In this proceeding , we present the results of a semi-analytic study of CDM substructure as a function of the primordial power spectrum .
We apply our method to several “ tilted ” models in the \Lambda \textrm { CDM } framework , with n \simeq 1.1 - 0.8 , or \sigma _ { 8 } \simeq 1.2 - 0.65 when normalized to COBE .
We also study a more extreme , warm dark matter-like spectrum that is sharply truncated below a scale of \sim 0.3 h ^ { -1 } Mpc ( \sim 10 ^ { 10 } h ^ { -1 } M _ { \odot } ) .
Contrary to some expectations , we show that the mass fraction of halo substructure is not a strong function of spectral slope , so it likely will be difficult to constrain tilt using flux ratios of gravitationally lensed quasars .
On the positive side , all of our CDM-type models yield projected mass fractions that are in good agreement with strong lensing estimates : f \approx 1.5 \% at M \sim 10 ^ { 8 } M _ { \odot } .
The truncated model produces a significantly smaller fraction , f \approx 0.3 \% , suggesting that warm dark matter-like spectra are disfavored and potentially may be distinguished from CDM spectra using lensing .
We also discuss the issue of dwarf satellite abundances , with emphasis on the cosmological dependence of the map between the observed central velocity dispersions of Milky Way satellites and the maximum circular velocities of their host halos .
In agreement with earlier work , we find that standard \Lambda \textrm { CDM } over-predicts the estimated count of Milky Way satellites at fixed V _ { max } by an order of magnitude , but tilted models do better because subhalos are less concentrated .
Interestingly , under the assumption that dwarfs have isotropic velocity dispersion tensors , models with significantly tilted primordial power spectra ( e.g . , n \lower 2.58 pt \vbox { \hbox { $ \buildrel \textstyle < \over { \sim } $ } } 0.85 , \sigma _ { 8 } \lower 2.58 pt \vbox { \hbox { $ \buildrel \textstyle < \over { \sim } $ } } 0.7 ) may underpredict the number of large Milky Way satellites with V _ { max } \lower 2.58 pt \vbox { \hbox { $ \buildrel \textstyle > \over { \sim } $ } } 40 km s ^ { -1 } .