# Impact of the finite life-time of UHECR sources

B. Eichmann11footnotetext: Corresponding author. and M. Kachelrieß

###### Abstract

The observational data on ultrahigh energy cosmic rays (UHECR), in particular
their mass composition, show strong indications for extremely hard spectra of
individual mass groups of CR nuclei at Earth. In this work, we show that such
hard spectra can be the result of the finite life-time of UHECR sources, if a
few individual sources dominate the UHECR flux at the highest energies. In
this case, time delays induced by deflections in the turbulent extragalactic
magnetic field as well as from the diffusive or advective escape from the
source environment can suppress low-energy CRs, leading to a steepening of the
observed spectrum. Considering radio galaxies as the main source of UHECRs, we
discuss the necessary conditions that few individual sources dominate over the
total contribution from the bulk of sources that have been active in the past.
We provide two proof-of-principle scenarios showing that for a turbulent
extragalactic magnetic field with a strength $B$ and a coherence length
$l_{\rm coh}$, the life-time of a source at a distance $d_{\rm src}$ should
satisfy ${t_{\rm act}\sim\left(B/1\,\text{nG}\right)^{2}\,\left(d_{\rm
src}/10\,\text{Mpc}\right)^{2}\,\left(l_{\rm
coh}/1\,\text{Mpc}\right)\,\text{Myr}}$ to obtain the necessary hardening of
the CR spectrum at Earth.

##  1 Introduction

The quest to reveal the sources of the highest energy particles observed has
been a key-driver in the field of cosmic ray (CR) physics since the first
observation of CRs with energy close to $10^{20}$ eV [1]. On the experimental
side, considerable progress has been made in the last 15 years as a result of
the observations made with the Pierre Auger Observatory (PAO) and the
Telescope Array (TA) experiment, for recent reviews see [2, 3, 4]. In
particular, the diffuse energy spectrum [5, 6, 7] of ultrahigh energy cosmic
rays (UHECRs) is now known rather precisely, and several deviations from a
simple power law have been determined: In addition to the second knee around
$E\simeq 5\times 10^{17}$ eV, the ankle at $E\simeq 5\times 10^{18}$ eV and
the flux suppression at the highest energies, a fourth feature, the so-called
instep, has been established by the PAO [7, 6]. The second important piece of
information, the mass composition of the UHECR flux, can be inferred only
indirectly from the number of particles an air shower is containing as
function of atmospheric depth $X$, relying on simulations for hadronic
interactions. Comparing the predicted $X_{\max}$ distributions for a mixture
of CR nuclei to the observed distribution, one can fit the relative fraction
of the CR nuclei: Above $10^{18}$ eV, the dominant component in the UHECR flux
changes successively from protons, to helium and nitrogen, a behaviour
suggestive for the presence of a Peters’ cycle. According to these fits, the
different dominating element groups are surprisingly well separated [7].

Adding some assumptions on the propagation of UHECRs (e.g., a model for the
extragalactic magnetic field) allows one to test if specific UHECR source
models can reproduce these data. The arguable simplest theoretical model
employs a continuous, uniform distribution of identical sources, combined
often with a negligible extragalactic magnetic field (EGMF). Inspired by the
idea of a Peters’ cycle, a rigidity dependent source spectrum with an
exponential cut-off,
$\mathrm{d}N_{i}/\mathrm{d}R=K_{i}R^{-\alpha}\exp(-R/\hat{R})$, is typically
used [8, 9, 10, 11, 12, 13]. Fitting then the common slope $\alpha$ and cut-
off $\hat{R}$ as well as the relative abundances $K_{i}$ of the different mass
groups, a satisfactory description of both the spectrum and the composition
data can be achieved despite the simplicity of this model. However, in order
to obtain a clean separation of the mass groups and thus to reproduce the
composition data, a very hard slope of the injection spectrum into
extragalactic space is required: For instance, the fits presented in Ref. [13]
use slopes222Note the sign. between $\simeq-1$ and $-2$. Moreover, the ankle
has to be explained by a second extragalactic population which does not show
the same Peters’ cycle but has a dominantly light composition. Alternatively,
there are also some approaches [8, 9] that used a slope much closer to 2 as
well as a dominant contribution from (identical) local sources. But in these
approaches, only the data above the ankle are taken into account yielding the
need of a sharp flux suppression of the sub-ankle contribution to keep the
variance of the chemical composition low at about 10 EeV. Still, these results
show some discrepancies with the data, which however, is unavoidably as there
is necessarily a mixture of different CR nuclei types at about 10 EeV in the
case of an initial $1/E^{2}$ spectrum.

Since the assumption of identical sources is not well justified, one may ask
how these fits change if one uses a source populations with, e.g., a
distribution of $\hat{R}$ values. Considering only the energy spectrum, it is
clear that a distribution of maximal rigidities $\hat{R}$ decreasing faster
than $1/\hat{R}$ will require even harder spectra of individual sources [14,
4]. Moreover, a distribution of $\hat{R}$ values will spread out the different
mass groups, even if they are well separated in the spectra of individual
sources. This qualitative argument was quantified in the recent work [15],
which showed that UHECR sources have to be close to “standard candles”, if
many sources contribute to the flux above $10^{18}$ eV.

One may also wonder how such hard escape spectra may be generated. While
first-order Fermi acceleration [16, 17, 18, 19, 20] predicts for strong, non-
relativistic shocks in the test-particle approximation the slope $\alpha\simeq
2$ for the source spectrum, the slope is modified by non-linear and
relativistic effects. However, the expected range of slopes arising from these
modifications is far from the values $-2\lesssim\alpha\lesssim-1$ obtained in
the fits. Alternative acceleration mechanisms, as e.g. unipolar acceleration
in pulsars, may however lead naturally to flatter energy spectra [21].

Another possibility is that an energy-dependent escape from the shock modifies
strongly the spectra. In the case of non-relativistic shocks around supernova
remnants, it was however argued that an acceleration spectrum that is flatter
than $1/E^{2}$ will result in an $1/E^{2}$ escape spectrum—thus this effect
works in this case in the opposite direction [22]. A final option is the
modification of spectra by threshold effects of photo-nuclear $A\gamma$
interactions in the source, see, e.g., Refs. [23, 24, 25, 26].

In this work, we suggest to use the magnetic horizon effect [27, 28, 29]
applied to a small number of source with finite life-time as explanation for
the apparently hard injection spectra obtained in fits to the observed
spectrum and composition of UHECR data. In its original version, the magnetic
horizon effect was used in Refs. [27, 28, 29] to explain the suppression of
the diffuse extragalactic cosmic ray flux below few $\times 10^{17}$ eV in the
“dip model” [30]. Later, the same effect was studied for the case of a mixed
composition: For instance, Ref. [31] showed that the suppression of low-energy
cosmic rays helps to reconcile the measured composition data with steeper
injection spectra. More recently, Ref. [32] studied the same effect for a
single source in the local Supercluster. Our starting point in this work is
the observation made in Ref. [33] that a very small number of local radio
galaxies can dominate the UHECR flux above the ankle. In this scenario, the
instep is just the most obvious of several irregularities which are caused by
the small number of sources contributing to the high-energy end of the energy
spectrum. Since in this model, UHECRs are accelerated in the extended jets of
radio galaxies, a modification of the escape spectra by $A\gamma$ interactions
cannot be used to explain the hard escape spectra. However, the small number
of sources opens up the possibility that the finite life-time of these sources
together with energy-dependent time delays in the extragalactic magnetic field
modifies the observed spectra. In particular, we show that the delay of low-
energy CRs leads to a flattening of the energy spectra of these sources, which
can reconcile the composition data with a standard acceleration spectrum close
to $1/E^{2}$.

This paper is structured as follows: In Sec. 2, we first introduce the model
used by us for the CR escape and the time evolution of the CR luminosity of
individual sources. In addition, we present how we treat the CR contribution
from the large-scale distribution of radio sources that has been active in the
past. In Sec. 3, we investigate the constraints on the currently active, local
sources in order that they dominate the UHECR flux above the ankle. In Sec. 4,
we provide some proof-of-principle examples, where we account for a single
local source with a finite life-time to explain the observed energy spectrum
and the composition data. Finally, we discuss in Sec. 5 our results and
provide some conclusions.

##  2 Cosmic ray sources with a finite life-time

Active galactic nuclei (AGN) show a varied and complex history [34, 35, 36],
so that the actual source properties can significantly deviate from the time-
integrated ones. For variations on sufficiently short time scales, one may
expect that time-delays due to diffusion in magnetic fields smooth out the
source variability in the UHECR flux. Here, we will concentrate on the
variability on the largest time scales, which we associate with the life-time
$t_{\rm act}$ of the source, which is of order of 1–100 Myr in the case of
radio galaxies.

In terms of the other characteristics of the CR ejecta of radio galaxies, we
stick to the approach that has been introduced in previous works [37, 38, 33]
and suppose that the fraction $g_{\rm m}<1$ of the jet power $L_{\rm jet}$ is
transferred to CRs, while the radio-to-jet power correlation determines
$L_{\rm jet}\propto L_{151}^{\beta_{\rm L}}$, where in general
$0.5\lesssim\beta_{\rm L}\lesssim 0.9$ is observed [39, 40, 41, 42]. Unless
otherwise stated, we will subsequently adopt a rather strong dependence with
$\beta=0.89$ as suggested from one of the most recent works [39]. Further, the
associated maximal rigidity of CRs is then given by $\hat{R}=g_{\rm
acc}\sqrt{(1-g_{\rm m})L_{\rm jet}/c}$, where the acceleration efficiency
$0.01\lesssim g_{\rm acc}\lesssim 1$ encapsulates all details of the
acceleration process.

###  2.1 Time-dependent luminosity

We will model the time variability of a UHECR source which in principle can
affect the energy spectrum in a complicated way by a sole parameter, the
overall CR luminosity $L_{\rm cr}$. In the simplest picture, implementing a
finite life-time as a step function, $L_{\rm cr}$ vanishes if the activity
time satisfies $t_{\rm act}<|t_{\rm obs}-[t(\varepsilon)-d_{\rm src}/c]|$ for
a source at distance $d_{\rm src}$ and the travel time $t(\varepsilon)$ of a
CR with energy $\varepsilon$. More realistically, the CR luminosity will
evolve during the life-time of the source. To be specific, we choose this time
dependence as a Gaussian normal distribution,

$L_{\rm cr}(t^{\prime})=\frac{L_{0}}{\exp\left[-(t_{\rm max}/\sigma_{\rm
act})^{2}/2\right]}\,\exp\left[-\frac{1}{2}\,\left(\frac{t^{\prime}+d_{\rm
src}/c-t_{\rm max}}{\sigma_{\rm act}}\right)^{2}\right]\,,$ (2.1)

where $L_{0}$ denotes the present CR luminosity of the source measured by the
observer, i.e. at the time $t^{\prime}=-d_{\rm src}/c$ with respect to the
observation time $t_{\rm obs}$. Further, $t_{\rm max}$ is the time of maximal
luminosity and the source’s active time is connected to the variance as
$\sigma_{\rm act}=t_{\rm act}/3$. Thus, we define the activity time of a
source as the time in which 99,7% of the total ejected CR energy is emitted.
Note that for $-t_{\rm act}\lesssim t_{\rm max}\ll 0$, the source showed a
much higher CR luminosity in the past that increases for a decreasing
difference of $t_{\rm act}-|t_{\rm max}|$. Moreover, there is an additional
modification of the actually observed source luminosity, if the CRs escape the
source environment by diffusive or advective transport as illustrated in the
following.

###  2.2 Escape from the source environment

We assume that UHECRs are accelerated by shocks in the jets or lobes of the
AGN, at large distances $\gtrsim 1\,\text{kpc}$ from its central engine. At a
distances of about $10\,\text{kpc}$ from the central nucleus the photo-
hadronic energy loss due to the presence of a strong AGN photon field (with a
bolometric luminosity of $\sim 10^{46}\,\text{erg s}^{-1}$) occurs already on
a timescale $\gtrsim 100\,\text{Myr}$ for CR energies $\lesssim
10^{19}\,\text{eV}$. In addition, proton synchrotron losses occur on a
timescale [43] $\tau_{\rm syn,p}\sim 14\,(E/10^{19}\text{eV})^{-1}\,(B_{\rm
jet}/1\text{mG})^{-2}\,\text{Myr}$, so that we obtain similar timescales for a
realistic magnetic field strength of $B_{\rm jet}\sim 0.1\,\text{mG}$ at these
energies. Hence, at these distances from the central nucleus energy losses can
be neglected for escape times $\lesssim 100\,\text{Myr}$, since even at the
highest energies CRs escape the source environment before loosing a
significant fraction of their energy.

While we remain agnostic about the details of the acceleration process, we
have to take into account that UHECRs do not immediately leave the source
region. The details of the escape process are in general complicated and
dependent on the jet dynamics as well as the spatial position of the
accelerator and the magnetic field structure. Apart from some additional
impact by streaming motions, the CR escape from the source is expected to be
dominated by diffusion or advection. If the relevant bulk flows that drive the
particle escape, e.g. the back-flowing jet material, are rather uniform and
reach a significant fraction of the speed of light [e.g. 44, 45], UHECRs can
escape the source on Myr time scales via advection. But in particular at large
distances from the shock, the bulk flows are typically slower, i.e. $v_{\rm
bulk}\sim 1000\,\mathrm{km/s}$, and less uniform so that the UHECR escape is
rather driven by diffusion. In addition, the UHECR sources can be embedded in
a galaxy cluster—which is often the case for radio-loud AGNs—so that in any
case the CRs scatter off the turbulent cluster magnetic field and diffuse out
of this structure. In the following, we characterize the escape from the
source region with the characteristic size $l_{\rm src}$ by the escape time
[46, 47]

$\tau_{\mathrm{esc}}\simeq\begin{cases}9.81~{}\mathrm{Myr}\left(\frac{l_{\mathrm{src}}}{100~{}\mathrm{kpc}}\right)\,\left(\frac{v_{\rm
bulk}}{10000\,\mathrm{km/s}}\right)^{-1},\,&\text{for advection,}\\\
9.77~{}\mathrm{Myr}\left(\frac{l_{\mathrm{src}}}{100~{}\mathrm{kpc}}\right)^{2}\left(\frac{l_{\mathrm{coh}}}{1~{}\mathrm{kpc}}\right)^{-1}\left[4\left(\frac{R}{R_{\rm
c}}\right)^{2}+a_{\rm I}\left(\frac{R}{R_{\rm c}}\right)+a_{\rm
L}\left(\frac{R}{R_{\rm c}}\right)^{2-m}\right],\,&\text{for
diffusion}\,,\end{cases}$ (2.2)

where we account for the quasi-rectilinear propagation at rigidities
significantly higher than the critical rigidity $R_{\rm c}=9\,(B_{\rm
src}/10\,\mu\text{G})\,(l_{\rm coh}/1\,\text{kpc})\,\text{EV}$. Moreover,
$B_{\rm src}$ denotes the strength of the turbulent magnetic field within the
source environment and $l_{\rm coh}$ is its coherence length. Dependent on the
spectral index $m$ of the turbulence spectrum, the coefficients are given by
$a_{\rm I}=0.9$ and $a_{\rm L}=0.23$ for Kolmogorov ($m=5/3$) and $a_{\rm
I}=0.65$ and $a_{\rm L}=0.42$ for Kraichnan turbulence ($m=3/2$).333Since the
work by Harari et al. does not investigate the case of Bohm diffusion ($m=1$),
we stick to the Kraichnan coefficients $a_{\rm I}=0.65$ and $a_{\rm L}=0.42$
in that case. Hence, the CR luminosity that escapes the sources is determined
by

$\displaystyle L_{\rm cr}^{\rm esc}(t^{\prime})$
$\displaystyle=\frac{1}{\tau_{\mathrm{esc}}}\int_{-\infty}^{t^{\prime}}\mathrm{d}t^{\prime\prime}\,\,L_{\rm
cr}(t^{\prime\prime})\,\exp\left(-\frac{(t^{\prime}-t^{\prime\prime})}{\tau_{\mathrm{esc}}}\right)$
$\displaystyle=\frac{L_{0}\,\sigma_{\rm
act}\,\sqrt{\pi/2}}{\tau_{\mathrm{esc}}\exp\left(-(t_{\rm max}/\sigma_{\rm
act})^{2}/2\right)}\,\exp\left(\frac{\sigma_{\rm
act}^{2}-2\tau_{\mathrm{esc}}(t^{\prime}+d_{\rm src}/c-t_{\rm
max})}{2\tau_{\mathrm{esc}}^{2}}\right)$ (2.3)
$\displaystyle\qquad\times\left[1-\mathrm{erf}\left(\frac{\sigma_{\rm
act}^{2}-\tau_{\mathrm{esc}}(t^{\prime}+d_{\rm src}/c-t_{\rm
max})}{\sqrt{2}\sigma_{\rm act}\tau_{\mathrm{esc}}}\right)\right]\,,$

where $\mathrm{erf}(x)$ denotes the error function. It can be shown that
$L_{\rm cr}^{\rm esc}(t^{\prime})\rightarrow L_{\rm cr}(t^{\prime})$ for
$\tau_{\mathrm{esc}}\rightarrow 0$. In the opposite limit,
$\tau_{\mathrm{esc}}\gg t_{\rm act}$, the CR contribution of the present,
individual sources becomes negligible. Hence, a present source with a life-
time of a few Myr and a characteristic size of about 100 kpc can only provide
a significant contribution of 10 EV CRs, if either a bulk flow with $v_{\rm
bulk}\gtrsim 10000\,\text{km/s}$ or a magnetic field with $B_{\rm src}\lesssim
10\,\mu\text{G}$ is present, cf. with Fig. 1.

Figure 1: The escape time of CRs with a rigidity of 10 EV in the case of
advection (_left_) and Kolmogorov diffusion (_right_) based on Eq. 2.2. In the
diffusion case a characteristic size of $l_{\rm src}=100\,\text{kpc}$ is
adopted. Note that Kraichnan or Bohm turbulence yields only small
modifications of the pattern.

Figure 2: The escaping luminosity $L_{\rm cr}^{\rm esc}$ in the case of
advection (blue), Bohm diffusion (orange) and Kolmogorov diffusion (green),
respectively, $l_{\rm src}=100~{}\mathrm{kpc}$, $B_{\rm
src}=10~{}\mu\mathrm{G}$, $l_{\rm coh}=10~{}\mathrm{kpc}$, $v_{\rm
bulk}=3000~{}\mathrm{km/s}$ and an activity time of $t_{\rm
act}=15\,\mathrm{Myr}$. The grey lines indicate the luminosity in case of an
instantaneous escape from the source.
_Left:_ The evolution of the source luminosity dependent on the emission time
($-t^{\prime}-d_{\rm src}/c$) for different $t_{\rm max}$ values for a CR
rigidity $R^{\prime}=1\,\mathrm{EV}$. _Right:_ The spectral behaviour of the
present ($-t^{\prime}=d_{\rm src}/c$) source luminosity for different $t_{\rm
max}$ values.

For a given initial CR luminosity $L_{0}$ at the present (with respect to the
light travel time distance) time, it is illustrated in Fig. 2 that the past
maximal value of the CR luminosity $L_{\rm cr}$ can differ by orders of
magnitude, depending on the given activity time $t_{\rm act}$ as well as the
time $t_{\rm max}$ of maximal emission. In the limit $t_{\rm act}\gg|t_{\rm
max}|$, this difference obviously vanishes. When we account for advective or
diffusive escape, respectively, the increased past source activity gets
delayed so that the present escaping luminosity $L_{\rm cr}^{\rm esc}(-d_{\rm
src}/c)$ becomes typically maximal if $t_{\rm max}\sim-\tau_{\mathrm{esc}}$.
In that case the luminosity $L_{\rm cr}^{\rm esc}(-d_{\rm src}/c)$ can also
exceed $L_{0}$, which does not account for any CR propagation effects since
this quantity is typically derived from the present luminosity in some
particular band of the electromagnetic spectrum. Since the diffusive escape of
the CRs from the source region depends on the particles’ rigidity, the
resulting emissivity does so too: The total number of escaping CRs typically
increases with increasing rigidity leading to a significant hardening of the
escaping energy spectrum, as can be seen in the right panel of Fig. 2. Note
that this is not the case if the CRs escape by advection. However, the
subsequent propagation through the extragalactic magnetic field can still lead
to a hardening of the CR energy spectrum at Earth if the sources have a finite
life-time. Moreover, for the limiting case of Bohm diffusion the hardening of
the spectrum only yields a spectral index of $\alpha-1$ at low rigidities.
Considering a spectral index of $\alpha=2$ at the acceleration site this
hardening alone is clearly not sufficient to reproduce the composition data.

Although, it is most reasonable to adopt a finite life-time of the considered
UHECR sources, this also introduces the difficulty of modelling those sources
that are currently inactive with respect to their electromagnetic emission but
still contribute UHECRs due to their past activity. At least on large scales,
we can use a redshift dependent luminosity function to account for those
additional sources.

###  2.3 Continuous CR sources

Based on the radio luminosity function of low- and high-luminosity radio
sources from Ref. [48] which provides the number per volume and luminosity
$\mathrm{d}N/\mathrm{d}V\,\mathrm{d}L_{\rm cr}$ of radio sources, we use the
so-called continuous source function (CSF),

$\Psi_{i}(R^{\prime},\,z)\equiv\frac{\mathrm{d}N_{\rm
cr}(Z^{\prime}_{i})}{\mathrm{d}V\mathrm{d}R^{\prime}\,\mathrm{d}t}=\int\left(\frac{\mathrm{d}N}{\mathrm{d}R^{\prime}}\right)_{\rm
sim}\,w_{\rm R}\big{(}R^{\prime},\hat{R}(L_{\rm
cr})\big{)}\,\frac{\mathrm{d}N}{\mathrm{d}V\,\mathrm{d}L_{\rm
cr}}\,\mathrm{d}L_{\rm cr}\,,$ (2.4)

that has already been introduced in previous works, e.g. Refs. [38, 49], to
obtain the time (redshift) dependent CR emissivity arising from the background
population of sources on large scales. Here, we suppose that the CSF is
homogeneous in space, but even for a discrete distribution of sources we can
still suppose that the universe is homogeneously filled with CRs if they
propagate at least the average source distance [50]. Moreover,
$\left(\mathrm{d}N/\mathrm{d}R^{\prime}\right)_{\rm sim}$ denotes the initial
rigidity distribution of UHECRs that have been simulated in 1D with the UHECR
propagation code CRPropa3 [51, 52, 53]. The weight $w_{R}$ has been introduced
to modify the initial spectrum afterwards (see the Appendix A.1 for more
details). Then the isotropic UHECR intensity is generally given by

$\displaystyle J_{\rm csf}(E+\Delta E,\,t_{\rm obs})$
$\displaystyle=\frac{c}{4\pi}\sum_{\varepsilon,i,j}\Delta
z_{j}\,\,\left|\frac{\mathrm{d}t}{\mathrm{d}z}\,\right|\,\frac{R^{\prime}(\varepsilon)\,\Psi_{i}(R^{\prime}(\varepsilon),\,z_{j})}{\Delta
E}\,,$ (2.5)

where we sum over all CR particles species $i$, all energies
$\varepsilon\in[E,E+\Delta E]$ within the energy bin of interest, and all
source distances $z_{j}\in[0,\,2]$. Since the source sample considered by us
consists of only a few sources, it is appropriate to extend the CSF up to
$z=0$. As shown in the Appendix A.2, the previous equation can be simplified
for a 1D distribution of sources uniform in light travel distance. In a
similar manner, but based on 3D simulations, also the flux $J(E+\Delta
E,\,t_{\rm obs})$ from individual sources with the temporal CR luminosity
given by Eq. (2.2) can be treated; for details see Appendix A.1.

##  3 Conditions for the dominance of individual sources

The contribution (2.5) from the CSF to the UHECR intensity is much softer than
the one from individual sources with a finite life-time, even if their source
spectral indices are comparable. This difference is a result of the magnetic
horizon [27, 29] which prevents that CRs with low rigidity from individual
sources do reach the observer if the source life-time is too short. We expect
that there exists a critical energy for a given source luminosity $L_{0}$
where the contribution from individual sources will start to dominate over the
one from the CSF. Based on the observed spectral features, the hardening above
the ankle at about 5 EeV could indicate this transition. In general, the value
of this critical energy depends on a multitude of parameters, such as the
properties of the extragalactic magnetic field, the UHECR composition, the
escape from the source environment as well as the temporal features of the
activity phase, and the source distance. In addition, the CSF contribution is
strongly dependent on the assumed radio luminosity function as well as the
radio-jet power correlation.

Figure 3: The necessary CR luminosity $L_{0}$ of an individual source for
different escape scenarios (instantaneous– _upper panel_ ; Kolmogorov
diffusion– _middle panel_ ; advection– _lower panel_) to dominate the observed
UHECR spectrum above the ankle (at $E=5\,\text{EeV}$). On the left hand side,
it is supposed that the ejected chemical composition is dominated by carbon
and the source is at a distance $d_{\rm src}=15\,\mathrm{Mpc}$. On the right
hand side, we check the dependence on those assumptions by keeping the
characteristics of the activity time fixed ($t_{\rm max}=-4\,\mathrm{Myr}$ and
$t_{\rm act}=10\,\mathrm{Myr}$). The adopted escape parameters for the
diffusive and advective scenario, respectively, are $l_{\rm src}=100$ kpc,
$B_{\rm src}=10\,\mu\mathrm{G}$ and $v_{\rm bulk}=3000\,\mathrm{km/s}$.

Despite this rather complicated dependencies, we would like to determine the
typical luminosity $L_{0}$ of an individual source at a given distance $d_{\rm
src}$ required to dominate the observed UHECR flux above the ankle. In order
to reduce the parameter space, we consider only cases where the flux is
dominated by a single nucleus type, as mixed scenarios can be interpolated
from that. Furthermore, we suppose a generic maximal rigidity of
$\hat{R}=50\,g_{\rm acc}/\sqrt{1-g_{\rm m}}$ EV for the considered source.
This ensures that this source can in principle explain the observed UHECR flux
up to the highest energies. Moreover, we adopt $g_{\rm m}=4/7$ and $g_{\rm
acc}=1$ both for the individual sources as well as for the CSF. Since both
contributions depend on these parameters in a similar way, the following
results depend only weakly on these values as long as the cutoff is well above
the ankle. Note that for these values the UHECR flux contribution from the CSF
is about a factor 2–10 smaller (dependent on the assumed composition) than the
flux at the ankle. Hence, we ensure that the total flux prediction by
individual sources and the CSF has also the right order of magnitude.

In Fig. 3, we show the necessary CR luminosity $L_{0}$ of an individual source
to dominate the observed UHECR flux above $E=5\,$EeV. All panels are for a
purely turbulent EGMF with a RMS field strength of 1 nG, a Kolmogorov spectrum
and a coherence length of $0.2\,$Mpc using wave modes between 60 and 800 kpc.
For these parameters, CR propagation takes predominantly place in the small-
angle scattering regime. Three different escape scenarios are compared:
instantaneous, advective and diffusive escape. For the latter we adopt
Kolmogorov diffusion, although Bohm diffusion that is caused by, e.g., the
Bell instability [54] could be present at the acceleration site. But since its
size is small compared to the whole source extension of about a few $\times
100\,\text{kpc}$, we expect that the diffusive escape from the source
environment is rather dominated by the Kolmogorov turbulence on large scales.
On the left hand side of Fig. 3, the UHECR flux consists of carbon and the
source is at a distance $d_{\rm src}=15\,\mathrm{Mpc}$. The contribution of
this source is strongest, i.e. the necessary $L_{0}$ value is the smallest, if
the source is close to its final stage, $t_{\rm max}\sim-t_{\rm act}$. In
contrast, young sources with an increasing CR luminosity evolution (i.e.
$t_{\rm max}>0$) need an extraordinarily high present luminosity to exceed the
CSF contribution and to explain the observed flux above the ankle. A larger
activity time decreases the necessary $L_{0}$, since then CRs with a larger
propagation delay introduced by the extragalactic magnetic field as well as
the escape process can still reach the observer. Before for large activity
times the influence of $t_{\rm max}$ vanishes—as all additional CRs have
already reached the observer in the past—there is a minimal CR luminosity
reached. Here, the time $t_{\rm max}$ of maximal CR emissivity corresponds to
the necessary propagation time of the majority of CRs at that energy.

In addition to the obvious decrease of $L_{0}$ with decreasing source
distance, we also notice that the source composition has a significant
influence: This effect is visible on the right hand side of Fig. 3, where we
check the dependence of $L_{0}$ on the composition and the distance keeping
the activity time fixed, choosing $t_{\rm max}=-4\,\mathrm{Myr}$ and $t_{\rm
act}=10\,\mathrm{Myr}$. A light composition yields a lower value of $L_{0}$
than a heavy one, which is predominantly due to the decreasing number of
sources in the CFS contribution that yield energies above the ankle. Hence,
$J_{\rm csf}$ increases with increasing $Z^{\prime}$ despite $J_{\rm
csf}\propto 1/\bar{Z}$, where the average initial charge number $\bar{Z}$
increases (for definition of $\bar{Z}$ see A.1). In contrast, the flux of an
individual source decreases with increasing $\bar{Z}$—especially if it is
close-by so that propagation effects are small—according to the flux
normalization (A.9). Consequently, distant sources ($d_{\rm src}\sim
100\,\text{Mpc}$) typically need an extraordinarily high CR luminosity $L_{0}$
in order to dominate the UHECR flux, in particular if the composition is
dominated by heavy elements.

Comparing the different escape scenarios, there are only minor differences
with respect to the general parameter dependence. However, the additional
delay by diffusive or advective escape generally requires higher $L_{0}$
values at small activity times, as less CRs manage to reach the observer in
the given time, and shifts the minimal $L_{0}$ value towards larger $t_{\rm
act}$. For all scenarios, we need at least $L_{0}\sim 10^{44}\,\mathrm{erg/s}$
to have an individual source dominating over the UHECR contribution from the
large scale population above the ankle. However, we note that the actual
transition energy has a significant impact on these results. Individual
sources with $t_{\rm act}\lesssim 10\,\text{Myr}$ can only provide the
necessary CR power to dominate the CR spectrum above the ankle, if these
source are rather old (i.e. $t_{\rm max}\sim-t_{\rm act}$) and located at a
distance of only a few $\times 10\,\text{Mpc}$.

Figure 4: The maximal distance $\hat{d}_{\rm src}$ of an individual CR source
(to dominate the UHECR flux above the ankle) with a CR luminosity of
$L_{0}=10^{44}\,\text{erg s}^{-1}$ (_left_) and $L_{0}=10^{45}\,\text{erg
s}^{-1}$ (_right_), respectively, dependent on its activity time. It is
adopted that this source is currently observed at its maximal emission stage
and emits only a single type of nucleus as indicated by the colours.

Based on an individual source with a given present CR luminosity, Fig. 4 shows
its maximal distance $\hat{d}_{\rm src}$ dependent on the activity time. Here,
we consider a middle-aged source—i.e. it is currently (under consideration of
the light travel time) in its maximal emission stage. For an optimal choice of
$t_{\rm max}$, which is typically $t_{\rm max}\sim-t_{\rm act}$ (as in that
case the source emissivity has been much higher in the past), the maximal
distance can significantly increase. For instance, for $t_{\rm
max}=-0.7\,t_{\rm act}$ we obtain for certain activity times a maximal
distance that is about twice as large as the one shown in Fig. 4. Independent
of $t_{\rm max}$ all curves necessarily converge towards large $t_{\rm act}$,
approaching the steady state scenario. This limit yields also an upper limit
of the source distance dependent on its CR composition and luminosity, if the
source’s activity has not been significantly higher in the past. Thus, Fig. 4
exposes clearly that the limited life-time of the sources constraints the
potential sources to a quite small number of close-by objects. Hence, in the
case of a strong, turbulent EGMF (of $1\,\text{nG}$ rms strength) and a
present CR luminosity of $L_{0}=10^{45}\,\text{erg s}^{-1}$, there are just a
very few local radio galaxies, such as Centaurus A, Virgo A, Fornax A and 3C
270, that manage to exceed the CSF contribution at the ankle for a heavy
initial composition and $t_{\rm act}\lesssim 1\,\text{Gyr}$. Larger local
samples might be still possible for an appropriate choice of the parameters,
such as a different acceleration efficiency (i.e. different values of $g_{\rm
m}$ and $g_{\rm acc}$) of the individual local sources and the CSF or a
significantly higher CR luminosity of the local source(s) in the past (i.e.
$t_{\rm max}\sim-t_{\rm act}$).

These results already show that the actual parameter space that allows a
dominant CR flux by individual local sources of a given CR luminosity is much
more limited than one may have expected in the first place. Moreover, it still
needs to be quantified, if a given set of a few local CR sources is actually
able to explain the observed UHECR spectrum and its mass composition.

##  4 Constraints from composition data

In this section we give some illustrative examples of the consequences of the
finite life-time of the sources with respect to the resulting energy spectrum
and the mass composition. Because of the large parameter space of this model,
a fit to all UHECR data is beyond the scope of this work. Therefore, we do not
use a dedicated fit algorithm to explain these data but provide different
scenarios that show the general consequences and challenges. As a result, the
proof-of-principle examples shown in Fig. 5 do not explain all data perfectly:
We recognize that in particular the combination of the almost vanishing
variance of $\ln A$ at $7\lesssim E\lesssim 10\,\text{EeV}$ and the rather
minor increase of the mean of $\ln A$ at $6\,\text{EeV}\lesssim E\lesssim
40\,\text{EeV}$ is challenging to reproduce. However, a dedicated examination
of the whole parameter space would clearly enable an improvement of the fit.
In the following, we only account for a single local source to explain the CRs
above the ankle—even though previous works [33] have shown that at least a few
sources are needed to account for the observed anisotropy—since this case
yields quite illustrative results and keeps the parameter space rather small.
However, we will discuss some consequences of multiple local sources in Sec.
5.

We first consider a scenario, hereafter referred to as _Scenario A_ , with an
individual source modelled similar to Fornax A, where UHECRs escape
diffusively and subsequently propagate through a strong turbulent EGMF (with a
RMS field strength of 1 nG, a Kolmogorov spectrum and a coherence length of
$0.2\,$Mpc). Moreover, we suppose that all sources eject UHECRs with an
initial spectral index of $\alpha=2$.

Figure 5: Two proof-of-principle fits to the observed energy spectrum [6] and
the mass composition [55] (small/large caps of the error bar indicated the
statistical/systematical uncertainty) using Fornax A as the single local
source. Some of the major differences between the panels are as follows:
Scenario A (_left panel_): A different source composition at Fornax A and a 1
nG rms EGMF strength. The mass composition data is shown as predicted from the
EPOS-LHC model.
Scenario B (_right panel_): An equal initial composition at all sources and a
0.5 nG rms EGMF strength. The mass composition data is shown as predicted from
the Sibyll2.3c model.

Using a finite life-time of $t_{\rm act}=4.1\,\text{Myr}$ where Fornax A is
currently about in its final stage ($t_{\rm max}\sim-t_{\rm act}$), the left
panel of Fig. 5 shows that the observational data can be explained by the
combination of two contributions: the background of low-luminosity radio
galaxies (predominantly below the ankle due to a rather low acceleration
efficiency $g_{\rm acc}=0.15$) and Fornax A (above the ankle). However, in
this case the initial mass composition of Fornax A and the background sources
needs to be different: While the fraction of hydrogen and helium is about
equal, the background (CSF) sources should eject at least about an order of
magnitude less of the heavier elements as the individual source (where we used
$\sim 10\%$ of carbon, $\sim 1\%$ of neon, $\sim 0.5\%$ of silicon, and $\sim
0.05\%$ of iron). Note that, we used the hadronic interaction model _EPOS-LHC_
[56, 57] to convert the observed $X_{\rm max}$ into $\ln A$, as provided by
Ref. [55]. But in case of Scenario A a similar fit is in principle possible
for any other hadronic interaction model after some adjustment of the initial
compositions.
The right panel of Fig. 5 shows the results for a different scenario,
hereafter referred to as _Scenario B_ , where an equal initial composition at
all sources is adopted that, however, is in comparison much heavier with $\sim
10\%$ of hydrogen, $\sim 72\%$ of helium, $\sim 14\%$ of carbon, $\sim 3\%$ of
neon, $\sim 0.1\%$ of silicon, and $\sim 0.01\%$ of iron. Moreover, a weaker
rms EGMF strength of $0.5\,\text{nG}$ has been used, so that we had to reduce
the source life-time of Fornax A to $t_{\rm act}=1.5\,\text{Myr}$ in order to
obtain a proper hardening of the individual CR nuclei spectra at Earth.
Another important difference with respect to Scenario A is the use of a
somewhat higher acceleration efficiency (with $g_{\rm acc}=0.27$) of the low-
luminosity radio galaxies, so that in total their contribution above the ankle
becomes significantly higher. Note that with respect to the small level of
anisotropy that is observed, a higher (apriori close to isotropic)
contribution by the bulk of radio galaxies reduces the need for a strong EGMF
or a large number of local sources. The composition data in the right panel of
Fig. 5 have been determined using the interaction model of _Sibyll2.3c_ [58],
since for _EPOS-LHC_ , that predicts a lighter chemical composition, the fit
to the composition data becomes significantly worse for Scenario B. Apart from
some minor discrepancies in the total energy spectrum the two scenarios differ
predominantly with respect to their prediction on the chemical composition:
Scenario A yields an almost constant $\text{Var}(\ln A)$ throughout the
considered energy range, that is in very good agreement with the observations
above $5\,\text{EeV}$. But on the downside, the small variance yields a strong
increase of $\langle\ln A\rangle$ around 10 EeV, which leads to some tension
with the observational data above 10 EeV. In contrast, Scenario B shows
generally a higher variance of the chemical composition—in particular below
about 10 EeV—yielding some discrepancies with the observational data, but on
the upside $\langle\ln A\rangle$ increases less around 10 EeV, so that it
agrees much better to the observational data above 10 EeV.
Apart from these differences both scenarios also show several similarities,
such as that Fornax A needs to be rather old, so that its past CR luminosity
has been significantly larger. Note that Fornax A has in comparison to other
close-by radio sources already a rather high present jet power. However, it
also shows indications of enhanced activity in the past [59] and in addition,
the present jet power estimate can also be enhanced under the assumption of a
different radio-to-jet-power correlation, such as the one from Godfrey and
Shabala [41]. Moreover, in both fit scenarios a diffusive escape time of about
$\tau_{\rm esc}\simeq t_{\rm act}/4$ at 10 EeV has been adopted. However, the
results are hardly sensible to the actual escape process, so that similar fits
could also be obtained for an advective scenario with similar values of
$\tau_{\rm esc}$.

Figure 6: The impact of an extended life-time $1.5\,f_{\rm act}\,\text{Myr}$
of Fornax A, where all other parameters are as in Scenario B. In addition to
the Gaussian luminosity profile (solid lines) according to Eq. 2.2, the dotted
lines indicate the simple case for a step function luminosity profile. _Left:_
The energy spectrum of $A=[5,20]$ CR nuclei. _Right:_ The variance of $\ln A$
of the Fornax A contribution. Here the grey region indicates where the total
$\text{Var}(\ln A)$ will be dominated by the contribution from the bulk of
radio galaxies.

Generally, it can be seen that taking into account the finite life-time of the
individual local source a quite accurate agreement with the observational data
at energies $\gtrsim 10\,\text{EeV}$ can be obtained: The hard spectra from
the individual initial elements that result from the finite life-time of the
sources—low energy CR are suppressed as they cannot reach the observer in the
given life-time—enable an increase of $\langle\ln A\rangle$ while keeping
$\text{Var}(\ln A)$ small. But to obtain the observed CR flux at Earth, even
the most powerful local radio sources provide typically not enough CR power in
the case of an $1/E^{2}$ source spectrum, unless theses sources had a
significantly higher CR power (of about an order of magnitude) in the past.
This is illustrated in the left panel of Fig. 6 where we modified $t_{\rm
act}$ by a factor $f_{\rm act}$, but keep all other parameters of Scenario B
fixed—including the time of maximal emission (which is only relevant in case
of the Gaussian luminosity profile) of $t_{\rm max}=-1.4\,\text{Myr}$. Hence,
using $f_{\rm act}\gg 1$ yields $t_{\rm act}\gg-t_{\rm max}$, so that the past
luminosity of the source hardly increases and the resulting flux contribution
converges towards the case of a step function luminosity profile, where the
source has a constant luminosity during $t_{\rm act}$. In case of $f_{\rm
act}\geq 500$ about all CRs manage to reach Earth in time, so that at low
energies the spectrum of CR nuclei with $A=[5,\,20]$ show the initial spectral
behavior of $1/E^{2}$. In addition, it can also be seen in Fig. 6 that nearby
sources such as Fornax A are constrained to have rather short life-times of
only a few Myr, if the EGMF strength is $\lesssim 1\,\text{nG}$, otherwise the
width of the flux contributions from the individual elements increases leading
to an increase of $\text{Var}(\ln A)$ around $10\,\text{EeV}$ by the
increasing contribution of heavy nuclei at lower energies.

We verified that also other powerful local sources could in principle be used
to explain the data above the ankle. However, this requires some adjustment of
the life-time and/or the EGMF strength: Supposing a very high EGMF strength of
the order of $10\,\text{nG}$, which could be present in the Supergalactic
plane, also very nearby sources such as Centaurus A could provide the proper
hardening of the individual CR nuclei spectra if their activity time is on the
order of a few Myr. More distant sources such as Centaurus B would need a much
longer activity time of the order of a few $\times 100\,\text{Myr}$. Note that
on timescales of few Gyr CRs with energies of a few tens of EeV suffer from
significant energy losses by the CMB and the EBL, yielding also an upper limit
on the possible source distance and relevant activity time, respectively. But
for significantly smaller timescales energy losses can be neglected in the
first place and there is a distinct correlation between the source distance
and life-time as well as the EGMF characteristics that enables a proper
description of the data: The average propagation delay $\langle t_{\rm
del}\rangle$ of CRs with a rigidity of about $5\,\text{EV}$ needs to equal the
source life-time, i.e.

$t_{\rm act}\sim\langle t_{\rm del}\rangle\simeq
1.2\,\left(\frac{B}{1\,\text{nG}}\right)^{2}\,\left(\frac{d_{\rm
src}}{10\,\text{Mpc}}\right)^{2}\,\left(\frac{l_{\rm
coh}}{1\,\text{Mpc}}\right)\,\text{Myr}\,,$ (4.1)

assuming the limiting case of small-angle scattering without energy losses
[60]. Hence, if the source is much longer active, $t_{\rm act}\gg\langle
t_{\rm del}\rangle$, there is no flux suppression at low rigidities and in the
opposite case, $t_{\rm act}\ll\langle t_{\rm del}\rangle$, this suppression
affects also the CRs at the highest rigidities leaving not enough CRs to
explain the data.

As a general rule of thumb we can conclude that: The closer the local source,
the shorter needs to be its life-time (or the stronger the EGMF), to provide a
low variance of the mass composition at about 10 EeV.

Figure 7: The possible activity time of UHECR sources as function of their
distance required to obtain a sufficient magnetic suppression at about the
ankle according to Eq. (4.1). We adopted $l_{\rm coh}=1\,\text{Mpc}$ and two
different rms EGMF strengths as indicated by color. Solid line shows the case
of $t_{\rm act}=\langle t_{\rm del}\rangle(5\,\text{EV})$ and the colored
bands refer to $t_{\rm act}>\langle t_{\rm del}\rangle(50\,\text{EV})$ and
$t_{\rm act}<\langle t_{\rm del}\rangle(1\,\text{EV})$. The dark grey band
indicates the typical duration of the active phase of radio galaxies.

The Fig. 7 shows the consequences of this relation on the possible activity
time of UHECR sources dependent on their distance to Earth. Moreover, the grey
bands indicate the range of typical outburst and activity timescales of radio
galaxies, e.g. [61, 62]. However, also shorter outburst intervals (light grey
band) up to $\sim 5\,\text{Myr}$ have been derived from studying X-ray
cavities [63]. Local radio sources with distances of a few $\times
10\,\text{Mpc}$ need therefore a rather strong EGMF and close-by sources like
Centaurus A, that shows indications of enhanced activity within the last $\sim
100\,\text{Myr}$ [59], can clearly not provide a sufficiently low variance of
the mass composition at about 10 EeV. Note that in our model we neglect any CR
contribution from past activity periods which is in particular at low energies
not accurate for $t_{\rm act}\ll 100\,\text{Myr}$, as the quiescent phase is
typically smaller than the active phase, i.e. $0.1-10\,\text{Myr}$ [61, 62].
However, even for a vanishing duration of the quiescent phase, past activity
phases can become negligible if the duration between the different maxima in
the luminosity evolution (in case of a Gaussian luminosity profile) becomes
sufficient long or the total CR power of the past activity/outburst phases is
much smaller than the most recent one. Still, multiple activity or outburst
phases can reduce the hardening of the individual CR nuclei spectra and
introduce an additional CR contribution at low energies.

##  5 Discussion and conclusions

In this work, we have examined the consequences of a finite life-time of UHECR
sources, using as specific example the case of radio galaxies. We have used
the simulations code CRPropa3 [51, 52, 53] to account for the energy losses
and the deflections of UHECRs by the turbulent EGMF. The main motivations to
account for the finite life-time of UHECR sources was to explain the hardness
of the observed UHECR spectra of individual mass groups as a propagation
effect rather than as a characteristics of the ejected source spectra. This
idea has already been investigated in previous works [64]. In contrast to this
reference, we have developed for the specific example of radio galaxies a
concrete realisation of this proposal, taking into account both the individual
properties of selected nearby sources as well as the background from the bulk
of radio galaxies at larger distances. On the downside, we neither address the
impact of the finite life-time on the arrival directions, as recently done by
Ref. [33, 64], nor do we perform a proper fit to the data. Instead we focus on
the basic principles, consequences and challenges of that idea, as summarized
in the following:

  1. $\bullet$

The finite life-time of the CR sources yields a hardening of their spectra at
low rigidities due to the so-called "magnetic horizon" suppression. We showed
in Fig. 5 that even for a standard acceleration spectrum of $1/E^{2}$, these
hardening effects are in principle sufficient to explain the composition data
at the highest energies.

  2. $\bullet$

An additional hardening of the CR spectra at low rigidities by $2-m$, where
$m$ refers to the spectral index of the turbulence spectrum, can result from
the diffusive escape from the source environment. However, this hardening
alone is not sufficient for a standard acceleration spectrum to explain the
data.

  3. $\bullet$

The condition that a present CR source with a given life-time contributes
significantly to the observed UHECRs leads to constraints on its size,
magnetic field properties and wind speeds, as shown in Fig. 1.

  4. $\bullet$

For a homogeneous distribution of CR sources, there is no hardening of the
resulting spectra even for a finite source life-time. Hence, only a few of
these (present or past) sources can be responsible for the CRs at the highest
energies.

  5. $\bullet$

If all source have the same values of the parameters $g_{\rm m}$, $g_{\rm
acc}$ and $\alpha$ (which does still allow for different source spectra
dependent on the given radio luminosity), an individual CR source at a
distance $d_{\rm src}$ needs a CR luminosity $\gtrsim 10^{45}(d_{\rm
src}/10\,\text{Mpc})^{2}\,\text{erg/s}$ at some stage of their evolution, as
shown in Fig. 3. Hence, individual sources at distances $\gtrsim
100\,\text{Mpc}$ are not powerful enough to exceed the CR contribution from
the bulk of radio sources.

  6. $\bullet$

A particular combination of the EGMF strength, the source distance, life-time
and time of maximal emission is needed to obtain the proper width of the
resulting spectra of the individual CR nuclei at Earth. Generally, young
sources whose luminosity evolution is currently still increasing do not
provide enough UHECRs. Including the previous result of a rather small source
distance, we can conclude that the source life-time needs to be $\lesssim
100\,\text{Myr}$. Generally, the closer the local source, the shorter needs to
be its life-time (or the stronger the EGMF), as illustrated by Fig. 7.

  7. $\bullet$

Although we could explain the observed extreme hard spectra of individual mass
groups as an effect of the finite life-times of a few dominating UHECR
sources, this approach also introduce a few challenges: (i) the resulting flux
contributions of (initial) helium and carbon CRs can become shifted to too low
energies to explain the observed spectral behavior at about 10 EeV; (ii) the
necessary elongation of the propagation length (due to EGMF deflections)
typically yields a sharp cut-off at an energy $\leq 100\,\text{EeV}$; (iii)
the observed vanishing variance of $\ln A$ around about 10 EeV is difficult to
explain: The need for hard spectra of the individual elements—that sum up to a
softer spectrum without any significant dips—leads unavoidably to rather large
values of $\mathrm{d}\langle\ln A\rangle/\mathrm{d}E$ that challenge the
observational data of $\langle\ln A\rangle$ around 10 EeV.

  8. $\bullet$

If the local EGMF is rather weak ($B_{\rm rms}\ll 1\,\text{nG}$), the typical
evolution timescales of radio galaxies indicate that either the recent
activity/outburst phase is already too long—especially for nearby sources—or
there is the change of a non-vanishing contribution from past activity phases,
leading to a too weak suppression of CRs at energies of about the ankle by the
magnetic horizon.

Note that the anisotropy data introduces the need for at least a couple of
individual local sources that contribute CRs above the ankle (see e.g. [33]).
The necessary number of sources to explain the observed high level of isotropy
depends strongly on the actual Galactic and extragalactic magnetic field as
well as the source direction and distance. Moreover, it needs to be taken into
account that not only the observed dipol anisotropy but in particular the
(current) absence of a significant quadrupole anisotropy as well as the
intermediate-scale anisotropies above $32\,\text{EeV}$ yield strong
constraints on the necessary amount and location of the sources—as e.g. shown
in Ref. [33]. But to keep the variance of $\ln A$ almost vanishing at about
$10\,\text{EeV}$, all sources need to show a similar spectral behavior of the
CR contribution from different types of nuclei, and hence, if the UHECR
sources are not standard candles—as indicated by the radio-to-jet power
correlation—the amount of relevant sources needs to be small. This conclusion
is in good agreement with what has been found recently in Ref. [15]. Still,
the contribution from multiple sources with slightly different spectral
properties can become useful to resolve the previously mentioned challenges
(i)-(iii), and it relaxes the need for a very high CR luminosity of the
individual sources at some stage of their evolution.

## Acknowledgments

BE acknowledges support by the DFG grant EI 963/2-1.

#### Software:

Some of the results in this paper have been derived using the software
packages Numpy [65], Scipy [66], Pandas [67], Matplotlib [68], Seaborn [69]
and the CR simulation tool CRPropa 3 [51, 52, 53].

##  Appendix A Weighting cosmic rays from sources with a finite life-time

###  A.1 Individual sources

In the following, we introduce the normalization of UHECR intensity based on
the simulation of individual CR candidates from a quasi-instantaneously
bursting source at a time $t^{\prime}$. We use a simulated, initial CR
rigidity distribution

$\left(\frac{\mathrm{d}N}{\mathrm{d}R^{\prime}}\right)_{\rm sim}=\frac{N_{\rm
sim}}{\ln(\hat{R}_{\rm sim}/\check{R}_{\rm sim})\,R^{\prime}}$ (A.1)

in the rigidity range $R^{\prime}\in[\check{R}_{\rm sim},\hat{R}_{\rm sim}]$,
where $N_{\rm sim}$ denotes the total number of simulated CRs. Based on
sufficient statistics this initial distribution can still be modified after
propagation by using the weight

$w_{\rm R}=\left(R^{\prime}/\check{R}_{\rm
sim}\right)^{1-\alpha}\,\exp(R^{\prime}/\hat{R})\,,$ (A.2)

so that we obtain a generic source spectrum with a spectral index $\alpha$ and
an exponential cut-off at $\hat{R}<\hat{R}_{\rm sim}$. The isotropic UHECR
intensity at the time $t_{\rm obs}$ at a (spherical) observer with radius
$r_{\rm obs}$ is generally given by

$J(E+\Delta E,\,t_{\rm
obs})\equiv\frac{\mathrm{d}N}{\mathrm{d}A\,\mathrm{d}E\,\mathrm{d}t\,\mathrm{d}\Omega}=\sum_{\varepsilon,i}\frac{w_{i}(\varepsilon,\,t_{\rm
obs})}{\Delta E\,4\pi\,r_{\rm obs}^{2}}\,,$ (A.3)

where all CR particles species $i$ and energies $\varepsilon\in[E,E+\Delta E]$
are summed using the normalized total weight

$w_{i}(\varepsilon,\,t_{\rm obs})=\frac{1}{4\pi}\,\frac{L_{{\rm
in},i}(\varepsilon,\,t_{\rm obs})}{E_{{\rm in},i}}\,w_{\rm R}(\varepsilon)\,.$
(A.4)

Here, the initial UHECR luminosity

$\displaystyle L_{{\rm in},i}(\varepsilon,\,t_{\rm obs})$ $\displaystyle\equiv
Z^{\prime}_{i}\int_{\check{R}_{\rm sim}}^{\hat{R}_{\rm
sim}}\mathrm{d}R^{\prime}\,\,R^{\prime}\left(\frac{\mathrm{d}N}{\mathrm{d}R^{\prime}}\right)_{\rm
sim}\,w_{\rm R}(\varepsilon)\,\delta(t^{\prime}-[t_{\rm obs}-t(\varepsilon)])$
(A.5) $\displaystyle=\frac{\hat{R}_{\rm
sim}^{2-\alpha}\,E_{\alpha-1}\left(\hat{R}_{\rm
sim}/\hat{R}\right)-\check{R}_{\rm
sim}^{2-\alpha}\,E_{\alpha-1}\left(\check{R}_{\rm
sim}/\hat{R}\right)}{\hat{R}_{\rm
sim}^{2-\alpha}\,E_{\alpha-1}\left(\hat{R}_{\rm
sim}/\hat{R}\right)-\check{R}^{2-\alpha}\,E_{\alpha-1}\left(\check{R}/\hat{R}\right)}\,\frac{f_{i}\,Z^{\prime}_{i}}{\bar{Z}}\,L_{\rm
cr}(t_{\rm obs}-t(\varepsilon))$ (A.6)

is normalized according to the total CR luminosity $L_{\rm cr}$ in the range
$R^{\prime}\in[\check{R},\hat{R}]$ at the time $t_{\rm obs}-t(\varepsilon)$,
where $t(\varepsilon)$ denotes the propagation time of the CR candidate.
Further, we have to account for the average initial charge number
$\bar{Z}=\sum_{i}f_{i}\,Z^{\prime}_{i}$ of the mass composition with a
normalized abundance $f_{i}$ of the initial charge number $Z^{\prime}_{i}$.
Here, $E_{n}(x)$ denotes the exponential integral function. In addition, the
total initial UHECR energy is determined by

$\displaystyle E_{{\rm in},i}$ $\displaystyle\equiv
Z^{\prime}_{i}\int_{\check{R}_{\rm sim}}^{\hat{R}_{\rm
sim}}\mathrm{d}R^{\prime}\,\,R^{\prime}\left(\frac{\mathrm{d}N}{\mathrm{d}R^{\prime}}\right)_{\rm
sim}\,w_{\rm R}(\varepsilon)$ (A.7)
$\displaystyle=\frac{Z^{\prime}_{i}\,N_{\rm sim}\,\left[\hat{R}_{\rm
sim}^{2-\alpha}\,E_{\alpha-1}\left(\hat{R}_{\rm
sim}/\hat{R}\right)-\check{R}_{\rm
sim}^{2-\alpha}\,E_{\alpha-1}\left(\check{R}_{\rm
sim}/\hat{R}\right)\right]}{\ln(\hat{R}_{\rm sim}/\check{R}_{\rm
sim})\,\check{R}_{\rm sim}^{1-\alpha}}\,$ (A.8)

so that the normalized total weight (A.4) yields

$w_{i}(\varepsilon,\,t_{\rm obs})=\frac{f_{i}\,L_{\rm cr}(t_{\rm
obs}-t(\varepsilon))\,\ln(\hat{R}_{\rm sim}/\check{R}_{\rm
sim})\,R^{\prime}{}^{1-\alpha}\,\exp\left(-R^{\prime}/\hat{R}\right)}{4\pi\,\bar{Z}\,N_{\rm
sim}\,\left[\hat{R}_{\rm sim}^{2-\alpha}\,E_{\alpha-1}\left(\hat{R}_{\rm
sim}/\hat{R}\right)-\check{R}_{\rm
sim}^{2-\alpha}\,E_{\alpha-1}\left(\check{R}_{\rm
sim}/\hat{R}\right)\right]}\,.$ (A.9)

###  A.2 Continuous sources

Based on the simulation of individual CR candidates from a uniform 1D
distribution of (discrete) sources in light travel distance between
$\check{d}_{\rm L}(t_{\rm obs})$ and $\hat{d}_{\rm L}$, the isotropic UHECR
intensity is generally given by

$\displaystyle J_{\rm csf}(E+\Delta E,\,t_{\rm obs})$
$\displaystyle=\frac{c}{4\pi}\sum_{\varepsilon,i,j}\Delta
z_{j}\,\,\left|\frac{\mathrm{d}t}{\mathrm{d}z}\,\right|\,\frac{R^{\prime}\,\Psi_{i}(R^{\prime},\,z_{j})}{\Delta
E}$ (A.10)
$\displaystyle\simeq\frac{1}{4\pi}\sum_{\varepsilon,i,j}\frac{\bigl{(}\hat{d}_{\rm
L}-\check{d}_{\rm L}(t_{\rm obs})\bigr{)}}{N_{\rm
sim}}\,\frac{R^{\prime}\,\Psi_{i}(R^{\prime},\,z_{j})}{\Delta E}\,.$ (A.11)

Note, that we need a sufficient number of statistic in the last step of
calculation in order to approximate that

$\Delta z_{j}=\frac{\Delta d_{{\rm
L},j}}{H_{0}^{-1}c}\,\mathcal{E}(z_{j})(1+z_{j})\simeq\frac{\bigl{(}\hat{d}_{\rm
L}-\check{d}_{\rm L}(t_{\rm obs})\bigr{)}}{N_{\rm
sim}\,H_{0}^{-1}c}\,\mathcal{E}(z_{j})(1+z_{j})\,,$ (A.12)

where we introduced the dimensionless Hubble parameter
$\mathcal{E}(z_{j})=H(z_{j})/H_{0}$.

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