Key words : spiked covariance, sub-critical regime, signal detection, sphericity tests, asymptotic power, contiguity, power envelope, Harish-Chandra/Itzykson- Zuber integral, torus scalar product, hypergeometric function. Much contemporary research in statistics concerns with situations where the di- mensionality of data is large and comparable to the number of observations (see special issues of the Philosophical Transactions of the Royal Society (2009) 367 and Annals of Statistics (2008) 36 ). Often, the goal is to estimate or detect a few signals contaminated by high-dimensional noise. One general conclusion that seems to emerge from this research is that, in the absence of a priori sparsity assumptions about signals, there is a lower limit for the signal-to-noise ratio below which statistical inference about the signals completely fails (Johnstone and Titter- ington, 2009, Nadakuditi and Edelman, 2008, Nadakuditi and Silverstein, 2010). This limit equals the phase transition threshold studied in Baik et al (2005). In a recent paper, Onatski et al (2012) show that not all is lost below the threshold. They consider the case of a single non-sparse signal in high-dimensional noisy data and establish sharp non-trivial limits for the asymptotic power, as both the data dimensionality and the number of observations go to infinity, of statistical tests for signal detection when the signal may be arbitrarily weak. This paper extends Onatski et al (2012) to the case of multiple non-sparse arbi- trarily weak signals when the data are complex-valued. Complex-valued data are of interest in signal processing (Schreier and Scharf, 2010), wireless communication (Telatar, 1999, Tulino and Verdu, 2004), and the spectral analysis of economic and financial time series (Onatski, 2009). Considering the case of multiple signals is important for applied work because the constraint that there is no more than one 2 1 Introduction