! ! ! Shinji ! !Tsujikawa ! ! ! ! ! ! !(Tokyo ! !University ! - PowerPoint PPT Presentation

� ! ! 2-nd APCTP-TUS workshop, August, 2015 ! ! Possibility of realizing weak gravity in red-shift space distortions ! ! ! Shinji ! !Tsujikawa ! ! ! ! ! ! !(Tokyo ! !University ! !of ! !Science) ! !

The problem of dark energy and dark matter is an interesting intersection between astronomy and physics! !

Planck constraints on the effective gravitational coupling and the gravitational slip parameter (Ade et al, 2015) DE-related Planck 1.0 Planck +BSH today ! Planck +WL Planck +BAO/RSD 0.5 Planck +WL+BAO/RSD Strong µ 0 − 1 gravity ! 0.0 Weak GR ! gravity ! − 0.5 − 1.0 − 1 0 1 2 3 η 0 − 1 − Φ / Ψ − 1 today !

Weak gravity ! The recent observations of redshift-space distortions (RSD) measured the lower growth rate of matter perturbations lower than that predicted by the LCDM model. ! Macaulay et al, PRL (2014) ! 0.55 Growth Rate, f(z) σ 8 (z) Planck LCDM fit ! 0.5 Tension between 0.45 Planck and RSD data ! 0.4 RSD fit ! 0.35 Planck Λ CDM RSD fit 6dFGS LRG BOSS 0.3 WiggleZ VIPERS 0 0.2 0.4 0.6 0.8 1 Redshift, z

One possibility for reconciling the discrepancy: Massive neutrinos ! Battye and Moss (2013) ! 0.84 75 0.80 H 0 [km s − 1 Mpc − 1 ] 70 0.76 σ 8 65 0.72 60 0.68 0.64 55 0.60 0.0 0.4 0.8 1.2 1.6 Σ m ν [eV] Increasing the neutrino mass m ν leads to the lower values of σ 8 , but it also decreases H 0 . Tension with the direct measurement of H 0

Another possibility: Interacting dark matter/vacuum energy ! Salvatelli et al There is an energy transfer between CDM and vacuum: ! (2014) ! ˙ ρ c + 3 H ρ c = − Q ˙ V = Q The coupling Q is usually taken in an ad-hoc way (like Q = − qHV ). 0.65 Λ CDM Bestfit q 34 0.6 6dFGRS [28] 2dFGRS [29] WiggleZ [30] 0.55 SDSS LRG [31] BOSS CMASS [32] VIPERS [33] 0.5 f σ 8 Lower growth rate 0.45 than in LCDM for 0.4 Q > 0. 0.35 0.3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 z However there is no concrete Lagrangian explaining the origin of such a coupling. It is likely that the low growth rate is associated with the appearance of ghosts. !

Another possibility: Modified gravity with a concrete Lagrangian ! In this case we can explicitly derive conditions for the absence of ghosts and instabilities. ! The question is ! Is it possible to realize the cosmic growth rate lower than that in LCDM in modified gravity models, while satisfying conditions for the absence of ghosts and instabilities? ! ! In doing so, we begin with most general second-order scalar-tensor theories with single scalar degree of freedom (Horndeski theories).

Horndeski theories ! ! Horndeski (1973) Deffayet et al (2011) Most general scalar-tensor theories Charmousis et al (2011) with second-order equations ! Kobayashi et al (2011) ! ! The Lagrangian of Horndeski theories is constructed to keep the equations of motion ! up to second order, such that the theories are free from the Ostrogradski instability. !

Cosmological perturbations in Horndeski theories ! ! The scalar degree of freedom in Horndeski theories can give rise to ! l the late-time cosmic acceleration at the background level ! l interactions with the matter sector (CDM, baryons) ! We take into account non-relativistic matter with the energy density ! ____ ! _______ ! Perturbations ! Background ! The perturbed line element in the longitudinal gauge is ! ds 2 = − (1 + 2 Ψ ) dt 2 + a 2 ( t )(1 + 2 Φ ) δ ij dx i dx j The four velocity of non-relativistic matter is !

Matter perturbations in the Horndeski theories ! ! ˙ δ is related with v . In RSD observations, the growth rate of matter perturbations is constrained from peculiar velocities of galaxies. ! obeys ! The gauge-invariant density contrast ! δ m + k 2 � � δ m + 2 H ˙ ¨ I + 2 H ˙ ¨ where ! a 2 Ψ = 3 I ___ ! Ψ is related with δ m through the modified Poisson equation: k 2 G e ff is the e ff ective gravitational a 2 Ψ � � 4 π G e ff ρ m δ m coupling with matter.

Effective gravitational coupling in Horndeski theories ! ! De Felice, Kobayashi, For the modes deep inside the Hubble radius ( k � aH ) we can employ S.T. (2011). ! ! the quasi-static approximation under which the dominant terms are those including k 2 /a 2 , δ m , and M 2 � � K , φφ . It then follows that Schematically ! G e ff = a 0 ( k/a ) 2 + a 1 7 ) ( k/a ) 2 − B 6 M 2 ] 2 M 2 pl [( B 6 D 9 − B 2 G e ff = 8 M 2 G b 0 ( k/a ) 2 + b 1 8 D 9 − 2 A 6 B 7 B 8 ) ( k/a ) 2 − B 2 ( A 2 6 B 6 + B 2 M corresponds to the mass of a scalar degree of freedom and A 6 = − 2 XG 3 ,X − 4 H ( G 4 ,X + 2 XG 4 , XX ) ˙ φ + 2 G 4 , φ + 4 XG 4 , φ X +4 H ( G 5 , φ + XG 5 , φ X ) ˙ φ − 2 H 2 X (3 G 5 ,X + 2 XG 5 , XX ) D 9 = − K ,X + derivative terms of G 3 , G 4 , G 5 In GR, G 4 = M 2 pl / 2, B 6 = B 8 = 2 M 2 G e ff = G pl , A 6 = B 7 = 0, D 9 = − K ,X In the massive limit ( M 2 � � ) with B 6 � B 8 � 2 M 2 pl we also have G e ff � G In the massless limit M 2 → 0 we have 2 M 2 pl ( B 6 D 9 − B 2 7 ) The effect of modified gravity G e ff = G manifests itself. ! A 2 6 B 6 + B 2 8 D 9 − 2 A 6 B 7 B 8

Conditions for the absence of ghosts and instabilities ! The second-order action for tensor perturbations γ ij is where ! We require q t > 0 and c 2 t > 0 to avoid ghosts and Laplacian instabilities. In GR we have q t = M 2 pl / 8 and c 2 t = 1. For scalar perturbations we also have corresponding quantities q s and c 2 s which must be positive.

ST (2015) ! Simple form of the effective gravitational coupling ! In the massless limit, the effective gravitational coupling in Horndeski theories reads ! Q and α W are functions of G i and their derivatives. ____ ! _______ ! Tensor Scalar contribution ! contribution ! This correspond to the intrinsic Always positive under the no-ghost modification of the gravitational part. ! and no-instability conditions: ! The necessary condition to realize weaker gravity than that in GR is ! This is not a su ffi cient condition for realizing G e ff < G . The scalar-matter interaction always enhances the effective gravitational coupling.

Examples ! Hu and Sawicki, ( R/R c ) 2 n (i) f(R) gravity: ! f ( R ) = R − µR c Starobinsky, ST. ! ( R/R c ) 2 n + 1 ✓ ◆ G e ff = G 1 + 1 f ,R 3 Typically, f ,R varies from 1 (matter era) to the value like 0.9 (today), so the scalar-matter interaction leads to G e ff > G . Deffayet et al. ! (ii) Covariant Galileons ! pl / 2 + c 4 X 2 , G 4 = M 2 G 5 = c 5 X 2 G 2 = c 2 X , G 3 = c 3 X , For late-time tracking solutions, it is possible to realize M 2 pl c 2 t / (8 q t ) < 1 ! due to the decrease of c 2 t ( < 1), but the scalar-matter interaction overwhelms this decrease. G e ff > G typically. De Felice, Kase, ST (2011) !

Two crucial quantities for the realization of weak gravity ! To recover the GR behavior in regions of the high density, the dominant contribution to the Horndeski Lagrangian is the M 2 pl / 2 term in G 4 . q t ' M 2 pl / 8 and c 2 t ' 1 during most of the matter era. The large variations of q 2 t and c 2 t from the end of the matter era M 2 pl c 2 t to today are required to satisfy the condition < 1. 8 q t If we go beyond the Horndeski domain, it is possible to realize c 2 t < 1 even in the deep matter era.

Horndeski Lagrangian in the ADM Language ! ! In the ADM formalism, we can construct a number of geometrical scalars: ! S ≡ K µ ν K µ ν , Z ≡ R µ ν R µ ν , U ≡ R µ ν K µ ν . K ≡ K µ R ≡ R µ µ , µ , where K µ ν and R µ ν are extrinsic and intrinsic curvatures, respectively. In the unitary gauge ( δφ = 0), the Horndeski Lagrangian on the FLRW background is equivalent to Gleyzes et al (2013) ! (Horndeski conditions) ! What happens if we do not impose these two conditions ? Gleyzes-Langlois-Piazza-Vernizzi (GLPV) theories (PRL, 2014)

A simple dark energy model in GLPV theories ! De Felice, Koyama, and ST(2015) ! ! where ! How about observational signatures in this model ? !

Model of constant tensor propagation speed ! For constant F ( φ ), c 2 t = constant. If c 2 t deviates from 1, this leads to the growth of c 2 s as we go back to the past. The c 2 s can remain constant for the scaling dark energy model: Provided that the oscillating mode of scalar perturbations is initially suppressed, the e ff ective gravitational coupling G e ff and the anisotropy parameter η = − Φ / Ψ are given by = 1 + 1 − c 2 G e ff during the scaling t G c 2 matter era s In the sub-luminal regime ( c 2 t < 1), the Laplacian instability associated with negative c 2 s can be avoided. G e ff > G (strong gravity), but the deviation from G is not large. η > 1 for c 2 t away from 1

The anisotropy parameter ! De Felice, Koyama and ST (2015) ! Planck constraints (2015) ! #! $ DE-related Planck $ %*%!"# ( ) 1.0 Planck +BSH Planck +WL #! Planck +BAO/RSD 0.5 Planck +WL+BAO/RSD $ %*%!"+ ( ) µ 0 − 1 � 0.0 #"! $ %*%!", ( ) GR ! − 0.5 − 1.0 !"#! − 1 0 1 2 3 !"# # #! #!! η 0 − 1 #%&%' It is possible to realize η > 1 in this model.