Constraints imposed on LHC commissioning currents due to transverse collective modes and chromaticity

There are two situations during commissioning in which it is desired to have less current than the nominal total current, but in which the operating conditions may be less than optimal. This report examines what constraints will be imposed on these situations by the growth rates of transverse instabilities when compared to Landau damping thresholds. The first situation is when there is only a single ``pilot'' bunch in the machine at injection energy, with a current which is significantly less than the nominal single-bunch current, the initial goal being from 3 to 10% of the nominal single-bunch current. The purpose of such a running condition is to measure and correct the lattice parameters. Thus, in this situation, one expects to have a chromaticity which is not well-corrected, and no feedback system. The second situation is when there are the nominal number of bunches in the ring, but with less current than the nominal current, the initial goal being 16% of the nominal current. The chromaticity will be more well-controlled, but the feedback system may not be in operation. This report computes what the coherent frequencies of transverse modes would be in these situations for various chromaticities and currents, and indicates what constraints will be placed on the machine operation due to transverse instabilities.