Impact of weak lensing on bright standard siren analyses

Gravitational waves from binary mergers at cosmological distances will experience weak lensing by large scale structure. This causes a (de)magnification, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>μ</a:mi></a:math>, of the wave amplitude, and a completely degenerate modification to the inferred luminosity distance <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msub><c:mi>d</c:mi><c:mi>L</c:mi></c:msub></c:math>. The customary method to address this is to increase the uncertainty on <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>d</e:mi><e:mi>L</e:mi></e:msub></e:math> according to the dispersion of the magnification distribution at the source redshift, <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msub><g:mi>σ</g:mi><g:mi>μ</g:mi></g:msub><g:mo stretchy="false">(</g:mo><g:mi>z</g:mi><g:mo stretchy="false">)</g:mo></g:math>. But this term is dependent on the cosmological parameters that are being constrained by gravitational wave “standard sirens,” such as the Hubble parameter <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msub><k:mi>H</k:mi><k:mn>0</k:mn></k:msub></k:math>, and the matter density fraction <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:msub><m:mi mathvariant="normal">Ω</m:mi><m:mi mathvariant="normal">m</m:mi></m:msub></m:math>. The dispersion <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mrow><q:msub><q:mrow><q:mi>σ</q:mi></q:mrow><q:mrow><q:mi>μ</q:mi></q:mrow></q:msub><q:mo stretchy="false">(</q:mo><q:mi>z</q:mi><q:mo stretchy="false">)</q:mo></q:mrow></q:math> is also sensitive to the resolution of the simulation used for its calculation. Tension in the measured value of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:msub><u:mi>H</u:mi><u:mn>0</u:mn></u:msub></u:math> from independent datasets, and the present use of weak-lensing fitting functions calibrated using outdated cosmological simulations, suggest <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"><w:msub><w:mi>σ</w:mi><w:mi>μ</w:mi></w:msub><w:mo stretchy="false">(</w:mo><w:mi>z</w:mi><w:mo stretchy="false">)</w:mo></w:math> could be underestimated. This motivates an investigation into the consequences of mischaracterizing <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"><ab:msub><ab:mi>σ</ab:mi><ab:mi>μ</ab:mi></ab:msub><ab:mo stretchy="false">(</ab:mo><ab:mi>z</ab:mi><ab:mo stretchy="false">)</ab:mo></ab:math>. We consider two classes of standard siren, supermassive black hole binary and binary neutron star mergers. Underestimating <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" display="inline"><eb:msub><eb:mi>H</eb:mi><eb:mn>0</eb:mn></eb:msub></eb:math> and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" display="inline"><gb:msub><gb:mi mathvariant="normal">Ω</gb:mi><gb:mi mathvariant="normal">m</gb:mi></gb:msub></gb:math> when calculating <kb:math xmlns:kb="http://www.w3.org/1998/Math/MathML" display="inline"><kb:msub><kb:mi>σ</kb:mi><kb:mi>μ</kb:mi></kb:msub><kb:mo stretchy="false">(</kb:mo><kb:mi>z</kb:mi><kb:mo stretchy="false">)</kb:mo></kb:math> increases the probability of finding a residual lensing bias on these parameters greater than <ob:math xmlns:ob="http://www.w3.org/1998/Math/MathML" display="inline"><ob:mn>1</ob:mn><ob:mi>σ</ob:mi></ob:math> by 1.5–3 times. Underestimating <qb:math xmlns:qb="http://www.w3.org/1998/Math/MathML" display="inline"><qb:msub><qb:mi>σ</qb:mi><qb:mi>μ</qb:mi></qb:msub><qb:mo stretchy="false">(</qb:mo><qb:mi>z</qb:mi><qb:mo stretchy="false">)</qb:mo></qb:math> by using low resolution/small sky-area simulations can also significantly increase the probability of biased results; the probability of a <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML" display="inline"><ub:mn>1</ub:mn><ub:mi>σ</ub:mi><ub:mo stretchy="false">(</ub:mo><ub:mn>2</ub:mn><ub:mi>σ</ub:mi><ub:mo stretchy="false">)</ub:mo></ub:math> bias in <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML" display="inline"><yb:msub><yb:mi>H</yb:mi><yb:mn>0</yb:mn></yb:msub></yb:math> and <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" display="inline"><ac:msub><ac:mi mathvariant="normal">Ω</ac:mi><ac:mi mathvariant="normal">m</ac:mi></ac:msub></ac:math> found from binary neutron star mergers is 54% (19%) in this case. For neutron star mergers, the mean bias on <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" display="inline"><ec:msub><ec:mi>H</ec:mi><ec:mn>0</ec:mn></ec:msub></ec:math> caused by magnification selection effects is <gc:math xmlns:gc="http://www.w3.org/1998/Math/MathML" display="inline"><gc:mrow><gc:mi mathvariant="normal">Δ</gc:mi><gc:msub><gc:mrow><gc:mi>H</gc:mi></gc:mrow><gc:mrow><gc:mn>0</gc:mn></gc:mrow></gc:msub><gc:mo>=</gc:mo><gc:mo>−</gc:mo><gc:mn>0.1</gc:mn><gc:mtext> </gc:mtext><gc:mtext> </gc:mtext><gc:mi>km</gc:mi><gc:mtext> </gc:mtext><gc:msup><gc:mrow><gc:mi mathvariant="normal">s</gc:mi></gc:mrow><gc:mrow><gc:mo>−</gc:mo><gc:mn>1</gc:mn></gc:mrow></gc:msup><gc:mtext> </gc:mtext><gc:msup><gc:mrow><gc:mi>Mpc</gc:mi></gc:mrow><gc:mrow><gc:mo>−</gc:mo><gc:mn>1</gc:mn></gc:mrow></gc:msup></gc:mrow></gc:math>. The spread around this mean bias—determined by assumptions on <kc:math xmlns:kc="http://www.w3.org/1998/Math/MathML" display="inline"><kc:msub><kc:mi>σ</kc:mi><kc:mi>μ</kc:mi></kc:msub><kc:mo stretchy="false">(</kc:mo><kc:mi>z</kc:mi><kc:mo stretchy="false">)</kc:mo></kc:math>—is <oc:math xmlns:oc="http://www.w3.org/1998/Math/MathML" display="inline"><oc:mrow><oc:mi mathvariant="normal">Δ</oc:mi><oc:msub><oc:mrow><oc:mi>H</oc:mi></oc:mrow><oc:mrow><oc:mn>0</oc:mn></oc:mrow></oc:msub><oc:mo>=</oc:mo><oc:mo>±</oc:mo><oc:mn>0.25</oc:mn><oc:mtext> </oc:mtext><oc:mtext> </oc:mtext><oc:mi>km</oc:mi><oc:mtext> </oc:mtext><oc:msup><oc:mrow><oc:mi mathvariant="normal">s</oc:mi></oc:mrow><oc:mrow><oc:mo>−</oc:mo><oc:mn>1</oc:mn></oc:mrow></oc:msup><oc:mtext> </oc:mtext><oc:mrow><oc:msup><oc:mrow><oc:mi>Mpc</oc:mi></oc:mrow><oc:mrow><oc:mo>−</oc:mo><oc:mn>1</oc:mn></oc:mrow></oc:msup></oc:mrow></oc:mrow></oc:math>, comparable to the forecasted uncertainty. These effects do not impact merging neutron stars’ utility for addressing the <sc:math xmlns:sc="http://www.w3.org/1998/Math/MathML" display="inline"><sc:msub><sc:mi>H</sc:mi><sc:mn>0</sc:mn></sc:msub></sc:math> tension, but left uncorrected they limit their use for precision cosmology. For supermassive black hole binaries, the spread of possible biases on <uc:math xmlns:uc="http://www.w3.org/1998/Math/MathML" display="inline"><uc:msub><uc:mi>H</uc:mi><uc:mn>0</uc:mn></uc:msub></uc:math> is significant, <wc:math xmlns:wc="http://www.w3.org/1998/Math/MathML" display="inline"><wc:mrow><wc:mi mathvariant="normal">Δ</wc:mi><wc:msub><wc:mrow><wc:mi>H</wc:mi></wc:mrow><wc:mrow><wc:mn>0</wc:mn></wc:mrow></wc:msub><wc:mo>=</wc:mo><wc:mo>±</wc:mo><wc:mn>5</wc:mn><wc:mtext> </wc:mtext><wc:mtext> </wc:mtext><wc:mi>km</wc:mi><wc:mtext> </wc:mtext><wc:msup><wc:mrow><wc:mi mathvariant="normal">s</wc:mi></wc:mrow><wc:mrow><wc:mo>−</wc:mo><wc:mn>1</wc:mn></wc:mrow></wc:msup><wc:mtext> </wc:mtext><wc:msup><wc:mrow><wc:mi>Mpc</wc:mi></wc:mrow><wc:mrow><wc:mo>−</wc:mo><wc:mn>1</wc:mn></wc:mrow></wc:msup></wc:mrow></wc:math>, but <ad:math xmlns:ad="http://www.w3.org/1998/Math/MathML" display="inline"><ad:mi mathvariant="script">O</ad:mi><ad:mo stretchy="false">(</ad:mo><ad:mn>200</ad:mn><ad:mo stretchy="false">)</ad:mo></ad:math> observations are needed to reduce the variance below the bias. To achieve accurate subpercent level precision on cosmological parameters using standard sirens, first much improved knowledge on the form of the magnification distribution and its dependence on cosmology is needed.