Stars and Solar System constraints in extended gravity theories

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1 Stas and Sola System constaints in extended gavity theoies Daniel Sunhede Univ of Jyväskylä / Helsinki Institute of Physics K Kainulainen & D Sunhede in pepaation

2 Intoduction 6D bane-wold with lage exta dimensions µm, tue scale of gavity M b 1 TeV Hieachy poblem Albecht et al Phys Rev D65 (22) Bugess et al (SLED) Nucl Phys B76 (25) 71 Pesence of enomalizable bulk scala fields induce a Casimi potential fo the adion as well as logaithmic coections to both kinetic tems and the potential Effective 4D action, ϕ log M b : S LED = d 4 x g [ M 2 b e2ϕ 2 (AR g +2B( ϕ) 2 ) CM 4 bu e 4ϕ + L m (g µν, ψ m ) A(ϕ) = 1 + aϕ, B(ϕ) = 1 + bϕ, C(ϕ) = 1 + cϕ, U 1, a, b, c 1 Induced potential is of a fom suitable fo addessing the Dak Enegy poblem. No small mass scale needed! ]

3 Intoduction In geneal somewhat difficult to stabilize exta dimensions Howeve if A(ϕ) = 1 + aϕ 1 Einstein fame potential become vey steep and easily Sunhede et al Phys Rev D73 (26) 8351 confines the adius: V (χ) U(φ) A 2 (ϕ) Second ode tems emain small if coections come not fom one, but seveal weakly coupled bulk scala fields V (χ) χ log V (χ) Cosmological evolution OK, close to ΛCDM

4 Sola System constaints Scala field coupling to matte must emain small: χ = V (χ) α(χ) T α Almost automatically povided by the new adius stabilization mechanism whee A(ϕ) 1, since α (2A + a) γ PPN =1 2α2 1+α 2 Cassini time delay Betotti et al Natue 425 (23) 374 Sunhede et al Phys Rev D73 (26) 8351 Nevetheless some fine-tuning needed... I. Could evolution of the field in the Sola System spoil the above pediction?

5 Sola System constaints Sola System constaints / Post-Newtonian paamete ds 2 g µν x µ x ν = e A() dt 2 + e B() d dω 2 ds 2 = (1 2GM )dt 2 + (1 + 2γ PPNGM )d dω 2 γ PPN 1 B A 1 = (2.1 ± 2.3) 1 5 Cassini time delay Betotti et al Natue 425 (23) 374 In GR exteio vacuum solution is always Schwazschild and egadless of inteio γ PPN =1 Extended gavity lack of unique vacuum solution equies that we compute the full metic to obtain γ PPN

6 gavity Pevious wok: Sola System constaints in metic f(r) gavity obtained by deiving and solving genealized Tolman-Oppenheime- 1.5 Volkoff equations γ PPN = 1 2 γ PPN f(r) γppn 1.5 GR f(r) Sunhede et al Phys Rev D76 (27) 242 axiv: (accepted in PRD) / Metic f(r) gavity is equivalent to an ω = JBD theoy, indeed giving γ PPN =1/2 in the massless limit LED scenaio at tee level coesponds to an ω = 1/2 JBD theoy giving γ PPN =1/3. II. Is passing Sola System tests eally possible? Diffeence to metic f(r)?

7 The scala-tenso TOV:s Scala-tenso gavity Fo a static, spheically symmetic metic we obtain Togethe with the consevation law and a given equation of state, these fom a complete genealization of the Tolman Oppenheime-Volkov equations S = L G = 1 2κ [F (ϕ)r Z(ϕ)g µν µ ϕ ν ϕ U(ϕ)] A = 1 1+γ B = 1 eb + 2F ( 1 e B eb F κ p + eb 2F U 2F Zϕ 2 + 4γ ( + eb F κ ρ + 1 ) 2ϖ F,ϕT 2 + eb 2F (Z +2F,ϕϕ 1 ) ϖ F,ϕϖ,ϕ ϕ 2 γa d 4 x g [L G (g µν, ϕ)+l m (g µν, ψ m )] ( 1 2 ϖ F,ϕ )U 2 + eb 2ϖ F,ϕU,ϕ p = A 2 κ = M 2 b 2ϖ ϕ = F,ϕ (κ T 2U)+FU,ϕ ϖ,ϕ ( ϕ) 2 2ϖ 3F 2,ϕ +2FZ ) (ρ + p), p = p(ρ)

8 The LED model Repoducing obseved stength of gavity: M 2 be 2ϕ M 2 Pl U κ ρ M be 4ϕ κ ρ Intoduce: 1 31 dφ dϕ 2ϖ F,ϕ Potential tems negligible eqn in weak field, Newtonian limit simplifies to ϕ ( 2 Φ ) = κ ρ 2 ( F,ϕϕ Φ() = 2ϖ F ),ϕϖ,ϕ (2ϖ) 2 (Φ ) 2 d 2G m( ) 2 + Φ m() d 4π 2 ρ Field stays vey close to cosmol backgound value Φ() Φ Φ() Φ Φ 1 35 ϕ() ϕ 1

9 The LED model Weak field, Newtonian limit, dopping negligible non-linea tems (B) ( 1 α 2) κ A B F ρ2 2α2 α 2 = F,ϕ 2ϖ Tee level, B G NM A G NM B 2G NM A 2G NM ω = 1/2 JBD α 2 =1/2 Φ() F κ /F κ /F =8πG N (1 + 2)G NM α small via adius stabilization, fo γ PPN 1 F Φ γ PPN 1 3 α ω(ϕ) 1 3

10 Summay Evolution of scala field in Sola System negligible compaed to cosmological backgound value α emains small and γ PPN =1 Field emains light in the LED model, Sola System tests fulfilled since fo α 1 the theoy no longe coesponds to an ω 1/2 JBD theoy. Howeve, metic f(r) gavity is invaiably ω = Successes of the 6D bane-wold (S)LED scenaio: 1. Addesses the hieachy poblem II. Induced potential natually yields obseved amount of DE III. Valid cosmological evolution close to ΛCDM IV. Radius stabilization mechanism also povides consistency with Sola System constaints Kainulainen & Sunhede in pepaation [see also Phys Rev D73 (26) 8351]

11 1.5 Obsevational w Impint log 1 R Tension between minima of α(χ) and V (χ) m Ω tot = 1. Ω Λ =.8 Sunhede et al Phys Rev D73 (26) V (χ) Ω Λ = Simulated SNAP data: Phys Rev Lett 86 (21) 1939 Ω Λ = z