D-Branes and The Disformal Dark Sector - Danielle Wills and Tomi Koivisto
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
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
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 . 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 . 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
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 and has ith in the 1st order formalism : • 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  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 ,⌫ ˜ - •  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  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
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: firstname.lastname@example.org 2 How to reconcile with observa7ons? 1. Make the ﬁeld very massive : no DE 2. Make α very small : uninteres7ng nic address: email@example.com 3. Make them species-‐dependent : coupled DE 4. Make them density-‐dependent : chameleon
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
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 . ˙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¯ 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 , 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: firstname.lastname@example.org 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
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]
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….
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
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
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]
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 . 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 . 11 self-coupling bythat triggers the For a point can breakwill address the eﬀects of. 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 . 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  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  using the U both 8⇡G = (ρ + C and 2 of (∂φ) ,2 CMBEasy  using the Union2factor C. comp Supernovae In k 3 ˙2 2 Φenergy . Th lation , WiggleZdata , 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 , Previous ap- ˙ aH + 2 focus on ), scale H + H 2 = − = (ρ + 2φ2 − 2V ) 2φ − 2V ) , mic microwave background angular scale  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. 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 , 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 , 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  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
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 email@example.com 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
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 . that triggers the transition provides a me disformal coupling to an accele D⇠e wn: • down dynamically . 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: firstname.lastname@example.org 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 +email@example.com full background MCMC analysis with a modiﬁed 1.0 2 H And+ = = 8πG (ρ + bounds on γ and β/γ: of CMBEasy  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  using the Union2 Supernovae 4πG lation , WiggleZ baryon acoustic sca ˙• + Since 2 ⌦−⇠(blue) (ρ diﬀerent2choices of ﬁeld coupling slope β. Highmic , 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 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 . obtained the HST early dark energy The and combined con- . 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  and Ω 0 from Big We see that for 2 .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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