  Vacuum bubbles may nucleate and expand during the cosmic inflation. When
inflation ends, the bubbles run into the ambient plasma, producing strong
shocks followed by underdensity waves, which propagate outwards. The bubbles
themselves eventually form black holes with a wide distribution of masses. It
has been recently suggested that such black holes may account for LIGO
observations and may provide seeds for supermassive black holes observed at
galactic centers. They may also provide a significant part or even the whole of
the dark matter. We estimate the spectral $\mu$-distortion of the CMB induced
by expanding shocks and underdensities. The predicted distortions averaged over
the sky are well below the current bounds, but localized peaks due to the
largest black holes impose constraints on the model parameters.
