It is well known that two types of gravitational wave memory exist in general relativity (GR): the linear memory and the non-linear, or Christodoulou memory. These effects, especially the latter, depend on the specific form of Einstein equation. It can then be speculated that in modified theories of gravity, the memory can differ from the GR prediction, and provides novel phenomena to study these theories. We support this speculation by considering scalar-tensor theories, for which we find two new types of memory: the T memory and the S memory, which contribute to the tensor and scalar components of gravitational wave, respectively. In particular, the former is caused by the burst of energy carried away by scalar radiation, while the latter is intimately related to the no scalar hair property of black holes in scalar-tensor gravity. We estimate the size of these two types of memory in gravitational collapses, and formulate a detection strategy for the S memory, which can be singled out from tensor gravitational waves. We show that (i) the S memory exists even in spherical symmetry, and is observable under current model constraints, and (ii) while the T memory is usually much weaker than the S memory, it can become comparable in the case of spontaneous scalarization.
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