Probing gravitational wave speed and dispersion with LISA observations of supermassive black hole binary populations

According to general relativity (GR), gravitational waves (GWs) should travel at the speed of light <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>c</a:mi> </a:math> . However, some theories beyond GR predict deviations of the velocity of GWs <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>c</c:mi> <c:mi>gw</c:mi> </c:msub> </c:math> from <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>c</e:mi> </e:math> , and some of those expect vacuum dispersion. Therefore, probing the propagation effects of GWs by comparing the wave format detectors against the one at emission excepted from GR. Since such propagation effects accumulate through larger distance, it is expected that supermassive black holes binary (SMBHB) mergers serve as better targets than their stellarmass equivalent. In this paper, we study with simulations on how observations on a population of SMBHs can help to study this topic. We simulate LISA observations on three possible SMBHB merger populations, namely PopIII, Q3-nod, and Q3-d over a 5-year mission. The resulting constraints on the graviton mass are 9.50, 9.33, and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mn>9.05</g:mn> <g:mo>×</g:mo> <g:msup> <g:mn>10</g:mn> <g:mrow> <g:mo>−</g:mo> <g:mn>27</g:mn> </g:mrow> </g:msup> <g:mtext> </g:mtext> <g:mtext> </g:mtext> <g:mi>eV</g:mi> <g:mo>/</g:mo> <g:msup> <g:mi>c</g:mi> <g:mn>2</g:mn> </g:msup> </g:math> , respectively. We also obtain the corresponding constraints on the dispersion coefficients assuming different dispersion scenarios. If the electromagnetic wave counterparts of SMBHB merger can be detected simultaneously, the <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>c</i:mi> <i:mi>gw</i:mi> </i:msub> </i:math> can be constrained waveform-independently to <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi mathvariant="normal">Δ</k:mi> <k:mi>c</k:mi> <k:mo>/</k:mo> <k:mi>c</k:mi> </k:math> to <n:math xmlns:n="http://www.w3.org/1998/Math/MathML" display="inline"> <n:msup> <n:mn>10</n:mn> <n:mrow> <n:mo>−</n:mo> <n:mn>13</n:mn> </n:mrow> </n:msup> <n:mi>–</n:mi> <n:msup> <n:mn>10</n:mn> <n:mrow> <n:mo>−</n:mo> <n:mn>12</n:mn> </n:mrow> </n:msup> </n:math> , corresponding to graviton mass constraints of <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"> <p:mrow> <p:msup> <p:mrow> <p:mn>10</p:mn> </p:mrow> <p:mrow> <p:mo>−</p:mo> <p:mn>26</p:mn> </p:mrow> </p:msup> <p:mi>–</p:mi> <p:msup> <p:mrow> <p:mn>10</p:mn> </p:mrow> <p:mrow> <p:mo>−</p:mo> <p:mn>24</p:mn> </p:mrow> </p:msup> <p:mtext> </p:mtext> <p:mtext> </p:mtext> <p:mi>eV</p:mi> <p:mo>/</p:mo> <p:msup> <p:mrow> <p:mi>c</p:mi> </p:mrow> <p:mrow> <p:mn>2</p:mn> </p:mrow> </p:msup> </p:mrow> </p:math> .