Wide Angle Redshift Distortions Revisited

We explore linear redshift distortions in wide angle surveys from the point of view of symmetries. We show that the redshift space two-point correlation function can be expanded into tripolar spherical harmonics of zero total angular momentum $S_{l_1 l_2 l_3}(\hat x_1, \hat x_2, \hat x)$. The coefficients of the expansion $B_{l_1 l_2 l_3}$ are analogous to the $C_l$'s of the angular power spectrum, and express the anisotropy of the redshift space correlation function. Moreover, only a handful of $B_{l_1 l_2 l_3}$ are non-zero: the resulting formulae reveal a hidden simplicity comparable to distant observer limit. The $B_{l_1 l_2 l_3}$ depend on spherical Bessel moments of the power spectrum and $f = \Omega^{0.6}/b$. In the plane parallel limit, the results of \cite{Kaiser1987} and \cite{Hamilton1993} are recovered. The general formalism is used to derive useful new expressions. We present a particularly simple trigonometric polynomial expansion, which is arguably the most compact expression of wide angle redshift distortions. These formulae are suitable to inversion due to the orthogonality of the basis functions. An alternative Legendre polynomial expansion was obtained as well. This can be shown to be equivalent to the results of \cite{SzalayEtal1998}. The simplicity of the underlying theory will admit similar calculations for higher order statistics as well.Comment: 6 pages, 1 figure, ApJL submitte

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R. v. Oakes 1986-1997: Back to the Drawing Board

The Supreme Court of Canada, in R. v. Oakes, identified two standards of justification in applying section 1. The first standard was normative. The second was methodological, called the Oakes test. The Court, until recently, applied the Oakes test mechanically and avoided the normative standard. More recently, in Egan v. Canada and RJR-MacDonald Inc. v. Canada (A.G.), it resorted to a normative analysis that is indeterminate and unpredictable. This article challenges both the mechanical application of the Oakes test and the Court\u27s new normative approach. It proposes, and illustrates, a preferable alternative that is both determinate and predictable. It is supported by appendices that analyze section 1 cases between 1986 and 1997

Staff perceptions on the effectiveness of GRiP-S, a new approach to clinical supervision incorporating safewards: An interpretive phenomenological analysis

This study explored the impact of an innovative approach to clinical supervision for mental health nurses which integrates Safewards, named Group Reflective integrated Practice with Safewards–GRiP-S. Qualitative data was collected through 10 individual semi-structured interviews with nursing staff who had participated within the clinical supervision approach. Interviews provided insights into the nursing staff’s perception and experience of the clinical supervision approach. Through interpretive phenomenological analysis six themes emerged (i) illuminating embodied practice of Safewards, (ii) building confidence through empowering connections, (iii) creating a culture of positive change, (iv) identifying internal motivation for and external barriers to supervision engagement, (v) navigating a global pandemic, and (vi) the transformative role of reflection. Findings demonstrated that the GRiP-S approach assisted mental health nurses’ adoption of Safewards interventions in practice, while supporting the development of a cohesive staff team. The impact of COVID-19 within the study setting was addressed and nurses identified how the Safewards model assisted in navigating challenges during this time. Findings further supported prior research on the role of the supervisor and supervisee relationship. This study supports the integration of Safewards within reflective clinical supervision for mental health nursing staff to assist in Safewards fidelity and nursing staff personal and professional development

Getting a grip on Safewards: The cross impact of clinical supervision and Safewards model on clinical practice

The Safewards model is used across various mental health settings to reduce incidents of conflict and containment and its efficacy in reducing the use of seclusion and restraint, improving patients\u27 experiences of care, and enhancing safety within clinical settings is well documented (Bowers, Journal of Psychiatric & Mental Health Nursing, 21, 2014, 499). However, there are barriers to successful implementation, including level of staff buy-in (Baumgardt et al., Frontiers in Psychiatry, 10, 2019, 340; Price et al., Mental Health Practice, 19, 2016, 14). This mixed-method study assessed the impact of adopting a Safewards model within a clinical supervision framework in an approach, named Group Reflective integrated Practice with Safewards (GRiP-S), which integrates Safewards theory within the clinical supervision framework. Both quantitative and qualitative data were collected using the questions derived from the Manchester Clinical Supervision Scale −26© (Winstanley & White, The Wiley International Handbook of Clinical Supervision. John Wiley & Sons Ltd, 2014). A total of 67 surveys and eight interviews were completed by nursing staff. Overall, the results showed that the GRiP-S approach improves the implementation of Safewards and nurses\u27 clinical practice. Nursing staff satisfaction with clinical supervision and Safewards improved post GRiP-S pre-GRIP-S- 69.54 (SD 16.059); post-GRIP-S 71.47 (SD 13.978). The survey also identified nursing staff\u27s perception of GRiP-S in the restorative and formative domains of clinical supervision improved. The restorative mean score pre-GRiP-S was 28.43 (SD 5.988) and post-GRiP-S 29.29 (SD 3.951). The formative mean score pre-GRiP-S was 20.10 (SD 5.617) and post-GRiP-S 20.63 (SD 13.978). The qualitative results further explained the satisfaction levels and the changes seen in perception domains. The GRiP-S approach reported (i) improved therapeutic relationships and patient centred care, (ii) improved staff communication and teamwork, (iii) barriers to GRiP-S engagement, and (iv) assistance with the change process. The results indicate that the GRiP-S approach had a positive impact on Safewards delivery and supports ongoing change of practice

An Inversion Method for Measuring Beta in Large Redshift Surveys

A precision method for determining the value of Beta= Omega_m^{0.6}/b, where b is the galaxy bias parameter, is presented. In contrast to other existing techniques that focus on estimating this quantity by measuring distortions in the redshift space galaxy-galaxy correlation function or power spectrum, this method removes the distortions by reconstructing the real space density field and determining the value of Beta that results in a symmetric signal. To remove the distortions, the method modifies the amplitudes of a Fourier plane-wave expansion of the survey data parameterized by Beta. This technique is not dependent on the small-angle/plane-parallel approximation and can make full use of large redshift survey data. It has been tested using simulations with four different cosmologies and returns the value of Beta to +/- 0.031, over a factor of two improvement over existing techniques.Comment: 16 pages including 6 figures Submitted to The Astrophysical Journa

Redshift distortions in one-dimensional power spectra

We present a model for one-dimensional (1D) matter power spectra in redshift space as estimated from data provided along individual lines of sight. We derive analytic expressions for these power spectra in the linear and nonlinear regimes, focusing on redshift distortions arising from peculiar velocities. In the linear regime, redshift distortions enhance the 1D power spectra only on small scales, and do not affect the power on large scales. This is in contrast to the effect of distortions on three-dimensional (3D) power spectra estimated from data in 3D space, where the enhancement is independent of scale. For CDM cosmologies, the 1D power spectra in redshift and real space are similar for wavenumbers $q<0.1h/Mpc$ where both have a spectral index close to unity, independent of the details of the 3D power spectrum. Nonlinear corrections drive the 1D power spectrum in redshift space into a nearly universal shape over scale $q<10h/Mpc$, and suppress the power on small scales as a result of the strong velocity shear and random motions. The redshift space, 1D power spectrum is mostly sensitive to the amplitude of the initial density perturbations. Our results are useful in particular for power spectra computed from the SDSS quasars sample.Comment: MNRAS in press. matches published versio

Radial Redshift Space Distortions

The radial component of the peculiar velocities of galaxies cause displacements in their positions in redshift space. We study the effect of the peculiar velocities on the linear redshift space two point correlation function. Our analysis takes into account the radial nature of the redshift space distortions and it highlights the limitations of the plane parallel approximation. We consider the problem of determining the value of \beta and the real space two point correlation function from the linear redshift space two point correlation function. The inversion method proposed here takes into account the radial nature of the redshift space distortions and can be applied to magnitude limited redshift surveys that have only partial sky coverage.Comment: 26 pages including 11 figures, to appear in Ap