Tomographic constraints on the production rate of gravitational waves from astrophysical sources

Using an optimal quadratic estimator, we measure the large-scale cross-correlation between maps of the stochastic gravitational-wave intensity, constructed from the first three LIGO-Virgo observing runs, and a suite of tomographic samples of galaxies covering the redshift range <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>z</a:mi><a:mo>≲</a:mo><a:mn>2</a:mn></a:math>. We do not detect any statistically significant cross-correlation, but the tomographic nature of the data allows us to place constraints on the (bias-weighted) production rate density of gravitational waves by astrophysical sources as a function of cosmic time. Our constraints range from <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mo stretchy="false">⟨</c:mo><c:mi>b</c:mi><c:msub><c:mover accent="true"><c:mi mathvariant="normal">Ω</c:mi><c:mo>˙</c:mo></c:mover><c:mi>GW</c:mi></c:msub><c:mo stretchy="false">⟩</c:mo><c:mo>&lt;</c:mo><c:mn>3.0</c:mn><c:mo>×</c:mo><c:msup><c:mn>10</c:mn><c:mrow><c:mo>−</c:mo><c:mn>9</c:mn></c:mrow></c:msup><c:mtext> </c:mtext><c:mtext> </c:mtext><c:msup><c:mrow><c:mi>Gyr</c:mi></c:mrow><c:mrow><c:mo>−</c:mo><c:mn>1</c:mn></c:mrow></c:msup></c:math> at <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>z</i:mi><i:mo>∼</i:mo><i:mn>0.06</i:mn></i:math> to <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mo stretchy="false">⟨</k:mo><k:mi>b</k:mi><k:msub><k:mover accent="true"><k:mi mathvariant="normal">Ω</k:mi><k:mo>˙</k:mo></k:mover><k:mi>GW</k:mi></k:msub><k:mo stretchy="false">⟩</k:mo><k:mo>&lt;</k:mo><k:mn>2.7</k:mn><k:mo>×</k:mo><k:msup><k:mn>10</k:mn><k:mrow><k:mo>−</k:mo><k:mn>7</k:mn></k:mrow></k:msup><k:mtext> </k:mtext><k:mtext> </k:mtext><k:msup><k:mrow><k:mi>Gyr</k:mi></k:mrow><k:mrow><k:mo>−</k:mo><k:mn>1</k:mn></k:mrow></k:msup></k:math> at <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>z</q:mi><q:mo>∼</q:mo><q:mn>1.5</q:mn></q:math> (95% confidence level), assuming a frequency spectrum of the form <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:msup><s:mi>f</s:mi><s:mrow><s:mn>2</s:mn><s:mo>/</s:mo><s:mn>3</s:mn></s:mrow></s:msup></s:math> (corresponding to an astrophysical background of binary mergers), and a reference frequency <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>f</u:mi><u:mrow><u:mi>ref</u:mi></u:mrow></u:msub><u:mo>=</u:mo><u:mn>25</u:mn><u:mtext> </u:mtext><u:mtext> </u:mtext><u:mi>Hz</u:mi></u:math>. Although these constraints are <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:mo>∼</w:mo><w:mn>2</w:mn></w:math> orders of magnitude higher than the expected signal, we show that a detection may be possible with future experiments. Published by the American Physical Society 2024