Chaos and complementarity in de Sitter space
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Springer
Received: February 25, 2020 Accepted: April 27, 2020 Published: May 28, 2020
Lars Aalsma and Gary Shiu Department of Physics, University of Wisconsin, 1150 University Ave, Madison, WI 53706, U.S.A.
E-mail: [email protected], [email protected] Abstract: We consider small perturbations to a static three-dimensional de Sitter geometry. For early enough perturbations that satisfy the null energy condition, the result is a shockwave geometry that leads to a time advance in the trajectory of geodesics crossing it. This brings the opposite poles of de Sitter space into causal contact with each other, much like a traversable wormhole in Anti-de Sitter space. In this background, we compute out-of-time-order correlators (OTOCs) to asses the chaotic nature of the de Sitter horizon and find that it is maximally chaotic: one of the OTOCs we study decays exponentially with a Lyapunov exponent that saturates the chaos bound. We discuss the consequences of our results for de Sitter complementarity and inflation. Keywords: AdS-CFT Correspondence, Black Holes, Cosmology of Theories beyond the SM ArXiv ePrint: 2002.01326
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)152
JHEP05(2020)152
Chaos and complementarity in de Sitter space
Contents 1 Introduction
1
2 Basics of de Sitter space 2.1 Coordinate systems 2.2 Wightman function 2.3 Shockwaves
3 3 4 5 7 7 10 14
4 Consequences for complementarity 4.1 Implications for inflation
16 18
5 Discussion
20
A Derivation of the shockwave geometry
21
1
Introduction
Over the last years, it has been realized that quantum chaos plays an important role in the physics of black holes. The key property that makes black holes chaotic is the large blueshift between an asymptotic and a freely falling observer. Any perturbation with a 2π t small energy E0 experiences a boost in energy given by E = E0 e β , where t is the Killing time used by an asymptotic observer and β is the inverse temperature of the black hole. One probe of chaos in quantum systems that already has been known for a long time is the double commutator of two generic operators V, W [1] C(t) = h−[V (0), W (t)]2 i ,
(1.1)
which measures the sensitivity of the operators W and V with respect to each other. For Hermitian and unitary operators V and W , we can write C(t) = 2 − 2 RehV (0)W (t)V (0)W (t)i ,
(1.2)
F (t) = hV (0)W (t)V (0)W (t)i ,
(1.3)
where is referred to as the out-of-time-order correlator (OTOC). Chaotic behaviour shows itself in an exponential growth of the double commutator C(t) or, equivalently, an exponential
–1–
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3 Out-of-time-order correlators 3.1 Geodesic approximation 3.2 Beyond the geodesic approximation 3.3 Stringy corrections
decay of the OTOC F (t). In some thermal systems with a large number of degrees of freedom N , such as holographic CFTs dual to black holes, F (t) behaves as [2–5] F (t) = 1 −
f0 λL t e + O(N −2 ) , N
β/2π t λ−1 L log(N ) ,
(1.4)
–2–
JHEP05(2020)152
such that C(t) ∼ N −1 eλL t . Here
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