Geodesic motion of test particles in the space-time of cosmic (super)strings

Geodesic motion of test particles in the space-time of cosmic (super)strings Parinya Sirimachan Jacobs University of Bremen Kosmologietag Bielefeld 5/5/ P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ /

This work Collaboration Betti Hartmann School of Engineering and Science, Jacobs University Bremen Claus Laemmerzahl References ZARM,University of Bremen Institut fuer Physik, Oldenburg University. Detection of cosmic superstrings by geodesic test particle motion Physical Review D 83, 457 () Betti Hartmann, Claus Laemmerzahl and Parinya Sirimachan Geodesic motion in the space-time of a cosmic string Journal of High Energy Physics JHEP 8 (). Betti Hartmann and Parinya Sirimachan P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ /

Table of contents Introduction The difference between Cosmic String and Cosmic Superstring? Reviews and Motivations Reviews Motivations 3 Model Assumptions ODEs and BCs 4 Geodesic equations 5 Results Effect of winding number Observable 6 Conclusions P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 3 /

Introduction The difference between Cosmic String and Cosmic Superstring? Cosmic Strings or Cosmic Superstring? Classical Field Theory Type IIB String Theory -d (spatially) topological defect due to series of SSB in the early time of the Universe. Their forms and the stabilities can be explained from homotopy group. Classify according to the type of symmetry. F and D strings were produced after the collision of a pair of D-3 branes (the end of inflation). They are BPS-object with different charges cosmic superstrings. P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 4 /

Introduction The difference between Cosmic String and Cosmic Superstring? From Cosmic string to Cosmic Superstring? When strings collide, Same type. Different type. They can pass through or reconnect. The probability of the intercommuting is close to due to the energetically favorable shortcut. They cannot intercommute but they exchange charges and form a junction because of charge conservation The junction of F-and D strings with winding number p and q respectively is called p-q strings. P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 5 /

Reviews and Motivations Reviews Reviews In the framework of field theoretical models Saffin(5) and Rajantie et al.(7): (p,q) Bound states have been investigated by using the different type of Abelian Higgs strings with a coupling of scalar field via potential. Salmi(8): Type-I Abelian Higgs strings can form a bound state. Hartmann & Urrestilla (8): (p,q) Bound stated have been studied in a curved space-time. Hartmann & Minkov (9): (p,q) Bound stated have been studied in anti-de-sitter space-time. Question What should be observational consequences of their existence? P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 6 /

Reviews and Motivations Motivations Objectives & Motivations The goals of this work Investigate (p,q) string space-times by solving geodesic equations numerically. Study the effect of binding parameter, winding numbers and the ratio of symmetry breaking and Plank mass scale on change the geodesics. Connect the results to the observable phenomena e.g. perihelion shift, light deflection. Why we need to study p-q strings? The observational part of the String Theory. Better understanding of the fundamental theory of nature. P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 7 /

Model Assumptions Model /3 Two U() Abelian Higgs strings with the minimal coupling to the gravity. S = d 4 x ( ) R g 6πG +L m where L m = D µφ(d µ φ) 4 FµνFµν +D µξ(d µ ξ) 4 HµνHµν V(φ,ξ), Fµν = µa ν νa µ, Hµν = µb ν νb µ D µφ = µφ ie A µφ, D µξ = µξ ie B µξ. and Interaction appears in potential via coupling constant λ 3 with V(φ,ξ) = λ 4 (φφ η ) + λ 4 (ξξ η ) λ 3(φφ η )(ξξ η ) λ λ > 4λ 3. P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 8 /

Model Assumptions Model /3 Cylindrically static symmetric source for the metric with boost invariance along z direction. ds = N() dt d L() dϕ N() dz Nielsen-Olesen ansatz for the matter field. φ(,ϕ) = η h()e inϕ ξ(,ϕ) = η f()e inϕ,a µdx µ = e (n P())dϕ,B µdx µ = e (m R())dϕ Rescaling x = e η, L(x) = e η L(), N(x) = e η N() γ = 8πGη = 8πη /M Pl, g = e, q = η ( ) MHi, β i = = λ i e η M W e, i =, P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 9 /

Model ODEs and BCs Model 3/3 From the variation of action with respect to matter fields (N Lh ) P h N = L L + V h, (N Lf ) N = R f L L + V f, ( ) L N P ( ) N = h L N R P, L N = f R L From the variation of action with respect to the metric (LNN ) N L (N L ) N L [ (P ) = γ L + (R ] ) g L u, [ h P = γ L + R f L + (P ) L + (R ] ) g L + u where u = 4 (h ) + β 4 (f q ) (h )(f q ). Boundaries conditions (regularity at = ) The finiteness of energy per unit length h() =, f() =, P() = n, R() = m, N() =, L() =, N() =, L() =. h( ) =, f( ) = q, P( ) =, R( ) =. P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ /

Geodesic equations Constants of Motion and Effective Potential From L g = dxµ gµν dτ dx ν dτ = ε, N() dt dτ V eff () = Effective Potential = E, L() dϕ dτ = Lz, N() dz dτ = Pz. [ P ( )] z N() + L z L() E N().4.35.3 γ =.3 β =.5 β =.5 β =. β = 7. From dv eff /d = we obtain E P z L z = N()3 N L L() 3..5 V eff..5 The existence of minimal point. N () >.5 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ /

Metric Solutions g tt From cylindrically static symmetric metric ( ds = N() dt d L() dϕ N() dz MH, ), where, = M W and β3 = λ 3 e. No strings interaction With bound state = β, =..8, =., =., =.5..8 =., β =., γ =.3 =. =.35 =.7.6 N.6 N.4.4...98 5 5.98 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ /

Metric Solutions g tt From cylindrically static symmetric metric ( ds = N() dt d L() dϕ N() dz MH, ), where, = M W and β3 = λ 3 e. No strings interaction With bound state = β, =..8, =., =., =.5..8 =., β =., γ =.3 =. =.5 =.95.6 N.6 N.4.4...98 5 5.98 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ /

Metric Solutions g tt From cylindrically static symmetric metric ( ds = N() dt d L() dϕ N() dz MH, ), where, = M W and β3 = λ 3 e. No strings interaction With bound state = β, =..8, =., =., =.5..8 =., β =., γ =.3 =. =.5 =.49.6 N.6 N.4.4...98 5 5.98 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ /

.988.986.984.98.98.9798.9796.9794.5.5 3 3.5 Metric Solutions g tt From cylindrically static symmetric metric ( ds = N() dt d L() dϕ N() dz MH, ), where, = M W and β3 = λ 3 e. No strings interaction With bound state = β, =., =.. =.5, β = 6., γ =.3.8.6 N, =., =.5.8.6 N =. =.3 =.7 =.5.4.4...98 5 5.98 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ /

.3...9999.9998.9997.9996..4.6.8 Metric Solutions g tt From cylindrically static symmetric metric ( ds = N() dt d L() dϕ N() dz MH, ), where, = M W and β3 = λ 3 e. No strings interaction With bound state = β, =..8.6 N, =., =., =.5..8.6 N =., β = 4.5, γ =.3 =. =. =.66 =.85.4.4...98 5 5.98 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ /

..999.998.5.5.5 3 3.5 4 Metric Solutions g tt From cylindrically static symmetric metric ( ds = N() dt d L() dϕ N() dz MH, ), where, = M W and β3 = λ 3 e. No strings interaction With bound state = β, =..8, =., =., =.5..8 =.5, β = 4., γ =.3 =. =.45 =.5.6 N.6 N =.6.4.4...98 5 5.98 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ /

..999.998.5.5.5 3 3.5 4 Metric Solutions g tt From cylindrically static symmetric metric ds = N() dt d L() dϕ N() dz, where, = ( MH, M W ) and β3 = λ 3 e. No strings interaction With bound state = β, =..8, =., =., =.5..8 =.5, β = 4., γ =.3 =. =.45 =.5.6 N.6 N =.6.4.4...98 5 5.98 5 5 Remarks Without interaction bound orbit exits when β i < (Type I). shifts up the asymptotical value of N(). P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ /

Effective Potential and Classification of Orbit V eff () = [ P z N() + L ( )] z L() E N(), dv eff () dx = E p z L z = N(x)3 L (x) N (x) L(x) 3. Effective Potential µ ν plot with µ = E P z and ν = L z.4.. =., β =., γ =.3 =. =.35 =.7.8 V eff.6.4.. 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 3 /

Effective Potential and Classification of Orbit V eff () = [ P z N() + L ( )] z L() E N(), dv eff () dx = E p z L z = N(x)3 L (x) N (x) L(x) 3. Effective Potential µ ν plot with µ = E P z and ν = L z.4.. =., β =., γ =.3 =. =.5 =.95.8 V eff.6.4.. 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 3 /

Effective Potential and Classification of Orbit V eff () = [ P z N() + L ( )] z L() E N(), dv eff () dx = E p z L z = N(x)3 L (x) N (x) L(x) 3. Effective Potential µ ν plot with µ = E P z and ν = L z.4.. =., β =., γ =.3 =. =.5 =.49.8 V eff.6.4.. 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 3 /

..5.5. Results Effective Potential and Classification of Orbit V eff () = [ P z N() + L ( )] z L() E N(), dv eff () dx = E p z L z = N(x)3 L (x) N (x) L(x) 3. Effective Potential µ ν plot with µ = E P z and ν = L z.4...8 =.5, β = 6., γ =.3 3.5 =. =.3 =.7 =.5 V eff.6.4.. 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 3 /

..8.6.4..5.5.5 3 Results Effective Potential and Classification of Orbit V eff () = [ P z N() + L ( )] z L() E N(), dv eff () dx = E p z L z = N(x)3 L (x) N (x) L(x) 3. Effective Potential µ ν plot with µ = E P z and ν = L z.4...8 =., β = 4.5, γ =.3 =. =. =.66 =.85 V eff.6.4.. 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 3 /

4 3 3.5.5.5 3 Results Effective Potential and Classification of Orbit V eff () = [ P z N() + L ( )] z L() E N(), dv eff () dx = E p z L z = N(x)3 L (x) N (x) L(x) 3. Effective Potential µ ν plot with µ = E P z and ν = L z.4...8 =.5, β = 4., γ =.3 x 3 =. =.45 =.5 =.6 V eff.6.4.. 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 3 /

4 3 3.5.5.5 3 Results Effective Potential and Classification of Orbit V eff () = [ P z N() + L ( )] z L() E N(), dv eff () dx = E p z L z = N(x)3 L (x) N (x) L(x) 3. Effective Potential µ ν plot with µ = E P z and ν = L z.4. =.5, β = 4., γ =.3 x 3 =. =.45.8.7 = β =., =.75, γ =.3 (µ 3,ν 3 ) M4..8 =.5 =.6.6.5 M M3 M4 V eff.6 ν.4.4.3... M. 5 5 (µ,) (µ,).98..4.6.8 µ P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 3 /

4 3 3.5.5.5 3 Results Effective Potential and Classification of Orbit V eff () = [ P z N() + L ( )] z L() E N(), dv eff () dx = E p z L z = N(x)3 L (x) N (x) L(x) 3. Effective Potential µ ν plot with µ = E P z and ν = L z.4. =.5, β = 4., γ =.3 x 3 =. =.45.8.7 = β =., =., γ =.3..8 =.5 =.6.6.5 V eff.6 ν.4.4.3... (µ 3,ν 3 ). 5 5 (µ,) (µ,).98..4.6.8 µ P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 3 /

4 3 3.5.5.5 3 Results Effective Potential and Classification of Orbit V eff () = [ P z N() + L ( )] z L() E N(), dv eff () dx = E p z L z = N(x)3 L (x) N (x) L(x) 3. Effective Potential µ ν plot with µ = E P z and ν = L z.4. =.5, β = 4., γ =.3 x 3 =. =.45.35.3 =., β = 3.6, =.5, γ =.35..8 V eff.6 =.5 =.6.5. ν.5.4...5. 5 5.98.99...3 µ P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 3 /

4 3 3.5.5.5 3 Results Effective Potential and Classification of Orbit V eff () = [ P z N() + L ( )] z L() E N(), dv eff () dx = E p z L z = N(x)3 L (x) N (x) L(x) 3. Effective Potential µ ν plot with µ = E P z and ν = L z.4. =.5, β = 4., γ =.3 x 3 =. =.45.35.3 =., β = 3.6, =.5, γ =.35..8 V eff.6 =.5 =.6.5. ν.5.4...5. 5 5.98.99...3 µ Remarks Bound state of p q strings bound orbit of massive test particle for all values of, strings. No bound orbit for massless test particle. P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 3 /

Metric solution g φφ From cylindrically static symmetric metric ds = N() dt d L() dφ N() dz, L() dφ ( 8πGµ) dφ where, = ( MH, M W ) and β3 = λ 3 e. =., γ =.35, p =, q = =., β = 3.6,γ =.35 6 o β = 3.6 =.5 5 x β = 4.9 =.5 4 + β = 6.4 8 =.38 =.96 3, E in L() 6 4 9 8 7.5.5 3 3.5 4 4 6 8 Remark Interaction between p q strings reduces the deficit angle in conical space-time. P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 4 /

Examples of orbits Bound orbits with the change of coupling constant Effective Potential =., β = 3.6,γ =.35, =...5. V eff.5.5 E =.995, p z =., L z =. =. =.5 =.. 4 6 8 Y x y plane Blue circle is the radius of gauge field /M W = /( e η ) Red circles are circle is the radius or scalar field /M H, = /(η λ, ) Z 4 3 d bound orbit Y P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 5 /

Examples of orbits Bound orbits with the change of coupling constant Effective Potential =., β = 3.6,γ =.35, =.5. E =.995, p z =., L z =. =. x y plane 3 d bound orbit.5 =.5 =.. V eff.5 Y 4.5. 4 6 8 Blue circle is the radius of gauge field /M W = /( e η ) Z Y Red circles are circle is the radius or scalar field /M H, = /(η λ, ) P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 5 /

Examples of orbits Bound orbits with the change of coupling constant Effective Potential =., β = 3.6,γ =.35, =.. E =.995, p z =., L z =. =. x y plane 3 d bound orbit.5 =.5 =.. V eff.5 Y 4.5. 4 6 8 Blue circle is the radius of gauge field /M W = /( e η ) Z Y Red circles are circle is the radius or scalar field /M H, = /(η λ, ) P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 5 /

Escape orbits with the change of coupling constant Effective Potential =., β = 3.6,γ =.35, =.5..5. V eff.5.5 E =.995, p z =., L z =. =. =.5 =.. 4 6 8 Y 5 5 x y plane 5 5 Blue circle is the radius of gauge field /M W = /( e η ) 5 5 5 5 Y Red circles are circle is the radius or scalar field /M H, = /(η λ, ) Z 3 d escape orbit 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 6 /

Escape orbits with the change of coupling constant Effective Potential =., β = 3.6,γ =.35, =.48. E =.995, p z =., L z =. =. x y plane 3 d escape orbit.5. =.5 =. 5 5 V eff.5 Y Z 5.5. 4 6 8 5 5 5 Blue circle is the radius of gauge field /M W = /( e η ) 5 5 Y Red circles are circle is the radius or scalar field /M H, = /(η λ, ) 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 6 /

Escape orbits with the change of coupling constant Effective Potential =., β = 3.6,γ =.35, =.96. E =.995, p z =., L z =. =. x y plane 3 d escape orbit.5. =.5 =. 5 5 V eff.5 Y Z 5.5. 4 6 8 5 5 5 Blue circle is the radius of gauge field /M W = /( e η ) 5 5 Y Red circles are circle is the radius or scalar field /M H, = /(η λ, ) 5 5 P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 6 /

Effect of winding number Bound orbits with the change of winding number p,q N() = g tt = 8., β =.5, =.99, p =, q = = 8., β =.5, =.99, γ =. x y plane 3 d bound orbit.8 p =,q =,E B = 3.86 p =,q =,E B =.77.6.4 N(). p = 3,q =,E B =.33 p =,q =,E B = 9.6 p =,q = 3,E B = 4.4 p =,q = 3,E B = 3.66 Y 5 Z.98.96 5 5 5 3 35 4 5 Y Blue circle is the radius of gauge field /M W = /( e η ) Red circles are circle is the radius or scalar field /M H, = /(η λ, ) P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 7 /

Effect of winding number Bound orbits with the change of winding number p,q N() = g tt = 8., β =.5, =.99, p =, q =.8 = 8., β =.5, =.99, γ =. p =,q =,E B = 3.86 x y plane 3 d bound orbit.6.4 N(). p =,q =,E B =.77 p = 3,q =,E B =.33 p =,q =,E B = 9.6 p =,q = 3,E B = 4.4 p =,q = 3,E B = 3.66 5 Y Z.98.96 5 5 5 3 35 4 5 Y Blue circle is the radius of gauge field /M W = /( e η ) Red circles are circle is the radius or scalar field /M H, = /(η λ, ) P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 7 /

Effect of winding number Bound orbits with the change of winding number p,q N() = g tt = 8., β =.5, =.99, p =, q =.8 = 8., β =.5, =.99, γ =. p =,q =,E B = 3.86 x y plane 3 d bound orbit.6 p =,q =,E B =.77 p = 3,q =,E B =.33 p =,q =,E B = 9.6.4 N() p =,q = 3,E B = 4.4 p =,q = 3,E B = 3.66 5. Y Z.98.96 5 5 5 3 35 4 5 Y Blue circle is the radius of gauge field /M W = /( e η ) Red circles are circle is the radius or scalar field /M H, = /(η λ, ) P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 7 /

Effect of winding number Escape orbits with the change of winding number p,q L() = g φφ = 8., β =.5, =.99, p =, q = 6 5 4 L() 3 = 8., β =.5, =.99, γ =. p =,q =,E B = 3.86 p =,q =,E B =.77 p = 3,q =,E B =.33 p =,q =,E B = 9.6 p =,q = 3,E B = 4.4 p =,q = 3,E B = 3.66 6 4 Y x y plane Z 5 3 d escape orbit 5 5 5 3 35 4 4 6 5 5 5 5 Y 5 5 5 Blue circle is the radius of gauge field /M W = /( e η ) Red circles are circle is the radius or scalar field /M H, = /(η λ, ) P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 8 /

Effect of winding number Escape orbits with the change of winding number p,q L() = g φφ = 8., β =.5, =.99, p =, q =3 L() 6 5 4 3 = 8., β =.5, =.99, γ =. p =,q =,E B = 3.86 p =,q =,E B =.77 p = 3,q =,E B =.33 p =,q =,E B = 9.6 p =,q = 3,E B = 4.4 p =,q = 3,E B = 3.66 6 4 Y x y plane Z 5 3 d escape orbit 5 5 5 3 35 4 4 6 5 5 5 5 Y 5 5 5 Blue circle is the radius of gauge field /M W = /( e η ) Red circles are circle is the radius or scalar field /M H, = /(η λ, ) P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 8 /

Effect of winding number Escape orbits with the change of winding number p,q L() = g φφ = 8., β =.5, =.99, p = 3, q = L() 6 5 4 3 = 8., β =.5, =.99, γ =. p =,q =,E B = 3.86 p =,q =,E B =.77 p = 3,q =,E B =.33 p =,q =,E B = 9.6 p =,q = 3,E B = 4.4 p =,q = 3,E B = 3.66 Y 6 4 x y plane Z 5 3 d escape orbit 5 5 5 3 35 4 4 6 5 5 5 5 Y 5 5 5 Blue circle is the radius of gauge field /M W = /( e η ) Red circles are circle is the radius or scalar field /M H, = /(η λ, ) P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 8 /

Observable Observable Perihelion Shift Light Deflection perihelion shift (rad).5.6.7.8.9 =., β =. =.4, β =.4 =.8, β =.8 =.8, β =. =.8, β =.4 =., β =.4 light deflection (rad)..9.8.7.6 =., β =. =.4, β =.4 =.8, β =.8 =.8, β =. =.8, β =.4 =., β =.4 3.5 3..45.5.55.6.65.7.75 λ 3.4.45.5.55.6.65.7.75 Remark The negative perihelion shift could be the sign of the detection of string. P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ 9 /

Conclusions Conclusions Interaction between strings has influence on the change of metric profile and the energy per unit length of strings. Bound state increases possibility of a test particle to have an interaction with strings. 3 In comparison to infinitely thin cosmic string, bound orbits are possible for a finite width of cosmic strings and perihelion shift can become negative. Thank you for your attention P. Sirimachan (Jacobs University Bremen) Geodesics Cosmic Superstring 5/5/ /