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Testing Modified gravity and gravitational waves

Testing Modified gravity and gravitational waves. Gravitational waves. Takahiro Tanaka (Dept. of Phys./YITP Kyoto university ). Motivation for modified gravity. 1) Incompleteness of General relativity. GR is non-renormalizabile Singularity formation after gravitational collapse.

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Testing Modified gravity and gravitational waves

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  1. Testing Modified gravity and gravitational waves Gravitational waves Takahiro Tanaka (Dept. of Phys./YITP Kyoto university)

  2. Motivation for modified gravity 1) Incompleteness of General relativity GR is non-renormalizabile Singularity formation after gravitational collapse 2) Dark energy problem 3) To test General relativity GR has been repeatedly tested since its first proposal. The precision of the test is getting higher and higher. ⇒ Do we need to understandwhat kind of modification is theoretically possible before experimental test? Yes, especially in the era of gravitational wave observation!

  3. Gravitation wave detectors TAMA300,CLIO 　⇒KAGRA eLISA(NGO) ⇒DECIGO/BBO LIGO⇒adv LIGO

  4. (Moore, Cole, Berry http://www.ast.cam.ac.uk/~rhc26/sources/)

  5. Inspiraling-coalescing binaries • Inspiral phase (large separation) • Merging phase Numerical relativity • Ringing tail -　quasi-normal oscillation of BH (Cutler et al, PRL 70 2984(1993)) Clean system: ～point particles Internal structure of stars is not so important Accurate theoretical prediction of waveform is possible. • for detection • for parameter extraction(direction, mass, spin,…) • for precision test of general relativity • EOS of nuclear matter • Electromagnetic counterpart

  6. Theoretical prediction of GW waveform Standard post Newtonian approximation ~ (v/c)expansion 4PN=(v/c)8 computation is ready (Blanchet,Living Rev.Rel.17:2 Damour et al. Phys.Rev. D89 (2014) 064058) Waveform in Fourier space for quasi-circular inspiral 1.5PN 1PN

  7. GR is correct in strong gravity regime? Many cycles of gravitational waves from an inspiraling binary 1 cycle phase difference is detectable • Precise determination of orbital parameters • Mapping of the strong gravity region of BH spacetime

  8. Typical modification of GR often discussed in the context of test by GWs Scalar-tensor gravity scalar charge: G-dependence of the gravitational binding energy Dipole radiation＝－1 PNfrequency dependence For binaries composed of similar NSs,

  9. Spontaneous scalarization More general model j is canonically normalized EOM Effective potential for a star with radius R. smaller radius larger radius j2/R2 j2/R2 8pTf (j) 8pTf (j) j j As two NS get closer, “spontaneous scalarization” may happen. Sudden change of structure and starting scalar wave emission.

  10. Scalar-tensor gravity (conti) Current constraint on dipole radiation: wBD＞2.4×104 J1141-6545 (NS(young pulsar)-WD) (Bhat et al. arXiv:0804.0956) Constraint from future observations: (Yagi & TT, arXiv:0908.3283) LISA-1.4M◎NS+1000M◎BH: wBD > 5×103 at 40Mpc corresponding to DECIGO-1.4M◎NS+10M◎BH： wBD > 8×107 collecting 104events at cosmological distances

  11. Einstein dilaton Gauss-Bonnet, Chern-Simons gravity Turu-turu Scalar-tensor theory BH no hair NS can have a scalar hair q×(higher curvature) • For constnat q,these higher curvature terms are topological invariant. Hence, no effect on EOM. • Higher derivative becomes effective only in strong field.

  12. Hairy BH - bold NS • NS in EDGB and CS do not have any scalar charge. • By contrast, BH solutions in EDGB and CS have scalar monopole and dipole, respectively. topological invariant, which vanishes on topologically trivial spacetime. EDGB： monopole chargedipole radiation (-1PN order) CS：dipole charge2PN order corrections (Yagi, Stein, Yunes, Tanaka (2012))

  13. Observational bounds (Amendola, Charmousis, Davis (2007)) Low mass X-ray binary, A0620-00 • EDGB Cassini (Yagi, arXiv:1204.4524) Future Ground-based GW observation SNR=20,6Msol＋12Msol (Yagi, Stein, Yunes, TT, arXiv:1110.5950) • CS Gravity Probe B, LAGEOS (Ali-Haimound, Chen (2011)) Future Ground-based GW observation with favorable spinalignment: 100Mpc,a～0.4M (This must be corrected…) (Yagi, Yunes, TT, arXiv:1208.5102)

  14. Test of GW generation Periastron advance due to GW emission Pulsar : ideal clock PSR B1913+16 Hulse-Taylor binary dPorb/dt=-2.423×10-12 Agreement with GR prediction Test of GR by pulsar binaries （J.M. Weisberg, Nice and J.H. Taylor, arXiv:1011.0718)

  15. We know that GWs are emitted from binaries. But, then what can be a big surprise when we first detect GWs? Is there any possibility that gravitons disappear during its propagation over a cosmological distance? Just fast propagation of GWscan be realized in Lorentz violating models such as Einstein Æther theory.

  16. Chern-Simons Modified Gravity Right-handed and left-handed gravitational waves are amplified/decreased differently during propagation, depending on the frequencies. (Yunes & Spergel, arXiv:0810.5541) The origin of this effect is clear in the effective action. (Flanagan & Kamionkowski, arXiv:1208.4871) The time variation of this factor affects the amplitude of GWs.

  17. Current constraint on the evolution of the background scalar field q : : J0737-3039(double pulsar, periastron precession) (Ali-Haimoud, (2011) But the model has a ghost for large w, and the variation of GW amplitude is significant only for marginally large w. In other words, modes are in the strong coupling regime, which are outside the validity of effective field theory.

  18. Bi-gravity (De Felice, Nakamura, TT arXiv:1304.3920)

  19. Massive gravity Simple graviton mass term is theoretically inconsistent → ghost, instability, etc. Bi-gravity Both massive and massless gravitons exist. → n oscillation-like phenomena? First question is whether or not we can construct a viable cosmological model.

  20. 1) Ghost-free bigravity model exists. 2) It has a FLRW background very similar to the GR case at low energy. 3) The non-linear mechanism seems to work to pass the solar system constraints. (Vainshtein mechanism) 4) Two graviton eigen modes are superposition of two metric perturbations, which are mass eigen states at low frequencies and dg and dg themselves at high frequencies. ~ 5) Graviton oscillations occur only at around the crossover frequency, but there is some chance for observation.

  21. Ghost free bi-gravity When gis fixed, de Rham-Gabadadze-Tolley massive gravity. only 5 possible terms including 2 cosmological constants. ~ ~ Even if gis promoted to a dynamical field, the model remains to be free from ghost. (Hassan, Rosen (2012))

  22. FLRW background (Comelli, Crisostomi, Nesti, Pilo (2012)) Generic homogeneous isotropic metrics branch 1 branch 2 branch 1：Pathological: Strong coupling Unstable for the homogeneous anisotropic mode. branch 2：Healthy

  23. Branch 2 background A very simple relation holds: is algebraically determined as a function of r. We consider only the branches with F >0, F’< 0. required for the absence of Higuchi ghost (Yamashita and TT) We further focus on low energy regime. x→xc for r →0.

  24. Branch 2 background We expand with respect to dx = x -xc. effective energy density due to mass term Effective gravitational coupling is weaker because of the dilution to the hidden sector. Effective graviton mass natural tuning to coincident light cones (c=1)at low energies (r→ 0)!

  25. Solar system constraint: basics • vDVZ discontinuity In GR, this coefficient is 1/2 current bound <10-5 To cure this discontinuity we go beyond the linear perturbation (Vainshtein) Schematically Correction to the Newton potential F

  26. Gravitational potential around a star in the limit c→1 Spherically symmetric static configuration: Erasing , , which can be tuned to be extremely large. Then, the Vainshtein radius can be made very large, even if m -1 << 300Mpc . Solar system constraint: v is excited as in GR.

  27. Excitation of the metric perturbation on the hidden sector: Erasing u, v and R ~ u is also suppressed like u. ~ v is also excited like v. The metric perturbations are almost conformally related with each other: ~ Non-linear terms of u (or equivalently u) play the role of the source of gravity.

  28. Gravitational wave propagation Short wavelength approximation： (Comelli, Crisostomi, Pilo (2012)) kc k mass term is important. C ≠１ is important. Eigenmodes are Eigenmodes are modified dispersion relation due to different light cone modified dispersion relation due to the effect of mass

  29. At the GW generation, both and are equally excited. X kc k Only the first mode is excited Only the first mode is detected X We can detect only h. Only modes with k～kcpicks up the non-trivial dispersion relation of the second mode. Interference between two modes. Graviton oscillations If the effect appears ubiquitously, such models would be already ruled out by other observations.

  30. Summary Gravitational wave observations open up a new window for modified gravity. Even the radical idea of graviton oscillations is not immediately denied. We may find something similar to the case of solar neutrino experiment in near future. Although space GW antenna is advantageous for the gravity test in many respects, more that can be tested by KAGRA will be remaining to be uncovered.

  31. Branch 2 background branch 1 branch 2 branch 2： x becomes a function of r. x→xc for r →0. effective energy density due to mass term Natural Tuning to c=1for r→ 0. Effective gravitational coupling is weaker because of the dilution to the hidden sector.

  32. Stability of linear perturbation Gradient instability Healthy branch Higuchi ghost JCAP1406 (2014) 037 De Felice, Gumrukcuoglu, Mukohyama, Tanahashi and TT

  33. Gravitational potential around a star (PTEP2014 043E01 De Felice, Nakamura and TT) Spherically symmetric static configuration: Erasing , and truncating at the second order can be tuned to be extremely large. Then, the Vainshtein radius can be made very large, even if m -1 << 300Mpc . Solar system constraint: ~ Both v and v are excited as in GR.

  34. Meaning of A>>1 Gradient instability Healthy branch

  35. Gravitational wave propagation Short wavelength approximation： (Comelli, Crisostomi, Pilo (2012)) kc k mass term is important. c ≠１ is important. Eigenmodes are Eigenmodes are modified dispersion relation due to different light cone modified dispersion relation due to the effect of mass

  36. Gravitational wave propagation over a long distance D Phase shift due to the modified dispersion relation: 1.0 0.5 becomes O(1) after propagation over the horizon distance log10x - 2 - 1 1 2 k =0.2 - 0.5 d F2 k =1 - 1.0 k =100 - 1.5 d F1

  37. At the GW generation, both and are equally excited. X kc k Only the first mode is excited Only the first mode is detected X We can detect only h. Only modes with k～kcpick up the non-trivial dispersion relation of the second mode. Graviton oscillations Interference between two modes If the effect appears ubiquitously, the model would be already ruled out.

  38. Gravitational wave oscillations B 4 2 kx 2=0.2 kx 2=1 log10x - 2 - 1 1 2 B1 kx 2=100 - 2 Detectable range of parameters by KAGRA, assuming NS-NS binary at 200Mpc. B2 - 4 At low frequencies only the first mode is excited. At high frequencies only the first mode is observed. (Phys. Rev. D91 (2015) 062007 Narikawa, Tagoshi, TT, Kanda and Nakamura)

  39. If Vainshtein mechanism works for GW detectors… At the GW generation, both and are equally excited. X kc k Only the first mode is excited X Only the first mode is detected X We detect both and . All modes with k > kcpick up the non-trivial dispersion relation of the second mode. ～ Graviton oscillations Interference between two modes

  40. Solar system constraint: For detection by aLIGO, aVirgo and KAGRA: No Vainshtein effect in the inter-galactic space: Window is narrow but open. Bare mass: Gradient instability Cosmological solution disappears

  41. Can Bigravity with large A be naturally realized as a low energy effective theory? KK graviton spectrum Only first two modes remain at low energy Higher dimensional model?! Matter on right brane couples to h. If the internal space is stabilized However, pinched throat configuration looks quite unstable…

  42. Induced gravity on the branes Dvali-Gabadadze-Porrati model (2000) • Induced gravity terms play the role of potential well. • Lowest KK graviton mass • KK graviton mass Critical length scale ?? D y Brane Brane Minkowski Bulk is required to reproduce bigravity.

  43. (JCAP 1406 (2014) 004 Yamashita and TT) • To construct a viable model, • the radion (=brane separation) must be stabilized. • Radions can be made as heavy as KK gravitons. • However, the energy density cannot be made large. Why? We know that self-accelerating branch has a ghost. Normal branch condition: cannot be made larger than in this model.

  44. Deriving bigravity without fine tuning (Yamashita and TT, in preparation) • Despite the fore-mentioned limitation, it would be interesting to see how bigravity derives from brane setup without fine tuning of coupling constants. • We neglect radion stabilization, for simplicity. • Thus, we consider a system of bigravity with radion Gradient expansion We solve the bulk equations of motion for given boundary metrics with the scaling assumptions: K :bulk extrinsic curvature KDy <<1 Dy :Brane separation Krc<1 ~ m 2 ~H 2~∂ 2~l -2~ 1/ rcDy

  45. Substituting back the obtained bulk solution into the action, we obtain at the quadratic order in perturbation with This looks very complicated but can be recast into the form of bigravity + radion, which is coupled to the averaged metric:

  46. Why do we have this attractor behavior, c→1 and x→xc, at low energies? ?? KK graviton mass spectrum ～ ～ ~1/d 2 d potential wells due to induced gravity terms DGP 2-brane model?! Only first two modes remain at low energy d→0 identical light cone

  47. Gravitational wave propagation over a long distance D Phase shift due to the modified dispersion relation: 1.0 0.5 becomes O(1) after propagation over the horizon distance log10x - 2 - 1 1 2 k =0.2 - 0.5 d F2 k =1 - 1.0 k =100 - 1.5 d F1

  48. Gravitational wave oscillations B 4 1) At the time of generation of GWs from coalescing binaries, both h and h are equally excited. ~ 2 2) When we detect GWs, we sense h only. - 2 - 1 1 2 log10x - 2 kx 2=0.2 - 4 kx 2=1 B1 kx 2=100 B2 At low frequencies only the first mode is excited. At high frequencies only the first mode is observed.

  49. B 4 kx 2=0.2 2 kx 2=1 B1 kx 2=100 - 2 - 1 1 2 log10x 1.0 B2 - 2 0.5 log10x - 4 - 2 - 1 1 2 k =0.2 Graviton oscillations occur only around the frequency - 0.5 k =1 - 1.0 k =100 - 1.5 Phase shift is as small as? No, x << 1 when the GWs are propagating the inter-galactic low density region.

  50. Summary Gravitational wave observations give us a new probe to modified gravity. Even graviton oscillations are not immediately denied, and hence we may find something similar to the case of solar neutrino experiment in near future. Although space GW antenna is advantageous for the gravity test in many respects, we should be able to find more that can be tested by KAGRA.

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