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Galaxy cluster gas temperatures ($T$) play a crucial role in many cosmological and astrophysical studies. However, it has been shown that $T$ measurements can significantly vary between different X-ray telescopes. These $T$ biases can propagate to several cluster applications in which $T$ can be used, such as measuring hydrostatic cluster masses and constraining the angular variation of cosmological parameters. Thus, it is important to accurately cross-calibrate X-ray instruments to account for systematic biases. In this work, we present the cross-calibration between Spectrum Roentgen Gamma/eROSITA (SRG/eROSITA) and Chandra /ACIS and between SRG/eROSITA and XMM-Newton /EPIC using for the first time a large sample of galaxy cluster $T$. To do so, we used the first eROSITA All-Sky Survey data and the preliminary extremely expanded HIgh FLUx Galaxy Cluster Sample, a large X-ray flux-limited cluster catalog. We spectroscopically measured X-ray $T$ for 186 independent cluster regions with both SRG/eROSITA and Chandra /ACIS in a self-consistent way for three energy bands: 0.7-7 keV (full), 0.5-4 keV (soft), and 1.5-7 keV (hard). We did the same with SRG/eROSITA and XMM-Newton /EPIC for 71 different cluster regions and all three bands. We find that SRG/eROSITA measures systematically lower $T$ than the other two instruments, with hotter clusters deviating more than cooler ones. For the full band, SRG/eROSITA returns 20$<!PCT!>$ and 14$<!PCT!>$ lower $T$ than Chandra /ACIS and XMM-Newton /EPIC, respectively, when the two other instruments each measure $k_ B T 3$ keV. The discrepancy respectively increases to 38<!PCT!> and 32<!PCT!> when Chandra /ACIS and XMM-Newton /EPIC each measure $k_ B T 10$ keV. On the other hand, the discrepancy becomes milder for low-$T$ galaxy groups. Moreover, a broken power law fit demonstrated that there is a break at the SRG/eROSITA- Chandra /ACIS scaling relation at $k_ B T 1.7-2.7$ keV, depending on the energy band. The soft band shows a marginally lower discrepancy compared to the full band. In the hard band, the cross-calibration of SRG/eROSITA and the other instruments show very strong differences. We tested several possible systematic biases (such as multiphase cluster gas, Galactic absorption, non-Gaussian scatter, and selection effects) to identify the reason behind the cross-calibration discrepancies, but none could significantly alleviate the tension. For now, it is most likely that the systematically lower SRG/eROSITA $T$ can be attributed to systematic effective area calibration uncertainties; however, the exact role of multiphase cluster gas in the observed $T$ discrepancies needs to be further investigated. Furthermore, we provide conversion factors between SRG/eROSITA Chandra /ACIS, and XMM-Newton /EPIC $T$ that will be beneficial for future cluster studies that combine SRG/eROSITA $T$ with data from other X-ray instruments. Finally, we also provide conversion functions between the official eRASS1 cluster catalog $T$ and the equivalent core and core-excised Chandra /ACIS and XMM-Newton /EPIC $T$.
Galaxy cluster gas temperatures ($T$) play a crucial role in many cosmological and astrophysical studies. However, it has been shown that $T$ measurements can significantly vary between different X-ray telescopes. These $T$ biases can propagate to several cluster applications in which $T$ can be used, such as measuring hydrostatic cluster masses and constraining the angular variation of cosmological parameters. Thus, it is important to accurately cross-calibrate X-ray instruments to account for systematic biases. In this work, we present the cross-calibration between Spectrum Roentgen Gamma/eROSITA (SRG/eROSITA) and Chandra /ACIS and between SRG/eROSITA and XMM-Newton /EPIC using for the first time a large sample of galaxy cluster $T$. To do so, we used the first eROSITA All-Sky Survey data and the preliminary extremely expanded HIgh FLUx Galaxy Cluster Sample, a large X-ray flux-limited cluster catalog. We spectroscopically measured X-ray $T$ for 186 independent cluster regions with both SRG/eROSITA and Chandra /ACIS in a self-consistent way for three energy bands: 0.7-7 keV (full), 0.5-4 keV (soft), and 1.5-7 keV (hard). We did the same with SRG/eROSITA and XMM-Newton /EPIC for 71 different cluster regions and all three bands. We find that SRG/eROSITA measures systematically lower $T$ than the other two instruments, with hotter clusters deviating more than cooler ones. For the full band, SRG/eROSITA returns 20$<!PCT!>$ and 14$<!PCT!>$ lower $T$ than Chandra /ACIS and XMM-Newton /EPIC, respectively, when the two other instruments each measure $k_ B T 3$ keV. The discrepancy respectively increases to 38<!PCT!> and 32<!PCT!> when Chandra /ACIS and XMM-Newton /EPIC each measure $k_ B T 10$ keV. On the other hand, the discrepancy becomes milder for low-$T$ galaxy groups. Moreover, a broken power law fit demonstrated that there is a break at the SRG/eROSITA- Chandra /ACIS scaling relation at $k_ B T 1.7-2.7$ keV, depending on the energy band. The soft band shows a marginally lower discrepancy compared to the full band. In the hard band, the cross-calibration of SRG/eROSITA and the other instruments show very strong differences. We tested several possible systematic biases (such as multiphase cluster gas, Galactic absorption, non-Gaussian scatter, and selection effects) to identify the reason behind the cross-calibration discrepancies, but none could significantly alleviate the tension. For now, it is most likely that the systematically lower SRG/eROSITA $T$ can be attributed to systematic effective area calibration uncertainties; however, the exact role of multiphase cluster gas in the observed $T$ discrepancies needs to be further investigated. Furthermore, we provide conversion factors between SRG/eROSITA Chandra /ACIS, and XMM-Newton /EPIC $T$ that will be beneficial for future cluster studies that combine SRG/eROSITA $T$ with data from other X-ray instruments. Finally, we also provide conversion functions between the official eRASS1 cluster catalog $T$ and the equivalent core and core-excised Chandra /ACIS and XMM-Newton /EPIC $T$.
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