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Stringent Constraints on Brans-Dicke Parameter using DECIGO Kent Yagi (Kyoto-U.) LISA/DECIGO Workshop @ Sagamihara Nov. 2008 Collaborators: T.Tanaka(YITP) N.Seto (Kyoto-U.)

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§1. Introduction ・ The aim of my work: Verification of G.R. in strong field region by detecting G.W.s from NS/BH binaries using DECIGO. ・ There exist many modified theories of gravity. e.g. scalar-tensor, vector-tensor, f(R) gravity, massive graviton ・ scalar-tensor theory gravity = metric + scalar

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(i) KK theory （ dilaton ） (ii) superstring theory(dilaton) (iii) braneworld (radion)…Garriga&Tanaka(2000) (iv) inflation (inflaton)…Steinhardt&Accetta(1990) (v) dark energy (quintessence)…Perrotta&Baccigalupi(1999) ・ Why do we consider such theory? Because it appears in many situations. Here are some lists (+corresponding scalar fields): ・ For now, we consider the simplest scalar-tensor theory, the Brans-Dicke theory.

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§2. Brans-Dicke Theory ・ Action of Brans-Dicke theory: ・ B.D.→G.R. (ω BD →∞) (in this case, φ becomes const.) ・ The current strongest constraint on ω BD : Shapiro time delay from Saturn probe Cassini. (ω BD =const. ) ω BD >40000 (Cassini Bound)

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・ EoM for the B.D. scalar field: ・ In the case of compact objects binary: This characterises the gravitational dipole radiation. Generating gravitational dipole radiation. ・ Gravitational Const. G: G is not a constant in this theory. T: trace of energy-momentum tensor

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§3. Waveforms of NS/BH in B.D. Assumptions: ・ Stationary phase approximation ・ Observation starts 1 year before the coalescence ⇒ We assume that the orbits are circular. ・ Neglect spin precessions (for simplicity) ・ Neglect the directions of binaries and the inclinations of the orbits (assuming we have averaged over those angles.)

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・ Gravitational waveform in Fourier component: Amplitude: : chirp mass D L : luminosity distance

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Phase: (up to 2PN) t c : coalescence time φ c : coalescence phase β ： spin-orbit coupling S=s 1 -s 2 ← φdependence of mass B.D. dipole(-1PN) G.R. quadrupole

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§4. Parameter Estimation ・ We want to determine the binary parameters from G.W. ・ 6 parameters: θ a = M, η, t c, φ c, β, ω BD -1 ・ Apply the usual matched filtering analysis.

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Fisher Matrix : Inner product ： ・ We assume the noise is stationary & Gaussian. ⇒ the probability distributions of parameters are Gaussian. S n (f):noise spectrum θaθa θ a true p(θ a ) estimation error ・ constraint for ω BD : ω BD > 1/Δ(ω BD -1 )

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§5. Results BH mass (M ◉ ) ω BD ×(SN/10) 50006486 100021257 40039190 ・ Fix M NS =1.4M ◉, tc=0, φc=0, β=0, ω=0, S=0.3 ・ Change BH mass and see how strongly we can constraint ω BD. LISA (Berti et al. 2004) c.f. Cassini Bound ω BD >40000

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BH mass (M ◉ ) ω BD ×(SN/10) 50 8.176×10 5 10 2.696×10 6 3 5.391×10 6 DECIGO ・ DECIGO can put 100 times stronger constraint than Cassini and LISA. ・ This constraint can be made even stronger by considering larger SN binaries. ・ Why lower mass puts stronger constraint? Lower the mass ⇒ slower the velocity ⇒ greater the contribution of dipole rad. ・ This result certainly becomes one of the big motivation for the DECIGO Project!

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§6. Future Works ・ To take the direction dependence into account, we are working on Monte Carlo simulations. We distribute 10 4 random direction of binaries and take the average of each error. We expect the constraint would be weakened by a few factors. ・ Take the spin precession into account. (Vecchio 2004, Lang & Hughes 2006 for G.R.) This effect make the estimation errors lower. ⇒ We expect these make the ω BD constraint stronger.

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§ Appendix ・ Staionary phase approximation ( dlnA/dt ≪ dψ/dt Amplitude ≒ const. d 2 ψ/dt 2 ≪ (dψ/dt) 2 Phase changing rate ≒ const.) stationary, weak field approximation ・ Gravitational Const. G

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Why do we consider NS/BH ? ・ To put strong constraint, we want large contribution of dipole rad. ⇒ We want S=s 1 -s 2 to be large. ・ s i is roughly binding energy per mass s WD ~10-3, s NS ~0.2, s BH =0.5 ⇒ We want binaries of different type of compact objects.

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