Pole skipping in two-dimensional de Sitter spacetime and double-scaled SYK model

We develop the pole-skipping structure in de Sitter (dS) spacetime and find that their leading frequencies satisfy the relation <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>ω</a:mi><a:mrow><a:mi>d</a:mi><a:mi>S</a:mi></a:mrow></a:msub><a:mo>=</a:mo><a:mi>i</a:mi><a:mn>2</a:mn><a:mi>π</a:mi><a:msub><a:mi>T</a:mi><a:mrow><a:mi>d</a:mi><a:mi>S</a:mi></a:mrow></a:msub><a:mo stretchy="false">(</a:mo><a:mn>1</a:mn><a:mo>−</a:mo><a:mi>s</a:mi><a:mo stretchy="false">)</a:mo></a:math>, where <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msub><e:mi>T</e:mi><e:mrow><e:mi>d</e:mi><e:mi>S</e:mi></e:mrow></e:msub><e:mo>=</e:mo><e:mn>1</e:mn><e:mo>/</e:mo><e:mn>2</e:mn><e:mi>π</e:mi><e:mi>L</e:mi></e:math> and <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>s</g:mi></g:math> denotes spin. In the two-dimensional dS spacetime, the pole-skipping points near the cosmic horizon <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mi>r</i:mi><i:mo>=</i:mo><i:mi>L</i:mi></i:math> for the scalar field of spin 0 and the fermionic field of spin <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mfrac><k:mn>1</k:mn><k:mn>2</k:mn></k:mfrac></k:math> correspond one to one with those in the classical limit as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>λ</m:mi><m:mo stretchy="false">→</m:mo><m:mn>0</m:mn></m:math> in double-scaled Sachdev-Ye-Kitaev model when the temperature is infinite (<p:math xmlns:p="http://www.w3.org/1998/Math/MathML" display="inline"><p:msub><p:mrow><p:mi>DSSYK</p:mi></p:mrow><p:mi>∞</p:mi></p:msub></p:math>). This provides a numerical correspondence between quantum gravity in the static patch of two-dimensional dS spacetime and a one-dimensional quantum system, which we consider as a realization of the DS/dS correspondence.