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String Regge trajectory on de Sitter space and implications to - - PowerPoint PPT Presentation

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Strings and Fields 2019 String Regge trajectory on de Sitter space and implications to inflation Siyi Zhou HKUST Stockholm U Based on arXiv: 1907.02535 [hep-th] w/Toshifumi Noumi, Toshiaki Takeuchi Outline Nave Expectation from Flat

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String Regge trajectory on de Sitter space and implications to inflation Siyi Zhou HKUST Stockholm U

Based on arXiv: 1907.02535 [hep-th] w/Toshifumi Noumi, Toshiaki Takeuchi

Strings and Fields 2019

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Outline

  • NaΓ―ve Expectation from Flat Space
  • Regge Trajectory in dS
  • High Energy Scattering
  • Summary
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dS in String Theory

  • Maldacena-Nunez no-go theorem
  • KKLT evades the no-go theorem
  • Non-perturbative effects
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Can we have a consistent world-sheet theory on de Sitter space?

  • Higuchi bound in de Sitter space
  • Modification of Regge trajectories
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Higuchi Bound

  • A unitarity bound on the mass of higher-spin

particles in de Sitter space

  • 𝑛2 β‰₯ 𝑑 𝑑 βˆ’ 1 𝐼2 for bosons
  • 𝑛2 β‰₯ 𝑑2 𝐼2 for fermions
  • Particles with masses violating the Higuchi

bound contain helicity modes with a negative norm β†’ Prohibited by unitarity

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Linear Regge Trajectory Violates the Bound

Prohibited by Higuchi Bound Linear Regge Trajectory Violates the Bound

𝑛𝑑 ≃ 𝑛𝑑

2

𝐼 𝑑𝑑 ≃ 𝑛𝑑

2

𝐼2 𝑛2 ≃ 𝑑 𝑁𝑑

2

A typical length of the string at the critical value is near the Hubble horizon scale

π‘š ∼ 𝑛𝑑 𝑁𝑑 π‘šπ‘‘ ∼ πΌβˆ’1 π‘šπ‘‘ ∼ 1/𝑁𝑑

See also Lust, Palti 19

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How the Regge Trajectory is modified in dS?

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Rotating Folded String in Flat Space

  • Centrifugal Force
  • String Tension
  • Balance of the two

forces requires the boundary have the speed of light

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Regge Trajectory on de Sitter

  • Static coordinate of de Sitter space

𝑒𝑑2 = 𝑆2 βˆ’ 1 βˆ’ 𝑠2 𝑒𝑒2 + 𝑒𝑠2 1 βˆ’ 𝑠2 + 𝑠2𝑒Ω2

2

  • Change the variable 𝑠 = sin 𝜍

𝑒𝑑2 = 𝑆2 βˆ’ cos2 πœπ‘’π‘’2 + π‘’πœ2 + sin2πœπ‘’Ξ©2

2

  • Wick Rotation 𝜍 β†’ βˆ’π‘—πœ 𝑒 β†’ 𝑗𝑒 𝑆2 β†’ βˆ’π‘†2
  • In this way we obtain the global coordinate on

anti-de Sitter space

See also de Vega-Egusquiza '96 Gubser-Klevanov-Polyakov '02 for AdS

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Rotating Folded String in dS Space

  • Centrifugal Force
  • String Tension
  • Hubble Expansion
  • Boundaries have

the speed of light

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Semiclassical Rotating String

  • 𝜚 = πœ•π‘’
  • The string Lagrangian (Nambu-Goto)

𝑀 = βˆ’4 𝑆2 2πœŒπ›½β€² ΰΆ±

𝜍0

π‘’πœ cos2𝜍 βˆ’ πœ•2 sin2 𝜍

  • String Energy

𝐹 = βˆ’4 𝑆2 2πœŒπ›½β€² ΰΆ±

𝜍0

π‘’πœ cos2 𝜍 cos2𝜍 βˆ’ πœ•2 sin2 𝜍

  • String Spin

𝑇 = βˆ’4 𝑆2 2πœŒπ›½β€² ΰΆ±

𝜍0

π‘’πœ πœ• sin2 𝜍 cos2𝜍 βˆ’ πœ•2 sin2 𝜍

See also de Vega-Egusquiza '96

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Leading Regge trajectory vs Higuchi bound

𝑆2/𝛽′ 𝑆2/𝛽′2

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Leading Regge trajectory vs Higuchi bound

  • Curved space effects modify the Regge

Trajectory to make it consistent with Higuchi bound

  • Existence of maximal spin on the trajectory
  • The longest string touching the horizon has a

speed of light even if πœ• = 0

  • The spectrum of long strings on de Sitter is

qualitatively different from the flat space and

  • AdS. The longest string has a vanishing spin and

a finite mass due to the accelerated expansion

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Energy Spin Relation

𝑆2/𝛽′2 𝑆2/𝛽′

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Short Strings

  • The energy and spin are the same as the flat

space ones

  • Linear Regge trajectory
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Long Strings

  • Spacetime curvature is not negligible
  • Touching the horizon 𝜍0 β†’

𝜌 2

  • πœ• β†’ 0
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High Energy Scattering

  • In string theory, existence of an infinite higher

spin tower (the Regge tower) is crucial to make mild the high-energy behavior of scattering amplitudes and to UV complete gravity in a weakly coupled regime

  • Existence of a maximum spin in the Regge

trajectory makes it nontrivial to maintain the mildness of high-energy scattering

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Two Possibilities

  • UV completion by the leading Regge trajectory
  • UV completion by multiple Regge trajectories
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UV completion: leading Regge trajectory

  • In order to UV complete gravity in a weakly

coupled regime, we would need sufficiently many higher-spin states from the string scale up to near the Planck scale.

  • The mass πΉβˆ—~𝑆/𝛽′~𝑁𝑑

2/𝐼 of the maximum

spin state in the leading Regge trajectory has to be bigger than the Planck scale πΉβˆ— > π‘π‘žπ‘š

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UV completion: leading Regge trajectory

  • This condition implies an upper bound on the

vacuum energy of inflation π‘Š = 3π‘π‘žπ‘š

2 𝐼2 < 𝑁𝑑 4

  • An upper bound on the tensor-to-scalar ratio

𝑠 = 0.01 Γ—

π‘Š 1016π»π‘“π‘Š 4 < 0.01 𝑁𝑑 1016π»π‘“π‘Š 4

  • π‘π‘žπ‘š can be the Planck scale in higher

dimension, which makes the bound weaker

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UV completion: multiple Regge trajectories

𝑆2/𝛽′2 𝑆2/𝛽′

N-folded closed string 𝑢 =1 𝑢 = πŸ‘ 𝑢 = πŸ’ 𝑢 = πŸ“ 𝑭𝑢 = 𝑢𝑭 𝑻𝑢 = 𝑢𝑻 𝑭𝑢

πŸ‘ = 𝑢 Γ— πŸ‘π‘»π‘Ά

πœ·β€²

We don’t know yet if scattering amplitudes are Reggeized.

(π‘­π‘Άβˆ—

πŸ‘ , π‘»π‘Άβˆ—) = (π‘ΆπŸ‘π‘­π‘Άβˆ— πŸ‘ , π‘Άπ‘»π‘Άβˆ—)

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Summary

  • String Regge trajectory is modified in de Sitter

space

  • There exists a maximum spin for each trajectory
  • Semiclassical string spectrum on de Sitter space is

consistent with the Higuchi bound

  • There may exist an upper bound on tensor to

scalar ratio under some assumptions 𝑠 = 0.01 Γ—

π‘Š 1016π»π‘“π‘Š 4 < 0.01 𝑁𝑑 1016π»π‘“π‘Š 4

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Next Step

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Thank you