D-Branes and The Disformal Dark Sector - Danielle Wills and Tomi Koivisto

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D-Branes and The Disformal Dark Sector - Danielle Wills and Tomi Koivisto

  1. 1.                                                                                                                                                                                      Cape  Town,  11.03.2013                        African  Ins7tute  for  Mathema7cal  Sciences  D-branes and the disformal dark sector Danielle Wills and Tomi Koivisto Institute for Theoretical Astrophysics University of Oslo Centre for Particle Theory Durham University
  2. 2. gµ⌫ = gµ⌫ + ˜ ,µ ,⌫ (5) 1 + 2X ) = 1 C( D( ) = D0 e ( 0) On the C( ) = 1 physical and relationV0 e  V( ) = between gravitational geometry D( ) = D0 e ( ) 0  V ( ) =Z V0 e  p R p S= d4 x g + g L (matter, gµ⌫ ) ¯ ¯ Z  16⇡G p R p•  For   d4 x S =simplicity,  let  us  take  the  rela3on  to  be  given  by  a  scalar  Φ   g + g L (matter, gµ⌫ ) ¯ ¯ (6) 16⇡G•  It  can  be  argued  that  the  most  general  consistent  rela3on  then  has  the  form     gµ⌫ = C( , X)gµ⌫ + D( , X) ,µ ,⌫ ¯ [Bekenstein, Phys.Rev.  D48  (1993)  ]   (7)  •  C  ≠  1  is  very  well  known  and  extensively  studied.  We’ll  focus  on  D  ≠  0.                                        The  outline:                                                                                                                -­‐  What  is  it  good  for?                                                                                                                      1)  Mo7va7ons,  Phenomenology:  the  screening                                                                                          -­‐  Where  does  it  come  from?                                                                                                              2)  The  DBI  string  scenario                                                                                          -­‐  How  to  detect  it?                                                                                                              3)  Cosmology:  background  expansion,  Large-­‐scale  structure                                                                                                    -­‐    So  what?                                                          An  outlook  and  conclusion  
  3. 3. Institute for Theoretical Astrophysics, University of Osloother between scalar degree of freedom and ter, which in turn could help to explain arXiv:1205.3167v2 Frames of gravity becomes dynamically important at the p There are myriad variations of such mod of them the coupling can be effectively dy as field-dependent mass of the dark matter p Yukawa-type couplings can be motivated b gµ⌫ = ˜ gµ⌫ relation to scalar-tensor theories, which inc f (R) class of modified gravity [2]. However, for any other type of gravity the relation between the matter and gravit  Brans-­‐Dicke  theory,  e.g.  f(R):         will be non-conformal. This can also be m type 1 in a DBI ~ ~scenario where matter is all 2 L = φ R + V (φ ) L = R − (∂ϕ ) + U (ϕ ) + Lm (ϕ ) 2 the additional dimensions [3]. When give field φ, the disformal relation can be param •  The  generalisa3on  of  conformal  mapping   gµν = C(φ)gµν + D(φ)φ,µ φ,ν , ¯ •  Is  contained  in  any  modified  gravity*  commas denote partial derivatives. where beyond  f(R)   •  and  in  any  scalar-­‐tensor  theory  the most general physical  case, Bekenstein [ beyond  Brans-­‐Dicke   both functions C and D may also depend *  The  generic  ghost  problem  of  higher  deriva3ve  theories  may  be  avoided  in  nonlocal  gravity  that  may  further  be  simpler case here. but we will focus on the asympto(cally  free!   [Biswas,  TK,  Mazumdar:  PRL  (2012)].     plications of such a relation to cosmology
  4. 4. further relations between scalar- 4  V( ) =  V0 eConsiderand itsfderivatives which gives rise to second or- tensor an (R) theory as C( ) = 1 der equations of motion in four space-time dimensions. Z The addition of a scalar degree of freedom provides = An example of both S a d x  p 4 D( ) = D0 e R p ( 0) g µ⌫ =+ gµ⌫ L (matter, g g˜ g ¯ ¯ e generous extension of the possibilities. The most general e derivatives which gives rise to second theory was first s gravitational sector for a scalar-tensor or- V ( ) 16⇡G 0 e  = V y of motionmodified  g[14] and has ith                                                    in  the  1st  order  formalism  [1]:   •  A   by four space-time dimensions. considerable derived in Horndeski ravity  w received , on of a scalarrecently [15–21]. It is provides the Horndeski gµ⌫g C( =X)gµ⌫ ++ Dv v ,µ ,⌫ attention degree of freedom given by a tnsion of the possibilities. The most general Z ¯ =µ⌫ , Cgµ⌫ D( , X) ⌫ ˜ µ Lagrangian s sector for a scalar-tensor theory was first S = 4 p R p d x g✓ + ◆ g L (matt ¯ l   [14] and has received5considerable orndeski X d 16⇡G X l LH = Li . (1) ently            è It iss  the  Levi-­‐Civita  connec3on  of     gµ⌫ dX f,R gµ⌫ + f> 0 µ⌫   [15–21]. Γ  i given by the Horndeski i=2 ˜ = C + XD ,Q R -  Up to total derivative terms that do not contribute to the µ⌫ = C( , X)gµ⌫ + D( , X) ,µ       g ¯ - equations ofX 5 motion, the di↵erent pieces can be written gµ⌫ = gµ⌫ + ,µ ,⌫ ˜ - •  [19]  H    =            L  i  .Horndeski    scalar-­‐tensor  theory,  “covariant  galileon”: as             L     (1)h - L 2 = G2   i=2(X, )  ,      (2)     ✓ ◆perivative terms that do not contribute to the G2 = G3 = G5 =X, G4 = > 0 2X d 0 1+wymotion, L              the    =          G  3  (X,pieces  ,  can    be    written                          (3)          3   di↵erent          )2                                                       dX C + XD ⇥ matter slow down: ⇤n -    L  4    =    G4  (X,  made    G  4,X      (2        2            ;µ⌫      ;µ⌫        ,      (4)                            is  the  E-­‐H  theory  for  [2]        Clocks      )R    + of    dark )                                        G2 (X,      )  ,                                                    ;µ⌫                                                          g  00                1 + D ˙ 2 ! 0 = 1 + D0 e (a      L5 = G5 (X, )Gµ⌫       1 G5,X h(2 )3 (2) ¯     =     - 6 gµ⌫ = gµ⌫ + ,µ ,⌫ ˜n G3 (X, )2 , (3) ;µ i ;µ⌫ ;⌫ ; - 3(2 ) ;µ⌫ ⇥ + 2 ;µ ⇤ ;⌫ ; . (5)G4 (X, )R + G4,X (2 ) 2 ;µ⌫ , (4) V ⇠e m    1.  [TK:  PRD  (2007)]                        2.    [Zumalacarregui,  TK,  Mota:  PRD  (2013,  to  appear)]   ;µ⌫ - Here R, Gµ⌫ are the Ricci scalar and the Einstein tensor, = G = G = 0 , G = (1 + 2X h G
  5. 5. ated e.g. The coupling will then generically involve second deriva- 0 ✓ ◆ e of dark matter slow down: the distortion of causal structure. to enter tives, which ) = 1 C( entail d X V ( ) = V0 e  >0 (8) a scalar g00 )= = 1 D D ˙ 2 ( 0 0 ) 1 + account ¯ + e ! taking + XD ) ˙2 For a point particle, and dX = Cinto D0 e ( the 0cor- (11) D( Interacting matter ed as 0 rect weight of the delta function, we have Z  √ ( ¯) = V0 e V  p R p (1) V g⇠µ=νg + ,µ ,⌫ S = ˜µ⌫ e µ⌫ (4) −¯Lm = −Σm −¯µν x x δ (x − x(λ)) . (4) g g ˙ ˙ d4 x g + ¯ (12) (9) g L (matter, gµ ¯ 16⇡G pnsidering ZFrom the point of view of the = G5 = 0frame, the proper  G2 = G3 physical , G4 = 1 + 2X p p D⇠e  (13) (10) ued that time the particleR 4 experiences is dilated by the conformalks made ,of dark matter slow down:+ the g L (matter, gµ⌫D gives ga = C( , X)gµ⌫ + D( , X) n (∂φ)2 = factor C. In 16⇡G S d x g addition, ¯disformal factor )¯ ¯µ⌫ (6) ,µ ,⌫vious ap- ⌦ ⇠ 2 direction-dependent effect proportional to theD0 e ( g00 = 1 + D ˙ 2 ! 0 = 1 + projection 0 ) ˙ 2 ¯ (14) (11) de vary- The  pthe four-velocity along the gradient ofrom  GR:   •  of hysical  proper  distances  differ  f the field: ✓ ◆k energy ↵2⇠ erm 2 V e   tensions ˙ 2 ¯ x ef ν x ≡ gµνGµ xf = C1 2 + D(x · ∂φ) G ¯ ˙ ˙ x+ ˙ ˙ . (5) (15) (12) 2n of cou- The  equivalence  principle  is  vthe particle along its path •  Extremising the proper time of iolated:   D⇠e  (13) -up that   in shows that xµ follows the x = µ geodesics: forceson (1) ¨ = ¯ µ x↵ disformal x↵ x + 5th it ↵ ˙ ˙ ↵ ˙ ˙ 2 (16) ¯ ˙B xβ ⌦ 0⇠ •  The  conformal  prototype,   xµ + Γµ xαrans-­‐Dicke  theory,  C(Φ)=exp(-­‐ακ(Φ-­‐Φ0)),  D=0:   ¨ ˙ = , (6) (14) αβ ✓ ◆ Effec3ve      ravita3onal  coupling                 g                                       ↵2 rm Newton’s  force  +  extra  5th  force   between  mager  par3cles   Gef f = 1+ e G mediated  by  scalar  par3cles  (15)    ress: tomi.koivisto@fys.uio.no 2                                                              How  to  reconcile  with  observa7ons?                                  1.  Make  the  field  very  massive              :          no  DE                                    2.  Make  α  very  small                                                :          uninteres7ng  nic address: tomi.koivisto@fys.uio.no                                3.  Make  them  species-­‐dependent  :          coupled  DE                                  4.  Make  them  density-­‐dependent  :          chameleon    
  6. 6. D( ) = 4 D0 e ( 0) V ( ) =  4 V0 e  Chameleonic screening Z  pR p S= d4 x g+ g L (matter, ¯ 16⇡G [Khoury and Weltman, PRL  (2004)  ]   gµ⌫ = C( , X)gµ⌫ + D( , X) ,µ ,⌫ ¯ •  Spherically symmetric NR point matter source Compton wavelength ✓ ◆ •  The scalar potential decays beyond the Compton wavelenght 1/m> 0 d XEffective potential •  High effective Suppose highscalar has a mass  mass in the density regions dX C + XD Consider non-relativistic matter 2 r gµ⌫ = g + ,µ ,⌫ (r) Comptonµ⌫wavelength ˜    m   4 G 2 p G = G3 = G5 = 0 , G4 = 1a mas 2 Suppose the scalar has + 2X Sphericallymade of dark solutions down: Clocks symmetric matter slow 2  m2  4 G g00 = ¯ 1 + D ˙2 ! 0 = 1 + D0 e ( d 2 2 d  Spherically symmetric solutions  2   m2    4 G  dr r dr   V ⇠e d 2 2 d   2   m2    4 G GM  dr Dr⇠ e   m 1 dr r  (r)   exp  mr  r GM  (r) ⌦  ⇠  exp  mr  2 r The scalar potential decays exponentially above✓the Compton ◆ The scalar potential rm ↵2 decays expo wavelength m 1 Gef f = 1 + wavelength 1 2 e G m
  7. 7. µν µν ensures that Einstein field equations have ◆ usual form ✓ the ch models, but in all µν µν d X s thatdescribed by a equations have the usual formcovariant conservation of tively µν Einstein field G = 8πGT . However, the >0 Disformal screening 8πGT . However, energy momentum does not hold formatter particle. Those the covariant conservation of dX C +the coupled compo- nents separately. Instead, we obtain that momentum does not hold for the coupled compo- XD vated by a conformal [TK, Mota & Zumalacarregui PRL  (2012)  ]   separately. Instead, we obtain that µ Tm ≡ −Qφµ⌫ ,= gµ⌫ + ,µ ,⌫ hich includes also the µν g ,ν ˜ (3) C D D Tm ≡ −Qφ ,ν , Q = µν gravity modification, µ where Tm − λ (3) ,µ Tm + φ µλ φ,µ φ,ν Tm µν = G3 =the effects , disformal couplings 1 2C G2Addressing C 5 = 0of G4 = (1 + 2X)therefore G 2C d gravitational metric C D D requires studying the field dynamics in high density, non- w µλ µν fi Q Clocks Tm −of dark matter m down: relativistic Tm involve second deriva- lso be= motivated e.g. λ The coupling will then µ φ,ν environments. This regime can be explored φ,µ T slow + φ, generically er is Spherically symmetric, static NRof causal structure.a station- •  allowed made 2C to enter C 2Cusing the general configuration: e tives, which entail the distortion scalar field equation (7) for t For involve tog00neglecting D ˙ =into + D0 ecurvature [31]. ˙hen given by thenterm proportionalparticle, densitythe remaining ρ(x)account the→cor-The aryoupling • will a scalar 1 distribution 1 in the limit ρ 0∞, 2 ¯D = deriva- 2 Each generically a point second and+takingidentically! vanishes spacetime ( ) and r inbe parametrized as distortion weight of the delta result follows from taking the limit ρ C/D, φ2 in which entail the rect of causal structure. same function, we have ˙ v •  High density Dρ>>1 limit: √ (8): t )φ point and g −¯L account ¨ −¯ 2˙ V˙ δ φ2  a ,µ φ,ν , particle,(1) taking into¯m = −Σmthe gµν˙xµ xν⇠ (4) (x 1− x(λ))β. ˙2 (4) cor- D e ˙ t eight of the delta function, we equation: •  The Klein-Gordon have φ≈− φ +C − =− φ , (15) 2D C 2D 2Mp s vatives. Considering From the point of view of the physicalgeneral andthe second applies frame, proper b nstein¯[4] arguedfield−¯µν xµ xν δ (4) (x − x(λ)) .example model. The by , expression •  The that indeed slows downour (4) dilated (e.g. if β>0) √ where the first equality is  D ⇠ e abovethe the departs sub- −¯Lm = −Σm g g time the particle experiences is ˙ ˙ to conformal m c •  The evolution is independent ρof ∞from theill-defined. Spatialcoupling, for which depend upon (∂φ)2 , the → the density factor C. In addition, the disformal conformal D gives a stantially limit is simple factor derivatives become tthehere. Previous ap- the physical frame, the proper are suppressed by projection se point of view of direction-dependent irrelevant, as they effect proportional to the a p/ρ factor w.r.t. phe •  The experiencesof the four-velocity along independent of importantly, the density, making particle 5 th force just isn’t there the gradient of the field: equation t mology include vary- is dilated by the time derivatives. More the local energy above conformal becomes foation [6], dark energy C. In addition,muchdisformalx2 ≡ of thefield evolution insensitive∂φ)2 D s •  Pretty the regardless gµν xthexνdetails+of Vhomogeneously, spatial gradi-[9, 10] and extensions ¯ factor ˙D ˙ gives xafield D(x ·and . ˙ ¯ bodies.= C ˙ 2 rolls˙ µ As the to the presence of massive (5) t on-dependent effect proportional to the projection ⇤ Electronic ents between separate objects, which would give rise to a generalization ofaddress: tomi.koivisto@fys.uio.no scalar force,thenot form. along its path cou- fo four-velocity along theExtremisingof the field: purely disformal case with exponential D, equa-s a simple set-up that gradient the proper time of do particle the In the in shows that it follows the disformal geodesics: tion (15) can be integrated directly
  8. 8. Potential signatures?Our assumptions are violated if we have:•  Matter velocity flows - Suppressed by v/c. Binary pulsars?•  Pressure - Potential instability if p>C/D-X. Astrophysics?•  Strong gravitational fields - Gravity coupling not suppressed by Dρ. Black holes?•  Spatial field gradients - Potential remnants of LSS formation. Even Solar system?Systematic study requires developing the PPN formalism [Work under progress with Kari Enqvist and Hannu Nyrhinen]    
  9. 9. Disformal couplings from DBI: Flux compactifications in Type IIB string theory  •  In  flux  compac3fica3ons  of  Type  IIB  string  theory,    warping  can  arise  from  the   backreac3on  of  fluxes/objects  onto  the  compact  space  →  warped  throats  •  Single  Dp-­‐branes  can  move  as  probes  in  this  geometry,    with  a  DBI  ac3on             Warped throat CY3   D3, h(r) wrapped D5…     D7  •  The  disformal  coupling  arises  generically  from  this  set-­‐up,  as  we  will  now  see….      
  10. 10. 0  V ( ) = V0 e Disformal couplings from DBI:  Z p R p FluxS compactificationsLin Type )IIB string theory = d4 x g 16⇡G + g (matter, gµ⌫ ¯ ¯•  Recall:   gµ⌫ = C( , X)gµ⌫ + D( , X) ,µ ,⌫ ¯   –  The  disformal  metric:  this  arises  from  the  pull-­‐back  of  the  10  dimensional  metric   on  the  worldvolume  of  a  moving  D-­‐brane       –  The  scalar  field:  the  changing  posi3on  coordinate  of  the  brane  is  a  scalar  field   from  the  four-­‐dimensional  point  of  view  (we  consider  mo3on  in  one  transverse   direc3on  only)   –  The  func7ons  C  and  D:  both  are  given  by  the  warp  factor,  1/C=D=√h   –  The  disformally  coupled  maVer:  whatever  stuff  resides  on  the  moving  brane      
  11. 11. Disformal couplings from DBI: Example – D3-brane in AdS5•  Consider  a  probe  D3-­‐brane  moving  in  the  radial  direc3on  of  an  AdS5-­‐type  geometry   induced  by  a  stack  of  D3-­‐branes     –  The  disformal  metric:     –  The  scalar  field  ac7on:         +  poten3al     +  charge       D-brane probe in AdS5 geometry from stack of D3- branes
  12. 12. Disformal couplings from DBI: Coupling to Matter •  Now  lets  couple  the  scalar  to  mager:   –  Open  string  endpoints  →  U(1)  vector  fields  on  the  world-­‐volume   –  These  can  acquire  masses  via  Stückelberg  couplings  to  bulk  2-­‐forms             →      The  massive  vectors  (or  their  decay  products)  are          -­‐  dark  to  our  standard  model                -­‐  disformally  coupled  to  our  metric  g   D-brane probe in AdS5 geometry from stack of D3-   branes •  Finally  lets  summarise  the  geometric  picture:     –  Transverse  open  string  oscilla7ons  →  scalar  field    →  dark  energy  in  cosmology?         –  Parallel  open  string  oscilla7ons              →  vector  field  →  dark  maVer  in  cosmology?    [Work under progress with Ivonne Zavala]
  13. 13. ν ν C−2DX der, partial differential equation. Its hyperbolic char- presence ofG2 = G3 energy 5 ede 0 , 3(1 G4 = 21[1 early dark = G Ω= = + w)/γ density keepsthe signaturemodifiedthe time derivative term the correct sign of gravity [2]. The new features nus unity asymptotically. Th˙ acter depends on class of of the tensor M , which f (R) µν appear when the disformal factor Dφ arXiv:12 involves the However, a large pressure cantensor. if D > 0. coupled matter energy-momentum flip the sign of the order avoid a singularitythat the eff grows towards to one. Then the clocks in tick fo However, for any introducingit, ofinstability.modification,, slow wherea de Sitt For a perfect fluid, in coordinates comoving spatial µ derivatives coefficient, type gravity ¯ universe ˙ enters1intoand make= other with an dark matter, g00 = −1 + Dφ2 down Q th Toy model: ΦDDM D Mµν = δνthe relation between the matter and gravitational state for dark matter approach m The present analysis focuses onderivative term effective equationsistance + 1 + 2X ,µ ,⌫also non-relativistic environ- gµ⌫ of metric pathology was ˜ = gµ⌫ to − C−2DX diag(−ρ, p, p, p). Positive energy density keeps the correct sign of the time Dp nus unity asymptotically. The field also begins to freez ments, However, a large pressure can flipwill sign of can also be a motivated the effectiveThe coupling if D > 0. will be non-conformal.theThisfurther to avoid singularity in e.g. scenariogµν , and th and hence C−2DX 1 be the assumed. self-coupling metric ¯ [7, 8]. spatial derivativesDBIaddress the effects of pressure, includ- Future work will type scenario where matter is allowed to de Sitter stage. Thiswhich re in a coefficient,on non-relativistic environ- universe enters into Thus, the disformal coupl introducing an instability. a enter tives, natural e The present analysis focuses under which the stability sistance to pathology was also observed in the disforma ing the circumstances condition C( ) = 1 The dark ingredients: ments, andthe additional will be further assumed.When given scenario [7, 8]. Dp hence C−2DX 1 dimensions [3]. 11 self-coupling bythat triggers the For a point can breakwill address the effects of[12]. down dynamically pressure, includ- a scalar 4 transition to Future work field φ,cosmological model. condition consider an disformal as  D0 steeper mechanism the disformal relation can beThus, the D( ) relatively rect weight of parametrized = e ( The coupling provides a the slop0) •  A canonical quintessence field Φ ing the circumstances under which the stability Let us An example can break down where the [12]. acts as quintessence and the the transition to an V0 e the √- VI. COSMOLOGY application dynamically field that triggers V ( ) the slopehigher  ratio β/ is, i.e. the 4 accelerated expansion =  then •  DDM living in disformal coupling is used gµν to trigger µν + D(φ)φ,µrelatively steeper (1)asofseendisformal−¯Lm An example cosmological model. =Let us consider an ¯ The φ , C(φ)g cosmic acceleration. happens, ,ν g¯ functio in FIG.1. Th ΦDDM cosmology application where the viability quintessence and the is, i.e. the higher a short “bump” in the equatio Having addressed the field acts asof the theory inthe the ratio β/γ, the faster the transitio The Friedmann used to trigger cosmic usual form happens, as seen in FIG.1. This transition also produce let us equations have the Z g •  2 extra parameters wrt ΛCDM, everything at Planck scale Solardisformal coupling is commas denote acceleration. System, where consider its cosmological implica- tions. Using the Einstein Framethe usual form the Fried- S = The Friedmann equations have description, From the poin partial derivatives.4 Considering of state, which may hav a short “bump” in the equation observational cons interesting p R pe mann equations have the general˙ 2 the most usual form 8πG physical case, Bekenstein d observational16⇡G + time L (matte ˙ φ2 interesting x full g consequences. Wethe analy background g performed [4] argued that with MCMC part ¯ The Friedmann equations: 2 Canonical field + DDM: , H +K = 8πG φ (ρ + + V ), full background MCMC analysis a modified versio- ● H 2 + K functions 3 + V ) , D may alsoofdepend upon CMBEasy [14] using the U both 8⇡G = (ρ + C and 2 of (∂φ) ,2 CMBEasy [14] using the Union2factor C. comp Supernovae In k 3 ˙2 2 Φenergy [18]. Th lation [15], WiggleZdata [16], cos H + 2˙ we will(⇢ + 4πG V the simpler case here. WiggleZ baryon acousticdirection-depe 2 baryon acd but = H 3 4πG − 2 + (ρ + , ˙ 2 (57) lation [15], Previous ap- ˙ aH + 2 focus on ), scale H + H 2 = − = (ρ + 2φ2 − 2V ) 2φ − 2V ) , mic microwave background angular scale [17] and bound ˙s plications of 3 3 gµ⌫ = C( ,microwaveof theX) ,µ ¯dark energy vary- + background an mic X)g µ⌫ 4⇡G such a relation to cosmology include[18]. The obtained constraints ar on early D( , four-vef ˙ H +H 2 = (⇢ + 2 ˙2 2V ) , (58) on early dark but the conservation equations for matter and the scalar but the ing speed 3 from (3), theories [5], and the scalar darkWe see in FIG.2. We see th conservation equations for matter inflation in FIG.2. shown that for steep slopes γ an field have to be computedof light (7): shown [6], energye The (non)conservation equations: field have to equations for from and the β, the background evolution becomes ◆ but the conservationbe computedmatter (3), (7):scalar ilar to ΛCDM. At this ✓ background higher bound [7, 8], ˙ gravitational alternatives to [9, 10] and extensions X β, the there are no evolution ˙2 increasingly sim x ¯ ¨ ˙ field have + 3Hρ computed from3H φ + V = −Q0 , ρ to be = Q0 φ , φ + (43, 45): d level ˙ (8) ΛCDM. At this0 >- of [11] dark, matter. 3H φ + V = −Q0 , χ2 (8) 538.79ofto C χ2 XD 538.91 level t ilar t on these parameters, and the model is completely viabl- ρ + 3Hρ = Q0 φ ˙ ˙ φ¨ + The disformal with disf = ˙ generalization versus + Extremising fi dX cou- ΛCDM = (best on these parameters, and theg ●- (Non)conservation equations: ⇢ + 3H⇢ = Q0 ˙ , ˙ were the background order coupling factor reads ¨ + 3H ˙ ˙+ V 0 = C Q0 , (59) WMAP7 parameters). However, the model is essentiall pled quintessence here introduced is a simple set-up that = 538.79 versus χ background order φ2 ) + D φ features different relation (60) withasχ2quite obvious when one look were the C useful to+studycoupling2 factor reads the from ΛCDM, (1) disfparameters). Howevis shows that it is − 2D(3H φ V + C ˙ ˙ generic ρ , (9) at the effective dark matterin of WMAP7equation of state in FIG.1., Q0 = gµ⌫ = g + ˜- different 2D(3H φ + V + matter + D φ2 Cosmological Perturbations.µ⌫A more,µ ,⌫ asdescrip scenarios. were ⇢ is the energy density of the φ2 ) ˙ ˙ 2 C + D(ρ − coupled C ˙ 2 com- ˙ different from ΛCDM, is qu realistics C − ponent and the background coupling factor reads ) C φ at the cosmological perturbations. e effective dark matter I Q0 = after solving away the higher derivatives. In the2 ρ , (9) tion requires considering- 0 ˙ 2 C +2D(ρ −˙ 2 ) 0 0 A ˙ 0 φ ˙ following the Newtonian gauge, the linearized field equation is
  14. 14. gµ⌫ = C( , X)gµ⌫ D( D( = X) D0 + ⌫ ¯ + ) dX  Cµ e, XD , , (7) 4  V( ) =  V0 e ΦDDM: the background story ✓ ◆ d Z X  gµ⌫ = gµ⌫ + ,µ ,⌫ ˜ p >0 R p (8) S = C 4 x XD g dX d+ + g L (matter, gµ⌫ ) ¯ ¯ (6) 16⇡G 1 G2 = G3 = G5 = 0 , G4 = (1 + 2X) Converging to the µ⌫ = gµ⌫ + ,µ  ,⌫ Radiation era, Matter era   Disformal “freezing”   gscaling attractor* ˜ (9) Sitter era   De gµ⌫ = C( , X)gµ⌫ + D( , X) ,µ ,⌫ ¯ (7) matter Practically slow down: The quintessence arbitrary initial0 , Gd = (1˙+ 2X)◆1 G2 = G3 = G5 = ✓ X= 1 + D e ( Acceleration (10) g00 = 1 + D ¯ 4 2 scaling solution0 ) ˙ 2 >0 0 (8) conditions dX C + XD down: 13 g00 = ¯ 1+D = V ⇠ e) ˙ 2 with exponential V   The coupling then triggers acceleration   The early evolution of 2 usual “self-tuning” scalar ˙ the ( 1 + µ⌫ 0= gµ⌫ + gD e ˜ 0 ,µ ,⌫ (11) (9) he * a saddle point if β>γ. Then as field gravitational metric also avoids problems with 10 108 n gravity tests and the subtleties related to the  1ce of di↵erent frames, hence e 3 , the the anal- rolls down the G2 ⇠ G = G5 = 0 , G4 = (1 + 2X)  V = simplifying D ⇠ e (12) Coupled Matter (10)cosmologicalto matter:   begins observations.rk matter slow down: 10 107 udy the dynamics within a particular example, Scalar Field D⇠e  s on a simple Disformally + D ˙ 2 ! 0 =Matter D e ( (13) g00 = 1 Coupled Dark 0) ˙2 4 ¯ 1+ 0 (11) Ρ MpcM) model, constructed with the following prescrip- 10 106 Γ 10 all •  The field slows rolling V ⇠ e  exponentially Β 5Γ (12) ark Matter disformally coupled to a canonical 10 105 Β 15 Γ alar• field, following Eq. (59-61). Eventually dark matter freezes tooi.koivisto@fys.uio.no Β 40 Γ n exponential parametrization is never flipped! •  However, the sign for the ⇠ e  D disformal 10 104 (13) 0.1 0.2 0.5 1.0 2.0 5.0 10.0ys.uio.no the scalar field potential: lation and a ( 0 )/Mp B = B0 e , (73) 0.5 /Mp
  15. 15. µ perfectD , X)g in D( p, p, p). δa g− = C( fluid, µ⌫ +coordinates ,⌫ Positive energy G2 = G3 = G5equation = (1 φ2˙2X) 1down and m ν ¯µ⌫ C−2DX diag(−ρ, , X) ,µ comoving with it, ˙ dark matter, g(7) =G4 1of D + ,2slow darkD e effective ¯ 000 = −1 + state = 1 + matt g , =µ µ D ¯ 00 + D for 0 eeps δν −correct sign of thep, p). derivative term effective equation of state for dark matter appr ν = the C−2DX diag(−ρ, p, time Positive energy nus unity asymptotically. The field also ✓Clocks made ◆ dX a + XD X ΦDDM: constraints However,da correct sign of of dark matter slowterm >0 the time the sign down: sity keeps the large pressure can flipderivative of the nus unity asymptotically. Thethe effective me > 0. However, C large pressure can flip the sign of the to avoid a singularity in field also begins erivatives coefficient, introducing an instability. to universe 2enters + in the effectiveemetric gµν , tial derivativesfocuses on introducing an instability. g00 = 1 + ¯ (8) a ˙ = 1 into ( V ⇠) ˙ 2  ¯ avoid D singularity D e a de Sitter stage. 0 0 ent analysis coefficient, non-relativistic environ- universe enters pathologySitter also observed sistance toMota & a de was stage. (2012)  ]na [TK, into Zumalacarregui PRL   This   Dp focuses on non-relativistic environ-e present analysisgµ⌫ + ,µ will be further assumed. sistance to pathology was also observed in the 3d nd hence gC−2DX ˜µ⌫ = Dp 1 ,⌫ self-coupling scenario [7, 8]. (9)nts, and hence C−2DX effects of pressure, includ- self-coupling scenario [7, 8]. D ⇠ e work will address the 1 will be further assumed. V ⇠e  We used the the effects data:   ircumstancesaddress following of 1 ure work will under which the stability condition 0.5 pressure, includ- Thus, the disformal coupling provid G2 = G3 = G5 = 0 , G4 = (1 + 2X) Thus, the (10) the circumstances under which the stability condition k down dynamically [12]. that triggers the transition provides a me disformal coupling to an accele D⇠e wn: •  down dynamically [12]. distance – redshift diagram The relatively steeper the slope of the ex break Supernovae Ia luminosity Let us consider an that triggers the transition to an accelerated di ample Baryon acousticmodel. •  cosmological oscillation Let The relatively steeper the slope ofβ/γ,disformalAn example ˙cosmological model. (scaleus2 consider an is, i.e. the higher the ratio the the fast 0.0 happens, ΦDDM cosmology 0) ˙ 0 = •  1 + D the microwave background angular scale on where !field actsD0 e 0 = 1 + as 2 Scalar quintessence and the Field licationCosmic the field acts as Matter (11) is, i.e. the higher the ratio β/γ, the faster the tr w where Coupled quintessence and the l coupling is used 0.5 trigger cosmic acceleration. happens, as seenseen in FIG.1. transition also p to •  BBN constraints onΓearly dark energy as in Φ: FIG.1. This This transiti dmann priorsV ⇠Hubble Βthe usual form 10 all a● Evolution of ormal coupling on used to trigger cosmic acceleration. (BBN)short “bump” in the equation of state, is •  +equationse have rate (HST) and baryon fraction a short “bump” in the equation of state, which m ⇤ 40 Γe Friedmann equations haveElectronic address: tomi.koivisto@fys.uio.no of self-tuningTRACKING" the usual form (12) The "EXACT Practically ARBITRARY ”Disformal freez interesting observational consequences. ● Β 15 Γ scalar field interesting observational consequences. We perf V B Β 5Γ initial conditions DE SITTER expa 8πG φ ˙ 2˙ 2 0.01 full background MCMC analysis with a D ⇠ e10  + φ +tomi.koivisto@fys.uio.no full background MCMC analysis with a modified 1.0 2 H And+ = = 8πG (ρ + bounds on γ and β/γ:   of CMBEasy [14] using the Union2 Su (13) 4 0.001 0.1 1 10 100 ⇤ Electronic address: + K obtained lower +V )),, a H 2 K 3 (ρ V 3 22 of CMBEasy [14] using the Union2 Supernovae 4πG lation [15], WiggleZ baryon acoustic sca ˙•  + Since 2 ⌦−⇠(blue) (ρ different2choices of field coupling slope β. Highmic [15], WiggleZbackground angular data [ 2 4πG + 2φ ˙ 2−scaling(red) and coupled mat- FIG. 2: Marginalizedbaryon two-sigma regions obtained FIG. 1: Equation of state for the H H + H = − for(ρ + 2φ − 2V the H = ter ˙ 2 during ˙the 2V ) , era lation (14) acoustic scale microwave one and angular scale [17]scale scale ), 3 β/γ (solid, dashed)preferredfit to observations, microwave background (Green), CMB angular and mic from Supernovae (Blue), BAO high values for γ are not produce enough acceleration.on + early dark energy bounds (Orange), The obtained values of 3 give a good while low values (dotted) do on early dark energyincluded [18]. obtained the HST early dark energy The and combined con- [18]. a prior on H0 constr the • conservation freezing for matter and the scalar shown in FIG.2. We Bang Nucleosynthesis from slopeconservation the equations is matter and the scalar Since equations for then swifter, straints. All contours shown inb HFIG.2. see that for steep stee [21] and Ω 0 from Big We see that for 2 [22].d have tocomputed from (3),β/γ are preferred equations β, β, the background evolution becomes i high values for continuity and Euler e to be be computed from (3),(7): while the perturbed (7): for the background evolution becomes increasin o.no •  The expansiondark mattera(t) then resembles ΛCDMto to ΛCDM.this level there are no higher coupled history are ilar ilar ΛCDM. At At this level there are n ˙Q0 φ , φ + + 3H φ Q0 V = −Q0 ,0 ˙ (8)˙ ˙ ¨ φ 3H θ˙ + V˙ = −Q 0 , ¨ ˙ φ ˙ + CDM ˙are3Hρ 3Hρ =φ ,though both Λ and φδ = 3Ψ + Qvery δQ φ , (11)on these parameters,the model model is c ρ + = Q0 ˙ (8) on δ+ + δ φ + different! these parameters, andthe late time dependence dur- For our example model, and the is complete a ρ ρ ρ ing dark energy domination produces a large enhance- Q0 ˙ Q0 with2 2 the matter versus efΛCDM 2 538.91 with mentχof = 538.79growth, δGχ2 /G ∼χ(γV /ρ)2 = χdisf disf = 538.79 versus ΛCDM 1, f = e the background order+couplingfactor kreads ρ δφ . background order coupling ρ factor reads ˙ θ θ H+ φ = 2 Φ+ (12) WMAP7 parameters).to avoid However, early is es WMAP7 parameters). the effects model dark as γ 10 is required However, the of the mo energy. Such behavior is in tension with large scale struc- The general coupling perturbation2 δQ is a much moredifferent from ΛCDM, quite obvious when o different observations, and also is as in conformally coupled from ΛCDM, as occurs is quite obviou ˙ ˙ C C ˙ 22˙ ˙2 ture

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