Gravitational waves in the presence of a cosmological constant

We derive the effects of a non-zero cosmological constant $\Lambda$ on gravitational wave propagation in the linearized approximation of general relativity. In this approximation we consider the situation where the metric can be written as $g_{\mu\nu}= \eta_{\mu\nu}+ h_{\mu\nu}^\Lambda + h_{\mu\nu}^W$, $h_{\mu\nu}^{\Lambda,W}<< 1$, where $h_{\mu\nu}^{\Lambda}$ is the background perturbation and $h_{\mu\nu}^{W}$ is a modification interpretable as a gravitational wave. For $\Lambda \neq 0$ this linearization of Einstein equations is self-consistent only in certain coordinate systems. The cosmological Friedmann-Robertson-Walker coordinates do not belong to this class and the derived linearized solutions have to be reinterpreted in a coordinate system that is homogeneous and isotropic to make contact with observations. Plane waves in the linear theory acquire modifications of order $\sqrt{\Lambda}$, both in the amplitude and the phase, when considered in FRW coordinates. In the linearization process for $h_{\mu\nu}$, we have also included terms of order $\mathcal{O}(\Lambda h_{\mu\nu})$. For the background perturbation $h_{\mu\nu}^\Lambda$ the difference is very small but when the term $h_{\mu\nu}^{W}\Lambda$ is retained the equations of motion can be interpreted as describing massive spin-2 particles. However, the extra degrees of freedom can be approximately gauged away, coupling to matter sources with a strength proportional to the cosmological constant itself. Finally we discuss the viability of detecting the modifications caused by the cosmological constant on the amplitude and phase of gravitational waves. In some cases the distortion with respect to gravitational waves propagating in Minkowski space-time is considerable. The effect of $\Lambda$ could have a detectable impact on pulsar timing arrays.Comment: 20 pages, 1 figur

Do GOLD stages of COPD severity really correspond to differences in health status?

The purpose of this study was to assess whether different stages of chronic obstructive pulmonary disease (COPD) severity defined according to the Global Initiative for Chronic Obstructive Lung Disease (GOLD) criteria correlate with meaningful differences in health status. A total of 381 COPD patients, aged 73¡6 yrs, were classified in the five GOLD stages. Disease-specific (St George Respiratory Questionnaire (SGRQ)) and generic indexes of health status were measured in all patients. Multivariate analysis of covariance or Kruskal Wallis tests were used to compare health status indexes across the spectrum of GOLD stages of COPD severity. GOLD stages of COPD severity significantly differed in SGRQ components and Barthel9s index, but not in the indexes assessing cognitive and affective status and quality of sleep. The largest variation in health status was observed at the transition from stage IIa to stage IIb, while there were no other significant differences between consecutive stages. Both female sex and comorbidity were associated with a greater impact of COPD on the health status. In conclusion, the upper limit of stage IIb (forced expiratory volume in one second of 49%) marks a threshold for dramatic worsening of health status. Progression of chronic obstructive pulmonary disease severity from stage 0 to stage IIa does not correspond to any meaningful difference in health status