Deflection of light and time delay in a hyperbolic Einstein-Straus–de Sitter solution

We analyze strong gravitational lensing by a spherically symmetric mass distribution within the Einstein-Straus–de Sitter framework in a spatially open universe with negative curvature (<a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>k</a:mi><a:mo>=</a:mo><a:mo>−</a:mo><a:mn>1</a:mn></a:math>). Applying the theory to the lensed quasar SDSS <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mrow><c:mi mathvariant="normal">J</c:mi><c:mn>1004</c:mn><c:mo>+</c:mo><c:mn>4112</c:mn></c:mrow></c:math>, we identify a critical threshold for the current scale factor <f:math xmlns:f="http://www.w3.org/1998/Math/MathML" display="inline"><f:msub><f:mi>a</f:mi><f:mn>0</f:mn></f:msub></f:math> of approximately <h:math xmlns:h="http://www.w3.org/1998/Math/MathML" display="inline"><h:mn>2.6</h:mn><h:mo>×</h:mo><h:msup><h:mn>10</h:mn><h:mn>27</h:mn></h:msup><h:mtext> </h:mtext><h:mtext> </h:mtext><h:mi mathvariant="normal">m</h:mi></h:math>, below which the effects of negative spatial curvature on lensing observables become significant, corresponding to a current curvature density of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mo stretchy="false">|</k:mo><k:msub><k:mi mathvariant="normal">Ω</k:mi><k:mrow><k:mi>k</k:mi><k:mn>0</k:mn></k:mrow></k:msub><k:mo stretchy="false">|</k:mo><k:mo>≳</k:mo><k:mn>0.0025</k:mn></k:math>. In particular, for <p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:msub><p:mi mathvariant="normal">Ω</p:mi><p:mrow><p:mi>k</p:mi><p:mn>0</p:mn></p:mrow></p:msub><p:mo>=</p:mo><p:mo>−</p:mo><p:mn>0.15</p:mn></p:math>, the light bending increases slightly by <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mo>∼</s:mo><s:mn>1</s:mn><s:mo>%</s:mo></s:math>, while the time delay exhibits a more substantial increase of <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"><u:mo>∼</u:mo><u:mn>10</u:mn><u:mo>%</u:mo></u:math>. Beyond this threshold, however, the lensing observables are found to be insensitive to the current scale factor and converge to those characteristic of a spatially flat universe. Importantly, our results indicate that, even in scenarios where spatial curvature would otherwise enhance lensing observables, the effect of the cosmological constant does remain present and acts to reduce light bending, corroborating the claim of Rindler and Ishak.