Back-reaction in AdSBraneworldNeven BilićRuđer Bošković InstituteZagrebBW2013, Vrnjačka Banja, April 2013
Basic ideaA 3-brane moving in AdS5 background of the secondRandall-Sundrum (RSII) model behaves effectively as atachyon with the inverse quartic potential.The RSII model may be extended to include the backreaction due to the radion field. Then the tachyonLagrangian is modified by the interaction with the radion.As a consequence, the effective equation of stateobtained by averaging over large scales describes awarm 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].
Outline1. Tachyons in the Randall–SundrumModel2. Backreaction Model3. Isotropic Homogeneous Evolution4. Warm Dark Matter5. Conclusins&Outlook
Unstable modes in string theory can be described by aneffective Born-Infeld type lagrangianfor the “tachyon” field θ.The typical potential has minima at . Of particularinterest is the inverse power law potential .For n > 2 , as the tachyon rolls near minimum it behaveslike pressure-less matter (dust) or cold dark matter. , ,1V g L nV Tachyon as CDMA. Sen, JHEP 0204 (2002); 0207 (2002).In general any tachyon model can be derived as a map fromthe motion of a 3-brane moving in a warped extra dimension.
x5x yL. Randall and R. Sundrum,Phys. Rev. Lett. 83 (1999)In RSII, the 5-dim bulk is ADS5/Z2 with the line elementObservers reside on the positive tension brane at y=00y y l5( )d x y 2 2 2(5) (5) ( ) M N kyMNds g X dX dX e g x dx dx dy 0 Second Randall – Sundrum (RSII) model
RadionThe Randall-Sundrum solution corresponds to an emptybrane at the fixed point y=0. Placing matter on thisobserver brane changes the bulk geometry; this is encodedin the radion field related to the variation of the physicalinterbrane distance d5(x).
2. Backreaction ModelThe naive AdS5 geometry is distorted by the radion. To seethis, consider the total gravitational action in the bulkbulk GH braneS S S S We choose a coordinate system such thatand we assume a general metric which admits Einsteinspaces of constant 4-curvature with the line element2 2 2 2(5) (5)= ( ) = ( , ) ( ) ( , ) ,M NMNds g X dX dX x y g x dx dx x y dy 5bulk 55 55122S d x g RK whereAt fixed xμ , is the distance along the fifth dimension.5d dy (5) 5 0, 0,1,2,3g
2 24 2 4b , , 551 ( )= 3 ( ) 62ulkRS d x g dy gK where R is the 4-d Ricci scalar made out of the 4-d metricWe have omitted the total y-derivative term because it iscancelled by the Gibbons – Hawking term in the actionThe 5-dim bulk action may be put in the formFor consistency with Einsten’s equations we requireIn addition, we impose the „Einstein frame‟ gaugecondition so the coefficient of R in Sbulk isa function of y only 5 50R 2 2( )W y g
We arrive at the 5-dimensional line element 222 2 2(5) 2( )( )( )W yds x W y g x dx dx dyx W y By choosing and neglecting the radion werecover the AdS5 geometry. Hence, the choicecorresponds to the RSII model.J.E. Kim, G. Tupper, and R. Viollier, PLB 593 (2004)( ) kyW y eThis metric is a solution to Einsten’s equationsprovidedprecisely as it is in the RSII model( ) kyW y ewhere the field ϕ represents the radion.2 5=6k
Integration over y fom 0 to ∞ yields, ,4b(5) (5) (5)3 6= (1 2 )2 4 (1 )ulkgR kS d x gkK kK K the last term may be canceled by the two brane actions0 RS(5)6= =lkK With the RSII fine tuning4 2 4 2 2b =0 b = 0| | = (1 ) ( )klrane y rane y l lS S d x g d x g e
Then, the bulk action takes a simple form4b , ,1=16 2ulkRS d x g gG where we have introduced the 4-dimensional gravitationalconstant5=8K kG 13= sinh4 G and the canonically normalized radion
is no longer infinite so the physical size of the 5-thdimension is of the order although its coordinatesize is infiniteThe appearance of a massless mode – the radion – causes2 effects1. Matter on observer’s brane sees the (induced) metric2. The physical distance to the AdS5 horizon at coordinateinfinity 1g g 1Plk l51 1ln2dk
0y y Consider a 3-brane moving in the 5-d bulk spacetimewith metric 222 2 2(5) 2= ( )kykykyeds e g dx dx dye Dynamical brane
The points on the brane are parameterized by. The 5-th coordinate Y is treated as adynamical field. The brane action ,MX x Y x22 2 3ind(5) , , , ,2 2 2( )= =( )kY kYM NMN kY kYe eg g X X g Y Ye e yields1/22 24 2 2b , ,2 3( )= ( ) 1( )kYkYrane kYeS d x g e g Y Ye with the induced metric4 ib = det ndraneS d x g
Changing Y to a new field we obtainthe effective brane action= /kYe kIn the absence of the radion field ϕ we have a pureundistorted AdS5 background andThis action describes a tachyon with inverse quarticpotential. Although p =L < 0, the steep potential drives adark matter attractor, so very quickly and onethus apparently gets cold geometric tachyon matter.R. Abramo and F. Finelli, PLB 575 (2003)2 2 2, ,4b 4 4 2 2 3(1 )= 1(1 )ranegkS d x gk k (0) 4b , ,4 4= 1raneS d x g gk 0p
E.g., for positive power law potential20( ) nV V One can get dark matter and dark energy as a single entityi.e., DE/DM unificationN. B., G. Tupper, R.Viollier, Cosmological tachyon condensation.Phys. Rev. D 80 (2009)Tachyon has been heavily exploited in almost anycosmological context: tachyon as inflaton, tachyon as DM,tachyon as DE
1. The geometric tachyon is seen on our brane as aform of matter and hence it affects the bulkgeometry in which it moves.2. The back-reaction qualitatively changes thegeometric tachyon: the tachyon and radion form acomposite substance with a modified equation ofstate.The dynamical brane causes two back-reaction effects
Instead of the tachyon field defined previously asit is convenient to introduce a new field= /kYe k22)(2)(3=3=)(xkex xkYThen the combined radion and brane Lagrangian becomes, ,2 2, , 2 31= 12gg L26= 2 6 , = , =6k k Field Equationswhere
Where we define canonical conjugate momentum fields,, 2 3, , , ,= = = =1 /gL Lgg For timelike and we may also define the norms= =g g Then the Lagrangian may be expressed as222 2 2 21 1=2 1 / ( ) L,,
The corresponding energy momentum tensormay be expressed as a sum of two ideal fluids1 1 1 1 2 2 2 2 1 2= ( ) ( ) ( )T p u u p u u p p g with22 2= 21 / ( )T g gg LL L1 2= =u u 221 2 2 2 2 21 1= =2 1 / ( )p p 22 2 2 21 2 21= = 1 / ( )2
The Hamiltonian may be identified with the total energydensity1 2= 3 =T H Lwhich yields22 2 2 221= 1 / ( )2 HIt may be easily verified that H is related to L through theLegendre transformation, , , ,( , ) = ( , ) H LHere the dependence on Φ and Θ is suppressed. The restof the field variables are constrained by Hamilton’sequations
Hamilton’s EquationsNow we multiply the first and second equations byand , respectively, and we take a covariant divergenceof the next two equations, ,, ,= == = H HL L2u1u
3. Isotropic Homogeneous EvolutionTo exhibit the main features we solve our equationsassuming spatially flat FRW spacetime with lineellement1 2=aH H Ha and we have in addition the equation for the scale a(t)8=3a GaHin which case2 2 2 2 2 2( )ds dt a dr r d We evolve the radion-tachyon system from t=0 withsome suitably chosen initial conditions
Evolution of the radion-tachyon system for λl2=1/3 with the initialconditions at t=0: 1.01, 0.1, 0 Time is taken in units of l
After the transient period the equation of state w=p/ρbecomes positive and oscillatory
Approximate asymptotic solution to second order in theamplitude A of tΦ23 2 24 22 cos3 2A tA 2 22 2231 2cos2A tAt In the asymptotic regime ( )22 3/22 | | /pw t so 23sin 28tw A cos 2A tt
Since the oscillations in w are rapid on cosmologicaltimescales, it is most useful to time average co-movingquantities. The effective equation of state is thenBy averaging the asymptotic w over long timescales wefind230.0098Aw =p w where we have estimated A by comparing the exact andapproximate solutions for tΦ
CDM is rather successful in explaining the large-scalepower spectrum and CMB spectrum.However there is some inconsistency of many bodysimulations with observations:1. Overproduction of satellite galaxies2. Modelling haloes with a central cusp.4. Warm Dark MatterIt is believed that these problems may be alleviatedto some extent with the so called Warm DMBesides, it has been recently argued thatcosmological data favour a dark matter equationof state wDM ≈ 0.01
We assume that our equation of state w≈0.009 correspondsto that of (nonrelativistic) DM thermal relics of mass mDM atthe time of radiation-matter equality teqDD=MMTwmHence, we take and identifyThis gives mDM ≈ 0.8 keVD e| =< >= 0.009M qw we eq= = 7.4 eV atqT T t tTo demonstrate that our asymptotic equation of state isassociated with WDM, we will show that the horizon mass atthe time when the equivalent DM particles just becomenonrelativistic is typically of the order of a small galaxy mass.
These DM particles have become non relativistic at thetemperature T≡TNR =mDM , corresponding to the scaleeqN e eNR< >R q qTa a w aT.The horizon mass before equality evolves as,3eeH qqaaMM where for a spatially flat universe.Thus, at a=aNR we obtain15e 2 10qM M 9H 10M Mthe mass scale typical of a small galaxy and thereforethe DM may be qualified as warm.
Conclusions&Outlook• We have demonstrated that back-reaction causes thebrane – radion system to behave as “warm” tachyonmatter with a linear barotropic equation of state• The ultimate question regards the clustering propertiesof the model. At the linear level one expects asuppression of small-scale structure formation: initiallygrowing modes undergo damped oscillations once theyenter the co-moving acoustic horizon• Perturbation theory is not the whole story – it would beworth studying the nonlinear effects, e.g., using thePress-Schechter formalism as in the pure tachyon modelof N. B., G. Tupper, R.Viollier, Phys. Rev. D 80 (2009)