Constraining the Spatial Curvature of the Local Universe with Deep Learning

Abstract We use the distance sum rule method to constrain the spatial curvature of the Universe with a large sample of 161 strong gravitational lensing systems, whose distances are calibrated from the Pantheon compilation of type Ia supernovae using deep learning. To investigate the possible influence of mass model of the lens galaxy on constraining the curvature parameter Ω k , we consider three different lens models. Results show that a flat Universe is supported in the singular isothermal sphere (SIS) model with the parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msubsup> <mml:mrow> <mml:mn>0.049</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>0.125</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>0.147</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> . While in the power-law (PL) model, a closed Universe is preferred at the ∼3 σ confidence level, with the parameter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>k</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:msubsup> <mml:mrow> <mml:mn>0.245</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>0.071</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mn>0.075</mml:mn> </mml:mrow> </mml:msubsup> </mml:math> . In the extended PL model, the 95% confidence level upper limit of Ω k is &lt;0.011. As for the parameters of the lens models, constraints on the three models indicate that the mass profile of the lens galaxy could not be simply described by the standard SIS model.