methods we are using here, however, are entirely new, as the Laplace ap- proximations considered in the single-spiked context do not extend to the multispiked case. Key words: sphericity tests, large dimensionality, asymptotic power, spiked covariance, contiguity, power envelope. In a recent paper, Onatski, Moreira and Hallin (2011) (hereafter OMH) analyze the asymptotic power of statistical tests in the detection of a signal in spherical real-valued Gaussian data as the dimensionality of the data and the number of observations diverge to infinity at the same rate. This paper generalizes OMH’s alternative of a single symmetry-breaking direction ( single-spiked alternative) to the alternative of multiple symmetry-breaking directions ( multispiked alternative), which is more relevant for applied work. Contemporary tests of sphericity in a high-dimensional environment (see Ledoit and Wolf (2002), Srivastava (2005), Schott (2006), Bai et al. (2009), Chen et al. (2010), and Cai and Ma (2012)) consider general alternatives to the null of spheric- ity. Our interest in alternatives with only a few contaminating signals stems from the fact that in many applications, such as speech recognition, macroeconomics, finance, wireless communication, genetics, physics of mixture, and statistical learn- ing, a few latent variables typically explain a large portion of the variation in high- dimensional data (see Baik and Silverstein (2006) for references). As a possible explanation of this fact, Johnstone (2001) introduces the spiked covariance model, where all eigenvalues of the population covariance matrix of high-dimensional data are equal except for a small fixed number of distinct “spike eigenvalues.” The alter- 2 1 Introduction