Breaking the baryon density-Hubble constant degeneracy in fast radio burst applications with associated gravitational waves

Fast radio bursts (FRBs) are a unique probe of the cosmos, owing to dispersion caused by free electrons in the intergalactic medium. Two of the main quantities of interest are degenerate: the density of matter <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi mathvariant="normal">Ω</a:mi> <a:mi mathvariant="normal">b</a:mi> </a:msub> <a:msub> <a:mi>f</a:mi> <a:mi mathvariant="normal">d</a:mi> </a:msub> </a:math> outside of galaxies and the Hubble constant <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"> <f:msub> <f:mi>H</f:mi> <f:mn>0</f:mn> </f:msub> </f:math> . Here, we present a new possibility of breaking the degeneracy without invoking early-Universe priors on <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"> <h:msub> <h:mi mathvariant="normal">Ω</h:mi> <h:mi mathvariant="normal">b</h:mi> </h:msub> </h:math> . Assuming some FRBs originate in compact object mergers, the combination of dispersion and luminosity distance from the gravitational wave (GW) can be used to measure <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mrow> <l:msub> <l:mi mathvariant="normal">Ω</l:mi> <l:mi mathvariant="normal">b</l:mi> </l:msub> <l:msup> <l:mi>h</l:mi> <l:mn>2</l:mn> </l:msup> <l:msub> <l:mi>f</l:mi> <l:mi mathvariant="normal">d</l:mi> </l:msub> </l:mrow> </l:math> (where <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>h</q:mi> </q:math> is the dimensionless Hubble constant). We show that this measurement can be combined with the abundant FRBs that have a redshift measurement. This combination breaks the degeneracy with the Hubble constant. We develop a Bayesian framework and forecast that third-generation GW detectors are required to obtain meaningful constraints. We forecast that 1 year of Einstein Telescope operations can constrain <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msub> <s:mi>H</s:mi> <s:mn>0</s:mn> </s:msub> </s:math> to <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:mrow> <u:mo>±</u:mo> <u:mn>6</u:mn> <u:mtext> </u:mtext> <u:mtext> </u:mtext> <u:mi>km</u:mi> <u:mtext> </u:mtext> <u:msup> <u:mrow> <u:mi mathvariant="normal">s</u:mi> </u:mrow> <u:mrow> <u:mo>−</u:mo> <u:mn>1</u:mn> </u:mrow> </u:msup> <u:mtext> </u:mtext> <u:msup> <u:mrow> <u:mi>Mpc</u:mi> </u:mrow> <u:mrow> <u:mo>−</u:mo> <u:mn>1</u:mn> </u:mrow> </u:msup> </u:mrow> </u:math> and <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" display="inline"> <x:mrow> <x:msub> <x:mi mathvariant="normal">Ω</x:mi> <x:mi mathvariant="normal">b</x:mi> </x:msub> <x:msup> <x:mi>h</x:mi> <x:mn>2</x:mn> </x:msup> <x:msub> <x:mi>f</x:mi> <x:mi mathvariant="normal">d</x:mi> </x:msub> </x:mrow> </x:math> to <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" display="inline"> <cb:mrow> <cb:mrow> <cb:mfrac linethickness="0"> <cb:mrow> <cb:mo>+</cb:mo> <cb:mn>0.0015</cb:mn> </cb:mrow> <cb:mrow> <cb:mo>−</cb:mo> <cb:mn>0.0016</cb:mn> </cb:mrow> </cb:mfrac> </cb:mrow> </cb:mrow> </cb:math> (68% credible interval). The method can also be used with luminosity distances obtained through other means than GWs.