Refined Swampland conjecture DBI from Two-field model Non-canonical theories Summary Hyperbolic Field Space and Swampland Conjecture for DBI Scalar Speaker: Yun-Long Zhang Yukawa Institute for Theoretical Physics, Kyoto University based on [arXiv:1905.10950] by Shuntaro Mizuno(Hachinohe), Shinji Mukohyama(YITP), Shi Pi(IPMU), Y. -L. Zhang (August 21, 2019) Y. -L. Zhang August 21, 2019 Swampland for DBI scalar 1 / 17
Refined Swampland conjecture DBI from Two-field model Non-canonical theories Summary Outline I Refined Swampland conjecture 1 Overview in the cosmological background Two-field Model with Hyperbolic Field Space 2 Attractor Behavior: From Two-Field to DBI Swampland Conjecture for Non-canonical Kinetic Terms 3 DBI scalar and P ( X , ϕ ) theory Summary 4 Y. -L. Zhang August 21, 2019 Swampland for DBI scalar 2 / 17
Refined Swampland conjecture DBI from Two-field model Non-canonical theories Summary De Sitter swampland conjecture • For a theory coupled to gravity with the potential V of scalar fields, L = − 1 2 G IJ ( φ K ) g µν ∂ µ φ I ∂ ν φ J + V ( φ I ) , (1) a necessary condition for the existence of a UV completion is |∇ V | ≥ c V . (2) I , J : indexes of scalar fields. [Obied-Ooguri-Spodyneiko-Vafa 1806.08362] • The refined de Sitter swampland conjecture min ( ∇ I ∇ J V ) ≤ − c ′ V , |∇ V | ≥ c V , or (3) where c ′ is another O (1) positive constant, and min ( ∇ I ∇ J V ) is the minimum eigenvalue of the Hessian of the potential in the local orthonormal frame. [Ooguri-Palti-Shiu-Vafa, 1810.05506] Y. -L. Zhang August 21, 2019 Swampland for DBI scalar 3 / 17
Refined Swampland conjecture DBI from Two-field model Non-canonical theories Summary How about theory with non-canonical kinetic term • The two-dimensional field model d 4 x √− g � − 1 2 g µν ∂ µ χ∂ ν χ − 1 � � 2 e 2 βχ g µν ∂ µ ϕ∂ ν ϕ − V ( χ, ϕ ) I = V ( χ, ϕ ) = T ( ϕ ) [cosh(2 βχ ) − 1] + U ( ϕ ) , (4) • How about the one field DBI scalar model? � � � � � d 4 x √− g � 2 X I eff = T ( ϕ ) − 1 − T ( ϕ ) + 1 − U ( ϕ ) , T ( ϕ ) = ϕ 4 X = − 1 2 g µν ∂ µ ϕ∂ ν ϕ , (5) λ One approach see: 1812.07670 by M. -S. Seo . Y. -L. Zhang August 21, 2019 Swampland for DBI scalar 4 / 17
Refined Swampland conjecture DBI from Two-field model Non-canonical theories Summary Covariant entropy bound in FRW background • The applicability of the covariant entropy bound S ≤ π/ H 2 , a sufficient condition � � ˙ and min m 2 H � � scalar � � c 1 � − c 2 , (6) � � H 2 H 2 � � � min m 2 scalar is the lowest among squared masses of perturbation modes of the scalar fields, and c 1 , 2 are positive numbers of O (1). • For example, the number of particle species N below the cutoff of an effective field theory is roughly given by N ∼ n ( φ ) e b φ with dn d φ > 0 . • The ansatz for the entropy of the towers of light particles in an accelerating universe S tower ( N , R ) ∼ N δ 1 R δ 2 , where R ∼ 1 / H is the radius of the apparent horizon, H is the Hubble expansion rate and δ 1 , 2 are positive numbers of O (1). Y. -L. Zhang August 21, 2019 Swampland for DBI scalar 5 / 17
Refined Swampland conjecture DBI from Two-field model Non-canonical theories Summary Refined Swampland Conjecture in FRW background • As a result, one obtains S ∼ S tower ( N , R ) and consider that � 1 � 2 − δ 2 N = n ( φ ) e b φ ∼ one obtains ln n ( φ ) ∼ − b φ − 2 − δ 2 2 δ 1 ln H 2 . δ 1 H � � � � c 1 and min m 2 ˙ H � − c 2 , from dn • Under the condition d φ > 0 scalar � � H 2 H 2 � d ( H 2 ) � � 1 c 0 ≡ 2 b δ 1 � � � � c 0 , . (7) � � H 2 d φ 2 − δ 2 � • It is therefore concluded that in FRW background � � ˙ min m 2 d ( H 2 ) � 1 � H � � � � scalar � � c 0 , or � � c 1 , or � − c 2 , (8) � � � � H 2 H 2 H 2 d φ � � � � For slow-roll models with canonical kinetic terms, the conjecture reduces to |∇ V | ≥ c V , or min ( ∇ I ∇ J V ) ≤ − c ′ V , where c ≡ min( c 0 , √ 2 c 1 ) and c ′ ≡ c 2 / 3 are still of O (1). Y. -L. Zhang August 21, 2019 Swampland for DBI scalar 6 / 17
Refined Swampland conjecture DBI from Two-field model Non-canonical theories Summary Two-field Model with Hyperbolic Field Space • We consider a two-dimensional hyperbolic field space, G IJ ( φ K ) d φ I d φ J = d χ 2 + e 2 βχ d ϕ 2 , (9) where β is a positive constant. • The action of the scalar fields { φ I } = { χ, ϕ } is then given by d 4 x √− g � − 1 2 g µν ∂ µ χ∂ ν χ − 1 � 2 e 2 βχ g µν ∂ µ ϕ∂ ν ϕ I = � − T ( ϕ ) [cosh(2 βχ ) − 1] − U ( ϕ ) , (10) where T ( ϕ ) ≡ A ( ϕ ) and U ( ϕ ) ≡ A ( ϕ ) + B ( ϕ ). Y. -L. Zhang August 21, 2019 Swampland for DBI scalar 7 / 17
Refined Swampland conjecture DBI from Two-field model Non-canonical theories Summary Attractor Behavior of χ • The equation of motion for χ leads ✷ χ + 2 β e 2 βχ X − 2 β T ( ϕ ) sinh(2 βχ ) = 0 , (11) where X ≡ − g µν ∂ µ ϕ∂ ν ϕ/ 2. • If β 2 is large then χ has a heavy mass, with ✷ χ dropped, i.e. 2 β e 2 βχ X − 2 β T ( ϕ ) sinh(2 βχ ) ≃ 0 , (12) • It is easily solved with respect to χ as � − 1 / 2 � χ ≃ 1 2 X 2 β ln γ , γ ≡ 1 − . (13) T ( ϕ ) Y. -L. Zhang August 21, 2019 Swampland for DBI scalar 8 / 17
Refined Swampland conjecture DBI from Two-field model Non-canonical theories Summary Attractor Behavior: From Two Field to DBI • The two-dimensional field space d 4 x √− g � − 1 2 g µν ∂ µ χ∂ ν χ − 1 � 2 e 2 βχ g µν ∂ µ ϕ∂ ν ϕ I = � − T ( ϕ ) [cosh(2 βχ ) − 1] − U ( ϕ ) , (14) • It is reduced to an effective one-dimensional field space spanned by ϕ with the effective action d 4 x √− g � − γ − 1 + 1 � � � � I eff = T ( ϕ ) − U ( ϕ ) , � − 1 / 2 � X = − 1 2 X 2 g µν ∂ µ ϕ∂ ν ϕ , = e 2 βχ . γ ≡ 1 − (15) T ( ϕ ) Y. -L. Zhang August 21, 2019 Swampland for DBI scalar 9 / 17
Refined Swampland conjecture DBI from Two-field model Non-canonical theories Summary Attractor Behavior of the two field system 10 2.0 e 2 β χ Two - field 9 1.5 DBI γ φ ( t ) 8 γ ( t ) 1.0 7 0.5 6 0.0 0 1 2 3 4 5 0 1 2 3 4 5 t t Figure: Non-relativistic attractor ( γ = 1) with the parameter choice U ( ϕ ) = 1 + 0 . 1 ϕ 2 , T ( ϕ ) ≡ ϕ 4 /λ , β = 20 and λ = 0 . 5. 20 - 16 e 2 β χ - 17 Two - field 15 DBI γ - 18 γ ( t ) 10 φ ( t ) - 19 5 - 20 - 21 0 0 1 2 3 4 5 0 1 2 3 4 5 t t Figure: Relativistic attractor ( γ = 10) with the parameter choice U ( ϕ ) = 7 . 5 ϕ 2 , T ( ϕ ) ≡ 1 /λ , β = 20 and λ = 10. Y. -L. Zhang August 21, 2019 Swampland for DBI scalar 10 / 17
Refined Swampland conjecture DBI from Two-field model Non-canonical theories Summary Equations of motion in the FRW background • “Two-field” denotes that we evaluate ϕ ( t ) based on the full equations of motion from the two-field action (14) χ − β e 2 βχ ˙ ϕ 2 + 2 β T ( ϕ ) sinh (2 βχ ) = 0 , χ + 3 H ˙ ¨ ϕ + T ′ ( ϕ ) + U ′ ( ϕ ) � � ϕ + 3 H ˙ ¨ ϕ + 2 β ˙ χ ˙ cosh(2 βχ ) − 1 e 2 βχ = 0 , e 2 βχ 3 H 2 = 1 χ 2 + e 2 βχ ˙ φ 2 ) + T ( ϕ ) � � 2( ˙ cosh(2 βχ ) − 1 + U ( ϕ ) , (16) • “DBI” denotes that we evaluate ϕ ( t ) based on the equation of motion for the single-field DBI model ϕ − T ′ ( ϕ ) ( γ − 1) 2 ( γ + 2) + U ′ ( ϕ ) γ 2 ¨ ϕ + 3 H ˙ = 0 , 2 γ γ � − 1 / 2 γ 2 ϕ 2 � ˙ 3 H 2 = ϕ 2 + U ( ϕ ) , ( γ + 1) ˙ γ = 1 − . (17) T ( ϕ ) Y. -L. Zhang August 21, 2019 Swampland for DBI scalar 11 / 17
Refined Swampland conjecture DBI from Two-field model Non-canonical theories Summary Geodesic Distance in the Field Space • It is given by integrating � G IJ ( φ K ) d φ I d φ J = � d χ 2 + e 2 βχ d ϕ 2 . d φ = (18) • For large enough β 2 , by using the attractor behavior χ ≃ 1 2 β ln γ , this is reduced to � 1 / 2 γ 2 � ˙ d ϕ ≃ √ γ d ϕ , d φ ≃ ϕ 2 + γ (19) 4 β 2 γ 2 ˙ • We have used the fact that the evolution of ϕ is well described by the γ 2 / ( γ 2 ˙ ϕ 2 ) remains finite in the β 2 → ∞ single-field model and thus ˙ � � d ( H 2 ) � 1 limit. Thus, the first inequality � � c 0 can be rewritten as � � H 2 d φ d ( H 2 ) � � 1 1 � � � � c 0 . (20) √ γ � � H 2 d ϕ � Y. -L. Zhang August 21, 2019 Swampland for DBI scalar 12 / 17
Refined Swampland conjecture DBI from Two-field model Non-canonical theories Summary Swampland conjecture for DBI scalar • The effective single-field DBI action � � � � � d 4 x √− g � 2 X I DBI = T ( ϕ ) − 1 − T ( ϕ ) + 1 − U ( ϕ ) . (21) • The swampland conjecture is written as � � ˙ d ( H 2 ) � � 1 1 H Ω � � � � � � c 0 , or � � c 1 , or H 2 � − c 2 . (22) √ γ � � � � H 2 d ϕ H 2 � � � � � • Where γ ≡ 1 / 1 − 2 X / T , X ≡ − g µν ∂ µ ϕ∂ ν ϕ/ 2 and T ′′ − [( γ + 3)( γ − 1) T ′ − 2 γ U ′ ] 2 γ 3 U ′′ + ( γ − 1) 2 Ω = 1 . (23) 2 γ 4 16 γ 4 T Y. -L. Zhang August 21, 2019 Swampland for DBI scalar 13 / 17
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