We constrain the spectral running by combining cosmic microwave background ( CMB ) data , baryon acoustic oscillation ( BAO ) data and the constraint from primordial black holes ( PBHs ) .
We find that the constraint from PBHs has a significant impact on the running of running of scalar spectral index , and a power-law scalar power spectrum without running is consistent with observational data once the constraint from PBHs is taken into account .
In addition , from the constraints on the slow-roll parameters , the derived tensor spectral index in the single-field slow-roll inflation model is quite small , namely |n _ { t } | \lesssim 9.3 \times 10 ^ { -3 } which will be very difficult to be measured by CMB data only in the future , and the absolute value of derived running of tensor spectral index is not larger than 2.1 \times 10 ^ { -4 } at 95 \% confidence level .