Emergent Dark Matter in Late Time Universe on Holographic Screen

Home , DGP model

JHEP10(2018)009 Springer April 9, 2018 : October 2, 2018 : September 3, 2018 September 26, 2018 : : Received Published Revised Accepted [email protected] d,e,f , Published for SISSA by https://doi.org/10.1007/JHEP10(2018)009 and Yun-Long Zhang c [email protected] , . 3 Sichun Sun 1712.09326 The Authors. a,b Gauge-gravity correspondence, Holography and condensed matter physics c

We discuss a scenario that the dark matter in late time universe emerges as , [email protected] E-mail: Asia Pacific Center for67 Theoretical Physics, Cheongam-ro, Hyoja-dong, APCTP Nam-gu, Headquarters, PohangCenter 790-784, for Korea Quantum Spacetime35 (CQUeST), Baekbeom-ro, Sogang Sinsu-dong, University, Mapo-gu, SeoulYukawa 121-742, Institute Korea for Theoretical Physics,Kitashirakawa Kyoto Oiwakecho, University, Sakyo-ku, Kyoto 606-8502, Japan Chinese Academy of Sciences, ZhongSchool Guan of Cun Physical Sciences, East University19(A) Street 55, of Yuquan Beijing Road, Chinese Shijingshan 100190, Academy District, of China Center Beijing Sciences, for 100049, Theoretical Sciences, China LeungDepartment Center of for Physics, Cosmology National andNo.1, Taiwan Particle University, Sec. Astrophysics, 4, Roosevelt Road, Taipei 10617, Taiwan, China CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, b c e d a f ArXiv ePrint: Open Access Article funded by SCOAP elasticity, the Tully-Fisher relationcan and be the derived. dark matterKeywords: distribution in the galaxy(AdS/CMT) scale present an alternative interpretation astoy the constraint holographic in universe. the It lateequation. can time be We universe, reduced also with to commentwhere the our on the new the dark parameterization possible matter of connectionSitter is the with treated Friedmann spacetime Verlinde’s as background. emergent the gravity, elastic We response show of that the from baryonic the matter holographic on the de de Sitter model with screen in the flatboth bulk. of the After dark matter addingtensor. and the dark From energy baryonic the can matter be Hamiltonian onrelation described constraint by the between equation the screen, Brown-York in the stress-energy we the darkcomparing assume flat with matter that bulk, the and we Lambda find baryonic coldholographic an dark embedding matter’s interesting matter of energy parameterization. emergent dark density We matter further parameters, with compare by traditional this braneworld scenario and Abstract: part of the holographic stress-energyspacetime. tensor Firstly on we construct the a hypersurface toy in model with higher a dimensional de Sitter flat hypersurface as the holographic holographic screen Rong-Gen Cai, Emergent dark matter in late time universe on JHEP10(2018)009 ] ], 7 5 ]. One 4 ]. More- 3 , 2 ]. The cold dark 1 15 17 8 15 4 9 ], which leads to the apparent 7 12 2 6 symmetry 5 2 8 Z ]. Although the Verlinde’s derivation in [ 9 , 8 – 1 – 19 1 ], which focuses on the small scale crisis that cold dark matter cannot 3 ], etc. – 6 1 Recently, E. Verlinde proposed an emergent modified gravity scenario from volume con- 4.2 Emergent Tully-Fisher relation from holographic elasticity 3.2 Braneworld scenario:3.3 DGP model Holographic with scenario: holographic FRW universe 4.1 Holographic stress tensor and Verlinde’s apparent dark matter 2.1 Hamiltonian constraint2.2 from hypersurface embedding Holographic de-Sitter2.3 screen in a Emergent flat dark bulk matter on holographic screen 3.1 Embedding FRW universe in a flat bulk Renzo’s Rule [ tribution of entanglement entropy in the dedark Sitter matter. spacetime [ It isnomenon also as the related entropy to force with the area idea law that [ Einstein gravity can be an emergent phe- over, the particle darkof matter the remains alternatives to elusiveified the from gravity cold the [ dark direct matterexplain. detection is so Although the those far modified modified [ the Newtonian gravity universe dynamics theories evolution or seem scenario to mod- can consistent be explain with less multiple CMB successful features and in in large producing galaxy scale rotational structure curves data, such they as Tully-Fisher relation [ components of the currentgalactic universe scale is to visible the to cosmicphenomena us. microwave to background test (CMB) At for scale, differentmatter various there scales, model models are ranging many which of from observed treats dark the and the matter large dark and scales, matter dark but as energy at collisionless [ the particles galactic is scale, successful some at discrepancies CMB were proposed [ 1 Introduction The origin of thein dark current high matter energy and physics the and dark cosmology. energy From is observations, only one 5% of of the the most energy important issues 5 Discussion and conclusion 4 Towards holographic de Sitter brane with elasticity 3 Holographic FRW universe and emergent dark matter Contents 1 Introduction 2 Embedding de-Sitter universe in a flat bulk JHEP10(2018)009 ] ]. 21 10 – 15 ]. and extrinsic 14 , we reproduce µν – g 4 12 ], we find several Verlinde’s 10 ]. We also compare our approach 11 – 2 – . 5 ]. Notice here that we adopt the novel idea of elasticity 24 – 22 of the baryonic matter and radiation, which is localized on the µν T , we generalize our toy model to the holographic FRW universe in a 3 , we firstly introduce the toy de-Sitter model in a flat bulk, which leads to 2 , which is embedded into a 4+1 dimensional flat bulk spacetime. After adding µν K In section To produce the galaxy rotational curves, we further sketch a holographic elastic model In this paper, we propose a new viewpoint beyond Verlinde’s emergent gravity, which We consider a 3+1 dimensional time-like hypersurfacecurvature with induced metric the stress-energy tensor hypersurface, we assume that the induced Einstein field equations on the hypersurface are assumptions. We briefly compareother and scenarios, discuss such the as connectionand between other summarize our our braneworld toy results models, model in holographic section and gravity, emergent gravity 2 Embedding de-Sitter universe in a flat bulk the relation between dark matter componentuniverse. and baryonic In matter component section of theflat current bulk and comparethe it Tully-Fisher with relation, the DGP with braneworld the scenario. help In of section holographic elasticity model and Verlinde’s of dark matter in thethat Verlinde’s the paper. apparent dark Because matter theIn is elasticity the only seems end, the to we response capture alsodifferent of the comment scenarios the nature on such presence the as of relation some thein braneworld of baryonic models the the matter. and current literature. holographic construction models in of this the paper universe with with a de Sitter boundaryWe and recover fix the an Tully-Fisher inconsistency relation in fromthe the the “de Verlinde’s first paper Sitter law proposed medium”. of in thermodynamics Theor [ elasticity and some can elasticity holographic also of models be [ realized in blackfold approaches [ parameterization. Furthermore, we generalize our toyRobertson-Walker model (FRW) universe to in the holographic a Friedmann- flatthe bulk holographic and propose model. a new Theidentified parameterization effective with from dark the matter Brown-Yorkto stress-energy and the tensor dark well [ energy studied are Dvali-Gabadadze-Porrati (DGP) emergent braneworld and model are [ can be considered asmensional a flat (3+1) spacetime. dimensional holographic Wetotal identify dark screen the components embedded including holographic into the stress-energy darkmodel, a energy tensor higher which and as provides di- dark that a matter.ergy, of constraint We and first relation the baryonic construct matter, among a in the the toy case densities of of considering the dark Lambda matter, cold dark dark matter en- (ΛCDM) key ideas rather inspiring. Oneand is gravity the emerge possibility from that our andevelopment macroscopic underlying of notions microscopic entanglement of description, entropy spacetime encouraged andthe dark by quantum matter the information. as recent merely Another aso gravitational one as response is of to the derive viewing baryonic the matter on dark the matter spacetime, distribution around the galaxy, the Tully-Fisher relation. received some doubts on the consistency in the literature [ JHEP10(2018)009 . 4 ), /κ . t (2.8) (2.4) (2.5) (2.6) (2.7) (2.1) (2.2) (2.3) 5 0 ), the ( t κ a 2.5 = = t ) as L . Since the in ( R 2.1 R . The Einstein will turn out to 2 + Ω µν /L . D through is the speed of light. 2 5 . hT i . . c κ 0 + Ω 2 0 t 4 B dΩ c = 2 . B ˜ H t Ω r | , cold dark matter density 4 µν . Λ 3 B + . κ + 4 T ˜ + Ω Ω ) ) 4 R D , 2 t = κ Λ ( ˜ µν Ω Ω = 2 µν c a kr g r ρ + B d − ) = + is the critical energy density of the Ω Λ hT i 1 − K 3 c 4 µν ˜ ⇒ Ω ) ρ B κ g t , ) represents the total dark components 2 ( µν c Ω ≡ 0 ) + t a . The Friedmann equation is given by ρ t K only includes the stress-energy tensor of 3 ( = 4 2.3 ( B − K and t ) c B a + µν | t 3 4 t L ( T ) is the Hubble constant today at µν Ω D 3 µν 1 4 κ + 4 in ( 0 a ) T ˜ D t Ω κ t K κ ] induced from higher dimensional space time. 2 ( ( ( t – 3 – Ω ) we have 1 = Ω = + µν is very small, in the late time universe we will a = d 11 = ≡ H 2 1 0 L 3 2 4 ) 2.5 c = 0, it contains a positive cosmological constant Λ, D + D t ≡ − hT i µν µν − ( − Ω Ω k Λ 0 a 10 Rg µν = H hT i , + 1 2 ν ∼ 0 = Ω ,H t Rg Λ x ) ) − 2 = t t 1 2 R d t ) ( ( 2 | 0 µ 1. After neglecting the radiation component Ω t ˙ a a µν ( Λ − x = Ω H ' ˜ R d Ω H ≡ 2 ) = 1, from ( is the Newton gravitational constant and µν ) Λ 0 2 µν 0 ) t = t R t g ( ( G ( H Λ a , H + Ω = 4 H Ω 2 4 B is related to the positive cosmological constant Λ = 3 s d L + Ω πG/c , and baryon density parameter Ω D is related to the higher dimensional coupling constant D = 8 L 4 κ We are going to consider the parameterization in ΛCDM model describing the evolution ) is the Hubble parameter and t ) is the derivative with respect to the time t ( ( ˙ We have introduced the time dependent notations with tilde, which satisify a universe. If requiring radiation density parameter Ω simply take Ω Friedmann equation of the late time spatial flat ΛCDM universe can be written as H In the spatial flat ΛCDMwhich model contributes with to the darkparameter energy Ω with density parameter Ω of the late timeassumes universe. that our In universe detail, is we uniform take and the isotropic FRW at metric large in scale, 3 with + scale 1 factor dimensions, which be the Brown-York stress-energy tensorWe will [ see At the cosmological scale,baryonic matter we and assume radiation. that While in our universe, such as the dark energy and dark matter. They are expected to govern the late time evolution of our universe. The length scale constant Equivalently, we can rewrite the above modified Einstein field equations in ( modified as JHEP10(2018)009 B = N (2.9) A (2.15) (2.10) (2.11) (2.12) (2.13) (2.14) A, B ], − N 7 B dimensional A d g = ˜ ) in the following ], with a bit priori B 2.9 A 26 ) are remarkably well g , are the indices on the , and the indices 25 , A , 2.10 µ, ν ) with minor difference in N 050 , . 003 ] . 0 is the covariant derivative as- . 2.10 0 ) µν ' ˜ ∇ B ) and ( , with ) and the Hamiltonian constraint . . ) B Ω ] ' − hT i can be embedded into one higher di- B B 2.9 Ω 2.1 ) − Ω N is the stress-energy tensor of baryonic + . B µν 4 3 hT i A D g , ( Ω µν µν + ˜ = ∇ (Ω T T 004 − [ . Λ 265 B T . 2 D 0 . d [ ν D 4 Ω 0 g κ 2 ' 1 2 d A /c (Ω ' µ – 4 – = d κ − Λ B g = 2 D d Ω Ω µν πG ≡ 2 D Ω 3 1 2 4 − R g − − µν . Even though there exists matter in the late time = = 8 , 1 2 K d 2 D 2 D is the effective dark components of our universe, which R Verlinde: Ω AB κ − Ω Ω 685 g . µν 0 CSZ: Ω ≡ ≡ µν R ), let us write down the Einstein equation in ' V hT i δ Λ CSZ 2.2 1), and δ Ω ) holds as well as the Verlinde’s ( − d 2.9 ( ,..., 1 , . We can define the induced metric on the hypersurface = 0 µ, ν Now we assume that the geometry with metric Based on the modified Einstein field equations ( , . . . , d 1 , mensional spacetime, as a hypersurface with0 the normal vector as well as thehypersurface, extrinsic which curvature depend onsociated the with coordinate the choices. bulk metric ˜ can include both of thethe dark Ricci energy scalar and dark matter. The trace of above equations yields with matter and radiation, and sections. 2.1 Hamiltonian constraintSimilar from hypersurface to embedding the formulaspacetime as ( Thus, our relationapproximation. ( We will show how to derive this constraint equation ( choice of the parameters as we can calculate the following differences, In the current universeobeyed. both Taking of the observation these values two within relations the ( ΛCDM model [ from the consistent embeddinginteresting in constraint higher relation between dimensional these flat parameters, bulk, we are going to show an Let us compare with the constraint relation in the Verlinde’s emergent gravity [ JHEP10(2018)009 , L =0 ν +1)- d (2.19) (2.20) (2.18) (2.23) (2.21) (2.22) (2.16) (2.17) N ], g ) can be A 11 ) becomes N 2.14 +1) 2.16 d ( AB T , = 2 . ν − ) = 0. N d g . A µν hT i dΩ , , 2 R. ) N i 2 i r + hT i ) = 0 − on the hypersurface. . X µ T ν + +1) ,X ( , 2 µ d 0 2 µν ∇ ( ) AB µν ) g 2 r = 0. Let us study the embedding + d X g G K d ( +1 µν 2 R 2 0 − d g µν 1 L ) − K +1) κ d X ( d d ν 2 ( AB = ) µ − K ct/L − K G − = A d K 2 ) lead to 2( +1 ( κ = e µν d N µν µ κ K B ( + ( 2.17 ≡ K ∇ x 2 hT i − t d – 5 – B = +1 d A 1 , µν d = 2 2 ν i N x c κ d A N =0. As a warm up, let us consider the hypersurface +1 dimensions. Replacing the extrinsic curvature by X µν − g d = N µν A + AB = − hT i + 1 dimensional flat spacetime T η ), we have 2 hT i µν N ν ) turn out to be +1) d 1 2 T d x = µν ( AB = 0. Thus, considering the Gauss equations, the Hamilto- d − − +1) 2.19 hT i G ) in more details. It is a hyperbolic spacetime with radius µ d 2.22 hT i +1 d B ( AB x = 2 d G d s 2 N − hT i d 2.21 ) vanish, A L µν 0 = 2 1 2 g ) into ( 0 = N − = 2.14 hT i d . The vacuum Einstein field equations associated with the ( +1) 2 d 2.15 d s ( AB d T = , . . . , d 2 B , N is the Einstein’s constant in dimensional de Sitter spacetime, A = 1 d +1 N i d Next we assume that the stress-energy tensor of the dark components in ( κ +1) is the covariant derivative associated with the metric d ( AB with dimensional flat bulk metric ( of de Sitter hypersurface ( which can be embedded into the 2.2 Holographic de-SitterFirstly screen we in set that a the flat stress-energystein bulk tensor field of equations the ( baryonic matteras and the radiation in the Ein- While the momentum constraint equations ( the Brown-York stress-energy tensor, the Hamiltonian constraint relation ( Then by plugging ( given by the Brown-York stress-energy tensor associated with the hypersurface [ and lead to ∇ G nian constraint equation of the hypersurface leads to On the other hand, the momentum constraint equations universe, we require them to be localized on the hypersurface, such that we still have JHEP10(2018)009 ), ) are 2.14 (2.28) (2.29) (2.24) (2.25) (2.26) (2.27) 2.21 ), the pure de ), one requires is the velocity , µ µν u h 2.19 2.21 R . Then we arrive at p in ( and µν + g ν ν µν 1 . . , the Brown-York stress- u g u L − µ 2 µ d µν . u u L g Λ − ) 2 hT i +1 1 1 L + d R d κ ρ + hT i ), after considering normal vec- . . µν = = − 1 2 T g 2 ), our assumption for the constraint c Λ µν = ( d c − = Λ d 1 d 2.21 2 +1 µν ) with the de Sitter metric ( Λ = ˜ ρ c d κ d κ µν R K hT i − Λ 2.26 κ µν Λ d − ˜ µν ρ = d )( 2.14 κ h = − ,T hT i µν Λ = ν 2.24 = µν when u Λ = µν in the constraint equation ( µ hT i – 6 – µν u ν , µν ) hT i R g µν u 4 B c g µ µν 1 2 µν Λ hT i ρ d g u hT i Λ κ d d − µν κ = ( = Λ µν Λ − hT i − ≡ − µν ˜ B R ρ = 0, we have the identity 1 2 Λ = − hT i T µν Λ − ) with the induced de Sitter metric 1 2 µν Λ hT i d − hT i will play the role of the dark energy. Notice that to balance the ,T hT i 2.14 d hT i = µν ) with 2) R − T d 2 µν ) turns out to be L + 2.20 2 1)( − hT i µν B d 2.18 ( ) we read out the dark energy density formula T ) becomes = = is the normal vector of the hypersurface pointing outwards. The cosmological 2.24 d is the mass density baryonic matter, µν 2.20 ) which leads to the extrinsic curvature A T B ρ N 2.23 dimensions. The dark energy and dark matter are all assumed to be related to the From ( Interestingly, for the pure de Sitter spacetime ( is the stress-energy tensors of baryonic matter and radiation, d µν where in extrinsic curvature of the hypersurface embedded in the higher dimensional flat bulk. We This is the main constraintT relation in this section. Since in Einstein field equations ( 2.3 Emergent darkNext, matter we on consider holographic to add screen form a and small isotropic amount distribution. of Considering baryonicrelation ( matter ( and radiation in with the uni- Thus, assuming Sitter spacetime satisfies theYork stress-energy above tensor identity plays automatically. thedark role Notice matter of here yet dark in that energy the the and set-up. there Brown- is no baryonic matter or After considering ( naturally satisfied energy tensor ( the stress-energy tensor of apparent dark energy, One can see that the Einstein field equations in ( constant Λ Einstein field equations ( either the cosmological constant or the apparent dark energytor term. ( where JHEP10(2018)009 = 0 (2.34) (2.35) (2.36) (2.31) (2.33) (2.37) (2.30) (2.32) D w . µν ), h = 0. Keeping . . ) D 2.9 R p Λ p ρ . + , despite being not 2 R 2 + +3 Λ c ν p c 2 D u Ω D c D p µ 1) ρ ρ R ) leads to u . , ρ ) − (2 c . ≡ D − D d 2 D /ρ ( ρ 2.32 1, we can also arrive at the c D B p B ˜ − ˜ w +Ω Ω 1 2 ρ = ( B R − − − ≡ ρ . µν D ) d d 05 ) . B . D , ) 2 B 0 ˜ Ω + − 2 ), we have w B D hT i c B ) by the squire of the critical energy 3 ˜ p ' Ω ρ , 2 + Ω 2 D R − , c 1) c B c − 2.28 p p − B , which is playing the role of dark energy 2.33 µν ) − /ρ (1 D 1) 1) Ω g µν D D d D ) ρ (Ω − 2 − − ˜ w Ω 2( c Λ 3 ˜ d d ), we arrive at hT i – 7 – D Λ ≡ ( ) Ω − ρ − D 1 2 ( D (Ω − 1 in the late time universe, then + ( D w 2.27 ˜ 1 3 − Ω (1 ρ R = D Λ ' D ρ = ρ 2) = ˜ Ω ρ 2 D B , ), as well as Ω − 2 D c − Ω ) µν + Λ Ω B d ). We also assume the emergent dark matter is pressureless /ρ D 2(1 + 3 ˜ ( + Ω ρ 2.11 B Λ ), we can obtain our main toy constraint in ( w ρ hT i ). On the other hand, since Ω ρ D 2.3 − = − Λ ρ + 2.8 Λ ≡ 2 D D , Λ 2.10 ρ ˜ + Ω Ω Λ dρ from ( µν D ˜ Λ Ω dρ 2(1 + 3 ˜ 2 ) B in ( Λ D − hT i = Λ ρ B ρ 5Ω ρ d and with equation ( + Ω 2 D + ), we obtain the generalized constraint relation 4 3 ρ ' Λ = ˜ of the dark matter in the constraint relation ( = Λ µν ρ Λ ρ 2.6 D ' D 2 D ( = p ρ = 4, the stress-energy tensor of radiation is traceless 2 D hT i Λ in ( d ρ for now and discuss the otherwise later in this paper. Through setting ˜ = 2 c ρ 0 ), and considering ( t µν = If we further use Ω We will take the assumption that the evolution of the late time universe is governed by When denotes the effective state equation of the emergent dark matter, which can be time 2.34 hT i t D ˜ Considering Ω Verlinde’s Ω at in ( which can be time dependent in general case. the ΛCDM parameterization, andYork the stress-energy total tensor dark in ( components are identified as the Brown- The components have been identified as w dependent in general.density Dividing both sides of ( the pressure If setting ˜ and dark matter, Putting them back into the constraint equation ( take the Brown-York stress-energy tensor JHEP10(2018)009 4 S (3.4) (3.1) (3.2) (3.3) + . The 5 µν S g . ) 0. y ( δ is the Lagrange , along with the B y < ν ˜ g AB M g A L µ , we review the usual ˜ g , . K M µν M g T 3.2 . , and L ν − g x and induced metric √ M ) = d − x 4 y µ ∂ 4 and metric ˜ for the region ( √ S x d δ x , we firstly review the consistent 5 d symmetry along the hypersurface. − 4 B S ν 2 M d µν ˜ g ∂ M g Z 3.1 Z A M µ . We also discuss its connection to the + ˜ ∂ 5 ˜ g 1 Z 2 κ y 3.2 = 0, which is the shared boundary of the + + µν with action y = d R g R Rg with action ˜ g – 8 – M B 1 2 − ∂ x − d √ M − p x A 4 x x µν 5 d d d R located at ) in our toy model can be recovered in late time universe. M 0 and the half bulk AB M ∂ g 4 , where Z Z M 1 2.9 4 κ ]. We assume that the stress-energy tensor of the total dark ∂ 5 4 = ˜ y > S 1 1 κ κ + 27 5 2 2 2 s + d , 5 = = S 5 4 , we have AB AB ˜ g S S g AB R g 1 2 − for the region , we further develop our viewpoint on the holographic FRW universe in a flat AB + R 3.3 M 5 1 κ The bulk equations of motion are given by the variation of the total action is the trace of extrinsic curvature of the hypersurface half bulk with the bulk metric ˜ of the bulk metric ˜ We assume the hypersurface K density of matter localized on the hypersurface. If choose the Gaussian normal coordinates Consider a 4 + 13 dimensional + 1 flat dimensional bulk timetotal like hypersurface action is given by bulk, using a different boundarywell studied condition DGP from braneworld model.choice, the In constraint particular, relation we ( show that under special parameter 3.1 Embedding FRW universe in a flat bulk components, including dark matter andenergy dark tensor energy, is on provided by theembedding the FRW holographic of hypersurface. stress- the In FRWparameterization hypersurface section in DGP in braneworld a modelIn with flat the section bulk. In section model in the next section. 3 Holographic FRW universe andIn emergent this section, dark we consider matter adimensional more flat consistent embedding spacetime of [ the FRW metric into one higher the dark matter in smallercurves, scales we around the need galaxies to andsame compare consider situation with in the galactic the effects rotational earlierthe of universe universe when and back-reaction matter can of or notcases, radiation baryonic be dominates this matter. treated in toy as the model perturbations This energy on turns is of the out the background to anymore. be In not such enough, we will resort to the more complicated so precise, the de Sitter background is still a good approximation. However, if we consider JHEP10(2018)009 , ], M 28 . L (3.9) (3.5) (3.6) (3.7) (3.8) I (3.12) (3.10) (3.11) , we will . . 3.3 2 µν g dΩ 2 − r + − K I, 2 , as well as the modified r + . − µν d ], − µν 2 . K ) 2 31 2 t = 0, ) M c 2 – is chosen as the normal vector ( , 5 ˙ 2 a y . L . κ −hT i 29 2 M 2 g y, t µν dΩ ) ( − 2 T t ] N − = symmetry symmetry along the brane similar = a r ( µν L √ a 2 = g 2 + + + µν + µν x is neglected. In section Z Z 2 4 r 2 µν t d y r , and I d 2 G d hT i 2 − K M 4 M c 1 − hT i ∂ ∂ 2 κ | ) 2 Z ) µν 2 t ) − µν ) ( B + K 2 t a c ( – 9 – N y, t ˙ symmetry along the brane in the bulk has been − µν a 5 µν ( A + δ 1 . 2 ( 2 κ n 2 y δg ) ˜ ≡ hT i t Z ∇ − ≡ g d B + ) ν 2 t 2 y, t ˜ K µν − hT i g − 2 2 c ( ( y µν ) A ˙ µ a a √ t ˜ g + µν − t ( ], the hT i ∂ , we will choose the − a = 4 dimensions, = d hT i = ≡ 14 hT i d = = 2 4 B – , s 3.2 µν x ) = 2 . The effective stress-energy tensor from extrinsic curvature is ) in Gaussian normal coordinates is d 12 ) M K µν d µν K T A µν y, t 3.3 ( x y, t hT i ( d n Rg directions, respectively. a 1 2 + AB y g − M µν T = ˜ µν 2 5 = R s d ≡ µν G µν 4 along the is the integration constant, with dimension of [ 1 G κ I In the following section We consider that our universe is uniform and isotropic at large scale, and take the M ∂ In the usual DGPimposed, model which [ leads toEinstein the field boundary equations condition on the brane as the usualconsider DGP the model, FRW brane as where ainterpretation the cutoff as hypersurface parameter the in holographic the flat FRW(hFRW) universe, bulk with and a present an3.2 non-zero alternative parameter Braneworld scenario: DGP model with Here The consistent embedding metric functions are solved as [ The consistent embedding in higherwhere dimensional the flat bulk spacetime metric has ( been discussed in [ The extrinsic curvature is of spatially flat FRW metric in We include the baryonic matter, radiationwhich and leads other to effective the terms stress-energy in the tensor Lagrangian where With the matching junction condition at the hypersurface JHEP10(2018)009 ) is 5 and 3.9 κ K (3.20) (3.17) (3.18) (3.19) (3.15) (3.13) (3.14) (3.16) ρ . . 2 . 2) 4 dΩ / ), we can read out ) 2 5 κ t r ( κ . ( K . 3.12 + p ] K 2 ≡ K 2 r c in ( d ) + M, t includes the baryonic matter hT i ( ), the coupling constant 2 K µν = . ) , M t + M µν # , ` ( p T )] ) ) 2 a 2.20 hT i 0 M t t t ( 2 ( ( 2 T H ) was chosen, then the metric ( [ c ˙ K 2 H H ) ) , ı ` 2 ρ ) + 3 t t t ( ( ( 2) 2 3.10 ˙ 4 = a a / , K ) + ) + | κ ` 5 ) /c t ρ t c y t ) ( ) is proportional to the Brown-York stress- ( κ | Ω ( t ( ( H M ı H ) + − 3 ρ p t 3 [ 3.12 " 1 ( = 0 in ( 3 = 4 – 10 – , c c c M 2 I ) + 5 5 ) 3 2 in ( 4 ρ t + t 0 µν K κ κ ( κ ( ı 2 H K µν t ∓ 4 ρ H correspond to the sDGP branch. The negative = c d hT i ` 6 4 2 ) 2 K c t Ω κ hT i ) K µν p ) = ) = ( t 2 t t ( − p ( ( H provides the effective dark energy. In the normal branch of ) ) K K 3 H t t + = p ρ ( ( − ˙ 2 − hT i 3 ¨ K µν a a ) ) | M t t ) into the stress-energy tensor 2 K ). Since c ( ). It is due to the double copies of the Brown-York stress-energy y ( | Ω ) = hT i 3 a ) in the DGP model. t H ( hT i ı 3.13 3.14 = 3.19 ˙ ρ 1 and negative 3.12 ) + 2 t ) K 2 0 ( t ρ − ( ) is satisfied and leads to ˙ at ( H 2 in ( H 2 H / y 5 K µν 2.16 κ correspond the nDGP branch, and the extra effective cosmological constant is = d ) is summarized as K hT i 5 2 p s d 3.14 Here we pay attention to the sDGP branch, where the modified Friedmann equa- The positive tensor in tion ( Compared with the constraint relationreplaced in by our toy model ( positive required in equation ( energy tensor, it isin natural the to bulk see ( that the Hamiltonian constraint equation on the brane Plugging the metric ( the effective energy density and pressure as well as the acceleration equation The energy conservation equations for each component are The above two equations will lead to the modified Friedmann equation on the brane, and dark matter, while the DGP model (nDGP), depending onto the parameterization, be a supplemented. cosmological constant In needs becomes the more usual simple, setup, In the self accelerating branch of the DGP model (sDGP), JHEP10(2018)009 ), 3.21 (3.24) (3.21) (3.22) (3.25) (3.23) ). , which is ]. I 2.9 ) again will . Compared 35 ` = 0 in ( , – . Ω ) ` t 3.20 32 M ( Ω √ K p + Ω . ), we can recover the − M ]. It will become clear − ) with ( t 0 ( 2 32 ) 3.10 H M 2 0 = 1 t H ( ` H ), M H Ω Ω , p = ). It is quite different from our 3.23 ` , which does not match with the ` D 3 . = Ω D ) Ω t ) is satisfied after putting back the Ω M t ) with that in the late time ΛCDM , c √ Ω ( K √ symmetry of the sDGP brane, which Ω 2 ρ − a ρ . / ) in the toy model, we need to give a − from ( 2 > − 1 2.34 ` 3.21 B ` ) Z ≡ ≡ ), we have 1 = Ω ) t B D t Ω 0 0 ( Ω ( ) (Ω 3 B 2.34 M w t H H H √ ) H √ ( 2 ˜ Λ t Ω Ω 3.20 ( K ` 2 − − a ) ) and eliminating Ω Ω Ω ) t ) t , including both of the components of baryonic ( t 0 0 = Ω ( in ( ( – 11 – + D H H H H 0 H 3.20 2 D 2 2 ` 3 . It is easy to see that if setting Ω 4 , H 0, which is a bit different with the ΛCDM model. If Ω ` − − R + Ω in the embedding function ( , Ω is the component of the effective dark energy. More ` 2 is due to the → B ), as well as ˜ I = t ) = + ) Ω ` − ) = ( t 0 ) L/ D t ) ˙ ( 3 t t ( ˜ − t H ) ( = Ω w ( , it can not recover our toy constraint relation ( M D ˙ ( t = ) H K ( t B H K w Ω ` M ). Thus, if taking the sDGP model with Friedmann equation ˙ ( a Ω ), at any cosmological time K ), because it requires Ω ), it is clear to see that only in the very late time universe that + Ω = Ω = Ω 2.11 ` ) 3.25 2.9 2 t D Ω ) in our toy model. M ( 3.25 and radiation Ω ˜ 1 Ω H = 2.9 3 K ) = 1, we arrive at Ω ) = 1 and 2 0 ) 0 , equating the right hand side of ( t 2 0 ) and ( − t t ( , ( B ( , we can have ˜ ) and ( 1 a ` H ` a H − Ω 3.22 = Ω 3.20 √ = Ω = ) at 0 ) in our toy model, M Λ D H 2.7 ˜ ) and setting Ω ˜ Ω w 2.24 → In the next section, we will give an alternative interpretation of the embedding scenario From ( In order to recover the constraint relation ( ) 3.21 t ( as the holographic FRW universe.usually In set particular, to we be will zerothat turn in only on the if the previous keeping parameter studies thisconstraint of parameter relation the ( DGP models [ H setting Ω model ( toy constraint relation ( observed parameters in ( in ( One can check that theexpressions general ( constraint relation ( In particular, taking thelead to derivative the of identical ( relation of detailed study of the phenomenology of the DGP modeldifferent can interpretation be of found these in parameters [ in the sDGP model, Notice that to make thespatial presentation curvature more Ω clear, we didwe not will include have the the contributionbranch from Friedmann of the equation DGP of model,matter the Ω and de-Sitter dark Universe. matter, In while the Ω self accelerating with ( includes the double copies of the Brown-York stress-energy tensor. Equivalently, If considering JHEP10(2018)009 . . 2 / M can 1 ∂ + (3.33) (3.30) (3.27) (3.28) (3.29) (3.31) (3.32) (3.26) 4 ) t I M ( a + 2 ) t 2 ( in the flat bulk, c H − . . M . . 5 4 , we have c 2 µν 0 κ κ 4 . ρ H ), and the Einstein field # c g ], the manifold H 4 4 − = 4 c ) H 3 ). It can be reduced to our 44 3 κ 3.5 t 4 I M, ( κ a = 3.3 − K hT i = = c + ρ + − µν 2 2 0 ,L ) K t , M 2 ( T L 1 5 H µν , ı 2 H 1 κ 2 c is 2 ) 2 c 4 4 , 5 ,H hT i 3 = κ κ 2 )+3 ) is automatically satisfied κ /c M t / )] ( ) + t ( 1 t ∂ ), such that the energy density and pressure − µν = ( ( ˙ − ı M H µν 2.20 H 4 Λ p ) ρ " = T hT i t 3.10 I ( – 12 – − = a = µν H ) + ], we can drop the manifold ) + t = t ) ( ), considering that + ı µν ( 5 , ρ 2 µν hT i ρ 43 ], or the blackfold approaches with higher dimensional = 0 plays the role of the holographic boundary of the S =0 2 ) G , 0 M ( t y 2 4 δg ρ H µν y

) 2.24 ( 2 δ c 41 1 H [ 42 t κ c 2 – ( g = 0 in the junction condition ( 4 H ) at L c − 2 ) 36 H 3 4 + µν 3 κ y,t √ = − hT i M ( y,t − = 3 ˙ ( ∂ − c a Λ = 2 H ˙ hT i y ρ a ρ 2 3 ∂ =0 = ) ) = y t t

= hT i + ( ( ) H µν ı ) Λ ˙ ρ H ) ) Ω y,t y,t hT i ( ]. In this subsection, we will give a new physical interpretation of the ( y,t a y,t ( a ( y 21 y a ∂ a – ∂ . Or from the viewpoint of membrane paradigm [ 3 2 15 + − M = = are given by H H H µν p ρ ) ) 5 2 We will choose the negative branch in ( c κ hT i 5 ( κ ( Again we use the same setting in ( The modified Friedmann equation is, And the energy conservation equation remains in Again the Hamiltonian constraint equation ( It is also equivalent toequation set becomes where the Brown-York stress-energy tensor on toy model with a de-Sitterholography hypersurface in in the the flat flatsuch spacetime bulk. that [ the From hypersurface the viewpointmanifold of the cutoff be replaced by the quasi-local Brown-York stress-energy tensor on the hypersurface In the previous subsection, weembedded have into studied the the higher dynamics dimensional of flattraditional a spacetime. braneworld FRW models The hypersurface, [ physical which picture is embedding is related [ to the FRW hypersurface in a flat bulk with the embedding metric ( 3.3 Holographic scenario: holographic FRW universe JHEP10(2018)009 ) 3.35 (3.38) (3.39) (3.34) (3.41) (3.37) (3.40) (3.35) (3.36) . 2 ). / ) again will 1 . 2.9 4 i 3.34 ) I . Λ t . t , ( Ω 2 ) 2 0 Ω a t . ( H Ic √ 2 + H / with ( − 1 ≡ Ω 2 ) 4 2 I 0 ), t ) I M I − t 4 ( Ω ( Ω ) , H and turning the parameter Ω 2 a t H ) Λ ( 2 2 0 B t 3.37 ) 2 a 0 t + ( Ω ( H . 2 H 2 / , ) H H ) + 2 1 Λ 0 ) is satisfied automatically after t = Ω 2 ( B Λ H / = 4 H 1 3 Ω Ω M ) I M , t ) 3 Λ = Ω ( 2.34 Ω t Ω ) from ( √ ) ) − 4 q a ( t M Ω Λ t ) I B h ( c a 1 3 ( H t − D Ω Ω ρ = 0, we can recover the usual Friedmann √ ( Ω D ρ a Ω 4 a 4 − I ) I w (Ω ) t − I + − t ≡ ≡ ( Ω i ( Ω Λ + a 4 a 2 Λ Λ ) ) D I ), at any cosmological time B 4 t t Ω ) and eliminating Ω 2 1 3 ( Ω Ω Ω ( + ) ˜ Ω 2 a 0 , and equalizing the right hand side of ( (Ω t and Ω − H √ 2 – 13 – ( − B ) H 3 2 3.39 2 I 0 t + Ω 3.34 D + i ( H − H Ω 2 Λ 3 = H ) 4 2 Λ 0 t ) ) I Ω ( 2 = Ω M +Ω t t I H ( / Ω ( q ), as well as ˜ , H Ω √ B 1 Λ a Ω t 2 a Λ ) and ( ( M = Ω , − q ˙ + Ω + H ), we has a different set of parameters in the hFRW model. 2 2 D Λ 4 + Ω = Ω 2 ) I ) − Λ 2 t 3.36 Ω 0 Ω t 3 2 ( Ω ) ( ), the modified Friedmann equation is summarized as 2.7 ) ) M H Ω a t M t t − H ( ( ( Ω ) + = H a H t 3.31 q ˙ ( 2 Ω 2 h ) 2 0 = ) t H 2 2 0 ( t ) H = Ω ( Ω 2 t H H ) ( 2 ) 0 t H D t ( ( q H H ) into ( ˜ Ω 1 h 3 ) = 1, we arrive at H H 0 − − t 3 ) of the sDGP model. While if setting Ω ( 3.29 ). If only setting Ω ), a = − , ) = 1 2.7 Λ D 3.21 t 2.34 ( ˜ − w ) at ˙ H = Ω = 2.7 Λ Now let us compare it with the late time evolution of ΛCDM model with Friedmann Firstly, we need to match these parameters in hFRW model with that in the constraint Putting ( D ˜ ˜ Ω , it can be shown that one is able to recover our toy constraint relation ( w I Thus, once taking plugging in above quantities ( equation ( and ( One can check that the general constraint relation ( In particular, taking thelead to derivative the of identical ( relation of Notice here that by setting Ω equation ( Ω relation ( We named this ScenarioΛCDM parameterization as in the ( holographic FRW(hFRW) model. Instead of using the Or equivalently, JHEP10(2018)009 ]. ) = 31 = 0, 0 t (3.46) (3.47) (3.43) (3.44) (3.45) (3.48) (3.42) ( D ). The 21 [ p . . /a ) i t 2.11 Λ ( = 0 and a Ω . M D √ 2 in terms of the / Ω − 1 c ). More detailed 4 0 ρ ) 4 z I ) I Ω = /H z 2.11 ) Ω (1+ z D ) from the Brown-York Λ ( and the state equations ρ + Ω . 0 2 H , (1 + ) Λ 2 0 2 3.30 z / ( ) for various models are Λ H + /H 1 )Ω z H ) in various models. The Fried- Ω ) from negative to positive in 2 ( 3 z ) with c z 3 ) z ( B √ D c ( q ) z M ρ z Ω w D H ) = 1, we have − ), with a special choice of the pa- ( 0 Ω Λ 2.30 .h 4 w t ` ) 4 Ω ( z I ) Ω (1 + ), with the parameters in ( a z I 3.35 Ω (1 + ' − Ω ). Plugging the ΛCDM parameteriza- (1+ + 2.7 H + (1+ + 2 Λ 1 3 2.9 2 ` 4 ) and pressure ( 2 4 Ω ) Ω 2 − 0 ), with the fitting parameter Ω z ( is related to the scale factor via i H Λ H , 3.29 , p + z ), we have + ) – 14 – Ω 3.21 B 3 3 q 3 , which can be easily included in the equations D ) ) √ ) 2 ), along with the values in ( z z ) and setting K z 3.40 B − B + Ω M 4 Ω + Ω Ω ) Ω D z I 3.35 Λ (1 + ) turns the value of ˜ (1 + − Ω (1 + Ω , )( (1+ (Ω ` + ` D c + + Ω 3.48 ρ Ω + ` Λ Λ 3.21 (Ω 2 2 2 √ . √ ) Ω Ω 2 3 Ω ' and fitting parameters in the hFRW model will appear in our 2 0 )( z 2 Λ − in ( ( − c H p I ) s s s H ) H = I z 2.7 z ρ 0 0 ( − ), the associated state equations ˜ ( = = = I H H H q H 2 = ) ) ) h Ω , , z z z 0 0 0 3.39 Λ − − ( ( ( B H H H Ω H H H c ρ ) = ) = ) = 0 , we plot the reduced Hubble parameters z z z = Ω and spatial curvature Ω ( ( ( = 1 ) and ( D D D R Λ M w w w in various models. The redshift ρ 3.25 ) into the effective energy density ( z ). Considering ( sDGP : hFRW : ΛCDM : z 2.7 ) of the emergent dark matter in terms of the redshift In figure Finally, we summarize the normalized Hubble parameters z ( sDGP : ˜ (1 + hFRW: ˜ D ΛCDM : ˜ / ˜ Friedmann equation of sDGP model isThe in Friedmann ( equation of ourrameters hFRW Ω model is instudies ( of this non-zerofuture Ω work. above. Here including Ω the late time universe, and thus effectively contributes to thew emergent dark matter. mann equation of spatial flat ΛCDM model is in ( Again in order toradiation make Ω the presentation simpler, we here neglected the contribution of Taking ( Thus, it is consistent withas the well ansatz as in our toy model ( redshift 1 tion ( stress-energy tensor, and considering ( we can recover our toy constraint relation in ( JHEP10(2018)009 , , ], d B 2.0 S 21 – (4.1) (4.2) + = Ω 15 +1 M ]; hFRW: ΛCDM sDGP hFRW ]. In this in various d S , 32 1.5 z 24 ); sDGP: the – K g 22 − 2.11 in various models. 21 in [ . √ z x z ], we have proposed 1.0 d 7 = 0 ]. In the holographic d 7 M M ∂ Z 0.5 . +1 1 d M κ L g , we considered the uniform and in terms of the redshift − ] + 3 0 φ √ 0.0 L x /H 1.0 0.5 0.0 d ) -1.0 -0.5 + d ]. We present a consistent derivation of z ), with the parameters in ( ( ) in terms of the redshift (z) 10 z +1 M D H ( d ∂ ˜ w 3.48 D ). Z w 2Λ and section – 15 – 2.11 − ), with a special choice of the parameters Ω ) + 2 d 2.0 , we have generalized the holographic de-Sitter sce- ) and ( +1 ), with the fitting parameter Ω 3 3.48 d 2Λ R 3.43 − [ 3.47 d ˜ g R − 1.5 ) and ( ( g p x − ) and ( 3.45 ]. In section √ +1 -dimensional Verlinde’s emergent gravity with elasticity into a 7 d x d 3.44 d d z 1.0 d , inspired by the emergent gravity by Verlinde in [ M M Z 2 ∂ Z +1 ), along with the values in ( d d 1 ΛCDM sDGP hFRW B 1 κ κ 0.5 2 2 Ω − = = . Left: the reduced Hubble parameters d D +1 S d (Ω 3 2 S 0.0 On the other hand, in both section 2 1 0 4 3 = 0 I H H(z) higher dimensional bulk spacetime,where we sketch the more general total action as Tully-Fisher relation in the frameoriginal of the Verlinde’s story. elastic model and try to resolve4.1 some issues in Holographic the stressIn tensor order and Verlinde’s to apparent embed dark the matter isotropic metric at thewith cosmological Verlinde’s derivation scale. on the While Tully-Fisherscales relation, in we of this focus the section, on de-Sitter the aimingspherically brane. response at at symmetric Instead a galactic baryonic of comparison matter. thein uniform We Verlinde’s baryonic are emergent matter, also gravity we pointed trying need out to to by reconcile add [ the the inconsistency setups still lack of themodels, elasticity the in elastic the property Verlinde’sor emergent can by gravity including usually [ the be effectivesection, realized mass we terms in will in the consider theeffective blackfold bulk the massive approaches of embedding gravity holographic [ terms, of models where a [ the de holographic Sitter elasticity hypersurface is implemented. in the flat bulk with In the above section the emergent dark matter on thesimilar de-Sitter mechanism hypersurface as in a in flat [ bulk,nario which to gives the rise time to evolution the case with a FRW hypersurface in a flat bulk. However, the above ΛCDM: the plotting functionsplotting are functions in are ( inthe ( plotting functions areΩ in ( 4 Towards holographic de Sitter brane with elasticity Figure 1 models. Right: the evolution of state equations ˜ JHEP10(2018)009 , we (4.7) (4.3) (4.4) (4.5) (4.6) 2 as the , which d d is caused S i n ]. However, . 1 with induced 24 – − = 0. The stress M , 22 , ) ∂ µ 2 2 ) becomes ν . u r ˜ L n 3 , . . . d B µ − 4.3 µν ( 2 represents the effective d − i , M r , ]. Although the detailed φ σ 2 h L µν 58 ≡ ∇ − = 1 i – d σ ) = 1 d ), we now consider the static h µν r 45 ( 2)Ω + πG spacetime and obtain dS 8 2.21 ν − in the boundary action u , f +1 d µ j d d , ε ( u x , i, j µν d D i . Like in our toy model in section 2 ε ˜ ρ x ν j − ≡ − which satisfies K 2 d d h ≡ . The Brown-York stress-energy tensor on B µ i ij µ µν ) h ) is given by dΩ ˆ h Φ i u 2 -dimensional de Sitter spacetime as the hyper- σ +1 r = + d µν h 4.1 d spacetime in massive gravity [ – 16 – 2 S + ij δg t ( , , the Lagrangian density 2 δ +1 )d r B d r g d AB ( velocity ) − 2 g f r d , ε 1 ( √ − are given by + 2Φ f µν − 2 i 2 = ij = r ε σ L = ν h j x ν j , we embed the − x d h µν 2 d µ µ i i x h x with metric ˜ d hT i d ≡ ) = 1 ij µν r M ˆ h g ( ij i f , along with the = = σ ν h + 1)-dimensional flat bulk in Einstein gravity. The usual holographic solid 1 2 d u d s − µ 2 d d -dimensional de Sitter hypersurface embedded in the higher dimensional flat = 0 in the bulk, and study the holographic elastic response of the screen after u s and strain tensor d d + is the effective holographic energy density induced from higher dimension, and , the trace of the extrinsic curvature is +1 ij i d is the induced metric on the spacial slice. One can also define the projection tensor µν D µν σ is the shift vector associated with the deformation. g h ρ g ij µ ˆ h n Now we take a detour to the Verlinde’s derivation, where the displacement ˜ For the The viscous or elastic response of the boundary theory in the holographic description ≡ µν and ˜ by the presence of the baryonic matter, and the metric solution in ( where ˜ we introduced the covarianttensor stress tensor Here h the boundary of the bulk with the action ( coordinate patch described by the metric for the rest ofonly this uses section the is theory actuallySitter that independent background. describes of the the elastic holographic solid construction with and different it modulusbulk values spacetime, in instead a of de the expanding coordinate in ( one may embed the dS-slicedboundary coordinates hypersurface. into the It AdS is foreseeablethe that case the with elastic solid the model deconstruction can Sitter be is boundary generalized still with into the leftthe tension to above term be [ action done can and capture we the leave feature it of to the a elastic theory. future work, In for any case, now, the we discussion assume of the boundary brane. is encoded in the transverse-tracelessIn tensor the toy mode model of in the section surface metric in perturbations the in ( themodel bulk. is on the flat boundary in AdS term which can providemetric the holographic elasticity.can set On Λ the boundary putting in the baryoniccan matter. contribute to One the may total also cosmological add constant Λ on the boundary theory, or the tension In the bulk manifold JHEP10(2018)009 i ij n ε 0 B a (4.8) (4.9) Φ ∼ (4.10) D = i . We can n 0 ij ε will be taken B as below, , ) ] avoids the facts 0 ij k 7 ε coming out of the k ]. ε 7 ( µ ], such that the bulk 1 + 2Φ ij 24 ≈ . λδ – ) ) k r + k 22 ) and reproduce the elastic ( 1). By tuning the effective ε . In the deep-MOND regime ij f ( is the constant total mass of simultaneously, since they are ), the spacial components are − B 4.1 ij 0 r µε . Thus, this is the inconsistency δ ij d r i ), the baryonic matter induces a ( 1 4.5 dr . The problem with this argument ) σ 0 i h 1 = 2 − µ/ 4.7 r ( ], is that the displacement ˜ ) d dA A j . + 2 10 +1 ij ) − j ] and ( 0 n d ˆ h λ ], such that r n ij S ij δ i ( 24 ( ε 3 σ n – , D δ = 4 in ( at the large 2 ρ 0 ij ∞ = [ g d 22 ε then only have the traceless part S 2 B − H /r 0 ij r L 0 − ij ≡ – 17 – 0 R √ 1 ε a ) 2 in [ r = = ( / and stress tensor = ε ) ij B , ε ij < r ˆ h F ε M 0 ij δ ]. In the following, we try to circumvent this issue and M g B , then the apparent dark matter surface density Σ ij − r − 10 B µ ε ), we expect the designed values of δε 1) = 0. We can define the traceless part r √ = and bulk modulus ]. ( 4.1 GM − = 2 7 δ ], where the ADM mass definition of the de Sitter spacetime is ij µ has to scale as 1 . When choosing 7 d g . However, to produce the Tully-Fisher relation or flat rotational ij 0 δε ( ij i r − 1 ∼ − ε σ µ/ cH and strain tensor h √ B ∼ ij = = ) as the largest eigenvalue of the traceless part of the strain tensor, i + 2 D 0 r σ ( λ ij a h i ε σ in the bulk ( h φ at the large L 2 is important in [ /r i n 0 B a Φ One issue in the Verlinde’s derivation raised by [ = i ˜ diagonalize the elastic strain tensor symmetric and linearly related. Theirvalues.We define eigenvalues are called the principal strain and stress This describes onlyThe the stress shear tensor deformation, without changing the volume of the body. with shear elastic modulus Lagrangian holographic engineering with onlymodulus the vanishes traceless stress tensor [ 4.2 Emergent Tully-Fisher relationConsidering from the holographic fact elasticity that such that the elasticity willfield emerge equations from will the be holographicby modified. brane, employing and a In the model effective thedark of Einstein following, matter holography we response with propose formula the as a bulk the way action baryonic to ( Tully-fisher resolve relation this in issue [ related to the strain tensor through is for a pure decan not Sitter have space the with elastic thethat property positive one on cosmological its needs own. to constant go in Thematter. beyond Einstein elaborated the gravity, derivation One it [ theory way of out Einstein is gravity to to have embed the the correct elastic de dark Sitter hypersurface in higher dimensional bulk, Newtonian potential Φ scales as 1 curves in galaxies, Σ in the Verlinde’s original storysee [ whether this assumptionn of the displacement can be abandoned. This displacement baryonic matter in the galaxy,we with are the looking characteristic at scale inin this the section, following derivations [ is identified, with We consider the simple case that JHEP10(2018)009 , we r (4.16) (4.11) (4.12) (4.17) (4.13) (4.14) (4.15) ], we start 7 . 2 and assuming the ) 0 r ij ( ε ε , ) 0 . ) ) with respect to r ) ). Similarly, we will only B r ( . ( ij in Verlinde’s [ 0 as the total apparent dark A M 0 ε A 4.8 is approximately , 0 i 2 0 4.14 r πG ij 0 0 n a d r π ≡ dr a r r. 16 ) 2 in ( ) of the de Sitter spacetime by gε ~ 0 ) 0 r . The change of the free energy B with a fixed background metric . r r − Z ( r F ( ( = = B M S √ B 1 2 B B A ) and ( 0 ( ) M 0 Σ a dS M M − − 0 r . This step is similar to the Verlinde’s µ δ ( a d d ε 4.12 ∝ D µ ) = = − , µ = ) ρ ], r ) , r ( r 7 2 & ij r ) 0 ( ) ,T ( S r R S r r ( δε within a radius B ( ij in [ ∆ ε 0 B ε ij Σ 0 ) i ε r F – 18 – ) = µ dS ~ 2 1 a r 1 σ ij r T ( 0 h − ( π M − − ε d g = A − 2 D V d d 1 2 ) − ) R as 0 will also be neglected. If we only consider the shear , and assume the surface density of the apparent dark − M 0 ) r r r √ µ − − ( r a ) = ( 2 = B . We will take the last equal sign to be approximately ( r 2 D d d A 0 ( ) = A = F − /L µ r ) = r M d F 2 2 d ( r r ) F 2 r ∆ S r ≡ g 0 − ( ) ∆ d Z − D r µ ( √ Σ ( D δ Σ ) = ) = Ω r r ) as the largest eigenvalue of the traceless part of ( ( r ( F A ε ) we can arrive at the response relation as below, ∆ ) and do not consider the variation of background metric, the free energy and baryonic matter Σ 4.15 is the Gibbons-Hawking temperature of de Sitter spacetime. Notice that this 4.9 D dS T ), where the effects of Now slightly different from assuming the displacement ˜ 4.7 then from ( One notices here that ifmatter we mass identify enclosed inside radius matter Σ other principal strains are equalderivation. and summed After to differentiatingarrive both at equations ( Here the area is true by defining modes in ( density formula becomes The change of total free energy from the variation of theconsider holographic free the energy leading density orderin contribution ( in terms of has the same entropygument formula that as nevertheless, the it purethen can de follows still the Sitter have volume space. ∆ law of We the can thermodynamics put of forth the a “de naive Sitter ar- medium” with the total baryonic matter within a radius where relation is somewhat ad-hoc here since it assumes that holographic elastic de Sitter brane We adopt the volume formula of the entropy change ∆ JHEP10(2018)009 ], B 7 ) is M (4.18) (4.19) 4.6 in ( . The last, , the equa- , we employ π 2 µν 2 in the bulk is 2 / /r 0 φ B a L ~ ) contributes to the GM = 4.9 dS T ) = in ( r ( ). B µν r ]. We may compare the dark g i , ( 2 σ f D , r 61 h v 2 – /r 59 . The strain tensor ) . ) = ) in the toy model. This leads to the r ], ) 2 r ( 7 r ( ( D 2.1 D B g ) is also related to the Brown-York stress- 0 GM 6 ), the above conclusion leads to the baryonic a . For the toy model in section r 4.1 ( = , g µν B – 19 – B ) = 2 g ) r ( r hT i ( has been chosen to be the same as in Verlinde’s [ . The stress tensor D GM D ) g 0 µν µ g r 6 a ( K D = g 4 f v ]. Most interestingly, the branes with tensions and dynamics, may react 63 = 4, and considering , d 62 ) leads to the same relation in Verlinde’s [ is the asymptotic velocity of the flattened galaxy rotation curve. f v 4.17 Notice here that we haven’t explicitly constructed the theory with baryonic mass When logical constant and the standardliterature model can [ be easily implementedto with the braneworld in matter the fieldsvature, we a put concept in, valid with onlyresponse” extra from nature terms higher of dimensional introduced. the spacetime, apparent Especially may dark the describe matter extrinsic the from “elastic cur- the Verlinde’s theory. vides the thermal baththe with apparent the Gibbons-Hawking dark temperature matterThe braneworld is scenario only may thebecomes offer a response something natural similar playground to for toof the the above-mentioned entropy Verlinde’s presence conditions can emergent gravity. be and of The explainedtreating the elastic from our medium higher baryonic full (3+1) dimensions matter. dimensional though spacetime additional as brane the dynamics, boundary by of the bulk spacetime. Cosmo- In this final section,approach and we some will well-studied furtheruniverse. scenarios, discuss especially Following the the the difference braneworld Verlinde’sthree model and and crucial for the conditions connections holographic the for between apparent hiswhich our dark construction distributes so matter, evenly far. there in First, seem the there to volume. is be the Second, background the entropy, positive cosmological constant, pro- questions in the previous sectionbraneworld, that although whether the we can answer realize is this not response clear in at a this model5 moment. with Discussion and conclusion total Brown-York stress-energy tensor the hypersurface displacement inwhereas extrinsic in the curvature holographic asintroduced. model the The in elastic boundary this displacementenergy term section, tensor, tensor in the in action extra the ( fields modified with Einstein equations ( where on top of thecan elastic become background. complicated, To do formatter so such density requires examples, more here see ingredients asrelated e.g. in to in [ the the the theory extrinsic and toy curvature it model in section In the deep-MOND regime that Tully-Fisher relation that, although we take a different ansatz for the pre-factortion in ( Σ The value of shear elastic modulus JHEP10(2018)009 , – ]. 2 − 36 64 d (5.2) (5.1) /r ], the gravity ]. The extra 27 69 , ..., is associated with ) itself is conserved 68 + . If we did not put µν 2.3 µν µν g W hT i ρσ µ K . ∇ 4 ρσ κ µν + − K ]. One not only needs to add tension hT i 2 µν K T + 67 , µ 1 2 µν ∇ 66 T 4 − κ 4 ν κ σ = ]. K = . Naively this does not work. While one may try – 20 – ) on the brane, in principle we can also define the 65 µν R /r µσ G . To have the observed flat galaxy rotational curves, µν µ µν 2 T g − K /r 1 2 ]. Modified gravity models are constrained from two as- ≡ ∇ µσ − g 64 µν K ], which emerges with a different underlining physical origin, 7 R + ( µν ≡ W B µν hT i ], the FRW metric is also embedded into one higher dimensional flat spacetime. = 0. Thus it is similar to the effects of real dark matter, and it does not conflict 28 µν Let us further compare our toy model to the other well studied braneworld models [ We comment on the possible constraints from gravitational wave observations. Re- hT i ] other than DGP. Except for the constraint equations, we also have the dynamical Ein- µ the dark matter response theory, seeterm for example to [ the D-brane actionits as the response cosmological with constant, the baryonicdimensions extra in matter scalar/vector the fields are braneworld and setups alsowaves may production needed, also and have see propagation. some for extra If effects example we on [ indeed the take gravitational a higher dimensional point of view, the bulk Weyl tensor.this It formula. might One be thing interesting we toleaks notice is derive to that the the in Tully-Fisher most extra relationcomparing of based to dimension the the on braneworld at models Newtonian force [ large 1 the scale, gravitational force so has the to scaleto gravitational like extend force 1 the scales DBI action like of 1 the branes with more fields to capture the elastic behaviors of effective stress-energy tensor from the braneworld model which includes the description of the hypersurface evolution, and 41 stein equations in highermodel dimensions. [ For example,After in including BDL the (Binetruy-Deffayet-Langlois) baryonic matter The Bianchi identity leadsadditional to sources in 0 the bulk,∇ the Brown-York stress-energy tensor ( with the observations from LIGO so far [ the ΛCDM models at the leadingTo order, be so more it specific, can pass thefills the extra the above-mentioned apparent space constraints [ dark as dark sector mediumgravity, as gravitational and the field interacts extra with equations term propagating are photons in modified in Einstein it. as equation In our induced gravity and suggest that the gravitationalthe wave weak-field should limit. satisfy linear Although equationsnamics Verlinde of as suggested motion in in similar MOND modifications theorythere [ of are Newtonian no dy- covariant equationsin of the motion previous for sections, thedark the gravitational matter induced waves. stress-energy dark For tensor our components and toy can they model be behave viewed the as same dark as energy the and particle dark matter in cently it isinconsistent argued with that observations [ twopects. relativistic models One is ofwhich the indicates modified constraint that Newtonian gravitational of waves dynamics theis should propagate energy the seem at observed loss the gravitational rate speed waveforms of from from light. ultra LIGO, The high which other energy are consistent cosmic with rays, Einstein’s JHEP10(2018)009 we 4 ]. The ], grav- 71 75 , – 70 72 ]. The entangle- ], there is an ef- 83 ]. In section 58 – 89 52 ]. It is named as Rindler 82 ]). – 97 – 76 , 94 ], the idea that considers the dark 43 , 88 42 , 10 – 21 – ]), as well as the emergent cosmology from quantum ], that the entangled pair in 3+1 dimensions can be 93 87 – – 90 84 ]. It is shown in [ ], in addition to the long searching constraints from colliders and precision 7 71 In summary, we construct a model for the dark components of our universe, where In the traditional holographic theories for our 3+1 dimensional universe [ matter as the responsefix of an baryonic inconsistency matter offuture, is Verlinde’s it still story is quite and interesting interestingmatter to re-derived models [ relate the (see for our Tully-Fisher example holographicentanglement relation. [ or model thermodynamical to In laws the the (see other for well-motivated example dark [ with the known covariantconstraint relativistic of formulas. the late Therelate time toy universe our components model toy from producesmatter. model ΛCDM one to parameterization. Moreover, additional the We wethe then DGP suggest late braneworld a time with new universeinconsistencies holographic the evolution. in the scenario new Although Verlinde’s with emergent interpretation it gravity a of [ has set the been of pointed dark parameters out for that there are some spacetime. In this holographiclate picture, time there universe is and onlythe dark baryonic matter matter geometric is and effects. considered radiationmodel as in In the the are response our partly of approach,choose baryonic borrowed the to matter from from start toy the from model Verlinde’s a and emergent holographic the de gravity more Sitter in screen developed a in hFRW subtle higher way. dimensional flat We spacetime, by Verlinde [ described by the wormholesee in the 4+1 entanglement through dimensional the bulk embeddings of spacetime. wormholes into Thus, a it higherthe dimensional is bulk. more dark clear sector to originates from the induced stress-energy tensor of higher dimensional fluid, which is described byhypersurface the in Brown-York a stress-energy flat tensor bulk. onconstraint the equation The on accelerating dynamics the cutoff is hypersurface. governedentanglement by What can the is happen more, Hamiltonian in it and our will momentum toyment be model between interesting with to two the see cosmological de how horizons Sitter the boundary may [ have an impact on the gravity as suggested consider the universe asfective contribution the from boundary the ofthe holographic stress-energy 4+1 stress-energy tensor tensor, dimensional of which dark AdSgraphic can energy screen [ and/or is be embedded dark identified into matter. as onethe In higher holographic dimensional our hydrodynamics flat in construction, spacetime, Rindler the spacetime which holo- [ is inspired by It is still quite interestingno to evidence ask from whether experiments we so canphysics far. probe in One extra [ recent dimensions, study although maymeasurements there come of is from gravity. gravitational waves ity in the bulk is encoded into the field theory on the boundary. For the models which extra breezing mode ismassive Kaluza-Klein constrained gravitational by modes the associatedthose with current massive the experiments. extra modes dimensions. decay There fastare Although are and constrained also may as multiple not well reachhole by signals. the the gravitational Our gravitational waves model waves detector, does signal they not templates necessarily from have observable the effects binary from extra black dimensions. we expect one extra breezing mode on top of two polarized propagating modes [ JHEP10(2018)009 D , Phys. (2017) , 11 J. Phys. ]. , Astron. Phys. Rev. , , ]. (2011) 029 SPIRE JHEP , IN (2017) 016 [ 04 2 SPIRE IN ][ JHEP (2012) 10 , ]. 15 ]. SciPost Phys. SPIRE , IN [ astro-ph/0311348 Modified Gravity and Cosmology , SPIRE gr-qc/9209012 ]. [ IN ]. ][ ]. SPIRE Living Rev. Rel. 4-D gravity on a brane in 5-D Minkowski space , (1983) 365 IN SPIRE ][ – 22 – SPIRE Dark matter direct-detection experiments IN IN 270 (1993) 1407 ][ ][ Friedmann Equations from Entropic Force ]. ]. D 47 Quasilocal energy and conserved charges derived from the hep-th/0005016 ]. ), which permits any use, distribution and reproduction in Modified Newtonian Dynamics (MOND): Observational [ ]. ]. Inconsistencies in Verlinde’s emergent gravity SPIRE A New method of determining distances to galaxies SPIRE IN IN Astrophys. J. ][ SPIRE , SPIRE SPIRE IN arXiv:1509.08767 Phys. Rev. IN IN arXiv:1106.2476 [ ][ , (2000) 208 [ arXiv:1001.3470 ][ ][ CC-BY 4.0 [ (1977) 661 [ Emergent Gravity and the Dark Universe On the Origin of Gravity and the Laws of Newton This article is distributed under the terms of the Creative Commons A Modification of the Newtonian dynamics as a possible alternative to the B 485 The Visible matter-Dark matter coupling 54 (2012) 1 (2016) 013001 513 arXiv:1710.00946 [ (2010) 061501 arXiv:1001.0785 arXiv:1611.02269 arXiv:1112.3960 007 gravitational action Phys. Lett. [ 81 Astrophys. [ hidden mass hypothesis G 43 [ Rept. Phenomenology and Relativistic Extensions J.D. Brown and J.W. York Jr., G.R. Dvali, G. Gabadadze and M. Porrati, D.-C. Dai and D. Stojkovic, E.P. Verlinde, R.-G. Cai, L.-M. Cao and N. Ohta, R.B. Tully and J.R. Fisher, R. Sancisi, E.P. Verlinde, M. Milgrom, T. MarrodánUndagoitia and L. Rauch, T. Clifton, P.G. Ferreira, A. Padilla and C. Skordis, B. Famaey and S. McGaugh, [8] [9] [5] [6] [7] [3] [4] [1] [2] [11] [12] [10] Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. References of Frontier SciencesY.L. of Zhang CAS. was S.funded supported Sun by by was the Young supported Ministryand Scientist Pohang by of Training City. MOST Science, Program and ICT in & NCTS APCTP, which Future in Planning(MSIP), is Taiwan. Gyeongsangbuk-do We thank Y.F. Cai,S.J. S.P. Wang, Kim, M. Yamazaki B.H. forvery Lee, helpful helpful T. comments conversations, and Liu, as suggestions.Science well C. Foundation R.G. as Park, of Cai the M. China was anonymousStrategic supported (No. Sasaki, referees Priority by 11690022, J. for Research the No. Program Soda, National 11375247, of Natural H. No. CAS Tye, (No. 11435006, XDB23030100), No.11647601), Key Research Program Acknowledgments JHEP10(2018)009 , , ] D D ]. 109 D 90 ] (2001) 3669 SPIRE ]. IN Phys. Rev. ]. Phys. Rev. , ][ , ]. (2001) 2099 B 32 (2001) 199 (2010) 5 Phys. Rev. SPIRE 40 arXiv:0902.0427 , SPIRE (2018) 043526 IN 13 [ IN SPIRE ][ Phys. Rev. Lett. ][ IN B 502 , Black hole elasticity and ][ arXiv:1708.08477 D 98 [ ]. hep-th/0105320 ]. ]. Solid Holography and Massive [ World-Volume Effective Theory for Essentials of Blackfold Dynamics New Horizons for Black Holes and (2009) 191301 Acta Phys. Polon. Phys. Lett. SPIRE , , Living Rev. Rel. Phys. Rev. SPIRE SPIRE IN (2018) 129 , , 102 IN ]. IN ][ Int. J. Theor. Phys. arXiv:1502.01589 01 , ][ ][ (2001) R1 arXiv:1601.03384 [ ]. [ ]. astro-ph/0105068 18 Accelerated universe from gravity leaking to extra SPIRE [ ]. ]. Dark Energy Survey year 1 results: Cosmological IN Planck 2015 results. XIII. Cosmological Nonconventional cosmology from a brane universe JHEP – 23 – SPIRE , Viscosity bound violation in holographic solids and ][ SPIRE IN IN SPIRE SPIRE ][ (2016) A13 Black Branes as Piezoelectrics (2016) 074 ][ IN IN Phys. Rev. Lett. ][ ][ , 07 594 hep-th/9905012 arXiv:1510.09089 arXiv:0912.2352 Brane-World Gravity [ (2002) 044023 ]. [ [ ]. Black Holes and Biophysical (Mem)-branes ]. ]. Relativistic Elasticity of Stationary Fluid Branes ]. JHEP D 65 , SPIRE Class. Quant. Grav. SPIRE arXiv:0910.1601 , IN SPIRE SPIRE [ IN SPIRE ][ (2000) 269 IN IN (2016) 114 hep-th/0208169 arXiv:1210.5197 (2010) 046 ][ IN ][ ][ [ [ ][ 02 arXiv:1209.2127 arXiv:1402.6330 04 [ [ Cosmology on a brane in Minkowski bulk Phys. Rev. B 565 Astron. Astrophys. , Essentials of classical brane dynamics , (2010) 063 collaboration, P.A.R. Ade et al., Brane worlds Standard cosmology in the DGP brane model ]. ]. JHEP JHEP Global structure of Deffayet (Dvali-Gabadadze-Porrati) cosmologies , 03 , collaboration, T.M.C. Abbott et al., (2003) 064004 (2013) 044058 SPIRE SPIRE hep-th/0110162 arXiv:1708.01530 arXiv:1004.3962 IN IN hep-th/0010186 gr-qc/0012036 [ 67 [ [ Nucl. Phys. Planck parameters DES constraints from galaxy clustering and weak lensing the viscoelastic response gapped transverse phonons in holography [ 87 (2014) 124022 Gravity Branes (2012) 241101 Higher-Dimensional Black Holes [ JHEP [ dimensions [ R. Dick, R. Dick, A. Lue, R. Maartens and K. Koyama, P. Binetruy, C. Deffayet and D. Langlois, L. Alberte, M. Baggioli and O. Pujolàs, L. Alberte, M. Ammon, M. Baggioli, A. and Jiménez O. Pujolàs, J. Armas and T. Harmark, L. Alberte, M. Baggioli, A. Khmelnitsky and O. Pujolàs, R. Emparan, T. Harmark, V. Niarchos and N.A. Obers, J. Armas, J. Gath and N.A. Obers, J. Armas and N.A. Obers, R. Emparan, T. Harmark, V. Niarchos and N.A. Obers, R. Emparan, T. Harmark, V. Niarchos and N.A. Obers, C. Deffayet, G.R. Dvali and G. Gabadadze, B. Carter, C. Deffayet, [29] [30] [31] [27] [28] [25] [26] [23] [24] [21] [22] [18] [19] [20] [16] [17] [14] [15] [13] JHEP10(2018)009 ] , ]. , ]. Phys. JHEP ]. , , Phys. (1999) (2006) 1 Phys. (2002) SPIRE , D 75 , (1998) SPIRE IN 83 AIP Conf. IN , ][ 423 SPIRE (2016) 106013 ][ IN D 65 D 58 ][ arXiv:1105.0038 [ D 94 Phys. Rev. , Phys. Rept. ]. Phys. Rev. , , Phys. Rev. Phys. Rev. Lett. , , gr-qc/0312059 (2011) 053 arXiv:0905.1112 [ Phys. Rev. SPIRE (2008) 103509 [ Challenges to the DGP Model , IN hep-ph/9803315 07 ]. ]. [ ][ D 78 The dS/dS correspondence ]. (2004) 7 ]. Wilsonian Approach to Fluid/Gravity From Navier-Stokes To Einstein SPIRE SPIRE ]. Holographic models of de Sitter QFTs JHEP 7 ]. IN The Hierarchy problem and new Phenomenology, astrophysics and , IN [ ]. ][ SPIRE SPIRE (1998) 263 (2009) 063536 Cosmological Constraints on DGP Braneworld SPIRE IN IN SPIRE ]. IN ][ ]. ][ IN Phys. Rev. SPIRE The Einstein equation on the 3-brane world ][ , ][ IN – 24 – D 80 Holographic self-tuning of the cosmological constant Holographic Preheating: Quasi-Normal Modes and B 429 arXiv:1007.3996 ][ SPIRE [ SPIRE IN IN ]. Large-Scale Tests of the DGP Model Living Rev. Rel. ][ ]. , ][ ]. arXiv:1006.1902 AdS/dS CFT Correspondence arXiv:1612.04394 ]. ]. [ A Large mass hierarchy from a small extra dimension An Alternative to compactification Phys. Rev. SPIRE ]. , Phys. Lett. , SPIRE An Action for black hole membranes IN hep-ph/9905221 , hep-ph/9807344 gr-qc/9910076 SPIRE IN [ [ [ ][ IN (2011) 105015 ][ SPIRE SPIRE ][ hep-th/0407125 SPIRE arXiv:1704.05075 IN IN [ [ 28 IN Holographic de Sitter Universe ][ ][ (2011) 141 ][ 03 astro-ph/0606286 arXiv:1101.2451 [ [ (1999) 3370 Brane world gravity (1999) 086004 (2000) 024012 Gauge/gravity correspondence in accelerating universe (2005) 393 83 (2017) 031 ]. JHEP gr-qc/9712077 hep-th/0203041 The phenomenology of dvali-gabadadze-porrati cosmologies , 09 [ [ hep-th/9906064 743 D 59 D 62 [ (2012) 146 SPIRE arXiv:1604.05452 IN arXiv:0808.2208 astro-ph/0510068 [ Holographic Renormalization JHEP Proc. [ 064011 Class. Quant. Grav. 125015 Duality 07 Rev. Lett. 4690 Rev. dimensions at a millimeter cosmology of theories with submillimeterRev. dimensions and TeV scale quantum gravity [ Gravity with Brane Tension [ (2007) 064003 from Horizon-Scale Growth and Geometry Y.-F. Cai, S. Lin, J. Liu and J.-R. Sun, C. Charmousis, E. Kiritsis and F. Nitti, M. Alishahiha, A. Karch, E. Silverstein and D. Tong, M. Li and Y. Pang, C.-S. Chu and D. Giataganas, D. Marolf, M. Rangamani and M. Van Raamsdonk, A. Buchel, I. Bredberg, C. Keeler, V. Lysov and A. Strominger, I. Bredberg, C. Keeler, V. Lysov and A. Strominger, M. Parikh and F. Wilczek, L. Randall and R. Sundrum, T. Shiromizu, K.-i. Maeda and M. Sasaki, N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, L. Randall and R. Sundrum, L. Lombriser, W. Hu, W. Fang and U. Seljak, R. Maartens, N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, Y.-S. Song, I. Sawicki and W. Hu, W. Fang, S. Wang, W. Hu, Z. Haiman, L. Hui and M. May, A. Lue, [50] [51] [47] [48] [49] [45] [46] [42] [43] [44] [40] [41] [38] [39] [35] [36] [37] [33] [34] [32] JHEP10(2018)009 , ]. (2017) ] ]. Ann. (2017) SPIRE Phys. , ]. 11 , IN (2001) 305 ][ SPIRE D 95 IN (2016) 095 SPIRE JHEP ][ (2002) 007 IN , (2016) 106011 B 507 05 ]. ][ 03 ]. (2018) 300 hep-th/9912001 ]. The DBI Action, [ D 94 SPIRE Phys. Rev. JHEP , IN SPIRE , JHEP [ , IN B 780 (2018) 307 Phys. Lett. arXiv:1112.0302 SPIRE ][ , [ ]. IN ]. (2017) 031102 ][ arXiv:0809.3505 Phys. Rev. hep-th/0511186 [ B 928 , [ 119 (2001) 084017 ]. SPIRE SPIRE IN IN Phys. Lett. , ][ ][ Cosmology from an AdS Schwarzschild black Holographic principle in the BDL brane D 63 SPIRE (2012) 181807 IN arXiv:1702.08590 hep-th/0104159 ][ , [ Nucl. Phys. Modified Gravity (MOG), the speed of gravitational (2009) 151301 108 , – 25 – (2006) 044005 Entropy Production, Hydrodynamics and Resurgence arXiv:0803.0982 Dark Energy and the Accelerating Universe Emergent Gravity from Vanishing Energy-Momentum [ ]. Phys. Rev. Lett. 102 Phys. Rev. , Brane world and holography , Holographic Dual of de Sitter Universe with AdS Bubbles D 73 ]. arXiv:1107.1491 SPIRE arXiv:1610.08521 The Spectrum of gravitational waves in Randall-Sundrum [ CFT and entropy on the brane ]. ]. ]. ]. ]. IN [ (2001) 064022 Inconsistencies in Verlinde’s emergent gravity (2008) 385 SPIRE ]. ]. Constraining Relativistic Generalizations of Modified Newtonian ][ hep-th/0205188 IN Spacetime as a topological insulator: Mechanism for the origin of [ Phys. Rev. Lett. 46 ][ SPIRE SPIRE SPIRE SPIRE SPIRE , Brane world effective action at low-energies and AdS/CFT Phys. Rev. D 64 IN IN IN IN IN , SPIRE SPIRE Phys. Rev. Lett. ][ ][ ][ ][ ][ (2012) 361 IN IN , (2017) 134 ][ ][ Covariant version of Verlinde’s emergent gravity 03 AdS/CFT and gravity B 855 (2002) 043526 Verlinde Gravity and AdS/CFT Ringing in de Sitter spacetime Phys. Rev. , ]. JHEP arXiv:1703.01415 , [ D 66 arXiv:1710.00946 [ SPIRE arXiv:1602.00699 arXiv:1704.05116 arXiv:1710.11177 arXiv:1707.01030 arXiv:1603.05344 hep-th/0105256 IN hep-th/0102042 124018 007 braneworld cosmology Higher-derivative Supergravity and Flattening Inflaton[ Potentials Tensor [ radiation and the event[ GW170817/GRB170817A Rev. Astron. Astrophys. the fermion generations Dynamics with Gravitational Waves [ in the Primordial quark-gluon[ Plasma from Holography hole via holography Nucl. Phys. [ Rev. [ [ cosmology D.-C. Dai and D. Stojkovic, T. Kobayashi and T. Tanaka, C.D. Carone, J. Erlich and D. Vaman, S. Hossenfelder, M.A. Green, J.W. Moffat and V.T. Toth, S. Bielleman, L.E.Pedro, I. Ibáñez,F.G. Valenzuela and C. Wieck, D.B. Kaplan and S. Sun, P.M. Chesler and A. Loeb, A. Buchel, M.P. Heller and J. Noronha, J. Frieman, M. Turner and D. Huterer, S. Kanno, M. Sasaki and J. Soda, A. Buchel, A. Buchel, T. Shiromizu, T. Torii and D. Ida, S. Kanno and J. Soda, P.S. Apostolopoulos, G. Siopsis and N. Tetradis, I. Savonije and E.P. Verlinde, N.J. Kim, H.W. Lee, Y.S. Myung and G. Kang, S.S. Gubser, [69] [70] [67] [68] [65] [66] [63] [64] [61] [62] [58] [59] [60] [55] [56] [57] [53] [54] [52] JHEP10(2018)009 , , ] (2013) ] (2012) 116 02 ]. (1993) 284 05 (2010) 021301 Phys. Rev. Lett. ]. (2017) 111102 ]. , JHEP , SPIRE ]. JHEP IN hep-th/9409089 119 D 81 , [ SPIRE arXiv:1104.5502 ][ SPIRE IN , C 930308 hep-th/0106113 IN ]. [ ][ arXiv:1612.09513 SPIRE ][ , IN [ ]. ]. ]. SPIRE ]. Effective Fluid Description of Phys. Rev. IN (1995) 6377 , ][ SPIRE The Holographic fluid dual to The relativistic fluid dual to vacuum (2001) 034 Conf. Proc. IN SPIRE 36 SPIRE SPIRE Phys. Rev. Lett. , How the Self-Interacting Dark Matter IN IN ][ , IN 10 Rindler Fluid with Weak Momentum ]. ][ ][ ][ arXiv:1707.09945 [ arXiv:1307.1132 ]. arXiv:1103.3022 [ [ JHEP SPIRE Holographic EPR Pairs, Wormholes and Petrov type-I Condition and Dual Fluid , arXiv:1612.09582 IN , SPIRE ][ – 26 – IN J. Math. Phys. arXiv:1201.2678 (2018) 242 , Holographic Bell Inequality Petrov type-I Spacetime and Dual Relativistic Fluids [ ][ (2011) 050 Signatures of extra dimensions in gravitational waves arXiv:1705.05078 Entanglement entropy in de Sitter space arXiv:1302.2016 arXiv:1308.3695 (2013) 211602 ]. arXiv:1401.7792 [ 07 Holography for Cosmology [ [ Perspective on MOND emergence from Verlinde’s “emergent [ B 776 The Relativistic Rindler Hydrodynamics ]. From Petrov-Einstein to Navier-Stokes ]. ]. 111 (2012) 076 SPIRE ]. JHEP Holographic Dual of an Einstein-Podolsky-Rosen Pair has a IN , 03 SPIRE ][ SPIRE SPIRE arXiv:1704.07392 (2018) 058 IN (2013) 118 (2013) 211 [ IN IN SPIRE arXiv:1307.6850 ][ Phys. Lett. 01 [ ][ ][ IN , 04 10 (2014) 041901 JHEP ][ , The dS/CFT correspondence The World as a hologram Dimensional reduction in quantum gravity JHEP Phys. Rev. Lett. D 90 JHEP JHEP Holographic Schwinger Effect and the Geometry of Entanglement , (2017) 048 , , , ]. ]. ]. ]. 06 arXiv:1210.7244 (2013) 211603 [ SPIRE SPIRE SPIRE SPIRE arXiv:1611.02716 IN IN arXiv:1201.2705 IN arXiv:0907.5542 gr-qc/9310026 IN Radiation Wormhole 111 the Dark Universe Model Explains the Diverse[ Galactic Rotation Curves [ gravity” and its recent test by weak lensing Phys. Rev. Relaxation 038 [ Dynamics vacuum Einstein gravity Einstein gravity [ [ [ JCAP [ [ M. Cadoni, R. Casadio, A. Giusti, W. and Mück M. Tuveri, A. Kamada, M. Kaplinghat, A.B. Pace and H.-B. Yu, J.-W. Chen, S. Sun and Y.-L. Zhang, M. Milgrom and R.H. Sanders, K. Jensen and A. Karch, J. Sonner, M. Chernicoff, A. Güijosaand J.F. Pedraza, S. Khimphun, B.-H. Lee, C. Park and Y.-L. Zhang, J. Maldacena and G.L. Pimentel, V. Lysov and A. Strominger, R.-G. Cai, L. Li, Q. Yang and Y.-L. Zhang, R.-G. Cai, Q. Yang and Y.-L. Zhang, G. Compere, P. McFadden, K. Skenderis and M. Taylor, C. Eling, A. Meyer and Y. Oz, A. Strominger, P. McFadden and K. Skenderis, G. Compere, P. McFadden, K. Skenderis and M. Taylor, G. ’t Hooft, L. Susskind, D. Andriot and G. Lucena Gómez, [89] [90] [87] [88] [84] [85] [86] [82] [83] [79] [80] [81] [77] [78] [74] [75] [76] [72] [73] [71] JHEP10(2018)009 ]. 42 ]. D 93 SPIRE SPIRE IN Phys. Lett. IN ][ , ][ Phys. Rev. Gen. Rel. Grav. , , ]. SPIRE hep-th/0501055 (2013) 115007 arXiv:1708.06811 [ IN [ ][ D 87 (2005) 050 (2018) 047 ]. 02 02 ]. Phys. Rev. gr-qc/0611071 SPIRE , ]. [ – 27 – IN JHEP ][ JCAP SPIRE , , IN Beyond Collisionless Dark Matter: Particle Physics SPIRE ][ IN ][ Dark Matter Superfluidity and Galactic Dynamics ]. (2007) 064008 Dark matter superfluid and DBI dark energy Unified first law and thermodynamics of apparent horizon in FRW First law of thermodynamics and Friedmann equations of Quantum computational complexity, Einstein’s equations and SPIRE Building up spacetime with quantum entanglement D 75 IN arXiv:1506.07877 ][ [ arXiv:1511.00627 [ arXiv:1005.3035 [ Phys. Rev. , (2016) 639 arXiv:1302.3898 Friedmann-Robertson-Walker universe universe (2010) 2323 accelerated expansion of the Universe Dynamics for Dark Matter[ Halo Structure B 753 (2016) 023515 R.-G. Cai and S.P. Kim, R.-G. Cai and L.-M. Cao, M. Van Raamsdonk, X.-H. Ge and B. Wang, L. Berezhiani and J. Khoury, R.-G. Cai and S.-J. Wang, S. Tulin, H.-B. Yu and K.M. Zurek, [96] [97] [94] [95] [92] [93] [91]