text:
\section{Introduction}
\label{Introduction}

According to the $\Lambda$CDM model, we live in a Universe that is composed of a cosmological constant ($\Lambda$), cold dark matter (CDM), baryons, neutrinos and photons. Today's observed large-scale structure is the consequence of the gravitational evolution of these components from small initial perturbations, and the theory governing the evolution is the coupled set of non-linear Einstein--Boltzmann equations. So far these relativistic equations have been solved not in full non-linearity but by approximate methods within the framework of cosmological perturbation theory (CPT; see e.g.\ \cite{Kodama:1985bj,Malik:2008im,Villa:2015ppa}), utilised by numerical Einstein--Boltzmann solvers. Popular Boltzmann codes that solve the equations to first order include {\sc camb} \cite{Lewis:1999bs} and {\sc class}~\cite{Blas:2011rf}, while several codes, including {\sc song}{}, solve the evolution up to second order \cite{Huang:2012ub,Su:2012gt,Pettinari:2013he}.

These Boltzmann codes determine the gravitational evolution to high accuracy as long as matter and radiation perturbations remain small. This is a good approximation for the first few billion years of the gravitational evolution, but the matter perturbations keep growing during matter domination, making the gravitational collapse of matter non-linear and eventually leading to a complete perturbative breakdown.
To solve for the fully non-linear gravitational collapse, one makes use of cosmological (N-body) simulations \cite{Springel:2005mi,Teyssier:2001cp,Hahn:2015sia} which themselves usually work in the Newtonian approximation (see however \cite{Adamek:2013wja,Adamek:2015eda,Giblin:2015vwq,Hahn:2016roq,Brandbyge:2016raj}), an approach which is justified only on small scales and at sufficiently late times, where the effects of radiation can be neglected.

To set up initial conditions for such N-body simulations, one has to use perturbative Boltzmann codes and pass the dynamical information about the system over to the Newtonian simulation.
Perhaps the simplest prescription is to use the Boltzmann codes to calculate the matter density field at high redshift, say at $z=50$, and use this information to set up the initial particle displacements and then evolve these forwards in the N-body simulation. We will refer to this as the ``forwards approach''. However at early times in the standard $\Lambda$CDM cosmology there is residual thermal radiation from the cosmic microwave background and the cosmic neutrino background. Although N-body simulations can include the effect of radiation in the background expansion, they do not describe the effect of inhomogeneous radiation, which tends to suppress the growth of structure. Thus in this naive ``forwards approach'', the large scale density field at the present day ($z=0$) does not match that calculated in perturbative Boltzmann codes which do consistently solve the relativistic evolution to first order.

A commonly employed work-around is to use a perturbative Boltzmann code to calculate the relativistic density field today and use the linear Newtonian growth factor, which neglects inhomogeneous radiation, to set up an initial density field which, although ``wrong'' at e.g. $z=50$, does enable Newtonian N-body simulations to reproduce the correct linear matter power spectrum on large scales at $z=0$. We refer to this as the ``backscaling approach'', as illustrated in figure~\ref{fig:backscaling}.
It is the purpose of this paper to analyse whether the methods currently employed in the literature provide a consistent relativistic framework for the analysis of Newtonian N-body simulations, using either forwards or backscaled initial conditions.
\begin{figure}[tb]
  \centering
    \includegraphics[width=0.6\textwidth]{Kazuya_CDM_xkcd2.pdf}
  \caption{Two different approaches to setting initial conditions for N-body simulations. The ``forwards approach'' (top line) evolves the linear Einstein--Boltzmann perturbations (green arrow) up until a fixed redshift from where the Newtonian N-body solver (red arrow) continues until the present time. In the ``backscaling approach'' (bottom line), the Einstein--Boltzmann system is evolved until $z=0$ (green arrow). The matter perturbations are then scaled back (yellow arrow) using the linear Newtonian theory and are then evolved non-linearly by the N-body simulation (red arrow) to the present day.}
  \label{fig:backscaling}
\end{figure}

This paper is organised as follows. We start by discussing the usual Newtonian framework and the related initial conditions in section~\ref{sec:Newt}.
In particular we discuss how a ``backscaling approach'' is used to incorporate the consequences of residual radiation perturbations in the initial conditions for Newtonian N-body simulations evolving pressureless dust.
We then briefly summarise the relativistic interpretation in the framework of Newtonian motion gauges in section~\ref{sec:NMrecap}, with the definition of a particular Newtonian motion gauge related to the backscaling method. In section~\ref{sec:rad} we discuss the general solution for the metric in both forwards and backscaled simulations and provide a numerical analysis of the underlying relativistic space-time. We provide an example of a non-standard cosmology with radiation in the late Universe in section~\ref{sec:decay} and conclude in~\ref{sec:conclude}.

\section{Newtonian backscaling}\label{sec:Newt}

Almost all current N-body simulations are performed employing Newtonian gravity. While this is thought to be a good approximation on small scales and at late times, the physics of the early Universe is fundamentally relativistic. This poses a challenge, namely to match relativistic solutions including radiation and matter in the early Universe to Newtonian late-time physics.
In addition, large scales close to the Hubble scale are subject to relativistic effects at all times and relativistic solutions (e.g., for the particle displacements at a given time) require the specification of a gauge, which is absent in a conventional Newtonian simulation.

In the commonly-used ``backscaling approach'',
one first generates the linear matter power spectrum today (at redshift $z=0$) using a first-order Boltzmann code to provide a target for the Newtonian simulation.
The power spectrum is typically calculated in the synchronous gauge and this sets a relativistic boundary condition for the largest (linear) scales in the simulation at $z=0$, by which time the radiation content of the universe is negligible compared to dark matter and dark energy.

\begin{figure}[tb]
  \centering
    \includegraphics[width=0.7\textwidth]{Kazuya_CDM_xkcd.pdf}
  \caption{{Comparison between a Newtonian simulation setting initial conditions based on the initial power spectrum for the relativistic density and one using the ``backscaling approach'', employing initial conditions based on the final power spectrum for the relativistic density. The relativistic evolution of the density in synchronous gauge follows the solid lines, from the initial time (red) to the present time (green).
When evolving using the Newtonian growth function, $D_+$, which neglects the effect of inhomogeneous radiation, the initial relativistic power spectrum (solid red line) leads to the wrong matter power spectrum at the present time (green dashed line). The backwards method, on the other hand, is designed to match the relativistic power spectrum in the linear regime at the present time (solid green line) by starting from a ``fictitious'' initial density field (red dashed line).}}
  \label{fig:figs_KazuyaPlot}
\end{figure}

The amplitude of the matter power spectrum is then scaled back using the \emph{Newtonian} linear growth function for matter perturbations, $D_+$, to set initial conditions when the nonlinear, but Newtonian N-body simulation is initialised, as shown in figure~\ref{fig:figs_KazuyaPlot}.  Matter-only simulations initialised in this way will end up, by construction, with the correct final matter power spectrum on the largest scales where the N-body simulation reproduces the Newtonian linear growth function.
Instead of performing a simulation of the actual Universe, backscaling can be understood as simulating an artificial radiation-free Universe that is designed to mimic our Universe on large scales and at the present time. On the other hand, small scales are assumed to be well-described in the Newtonian theory and they should remain unaffected as long as there is no significant transfer to small scale physics from large scales, where the actual relativistic solution has been replaced by the Newtonian backscaling solution.

The linearised Newtonian equations of motion employed for backscaling correspond to the Poisson, continuity and Euler equations for the Newtonian potential, $\Phi^{\rm N}$, the Newtonian density contrast, $\delta_{\rm cdm}^{\rm N}$, and the Newtonian velocity, $v_{\rm cdm}^{\rm N}$:
\begin{subequations}\label{eq:Newton}
\begin{align}
	 k^2 \Phi^{\rm N} &= 4\pi G a^2 \bar{\rho}_{\rm{cdm}}^{\rm N}  \delta_{\rm cdm}^{\rm N} \,, \label{NPoisson}\\
	 \dot{\delta}_{\rm cdm}^{\rm N} + k v_{\rm cdm}^{\rm N} &= 0 \,, \\
   \left[ \partial_\tau + {\cal H} \right] v_{\rm cdm}^{\rm N} &= -k \Phi^{\rm N} \label{NEuler}\,,
\end{align}
\end{subequations}
These three equations can be combined into the well-known ordinary differential equation for the Newtonian growth of structure in an expanding cosmology with scale factor $a(\tau)$:
\begin{equation}
\label{eq:Newtoniangrowth}
\left(\partial_{\tau} +\mathcal{H}\right)\dot{\delta}_{\rm cdm}^{\rm N} = 4 \pi G a^2 \bar \rho_{\rm cdm}\delta_{\rm cdm}^{\rm N} \,.
\end{equation}
We use the conformal time $\tau$ defined by $a {\rm{d}} \tau = {\rm{d}} t$, where $t$ is the cosmic time. We denote partial derivatives with respect to conformal time with $\partial_\tau$ or an overdot, and the conformal Hubble rate ${\cal H} \equiv \dot a/a$.

Being a second-order differential equation, it has two linearly-independent solutions and we label these as the growing mode, $D_{+}$, and the decaying mode, $D_{-}$. We normalise both to unity at the present time: $D_{\pm}(z=0) = 1$.
The general solution can be given in terms of these two modes as
\begin{equation}\label{eq:decNew}
\delta^{\rm N}(\tau) = C_{+}^{\delta^{\rm N}} D_+(\tau) + C_{-}^{\delta^{\rm N}} D_-(\tau) \,,
\end{equation}
where $C_{\pm}^{\delta^{\rm N}}$ are the remaining integration constants fixed by the boundary conditions (typically by the initial displacement field in the particle simulation).
The Wronskian of the two modes is defined by
\begin{equation}
\label{Wronskian}
W = D_+ \dot{D}_- - D_- \dot{D}_+ .
\end{equation}
It has the particularly simple equation of motion $\dot{W} = -\mathcal{H} W \,,$
and is thus required to decay as $a^{-1}$. This provides a simple but powerful numerical test for the linear independence of the two modes.

Backscaling, using a pure growing mode, from the present relativistic synchronous gauge density, $\delta^{\rm syn}(z=0)$, thus corresponds to setting the initial density contrast in the Newtonian simulation
\begin{equation} \label{eq:backIcs}
\delta^{\rm N}(\tau_{\rm ini})
= D_{+}(\tau_{\rm ini}) \,\delta^{\rm syn}(z = 0) \,.
\end{equation}
In the absence of radiation an analytic solution for the growing mode exists \cite{Demianski:2005is}
\begin{equation} \label{eq:Newgrowth2}
D^{\text{analytic}}_+
\propto  \hypF{\frac{1}{3}}{1}{\frac{11}{6}}{-\frac{\Omega_\Lambda}{\Omega_{\rm m}}a^3} \,,
\end{equation}
where
${}_2F_1$ is Gauss' hypergeometric function. In the presence of radiation, equation~\eqref{eq:Newgrowth2} no longer holds. However, one may approximate $D_+$ by the integral
\begin{equation} \label{eq:Newgrowth1}
D^\text{approx}_+
\propto \frac{\mathcal{H}}{a} \int {\rm{d}} a'\, \mathcal{H}^{-3}(a') \,,
\end{equation}
including the energy density of radiation in the Hubble expansion $\mathcal{H}$.\footnote{This quantity was computed (but not used internally) by {\sc class}{} \textsc{2.5.1} where it was denoted ``\texttt{gr.fac.\ D}''. {\sc class}{} \textsc{2.6.0} now computes $D_+$ and the growth factor $f$ correctly by directly solving the differential equation.}

\begin{figure}[tb]
  \centering
    \includegraphics[width=.95\textwidth]{GrowingDecayingBackground.pdf}
  \caption{The Newtonian growing and decaying modes, $D_+$ and $D_-$ in \eqref{eq:decNew}, calculated in a standard $\Lambda$CDM cosmology with massless neutrinos $\left(H_0=67.6\frac{\text{km}}{s \text{Mpc}},\Omega_m=0.29,\Omega_\Lambda=0.71 \right)$. We show for comparison the integral approximation, $D^\text{approx}_+$ in \eqref{eq:Newgrowth1}, and the analytic growth factor, $D^\text{analytic}_+$ in \eqref{eq:Newgrowth2}, for the same cosmology. The error is at the \% level at $z=49$, and the same holds for the logarithmic derivative, $f\equiv {{\rm{d}}\log D_+}/{{\rm{d}}\log a}$.
  The Wronskian $W$ of $D_+$ and $D_-$ is proportional to $a^{-1}$ and shows that our two modes are linearly independent.}
   \label{fig:figs_GrowingDecayingBackground}
\end{figure}

Our numerical computations of $D_+$ and $D_-$ are shown in figure~\ref{fig:figs_GrowingDecayingBackground}. This demonstrates that $D^{\rm approx}_+$, $D^\text{analytic}_+$ and $D_+$ disagree at the percent level before redshift $z=49$. The same holds for the logarithmic derivative, $f\equiv {{\rm{d}}\log D_+}/{{\rm{d}}\log a}$, used to set the initial velocity field. The analytic expression~\eqref{eq:Newgrowth2} works better than the approximate form~\eqref{eq:Newgrowth1}, but even a 1\% error in the initial power spectrum is unacceptable and one should solve the second order differential equation \eqref{eq:Newtoniangrowth} directly when radiation cannot be neglected.

In a numerical simulation one needs to set up particles to represent the density $\delta^{\rm N}$ and the three velocity $v^{{\rm N},i}$ in a real space. In practice, the density field is constructed by displacing initially homogenous N-body particles according to the Zel'dovich displacement field
\begin{equation}
\fett{\nabla}^i \Psi_i(\tau_{\rm ini}) =- \delta^{\rm N}(\tau_{\rm ini})
= - D_{+}(\tau_{\rm ini}) \,\delta^{\rm syn}(z = 0)\,.
\end{equation}
Assuming a pure growing mode, the velocities may be obtained directly from the density: $\fett{\nabla}^i v_i^{\rm N} = -\frac{\dot{D}_+}{\mathcal{H} D_+} \delta^{\rm N}$. Employing the displacement field we find
\begin{equation}
v^{\rm N}_i(\tau_{\rm ini}) = f(\tau_{\rm ini}) \Psi_i(\tau_{\rm ini})\,.
\end{equation}
This allows for an efficient way to generate both, the initial velocities and the initial particle distribution, based only on the displacement field.

\section{The Newtonian motion gauge framework} \label{sec:NMrecap}

The conventional Newtonian backscaling lacks a consistent relativistic interpretation to quantify the underlying approximations and limitations.
This paper aims to solve these open issues and provide a fully relativistic framework, making use of the recently introduced Newtonian motion gauges~\cite{Fidler:2016tir}.

The idea of the Newtonian motion (Nm) approach is to choose a gauge in which, by construction, the CDM component
follows Newtonian trajectories, whilst incorporating relativistic effects in the underlying space-time. The relativistic trajectories
will then coincide with the Newtonian ones, which implies that
we can identify a relativistic space-time complementing a Newtonian simulation.
The challenge of relativistic particle evolution is thus separated into a purely Newtonian, but highly non-linear part and a relativistic, but perturbative part. Particle trajectories can be solved by conventional Newtonian N-body simulations, while a standard linear Einstein--Boltzmann code is used for the computation of the underlying relativistic space-time.

\subsection{Notation and conventions}

We adopt the same conventions as ref.~\cite{Fidler:2016tir}, and use
the following Fourier metric perturbations about a homogeneous and isotropic Friedmann--Lema\^itre--Robertson--Walker background,
\begin{subequations}
\label{metric-potentials}
\begin{align}
  g_{00} &= -a^2 \left( 1 + 2A \right) \,, \\
  g_{0i} &=  a^2\, {\rm{i}} \hat k_i B \,, \\
  g_{ij} &= a^2 \left[ \delta_{ij} \left( 1 + 2 H_{\rm L} \right) + 2 \left( \delta_{ij}/3 - \hat k_i \hat k_j \right) H_{\rm T} \right] \,,
\end{align}
where $\hat k_i \equiv k_i/k$, with $k \equiv |\fett{k}|$,
 $A$ is the perturbation of the lapse function, $B$ is a scalar perturbation in the shift, and $H_{\rm L}$ and $H_{\rm T}$ are respectively the trace and trace-free scalar perturbations of the spatial metric.
The matter and radiation content is characterised by the stress-energy tensor
\begin{align}
	T^{0}_{\phantom{0}0} &= - \sum_\alpha \bar \rho_\alpha \left( 1  + \delta_\alpha \right) \equiv - \bar \rho \left( 1  + \delta \right) \,, \\
	T_{{\phantom{0}}0}^i &=  \sum_\alpha (\bar \rho_\alpha +\bar p_\alpha ) \,{\rm{i}} \hat k^i v_\alpha \equiv (\bar \rho +\bar p )\, {\rm{i}} \hat k^i v \,, \\
	T^{i}_{\phantom{i}j} &= \sum_\alpha (\bar p_\alpha+\delta p_\alpha ) \delta^i_j + \bar p_\alpha \left( \delta_j^i/3 - \hat k^i \hat k_j \right) \Pi_\alpha
  \equiv (\bar p+\delta p ) \delta^i_j + \bar p \left( \delta_j^i/3 - \hat k^i \hat k_j \right)  \Pi  \,, \label{Tmunu}
\end{align}
\end{subequations}
where the summation over $\alpha$ runs over all fluid species, i.e.,
CDM, baryons, photons and neutrinos; $\delta \equiv (\rho - \bar \rho) / \bar \rho$
is the density contrast and $\bar \rho$ the background density,
$\overline p$ the background pressure and its perturbation $\delta p$, and $\Pi$ is the anisotropic stress of radiation.

\subsection{Relativistic dynamics}

The Einstein--Boltzmann equations provide the following equations of motion for the CDM component to first order in a yet unfixed gauge \cite{Hu:2004xd},
\begin{subequations}
\begin{align}
 	 k^2 \Phi &= 4 \pi G a^2 \left[ \bar \rho \delta + 3 {\cal H}  \left( \bar \rho+ \bar p \right)  ( v - B ) /k \right]  \,, \label{eq:EE2}\\
\dot{\delta}_{\rm{cdm}} + k v_{\rm cdm} &= - 3 \dot H_{\rm L} \,, \label{eq:NMconti} \\
 \left[ \partial_\tau + {\cal H} \right] v_{\rm cdm} &= -k ( \Phi +  \gamma ) \,, \label{eq:NMvelo}
\end{align}
\end{subequations}
where $\Phi$ is the gauge-invariant Bardeen potential \cite{Bardeen:1980kt}
and $\gamma$ is a relativistic correction describing the forces acting on the dark matter particles,
\begin{equation} \label{def:gammaFourier}
 - k^2  \gamma \equiv  \left( \partial_\tau + {\cal H} \right) \dot{ H}_{\rm T} - 8\pi G a^2 \bar p \Pi \,.
\end{equation}
The Newtonian motion gauges are defined by the spatial gauge condition~\cite{Fidler:2016tir}
\begin{equation} \label{eq:NMgauge}
\gamma + \Phi= \Phi^{\rm N} \, ,
\end{equation}
which guarantees a Newtonian motion for the dark matter particles. That is, the relativistic Euler equation~(\ref{eq:NMvelo}) has the same form as the Newtonian Euler equation (\ref{NEuler}), where the Newtonian potential, $\Phi^{\rm N}$, is given by the Newtonian Poisson equation~(\ref{NPoisson}). Note that the Newtonian density $\delta^{\rm N}_{\rm cdm}$ is the coordinate density and does not include the relativistic volume deformation of the underlying space-time; it is related to the full (relativistic) matter density, $\delta_{\rm cdm}$, via
\begin{equation} \label{eq:gemoetry}
\delta^{\rm N}_{\rm cdm} = \delta_{\rm cdm} + 3 H_{\rm L}.
\end{equation}
Thus, although the particle displacements in a Newtonian simulation now agree with the relativistic particle displacements in a Newtonian motion gauge, the Newtonian density calculated in an N-body simulation does not in general match the relativistic density.

Note that the temporal gauge is not fixed by the Newtonian motion gauge and we leave it unspecified for now.

\subsection{Relation to the N-body gauge}
\label{forwards}

Another useful and related gauge is the N-body gauge~\cite{Fidler:2015npa} (denoted in the following equations by the ``Nb'' superscript). It is defined with a gauge condition such that $H_{\rm L}^{\rm Nb} = 0$, so that the physical volumes are not modified by relativistic volume deformations. As a result the relativistic matter density corresponds to the simulation density computed by a naive counting of particles.

The temporal gauge condition for the N-body gauge is $B^{\rm Nb} = v^{\rm Nb} $, so that the constant time hyper-surfaces are orthogonal to the total matter and radiation 4-velocity. As a result, the N-body gauge further satisfies the spatial constraint
$H_{\rm T}^{\rm Nb} = 3\zeta$,
where $\zeta$ is the
gauge-independent
comoving curvature perturbation~\cite{Hu:2004xd}
\begin{equation}
\zeta = H_{\rm L} +\frac{1}{3} H_{\rm T} + \mathcal{H} k^{-1} (B - v) \,.
\end{equation}

We have previously shown in ref.~\cite{Fidler:2016tir} that the Newtonian motion gauge can be completely fixed by identifying it with the N-body gauge at a particular time. In that work, we focus on matching with the N-body gauge at an early epoch, and evolving forward to see how the gauge choices diverge. In the absence of radiation, the two gauges remain synchronised, but they quickly diverge if the matching occurs sufficiently early when radiation is important. We refer to this type of gauge fixing as \emph{forwards Newtonian motion gauges.}

\subsection{Backscaling and Newtonian motion gauges}

Instead of connecting the Newtonian motion gauges to the N-body gauge at the initial time, we may alternatively match the Newtonian motion gauge to the N-body gauge at the present time. As a Newtonian simulation is interpreted in the corresponding Newtonian motion gauge, this implies that the output of the Newtonian simulation must match the present day power spectrum in the N-body gauge on the linear scales.

The Newtonian motion gauge and N-body gauge remain synchronised in the absence of radiation;
working backwards in time from the present, this correspondence holds
in the standard $\Lambda$CDM cosmology until the beginning of the matter era at which time
radiation becomes important and the two gauges begin to diverge.
Therefore, the initial conditions in this Newtonian motion gauge no longer match the N-body gauge initial power spectrum in the presence of radiation.

We call this type of Newtonian motion gauge the \emph{backwards Newtonian motion gauges}. In the following we show that they provide a relativistic embedding of the usual Newtonian ``backscaling approach''.

\section{Solutions to the Newtonian motion gauge condition}\label{sec:rad}

The Newtonian motion gauge condition (eq.\,\ref{eq:NMgauge}) is equivalent to a second-order differential equation for $H_{\rm T}$, as can be seen from the explicit form of $\gamma$ in eq.\,(\ref{def:gammaFourier}).
Without specifying the cosmology, the metric potential $H_{\rm T}$ evolves according to:
\begin{equation}\label{eq:metricfull}
\left(\partial_{\tau} +\mathcal{H}\right)\dot{H}_{\rm T} - 4 \pi G a^2 \bar \rho_{\rm cdm}\left( H_{\rm T} -3\zeta \right)= S
\end{equation}
with a source term $S$ that is non-zero in the presence of radiation,
\begin{equation}
S = 4 \pi G a^2 \left(\bar \rho_{\rm other}\delta_{\rm other} +  3 \mathcal{H}k^{-1} (\bar \rho+\bar p)_{\rm other}(v-B) + 2 \bar p \Pi\right) \,,
\end{equation}
where $\bar{\rho}_{\rm other}$, etc, represent fluid species other than matter.

In the radiation-free limit, we can ignore the source term to find
\begin{equation} \label{eq:Metricgrowth}
\left(\partial_{\tau} +\mathcal{H}\right)\dot{H}_{\rm T} = 4 \pi G a^2 \bar \rho_{\rm cdm}\left( H_{\rm T} -3\zeta \right).
\end{equation}
The homogenous part of the differential equation for $H_T$ (eq.\,\ref{eq:Metricgrowth}) is identical to the equation for the Newtonian density (eq.\,\ref{eq:Newtoniangrowth}). It therefore shares the same linearly independent solutions $D_{+}(\tau)$ and $D_-(\tau)$. The particular solution is driven by the term proportional to $\zeta$ on the right-hand side.
Since $\zeta$ is constant in the absence of radiation, we find the simple particular solution $H_{\rm T} = 3 \zeta$, and
the full solution is
\begin{eqnarray}\label{eq:metricdeco}
H_{\rm T}(\tau) &=& C_{+}^{H_{\rm T}} D_+(\tau) + C_{-}^{H_{\rm T}} D_-(\tau) + 3 \zeta, \label{eq:Newtondeco}
\end{eqnarray}
where the integration constants $C_{+}^{H_{\rm T}}$ and $C_{-}^{H_{\rm T}}$ encode the two parameter residual gauge freedom of the Newtonian motion gauges.
The spatial gauge condition of the N-body gauge is equivalent to $H_{\rm T}(\tau) = 3 \zeta$ and thus setting $C_{+}^{H_{\rm T}} = C_{-}^{H_{\rm T}} = 0$ corresponds to fixing the Newtonian motion gauge to the N-body gauge.\footnote{We remark that it is further possible to set $C_{+}^{H_{\rm T}} = -C_{-}^{H_{\rm T}}\neq0$, but this choice is only equal to the N-body gauge momentarily at the present time and rapidly diverges to a different gauge. Mathematically it would correspond to backscaling methods including decaying modes.}

However, in the early Universe radiation plays an important role and we must solve the full equation~(\ref{eq:metricfull}).
Using the method of variation of constants, a solution of the space-time can be constructed from equation~(\ref{eq:metricdeco}) by adding a time dependence to the coefficients,
\begin{equation}
H_{\rm T} = C_{+}^{H_{\rm T}}(\tau) D_+(\tau) + C_{-}^{H_{\rm T}}(\tau) D_-(\tau) + 3 \zeta \,.
\end{equation}
For this ansatz to solve equation~(\ref{eq:metricfull}) the coefficients must fulfil the conditions
\begin{align}
	 \dot{C}_{+}^{H_{\rm T}} D_+ + \dot{C}_{-}^{H_{\rm T}} D_- &= 0 \,,\\
	 \dot{C}_{+}^{H_{\rm T}} \dot{D}_+ + \dot{C}_{-}^{H_{\rm T}}\dot{D}_- &= S - 3\ddot{\zeta} - 3 \mathcal{H}\dot{\zeta} \equiv \tilde{S} \,,
\end{align}
where the corrections included in $\tilde{S}$ reflect that the comoving curvature is no longer conserved in the presence of relativistic species.

In the backwards Newtonian motion gauge, the equations above can be integrated with the remaining integration constants fixed by the specification of the gauge at the final time:
$C_{+}^{H_{\rm T}}(\tau_{\rm final}) = C_{-}^{H_{\rm T}}(\tau_{\rm final}) = 0$~(corresponding to the N-body gauge).
Using the Wronskian $W$, defined in \eqref{Wronskian}, the time-dependent coefficients can be found to be
\begin{equation}
	C_{\pm}^{H_{\rm T}}(\tau) = \pm \int \limits_{\tau}^{\tau_{\rm final}} \,\tilde{S}(\tilde \tau) D_\mp(\tilde \tau) \,W(\tilde \tau)^{-1} {\rm{d}} \tilde \tau \,.
\end{equation}
These coefficients are zero as long as radiation does not have an impact after time $\tau$. This means that even including radiation at early times we will have $H_{\rm T} = 3\zeta = \text{const.}$ for most of the cosmic evolution. Only at early times will the gauge depart from N-body gauge due to the impact of radiation.

\begin{figure}[tb]
  \centering
    \includegraphics[width=\textwidth]{H_T_BS_diff.pdf}
  \caption{Comparison of the metric potentials, $H_{\rm T}$, in the backward and the forward Newtonian motion gauge approach in a standard $\Lambda$CDM cosmology. \emph{Left panel:} $H_{\rm T}^\text{Nb}=3\zeta$ in the N-body gauge. \emph{Centre panel:} The deviation from the N-body gauge metric potential of $H_{\rm T}^{\text{Nm}100}$ in the ``forwards approach'' when the Newtonian motion gauge is initiated at $z=100$. The modes starts to deviate due to the residual radiation. \emph{Right panel:} The deviation from the N-body gauge metric potential of $H_{\rm T}^{\text{NmB}}$ in the ``backscaling approach''. All modes initially differ from the N-body gauge potential but they are quickly pushed towards the N-body gauge due to the residual radiation. The modes are normalised to $\zeta=-1$ at super-horizon scales according to the {\sc class}{}-convention.}
  \label{fig:figs_H_T_decaying2}
\end{figure}

We have implemented these equations in {\sc class}{} and present our results for a standard $\Lambda$CDM cosmology in figure~\ref{fig:figs_H_T_decaying2}. The metric potential $H_{\rm T}$ stays close to the N-body gauge value for the entire late-time evolution, while it changes significantly at the early times. This is opposed to the ``forwards approach'' discussed in subsection~\ref{forwards} and presented in our previous paper \cite{Fidler:2016tir}; this case is shown in the second panel of the figure. In figure~\ref{fig:figs_H_T_transfer} we show the same quantities but now as function of $k$ for eight different redshifts. The ``backscaling method'' evidently works extremely well for the analysed redshifts and scales. The corrections are still negligible at $z=20$, but quickly become relevant at earlier times.

\begin{figure}[tb]
  \centering
    \includegraphics[width=\textwidth]{H_T_transfers3.pdf}
  \caption{Comparison of the metric potentials, $H_{\rm T}(k)$, in the backward and the forward Newtonian motion gauge approach in a standard $\Lambda$CDM cosmology. \emph{Top row:} $H_{\rm T}^\text{Nb}=3\zeta$ in the N-body gauge. \emph{Centre row:} The deviation from the N-body gauge metric potential of $H_{\rm T}^{\text{Nm}100}$ in the ``forwards approach'' when the Newtonian motion gauge is initiated at $z=100$. The scale dependence is fixed by the early radiation and the deviation then grows with $D_+$.  \emph{Bottom row:} The deviation from the N-body gauge metric potential of $H_{\rm T}^{\text{NmB}}$ in the ``backscaling approach'', where the y-axis in the bottom left panel is rescaled by a factor of $10000$. For the observationally relevant redshifts the difference is below $10^{-3}$ on all scales, and the difference is always below $0.01$ for redshifts $z<20$. All modes are normalised to $\zeta=-1$ at super-horizon scales according to the {\sc class}{}-convention.}
  \label{fig:figs_H_T_transfer}
\end{figure}

Our results can be understood in the following way. In the forward Newtonian motion gauges we start on the N-body gauge, utilising the N-body gauge initial displacements. However, due to the impact of radiation we are pushed away from the N-body gauge.
By contrast, in the backwards approach we start away from the N-body gauge, using different initial displacements. Then, due to the impact of radiation, we are pushed exactly onto the N-body gauge. As the N-body gauge is already a Newtonian motion gauge in the absence of radiation, we simply stay on that space-time for the remaining cosmic evolution in the standard $\Lambda$CDM cosmology.

The relativistic initial matter displacements in the backwards gauge could be directly obtained from a linear Boltzmann code. Alternatively, we know that in the backwards Newtonian motion gauge the present day linear Newtonian density agrees with the linear N-body gauge density, since the volume deformation $H_{\rm L}$ is vanishing at that time, and we can scale back the present day Newtonian density to the initial time using $D_+$, even in the presence of radiation in the early Universe,
\begin{equation}
\delta^{\rm N}(\tau_{\rm ini}) =  D_+(\tau_{\rm ini}) \,\delta^{\rm Nb} (\tau_{\rm final}) \,.
\end{equation}
This relation corresponds exactly to the computation of the initial conditions in the
conventional backscaling method~(\ref{eq:backIcs}), only that instead of the synchronous gauge density we use the N-body gauge at the present time. Note that the density in the synchronous gauge and N-body gauge are almost identical at the present time in a standard cosmology due to the absence of radiation.

This backwards gauge is not the only possible Newtonian motion gauge fixing. However it is the unique choice corresponding to a simple interpretation in terms of the standard back-scaling approach to set initial conditions. Only the choice of $C_{\pm}^{H_{\rm T}} = 0$ forces a constant metric potential $H_{\rm T} = 3\zeta$ in the absence of radiation. In addition the identification of the relativistic density and the simulation density at the final time requires a vanishing volume deformation $H_{\rm L} = 0$. These two conditions together uniquely select the N-body gauge as the metric at the present time. This means that any other gauge fixing will lead to a more complex dynamics for the relativistic metric potentials. The N-body gauge is thus uniquely suited for the ``backscaling approach''. The commonly used synchronous gauge density is suitable for backscaling only because it is identical to the N-body gauge density in the absence of radiation.

By gauge fixing the Newtonian motion gauge to the N-body gauge at the final time, we have provided the first fully relativistic interpretation of the backscaling method. We have explained why it is crucial to utilise the N-body gauge density at the present time and have constructed the backwards Newtonian motion gauge in which conventional N-body simulations can be interpreted and observables can be constructed in agreement with general relativity at first order.

\subsection{Summary of the relativistic backscaling prescription}

We briefly summarise our backscaling prescription that enables a consistent relativistic interpretation of Newtonian simulations in a $\Lambda$CDM Universe:
\begin{itemize}
\item Using a linear relativistic Boltzmann code, generate the present day matter power spectrum in the N-body gauge. (For practical purposes the power spectrum in synchronous gauge is a good approximation and almost identical the one in N-body gauge at the present time).
\item Scale the power spectrum back to the time of initialisation using the linear Newtonian growing mode, $D_+$ in eq.~(\ref{eq:backIcs}), consistently including radiation in the background.
\item Compute initial displacements from this power spectrum and utilise the logarithmic derivative of the growing mode, $f$, to set the initial velocities.
\item Evolve these initial conditions under Newtonian gravity to the present time, using a Newtonian N-body simulation containing pure pressureless dust, while including radiation in the background evolution.
\item Interpret the output in the backwards Newtonian motion gauge coordinates, which can be computed using a linear Boltzmann code. For observables that focus on the late evolution this step simplifies as the corresponding Newtonian motion gauge is identical to the N-body gauge in the late Universe. Note that the N-body gauge metric and displacements are not identical to those in the synchronous gauge.\footnote{N-body gauge perturbations can be extracted from {\sc class}{}. An implementation including Newtonian motion gauge perturbations is available on request.}
\end{itemize}

\section{Decaying dark matter}\label{sec:decay}

We have shown that the backwards Newtonian motion gauge is well suited for the analysis of standard $\Lambda$CDM models where the Universe is dominated at late-times by cold dark matter and a cosmological constant.  This is no longer true in the presence of radiation or other relativistic effects at late times, and we demonstrate this explicitly by investigating a decaying dark matter model~\cite{Audren:2014bca,Ichiki:2004vi,Adams:1998nr,DeLopeAmigo:2009dc,Gong:2008gi}, described in detail in appendix~\ref{app:ddm}.

In the Newtonian picture, decaying dark matter primarily affects the cosmic evolution through its impact on the background.
A significant fraction of the dark matter density is transformed into decay radiation, changing the expansion of the Universe.  This effect can be partially modelled in a Newtonian simulation by reducing the mass of the dark matter particles as a function of the conformal time.   In this Newtonian limit, the decay is assumed to occur homogeneously.

However, in the full relativistic theory, the impact of decaying dark matter is more complex. The decay radiation affects the gravitational forces acting on the matter particles and furthermore, the decay of dark matter is described by its local clock (comoving proper time). Observed in conformal time, this implies that the decay is faster or slower in different regions of space-time, which induces perturbations in the dark matter density.

We will analyse this behaviour using the Newtonian motion gauge framework.
For the decaying fluid, described by the density contrast $\delta_{\rm dcdm}$ and the velocity $v_{\rm dcdm}$, we find the equations of motion, \begin{align}
\dot{\delta}_{\rm dcdm} + k v_{\rm dcdm} &= - 3 \dot{H}_{\rm L} + a \Gamma A  \,, \\
\left(\partial_{\tau} + \mathcal{H} \right)v_{\rm dcdm} &= - k (\Phi + \gamma) \,.
\end{align}
Here, the metric potential $A$, defined in \eqref{metric-potentials}, describes the inhomogeneous decay of the matter density.
In addition we add a full Boltzmann hierarchy for the induced decay radiation contributing to $\delta_{\rm rad}$, $v_{\rm rad}$, $p$ and $\Pi$.
We define a Newtonian motion gauge in the usual way by enforcing $\Phi +\gamma = \Phi^{\rm N}$.

Employing the continuity equations we obtain the relation between the Newtonian density and the relativistic density,
\begin{equation} \label{eq:deltaNdiff}
\delta_{\rm dcdm}^{\rm N} - \delta_{\rm dcdm}  = 3 H_{\rm L}  - \Gamma \int {\rm{d}}\tau \, a A \,,
\end{equation}
The Newtonian matter density, $\delta_{\rm dcdm}^{\rm N}$, is a pure coordinate density and does not include the impact of the relativistic volume deformation. The relativistic density on the other hand is based on the same particle positions (as we are in a Newtonian motion gauge), but evaluates them on the non-trivial space-time, inducing the correction of $3 H_{\rm L}$.
The second term describes the generation of dark matter perturbations by particle decays in regions where cosmic and dark matter times do not coincide.

This latter term is not implemented in the Newtonian simulation, but should not be ignored.  In particular, this contribution can be included in the relativistic density and added to the output of the Newtonian simulation as a post-processing.
To do this, we define the density contrast $\xi$, describing the fraction of mass lost by matter particles based on their spatial position,
\begin{equation}
\label{eq:xi}
\xi = - \Gamma \int \limits_{\tau_{\rm ini}}^{\tau} {\rm{d}}\tau  \,a A.
\end{equation}
where we have chosen to set $\xi = 0$ initially, corresponding to no decays of dark matter prior to the initial time.

As described in section~\ref{sec:NMrecap}, the Newtonian motion gauge condition is enforced by the evolution of $H_{\rm T}$, and its evolution in this model is determined by
\begin{align}
\left(\partial_{\tau} +  \mathcal{H}\right) & \dot{H}_{\rm T} -  4 \pi G a^2 \bar \rho_{\rm dcdm}\left( H_{\rm T} -3\zeta \right)=  \nonumber \\
& 4 \pi G a^2 \left( \bar \rho_{\rm other}\delta_{\rm other}   +  3 \mathcal{H}k^{-1} ( \bar \rho + \bar p)_{\rm other}(v-B) - \bar \rho_{\rm dcdm} \xi + 2 \bar p \Pi\right).
\end{align}

We see that the decay of dark matter introduces an extra source term proportional to the decay fraction $\xi$. In addition, the late-time presence of the decay radiation introduces non-vanishing terms: $\delta_{\rm other}$, $p_{\rm other}$ and $\Pi$.

\subsection{Numerical results}

We perform an explicit calculation of the Newtonian motion gauge space-time for the relativistic interpretation of Newtonian simulations including decaying dark matter. In this section we work in the comoving temporal gauge $B=v$.
We have implemented the relevant equations in {\sc class}{} and solve for the metric potentials and the decay fraction \eqref{eq:xi} in linear theory. We study two scenarios including dark matter that are compatible with current observations. In the first scenario, called ``DCDM'' in our plots, the entirety of the dark matter is decaying, while in the second scenario, called ``CDM+DCDM'', dark matter is made up by two species of which only one is decaying. For further details on the models we refer to appendix~\ref{app:ddm}.

\begin{figure}[tb]
  \centering
    \includegraphics[width=0.95\textwidth]{DecayMetricPotentialsk0002.pdf}
  \caption{Metric potentials in the forward Newtonian motion gauge approach in a cosmology where dark matter decays with a lifetime of $\Gamma_\text{dcdm}^{-1}=160\text{Gyr}$, leading to the presence of radiation at late times. The curves represent different initialisation times of the forward Newtonian motion gauge ($z=100$ and $z=500$). Here we show the time evolution of all four metric potentials for the mode $k=0.002 \text{Mpc}^{-1}$. All the forward-Nm gauge metric potentials remain perturbative. The mode is normalised to $H_{\rm T}=3\zeta=-1$ at super-horizon scale.}
  \label{fig:figs_DecayMetricPotentialsk001}
\end{figure}

Our results for the forward Newtonian motion gauge are summarised in fig.~\ref{fig:figs_DecayMetricPotentialsk001}, where we show a typical initialisation redshift of $z=100$ and in addition a very early initialisation at $z=500$ where the metric in comparison is affected mostly by the early radiation effects.
The metric potential $A$ vanishes in the absence of radiation. In the presence of dark matter decays we find that $A$ is growing due to the induced decay radiation. We see that $H_{\rm T}$ starts on the $N$-body gauge value, but is quickly driven away from the N-body gauge by residual radiation at early times. The presence of the decay radiation further enhances this at late times. At the same time the volume deformation $H_{\rm L}$, which vanishes in the $N$-body gauge, is induced.
Nonetheless the metric perturbations remain of the same order of magnitude as the comoving curvature perturbation $\zeta$, suggesting a perturbative approach to calculating the space-time remains valid.

A Newtonian simulation with decaying dark matter can thus be given a relativistic interpretation, by embedding it in this particular space-time and changing the final particle masses according to the decay fraction $\xi$. In fig.~\ref{fig:figs_L_and_M_decaying} we show the resulting difference between the simulation density and the relativistic density by comparing the volume deformation $3H_{\rm L}$ and the decay fraction $\xi$, the two corrections appearing in equation~\eqref{eq:deltaNdiff}. The left panel describes a standard $\Lambda$CDM cosmology without decays, and thus $\xi = 0$. The volume deformation at the present time, $3H_{\rm L}$, is vanishing if the simulation is initiated at sufficiently late times when radiation may be neglected. The right panels show the results in our two models including decaying dark matter. The decay fraction, $\xi$, is the dominant effect on the smaller scales and does not depend significantly on the initialisation time.
The volume deformation, $3H_{\rm L}$, contains the oscillations from early residual radiation, in agreement with the non-decaying case. On top of this, there is a smooth contribution from late decay radiation and, on the small scales, the impact of the inhomogeneous decay fraction $\xi$.

\begin{figure}[tb]
  \centering
    \includegraphics[width=0.95\textwidth]{L_and_M_decaying5a.pdf}
  \caption{We plot the two relativistic corrections to the Newtonian matter density, the volume deformation~$3H_{\rm L}$ (top row) and the decay fraction $\xi$ in Eq.~\eqref{eq:xi}, due to inhomogeneous dark matter decay (bottom row). The various lines correspond to different initialisation times for the Newtonian motion gauge in the ``forwards approach''. The top left panel shows the CDM case without decaying dark matter, hence $\xi=0$ in this case. The deviation of $3H_{\rm L}$ from zero is due to residual radiation at early times, so initialising later reduces $3H_{\rm L}$. For the two models including decaying dark matter (as described in the appendix), $3H_{\rm L}$ is mostly dominated by the late-time radiation from the dark matter decay, so it depends rather weakly on the initialisation redshift. The $\xi$ correction has almost no dependence on the time of initialisation as long as it is before $z=5$ since most of the dark matter decays happen later.}
  \label{fig:figs_L_and_M_decaying}
\end{figure}

We also analyse our models in the backwards Newtonian gauge.
The resulting metric is plotted in fig.~\ref{fig:figs_H_T_decaying}, where the late-time presence of radiation induces evolution in the metric over the whole lifetime of the Universe.
In order to finish on the $N$-body gauge space-time, the metric potentials have to start at very large initial values and only approach the simple $N$-body gauge at the final time.
It is thus no longer consistent to analyse the metric potentials in a perturbative approach
and the backscaling procedure should be avoided.

\begin{figure}[tb]
  \centering
    \includegraphics[width=0.95\textwidth]{H_T_decaying_diff.pdf}
  \caption{\emph{Top row:} The N-body gauge metric potential, $H_{\rm T}^\text{Nb}$, for the three scenarios, standard cold dark matter (CDM), decaying dark matter (DCDM) and mixed CDM+DCDM. \emph{Bottom row:} The difference between the metric potential in the backwards Newtonian gauge, $H_{\rm T}^\text{NmB}$, and the $N$-body gauge potential,  $H_{\rm T}^\text{Nb}$. In the CDM case all modes quickly converge on the $N$-body gauge value as shown previously. However, for the decaying dark matter scenarios the perturbative description breaks down in the ``backscaling approach''; the gauge potentials become large at early times and cannot be reliably computed in a first-order Boltzmann code.}
  \label{fig:figs_H_T_decaying}
\end{figure}

By comparison, the metric perturbations in the ``forwards approach'' remain small at all times. It is particularly efficient for including late-time sources of radiation, which only affect the metric evolution for a short period of time. Early radiation can also be included in this framework so long as the sources are relatively weak. In contrast, in the ``backscaling approach'' the early metric potentials are affected by all sources that will be relevant in the future, making late-time sources problematic, while it is more reliable for early sources.

We conclude that backscaling is well-suited to including early, residual radiation in the Universe, but late-time relativistic effects such as decaying dark matter should be avoided. For these the forward Newtonian motion approach is better suited as the metric perturbations stay small.

\section{Conclusions}\label{sec:conclude}

N-body simulations are necessarily performed using only an approximate physical model; they are intrinsically Newtonian rather than fully relativistic and they do not include all fluid species, including radiation at most through its effect on the background expansion. In addition, they treat the baryons and dark matter particles as a single pressureless fluid, neglecting the fact that the coupling to radiation in the early Universe leads to different initial positions and velocities~\cite{Tseliakhovich:2010bj}.

Some of these deficiencies are addressed by the
``backscaling approach'', which compensates for the lack of radiation and mixed baryon+CDM initial conditions
by starting with a set of \emph{fictitious} initial displacements designed to reproduce the linear-scale synchronous gauge total matter density at the present.  However, a number of questions remain regarding this approach, in particular how the late-time non-linear evolution is affected by missing elements, such as radiation perturbations and anisotropic stress arising from matter on non-linear scales.
Moreover, there is no relativistic interpretation of gauge-dependent quantities such as displacements and densities in a purely Newtonian framework.

We have addressed these questions by constructing a consistent relativistic treatment based on the Newtonian motion gauge approach~\cite{Fidler:2016tir},
where we construct a relativistic space-time where the matter follows the Newtonian trajectories calculated in the N-body code.
By matching the linear matter density to the relativistic boundary conditions at the final time, we reproduce the ``backscaling approach'' in the matter sector.  The physical interpretation of these results requires also the metric potentials in the Newtonian motion gauge, in particular the trace-free spatial metric perturbation $H_{\rm T}$, whose evolution consistently accounts for the presence of inhomogeneous radiation so long as it remains perturbative.

These solutions are particularly simple in the radiation-free limit, where the Newtonian motion gauge reduces to the N-body gauge and the metric perturbation is constant, $H_{\rm T} = 3 \zeta$, where $\zeta$ is the
comoving curvature perturbation.  The presence of radiation in the early Universe causes the N-body gauge and Newtonian motion gauges to diverge, but by matching them at the present, we have shown that they coincide for most of the period where non-linear structures are growing.

Using Newtonian motion gauges, we specify the underlying space-time and
show that in standard $\Lambda$CDM cosmologies the metric perturbations remain small, and can consistently be solved using linear Boltzmann codes.
Newtonian motion gauges provide
the fully relativistic space-time, enabling, for example, the use of relativistic ray-tracing.

In theories beyond the standard model backscaling must be performed with great care. The space-time may not have a simple late-time limit and the metric potentials may become large.
We have presented an example including decaying dark matter and conclude that backscaling is not an appropriate method to generate initial conditions for simulating such cosmologies. Instead the forward Newtonian motion approach may be employed and remains
well defined. In general we conclude that modifications to the late-time dynamics typically invalidate backscaling. For these models only a direct analysis in the Newtonian motion gauge framework can show whether backscaling can be used.

Backscaling could also be applied for simulating two distinct baryon and cold dark matter fluids. These fluids follow an identical velocity distribution at the present time. Employing backscaling we thus obtain initial conditions for a single fluid Universe that is designed to reproduce the present day power spectrum when fully including baryons and cold dark matter.
This approach is discussed in \cite{Valkenburg:2016xek}, comparing the backscaling method to setting initial conditions for both baryons and cold dark matter in the N-body gauge, however neglecting the impact of radiation.
Employing the Newtonian motion gauges, we can use exact initial conditions for both species in the ``forwards approach'' and thus fully include the baryon physics in the simulation. With backscaling, however, the decreasing difference between the baryon and cold dark matter velocities and densities resemble a decaying mode and may cause the metric potentials to diverge, comparable to the dynamics we observe in the decaying dark matter case.
Thus, while a ``forwards approach'' may be employed, detailed follow-up studies are required to investigate the performance of backscaling and baryons, which we leave for future work.

Another highly important application for backscaling
is generating initial conditions of N-body simulations in the presence of massive neutrinos (see \cite{Zennaro:2016nqo} for a Newtonian perspective), or modified gravity \cite{Valkenburg:2015dsa}. Massive neutrinos in particular resemble dark matter at late times, whilst they contribute to the radiation content at early times. In the language of Newtonian motion gauges this represents a modification to the ``Newtonian'' dynamics at the early times. In a forthcoming study we will investigate which type of gauges are suitable for this problem.

\acknowledgments

CF is supported by the Wallonia-Brussels Federation grant ARC11/15-040 and the Belgian Federal Office for Science, Technical \& Cultural Affairs through the Interuniversity Attraction Pole P7/37.
TT, RC, KK, and DW are supported by the UK Science and Technologies Facilities Council grants ST/N000668/1.
CR is supported by the DFG through the SFB-Transregio TRR33 ``The Dark Universe''.
KK is supported by the European Research Council through grant 646702 (CosTesGrav).
\\[1.5cm]

redpajama_set_name: RedPajamaArXiv


Tuesday night the Voice results were set to give us an end to the Top 10 as another two artists are cut from the 2014 season as the Live Shows continue.
Last week Tess Boyer was rescued by America's vote when she landed in the Bottom 3. Will she be safe this week or face the same threat of elimination again tonight? Which ever three singers have the fewest votes will have to give one last chance performance and fight for America's support.
Be sure to stick with the live show so you can vote as the results come out in case your favorite needs another chance.
Join us on Facebook & Twitter for the latest Voice updates all season long!
Host Carson Daly reveals the results as we meet the season's Top 8 contestants.
What do you think of this week's outcome? Did America get the votes right? Share your thoughts!


Last week was artichokes, this week, I feel the need to downsize and talk about something that's a little smaller and has several different names.
This cruciferous little mama is infamously known for it's antioxidants and potent anti-cancer qualities (some, like Harvard Medical School's Diana Post say this isn't scientifically proven).
The keyword there is little, as arugula comes with huge flavor in small doses – so while you may not cure leukemia with one bite, it remains a healthy vegetable.
In addition, Produce for Better Health notes that arugula is also low fat, cholesterol free, very low sodium, good source of folate and calcium, excellent source of vitamins A and C.
Vitamin A – Good for those night readers, vitamin A promotes good vision in low light, in addition to maintaining healthy teeth, skin and soft tissue. This vitamin is predominantly found in either animal products or beta carotene-packed leafy greens.
I would note that arugula has high vitamin K as well, but apparently most leafy green vegetables and oils flaunt a high vitamin K value.
Chlorophyll– if you combine Mr. Clean with the Energizer Bunny, that's comparable to what chlorophyll apparently does for our blood — leaves it clean and energized by bringing epic amounts of oxygen to all parts of body, as well as anti-germ environments to prevent bacteria from camping out. Other sources report that chorophyll can help treat bad breath and support healthy skin.
Arugula is also known as rocket, roquette, rugula and rucolarocket, or roquette.
Arugula tends to grow year round, so you're in luck!
Color: Leaves should be crisp and dark green with stems and roots still attached. Pass on any bunches that have yellow tinges or the leaves look slimy.
Does size matter? Yes! Bittman notes in his How to Cook Everything Vegetarian that baby arugula tends to be much milder than its fuller-leafed counterpart. So if you're looking for something less spicy than opt for the wee-sized arugula bunches.
Dunk the stem end in a glass half full of water and wrap the whole thing, glass and all, in a plastic bag. Store this cool tropical mini-environment in the fridge.
Remove the stems – while those leaves are pretty frail, stems can be snapped off by hand.
Wash arugula leaves in a bowl of cold water, changing the water several times, until clean, and then let it either air dry, use a salad spinner (no thanks), or blot dry any remaining moisture with a tea towel.
Keep leaves whole or tear them by hand.
According to just about everyone, arugula has a strong, peppery flavor, which is far more roaring when wild. Arugula is often a nice kickinthepants to any salad, bed of grilled vegetables (as it will slightly wilt while absorbing the grilling juices) or atop a bowl of soup, or something coming out of the oven — like pizza.
Margie King, in her related post, "Arugula is a nutritional rock star," over on Philadelphia Nutrition Examiner notes that arugula can pinch hit for spinach in many recipes and basil in several pesto sauces.
Anyone who mocks greens as rabbit food clearly hasn't had this flavorful, peppery green served up in the right way. Experiment with it and swap it out for some of your expected greens to see how it changes the flavor.
vegetable-101 Arugula, Vegetable 101, Year-Round Vegetables. permalink.


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How do I install Soundsets for Studio One (2 and 3)?
Installation of Studio One soundsets is more a convenience, not a necessity. That is to say, It's not an "installation" in the more conventional sense, it's more just an automated location pointer. Once you've installed Studio One and set the path for it to scan for soundsets using Options > Locations > Sound Sets, there's nothing else to do going forward but to download any new soundsets directly to that folder location. That folder or folders can be anywhere on your system.
In the image below you see a soundset package about to be downloaded to an external drive \Soundsets folder. In reality, all that install does is download the soundset to the install path you set in Options > Locations > Sound Sets. After the package downloads it will automatically show up in Studio One.
The same is true if it was downloaded with a web browser from your My Presonus user account as long as the file download was directed to a folder that Studio One is already setup to scan. In other words, if you download new soundsets to a folder that Studio One already scans for soundsets, there's nothing else to do but double click them if Studio One is currently running to put them online immediately. With a web browser this is typically "Save As..." to direct a file download to a specific location.
Only if a soundset is a commercial package and/or not included with your product version by default (Example: Core XT Instrument package and you have S1 Artist) will it also throw up the Activate dialog for you to activate or verify it's license.
asked Oct 22, 2017 in Studio One 3 by KeevinD (150 points) Half of what I've bought shows up in Studio One Pro; half does not.
Soundsets appear to be installed but I cannot access them.
Can I install Soundsets within multiple accounts?
Just bought Presonus i2 Studio and cant download a few of the larger soundset files as keeps on cutting out.


Being A Parent is hard. That's why you need to learn as many raising a child skills and techniques as you can, as early as possible. The more knowledge and information you have, including understanding what your child is going through at various stages, the more successful you'll be at bringing up a child.
When you travel with a child, strive to keep his eating and sleeping schedule the same as it is at home. Travel can seriously stress out children, especially very young children. Maintaining their routines will comfort your kid, and he is more likely to sleep soundly.
Your children need your attention, but you deserve some time for yourself, also. This helps you keep your individuality, while also being a great parent.
TIP! Always make time for yourself to do things that you like to do, without your children. Even getting a few hours to yourself while a friend or relative looks after your children can have a beneficial impact.
TIP! Smoking with children in your home can really have negative consequences on their health, so try not to do it in your home. It is actually a good idea to quit smoking altogether, especially if you have children.
TIP! If you find yourself driving a great distance, but have children in the vehicle for the trip, don't be afraid to pull over from time to time and get a breath of fresh air. Rather then dash to your destination, taking a leisurely route will often result in a more peaceful trip.
Regardless of their age, children who walk to or from school need to have retro-reflective material affixed to their person in a highly conspicuous spot, such as on their backpack or clothing. This reflective material comes in convenient Velcro strips. This material can improve your child's safety when out on the road at night, reflecting their backpack off car lights.
No two children are the same. Specific actions you take may yield results in one child, but get an opposite response from another child. This includes rewards as much as punishment. Remember which techniques work well, and which ones do not.
TIP! Adoptive parents need to be prepared for the day when the questions start. Adopted children are naturally curious about their birth parents and even why they were adopted.
As a parent, it is important to remember to take care of yourself also. It is important that you take time out each day, even if it is only a few minutes, to de-stress and revitalize your energy reserves. When you are relaxed and happy, your children will feel that way too.
Toddlers have short attention spans, and are easily amused, so switching up the toys in their toy boxes can help to keep them amused with the same old toys. Except for a few select items, small kids often become uninterested in individual toys quickly. By regularly rotating toys, you ensure that your child is always stimulated and interested by every object. It also eliminates the urge to constantly buy toys on every shopping trip.
TIP! It is important that parents of children with ADHD keep them occupied with positive activities to avoid the occurrence of negative behavior. The children that suffer from these disorders are full of energy and need to be doing something to keep them occupied so that they don't act out in an undesirable way.
Boost your kid's self-esteem by having them play team sports. Supporting and emboldening your child by being present at games will build solid character traits and memories for a lifetime.
This information served as a great tutorial regarding discover more here. You have found the information you require to get going, right here in this article. Now is the time to take the knowledge you have gained and apply it to your life!


While there are a variety of problems that your dryer can experience, a common occurrence is related to a failure of the dryer vent hood outside your home. When this cover gets damaged—perhaps becoming cracked and brittle due to the weather—critters such as squirrels, mice, and chipmunks can get inside the vent. These pests then make nests that impede the flow of air and possibly cause your dryer to overheat or, worse, result in a fire. While you can certainly call an appliance repair service to replace a broken vent hood, it's also possible to do this work yourself. Here's how to approach this task.
On some homes, the dryer vent hood outside the home will be at ground level; for other homes, it will be high enough that you'll need to access it with a ladder. If you're dealing with an elevated vent hood, you'll need to either use a ladder and have a family member or friend hold the bottom, or else rent one or two flights of scaffolding from your local home improvement center so that you can complete the job safely.
Depending on the degree of damage to the old vent hood, it may be possible to pull off with your hands. Otherwise, you'll typically be able to remove it with a screwdriver; although vent hoods differ in style and manufacturer, they're often just attached to a bracket with a set of screws. When you remove the screws, the vent hood will come off in your hands. You can then visit your local home improvement center and buy a replacement. It's worthwhile to take the old one so that you can buy a new one of a comparable size and with the mounting holes in the same location.
While you should always check the installation instructions on the packaging, installing your new vent hood is a simple process. Hold it up against the opening in the wall where the old one was situated, and make sure that you line up the screw holes. Then, it's simply a matter of driving the screws through the cover's mounting holes and into the bracket in the wall. When it's mounted correctly, the new vent hood's cover should hang down and cover the opening. Have a family member turn on the dryer, and you should see the force of the hot air blowing the cover open, but when the dryer is turned off, the cover should return to the closed position.


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class Google_Service_Firestore_Projection extends Google_Collection
{
  protected $collection_key = 'fields';
  protected $fieldsType = 'Google_Service_Firestore_FieldReference';
  protected $fieldsDataType = 'array';

  /**
   * @param Google_Service_Firestore_FieldReference
   */
  public function setFields($fields)
  {
    $this->fields = $fields;
  }
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   * @return Google_Service_Firestore_FieldReference
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  public function getFields()
  {
    return $this->fields;
  }
}

redpajama_set_name: RedPajamaGithub


How much does a Registered Nurse make in Midland, ON?
The average salary for a Registered Nurse is $32.03 per hour in Midland, ON, which is 14% below the national average. Salary estimates are based on 22 salaries submitted anonymously to Indeed by Registered Nurse employees, users, and collected from past and present job advertisements on Indeed in the past 36 months. The typical tenure for a Registered Nurse is 1-3 years.


BOUNTIFUL — Some people just know when they find a hobby to last a lifetime. For Gary Johnson, it's being a HAM radio operator. "I love it," he said. "There's a fire in my belly. I do love this hobby a lot."
There are times, though, when his hobby stops being a hobby and becomes something so much more. When severe winds hit northern Utah Thursday, Johnson and his HAM radio friends were called into action.
"We were used to pass information, health and welfare," he said; "be the eyes and the ears of the police department."
Johnson, or N7DND to his friends, was at home in Bountiful when he got the call from the Davis County sheriff. Communications were breaking down because of all the commotion, and the sheriff was wondering if some HAM radio operators could help get messages through to emergency personnel.


text:
const { encrypt, decrypt, generateNewPrivateKey, generateNewHmacKey } = require('./');

describe('Crypto Features', () => {
    it('should keep the consistancy between encryption & decryption', () => {
        const data = 'SOME VERY PRIVATE INFO';
        const privateKey = generateNewPrivateKey();
        const hmacKey = generateNewHmacKey();
        const encryptedData = encrypt(data, privateKey, hmacKey);

        expect(encryptedData).not.toBe(data);

        const decryptedData = decrypt(encryptedData, privateKey, hmacKey);

        expect(decryptedData).toBe(data);
    });

    it('should not return an identical signature twice for the same given entry and private key', () => {
        const data = 'SOME VERY PRIVATE INFO';
        const privateKey = generateNewPrivateKey();
        const hmacKey = generateNewHmacKey();

        const encryptedData = encrypt(data, privateKey, hmacKey);
        const encryptedData2 = encrypt(data, privateKey, hmacKey);

        expect(encryptedData).not.toBe(encryptedData2);

        const decryptedData = decrypt(encryptedData, privateKey, hmacKey);
        const decryptedData2 = decrypt(encryptedData2, privateKey, hmacKey);

        expect(decryptedData).toBe(data);
        expect(decryptedData2).toBe(data);
    });

    it('should throw an error if the data is tampered', () => {
        const data = 'SOME VERY PRIVATE INFO';
        const privateKey = generateNewPrivateKey();
        const hmacKey = generateNewHmacKey();
        const encryptedData = encrypt(data, privateKey, hmacKey);

        const [algorithm, cipherText, iv, signature] = encryptedData.split(':');
        const tamperedData = `${algorithm}:${cipherText}:${iv}:${signature}tampered`;

        expect(() => {
            decrypt(tamperedData, privateKey, hmacKey);
        }).toThrow(/tampered/);

        expect(() => {
            decrypt(encryptedData, privateKey, hmacKey);
        }).not.toThrow();
    });
});


text:
Q: Zip multiple folders indvidually, automatically I have folders full of photos within category folders
eg

*

*holidays

*

*spain

*france

*italy

*family

*

*son

*daugher

*christmas

*

*2010

*2011

... except much longer ... there are about 100 folders.
I need to compress each sub folder ( ie spain, france italy, son, daughter etc) individually; so that, for the holidays category folder for example I would end up with:

*

*holidays

*

*spain

*spain.zip

*france

*france.zip

*italy

*italy.zip

... and so on. ( or using rar files )
I want to avoid doing this manually if possible, and I think it's possible from the command line using WinRar, (though I could use something else if that was better). Could someone give me (and briefly explain to me) the code I'd use, and how to use it. I know a little about the command line but I have fairly limited experience.
I am using Windows 7

A: Check out my answer to the similar SU question titled "How do I create seperate 7z files from each selected directory with 7zip command line?". It could easily be adapted to use a different command-line zip/archive tool, just change the part after the DO in the statement.
I personally prefer 7-Zip over WinRAR and the couple of other compression utilities I tried. It's powerful, quick, and free. Note that it's unlikely that your image files will be able to be compressed much since by the nature of their format they're usually already compressed with sophisticated image-oriented algorithms which a generally better than any generic one used in a compression utility -- although the non-free WinZip program claims can further reduce JPEG image files by 20-25%.

A: I thought it would be better to answer my own question and accept that, despite the helpful answers, (giving upvotes to everyone instead of picking an answer) ... so that future visitors would get a concise answer. This is because I ended up getting my result from a combination of the four answers so here was tested solution I ended up with.
I changed to 7zip ( downloading the command line version and placing it in my user folder) because it seems to have better documentation on command line operation than Winrar.
Then I created a batch file, which I ran from the folder whose subfolders I wished to zip; and put into it the following command which I derived from the other answers:
for /d %%i in (*.*) do "c:\users\user\7za" a -tzip "c:\path\to\output\folder\%%i.zip" "%%i"
(My username is 'user' so this would be adjusted according to an individual's Windows 7 username)
I tested the command removing various aspects eg the /d and the -tzip and I didn't find it to work unless it was all in place.
I also found this to be a good resource http://www.dotnetperls.com/7-zip-examples
NB I was not concerned with folder depth and in my case all the sub folders are only one level below the top level folders. Also these folders only contain files, not more folders. I doubt if the above code would work for folders contained within folders but i'm sure it could be adapted easily.

A: I don't think this will be easy using a batch file, but it should be possible.
Batch allows you to iterate through directories using a for loop like this:
for /d %%X in (directorySet) do ( command )

/d is required to specify that you will be looping through directories
%%X is a directory in the list
command is what to run like 'echo "hello"'
Source: http://www.robvanderwoude.com/for.php
This is all I really know that might help you, what you use in place of command is very dependant on the program you are using.
It would be easier in Powershell as this has more bash-like notation, and is much more flexible, but using batch is completely fine.
Sorry that's all the help I could give, I have very minimal knowledge of batch...

A: I don't have a Windows system handy to cook up your command,
But, here is a rough list of things,

*

*you would use the dir command in a for loop (or, maybe you need a more recent manual page)

*limit your dir listing to depth=1 directories

*

*/A:D  Folder switch (?)

*some way to not-recurse or use depth==1 (I've forgotten windows batch files now)

*Check 7z too

*If you are often into windows scripting check PowerShell

A: In DOS, change to the parent folder that has your 100+ folders, then run one "for" command, like this:
for %v in (*.*) do pkzip %v.zip %v

Replace pkzip with your command-line zip/archive tool. The 2nd and 3rd "%v" expand to the folder names.

redpajama_set_name: RedPajamaStackExchange


What was the political stance of most conservatives toward the Civil Rights Movement?
This is also a puzzling statement, given that the Civil Rights Movement was a series of inherently political acts: Rosa Parks refusing to give up her seat on the bus; sixteen year old Barbara Johns organizing black high school students to protest the tar paper shack schools they were forced to attend; student organizers risking their lives to register black voters. These actions led to other political acts, such as the 1954 Supreme Court Brown v. Board of Education ruling, and Congress passing the landmark Civil Rights Act of 1964 and the Voting Rights Act of 1965.
Apparently, in Beck's mind, the Civil Rights Movement was a warm, misty-eyed love of country and "individual rights" which emanated from the swelling hearts of conservative patriots and transformed the country with no significant political action or legislation required.
Amidst the controversy surrounding Beck's remarks, little has been said about what conservatives actually did during the civil rights era and the years following it. One does not need to look too far to see that conservatives were largely against the Civil Rights movement. Two of the greatest icons of the conservative movement, Barry Goldwater and Ronald Reagan, exemplified conservative opposition to the movement.
Barry Goldwater, the Republican senator from Arizona and Lyndon Johnson's opponent in the 1964 presidential election, was in the vanguard of a new movement within the Republican Party. Alienated from the eastern, old guard of the party, which had made peace with some aspects of the New Deal, Goldwater's caustic conservatism appealed to a rising elite from newer industries, real estate, and car dealerships in the "Sunbelt" of the South, Southwest, and West. The senator had also endeared himself to racist whites in the South and elsewhere by voting against the Civil Rights Act of 1964, which had illegalized racial discrimination in employment, schools, hospitals, restaurants, hotels and theaters.
Goldwater's harsh persona, however, frightened many Americans, and he was easily crushed by Johnson in the 1964 election. Still, he had won five states in the South, and the lesson of the electoral benefits of exploiting the white backlash against civil rights was not lost on the Republican Party. Richard Nixon would employ the "Southern Strategy," a plan designed to harvest the votes of whites disaffected by civil rights and the 1960s counter culture, to retake the White House in 1968.
Ronald Reagan, a genial television personality and former film actor, launched his political career at the 1964 Republican Convention, speaking in support of Goldwater and individual rights, a term frequently invoked to justify the rights of individuals to discriminate against racial groups they despised. Elected Governor of California in 1966, Reagan became the darling of conservatives in the sixties and seventies.
When Reagan ran for president in 1980, he kicked off his campaign in Philadelphia, Mississippi, where three civil rights workers had been murdered in 1964, and spoke in favor of states' rights and limiting the power of the federal government. In retrospect, it is astonishing that a presidential candidate in 1980 could oppose federal power in connection with civil rights, given the crucial, if imperfect role the federal government had played in codifying civil rights with Brown v. Board of Education and the Civil Rights Act of 1964.
Conservative politicians were not devising their strategies on civil rights in a vacuum. Conservative intellectuals and grassroots political organizations were playing the tune of "individual rights" and speaking out against the Civil Rights Movement as well. The National Review, a magazine founded by conservative ideologue William F. Buckley, Jr., had supported black disenfranchisement in the South on the basis that "the claims of civilization supersede those of universal suffrage." The Young Americans for Freedom, a conservative student group founded in 1960 awarded its "Freedom Award" to arch-segregationist Senator Strom Thurmond in 1962.
In earlier years Democrats had been anything but heroic on civil rights. John F. Kennedy dragged his feet on civil rights legislation out of fear of alienating the party's southern brethren. FDR had refused to support a national anti-lynching law for the same reason. It should also be noted that most northern Republicans voted for the Civil Rights Act of 1964 along with most northern Democrats, and almost all southern Democrats, many of whom would become Republicans in the following years, voted against it. But Lyndon Johnson's push for civil rights legislation drove the white South into his opponents' camp, and the Republican Party was not about to question the racial motivations of their new voters.
In the future, if Glen Beck wants to honor the Civil Rights Movement, he might consider staying home, and watching the video series "Eyes on the Prize," which will teach him about the heroic political actions of people like Barbara Johns, Charles Houston, Medgar Evers, and others.


text:
Bostadsrättslagen  är en svensk lag som innehåller regler om bostadsrättsföreningar och bostadsrätter.
Den första bostadsrättslagen kom 1930, den har sedan uppdaterats 1971 och 1991
. Nu gällande är bostadsrättslag (1991:614) med förändringar.

Noter

Externa länkar
lagen.nu Bostadsrättslag (1991:614)
Boverket kräver ny bostadsrättslag

1991 i Sverige
Politikåret 1991
Juridikåret 1991
Ekonomiåret 1991
Fastighetsrätt
Sveriges lagar

redpajama_set_name: RedPajamaWikipedia


text:
create table patient(
	name		varchar(10),
	ahcn		char(9),
	dob		date,
	livestatus	boolean,
	doctor	varchar(15),
	pid		char(10),
	primary key (pid),
	unique key (ahcn)
)
create table ecg(
	pid			char(10),
	mv			int,
	pulse			int,
	o2			int,
	diastolicbp	int,
	systolicbp		int,
	map			int,
	timestamp		time,
	session_id		int,
	devicetype		char(10),
	primary key (pid),
	foreign key (pid) references patient
)
create table oximiter(
	pid			char(10),
	pulse			int,
	oxygen		int,
	timestamp		time,
	session_id		int,
	devicetype		char(10),
	primary key (pid),
	foreign key (pid) references patient
)
create table bpressure(
	pid			char(10),
	diastolicbp	int,
	systolicbp		int,
	map			int,
	timestamp		time,
	session_id		int,
	devicetype		char(10),
	primary key (pid),
	foreign key (pid) references patient
)
create table account(
	username	varchar(10),
	password	varchar(20),
	primary key (username)
)

(Unsure if session is relevant or not.  Feel free to ignore or delete it if necessary.)
create table session(
	ecgpeak	float,
	interval	float,
	sample	float,
	systolic	float,
	dystolic	float,
	map		float,
	o2		float,
	bloodo2	float,
	timestamp	time,
	datestamp	date,
	primary key (timestamp)
)


text:
Current Page Parent News
US Election: Esther Agbaje, Nigerian-American Wins Legislative Seat by an Overwhelming Margin
by Blessing Falade · November 4, 2020
A Nigerian-American, Esther Agbaje has been elected into the Minnesota House of Representatives to represent District 59B in the United States general election held on November 3, 2020.
Agbaje won by an overwhelming margin, scoring a total of 17,396 votes, which represents 74.7% of the total ballots cast. Her closest rival, Alan Shilepsky, a nominee for the Republican Party, scored 4,126 votes, which represents 17.7% of the total votes cast.
She was born in St. Paul the state capital of Minnesota, to Nigerian immigrant parents, lived in Brainerd, and went to high school in Faribault. Agbaje has a law degree from Harvard University, a Master's degree in Public Administration from the University of Pennsylvania, and has served in the U.S. Department of State, among others.
After law school, she moved to downtown Minneapolis in 2017 to contribute to the place she calls home.
Agbaje confirmed that she would be representing District 59B in the Minnesota House of Representatives on the platform of the Minnesota Democratic–Farmer–Labor Party (DFLP), an affiliate of the U.S. Democratic Party, on her website.
She defeated incumbent Raymond Dehn and Isaiah Whitmore in the Democratic Primary for Minnesota House of Representative District 59B on August 11, 2020.
Tags: 2020 US ElectionsEsther AgbajeMinnesota House of ReprentativeNigerian-American
'Christmas Is An Abomination And The Bible Warned Us About It' – Reno Omokri
`Army Calls Out TY Danjuma For Asking Nigerians To Defend Themselves Against Killer Herdsmen
Footballer, Adam Johnson Sentenced To Six Years Jail Term
Next story US Elections: Adeoye Owolewa Becomes First Nigerian to Hold Congressman Seat
Previous story Alimosho FC Ladies Signs A Deal With Gidi Burger

redpajama_set_name: RedPajamaCommonCrawl


The evidence that Nutrition is one of the most important factors of optimum health and overall wellbeing is stacking up and nutritional awareness is now at an all time high.
Our first and most popular course, our Nutritional Advisor course is for anyone who has an interest in nutrition and health and wants to set up their own business in the health and wellness industry.
Over 13 comprehensive modules, our nutritional advisor course will teach you the fundamentals of nutrition, nutritional science, nutritional changes through the many stages of life, teaches you how our lifestyle contributes to our health, how to read food labels, business and marketing to help you set up and grow your nutrition business and more. You will also benefit from mentoring during and after your course.
We know that our lifestyles can heavily impact our health, people are turning to natural approaches more than ever, this means that demand for knowledgeable nutrition experts is rising. Our Nutritional Advisor course will place you in excellent position to help those who need nutritional support.


Fabolous and Jadakiss are two of the greatest freestyle rappers that always bring their lyrical best to a beat. They are even a deadlier pair when they team up on the mixtape Friday on Elm Street. One of the best tracks from the tape is "Theme Music" with its soul-sample and hot bars from both rappers. Check it out below.


Contact lenses are a great option for patients who seek the freedom from having to wear glasses. Our Optometrist, Dr. Dash, is experienced and certified to properly fit contact lenses that give patients the vision they desire. Refractive errors such as myopia, hyperopia, astigmatism and presbyopia can be corrected with contact lenses.
To ensure successful contact lens wear, we require that contact lens fitting exams are offered only to patients that have had a comprehensive eye exam within the last 12 months from an eye care professional. We want to be certain that patients have eyes that are suitable for contact lenses before initiating contact lens wear. For first time contact lens patients and those that may need a refresher, our staff will gladly train and educate patients about the correct contact lens insertion, removal and care techniques. We would like our patients to understand that contact lenses are medical devices that can cause harm to your eyes if they are not professionally fitted or used properly.
Even if patients are able to wear contact lenses, they often need specialty lenses. Fittings for these lenses require special training and can require several visits as well as follow up visits once a lens has been chosen. Dr. Dash has specialized training and experience to help these hard to fit patients. The most frequently used contact lenses for keratoconus patients are gas permeable lenses.
And various specialty lenses for keratoconus and other challenging corneas.
Copyright 2017 - EyeSight Hawaii - All Rights Reserved.


text:
Lincoln Leads
Series 7 (2023)
Specialised Information
What's Love Got to Do with It?
The Modern Museum
'Seeing Double'
AI and Modern Medicine
Truth and Evidence
Gender (collab. Lincoln Unites)
WHERE HAS POWER GONE?
Thursday February 18th 2021, 5:30pm – 7pm UK time, on Zoom webinar
Registration: please contact jane.mitchell@lincoln.ox.ac.uk to obtain the registration link.
With the (former) Trump administration's sustained attempts to retract the US from its prominent role in multilateralism and China increasing efforts to extend its soft and hard power, the old balance of power has begun to shift. Let's add to the equation a growingly assertive Russia, a struggling but always more integrated EU and the rise of tech companies worth more than many countries' GDP, and we should have all the ingredients for an animated discussion.
Neil Bush is Her Majesty's Ambassador to the Organisation for Security and Cooperation in Europe (OSCE) – the world's largest regional security organisation covering 57 participating States, including the US, Russia, EU countries and the UK. Prior to this role, he was responsible for the Gulf, Iran and Yemen in the Foreign and Commonwealth Office. Mr Bush started his career in the Bank of England and has worked for the World Bank, including in the Western Balkans, as well as the private sector and as Head Energy Economist for the UK government. Mr Bush has a BA at University of Oxford in Philosophy, Politics and Economics and a Master's Degree from Columbia University and the London School of Economics.
DR JODY LAPORTE
Dr. Jody LaPorte is the Gonticas Fellow and Tutorial Fellow in Politics & International Relations at Lincoln College. Her research investigates the dynamics of politics and policymaking in post-Soviet Eurasia. Her work seeks to better understand politics in authoritarian regimes. She completed her undergraduate degree at Yale University, and her MA and PhD at the University of California, Berkeley. Prior to coming to Lincoln, she was a lecturer at the Blavatnik School of Government and Oxford's Department of Politics & International Relations, and a fellow at the Woodrow Wilson International Center for Scholars, a leading think tank in Washington, DC.
JELANI MUNROE
Jelani is a DPhil student in Geography, having previously completed an MPhil in Development Studies. He studies the political economy of austerity, with a current research focus on the implications of economic austerity on politics, public services, and the justice system in Jamaica. Before coming to Oxford, he completed a B.A in Economics and Public Policy at Stanford University, with interdisciplinary honours in Democracy, Development and the Rule of Law.
NICCOLO AYLWARD
Niccolo Aylward is in his third year of study for an undergraduate degree in Politics, Philosophy, and Economics at Lincoln College. He co-founded and helped run the Coronavirus Tutoring Initiative together with other Lincoln students in response to the school closures and educational disruption cause by the Covid-19 pandemic. Niccolo will be the first JCR member to chair a Lincoln Leads panel.


text:
The Life of John Milton
By Charles Symmons
ken, is the larg
These attacks were not calculated to occadamission much disturbance to the republican author: but he could not feel equally easy on the near approach of that thunder-cloud, which was just ready to burst upon him and his party. His spirit, however, did not desert him; and, while there remained a possibility of upholding his falling cause, he was resolute and active in its support. Bold in the anticipation of their triumph, the Royalists had already seized upon the press and the pulpit for the diffusion of their tenets and their resentments; and Dr. Matthew Griffith, one of the late king's chaplains, desirous of making a professional display of his loyalty, at a crisis when it might be especially beneficial to him, published a sermon, which he had preached at Mercer's Hall, on (Proverbs xxiv. 21.) My Son, fear the Lord and the King, and meddle not with them that are given to change." On this provocation Milton instantly kindled ; and, in a short but forcible commentary on the Doctor's sermon, renewed his strong
Strange war
Hope that
s much Istan
I were sure I sh
es and stone
twith the ." to tell the
hat I lave
ou suffer
ce! nor thr
s of our
on's was mad a serious
mphlet af republicanci
the Rota 1. "The free Comm
* Milton's severity on this intrusion of the pulpit into the province of politics, reminds us of the asperity with which Mr. Burke reprehended a similar invasion by a modern divine. Dr. Price differed as essentially, in his political principles, from the chaplain of Charles, as Milton did from the Marquis of Rockingham's secretary: yet the two doctors experienced the same treatment, and the two statesmen concurred in the same sentiments of reprobation. The politics of the pulpit may, at
could not be pardoned without powerful intercession. We may conclude that his friend, Andrew Marvell, the member for Hull, made what interest for him he could in the House; and we are told that Sir Thomas Clarges united his exertions with those of Secretary Morrice for the preservation of this valuable life. But Milton seems to have been saved principally by the earnest and grateful interposition of Sir William D'Avenant. When D'Avenant, who had been captured by the fleet of the Commonwealth on his passage from France to America, had been ordered by the Parliament, in 1651, to his trial before the High Court of Justice, the mediation of Milton had essentially contributed to snatch him from his danger; and, urged by that generous benevolence which shone conspicuously in his character, he was now eager to requite, with a gift of equal value, the life which he had received. For the existence of D'Avenant's obligation to Milton we have the testimony of Wood, and for the subse quent part of a story, so interesting in itself and so honourable to human nature, the evidence is distinctly and directly to be traced from Richardson to Pope; and from Pope to Betterton, the immediate client and intimate of D'Avenant.
i Athenæ Oxon. vol. ii. 412.
On the passing the full grace of w
cluded, Milton lef
had continued for 1
which he had heard
the Commons, the tion, and his two "Iconoclastes" an
People of England
by the hands of th species of insult, affected on the pre formerly been by
same publications and he probably,
the purpose of incr
of course, his dang fortune, the malig the abuse and c Tented against hi But those scenes
LIFE OF MILTON.
On the passing of the Act of Oblivion,* in grace of which he found himself included,' Milton left the retirement, where he had continued for nearly four months; and in which he had heard himself made, by a vote of the Commons, the object of a public prosecufor the presention, and his two great political works, the
: Milton seens
"Iconoclastes" and the "Defence of the People of England," condemned to be burnt by the hands of the hangman. By this last species of insult, he was probably no more affected on the present occasion than he had formerly been by its infliction on one of the same publications at Thoulouse and at Paris: and he probably, also, only smiled, when, for the purpose of increasing his unpopularity and, of course, his danger at this delicate crisis of his fortune, the malignity of his enemies published the abuse and calumnies, which had been vented against him by the dying Salmasius. But those scenes of sanguinary execution,"
mell, the memi rest for him ber
are told that S
by the ears of Sir William D , who bad be
Commogra
to America, t, in 1651.
Justice, ther ally contrib
and, urge
ger; e which shore
; he was none jual value,
For the ex • Milton wel id for the s teresting i nature, the
thly to be ta and from F client and g
* On the 29th of august.
John Goodwin, a divine and a writer of no celebrity, who had justified, without ability or effect, the murder of the king, was not beneath the condescension of this act of the legislature. He was incapacitated by it from holding any public office: and be is said to have owed his life only to the circumstance of his Arminian principles, which had conciliated the favour of some of the leading clergy of the church of England. His obnoxious work, which was called "The Obstructors of Justice," had the honour of burning with Milton's superior publications.
" I must here, with some shame and much regret, remark
victims, which the perfidiousness of party, in expiation of its own offences, was so base ás to offer to him; and he glutted the nation, as far as he durst, with an effusion of blood, not more guilty than that of thousands, perhaps, who were present to behold it; for they who, from their office, were more personally engaged in the trial and the execution of the king, were unquestionably not more criminal than were all those who had voted for these violences in Parliament; or, in the army, had first planned, and then imperiously carried them into effect. More, however, than they who were regarded as the actual regicides were exempted from the benefit of the amnesty. Neither Vane, nor Peters, nor Lambert was immediately implicated in the murder of the king; yet the two former of these were slaughtered (and Vane in violation of the royal promise to the Parliament for his pardon) while the last, the most guilty of the three, was indeed permitted to live, but to live only in a state of miserable exile.
But not limited to the sufferings of the living, the vengeance of Charles extended itself to mean and atrocious outrages on the dead. It broke the hallowed repose of the tomb, and exhibited that last infirmity of our mortal nature, the corruption through which it is doomed to pass into its kindred
earth, to the deris
dies of the illust
potent malignity.
It is well known th
the place in which
bas been advanced, on aut that his corpse was rem
his death, and buried i goes further and affirms t would probably, in a chan er's body, his friends su Charles; and that it was pored on the gallows at will present them with erved in Lord Sommers's that, as eleven years ha Charles, it is difficult to
tenance, or of seam ab
teed: unless, indeed,
pended by the arts of en
to the bones, must now
ments.
"A counter-interment of and ready to be de Barkstead, who
within Temple-Bar, that was executed as being, at the time of age of fifteen years. "That the said regi Tower of London, and among other such confi ness, desire to know w answered, where he had
earth, to the derision and the disgust of impotent malignity. When we behold the bodies of the illustrious usurper and of the
in It is well known that many doubts have existed respecting the place in which Oliver Cromwell was interred: and it has been advanced, on authority which cannot easily be rejected, that his corpse was removed, on the day succeeding to that of his death, and buried in the field of Naseby. The account goes further and affirms that, suspicious of the indignities which would probably, in a change of things, be offered to the Usurper's body, his friends substituted for it in the coffin that of Charles; and that it was this corpse which was afterwards exposed on the gallows at Tyburn. To entertain my readers I will present them with a curious paper on this subject, preserved in Lord Sommers's collection. I must premise, however, that, as eleven years had nearly elapsed since the death of Charles, it is difficult to conceive how any distinction of countenance, or of seam about the neck could, at this period, be traced unless, indeed, the process of dissolution had been suspended by the arts of embalming, the corpse, with exception. to the bones, must now have been resolved into its original ele
"A counter-interment of the aforesaid arch-traytor,* as averred, and ready to be deposed (if occasion required) by Mr. Barkstead, who daily frequents Richard's coffee-house, within Temple-Bar, being son to Barkstead, the regicide, that was executed as such, soon after the restoration, the son being, at the time of the said arch-traytor's death, about the age of fifteen years.
"That the said regicide Barkstead, being lieutenant of the Tower of London, and a great confident of the usurper, did among other such confidents, in the time of the usurper's sickness, desire to know where he would be buried: to which he answered, where he had obtained the greatest victory and glory,
Oliver Cromwell.


Aqueous is one of Femme Fatale's newest shades - its 'a medium bright blue crelly filled with teal, navy, blue, magenta and pink.glitters'.
It should come as no surprise that I love this polish :) FFC can do no wrong! :P But seriously its so nice when you have high expectations from a brand and they consistently live up to them.
The pictures show 3 coats of Aqueous with 1 coat of topcoat. Application was about what you would expect from this type of polish - a little thick due to the amount of glitter but nothing troublesome. No fishing was required for the glitter either, which is always a bonus!
Femme Fatale lacquers are available from their site for Australians, or Llarowe and Harlow & Co internationally. It seems to be sold out on both Llarowe and Harlow at the moment so make sure you keep an eye out on their facebook pages if you're trying to snag it!
This one is great - so vibrant!