Stability and quasinormal modes for black holes with time-dependent scalar hair

We investigate black hole solutions with time-dependent (scalar) hair in scalar-tensor theories. Known exact solutions exist for such theories at the background level, where the metric takes on a standard general relativity (GR) form (e.g., Schwarzschild–de Sitter), but these solutions are generically plagued by instabilities. Recently, a new such solution was identified by Bakopoulos [], in which the time-dependent scalar background profile is qualitatively different from previous known exact solutions—specifically, the canonical kinetic term for the background scalar <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>X</a:mi></a:math> is not constant in this solution. We investigate the stability of this new solution by analyzing odd parity perturbations, identifying a bound placed by stability and the resulting surviving parameter space. We extract the quasinormal mode spectrum predicted by the theory, finding a generic positive shift of quasinormal mode frequencies and damping times compared to GR. We forecast constraints on these shifts (and the single effective parameter <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mover accent="true"><c:mi>β</c:mi><c:mo stretchy="false">^</c:mo></c:mover></c:math> controlling them) from current and future gravitational wave experiments, finding constraints at up to the <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi mathvariant="script">O</g:mi><g:mo stretchy="false">(</g:mo><g:msup><g:mn>10</g:mn><g:mrow><g:mo>−</g:mo><g:mn>2</g:mn></g:mrow></g:msup><g:mo stretchy="false">)</g:mo></g:math> and <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"><l:mrow><l:mi mathvariant="script">O</l:mi><l:mo stretchy="false">(</l:mo><l:mrow><l:msup><l:mrow><l:mn>10</l:mn></l:mrow><l:mrow><l:mo>−</l:mo><l:mn>4</l:mn></l:mrow></l:msup></l:mrow><l:mo stretchy="false">)</l:mo></l:mrow></l:math> level for LIGO-Virgo-KAGRA and Laser Interferometer Space Antenna/TianQin, respectively. All calculations performed in this paper are reproducible via a companion notebook.