Pole skipping in two-dimensional de Sitter spacetime and double-scaled SYK model
Haiming Yuan, Xian-Hui Ge, Keunyoung Kim
Abstract
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.