Testing Alternative Theories of Gravity Using LISA and Pulsar Timing Detections of Gravitational Waves

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1 Testing Alternative Theories of Gravity Using LISA and Pulsar Timing Detections of Gravitational Waves Márcio E. S. Alves Universidade Federal de Itajubá Massimo Tinto Jet Propulsion Laboratory, California Institute of Technology M.E.S.ALVES is supported by Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

2 OUTLINE 1 Why alternative theories of gravity? 2 Extra polarization modes of GWs 3 One-way Doppler Response 4 LISA Sensitivities 5 Pulsar Timing Sensitivities 6 Results and Conclusion Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

3 Why alternative theories of gravity? Why alternative theories of gravity? General Relativity: Elegant and Simple! All attempts to quantize gravity and to unify it with the other forces suggest that standard GR is not likely to be the last word; Low energy string theory; Inflationary scenarios; Recent phase of accelerated expansion of the Universe; Corrections to gravity seem to be needed at some level! Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

4 Why alternative theories of gravity? Examples of alternative theories Scalar-tensor theories (e.g., Brans-Dicke theory); f (R) theories: very popular in cosmology; Quadratic gravity: quantum corrections; Theories which consider non-null graviton mass; Higher dimensional theories; And so on... Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

5 Extra polarization modes of GWs Extra polarization modes of GWs General Relativity predicts two polarization states for GWs with helicity 2 Figure: Effect of GWs on a ring of test particles. Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

6 Extra polarization modes of GWs The Newman-Penrose formalism GWs in metric theories of gravitation g µν and other fields... Riemann tensor is the only measurable field relative accelerations: a i = R i0j0 x j (1) Eardley et al. (1973) have used the Newman-Penrose (NP) formalism to write the irreducible parts of the linearized R λµκν ; Classification of GWs in AT; Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

7 Extra polarization modes of GWs The Newman-Penrose formalism The following set describes linearized GWs in any theory: Ψ 2 = 1 6 R lklk, (2) Ψ 3 = 1 2 R lklm, (3) Ψ 4 = R lmlm, (4) Φ 22 = R lmlm. (5) Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

8 Extra polarization modes of GWs The Newman-Penrose formalism Helicity of GWs Analyzing the behaviour of the set {Ψ 2, Ψ 3, Ψ 4, Φ 22 } under rotations: M = exp(isϕ)m, we find the helicity values: Ψ 2 s = 0, Ψ 3 s = 1, Ψ 3 s = +1 Ψ 4 s = 2, Ψ 4 s = +2 Φ 22 s = 0. (6) Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

9 Extra polarization modes of GWs The six polarization states of GWs Figure: Non-null polarization modes for some theories of gravity. Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

10 One-way Doppler Response One-way Doppler Response How can one detect the GW polarizations? Consider the following line element: ds 2 = dt 2 + ( δ ij + h ij (t z/v) ) dx i dx j, h ij << 1, (7) where h ij (t z/v) = 6 r=1 ɛ (r) ij h (r) (t z/v) (8) ɛ (r) ij polarization tensors Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

11 One-way Doppler Response Polarization matrices (the GW propagates in the +z direction) ɛ (1) = ɛ (2) = ɛ (3) = ɛ (5) = ɛ (4) = ɛ (6) = (9) Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

12 One-way Doppler Response One-way Doppler Response Photon 4-momentum P µ = ν 0 ( δ t µ + n µ h µνn ν ) By demanding P µ K µ = P µk µ Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

13 One-way Doppler Response One-way Doppler Response We find the one-way Doppler response measured at spacecraft 2 y(t) = (1 + ˆk [ ˆn) Ψ(t L) Ψ(t ˆk ] ˆnL) (10) where Ψ(t) ni h ij (t)n j 2[1 (ˆk ˆn) 2 ], and y(t) (ν (t) ν 0 )/ν 0 y (t) at spacecraft 1 can be obtained by changing ˆn ˆn and further delaying the waveforms by (ˆk ˆn)L y (t) = (1 ˆk [ ˆn) Ψ(t (1 + ˆk ] ˆn)L) Ψ(t) (11) Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

14 LISA Sensitivities LISA Responses 10 1 (a) Tensor Vector Scalar L Scalar T 10 0 X RMS (f)/h f(hz) Figure: RMS of the X response. For each polarization mode, we have assumed an ensemble of sinusoidal signals uniformly distributed on the celestial sphere and randomly polarized. Since the LISA arms differ by a few percent, we have analyzed an equilateral configuration of nominal arm-length L 16.7 light seconds. Note that the response increases with frequency for the scalar-longitudinal GW mode. Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

15 LISA Sensitivities LISA Sensitivities The Sensitivity is given by 5 S k (f )B/(RMS of GW response for data combination k) where S k is the total noise power spectrum. The bandwidth B = one cycle/year = Hz. Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

16 LISA Sensitivities LISA Sensitivities Tensor X Sensitivity Scalar T Vector Scalar L (a) f(hz) Figure: LISA Sensitivities for the X combination. For each polarization mode, we have assumed an ensemble of sinusoidal signals uniformly distributed on the celestial sphere and randomly polarized. Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

17 LISA Sensitivities Amplification of the Doppler signal by the Scalar Longitudinal (SL) mode For tensor modes, the maximum one-way Doppler shift measured is On the other hand, for the SL mode we have lim ỹ(f ) sin(πfl) (12) ˆk ˆn 0 lim ỹ(f ) (πfl) (13) ˆk ˆn 1 Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

18 Pulsar Timing Sensitivities Pulsar Timing Sensitivities Therefore, pulsar timing responses would experience a rather significant amplification to SL waves Frequency band Hz and typical pulsar-earth distance 1 kpc x kˆ Earth Figure: The radio pulses emitted by the pulsar are received at Earth by a radio telescope. z nˆ L PSR y Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

19 Pulsar Timing Sensitivities Pulsar Timing Sensitivities Pulsar Timing Sensitivities Tensor Scalar T log(h r.m.s. (f)) Vector Scalar L log(f (Hz)) Figure: Pulsar timing sensitivity to scalar-longitudinal waves is better than to tensor waves by more than one order of magnitude at f = 10 9 Hz and three orders of magnitude at f = 10 6 Hz if we assume a pulsar out to a distance of 1 kpc Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

20 Pulsar Timing Sensitivities Pulsar Timing responses 10 3 Pulsar Timing Responses 10 2 f = 10 9 Hz φ = 0 Scalar L log(rms of the response) Tensor Scalar T Vector θ Pulsar timing r.m.s. responses for f = 10 9 Hz. Note the rather large interval of θ values over which the response of the scalar-longitudinal polarization is better than the other polarizations. Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

21 Results and Conclusion Results and Conclusion LISA is more sensitive to scalar-longitudinal and vector signals than to tensor waves in the high-part of its frequency band. At low-frequencies its sensitivity to tensor and vector signals are equal and somewhat better than to scalar waves. Pulsar timing experiments are more sensitive to scalar-longitudinal and vector signals than to scalar-transverse and tensor waves over their overall accessible frequency band. Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

22 Results and Conclusion Results and Conclusion Due the fact that L is much larger to Pulsars, Pulsar Timing technique will be a powerful tool to test alternative theories. This will result in a dynamical test to Einstein s theory of relativity significantly more stringent than that based on monitoring the decay of the orbital period of a binary system. These results should provide LISA and Pulsar Timing detection with the capability for assessing the polarization of the waves it will detect. Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

23 Results and Conclusion Thank you! More details look at Tinto, M., Alves, M.E.S. Phys. Rev. D 82, (2010) Alves, M.E.S., Tinto, M. Phys. Rev. D 83, (2011) Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

24 Results and Conclusion One-way Doppler Response The speed of GWs can be frequency-dependent: ( m ) 2 v(ω) = 1 ω Operational frequency band of LISA = ( Hz) From solar-system dynamics: v = v c /c 10 8 (m < g) From dynamical properties of a galactic disk: v (m < g) LISA will not be able to resolve the propagation speeds of GWs. Márcio E. S. Alves (XIII Marcel Grossmann) LISA and Pulsar Timing Sensitivities JUL / 24

The International Pulsar Timing Array. Maura McLaughlin West Virginia University June

The International Pulsar Timing Array Maura McLaughlin West Virginia University June 13 2011 Outline Pulsar timing for gravitational wave detection Pulsar timing arrays EPTA, NANOGrav, PPTA The International