analysis we use the IceCube gamma-ray follow-up (GFU) data sample (Aartsen et al. 2017c), which contains well-reconstructed muon tracks with 4π sky coverage. The track selection is performed separately for both hemispheres due to their differing background characteristics. The background for astrophysical neutrinos originates from cosmic-ray-induced particle cascades in Earth's atmosphere. Atmospheric neutrinos produced in such cascades are able to reach the detector from both hemispheres, leading to a background event rate at the mHz level. Compared to astrophysical neutrinos, atmospheric neutrinos have a relatively soft spectrum that dominates at energies ≲100 TeV (Abbasi et al. 2011, 2020a; Aartsen et al. 2015, 2016, 2020b). Atmospheric muons produced in the Northern hemisphere are attenuated by Earth before reaching IceCube. In the Southern hemisphere, however, they are able to penetrate the Antarctic ice and leave track signatures in the detector. These atmospheric muons have a median event rate of 2.7 kHz, requiring a more stringent event selection in the Southern sky. The event selection criteria result in a GFU track sample at the atmospheric neutrino level, with an all-sky event rate of 6.6 mHz. Note that this is still various orders of magnitude above the expected rate of astrophysical neutrinos at the μHz level. The selection efficiency of the GFU sample is determined by simulating signal neutrinos according to an unbroken power-law spectrum ${{\rm{\Phi }}}_{\nu +\bar{\nu }}\propto {E}_{\nu }^{-2}$ between 100 GeV < E_ν < 1 EeV. In the Northern hemisphere, the selection efficiency is ∼50% at E_ν ∼ 100 GeV and reaches ∼95% for E_ν ≳ 100 TeV. In the Southern hemisphere, the selection efficiency is ∼5% at E_ν ∼ 10 TeV and exceeds 70% for E_ν ≳ 1 PeV. The GFU data used in this search contains over 1.5 × 10⁶ events, spanning a livetime of 2615.97 days ∼7.5 yr of the full 86-string IceCube configuration between 2011 and 2018.

In this work, we search for an astrophysical component in the IceCube data that is spatially correlated with our representative sample of 75 ULIRGs. This astrophysical signal would be characterized by an excess of neutrino events above the background of atmospheric muons and atmospheric neutrinos. To search for this excess, a time-integrated unbinned maximum likelihood analysis is performed (Braun et al. 2008). In addition, the ULIRGs are stacked in order to enhance the sensitivity of the analysis (see Achterberg et al. 2006).

The unbinned likelihood for a search stacking M candidate point sources using N data events is given by

$$\begin{eqnarray}&&{ \mathcal L }({n}_{s},\gamma )=\prod _{i=1}^{N}\left[\displaystyle \frac{{n}_{s}}{N}\sum _{k=1}^{M}{w}_{k}{{ \mathcal S }}_{i}^{k}(\gamma )+\left(1-\displaystyle \frac{{n}_{s}}{N}\right){{ \mathcal B }}_{i}\right].\end{eqnarray} \tag{ 1 }$$

Here, n_s denotes the number of signal events, and γ denotes the spectral index of the signal energy distribution, which is assumed to follow an unbroken power-law spectrum, ${{\rm{\Phi }}}_{\nu +\bar{\nu }}\propto {E}^{-\gamma }$. These fit parameters are restricted to n_s ≥ 0 and γ ∈ [1, 4] but allowed to float otherwise. For each event i, the probability density functions (PDFs) of the background, ${{ \mathcal B }}_{i}$, and the signal for source k, ${{ \mathcal S }}_{i}^{k}$, are evaluated. ⁶⁷ The parameter w_k represents the stacking weight of source k.

Both the signal PDF and background PDF are separated into spatial and energy components. The background PDF is evaluated as ${{ \mathcal B }}_{i}=B({\delta }_{i})/2\pi \times {{ \mathcal E }}_{{ \mathcal B }}({E}_{i})$, with δ_i and E_i the reconstructed decl. and energy of event i, respectively. The spatial component is uniform in R.A. due to the rotation of the IceCube detector with Earth's axis, resulting in a factor 1/2π. The spatial PDF, B, and the energy PDF, ${{ \mathcal E }}_{{ \mathcal B }}$, are constructed from experimentally obtained background data in reconstructed decl. and energy. Analogously, the signal PDF of candidate point source k is evaluated as ${{ \mathcal S }}_{i}^{k}(\gamma )=S({{\boldsymbol{x}}}_{k},{{\boldsymbol{x}}}_{i},{\sigma }_{i})\times {{ \mathcal E }}_{{ \mathcal S }}({E}_{i},\gamma )$. For the spatial PDF of the signal, S, we take a two-dimensional Gaussian centered around the location of the source, x _k . The Gaussian is evaluated using the reconstructed location of event i, x _i , and its angular uncertainty, σ_i , as the standard deviation. The estimation of the angular uncertainty does not take into account systematic effects, such that we follow Aartsen et al. (2020a) and impose a lower bound σ_i ≥ 02. Since σ_i is much larger than the uncertainty on the source location, the latter may safely be neglected. The energy PDF of the signal, ${{ \mathcal E }}_{{ \mathcal S }}$, is modeled using a power-law spectrum with spectral index γ ∈ [1, 4], where we fit a single value of γ for the stacked flux of all ULIRGs. This modeling is motivated by the diffuse neutrino observations, which are currently best described by a power law (e.g., Abbasi et al. 2021a; Stettner 2020).

The stacking weight of source k ∈ {1, 2,...,M} is constructed as ${w}_{k}={r}_{k}{t}_{k}/{\sum }_{j=1}^{M}{r}_{j}{t}_{j}$. First, it depends on the detector response for an unbroken E^−γ power-law spectrum at the source decl. δ_k . This dependence is reflected by the weight term r_k ∝ ∫A_eff(E, δ_k )E^−γ dE, where A_eff is the effective area of the detector (Aartsen et al. 2017c). Second, the stacking weight depends on a theoretical weight t_k , which is modeled according to the hypothesis tested by the analysis. In this work, we make the generic assumption that the total IR luminosity between 8 and 1000 μm, L_IR, is representative for the power of a possible hadronic accelerator in ULIRGs. This assumption is motivated by the fact that the total IR luminosity is directly related to the star formation rate (see e.g., Rieke et al. 2009), and that a possible AGN contribution likely increases with IR luminosity (see e.g., Nardini et al. 2010). Additionally, we assume that the neutrino production is identical in all candidate sources. Hence, we set t_k = F_IR,k , which is the total IR flux (8–1000 μm) of ULIRG k. The total infrared flux is computed as ${F}_{\mathrm{IR}}={L}_{\mathrm{IR}}/(4\pi {d}_{L}^{2})$, where L_IR is taken from the respective ULIRG catalog (see Section 2.1), and where d_L is the luminosity distance determined from the redshift measurements. We note that the theoretical weight is predominantly driven by the luminosity distance, since the IR luminosity values span less than an order of magnitude.

Subsequently, a test statistic is constructed as $\mathrm{TS}=2\mathrm{log}[{ \mathcal L }({n}_{s}={\hat{n}}_{s},\gamma =\hat{\gamma })/{ \mathcal L }({n}_{s}=0)]$. Here, ${\hat{n}}_{s}$ and $\hat{\gamma }$ are those values that maximize the likelihood given in Equation (1), where a single $\hat{\gamma }$ is fitted for the full ULIRG sample. This TS is used to differentiate data containing a signal component from data being compatible with background. The background-only TS distribution is determined by randomizing the GFU data in R.A. 10⁵ times, and calculating the TS in each iteration, which is called a trial. However, ∼ 10⁸ trials are required to obtain a background-only TS distribution that is accurate up to the 5σ significance level. As this is computationally unfeasible, we fit a χ² PDF to the 10⁵ background-only trials in order to approximate the background-only TS distribution (Wilks 1938). This PDF is then used to compute the one-sided p-value corresponding with a certain observed TS_obs, which is the unique TS value associated with the unscrambled GFU data. The p-value reflects the compatibility of TS_obs with the background-only scenario. This p-value is determined by evaluating the background-only survival function at TS_obs, i.e., the integral of the background-only PDF over all TS ≥ TS_obs.

To test the performance of the analysis, we simulate muon neutrinos with a true direction exactly at the location of our 75 selected ULIRGs. The relative number of neutrinos simulated from an ULIRG location is proportional to the stacking weight of that source. These neutrinos, with energies 100 GeV <E_ν < 100 PeV, are generated from an unbroken power-law spectrum, ${{\rm{\Phi }}}_{{\nu }_{\mu }+{\bar{\nu }}_{\mu }}{({E}_{\nu })={{\rm{\Phi }}}_{{\nu }_{\mu }+{\bar{\nu }}_{\mu }}({E}_{0})({E}_{\nu }/{E}_{0})}^{-\gamma }$, which is normalized at an energy ⁶⁸ E₀ = 10 TeV. Here we assume that all sources have identical accelerator properties, such that we use a single spectral index γ to represent the flux of all ULIRGs. In such a simulation, the neutrino produces a track signature in the detector, i.e., a pseudosignal event. The angle between the reconstructed direction of the track and the true direction of the neutrino creates a spread of these pseudosignal events centered around the ULIRG locations. The number of pseudosignal events are drawn from a Poisson distribution with a given mean. For various values of this mean pseudosignal, we determine the corresponding TS distribution by performing 10³ trials. We define the sensitivity of the analysis at 90% confidence level as the mean number of pseudosignal events required to obtain a p-value ≤ 0.5 in 90% of the trials. In addition, we define the 3σ (resp. 5σ) discovery potential as the mean number of pseudosignal events required to obtain a p-value ≤ 2.70 × 10⁻³ (resp. ≤ 5.73 × 10⁻⁷) in 50% of the trials. Panel (a) of Figure 2 shows these quantities in terms of the stacked muon-neutrino flux evaluated at ⁶⁹ E₀ = 10 TeV as a function of the spectral index γ. Panel (b) shows the same quantities in terms of the total mean number of pseudosignal events. We determine this number by integrating the injection spectrum, after convolving it with the detector effective area, over the energy and detector lifetime. We find that the sensitivity and discovery potentials are more competitive for harder spectra. This dependence is expected, since harder spectra are more easily distinguished from the atmospheric background, which is well described by an E^−3.7 power-law spectrum (Abbasi et al. 2011). In a more realistic scenario, explored by Ambrosone et al. (2021), the spectral index of the power-law spectrum may vary from object to object. For ULIRGs, such a variation could e.g., be the result of a different relative starburst-versus-AGN contribution to the neutrino flux, since the AGN contribution seems to increase with IR luminosity (Nardini et al. 2010). The stacked neutrino flux of our selected ULIRGs would therefore be a superposition of these spectra. This effect is also referred to as spectral-index blending (Ambrosone et al. 2021). In this scenario, the single spectral index $\hat{\gamma }$ fitted in our analysis is therefore slightly biased toward the sources providing most of the neutrino flux at Earth. Our sensitivity in the spectral-index blending scenario will worsen slightly compared to the values given in Figure 2. However, this reduction of the sensitivity is mitigated by the fact that the the fitted spectral index $\hat{\gamma }$ as well as the sensitivity are mostly driven by the strongest neutrino sources.

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The upper limits ${{\rm{\Phi }}}_{{\nu }_{\mu }+{\bar{\nu }}_{\mu }}^{90 \% }$ of the ULIRG stacking analysis can be translated into an upper limit on the diffuse muon-neutrino flux originating from all ULIRGs with redshift z ≤ 0.13, computed as ${{\rm{\Phi }}}_{{\nu }_{\mu }+{\bar{\nu }}_{\mu }}^{z\leqslant 0.13}={\epsilon }_{c}{{\rm{\Phi }}}_{{\nu }_{\mu }+{\bar{\nu }}_{\mu }}^{90 \% }/(4\pi \,\mathrm{sr})$. Here, _c = 1.1, which corrects for the completeness of the ULIRG sample (see Appendix A for more details). This result can be extrapolated to an upper limit on the contribution of the total ULIRG source population to the diffuse neutrino flux. For this extrapolation we assume that all ULIRGs have identical properties of hadronic acceleration and neutrino production over cosmic history. We follow the method presented in Ahlers & Halzen (2014) to estimate the neutrino flux of all ULIRGs up to a redshift z_max as

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{{\nu }_{\mu }+{\bar{\nu }}_{\mu }}^{z\leqslant {z}_{\max }}=\displaystyle \frac{{\xi }_{z={z}_{\max }}}{{\xi }_{z=0.13}}{{\rm{\Phi }}}_{{\nu }_{\mu }+{\bar{\nu }}_{\mu }}^{z\leqslant 0.13}.\end{eqnarray} \tag{ 2 }$$

The redshift evolution factor ξ_z effectively integrates the source luminosity function over cosmic history up to a redshift z. For an unbroken E^−γ power-law spectrum, ξ_z becomes energy-independent. ⁷⁰ It can then be computed directly given a parameterization ${ \mathcal H }(z)$ of the source evolution with redshift (Vereecken & de Vries 2020). More details and numerical values of ξ_z are provided in Appendix B.

The ULIRG luminosity function has been a topic of several studies (Kim & Sanders 1998; Blain et al. 1999; Genzel & Cesarsky 2000; Cowie et al. 2004; Chapman et al. 2005; Le Floc'h et al. 2005; Hopkins et al. 2006; Lonsdale et al. 2006; Caputi et al. 2007; Reddy et al. 2008; Magnelli et al. 2009, 2011; Clements et al. 2010; Goto et al. 2011; Casey et al. 2012; Gruppioni et al. 2013). Qualitatively, it is observed that ULIRGs have a rapidly increasing source evolution up to a redshift z ∼ 1, followed by a relative flattening of the source evolution at higher redshifts. Unless stated otherwise, in this work we follow Vereecken & de Vries (2020) by parameterizing the ULIRG redshift evolution as ${ \mathcal H }{(z)\propto (1+z)}^{m}$, with m = 4 for z ≤ 1 and m = 0 (i.e., a flat source evolution) for 1 < z ≤ 4.

Using Equation (2), we are now able to determine the upper limits at 90% CL on the diffuse muon-neutrino flux of the ULIRG population up to a redshift ${z}_{\max }=4.0$, which is chosen following Palladino et al. (2019). The integral limits for unbroken E^−2.0, E^−2.5, and E^−3.0 power-law spectra are shown in panel (a) of Figure 3. Each of these limits is plotted in their respective 90% central energy region, defined as the energy range in which events contribute to 90% of the total sensitivity. ⁷¹ We find that the limits under the E^−2.0 and E^−2.5 assumptions exclude ULIRGs as the sole contributors to the diffuse neutrino observations up to energies of ∼3 PeV and ∼600 TeV, respectively. The E^−3.0 limit mostly constrains the diffuse ULIRG flux at energies below the diffuse observations. Panel (b) of Figure 3 shows the quasi-differential limits on the diffuse neutrino flux from ULIRGs. These quasi-differential limits are determined by computing the E^−2.0 limit in each bin of energy decade between 100 GeV and 10 PeV. We find that the quasi-differential limits exclude ULIRGs as the sole contributors to the diffuse neutrino observations in the 10–100 TeV and 0.1–1 PeV bins, respectively.

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The above results do not constrain the possible contribution of LIRGs (10¹¹ L_⊙ ≤ L_IR < 10¹² L_⊙) to the diffuse neutrino flux. Within z < 1, the IR luminosity density of LIRGs evolves roughly the same with redshift as the IR luminosity density of ULIRGs, although the former is ∼10–50 times larger (Le Floc'h et al. 2005; Caputi et al. 2007; Magnelli et al. 2009; Goto et al. 2011; Casey et al. 2012). Assuming that the correlation between the total IR and neutrino luminosities holds down to L_IR ≥ 10¹¹ L_⊙, the contribution of LIRGs to the neutrino flux is expected to dominate with respect to ULIRGs by a factor ∼10–50. Under this assumption, the diffuse neutrino observations would likely provide more stringent constraints than our stacking analysis. Nevertheless, we note that the AGN contribution to the total IR luminosity seems to be smaller for LIRGs than for ULIRGs (Nardini et al. 2010). This difference of the AGN contribution suggests that the hadronic acceleration properties of LIRGs may differ from those of ULIRGs. A dedicated study searching for high-energy neutrinos from LIRGs would provide further insights on the possible acceleration mechanisms within these objects.

First, we compare our results with the cosmic-ray reservoir model developed by He et al. (2013). They consider hypernovae as hadronic accelerators that are able to produce ${ \mathcal O }(100\,\mathrm{PeV})$ protons. He et al. (2013) suggest that the enhanced star formation rate in ULIRGs leads to an enhanced hypernova rate. This enhanced hypernova rate results in ULIRG neutrino emission consistent with an E^−2.0 power-law spectrum, with a cutoff at ∼2 PeV. They predict a diffuse neutrino flux originating from ULIRGs up to a redshift ${z}_{\max }=2.3$. Hence, we compare the prediction by He et al. (2013) to our E^−2.0 upper limit (90% CL) on the diffuse ULIRG neutrino flux up to a redshift ${z}_{\max }=2.3$. This comparison is presented in panel (a) of Figure 4. We find that our upper limit at 90% CL using 7.5 yr of IceCube data is at the level of the predicted flux of He et al. (2013). A follow-up study using additional years of data is required to further investigate the validity of this model.

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Second, we consider the work of Palladino et al. (2019). Their study applies a multimessenger approach to construct a generic cosmic-ray reservoir model of hadronically powered gamma-ray galaxies (HAGS). Candidate HAGS include ULIRGs and starburst galaxies with L_IR < 10¹² L_⊙. Palladino et al. (2019) model the diffuse neutrino and gamma-ray emission of a population of HAGS up to a redshift ${z}_{\max }=4.0$ according to a power-law spectrum with an exponential cutoff. This spectrum is fitted to the diffuse IceCube observations given in Haack & Wiebusch (2018). They find that spectra with indices γ ≤ 2.12 are able to explain a significant fraction of the diffuse neutrino observations without violating the nonblazar EGB bound above 50 GeV (Ackermann et al. 2016; Lisanti et al. 2016; Zechlin et al. 2016). In panel (b) of Figure 4 we compare the baseline HAGS model with spectral index γ = 2.12 to our E^−2.12 upper limit (90% CL) on the diffuse ULIRG neutrino flux up to a redshift ${z}_{\max }=4.0$. To determine this limit, we follow Palladino et al. (2019) and parameterize the ULIRG source evolution according to the star formation rate, ${ \mathcal H }{(z)\propto (1+z)}^{m}$, with m = 3.4 for z ≤ 1 and m = − 0.5 for 1 < z ≤ 4 (Hopkins & Beacom 2006; Yüksel et al. 2008). We find that our limits exclude ULIRGs at 90% CL as the sole HAGS that can be responsible for the diffuse neutrino observations. However, it should be noted that this result does not have any implications for other candidate HAGS, such as starburst galaxies with L_IR < 10¹² L_⊙.

Last, we make a comparison with the beam-dump model of Vereecken & de Vries (2020). This model considers Compton-thick AGN, i.e., AGN obscured by clouds of matter with hydrogen column densities N_H ≳ 1.5 × 10²⁴ cm⁻² (Comastri 2004), as hadronic accelerators. The neutrino production is modeled through the pp-interactions of accelerated hadrons with cloud nuclei. Vereecken & de Vries (2020) predict a diffuse neutrino flux from ULIRGs, for which they consider two methods to normalize the proton luminosity. In one approach, it is normalized to the IceCube observations of Kopper (2018). The authors find that ULIRGs can fit the IceCube data without exceeding the Fermi-LAT nonblazar EGB bound above 50 GeV (Ackermann et al. 2016) for column densities N_H ≳ 5 × 10²⁵ cm⁻². A second approach is found by normalizing the ULIRG proton luminosity to the radio luminosity using the observed ULIRG radio-infrared relation (Sargent et al. 2010). In this case, the modeling mainly depends on the electron-to-proton luminosity ratio, f_e (see e.g., Merten et al. 2017).

Here, we fit the results of Vereecken & de Vries (2020) for an E^−2.0 spectrum to our E^−2.0 upper limit (90% CL) on the diffuse neutrino flux from ULIRGs up to a redshift ${z}_{\max }=4.0$. Subsequently, we use the method presented in Vereecken & de Vries (2020) to estimate the ULIRG radio luminosity using the radio-infrared relation of Sargent et al. (2010). For this estimation, we take a conservative value L_IR = 10¹² L_⊙, and we assume that AGN contribute to 10% of this infrared luminosity based on the results of Nardini et al. (2010). This allows us to set a lower limit on the electron-to-proton luminosity ratio, f_e ≳ 10⁻³, by fixing all other parameters in the model of Vereecken & de Vries (2020). It should be noted that among these parameters are quantities with large uncertainties, such as the electron-to-radio luminosity ratio (Becker Tjus et al.2014). Hence, the lower limit f_e ≳ 10⁻³ should be regarded as an order-of-magnitude estimation. Nevertheless, we point out that this limit on the electron-to-proton luminosity ratio lies within the same order of magnitude as the lower limits provided by Vereecken & de Vries (2020) for several obscured flat-spectrum radio AGN that were also studied with IceCube (Maggi et al. 2016, 2018).