Ultra-high energy cosmic rays and neutrinos from light nuclei composition

Ultra-high energy cosmic rays and neutrinos from light nuclei composition

Saikat Das saikatdas@rri.res.in    Soebur Razzaque srazzaque@uj.ac.za    Nayantara Gupta nayan@rri.res.in Astronomy & Astrophysics Group, Raman Research Institute, C.V. Raman Avenue, Bengaluru 560080, India Centre for Astro-Particle Physics, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa Department of Physics, University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa
September 29, 2019
Abstract

The baryonic mass composition of ultra-high energy ( eV) cosmic rays (UHECRs) at injection along with other factors directly governs the UHECR flux on the earth. High energy neutrinos produced from UHECR interactions on cosmological photon backgrounds can further serve as crucial astrophysical messengers of the most powerful particle accelerators in the Universe. The latest measurements at the Pierre Auger Observatory (PAO) suggest a mixed element composition of UHECRs with the sub-ankle spectrum being explained by a different class of sources than the super-ankle region ( eV). In this work, we achieve a fit to the UHECR spectrum with a single population of sources over an energy range commencing at eV. We test the credibility of p+He composition by considering various abundance fractions at injection and a simple power-law evolution in redshift for source emissivity. We use CRPropa 3, a Monte Carlo simulation tool, for propagating primary and secondary ultra-high energy particles through extragalactic space and for calculating UHECR and cosmogenic neutrino fluxes on the earth. Many good fits are found corresponding to a range of parameter values, that well explains the UHECR spectrum. We place limits on the source spectral index and cut-off rigidity of the source population. Cosmogenic neutrino fluxes can further constrain the abundance fraction and maximum source redshift in this light nuclei injection model.

pacs:
95.85.Ry, 98.70.Sa
preprint: APS/123-QED

I Introduction

The most powerful astrophysical accelerators in the Universe produce particles with energies at least up to few times eV. These are the highest energy particles observed in nature, called the ultra-high energy cosmic ray (UHECR) particles with energies eV. Interpretation of the origin of UHECRs is a problem of foremost importance in astroparticle physics and a long-standing one (sigl01; anchordoqui18). The existence of UHECRs was not known until 1962, when particles with energies eV were detected at the Volcano Ranch experiment in New Mexico (linsley2; linsley1). Ever since the first discovery, an enormous amount of theoretical and experimental study has been done to understand their origin and to detect them. Even after several decades of investigation, the nature and spatial distribution of sources, as well as the acceleration mechanism leading to the production of such high energy particles, remain elusive Kotera11. The leading experiments to observe UHECRs are done at present by the Pierre Auger Observatory (PAO) in Argentina (PAO1; PAO2) and the Telescope Array (TA) experiment in the United States (TA1; TA2). These experiments are expected to reach necessary sensitivity in upcoming years, that can unveil these mysteries.

UHECRs cannot be confined by the Galactic magnetic field at the highest energies, motivating for a search in extragalactic sources (PAO3; hackstein). The possible sources in astrophysical provenances include active galactic nuclei (AGNs) (Berezinsky06; Dermer2009; wang17), gamma-ray bursts (GRBs) Waxman:1995vg; zhang18, low-luminosity GRBs Murase:2006mm, hypernovae Wang:2007ya, starburst galaxies (attallah18), gravitational accretion shocks (kang96; berezinsky97), neutron stars, etc. that can confine particles in their magnetic field up to a specific maximum energy . The increase in Larmor radius of the particle beyond the typical length scale of the acceleration site leads to the escape of the particle in a timescale determined by the ambient medium Hillas84. Direct identification of UHECR sources is hindered by the deflection of charged particles in the Galactic and extragalactic magnetic fields.

UHECRs propagate from their sources through the Universe and interact with cosmic microwave background (CMB) and extragalactic background light (EBL). These interactions can modify the observed spectrum. Indeed the UHECR spectrum at energies EeV exhibits interesting and prominent features. These include the ankle at around eV where a hardening of the spectrum has been observed. The ankle is assumed to be a feature resulting from the transition of Galactic to extragalactic cosmic rays (aloisio12). Another compelling possibility of interpreting the ankle is the pair production dip caused by the interaction of cosmic ray protons with the CMB photons. UHECR model with pure proton composition explaining the pair-production dip has been studied earlier in great details Berezinsky06; Aloisio07; aloisio15. The most prominent feature in the UHECR spectrum is the flux suppression at eV, beyond which there is a steep decline in the number of observed events. This suppression may be a consequence of the interaction of UHECRs with CMB photons, called the GZK cut-off greisen; zatsepin or it can also be a manifestation of the maximum acceleration energy at the sources. The effect of UHECR interactions with the EBL photons is important below the GZK cut-off energy Dermer2009, where the uncertainties due to various EBL models is appreciable Batistaebl15. The UHECRs may also encounter interactions with cosmic magnetic fields which cause deflections and energy loss in the form of radiation. But, this has negligible or no effect on the energy spectrum or composition study.

Charged and neutral pions produced in UHECR interaction with the CMB and EBL decay to give high energy neutrinos and photons. Neutrinos are also produced from neutron beta decays. The flux of these cosmogenic neutrinos depends highly on the injection spectrum of UHECRs, the density and redshift evolution of the sources, the mass composition of UHECRs and also on the EBL models (berezinsky69; engel01). Unlike high energy photons and charged cosmic rays, high energy neutrinos can travel through cosmological distances unimpeded by interactions with other particles and undeflected by magnetic fields, providing a means to identify and study the extreme environments producing UHECRs. Thus cosmogenic neutrinos are a definite probe to study UHECRs. Current neutrino detectors Achterberg:2006md; Collaboration:2011nsa have limited sensitivities to neutrinos at energies eV but plans are underway to construct bigger and more sensitive experiments to detect such energetic neutrinos Aartsen:2014njl; Adrian-Martinez:2016fdl; ara; arianna; poemma1; grand1.

The exact mass-composition of UHECRs is not known to a high precision. The atmospheric depth where the number of particles in the cascade reaches it’s maximum is studied for the purpose (gaisser13). But reconstruction of from shower simulations for an UHECR with given energy depends on the hadronic interaction models, which are uncertain at these extreme energies (pierog15). The average shower depth distribution , created by primary cosmic rays, indicate that UHECRs become heavier with increasing energy above eV Aab17. Depending on the UHECR-air interaction model, this corresponds to a mass composition between p and He at eV and between He and N at eV. In some models, the in the highest-energy bin ( eV) is intermediate between N and Fe. The fluctuation in that is , which is another indicator of the mass composition, varies between H and He up to eV for all UHECR-air interaction models and is in between He and N in the eV energy bin Aab17.

Recent measurements at the PAO also shed light on the combined fit of a simple astrophysical model of UHECR sources to both the energy spectrum and mass composition data Aab17. The fit has been carried out for energies eV, which is the region of all-particle spectrum above the ankle. Here, the ankle is interpreted as the transition between two (or more) different populations of sources. The astrophysical model considered consists of identical extragalactic sources uniformly distributed in a comoving volume, where nuclei are accelerated through a rigidity-dependent mechanism. A noteworthy assumption is that sources inject five representative stable nuclei: H, He, N, Si and Fe. Two public codes for propagation of UHECRs viz. CRPropa 3 CRPropa3 and SimProp SimProp along with various models for EBL and photodisintegration cross sections are accounted for in different combinations for the study. Magnetic field interactions having negligible effect in spectral analysis are neglected. The results of the Auger fit puts forward a hard spectrum favoring low spectral indices ( with ), making it difficult to conform with most particle acceleration models.

In a more recent study batista18, the combined fit of the energy spectrum and mass composition made by the PAO is extended to the specific cases of source evolutions corresponding to AGN, SFR, GRB and power-law redshift dependence in the form , where is a free parameter. The latter results in slightly better fits. They obtained best fits for hard spectral indices (1.5) and low maximal rigidities ( eV) for compositions at injection dominated by intermediate-mass nuclei (nitrogen and silicon groups). They show that negative source emissivity evolution is preferred considering source evolution as a free parameter, with the best fit for , ensuing hardest spectral indices provide the lowest possible cosmogenic fluxes for the source redshift evolution.

In this paper, we model the latest UHECR spectrum starting from eV which is well below the ankle, up to the highest energy data point observed by the PAO, with a single population of sources. This indicates the same mass composition for the whole energy range from EeV to the cut-off region. Here, we explore one of the best fit models named “CTD”, mentioned in (Aab17) using CRPropa 3 propagation code and aim to constrain model parameters with UHECR data from the PAO and future cosmogenic neutrino data. We exploit the possible composition for the “CTD” model and vary the element fractions at injection to obtain a fit from lower energy. We consider H and He as the only injected nuclei at the sources with no contribution from N/Si/Fe. After fitting UHECR data, we discuss whether cosmogenic neutrinos can constrain the mass composition near the ankle, as well as other model parameters such as the redshift distribution of sources and the maximum UHECR energy, in order to distinguish between favorable scenarios.

We discuss UHECR propagation, interactions and fluxes in general in Sec. II and in details in Sec. III for CRPropa 3. Our results are presented in Sec. IV and discussed in Sec. V. We draw our conclusions in Sec. VI.

Ii UHECR propagation and
cosmogenic fluxes

UHECRs propagate through the intergalactic space, after escaping from their acceleration sites, to interact with the cosmic background photons primarily via the -resonance channel and lose their energies through secondary particle productions as

(1)

The neutral pion decays to give photons (), and the charged pion decays to produce neutrinos (). Additionally, there can be double pion production and multipion production processes with much lower cross-sections anita00; murase06. These photonuclear processes lead to the formation of the GZK feature in the cosmic ray spectrum, thereby causing a sharp decay at eV greisen; zatsepin, which is near to the threshold for the photopion production with the CMB photons ( eV). Cosmic rays interact dominantly with low energy CMB photons of energy eV, during their propagation. EBL photons having energy higher than the CMB photons, allow protons with energies lower than the threshold of photopion production with the CMB to interact with the EBL photons and generate neutrinos. Although the number of EBL photons is much smaller than the CMB, they have a significant effect on the neutrino flux. The CMB photon density increases with redshift as . The spectral shape and cosmological evolution of the infrared, ultraviolet and optical backgrounds comprising the EBL are not as well known as the CMB. With the redshift evolution of the photon background, the interaction length of cosmic rays also evolves with redshift.

Beta decay contributes to cosmogenic neutrino flux through the decay of neutrons resulting from the charged pion production,

(2)

Heavier nuclei with a higher atomic number () also undergo beta decay and give rise to photopion production. Photodisintegration of nuclei due to irradiation with photons of energy between 8 to 30 MeV is the dominant energy loss mechanism for UHECR nuclei,

(3)

In this process a nucleus interacts inelastically with a cosmic background photon which leads to partial fragmentation of the nucleus producing stripped nucleons (protons). The de-excitation of an excited nucleus can give high energy photons. UHECRs can also undergo Bethe-Heitler pair production to generate pairs. Electron pair production has the largest cross-section among the photo-hadronic interactions, the threshold energy being two orders of magnitude smaller than that of pion production. The electrons and positrons produced in various processes can induce electromagnetic cascades down to GeV energies and thus contribute to the photon flux.

The energy loss rate of protons with energy due to cosmic expansion is expressed as,

(4)

where is the scale factor and CDM cosmology is considered here with km s Mpc, , olive. Neutrinos, being weakly interacting, propagate unhindered through the cosmos and experience only adiabatic energy loss due to the cosmic expansion. The photons interact with cosmic background radiations and the universal radio background (URB) to produce electromagnetic cascades through various processes such as Breit-Wheeler pair production, double pair production, resulting in pairs (heitercrp). The relativistic cascade electrons lose energy by triplet pair production, synchrotron radiation on deflection in magnetic fields and up-scattering background photons by inverse Compton scattering. For UHECRs the energy loss via interaction with the Galactic and extragalactic magnetic fields is negligible.

Iii Setup for CRPropa simulations

CRPropa 3 is a public astrophysical simulation framework to propagate ultra-relativistic particles from their sources to the observer through the Galactic and extragalactic space. Primary and secondary cosmic messengers such as protons, pions, nuclei, charged leptons, neutrinos and photons are produced as output (CRPropa3; heitercrp). The code is written in C++ but can be steered with Python using SWIG. CRPropa 3 enables shared-memory multiprocessing using OpenMP for easy multi-core simulations, which can be customized by combining existing simulation modules with own modules written in Python or C++. The SWIG interface allows cross-language polymorphism, which can be used to extend a CRPropa simulation directly from the Python script

We use CRPropa 3 to propagate primary and secondary UHECR protons and nuclei to get the particle yields obtained at the earth. Secondary neutrinos produced by photopion production of UHECRs on background photons and beta decay of neutrons are also propagated. However, we do not investigate the fluxes of cosmogenic photons or secondary cascade electrons, and hence they are not propagated in our simulation. We include all possible energy loss processes for primary UHE protons and nuclei in the simulation, viz. photopion production, Bethe-Heitler pair production, photo-disintegration (for ), nuclear decay and adiabatic energy losses due to the expansion of the Universe. We assume an injection spectrum of primary particles at the UHECR source of the following form,

(5)

where is the abundance fraction of the th nuclei at injection, is the energy of the particles, is an arbitrary normalization flux, is the spectral index, is the charge of the primary cosmic ray and is the cut-off rigidity. We assume particles are injected with energies between EeV and . The evolution of the source emissivity with redshift is of the form , where is a free parameter.

Parameter Description Values
Source spectral index
Cut-off rigidity EV
Minimum redshift
Cut-off redshift
Source evolution index
Helium fraction
Flux normalisation Best-fit to Auger
Table 1: UHECR parameters used for simulations

At ultra-high energies, the cosmic rays interact with background radiation comprised of CMB and EBL. The CMB spectrum is well known to a high precision and can be characterized by an isotropic blackbody spectrum with K planck15. The EBL models implemented in CRPropa 3 are Kneiske et al. kneiske04, Stecker et al. stecker06, Franceschini et al. frances08, Finke et al. finke10, Dominguez et al. dominguez11, Gilmore et al. gilmore12 and also the upper and lower bounds determined by Stecker et al. steckerup. We investigate the possible mass composition of the “CTD” model mentioned in the PAO paper Aab17. The energy loss interactions of UHECRs and photon backgrounds (CMB and EBL) are considered in the simulations, with the TALYS 1.8 photodisintegration model koning05 and the Dominguez et al. dominguez11 EBL model. Interactions with magnetic fields are relevant for charged particles, mainly electrons and positrons produced in electromagnetic cascades. UHECR protons and nuclei being heavier than electrons has no significant energy loss via synchrotron emission. Allowing for magnetic field deflections increases the complexity of the simulation and makes it dependent on the magnetic field model. Since we are interested in composition and energy spectrum study, we do not take into account any interactions with the Galactic or extragalactic magnetic field, which is particularly small. Hence our simulations are effectively one-dimensional.

Iv Results

Figure 1: UHECR spectra (left) and cosmogenic neutrino spectra (right) for . The top (case 1) and middle (case 8) panels show the best-fit cases listed in Appendix A for which the difference in the cosmogenic neutrino flux is the maximum. The bottom panel (case 2) shows the lowest case.
Figure 2: UHECR spectra (left) and cosmogenic neutrino spectra (right) for . The top (case 10) and middle (case 26) panels show the best-fit cases listed in Appendix A for which the difference in the cosmogenic neutrino flux is the maximum. The bottom panel (case 14) shows the lowest case.
Figure 3: UHECR spectra (left) and cosmogenic neutrino spectra (right) for . The top (case 28) and middle (case 36) panels show the best-fit cases listed in Appendix A for which the difference in the cosmogenic neutrino flux is the maximum. The bottom panel (case 31) shows the second lowest case.

We study the parameter space of UHECR spectrum and thus, of cosmogenic neutrino flux with only proton and Helium as the primary composition at injection. The cut-off rigidity of injected primaries is varied through 150, 200 and 250 EV. The source evolution index is varied through 0, 1, 2, 3. We assume that contributing sources are restricted to , where the minimum source redshift is fixed at , corresponding to the distance to Centaurus A (the nearest AGN), and vary through 2, 3 and 4. The maximum redshift in the simulations are thus well above , the GZK horizon. We investigate three particular cases of source spectral index at injection, viz.  2.2, 2.4, 2.6 and aim to fit the UHECR spectrum, starting from 1 EeV, which is well below the ankle. The parameters in the simulation and their ranges used are given in Table 1. For each value of , there are thus 36 possible combination of the parameter set {, , }, in keeping with our choice of discrete parameter values. For each of these combinations, we vary the ratio at injection, in small steps over a wide range from 0.0 to 16.00. The UHECR flux obtained from CRPropa 3 with these parameter choices, at the point of the observer is normalized to fit the Auger data.

The goodness of fit to the Auger data is computed using a standard analysis,

(6)

Here, is the observed value of the UHECR flux at specific energies and are the standard errors of each measurement, as given by the PAO. We take to be , where no data for errors is given. The UHECR flux predicted from the “CTD” model is given by , where are the values of parameters in the simulations. The number of degrees of freedom is expressed as number of data points number of parameters to fit . In our case, for a specific value of , we vary , , and . Hence, we obtain . A best-fit cannot be found for many combinations, owing to low sensitivity of on variation of or, due to a saturation of at some value of . Variation over the three values of gives 108 combinations of the set of values {, , , }. After discarding those combinations where no best-fit can be found by varying , we apply the criterion of reduced to select the cases that best explains the UHECR energy spectrum as measured by the PAO. Thus, we obtain the best-fit parameter sets {, , , } for each value of . We list these 36 best-fit cases having reduced in Appendix A.

iv.1 Fits to the UHECR spectrum

In the left panels of Figs. 1, 2 and 3 we show some of our fits to the UHECR spectrum in units for the index of the injection spectrum 2.2, 2.4 and 2.6, respectively. For each , we choose the two best-fit UHECR scenarios (top two panels) for which the difference in the cosmogenic neutrino fluxes is the maximum. The bottom panel in each figure corresponds to the fit with the lowest . For (Fig. 3), the parameter set for the lowest neutrino flux is the same as that for minimum . Hence in the bottom panel of Fig. 3 we show the UHECR spectrum and neutrino flux for the second minimum value of . The injection spectrum is modeled with an exponential cut-off at a certain energy . We find that changing and hence leaves the position of the bump (at EeV) unchanged and affects only the declining part of the UHECR spectrum, with slower steepening for higher and somewhat rapid steepening for lower ; provided the sources are distributed up to well beyond the path lengths of photopion production at these energies. The highest energy Auger data points beyond 40 EeV are well covered for our chosen range of .

We make the following observations from our study of the model parameters fitting data:

  • The required Helium to proton fraction decreases with increasing value of and essentially no He injection is required for (see Fig. 3).

  • The He flux falls off sharply beyond a few EeV due to increased photo-disintegration on EBL, conforming with the predictions by Gerasimova and Rozental (gerasimova). This elucidates the proton dominance at the highest energies making GZK cut-off a conspicuous phenomenon.

  • For , the fraction and the He flux is comparable or dominating over the proton flux at  EeV. As such, a changing composition contribute to the dip feature. For the dip is purely due to pair production with the CMB photons Berezinsky06.

  • A uniform distribution of sources lacking any evolution, i.e.  in redshift evolution , scenario is able to fit data only in case .

  • Increasing causes hardening of the spectrum in the region of energy below the ankle. For a particular and , the fit improves on increasing and thereby lowering , implying a lower He abundance at the sources for higher values.

  • There is no significant change in the UHECR spectral models on the variation of beyond a redshift of 2.0.

However, the values suggest a better fit for lower values. But lowering below 2.2 makes it difficult to obtain a fit from 1 EeV with only proton and Helium at injection. This explains our choice of spectral indices in the range .

iv.2 Cosmogenic neutrino fluxes

We calculate the neutrino flux with CRPropa by taking into account all possible production channels and by using the same normalization factor used for fitting UHECR data in different cases (see table in Appendix A). We present the whole range of cosmogenic neutrino flux summed over all flavors, in units, possible within the p+He model in the right panels of Figs. 13. A double peak shape is a common feature to all neutrino spectra. The higher-energy bump at around eV is due to decay of pions produced in interactions of UHECRs with the CMB photons. The lower-energy bump at eV is due to a combination of neutron beta decay and decay of pions produced in interactions of UHECRs with the EBL photons. The flux values at the higher-energy peak for different cases are listed in the table in Appendix A. The main results from our study of all flavor cosmogenic neutrino fluxes are below.

  • Although, no significant change in UHECR spectrum is seen on variation of beyond a redshift of 2.0, the cosmogenic neutrino flux on the other hand increases with increasing , keeping all other parameters fixed.

  • The flux at the higher-energy peak is generally higher for harder () injection spectrum. The lower-energy peak becomes more pronounced for softer () injection spectrum. The flux ratio between the two peaks is generally higher, reaching up to an order of magnitude, for harder injection spectrum.

  • The exact position of the peaks and the flux values depend on the maximum distance and redshift evolution of sources as well as the relative abundance of proton and Helium.

The neutrino fluxes in Figs. 13 are compared to the current and upcoming detector sensitivities as well. We show the detection sensitivity curves for Auger (augerneu1; augerneu2) and the flux upper limits from IceCube (aartsen15; aartsen16; aartsen18) along with the extrapolated 3-year sensitivities for the proposed detectors ARIANNA (arianna), ARA (ara), POEMMA (poemma1; poemma2) and GRAND (grand1; grand2). Upcoming Mediterranean detector KM3NeT Adrian-Martinez:2016fdl and proposed extension of the IceCube detector called IceCube-Gen2 vanSanten:2017chb can also probe cosmogenic neutrino fluxes in near term.

The PAO is effective at searching for neutrinos of energies exceeding 0.1 EeV by selecting inclined showers that have significant electromagnetic component. The range of neutrino fluxes obtained in our simulations are clearly below the differential upper limit GeV cm s sr imposed by Auger at 0.6 EeV. We multiply the single-flavor neutrino flux limit of Auger by a factor of 3 to obtain the all-flavor neutrino flux limit, assuming an equal flavor ratio.

The 90% C.L. all-flavor differential flux upper limit from 9-years of IceCube data sample based on extreme high energy (EHE) neutrino events above GeV is shown in brown solid line. Two EHE events were observed in the 9-yr analysis, which are compatible with a generic astrophysical origin and inconsistent with the cosmogenic hypothesis (for details, see (aartsen18)). In our calculations, the limit by IceCube just touches the CMB peak of the cases 8 and 26 corresponding to maximum neutrino flux for 2.2 and 2.4 respectively. This indicates a detection should be possible in near future, with further increase in exposure time. For the pure proton injection model (), the cosmogenic neutrino fluxes are too low to be detected by IceCube in future.

The sensitivities for the upcoming detectors are calculated from simulation of antenna response. ARA and ARIANNA, proposed to be built in Antarctica, aims at using Askaryan effect to detect interactions of the cosmogenic neutrinos above 1 EeV with ice. With comparable 3-yr sensitivities, both detectors would be able to probe few of our harder () injection spectrum cases (e.g., cases 2, 8, 14, 26). The sensitivities of POEMMA and GRAND are expected to be much better and would be able to probe cosmogenic fluxes for all the cases we explored. In particular, GRAND plans to reach an all-flavor sensitivity of GeV cm s sr above eV and a sub-degree angular resolution.

iv.3 Cosmogenic neutrino flux components

Figure 4: Ratio of the neutrino flux of different flavors for the best fit cases. The top panels, middle panels and bottom panels are for = 2.2, 2.4 and 2.6 respectively. From left to right the figures correspond to the case with lowest neutrino flux, maximum neutrino flux, and the fit with the lowest (second lowest for ), respectively for each , i.e., for the cases plotted in Figs. 13

After propagation over astrophysical distances, the probability of neutrino flavor conversion from to is given by , where and is the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix between the neutrino flavor and mass eigenstates. We use the current best-fit values of the mixing angles (for the normal mass hierarchy): , , and the CP-violating phase Capozzi:2017ipn. The corresponding probability matrix is

(7)

which is the same for neutrinos and antineutrinos with . Interestingly, for , there is a change in the probabilities: , , and . The probabilities and remain unchanged. In principle, can be probed with precise knowledge of the mixing angles and cosmogenic fluxes.

We calculate the cosmogenic neutrino fluxes of different flavors on the earth from the fluxes generated by the CRPropa code as

(8)

The main discriminator for neutrino flavors at the ice/water Cherenkov detectors is event topology, namely tracks for charged-current events and showers for and charged-current events and for all neutral-current events. At PeV range, it is however, possible to discriminate events Learned:1994wg; Beacom:2003nh; DeYoung:2006fg and flavor identification for all charged-current events could be possible. In such a case the ratios of cosmogenic fluxes of different flavors can be written as

(9)

For typical initial flavor ratios, the expected ratio on the earth is just ratio of the probabilities given as

(10)

In Fig. 4, we plot the ratios , and obtained from CRPropa simulations. The upper, middle and bottom panels show the ratio of component fluxes for 2.2, 2.4, 2.6, respectively. From left to right the plots in Fig. 4 represent the same cases as shown accordingly in Figs. 13 from top to bottom. At the highest energy end, the ratios are not well defined due to few particles involved in the simulations. Below  eV, the ratios are roughly constant for a number of decades in energy depending on different cases. These constant ratio parts are roughly consistent with the expected values of (red lines), (blue lines) and (green lines); from typical pion-decay flavor ratios at production. A shift from these values at low energies is due to neutron beta decays and is an indicator of He/p ratio of the UHECR flux at injection. For example, in cases requiring larger He/p ratio, the deviation from constant flavor ratios happen at energies  eV, while for the pure protons injection cases (), the flavor ratios are constant down to  eV.

V Discussions

The reference model “SPG” of the PAO fit Aab17 considers SimProp propagation with Puget, Stecker and Bredekamp model of photo-disintegration Puget76 and Gilmore et al. EBL model gilmore12. The best-fit parameters for this model indicates two minima. One minimum corresponds to a low value of and a spectral index . The other minimum corresponds to and . The former corresponds to a hard spectrum and composition dominated by He, N, Si with no contribution from H or Fe at the best fit. An additional class of extragalactic sources is required to fit the spectrum below ankle in this scenario, owing to the inability of Galactic SNRs to accelerate particles to this energy. A variation of the source evolution index reveals that a reconciliation with the Fermi acceleration mechanism is possible only if is allowed to be negative. Such negative source evolution is also suggested in (taylor15). For a particular model “CTD” the PAO best-fits (also, only above the ankle) require the element fractions as , , and ; at the injection Aab17.

In this work we have addressed the prospects for an explanation of the UHECR spectrum with the “CTD” model over an energy range starting at eV with a single population of sources, that requires no additional sub-ankle component and is compatible with the most prevalent source redshift evolution , with (gelmini11). Since the fractions and are very small in the PAO results, we have neglected any contribution from them in our simulations. The p+He mass composition at injection is also studied in (aloisio17), but considering an older set of Auger data (pierre14) and with no cosmological evolution of the sources. A lower limit to the proton-to-Helium ratio is also given in (karpikov18) based on the study of shower depth distributions and hadronic interaction models. In our work we find that positive source evolution in redshift is equally capable of explaining the Auger data within the framework of the “CTD” model. The possible values of parameter set considered (see Appendix A) allow us to constrain the source spectral index to lie between . Outside this range the fit becomes poor and no suitable parameter values conform with the Auger data. For , a pure proton composition is obtained for all the best-fit cases, although there are arguments over the viability of such a scenario (heinze16). For the addition of other nuclei composition becomes indispensable.

Since the best-fit value of for the “SPG” model is found to be low in the PAO analyses, the observed spectrum steepens as a result of limited maximum source energy and photo-disintegration Aab17. Whereas, the GZK cut-off requires the primary proton energy to be at least comparable to the threshold for pion-production with the CMB photons. Thus the cosmogenic neutrino flux at EeV energy is predicted to be negligible in the PAO analysis. We assume, on the other hand, the steepening of the spectrum is due to the GZK effect via increased energy loss of primaries by copious production of charged and neutral pions. The neutrino fluxes obtained in our calculations vary due to composition, injection spectrum and the maximum distance up to which the sources accelerate cosmic rays. We find that our cosmogenic neutrino flux predictions are compatible with the plausible range of models studied in (kotera10).

Currently operating neutrino detectors do not reach yet the necessary sensitivity level for detecting cosmogenic fluxes. The most stringent upper limits on the flux come from analyzing 9-years of IceCube data (aartsen18). These limits are about an order of magnitude higher than the maximum flux level expected for our pure proton dominated cases with the injection index . For harder injection indices, we find that the cosmogenic neutrino fluxes can reach the IceCube upper limits at EeV in some cases. Among the future detectors, the prospect for detection of cosmogenic fluxes is particularly good for POEMMA (poemma1; poemma2) and GRAND (grand1; grand2). These detectors, with a combined energy coverage of 10 PeV–100 EeV, will be able to probe most of our models. Detection of cosmogenic neutrino flux together with flavor identification will be crucial to constrain UHECR composition and their sources.

Vi Conclusions

UHECR mass composition depends on various factors such as redshift evolution of sources, maximum energy of primary particles and also on the injection spectrum which is determined by the acceleration mechanism. The much acknowledged choice of power-law injection with source spectral index at originates in the well known Fermi mechanism. Latest measurements at the Pierre Auger Observatory favors low spectral index and present a fit to the UHECR spectrum starting at eV. PAO assumes a mixed composition at injection, with the precise element fractions being determined by specific propagation model, photodisintegration cross-section and EBL spectrum. In this paper, we have studied a model called “CTD”, with CRPropa 3 propagation, TALYS 1.8 photodisintegration cross-section, Dominguez et al. EBL model, assuming sources inject only H and He nuclei. We constrain the range of injection spectral index and the cut-off rigidity feasible in this light nuclei injection model. We vary the abundance fraction at injection to find the best-fit cases for a wide range of UHECR parameter values. We find, a fit to the entire spectrum is possible starting from E eV with a single population of sources of extragalactic origin. We have also calculated the cosmogenic neutrino fluxes from all production channels. We plot the spectra of best-fit cases which represent the entire range of neutrino fluxes possible for 2.2, 2.4, 2.6. All the best-fit cases yield neutrino spectra consistent with the flux upper limits imposed by present detectors. This suggests that the abundance fraction of H and He considered in the best-fit cases are plausible. The ratio of fluxes of different flavors obtained on earth after neutrino oscillation are consistent with our expectations. The value obtained in fitting procedure of UHECR spectrum favors over the pure proton cases. Source redshift evolution is found to play a significant role in determining the flux as well as the position of peak in the cosmogenic neutrino spectrum. We also note, with increase in maximum source redshift there is an increase in neutrino flux, due to increased propagation length of primary particles. Future neutrino telescopes with higher sensitivities at PeV energies, will be able to probe a range of flux models we predict. A measurement will be able to constrain the maximum redshift of the UHECR source distributions. Furthermore, neutrino flavor identification will shed light on the abundance fraction of nuclei in the UHECR spectrum at injection, as shown in neutrino flavor ratios for our flux models. While we show that a p+He composition model is capable of fitting UHECR data, future cosmogenic neutrino data will provide a robust test for this scenario.

Appendix A UHECR parameter sets

Case Neutrino flux
2.2 2 2 150 6.00 1
3 150 2.40 2
3 200 10.00 3
3 2 150 10.00 4
3 150 3.00 5
3 200 5.25 6
4 2 150 5.50 7
3 150 2.50 8
3 200 6.50 9
2.4 2 1 150 0.60 10
2 150 0.30 11
2 200 0.70 12
2 250 0.90 13
3 150 0.05 14
3 200 0.25 15
3 250 0.35 16
3 2 150 0.30 17
2 200 0.60 18
2 250 0.90 19
3 150 0.05 20
3 200 0.20 21
3 250 0.35 22
4 2 150 0.30 23
2 200 0.60 24
3 150 0.10 25
3 200 0.20 26
3 250 0.35 27
2.6 2 0 150 0.00 28
0 200 0.00 29
1 200 0.00 30
3 0 150 0.00 31
0 200 0.00 32
1 250 0.00 33
4 0 150 0.00 34
0 200 0.00 35
1 250 0.00 36
Table 2: Best-fits to UHECR spectrum

Here we give the best-fit combinations of the values of parameter set considered for CRPropa simulations. We list them in Table 2. There are 36 cases: 9 for , 18 for and 9 for . They are further subgrouped according to the maximum source redshift. For each , we select two cases having the lowest and highest cosmogenic neutrino flux at the higher energy peak for display. Also, we select one case from each corresponding to the lowest value. These are cases 1, 8 and 2 respectively for ; cases 10, 26 and 14 for . These are shown accordingly from top to bottom in Figs. 1 and 2 with the UHECR spectrum on the left and cosmogenic neutrino flux on the right panels. For , the lowest corresponds to the case of the lowest neutrino flux. Hence, in Fig. 3, we show the second lowest case in the bottom panel. The lowest neutrino flux, highest neutrino flux and the second lowest cases for correspond to the cases 28, 36 and 31 respectively. Thus, the top panels in each of the Figs. 13 show the minimum cosmogenic flux; the middle panels show the highest cosmogenic flux, and the bottom panels show the case corresponding to the lowest (second lowest for ) value.

References

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