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

Polychemotherapy for early breast cancer: an overview of the randomised trials

Background There have been many randomised trials of adjuvant prolonged polychemotherapy among women with early breast cancer, and an updated overview of their results is presented. Methods In 1995, information was sought on each woman in any randomised trial that began before 1990 and involved treatment groups that differed only with respect to the chemotherapy regimens that were being compared. Analyses involved about 18 000 women in 47 trials of prolonged polychemotherapy versus no chemotherapy, about 6000 in 11 trials of longer versus shorter polychemotherapy, and about 6000 in 11 trials of anthracycline-containing regimens versus CMF (cyclophosphamide, methotrexate, and fluorouracil). Findings For recurrence, polychemotherapy produced substantial and highly significant proportional reductions both among women aged under 50 at randomisation (35% [SD 4] reduction; 2p<0·00001) and among those aged 50–69 (20% [SD 3] reduction; 2p<0·00001); few women aged 70 or over had been studied. For mortality, the reductions were also significant both among women aged under 50 (27% [SD 5] reduction; 2p<0·00001) and among those aged 50–69 (11% [SD 3] reduction; 2p=0·0001). The recurrence reductions emerged chiefly during the first 5 years of follow-up, whereas the difference in survival grew throughout the first 10 years. After standardisation for age and time since randomisation, the proportional reductions in risk were similar for women with node-negative and node-positive disease. Applying the proportional mortality reduction observed in all women aged under 50 at randomisation would typically change a 10-year survival of 71% for those with node-negative disease to 78% (an absolute benefit of 7%), and of 42% for those with node-positive disease to 53% (an absolute benefit of 11%). The smaller proportional mortality reduction observed in all women aged 50–69 at randomisation would translate into smaller absolute benefits, changing a 10-year survival of 67% for those with node-negative disease to 69% (an absolute gain of 2%) and of 46% for those with node-positive disease to 49% (an absolute gain of 3%). The age-specific benefits of polychemotherapy appeared to be largely irrespective of menopausal status at presentation, oestrogen receptor status of the primary tumour, and of whether adjuvant tamoxifen had been given. In terms of other outcomes, there was a reduction of about one-fifth (2p=0·05) in contralateral breast cancer, which has already been included in the analyses of recurrence, and no apparent adverse effect on deaths from causes other than breast cancer (death rate ratio 0·89 [SD 0·09]). The directly randomised comparisons of longer versus shorter durations of polychemotherapy did not indicate any survival advantage with the use of more than about 3–6 months of polychemotherapy. By contrast, directly randomised comparisons did suggest that, compared with CMF alone, the anthracycline-containing regimens studied produced somewhat greater effects on recurrence (2p=0·006) and mortality (69% vs 72% 5-year survival; log-rank 2p=0·02). But this comparison is one of many that could have been selected for emphasis, the 99% CI reaches zero, and the results of several of the relevant trials are not yet available. Interpretation Some months of adjuvant polychemotherapy (eg, with CMF or an anthracycline-containing regimen) typically produces an absolute improvement of about 7–11% in 10-year survival for women aged under 50 at presentation with early breast cancer, and of about 2–3% for those aged 50–69 (unless their prognosis is likely to be extremely good even without such treatment). Treatment decisions involve consideration not only of improvements in cancer recurrence and survival but also of adverse side-effects of treatment, and this report makes no recommendations as to who should or should not be treated