Testing Lorentz invariance with the ultrahigh energy cosmic ray spectrum

The Greisen-Zatsepin-Kuzmin cutoff predicted at the ultrahigh energy cosmic ray spectrum has been observed by the HiRes and Auger experiments. The results put severe constraints on the effect of Lorentz invariance violation (LIV), which has been introduced to explain the absence of the Greisen-Zatsepin-Kuzmin cutoff indicated in the AGASA data. Assuming a homogeneous source distribution with a single power-law spectrum, we calculate the ultrahigh energy cosmic ray spectrum observed on the Earth by taking photopion production, $e^{+}{e}^{-}$ pair production, and the adiabatic energy loss into account. The effect of LIV is also taken into account in the calculation. By fitting the HiRes monocular spectra and the Auger combined spectra, we show that the LIV parameter is constrained to $\xi =-{0.8}_{-0.5}^{+3.2}\times {10}^{-23}$ and ${0.0}_{-0.4}^{+1.0}\times {10}^{-23}$, respectively, which is very consistent with strict Lorentz invariance up to the highest energy.

Figure 1

Left panel: the expected UHE proton energy spectrum (magenta curve) compared with the observational data from the AGASA [15], HiRes [19], and Auger [20] experiments. The source spectrum of the theoretical curve is adopted as ${\gamma }_g=2.67$, and the flux is normalized to the data of HiRes above ${10}^{18}\mathrm{eV}$. Right panel: the fitting ${\chi }^2$ distribution as a function of the parameter ${\gamma }_g$ for HiRes data. The three horizontal lines show the uncertainty ranges of ${\gamma }_g$ at $1\sigma$, $2\sigma$, and $3\sigma$ confidence levels, respectively.

Figure 2

The inelasticity as a function of the proton energy $E_p$ in the LS and the CMB photon energy ${ϵ}^{\prime }$ in the proton rest system, for the standard model (upper panel), LIV modified models with $\xi =1\times {10}^{-23}$ (middle panel), and $\xi =-1\times {10}^{-23}$ (lower panel), respectively.

Figure 3

The UHE proton spectra modified by LIV compared with the one predicted by the standard model. Left panel: $\xi >0$; right panel: $\xi <0$. The observational data are from HiRes [19].

Figure 4

The $1\sigma$, $2\sigma$, and $3\sigma$ confidence regions (from inside to outside) on the $\xi –{\gamma }_g$ parameter plane using HiRes (left panel [19]) and Auger (right panel [20]) data above ${10}^{18}\mathrm{eV}$. The red cross shows the best fitting values.