Influence of cosmic voids on the propagation of TeV gamma-rays and the puzzle of GRB 221009A

ABSTRACT

The recent detection of gamma-ray burst GRB 221009A has attracted attention due to its record brightness and first-ever detection of |$\gtrsim 10$| TeV gamma-rays from a GRB. Despite being the second-nearest GRB ever detected, at a redshift of |$z=0.151$|⁠, the distance is large enough for severe attenuation of gamma-ray flux at these energies due to |$\gamma \gamma \rightarrow e^\pm$| pair production with the extragalactic background light (EBL). Here, we investigate whether the presence of cosmic voids along the line of sight can significantly impact the detectability of very high energy (VHE, |$\gt $|100 GeV) gamma-rays from distant sources. Notably, we find that the gamma–gamma opacity for VHE gamma-rays can be reduced by approximately 10 per cent and up to 30 per cent at around 13 TeV, the highest-energy photon detected from GRB 221009A, for intervening cosmic voids along the line of sight with a combined radius of 110 Mpc, typically found from void catalogues, and 250 Mpc, respectively. This reduction is substantially higher for TeV photons compared to GeV photons, attributable to the broader target photon spectrum that TeV photons interact with. This finding implies that VHE photons are more susceptible to variations in the EBL spectrum, especially in regions dominated by cosmic voids. Our study sheds light on the detection of |$\gtrsim 10$| TeV gamma-rays from GRB 221009A in particular, and on the detection of extragalactic VHE sources in general.

1 INTRODUCTION

Gamma-rays from sources located at a cosmological distance and with energies greater than the threshold energy for electron–positron (⁠|$e^\pm$|⁠) pair production can be annihilated due to |$\gamma \gamma$| absorption by low-energy extragalactic background photons. These intergalactic |$\gamma \gamma$| absorption signatures have attracted great interest in astrophysics and cosmology due to their potential to indirectly measure the extragalactic background light (EBL; e.g. Finke & Razzaque 2009; Sánchez-Conde et al. 2009; Domínguez et al. 2011a; Domínguez, Sánchez-Conde & Prada 2011b; Sinha et al. 2014; Abdalla et al. 2017, 2021), and thereby probe the cosmic star formation history (e.g. Dwek & Krennrich 2013; Abdollahi et al. 2018). Extensive research has been conducted to study the |$\gamma \gamma$| absorption imprints on the gamma-ray fluxes from distant sources (e.g. Aharonian et al. 2006; Razzaque, Dermer & Finke 2009; Finke, Razzaque & Dermer 2010; Kneiske & Dole 2010; Gilmore et al. 2012; Biteau & Williams 2015; Domínguez & Ajello 2015; Tavecchio & Bonnoli 2016; Dzhatdoev et al. 2017, 2019; Martínez-Huerta, Lang & de Souza 2018; Korochkin, Rubtsov & Troitsky 2019; Lang, Martínez-Huerta & de Souza 2019; Long et al. 2020; Li & Ma 2021; Terzić, Kerszberg & Strišković 2021; Troitsky 2021; Finke et al. 2022; Alves Batista et al. 2023; Azzam & Hasan 2023; Pérez de los Heros & Terzić 2023; Qin et al. 2023). Observations have revealed that the spectra of some distant blazars at very high energies (VHE; E|$\gt $| 100 GeV) appear to be unexpectedly harder after correcting for |$\gamma \gamma$| absorption by the EBL (e.g. Furniss et al. 2013, 2015; Abdalla & Böttcher 2017). One potential explanation for the observed spectral hardening could be a lower density of the EBL than what is currently estimated by the existing models. The EBL might exhibit an inhomogeneous distribution, with the presence of cosmic voids along the line of sight to distant blazars. These voids could feasibly lessen the attenuation of gamma-rays emitted by these blazars, thereby rendering them detectable at greater redshifts (e.g. Furniss et al. 2013; Abdalla & Böttcher 2017; Kudoda & Faltenbacher 2017, 2018).

The recent detection of VHE gamma-rays from the gamma-ray burst GRB 221009A at a redshift of |$z = 0.151$| (Aharonian et al. 2023; Burns et al. 2023; Frederiks et al. 2023; Klinger et al. 2023; Lesage et al. 2023; Malesani et al. 2023; Mbarubucyeye et al. 2023; Wang et al. 2023; Williams et al. 2023; Zhao, Zhou & Wang 2023; Banerjee et al. 2024; Castro-Tirado et al. 2024; Khangulyan, Aharonian & Taylor 2024; Ror et al. 2024; Xia et al. 2024; Zheng et al. 2024; Zhang et al. 2024a) by the Large High Altitude Air Shower Observatory (LHAASO; Huang et al. 2022; Cao et al. 2023b) has increased focus on EBL models and fundamental physics, since the |$\gamma \gamma$| absorption effect is expected to reduce the source flux by a factor as large as |$\sim 2.2\times 10^{4}$| at 18 TeV (Finke & Razzaque 2023), the energy of the highest-energy photon detected by LHAASO. In its recent publication (Cao et al. 2023a), the LHAASO Collaboration conservatively estimated the highest-energy photon to be a 13 or 18 TeV photon, depending on whether the intrinsic spectrum is a power-law function with an exponential cut-off or a log-parabola, respectively, after considering the systematic uncertainties in the photon-energy determination. In order to rigorously evaluate various EBL models, the study detailed in Cao et al. (2023a) adopts a methodological approach. This involves initially characterizing the intrinsic spectrum of gamma-ray bursts (GRBs) utilizing a log-parabolic (LP) function, |$dN/dE = J_{0} (E/1\rm TeV)^{-a- b\log (E/1\rm TeV)}$|⁠, or a power-law function with exponential cut-off (PLEC), |$dN/dE = J_{0} (E/1\rm TeV)^{-\alpha } e^{-E/E_{\rm cut}}$|⁠, and subsequently, obtaining the parameters of the functions from fitting LHAASO data. This systematic procedure facilitates a comprehensive and quantitative analysis of the different EBL models under consideration. Precise parameters were obtained by Cao et al. (2023a) from the fitting process for the time intervals 230–300 and 300–900 s after |$T_{90}$|⁠.

Although there is a non-zero probability that the highest-energy photon originated from a galactic HAWC source (e.g. Fraija, Gonzalez & HAWC Collaboration 2022). Nevertheless, explaining even an |$\gtrsim 10$| TeV photon from this source is difficult. One possibility is via ultra-high energy cosmic rays (UHECRs) emitted from GRB 221009A which produce gamma-rays by interacting with CMB and EBL photons along our line of sight (e.g. Alves Batista 2022; Das & Razzaque 2023; Huang et al. 2023b; He, Zhang & Fan 2024; Kalashev et al. 2024). Exotic physics involving Lorentz invariance violation (LIV; e.g. Baktash, Horns & Meyer 2022; Dzhappuev et al. 2022; Finke & Razzaque 2023; Li & Ma 2023; Zheng et al. 2023; Abdalla et al. 2024; The LHAASO collaboration 2024; Yang, Bi & Yin 2024) or mixing of gamma-rays with Axion-like particles (ALPs; e.g. Baktash et al. 2022; Troitsky 2022; Galanti et al. 2023; Gonzalez et al. 2023; Nakagawa et al. 2023; Wang & Ma 2023; Gao et al. 2024; Rojas et al. 2024; Troitsky 2024) or radiative decay during the propagation of a hypothetical heavy neutrino produced by GRB 221009A (e.g. Cheung 2022; Brdar & Li 2023; Guo et al. 2023; ShivaSankar K. et al. 2023; Smirnov & Trautner 2023; Huang et al. 2023a; Aiello et al. 2024; Dhuria 2024) have also been proposed to evade |$e^\pm$| pair-production with EBL photons.

In this work, we seek to explain the detection of tens of TeV gamma-rays from GRB 221009A as a result of a reduced EBL intensity, and hence weaker |$\gamma \gamma$| absorption, due to presence of a cosmic void, regions devoid of stars and dust, along the line of sight. In previous works (Abdalla & Böttcher 2017, 2018a, b), we investigated the effect of such a cosmic void along the line of sight to a distant gamma-ray source on the resulting angle-averaged EBL energy density by considering the inhomogeneity of the direct starlight component of the EBL model by Razzaque et al. (2009) in the optical-ultraviolet. In this work, we also consider the inhomogeneity of the dust component of the EBL in the infrared-optical and accordingly modify the model by Finke et al. (2010). We also considered the potential existence of cosmic voids, incorporating Domínguez et al. (2011a), Saldana-Lopez et al. (2021), and Finke et al. (2022) EBL models. Specifically, we examined the effect of a reduced gamma-ray opacity due to the presence of such voids on the intrinsic spectra of GRB 221009A. We organize the paper in the following way. In Section 2, we summarize observations of voids from large-scale surveys and perform a Monte Carlo study to estimate the potential amount of void that could exist between us and a source located at the same redshift as GRB 221009A. In Section 3, we describe the model set-up and the method used to evaluate the EBL characteristics and the resulting |$\gamma \gamma$| opacity. The results and conclusions are presented in Sections 4 and 5, respectively. Throughout this paper the following cosmological parameters are assumed: |$H_{0} = 70$| km s|$^{-1}$| Mpc|$^{-1}$|⁠, |$\Omega _{\mathrm{ m}} = 0.3$|⁠, and |$\Omega _{\Lambda } = 0.7$|⁠.

2 THE PRESENCE OF COSMIC VOIDS IN LARGE-SCALE COSMIC SURVEYS

2.1 Observational considerations

In recent years, there has been an intensive study of the large-scale structure and dynamics of galaxy distribution. The role of cosmic voids as dynamically distinct elements in the universe’s large-scale structure has been firmly established through both theoretical and observational frameworks (e.g. Hoffman & Shaham 1982; Abazajian et al. 2009; Pan et al. 2012; Tavasoli, Vasei & Mohayaee 2013; Sutter et al. 2015; Wojtak, Powell & Abel 2016; Douglass, Veyrat & BenZvi 2023). The most extensive void catalogue currently available is designed to enable precise cosmological experiments using voids and to facilitate more accurate testing of galaxy formation theories is SDSS DR7 (e.g. Sutter et al. 2015; Douglass et al. 2023). Characterizing cosmic voids provides distinct insights into the large-scale distribution of galaxies, their evolution, and the prevailing cosmological paradigm (Micheletti et al. 2014). Traditionally, the quest to identify voids within galactic distributions has employed one of two methodologies: first, by identifying empty spherical regions amid the galactic expanse, or second, through the implementation of a watershed algorithm designed to interconnect regions of low density. The preeminent catalogues of cosmic voids, utilized extensively in contemporary astrophysical research, have been derived primarily from the seventh data release of the Sloan Digital Sky Survey SDSS DR7 (see Abazajian et al. 2009; Sutter et al. 2015). Recently, Douglass et al. (2023) developed a series of cosmic void catalogues, utilizing data from the SDSS DR7. Their approach incorporated three sophisticated void-finding algorithms: VoidFinder and two algorithms based on the ZOBOV framework (Neyrinck 2008), namely VIDE and REVOLVER. This comprehensive analysis led to the identification of over 500 cosmic voids, each exhibiting radii in excess of 10 Mpc h⁻¹. The research delved into various characteristics of these voids, including their radial density profiles and the spatial distribution of galaxies within them. Furthermore, this study’s results were cross-referenced with simulated data from the Horizon Run 4 N-body simulation’s mock galaxy catalogues, demonstrating a remarkable congruence that substantiated the efficacy of their methodologies and the validity of their findings.To provide a clear visual representation, we have plotted the locations of voids of different sizes in the sky in Fig. 1 that demonstrates the quantity of voids within each grid cell. These cells specifically fall within the ranges of 140|$^\circ$| to 240|$^\circ$| in Right Ascension (RA) and 0|$^\circ$| to 60|$^\circ$| in Declination (Dec.). Each cell depicted in Fig. 1, encompasses the cumulative voids (total effective radius) up to a redshift of |$z=0.114$|⁠. This value represents the maximum redshift encompassed by the VoidFinder updated Void Catalogs from Douglass et al. (2023). It is important to note that the sky region encompassing the extraordinarily luminous GRB 221009A, positioned at RA = 288.264|$^\circ$| and Dec. = 19.768|$^\circ$|⁠, is not included in these catalogues. Consequently, accurately estimating the extent of voids along our line of sight to GRB 221009A is currently unfeasible. To visualize the variations of cumulative void sizes in different locations of the sky, we present Fig. 1, which illustrates the distribution of relative void sizes within a region covered by the updated Void Catalogs SDSS DR7 (Douglass et al. 2023) using the VoidFinder algorithm. In Fig. 2, we computed the accumulated comoving effective radii contained in the volume of the solid angle subtended by each cell within the celestial coordinates of 140° to 240° in Right Ascension and 0° to 60° in Declination. A notable finding is that there are regions significantly underdense, with some cells containing around three times more voids than others. Our choice of a 10° square grid is arbitrary. At smaller scales, such as within 5° square cells, variations become more pronounced. We found that the minimum accumulated effective radii within a grid cell can be as low as zero, while the maximum, for the grid cell containing the most voids, can reach up to approximately 250 Mpc. The goal of using a 10° grid in Fig. 2 is to demonstrate that significant variations in void quantities can be found even at such large scales. Therefore, it is conceivable that GRB 221009A might have been situated in a direction where an underdense region exists between us and the source. Such a condition could result in reduced EBL energy density, potentially facilitating the arrival of TeV photons. In Section 3, we present detailed calculation by assuming an arbitrary possible void sizes, using the Finke et al. (2010) EBL model. We characterize the voids in the rest of this section.

2.2 Monte Carlo study on the average radii of cosmic voids

The objective of Figs 1 and 2 is to provide a general understanding based on observations that the distribution of cosmic voids varies; certain areas of the sky possess a higher concentration of voids in comparison to others. Here, we proceed to obtain a more precise quantitative estimate of the cumulative void radius within a spherical region in the direction towards a source, from an observational perspective. This involves assessing the likelihood of encountering such amount of voids in a specific area of the sky where a particular source might be situated behind these voids. To estimate the potential cumulative void radius of voids between us and a source at redshift 0.151, such as GRB 221009A, we initially extrapolated observed data beyond |$z = 0.114$|⁠, the limit covered by the most recent Void Catalogs from the VoidFinder SDSS DR7, extending up to |$z = 0.151$|⁠. This was done by assuming that the distribution of voids follows the same pattern up to this redshift. The properties of voids are not expected to change significantly over such low redshifts (see Sheth & van de Weygaert 2004; van de Weygaert & Platen 2011; Paz et al. 2013). The primary goal of this work is to determine if voids exist along the line of sight to a VHE gamma-ray source, potentially leading to a reduction in the EBL density and, consequently, the optical depth for TeV photons. In Section 3, for simplicity, we assumed the possibility of multiple voids with typical radii distributed in a large spherical region between the Earth and the source. This section explores the feasibility of this assumption from an observational consideration. However, the void catalogues of SDSS do not contain the portion of sky where GRB221009A is located; therefore, we cannot obtain an exact realistic estimation of the total effective radius of voids towards the source. Therefore, we estimate the likelihood of such an amount of voids being present in a large spherical region between us and the source at |$z = 0.151$|⁠. To achieve this, we conducted a comprehensive Monte Carlo simulation using the following approach: We randomly selected points within specified celestial coordinates, ranging from 140° to 230° in RA and 10° to 60° in Dec. at redshift z = 0.15. For each iteration, we draw a spherical region with its diameter extending from the observer to one of these points. Within this spherical comoving volume, we calculated the cumulative comoving effective radius (⁠|$\Sigma R_{i}$|⁠), which is found in the VoidFinder SDSS DR7 void catalogue and our extrapolated data beyond redshift z = 0.114. This procedure was repeated across a significant sample size of |$10^5$| iterations, creating an extensive data set. Subsequently, we calculated the mean and standard deviation of these cumulative effective radii to assess the distribution and variability across various directions. Our analysis indicated an average |$\Sigma R_{i}= 113$| Mpc, with a standard deviation of 50 Mpc. Furthermore, we evaluated the probabilities of |$\Sigma R_{i} \ge$| 150, 200, and 250 Mpc, which were found to be |$23.6~{{\ \rm per\ cent}}$|⁠, |$4.6~{{\ \rm per\ cent}}$|⁠, and |$0.6~{{\ \rm per\ cent}}$|⁠, respectively. Fig. 3 visually represents the possible cumulative effective radii of voids within spherical regions across specific areas of the sky. It features a histogram that illustrates the probability of encountering varying amounts of voids in these regions, The red line depicts the probability density function (PDF) smoothly summarizing the distribution trend. This detailed visualization aids in understanding the distribution, variation, and overall density of cosmic voids. Additionally, we conducted a Monte Carlo simulation using the same region of the sky to calculate the fraction of the line of sight that traverses a void. Our results showed that the average cumulative effective radius is 12.97 Mpc, with a standard deviation of 19.33 Mpc. Therefore, on average, approximately 2 per cent of the distance between us and the source is traversed through a void.

3 EBL IN THE PRESENCE OF COSMIC VOIDS

Our calculations of the inhomogeneous EBL are based on a modified version of the formalism presented in Finke et al. (2010), considering the direct starlight and the re-processing of starlight by dust. For the purpose of a generic study of the effects of cosmic voids along the line of sight to the gamma-ray source (i.e. GRB 221009A), we assume that a spherical cosmic void is located with its centre at redshift |$z_v$| and radius R between the observer and the gamma-ray source at redshift |$z_{s}$|⁠. The geometry is illustrated in Fig. 4. We calculate the angle- and photon-energy-dependent EBL energy density at each point between the observer and the source by using comoving coordinates, converting redshifts z to distances |$l (z)$|⁠. However, the voids are vast areas that are almost empty of luminous matter (see Foster & Nelson 2009; Ricciardelli, Quilis & Varela 2014). The matter density at the centre of the void can be less by up to 80 per cent compared to its surrounding areas (see e.g. Ricciardelli et al. 2014). Therefore, for simplicity, we assumed that the contribution to the EBL energy density produced within cosmic voids is negligible. In this work, we set the total emissivity (including both stellar and dust components) within these voids to zero. From Fig. 4 we can find the distances |$\ell _{1,2}$| where the EBL photon travel direction |$\Omega = (\theta , \phi)$| crosses the boundaries of the void, in the case where the gamma-ray photon is not inside the void, as

where |$\mu = \cos \theta$|⁠, |$\theta$| being the angle between the line of sight and the EBL photon travel direction, and |$\ell _v$| is the distance from the position of the photon at redshift z to the centre of the void. The maximum angle |$\theta _{\rm max}$| at which the EBL photon travel direction still crosses the boundary at one point, is given by

and the corresponding distance to the tangential point, |$\tilde{\ell } _{\rm max}$|⁠, is given by

To calculate the EBL energy density inside the void, there is exactly one point where an EBL photon crosses the boundary of the void, at a distance |$\ell _1$| from z. Note that z represents an arbitrary redshift along the line of sight between the observer and the source, where it equals zero at the observer’s location and |$z_s$| at the source. To account for the EBL inhomogeneity, we introduce a step function that sets the emissivities inside the void to zero. For a point at a distance |$\tilde{\ell }$| from the location of z (as a function of |$\theta$|⁠), this step function can be written as

for points z along the line of sight that are located outside the void, and

We adapt the expression presented in Finke et al. (2010) to modify the comoving luminosity density (luminosity per unit comoving volume), also termed as emissivity. This expression allows us to investigate the dependence of the luminosity density on comoving energy |$\epsilon$| at a specific redshift |$z_e$|⁠:

where the initial mass function (IMF) is represented by |$\xi (m)$|⁠, the comoving star formation rate (SFR) density (i.e. the SFR per unit comoving volume) is denoted as |$\psi (z_{1})$|⁠, |$f_{esc} (\epsilon)$| is the fraction of photons that manage to escape a galaxy without being absorbed by interstellar dust, and |$\dot{N} (\epsilon , m, t_{*}(z_e, z_{1}))$| is the total number of photons emitted per unit energy per unit time. The connection between cosmic time and redshift can be expressed as

To incorporate the effects of dust and account for the inhomogeneity in the comoving luminosity density, we have made modifications to the expression introduced in Finke et al. (2010),

where |$f_{n}$| represents the fraction of the absorbed emissivity reradiated within a specific dust component, |$\Theta _n = k_BT_n/m_e c^2$| denotes the dimensionless temperature of the respective dust component with subscripts n  = 1, 2, and 3 corresponding to the warm dust, hot dust, and PAHs, respectively (for more details, see Finke et al. 2010). The differential photon number density of the EBL at a given redshift z along the travel direction of the gamma-ray photon consists of the direct contribution from stars throughout cosmic history and the indirect contribution from the absorption and re-emission by dust. The energy density |$u_{\rm EBL}$| of the EBL in the comoving frame per unit comoving volume as a function of |$\epsilon$| can be written as

The quantity |$j(\epsilon ^{\prime }, \tilde{z})$| is the stellar and dust emissivity function defined as |$j(\epsilon ^{\prime }, \tilde{z}) = j^{\rm stars}(\epsilon ^{\prime }, \tilde{z}) + j^{\rm dust}(\epsilon ^{\prime }, \tilde{z})$| given by equations (6) and (8). |$dt/d \tilde{z}$| is evaluated using equation (7). An observer at a redshift |$z \gt 0$| would perceive a universe that is |$(1+z)^3$| times smaller than its current size. Photons that we observe today with energy |$\epsilon$| would have been seen by them with a proper energy |$\epsilon _{p} = (1+z) \epsilon$|⁠. Taking into account the cosmic voids described earlier, the energy density expected to be observed can be expressed as

where |$f_{\rm void}(z, \tilde{z}, \Omega)$| is the step function set to be zero inside the void (equations 4 and 5), and

When including the effects of voids, the EBL energy density exhibits an angular dependence. Consequently, we calculate the optical depth for a gamma-ray photon from a source at redshift |$z_{s}$| with observed energy E, using the angle-dependent EBL photon energy density from equation (10) as (see e.g. Razzaque et al. 2009):

The |$\gamma ~\gamma$| pair-production cross-section |$\sigma _{\gamma \gamma }(s)$| can be written as

where |$r_{e}$| is the classical electron radius and |$\beta _{cm} = (1-\frac{1}{s})^{1/2}$| is the electron–positron velocity in the centre-of-momentum (c.m.) frame of the |$\gamma \gamma$| interaction, |$s = \frac{s_{0}}{2} (1-\cos \theta)$| is the c.m. frame electron/positron energy squared and |$s_{0} = \frac{\epsilon E}{m_e^2 c^4}$|⁠. In the case of a homogeneous and isotropic EBL (with which we will compare our results for the inhomogeneous EBL case), equation (11) can be simplified using the dimensionless function |$\bar{\varphi }$| defined by Gould & Schreder (1967):

where |$\bar{\sigma }(s) = \frac{2 \sigma (s)}{\pi r_{e}^{2}}$| and |$s_{0}(\epsilon) = E (1+z) \epsilon /m_{e} ^2 c^{4}$|⁠, so that equation (11) can be written as

Knowing the optical depth |$\tau _{\gamma \gamma }$|⁠, we can calculate the attenuation of the intrinsic photon flux |$F_{\nu }^{int}$| as

where |$F_{\nu }^{\rm obs}$| is the observed spectrum. In Fig. 5, we show the |$\gamma \gamma$| opacity for |$z=0.151$|⁠, the distance to GRB 221009A, and contributions to the opacity by EBL photons of different wavelengths. It is clear that there would be significant attenuation of gamma-ray flux at energies beyond 10 TeV. In Section 4, we will explore to what extent the presence of voids may decrease the impact on the EBL |${\gamma \gamma }$| absorption in this energy range. The primary objective is to examine whether the presence of a cosmic void along the line of sight to GRB 221009A may explain the detection of |$\gtrsim 10$| TeV photons.

4 RESULTS AND DISCUSSION

In this work, we hypothesize that the region along our line of sight to GRB 221009A is dominated by voids of various sizes. We assume that the centre of the cumulative void is located at a redshift of |$z_{v}=0.075$|⁠, given that the redshift of GRB 221009A is |$z=0.151$|⁠. This assumption allows us to simplify the calculation using a single void where the centre of the hypothetical large sphere, is at the mid-point between us and GRB 221009A. In Abdalla & Böttcher (2017), we found that the EBL deficit is proportional to the void’s size. The impact of several small voids, each with radius |$R_{i}$|⁠, is roughly equivalent to the effect of a single large void with a radius R equal to the sum of all the small voids radii |$\Sigma R_{i}$|⁠. Therefore, the single, arbitrarily void radius R used in our calculations in Section 3 is effectively equivalent to the cumulative radius |$\Sigma R_{i}$| used in Section 2. We consider cases with void diameters between 15 per cent (corresponding to |$R = 50$| Mpc) and 80 per cent (corresponding to |$R = 250$| Mpc) of the distance to GRB 221009A.

In the top panel of Fig. 6, we show the effect of voids of size 50, 100, 150, 200, and 250 Mpc on the |$\gamma \gamma$| optical depth compared to the homogeneous case (black solid line, Finke et al. 2010). The bottom panel of Fig. 6 shows that the optical depth |$\tau _{\gamma \gamma }$| can be reduced by up to around 10 per cent and 30 per cent at 13 TeV for the cases of |$R = 110$| Mpc and |$R = 250$| Mpc, respectively. Note that the average combined radius that we obtain from our calculations in Section 2.2 is R|$\sim$| 113 Mpc. The deficit due to cosmic voids is much higher for TeV photons than for GeV photons. This can be explained by the fact that the TeV photons interact with a broader target photon spectrum, which means that the deficit for VHE photons is expected to be due to the deficit in the entire EBL spectrum, while the deficit for GeV photons is only due to the deficit in the optical-ultraviolet EBL component. This explanation is clearly depicted in Fig. 5, where we show |$\tau _{\gamma \gamma }$| calculated for EBL photons of different wavelengths. For the same reason, higher energy photons are more sensitive to the values of the Hubble constant when EBL |$\gamma \gamma$| absorption is used to constrain cosmology (e.g. Domínguez et al. 2019,2024).

4.1 Cosmic voids using different EBL models

In Section 3, we formulated a methodology for incorporating the concept of EBL inhomogeneity, or the potential existence of cosmic voids, within the framework of the EBL model proposed by Finke et al. (2010). This involved conducting meticulous calculations, assuming that the cumulative emissivity within the void regions could be considered negligible. Incorporating the notion of cosmic voids or underdense regions within various EBL models is crucial for conducting a thorough analysis and facilitating comparisons with the EBL models featured in Cao et al. (2023a). In the top panel of Fig. 6, we show the optical depth calculated for the Finke et al. (2010) EBL model in the homogeneous case and in the case of cosmic voids of various sizes for a source at |$z=0.151$|⁠, the redshift of GRB 221009A. The bottom panel shows the relative deficit of optical depth without and with voids. It is noteworthy that the anticipated effects of cosmic voids are expected to remain consistent across various EBL models.

To incorporate the effect of cosmic voids into other EBL models, we define |$\tau _{Finke_{10}}^{hom}$| and |$\tau _{Finke_{10}}^{void}$| as the optical depths calculated under two scenarios: one based on the standard homogeneous model and the other considering the presence of cosmic voids, both utilizing Finke et al. (2010) EBL model. Similarly, |$\tau _{model_x} ^{hom}$| and |$\tau _{model_x} ^{void}$| represent analogous optical depths when employing an alternative EBL model denoted as ‘|$\text{model}_x$|’ under the same two scenarios (homogeneous and void-inclusive). Therefore, the optical depth considering voids for any EBL model can be calculated as

Based on the hypothesis that the expected optical depth deficit remains consistent regardless of the chosen EBL model, we can introduce EBL inhomogeneity into various EBL models using equation (14). Fig. 7 displays the optical depths for photons originating from GRB 221009A as calculated using the Domínguez et al. (2011a), Saldana-Lopez et al. (2021), and Finke et al. (2022) EBL models. The amount by which the cosmic voids lower the optical depth depends on the amount of potential voids that exist along the line of sight to the source. Next we study how this change in the optical depth could potentially impact the observed flux for each EBL model, which amount of voids could provide an explanation for the LHAASO data, and how much this could increase the detectability of VHE gamma-rays from these sources.

4.2 GRB 221009A: observed and intrinsic fluxes

The LHAASO-KM2A detector has observed an unexpected abundance of 140 gamma-ray events with energies exceeding 3 TeV within a time span of 230–900 s following the initial trigger. A log-parabola function (referred to as LP) and a power law with an exponential cut-off function (referred to as PLEC) has been employed to describe the intrinsic energy spectrum of these gamma-rays, incorporating essential corrections to account for the effects of EBL absorption (Cao et al. 2023a). A detailed analysis of GRB 221009A, accounting for the systematic uncertainties in photon-energy determination, indicates that the maximum energy observed from this source is, depending on the assumed spectrum, approximately 18 TeV (LP) or 13 TeV (PLEC), as reported by Cao et al. (2023a).

In this study, we adopted an arbitrary power-law intrinsic flux and incorporated EBL attenuation, represented by the equation |$dN/dE = N (E/TeV)^{-\alpha } \times e^{- \tau _{\gamma \gamma }}$| utilizing the EBL models from Finke et al. (2010, 2022), Domínguez et al. (2011a), and Saldana-Lopez et al. (2021). Specifically, we employed a fitting procedure to match the attenuated flux from the adopted EBL model to the observed data from Cao et al. (2023a), thereby effectively constraining the values of the normalization and index of the power-law function for the intrinsic flux. For each EBL model, we calculated the fitting parameters in the standard homogeneous case first. We then incorporated the cosmic voids of different sizes for each EBL model and identified the best-fitting parameters for each case. The optimal values of the power-law index |$\alpha$| for each scenario are presented in Table 1. It can be noted that |$\alpha$| becomes smaller (the spectrum becomes harder) with increasing void size, as expected from a decreasing opacity. However, the value is not significantly different from the spectral index in the homogeneous case (⁠|$\boldsymbol{\alpha }_{hom}$|⁠).

Figs 8 and 9 showcase the intrinsic spectra of GRB 221009A, which have been derived using the power-law model based on the fitted parameters detailed in Table 1. These figures also present the anticipated observable flux as estimated by various EBL models. These spectra are shown for two distinct time intervals: |$T_{0} + 230$| s to |$T_{0} + 300$| s (Fig. 8) and |$T_{0} + 300$| s to |$T_{0} + 900$| s (Fig. 9) as observed by LHAASO (see Cao et al. 2023a). The expected EBL-induced absorption, as estimated by the EBL models in the homogeneous case, is presented as a solid black line. The potential influence of cosmic voids of various sizes (dashed lines) is investigated. The observed flux by LHAASO, especially in the |$T_0 + 300$| s to |$T_0 + 900$| s (Fig. 9) exhibits a hardening of the spectrum above 10 TeV and the fit seems better with some EBL models in the presence of voids. Although, we notice that no single void model can explain the whole spectrum.

For voids with accumulated size |$R = 250$| Mpc, the EBL-attenuated flux at 13 TeV can be greater than a factor of ten than the expected one in the case of a homogeneous EBL. Although illustrative, we note that the LHAASO flux is an optimistic estimate of the source-intrinsic |$\gtrsim 10$| TeV flux from GRB 221009A, but the reduction of the optical depth is independent of the gamma-ray emission model and intrinsic gamma-ray spectrum. When combined with our scenario of cosmic voids along the line of sight, physical models such as those presented by Das & Razzaque (2023), Zhang et al. (2023), Cao et al. (2023b), and Zhang et al. (2024b) or proton synchrotron emission (Zhang et al. 2023) would require a lower total energy output from GRB 221009A to explain the detection of >10 TeV photons by LHAASO.

4.3 Likelihood analysis

Our goal is to determine how well various EBL models, featuring different void sizes, align with LHAASO’s observational data as presented in Figs 8 and 9. We calculated the reduced chi-square statistic for each model with different void sizes, and convert these values to likelihoods for a probabilistic evaluation using the formula |$\exp \left(-\frac{\chi _{\text{reduced}}^2}{2}\right)$|⁠. See, Fig. 10 and Table 2. Based on our estimates from the Monte Carlo study (see Section 2.2), all these various quantities of voids, as required by each model, are possible. However, in certain scenarios, the quantity of voids needed to get the best fit with the LHASSO data appears less likely. Specifically, for Domínguez et al. (2011a) and Saldana-Lopez et al. (2021) EBL models, the accumulated radii of voids |$\Sigma R$| necessary to achieve the best fit can exceed 200 Mpc. The likelihood of such a mount of voids existing for an object at a redshift of |$z = 0.151$| is less than 5 per cent.

5 SUMMARY AND CONCLUSIONS

Cosmic voids remain an active area of research, and their precise sizes remain uncertain from an observational perspective. Determining the minimum size required for a region to be classified as a void, as well as quantifying the matter density compared to the average, poses significant challenges. Estimating the exact reduction in matter density within a spherical region, assumed to be a void, is particularly complex (e.g. Douglass et al. 2023). Further in-depth analysis will be crucial in the future to rigorously test our hypothesis and refine our understanding of cosmic voids and their implications

In this work, we have explored implications of cosmic voids on the gamma-ray propagation along the line-of-sight and through a region of reduced EBL opacity, using TeV flux detected by the LHAASO experiment from GRB 221009A. We have computed the optical depth for |$\ge 0.1$| TeV photons propagating to the Earth from |$z=0.151$|⁠, the redshift of GRB 221009A. We found that the presence of cosmic voids can reduce the optical depth significantly in the |$\gtrsim 1$| TeV range. Correspondingly, assuming the same intrinsic power-law spectrum, the observable gamma-ray flux from GRB 221009A can be greater than a factor of ten for the cases with the larger combined radius compared to the homogeneous EBL models. This can help explain detection of |$\gtrsim 10$| TeV photons from GRB 221009A.

We show that the optical depth may be reduced by about 10 per cent and 30 per cent at 13 TeV, in the cases of |$R = 110$| Mpc and |$R = 250$| Mpc, respectively. Furthermore, our findings suggest that the fit of all EBL models is more accurate within the context of possible voids present along our line-of-sight to GRB 221009A. The quantity of voids needed to achieve optimal fitting falls within the realm of feasibility, as suggested by our Monte Carlo study employing an updated void catalog from VoidFinder SDSS DR7. Our results, using EBL de-absorption in the presence of void shows a harder intrinsic spectrum than for the homogeneous EBL. Therefore, prospects for future detection of TeV from extragalactic sources such as GRBs or blazars are enhanced, if those take place in the directions where cosmic voids are present.

ACKNOWLEDGEMENTS

We thank the anonymous referee for a careful reading of the manuscript and helpful suggestions, which have led to significant improvements in the manuscript. The work of HA is supported through the María Zambrano researcher fellow funded by the European Union, the AfAS Seed Research Grant, SA-GAMMA ST/S002952/1 and by the University of Johannesburg (UJ) URC grant. SR was supported by grants from the National Research Foundation (NRF), South Africa, NITheCS, UJ URC. JDF was support by NASA through contract S-15633Y and the Office of Naval Research. AD is thankful for the support of the Ramón y Cajal program from the Spanish MINECO, Proyecto PID2021-126536OA-I00 funded by MCIN/AEI/10.13039/501100011033.

DATA AVAILABILITY

The codes used to produce the results featured in this article are available upon reasonable request to the corresponding author.

REFERENCES