Constraining the propagation speed of gravitational waves with compact binaries at cosmological distances

In testing gravity a model-independent way, one of crucial tests is measuring the propagation speed of a gravitational wave (GW). In general relativity, a GW propagates with the speed of light, while in the alternative theories of gravity the propagation speed could deviate from the speed of light due to the modification of gravity or spacetime structure at a quantum level. Previously we proposed the method measuring the GW speed by directly comparing the arrival times between a GW and a photon from the binary merger of neutron stars or neutron star and black hole, assuming that it is associated with a short gamma-ray burst. The sensitivity is limited by the intrinsic time delay between a GW and a photon at the source. In this paper, we extend the method to distinguish the intrinsic time delay from the true signal caused by anomalous GW speed with multiple events at cosmological distances, also considering the redshift distribution of GW sources, redshift-dependent GW propagation speed, and the statistics of intrinsic time delays. We show that an advanced GW detector such as Einstein Telescope will constrain the GW propagation speed at the precision of ~10^{-16}. We also discuss the optimal statistic to measure the GW speed, performing numerical simulations.Comment: 9 pages, 9 figure

Transient Raman spectroscopy of 15N-substituted bacteriochlorophyll a. An empirical assignment of T1 Raman lines

15N-substituted bacteriochlorophyll a (BChl a) was extracted from the cells of Rhodobacter sphaeroides 2.4.1 grown in a medium containing 15N-ammonium sulfate and yeast concentrate. The T1 Raman spectra of 14N-and 15N-BChl a were obtained as the difference spectra of high-power minus low-power of one-color, pump-and-probe measurements using 420 nm, 5 ns pulses. A set of empirical assignments of the T1 Raman lines was made, based on shifts upon 14N→15N substitution. The S0 Raman spectra of the two BChls were also obtained by using the 457.9 nm cw beam, and a set of assignments of the S0 Raman lines was given for comparison

Weak value amplification and beyond the standard quantum limit in position measurements

In a weak measurement with post-selection, a measurement value, called the weak value, can be amplified beyond the eigenvalues of the observable. However, there are some controversies whether the weak value amplification is practically useful or not in increasing sensitivity of the measurement in which fundamental quantum noise dominates. In this paper, we investigate the sensitivity limit of an optical interferometer by properly taking account quantum shot noise and radiation pressure noise. To do so, we formulate the weak value amplification in the Heisenberg picture, which enables us to intuitively understand what happens when the measurement outcome is post-selected and the weak value is amplified. As a result, we found that the sensitivity limit is given by the standard quantum limit that is the same as in a standard interferometry. We also discuss a way to circumvent the standard quantum limit.Comment: 12 pages, 5 figure

Measuring Speed of Gravitational Waves by Observations of Photons and Neutrinos from Compact Binary Mergers and Supernovae

Detection of gravitational waves (GW) provides us an opportunity to test general relativity in strong and dynamical regimes of gravity. One of the tests is checking whether GW propagates with the speed of light or not. This test is crucial because the velocity of GW has not ever been directly measured. Propagation speed of a GW can deviate from the speed of light due to the modification of gravity, graviton mass, and the nontrivial spacetime structure such as extra dimensions and quantum gravity effects. Here we report a simple method to measure the propagation speed of a GW by directly comparing arrival times between gravitational waves, and neutrinos from supernovae or photons from short gamma-ray bursts. As a result, we found that the future multimessenger observations of a GW, neutrinos, and photons can test the GW propagation speed with the precision of ~10^{-16} improving the previous suggestions by 8-10 orders of magnitude. We also propose a novel method that distinguishes the true signal due to the deviation of GW propagation speed from the speed of light and the intrinsic time delay of the emission at a source by looking at the redshift dependence.Comment: 8 pages, 6 figure

Universally Valid Error-Disturbance Relations in Continuous Measurements

In quantum physics, measurement error and disturbance were first naively thought to be simply constrained by the Heisenberg uncertainty relation. Later, more rigorous analysis showed that the error and disturbance satisfy more subtle inequalities. Several versions of universally valid error-disturbance relations (EDR) have already been obtained and experimentally verified in the regimes where naive applications of the Heisenberg uncertainty relation failed. However, these EDRs were formulated for discrete measurements. In this paper, we consider continuous measurement processes and obtain new EDR inequalities in the Fourier space: in terms of the power spectra of the system and probe variables. By applying our EDRs to a linear optomechanical system, we confirm that a tradeoff relation between error and disturbance leads to the existence of an optimal strength of the disturbance in a joint measurement. Interestingly, even with this optimal case, the inequality of the new EDR is not saturated because of doublely existing standard quantum limits in the inequality.Comment: 11 pages, 2 figure

The multiple gamma function and its q-analogue

We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product formula) of the Vign\'{e}ras multiple gamma function by considering the classical limit of the multiple q-gamma function.Comment: Latex, 15 page