Toward testing gravity with LSST using <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>G</mml:mi> </mml:msub> </mml:math>

<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>E</a:mi> <a:mi>G</a:mi> </a:msub> </a:math> is a summary statistic that combines cosmological observables to achieve a test of gravity that is relatively model independent. Here, we consider the power of a measurement of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>E</c:mi> <c:mi>G</c:mi> </c:msub> </c:math> using galaxy-galaxy lensing and galaxy clustering with sources from the Vera C. Rubin Observatory’s Legacy Survey of Space and Time (LSST), and lenses from the Dark Energy Spectroscopic Instrument (DESI). We first update the theoretical framework for the covariance of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>E</e:mi> <e:mi>G</e:mi> </e:msub> </e:math> to accommodate this Stage IV scenario. We then demonstrate that <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msub> <g:mi>E</g:mi> <g:mi>G</g:mi> </g:msub> </g:math> offers in principle a model-agnostic test of gravity using only linear-scale information, with the caveat that a careful treatment of galaxy bias is required. We finally address the persistent issue of <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>E</i:mi> <i:mi>G</i:mi> </i:msub> </i:math> ’s theoretical dependence on the measured value of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msubsup> <k:mi mathvariant="normal">Ω</k:mi> <k:mi mathvariant="normal">M</k:mi> <k:mn>0</k:mn> </k:msubsup> </k:math> . We propose a framework that takes advantage of the posterior predictive test to consistently incorporate uncertainty on <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:msubsup> <o:mi mathvariant="normal">Ω</o:mi> <o:mi mathvariant="normal">M</o:mi> <o:mn>0</o:mn> </o:msubsup> </o:math> in tests of gravity with <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msub> <s:mi>E</s:mi> <s:mi>G</s:mi> </s:msub> </s:math> , which should be of general use beyond the LSST + DESI scenario. Our forecasting study using this method shows that the prior information available for <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msubsup> <u:mi mathvariant="normal">Ω</u:mi> <u:mi mathvariant="normal">M</u:mi> <u:mn>0</u:mn> </u:msubsup> </u:math> is instrumental in determining the power of <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:msub> <y:mi>E</y:mi> <y:mi>G</y:mi> </y:msub> </y:math> in the LSST + DESI context. For the full survey dataset, with priors on <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:msubsup> <ab:mi mathvariant="normal">Ω</ab:mi> <ab:mi mathvariant="normal">M</ab:mi> <ab:mn>0</ab:mn> </ab:msubsup> </ab:math> from existing cosmic microwave background data, we find that for some modified gravity scenarios considered, we are likely to be able to reject the general relativity null hypothesis.