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Roberto Ricci

Publications and source records attributed to Roberto Ricci.

At least 19 recordsLinked to original sources

Bivariate incomplete-Bessel kernels for the first nonlinear Vlasov-Maxwell response

The weakly nonlinear response of a homogeneous magnetised plasma is usually written as a double cyclotron-harmonic expansion. This representation is explicit, but the repeated use of the Jacobi-Anger expansion produces long sums of Bessel functions and nested resonance denominators. In the linear problem we recently obtained an alternative formulation by keeping the Larmor phase unexpanded and by evaluating the characteristic integral in terms of the newly introduced incomplete-Bessel function $G_\mu(z,\psi)$. In this article we apply the same idea directly to the first nonlinear Vlasov-Maxwell equation. The nonlinear source contains the linear response of an inner mode. When this linear response is written in incomplete-Bessel form, the outer characteristic integral produces a bivariate orbit-resolvent $G_{\mu,\nu}^{(r)}(z,\psi;w,\chi)$. This function emerges therefore as the natural orbit integral generated by the first nonlinear characteristic problem. We derive the nonlinear distribution function in terms of these bivariate functions, collect the identities needed for the current projection, recover an extension of the classical double-harmonic Liu-Tripathi formula by expansion, and indicate how the nonlinear susceptibility tensor is obtained from bivariate angular contractions.

physics.plasm-ph

A Resurgent Analytic Framework for Indicial Umbral Calculus via Mellin-Barnes and Borel-Laplace Theories

Indicial umbral calculus offers an effective operational framework for manipulating transcendental functions, yet its analytic foundations have long remained only partially understood. In this work, we provide a rigorous analytic realisation of the theory grounded in Mellin-Barnes integrals, Borel-Laplace summation, and resurgent analysis. By elevating umbral operators from formal algebraic symbols to continuous linear functionals, we establish a topological duality akin to Gelfand-Shilov theory. Formal substitutions are thereby replaced by well-defined, continuous pairings between geometric kernels and suitably topologised admissible ground states. Within this framework, divergent umbral evaluations acquire a precise meaning: they emerge as sectorial asymptotic expansions of analytic functions reconstructed exactly by Mellin-Barnes integrals. The associated Stokes phenomena are natively encoded by jump functions in the spectral variable. This leads to a central spectral transmutation law relating entire and rational kernels directly through Gamma regularisation, proving that classical algebraic umbral identities are merely local expansions of a global analytic correspondence. The construction is systematically extended to general umbral functionals via the P\'olya representation of entire functions of exponential type. Explicit examples - including Hankel contours, genuinely Barnes-type integrals, and Lerch transcendents - demonstrate the exact geometric resolution of formal algebraic obstructions. Ultimately, this approach embeds indicial umbral calculus within a unified functional-analytic and resurgent framework, where formal series, analytic continuation, and topological spectral data are intrinsically linked.

math.CA

EP260321a/SN 2026gzf: The Faintest Shock Breakout Associated with a Broad-Lined Supernova

The explosion of a star is first marked by the shock wave breaking out of the stellar surface, producing a burst of ultraviolet and X-ray radiation. These events are observationally rare, despite likely accompanying the majority of supernovae. Here, we report on our multi-wavelength observing campaign of the closest Einstein Probe fast X-ray transient EP260321a at $z=0.0344$. The thermal ($kT=130$ eV) X-ray emission with peak luminosity $1.0\times10^{45}$ erg s$^{-1}$ points to a shock breakout origin. We demonstrate that EP260321a is accompanied by a broad-lined Type Ic supernova, SN 2026gzf. The supernova properties, including its spectral evolution, lightcurve evolution, and expansion velocities, are all typical of the energetic stripped-envelope supernovae associated with gamma-ray bursts. However, deep X-ray upper limits obtained with the \textit{Chandra X-ray Observatory} do not detect an X-ray afterglow, and instead exclude the afterglow of known gamma-ray bursts or fast X-ray transients. If the stellar explosion launched a successful relativistic jet, we require that it had both a low Lorentz factor $\Gamma_0$\,$<$\,$30$ and a kinetic energy $E_\textrm{kin}$\,$<$\,$10^{49}$ erg for a stellar wind density of $A_*$\,$\gtrsim$\,$1$. We propose that EP260321a originated from a mildly relativistic, weak outflow that was choked by the progenitor star. This scenario is capable of naturally explaining its low X-ray luminosity and lack of prompt gamma-ray emission. EP260321a bridges the gap between SN 2008D and low-luminosity GRBs, suggesting a greater diversity in the physical parameters of stripped stars as they undergo terminal collapse.

astro-ph.HE

Umbral methods, function factorisation and generalisation of the Fourier transform method

We propose a systematic way to construct trigonometric-like functions beyond the classical sine--cosine pair by factorising rational umbral operators. The guiding idea is simple: the usual trigonometric functions may be viewed as cyclic components arising from a finite factorisation, and the same principle can be extended to an $n$-fold decomposition of rational umbral expressions. For each integer $n\geq 2$, the construction produces $n$ functions which play the role of higher-order trigonometric components: their sum reconstructs the corresponding umbral function, while the individual components isolate the different cyclic sectors of its expansion. The construction is developed first in the formal umbral setting. The quadratic case $n=2$ gives the Gaussian trigonometric functions, in which the cosine-like component is a Gaussian and the sine-like component is its natural umbral companion. The cubic case $n=3$ yields a three-component cyclic system and shows how the same idea extends beyond the usual even--odd decomposition. These examples suggest that trigonometric factorisation is not restricted to ordinary rotations, but belongs to a broader cyclic principle in umbral calculus. We then reinterpret the same formal identities through the recently developed analytic umbral framework. In this second step, the cyclic components are realised by Mellin--Barnes pairings, and the root-of-unity decomposition is related to the splitting of the corresponding spectral kernel. This analytic formulation provides contour representations, local expansions, and sectorial asymptotics for the functions obtained formally. Finally, we indicate how the same cyclic kernels act on Fourier transforms. The resulting framework presents higher-order umbral trigonometric functions as natural cyclic components of factorised rational or exponential umbral operators.

math.GM

Analytic umbral transmutations and Bessel moments

We develop an analytic umbral approach to Bessel moments, using them as a concrete testbed justifying the passage from formal indicial umbral calculus to Mellin--Barnes umbral transmutation theory. [...] While the formal procedure reproduces the correct results in suitable convergence chambers, it may lead to non-admissible hypergeometric expansions at physically relevant parameter values. The cubic moment provides the basic example [...] We show that this obstruction is removed by replacing the purely formal expansion with an analytic umbral transmutation. In this setting, exponential umbral pairings are interpreted through Mellin--Barnes integrals, and Ramanujan's Master Theorem acts as an inverse selection principle for the spectral ground state, or clock, associated with a given Bessel product. The factorisation \(J_0^3=J_0J_0^2\) produces two distinct clocks and reduces the cubic full-line moment to a one-dimensional Barnes integral, equivalently to a Meijer \(G\)-function. This gives the classical value of the cubic Bessel moment and clarifies why the divergent Appell realisation is only a local representation of a globally meaningful umbral identity. The same mechanism is then applied to scaled cubic products and to the fourth Bessel moment. [...] The fifth moment marks the first genuinely higher-rank case: the natural umbral grouping leads to a bivariate Barnes transmutation rather than to an ordinary Meijer \(G\)-function. Finally, we discuss real fractional powers \(J_0^\alpha\), \(\alpha>2\), showing that the same interpretation persists beyond integer moments. [...] The resulting picture identifies Bessel moments as values of effective umbral transmutations and separates the global analytic meaning of the umbral representation from the local convergence properties of its hypergeometric residue expansions.

math.GM

An extremely bright slow-rising afterglow from an off-axis jet in GRB 260310A

We present a multi-wavelength study of GRB 260310A, a nearby long-duration gamma-ray burst at $z\simeq0.153$ associated with a broad-lined Type Ic supernova. Despite its modest prompt gamma-ray output, $E_{\gamma,\rm iso}\simeq3.5\times10^{50}$ erg, GRB\,260310A exhibits one of the brightest afterglows ever observed in the X-ray, optical, and radio bands. Its apparent brightness is not its only remarkable feature. The optical afterglow displays a delayed onset, characterized by a slow rising phase, with slope $\alpha\approx-1$, and a late peak at $\approx$0.1 d. We argue that the combination of weak prompt emission, hard peak energy, and late afterglow onset is naturally explained by a GRB jet viewed off-axis. The radio spectral energy distributions are consistent with synchrotron radiation and indicate the presence of both reverse- and forward-shock components, thus providing a first test of reverse-shock models in an off-axis geometry. The X-ray afterglow displays a prominent rebrightening, monitored for up to $\approx$68 d with no evidence of spectral evolution. A low level of linear polarization, $\Pi\approx1.7\%$, is measured at 15 GHz at $T_0+55$ d and suggests that, at these late times, the forward-shock is the dominant emission component from radio to X-rays. This late-time rebrightening represents a critical test for the two-component jet model. If interpreted as the emergence of a narrow jet core viewed further off-axis, it would imply extreme luminosities and energetics for an on-axis observer.

astro-ph.HE

A High-frequency Geodetic VLBI Experiment for Optical Clock Comparison

An intercontinental metrological clock comparison between Italy and the Republic of Korea was performed by means of geodetic K-band VLBI observations. The comparison involved the hydrogen masers (H-masers) used at Medicina and Sejong radio telescopes. The same clocks were simultaneously compared by a satellite link and by high-precision optical clocks maintained at the National Metrology Institutes, KRISS in Korea and INRIM in Italy, and delivered to VLBI antennas via optical fiber. The H-masers frequency difference was estimated by extrapolating the clock rate from VLBI data using two geodetic VLBI software. This was subsequently compared with clock differences derived by satellite link and by local optical clocks. Results obtained with different approaches were in agreement at the level of $10^{-15}$ s/s. This pilot study demonstrates that standard high-frequency (K-band) geodetic VLBI campaigns could be a viable approach to conduct intercontinental clock comparisons, now only possible via satellite links. This uncertainty can be reduced thanks to the planned installation of new-generation, broadband, high-frequency receivers on the involved telescopes. K/Q/W-band geodetic observations will allow an improvement of the accuracy of the resulting group delays through broad bandwidth synthesis from 20 to 100 GHz. Furthermore, the Frequency Phase Transfer (FPT) method will also be explored together with the use of PCAL systems installed at the radio telescopes to improve phase stability and thus allow a better estimation of the station clock parameters.

astro-ph.IM

Le Roy, Lerch and Legendre chi functions and generalised Borel-Le Roy transform

The Le Roy function has been the focus of intensive research in recent years, owing both to its relevance in analysis and its versatility in applications involving fractional differential operators. Other special functions - such as the Lerch transcendent and the Legendre chi function - have found applications ranging from Bose-Einstein and Fermi-Dirac statistics in physics to pure mathematical investigations involving polylogarithms and Dirichlet L-series. In this article, we present a unified framework based on a recent reformulation of Indicial Umbral Theory (IUT) grounded in the formal theory of power series. Within this setting, we study the properties and generalisations of these special functions. In particular, we build upon the revised formulation of IUT to incorporate the role of the Borel-Le Roy transform, and to explore the extension of the formalism to divergent series via appropriate resummation techniques.

math.CA

Minutes-long soft X-ray prompt emission from a compact object merger

Compact object mergers are multi-messenger sources and known progenitors of some gamma-ray bursts, bright flashes of high-energy radiation powered by a central engine, either an accreting black hole or a neutron star. Our understanding of these events has so far been shaped primarily by observations in the gamma-ray band, leaving their prompt phase poorly constrained at lower energies. A long-lasting ($\approx$100 s) engine-driven X-ray emission was discussed to explain rapidly fading X-ray afterglows following several ($\approx$30%) bursts of short ($\lesssim$2 s) duration. However, this prompt X-ray component was not directly observed and past candidates were not confirmed. Here we report the discovery of EP250704a containing a minutes-long ($\sim$560 s) flash of soft (0.5--4 keV) X-rays immediately following the short ($\sim$0.4 s) GRB 250704B. The variability and spectral shape of this emission are inconsistent with the canonical picture of a hard, accretion-powered spike followed by a standard external-shock afterglow. Instead, the long-soft bump points to a distinct phase of prompt emission in X-rays, which would not have been detected without the soft X-ray coverage of Einstein Probe. The detection of a prompt soft X-ray counterpart in an otherwise ordinary short GRB shows that long-lasting X-ray emission is likely a common feature of merger-driven bursts and a promising electromagnetic counterpart to gravitational wave sources.

astro-ph.HE

A new class of special functions arising in plasma linear susceptibility tensor calculations

We investigate some fundamental properties of a peculiar class of special functions strictly related to Bessel, Anger and Weber functions, whose introduction was originally motivated by linear susceptibility tensor calculations in a hot, magnetised plasma. We show that these functions are solutions of an inhomogeneous Bessel ODE, with specified initial conditions and a distinct right-hand-side term fulfilling the Nielsen's requirement. Beside deriving recurrence relations and an alternative representation involving incomplete Anger-Weber functions, we show that these functions admit a simple series expansion in terms of Bessel functions of integer order, obtained by resorting to the Jacobi-Anger formula. In plasma applications this eventually leads to expressions involving infinite sums of products of Bessel functions, not particularly apt to numerical evaluation ought to their slow convergence rate when the particle's gyro-radius is larger than the wavelength. By exploiting the previously determined recurrence properties of the new class of functions we present a particularly simple derivation of the linear susceptibility tensor that enables to avoid this inconvenience.

physics.plasm-ph

Umbral theory and the algebra of formal power series

Umbral theory, formulated in its modern version by S. Roman and G.~C. Rota, has been reconsidered in more recent times by G. Dattoli and collaborators with the aim of devising a working computational tool in the framework of special function theory. Concepts like umbral image and umbral vacuum have been introduced as pivotal elements of the discussion, which, albeit effective, lacks of generality. This article is directed towards endowing the formalism with a rigorous formulation within the context of the formal power series with complex coefficients $(\mathbb{C}[[ t ]], \partial)$. The new formulation is founded on the definition of the umbral operator $\operatorname{\mathfrak{u}}$ as a functional in the "umbral ground state" subalgebra of analytically convergent formal series $\varphi \in \mathbb{C}\{t\}$. We consider in detail some specific classes of umbral ground states $\varphi$ and analyse the conditions for analytic convergence of the corresponding umbral identities, defined as formal series resulting from the action on $\varphi$ of operators of the form $f(\zeta \operatorname{\mathfrak{u}}^\mu)$ with $f \in \mathbb{C}\{t\}$ and $\mu, \zeta \in \mathbb{C}$. For these umbral states, we exploit the Gevrey classification of formal power series to establish a connection with the theory of Borel-Laplace resummation, enabling to make rigorous sense of a large class of -- even divergent -- umbral identities. As an application of the proposed theoretical framework, we introduce and investigate the properties of new umbral images for the Gaussian trigonometric functions, which emphasise the trigonometric-like nature of these functions and enable to define the concept of "Gaussian Fourier transform", a potentially powerful tool for applications.

math.CA

AT2025ulz and S250818k: Deep X-ray and radio limits on off-axis afterglow emission and prospects for future discovery

The first joint electromagentic (EM) and gravitational wave (GW) detection, known as GW170817, marked a critical juncture in our collective understanding of compact object mergers. However, it has now been 8 years since this discovery, and the search for a second EM-GW detection has yielded no robust discoveries. Recently, on August 18, 2025, the LIGO-Virgo-KAGRA collaboration reported a low-significance (high false alarm rate) binary neutron star merger candidate S250818k. Rapid optical follow-up revealed a single optical candidate AT2025ulz ($z=0.08484$) that initially appeared consistent with kilonova emission. We quickly initiated a set of observations with \textit{Swift}, \textit{XMM-Newton}, \textit{Chandra}, and the Very Large Array to search for non-thermal afterglow emission. Our deep X-ray and radio search rules out that the optical rebrightening of AT2025ulz is related to the afterglow onset, reinforcing its classification as a stripped-envelope supernova (SN 2025ulz). We derive constraints on the afterglow parameters for a hypothetical binary neutron star merger at the distance of AT2025ulz ($\approx 400$ Mpc) based on our X-ray and radio limits. We conclude that our observational campaign could exclude a GW170817-like afterglow out to viewing angles of $\theta_\textrm{v}\approx 12.5$ degrees. We briefly discuss the prospects for the future discovery of off-axis afterglows.

astro-ph.HE

Evidence of Energy Injection in the Short and Distant GRB 250221A in a High Density Environment

We present the photometric and spectroscopic analysis of the short-duration GRB 250221A ($T_{90}=1.80\pm0.32$ s), using a data set from the optical facilities COLIBR\'I, the Harlingten 50~cm Telescope, and the Very Large Telescope. We complement these observations with data from the Neil Gehrels Swift Observatory and the Einstein Probe, as well as radio observations from the Very Large Array. GRB 250221A is among the few short GRBs with direct afterglow spectroscopy, which gives a secure redshift determination of $z=0.768$ and allows the unambiguous identification of the host as a galaxy with a star-formation rate of $\sim3\,M_\odot\,{\rm yr}^{-1}$. The X-ray and optical light curves up to $T_0+3\times 10^4$ s (where $T_0$ refers to the GRB trigger time) are well described by forward-shock synchrotron emission in the slow-cooling regime within the standard fireball framework. However, at $T_0 \sim 5\times 10^4$ s, both the X-ray and optical bands exhibit an excess over the same interval, which we interpret as evidence of energy injection into a jet with a half-opening angle of $\theta_j=11.5^{\circ}$ through a refreshed shock powered by late central engine activity or a radially stratified ejecta. The burst properties (duration, spectral hardness, peak energy, and location in the Amati plane) all favour a compact binary merger origin. However, our modelling of the afterglow suggests a dense circumburst medium ($n\sim80$ cm$^{-3}$), which is more typical of a collapsar environment.

astro-ph.HE

Exploring the connection between compact object mergers and fast X-ray transients: The cases of LXT 240402A and EP250207b

The connection between compact object mergers and some extragalactic fast X-ray transients (FXRTs) has long been hypothesized, but never ultimately established. In this work, we investigate two FXRTs, the LEIA X-ray Transient LXT 240402A and the Einstein Probe EP250207b, whose precise positions lie close to nearby ($z\!\lesssim\!0.1$) quiescent galaxies with negligible probability of chance coincidence, identifying them as particularly promising cases of merger-driven explosions in the local Universe. We used Chandra to derive accurate localizations for both events and secure otherwise ambiguous associations with their optical counterparts. Deep optical and near-infrared observations with VLT, GTC, and LBT were performed to characterize the surrounding environment and search for kilonova emission, the hallmark of neutron star mergers. Complementary early-time X-ray monitoring with Swift and Einstein Probe was used to constrain the non-thermal afterglow. We find that both FXRTs remain compatible with a compact binary merger progenitor, which produced low-mass ejecta and kilonova emission subdominant to the afterglow. However, alternative explanations such as a distant ($z\!\gtrsim\!1$) core-collapse supernova cannot be conclusively ruled out.

astro-ph.HE

Radio observations point to a moderately relativistic outflow in the fast X-ray transient EP241021a

Fast X-ray transients (FXRTs) are short-lived X-ray outbursts with diverse progenitor scenarios, including compact object mergers, stellar core-collapses and tidal disruption events. The Einstein Probe (EP) has enabled the rapid discovery and follow-up of dozens of FXRTs, revealing that while some of them overlap with traditional gamma-ray bursts (GRBs), a larger fraction of FXRTs have no associated gamma-ray counterpart down to deep limits. The origin of these gamma-ray dark FXRTs and their connection to the diverse landscape of stellar explosions remains an open question, which can be tackled through the study of their multi-wavelength counterparts and environment. In this paper, we present long-term radio observations of the gamma-ray dark EP241021a, which exhibits sustained radio emission for over 100 days, placing it among the longest-lived radio afterglows. We detect signature of interstellar scintillation in early epochs, allowing us to constrain the angular size and Lorentz factor of the emitting region. Our observations point to an outflow that is at least mildly relativistic with Lorentz factor > 4. Afterglow modeling favors a moderately relativistic and collimated outflow interacting with a low-density interstellar medium. The derived beaming-corrected kinetic energy and low radiative efficiency are consistent with a standard relativistic explosion which did not produce bright gamma-rays. Alternatively, a highly-relativistic structured jet remains consistent with our observations if seen substantially off-axis. In the latter case, the initial X-ray flare detected by EP would be caused by the slower ejecta from the lateral wings intercepting our line of sight rather than by traditional prompt-emission mechanisms within the jet core.

astro-ph.HE

The curious case of EP241021a: Unraveling the mystery of its exceptional rebrightening

Fast X-ray Transients (FXTs) are a rare and poorly understood phenomenon with a variety of possible progenitors. The launch of the Einstein Probe (EP) mission has facilitated a rapid increase in the real-time discovery and follow-up of FXTs. We focus on the recent EP discovered transient EP241021a, which shows a peculiar panchromatic behavior. We obtained optical and near-infrared multi-band imaging and spectroscopy with the Fraunhofer Telescope at Wendelstein Observatory, the Hobby-Eberly Telescope, and the Very Large Telescope over the first 100 days of its evolution. EP241021a was discovered by EP as a soft X-ray trigger, but was not detected at gamma-ray frequencies. The observed soft X-ray prompt emission spectrum is consistent with non-thermal radiation, which requires at least a mildly relativistic outflow with bulk Lorentz factor $\Gamma \gtrsim 4$. The optical and near-infrared lightcurve has a two component behavior where an initially fading component $\sim t^{-1}$ turns to a rise steeper than $\sim t^{4}$ after a few days before peaking at $M_r\approx -21.8$ mag and quickly returning to the initial decay. The peak absolute magnitude is the most luminous optical emission associated to an FXT, superseding EP240414a. Standard supernova models are unable to reproduce either the absolute magnitude or rapid timescale ($<2$ d) of the rebrightening. The X-ray, optical and near-infrared spectral energy distributions display a red color $r-J\approx 0.8$ mag, and point to a non-thermal origin ($\nu^{-1}$) for the broadband emission. By considering a gamma-ray burst as a plausible scenario, we favor a refreshed shock as the cause of the rebrightening. This is consistent with the inference of an at least mildly relativistic outflow based on the prompt trigger. Our results suggest a link between EP discovered FXTs and gamma-ray bursts.

astro-ph.HE

COMAP Galactic Science I: Observations of Spinning Dust Emission at 30GHz in Dark Clouds Surrounding the λ-Orionis Hii Region

Anomalous Microwave Emission (AME) is a major component of Galactic emission in the frequency band 10 to 60 GHz and is commonly modelled as rapidly rotating spinning dust grains. The photodissociation region (PDR) at the boundary of the $λ$-Orionis Hii region has been identified by several recent analyses as one of the brightest spinning dust emitting sources in the sky. We investigate the Barnard 30 dark cloud, a dark cloud embedded within the $λ$-Orionis PDR. We use total-power observations of Barnard 30 from the CO Mapping Array Project (COMAP) pathfinder instrument at 26 to 34GHz with a resolution of 4.5 arcminutes alongside existing data from Planck, WISE, IRAS, ACT, and the 1.447GHz GALFACTS survey. We use aperture photometry and template fitting to measure the spectral energy distribution of Barnard 30. We find that the spinning dust is the dominant emission component in the 26 to 34GHz range at the $7 σ$ level ($S_{30GHz} = 2.85\pm0.43$Jy). We find no evidence that polycyclic aromatic hydrocarbons are the preferred carrier for the spinning dust emission, suggesting that the spinning dust carriers are due to a mixed population of very small grains. Finally, we find evidence for variations in spinning dust emissivity and peak frequency within Barnard 30, and that these variations are possibly driven by changes in dust grain population and the total radiation field. Confirming the origin of the variations in the spinning dust spectrum will require both future COMAP observations at 15GHz combined with spectroscopic mid-infrared data of Barnard 30.

astro-ph.GA

Modelling of Long-Term Afterglow Counterparts to Gravitational Wave Events: The Full View of GRB 170817A

The arrival of gravitational wave astronomy and a growing number of time-domain focused observatories are set to lead to a increasing number of detections of short gamma-ray bursts (GRBs) launched with a moderate inclination to Earth. Being nearby events, these are also prime candidates for very long-term follow-up campaigns and very-long-baseline interferometry (VLBI), which has implications for multi-messenger modelling, data analysis, and statistical inference methods applied to these sources. Here we present a comprehensive modelling update that directly incorporates into afterglowpy astrometric observations of the GRB position, Poissonian statistics for faint sources, and modelling of a trans-relativistic population of electrons. We use the revolutionary event GW170817 to demonstrate the impact of these extensions both for the best-fit physics parameters and model selection methods that assess the statistical significance of additional late-time emission components. By including in our analysis the latest Chandra X-ray observations of GRB 170817A, we find only weak evidence (less than two sigma) for a new emission component at late times, which makes for a slightly more natural fit to the centroid evolution and prediction for the external medium density.

astro-ph.HE