  Causality - the principle stating that the output of a system cannot precede
the input - is a universal property of nature. Here, we extend the concept of
causality, and its implications, from the temporal to the spectral domain,
leveraging the peculiar properties of time-modulated non-Hermitian wave-physics
systems, with particular emphasis on photonic systems. Specifically, we uncover
the existence of a broad class of complex time-modulated metamaterials which
obey the time-domain equivalent of the well-established frequency-domain
Kramers-Kronig relations. We find that, in the scattering response of such
time-modulated systems, the output frequencies are inherently prohibited from
spectrally preceding the input frequencies, hence we refer to these systems as
'spectrally causal'. We explore the consequences of this newly introduced
concept for several relevant applications, including broadband perfect
absorption, temporal cloaking of an 'event', and truly unidirectional
propagation along a synthetic dimension. By extending the concept of causality
into the spectral domain and providing new tools to extend the field of
temporally modulated metamaterials ("chrono-metamaterials") into the complex
realm, our findings not only deepen our understanding of spectral scattering
dynamics, but may also open unexplored opportunities and enable relevant
technological advances in various areas of photonics and, more broadly, of wave
physics and engineering.
