We explore the detection, with upcoming spectroscopic surveys, of three-dimensional power spectra of emission line fluctuations produced in different phases of the Interstellar Medium (ISM) by ionized carbon, ionized nitrogen and neutral oxygen at redshift z>4. The emission line [CII] from ionized carbon at 157.7 micron, and multiple emission lines from carbon monoxide, are the main targets of planned ground-based surveys, and an important foreground for future space-based surveys like the Primordial Inflation Explorer (PIXIE). However, the oxygen [OI] (145.5 micron) line, and the nitrogen [NII] (121.9 micron and 205.2 micron) lines, might be detected in correlation with [CII] with reasonable signal-to-noise ratio (SNR). These lines are important coolants of both the neutral and the ionized medium, and probe multiple phases of the ISM. We compute predictions of the three-dimensional power spectra for two surveys designed to target the [CII] line, showing that they have the required sensitivity to detect cross-power spectra with the [OI] line, and the [NII] lines with sufficient SNR. The importance of cross-correlating multiple lines is twofold. On the one hand, we will have multiple probes of the different phases of the ISM, which is key to understand the interplay between energetic sources, the gas and dust at high redshift. This kind of studies will be useful for a next-generation space observatory such as the NASA Far-IR Surveyor. On the other end, emission lines from external galaxies are an important foreground when measuring spectral distortions of the Cosmic Microwave Background spectrum with future space-based experiments like PIXIE; measuring fluctuations in the intensity mapping regime will help constraining the mean amplitude of these lines, and will allow us to better handle this important foreground.
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