In this paper we investigate the power spectrum of the unresolved 0.5–2 keV cosmic X-ray background (CXB) with deep Chandra 4-Msec (Ms) observations in the Chandra Deep Field South (CDFS). We measured a signal that, on scales >30 arcsec, is significantly higher than the shot noise and is increasing with angular scale. We interpreted this signal as the joint contribution of clustered undetected sources like active galactic nuclei (AGN), galaxies and the intergalactic medium (IGM). The power of unresolved cosmic source fluctuations accounts for ∼12 per cent of the 0.5–2 keV extragalactic CXB. Overall, our modelling predicts that ∼20 per cent of the unresolved CXB flux is produced by low-luminosity AGN, ∼25 per cent by galaxies and ∼55 per cent by the IGM. We do not find any direct evidence of the so-called ‘warm hot intergalactic medium’ (i.e. matter with 105 < T < 107 K and density contrast δ < 1000), but we estimated that it could produce about 1/7 of the unresolved CXB. We placed an upper limit on the space density of postulated X-ray-emitting early black holes at z > 7.5 and compared it with supermassive black hole evolution models.