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- 1. Back-reaction in AdS Braneworld Neven Bilić Ruđer Bošković Institute Zagreb BW2013, Vrnjačka Banja, April 2013
- 2. Basic idea A 3-brane moving in AdS5 background of the second Randall-Sundrum (RSII) model behaves effectively as a tachyon with the inverse quartic potential. The RSII model may be extended to include the back reaction due to the radion field. Then the tachyon Lagrangian is modified by the interaction with the radion. As a consequence, the effective equation of state obtained by averaging over large scales describes a warm dark matter. Based on the work in colaboration with Garry Tupper “Warm” Tachyon Matter from Back-reaction on the Brane” , arXiv:1302.0955 [hep-th].
- 3. Outline 1. Tachyons in the Randall–Sundrum Model 2. Backreaction Model 3. Isotropic Homogeneous Evolution 4. Warm Dark Matter 5. Conclusins&Outlook
- 4. Unstable modes in string theory can be described by an effective Born-Infeld type lagrangian for the “tachyon” field θ. The typical potential has minima at . Of particular interest is the inverse power law potential . For n > 2 , as the tachyon rolls near minimum it behaves like pressure-less matter (dust) or cold dark matter. , , 1V g L n V Tachyon as CDM A. Sen, JHEP 0204 (2002); 0207 (2002). In general any tachyon model can be derived as a map from the motion of a 3-brane moving in a warped extra dimension.
- 5. x 5 x y L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) In RSII, the 5-dim bulk is ADS5/Z2 with the line element Observers reside on the positive tension brane at y=0 0y y l5( )d x y 2 2 2 (5) (5) ( ) M N ky MNds g X dX dX e g x dx dx dy 0 Second Randall – Sundrum (RSII) model
- 6. Radion The Randall-Sundrum solution corresponds to an empty brane at the fixed point y=0. Placing matter on this observer brane changes the bulk geometry; this is encoded in the radion field related to the variation of the physical interbrane distance d5(x).
- 7. 2. Backreaction Model The naive AdS5 geometry is distorted by the radion. To see this, consider the total gravitational action in the bulk bulk GH braneS S S S We choose a coordinate system such that and we assume a general metric which admits Einstein spaces of constant 4-curvature with the line element 2 2 2 2 (5) (5)= ( ) = ( , ) ( ) ( , ) ,M N MNds g X dX dX x y g x dx dx x y dy 5 bulk 55 5 5 1 2 2 S d x g R K where At fixed xμ , is the distance along the fifth dimension.5d dy (5) 5 0, 0,1,2,3g
- 8. 2 2 4 2 4 b , , 5 5 1 ( ) = 3 ( ) 6 2 ' ulk R S d x g dy g K where R is the 4-d Ricci scalar made out of the 4-d metric We have omitted the total y-derivative term because it is cancelled by the Gibbons – Hawking term in the action The 5-dim bulk action may be put in the form For consistency with Einsten’s equations we require In addition, we impose the „Einstein frame‟ gauge condition so the coefficient of R in Sbulk is a function of y only 5 5 0R 2 2 ( )W y g
- 9. We arrive at the 5-dimensional line element 2 2 2 2 2 (5) 2 ( ) ( ) ( ) W y ds x W y g x dx dx dy x W y By choosing and neglecting the radion we recover the AdS5 geometry. Hence, the choice corresponds to the RSII model. J.E. Kim, G. Tupper, and R. Viollier, PLB 593 (2004) ( ) ky W y e This metric is a solution to Einsten’s equations provided precisely as it is in the RSII model ( ) ky W y e where the field ϕ represents the radion. 2 5 = 6 k
- 10. Integration over y fom 0 to ∞ yields , ,4 b (5) (5) (5) 3 6 = (1 2 ) 2 4 (1 ) ulk gR k S d x g kK kK K the last term may be canceled by the two brane actions 0 RS (5) 6 = =l k K With the RSII fine tuning 4 2 4 2 2 b =0 b = 0| | = (1 ) ( )kl rane y rane y l lS S d x g d x g e
- 11. Then, the bulk action takes a simple form 4 b , , 1 = 16 2 ulk R S d x g g G where we have introduced the 4-dimensional gravitational constant 5 = 8 K k G 13 = sinh 4 G and the canonically normalized radion
- 12. is no longer infinite so the physical size of the 5-th dimension is of the order although its coordinate size is infinite The appearance of a massless mode – the radion – causes 2 effects 1. Matter on observer’s brane sees the (induced) metric 2. The physical distance to the AdS5 horizon at coordinate infinity 1g g 1 Plk l 5 1 1 ln 2 d k
- 13. 0y y Consider a 3-brane moving in the 5-d bulk spacetime with metric 22 2 2 2 (5) 2 = ( ) ky ky ky e ds e g dx dx dy e Dynamical brane
- 14. The points on the brane are parameterized by . The 5-th coordinate Y is treated as a dynamical field. The brane action ,M X x Y x 22 2 3 ind (5) , , , ,2 2 2 ( ) = = ( ) kY kY M N MN kY kY e e g g X X g Y Y e e yields 1/22 2 4 2 2 b , ,2 3 ( ) = ( ) 1 ( ) kY kY rane kY e S d x g e g Y Y e with the induced metric 4 i b = det nd raneS d x g
- 15. Changing Y to a new field we obtain the effective brane action = /kY e k In the absence of the radion field ϕ we have a pure undistorted AdS5 background and This action describes a tachyon with inverse quartic potential. Although p =L < 0, the steep potential drives a dark matter attractor, so very quickly and one thus apparently gets cold geometric tachyon matter. R. Abramo and F. Finelli, PLB 575 (2003) 2 2 2 , ,4 b 4 4 2 2 3 (1 ) = 1 (1 ) rane gk S d x g k k (0) 4 b , ,4 4 = 1raneS d x g g k 0p
- 16. E.g., for positive power law potential 2 0( ) n V V One can get dark matter and dark energy as a single entity i.e., DE/DM unification N. B., G. Tupper, R.Viollier, Cosmological tachyon condensation. Phys. Rev. D 80 (2009) Tachyon has been heavily exploited in almost any cosmological context: tachyon as inflaton, tachyon as DM, tachyon as DE
- 17. 1. The geometric tachyon is seen on our brane as a form of matter and hence it affects the bulk geometry in which it moves. 2. The back-reaction qualitatively changes the geometric tachyon: the tachyon and radion form a composite substance with a modified equation of state. The dynamical brane causes two back-reaction effects
- 18. Instead of the tachyon field defined previously as it is convenient to introduce a new field = /kY e k 22 )(2 )( 3 =3=)( xk ex xkY Then the combined radion and brane Lagrangian becomes , ,2 2 , , 2 3 1 = 1 2 g g L 2 6 = 2 6 , = , = 6k k Field Equations where
- 19. Where we define canonical conjugate momentum fields , , 2 3 , , , , = = = = 1 / gL L g g For timelike and we may also define the norms = =g g Then the Lagrangian may be expressed as 2 2 2 2 2 2 1 1 = 2 1 / ( ) L ,,
- 20. The corresponding energy momentum tensor may be expressed as a sum of two ideal fluids 1 1 1 1 2 2 2 2 1 2= ( ) ( ) ( )T p u u p u u p p g with 2 2 2 = 2 1 / ( ) T g g g L L L 1 2= =u u 2 2 1 2 2 2 2 2 1 1 = = 2 1 / ( ) p p 2 2 2 2 2 1 2 2 1 = = 1 / ( ) 2
- 21. The Hamiltonian may be identified with the total energy density 1 2= 3 =T H L which yields 2 2 2 2 2 2 1 = 1 / ( ) 2 H It may be easily verified that H is related to L through the Legendre transformation , , , ,( , ) = ( , ) H L Here the dependence on Φ and Θ is suppressed. The rest of the field variables are constrained by Hamilton’s equations
- 22. Hamilton’s Equations Now we multiply the first and second equations by and , respectively, and we take a covariant divergence of the next two equations , , , , = = = = H H L L 2u 1u
- 23. We obtain a set of four 1st order diff. equations = 2 2 2 = 1 / ( ) 2 2 2 1 2 2 2 2 3 4 3 3 = 1 / ( ) H 2 2 2 2 2 2 2 2 1 4 3 3 = 1 / ( ) H 4 4 = sinh 2 3 3 G G where 1 1 ; 2 2 ;3 = 3 =H u H u 1 , 2 , 1 , 1 ,, , ,u u u u
- 24. 3. Isotropic Homogeneous Evolution To exhibit the main features we solve our equations assuming spatially flat FRW spacetime with line ellement 1 2= a H H H a and we have in addition the equation for the scale a(t) 8 = 3 a G a H in which case 2 2 2 2 2 2 ( )ds dt a dr r d We evolve the radion-tachyon system from t=0 with some suitably chosen initial conditions
- 25. Evolution of the radion-tachyon system for λl2=1/3 with the initial conditions at t=0: 1.01, 0.1, 0 Time is taken in units of l
- 26. After the transient period the equation of state w=p/ρ becomes positive and oscillatory
- 27. Approximate asymptotic solution to second order in the amplitude A of tΦ 2 3 2 24 2 2 cos 3 2 A t A 2 2 2 2 2 3 1 2cos 2 A t A t In the asymptotic regime ( ) 2 2 3/2 2 | | / p w t so 23 sin 2 8 t w A cos 2 A t t
- 28. Since the oscillations in w are rapid on cosmological timescales, it is most useful to time average co-moving quantities. The effective equation of state is then By averaging the asymptotic w over long timescales we find 2 3 0.009 8 A w =p w where we have estimated A by comparing the exact and approximate solutions for tΦ
- 29. CDM is rather successful in explaining the large-scale power spectrum and CMB spectrum. However there is some inconsistency of many body simulations with observations: 1. Overproduction of satellite galaxies 2. Modelling haloes with a central cusp. 4. Warm Dark Matter It is believed that these problems may be alleviated to some extent with the so called Warm DM Besides, it has been recently argued that cosmological data favour a dark matter equation of state wDM ≈ 0.01
- 30. We assume that our equation of state w≈0.009 corresponds to that of (nonrelativistic) DM thermal relics of mass mDM at the time of radiation-matter equality teq D D =M M T w m Hence, we take and identify This gives mDM ≈ 0.8 keV D e| =< >= 0.009M qw w e eq= = 7.4 eV atqT T t t To demonstrate that our asymptotic equation of state is associated with WDM, we will show that the horizon mass at the time when the equivalent DM particles just become nonrelativistic is typically of the order of a small galaxy mass.
- 31. These DM particles have become non relativistic at the temperature T≡TNR =mDM , corresponding to the scale eq N e e NR < >R q q T a a w a T . The horizon mass before equality evolves as , 3 e eH q q a a MM where for a spatially flat universe. Thus, at a=aNR we obtain 15 e 2 10qM M 9 H 10M M the mass scale typical of a small galaxy and therefore the DM may be qualified as warm.
- 32. Conclusions&Outlook • We have demonstrated that back-reaction causes the brane – radion system to behave as “warm” tachyon matter with a linear barotropic equation of state • The ultimate question regards the clustering properties of the model. At the linear level one expects a suppression of small-scale structure formation: initially growing modes undergo damped oscillations once they enter the co-moving acoustic horizon • Perturbation theory is not the whole story – it would be worth studying the nonlinear effects, e.g., using the Press-Schechter formalism as in the pure tachyon model of N. B., G. Tupper, R.Viollier, Phys. Rev. D 80 (2009)
- 33. Thank you

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