text:
\section{Introduction}

Modern cosmology is based on the assumption of a background FLRW model
with small perturbations. With only six free parameters, the resulting
$\Lambda$CDM model successfully accounts for all cosmological data,
including the CMB and its anisotropies, structure formation and
galactic dynamics, redshift-luminosity relations, and the
nucleosynthesis of light elements. It stands as a truly remarkable
achievement of modern physics.

Nevertheless, there are grounds for questioning the validity---and,
perhaps, even the mathematical consistency---of using the $\Lambda$CDM
model to describe our universe. The observed density contrasts in our
universe are huge on scales of $\sim 1$ Mpc and smaller, and certainly
cannot be described as ``small perturbations'' of an FLRW model in the
usual sense. It might therefore appear that some sort of ``averaging''
over the small scale inhomogeneities must be used in order to even
define the FLRW background model that is supposed to represent our
universe in a $\Lambda$CDM model. If one then succeeds in defining an
averaged FLRW model, how does one know that its dynamics will be
properly described by the dynamical equations applicable to an exact
FLRW model?  Therefore, it is appropriate to ask the question posed as
the title of this paper: ``How well is our universe described by an
FLRW model?''

The issue of how best to approximate our lumpy universe by an FLRW
model---the ``fitting problem''---was discussed in classic papers of
Ellis~\cite{Ellis:1984} and Ellis and Stoeger~\cite{Ellis:1987zz}.
Ellis envisioned defining an operator that would take an inhomogeneous
metric and stress-energy tensor into successively smoother and
smoother versions (see \fref{fig:ellis}).
\begin{figure}
  \centering
  \includegraphics[width=0.7\textwidth]{figure1.pdf}
  \caption{Schematic of a smoothing map, reproduced from
    \cite{Ellis:1984} with kind permission from Springer Science and
    Business Media. Original caption: A comparison of models of the
    same region of the universe on three scales showing different
    amounts of detail.  Scale 1 represents all details down to stars.
    Scale 3 represents all details down to galaxies.  Scale 5
    represents only large scale features.  The models at different
    scales are related by (1) identification of which points at the
    different scales correspond, through maps $S^\ast$; (2) maps $S$
    determining the matter tensor and maps $S'$ determining the metric
    tensor of the smoothed-out representation from the more detailed
    one.  The usual models of cosmology are at Scale 5; but models for
    galaxy formation must be at Scale 3. }
  \label{fig:ellis}
\end{figure}
Given that the Einstein equation holds at the most inhomogeneous scale
(his ``Scale 1''), Ellis raised the very important question of whether
the Einstein equation holds at the level of the large scale averaged metric
and stress-energy tensor.  The averaged Einstein equation is
\begin{equation}
  G_{ab}(g^{(0)})+\Lambda g_{ab}^{(0)} = 8\pi \left( T_{ab}^{(0)} + t^{(0)}_{ab} \right)  .
\label{avgein}
\end{equation}
Here $g_{ab}^{(0)}$ and $T_{ab}^{(0)}$ represent, respectively, the
large scale averaged metric and stress-energy tensor, and
$t_{ab}^{(0)}$ is the average of the (nonlinear) contributions to
Einstein's equation arising from the departures of the actual metric
and stress-energy tensor from $g_{ab}^{(0)}$ and $T_{ab}^{(0)}$.
Thus, $t_{ab}^{(0)}$ represents the {\em backreaction} effects of the
inhomogeneities on the large scale averaged metric, and we will refer
to it as the {\em effective stress-energy tensor} produced by the
inhomogeneities. The issues raised in the previous paragraph then
reduce to the following: (i) How does one define the FLRW quantities
$g_{ab}^{(0)}$ and $T_{ab}^{(0)}$ for the actual universe? (ii) Is
$t_{ab}^{(0)}$ negligibly small?

One of the main points that we wish to emphasize in this article is
that the first question has an extremely simple answer. As we shall
elucidate further below, it is akin to asking how to represent the
surface of a real ball bearing by a perfect sphere. The surface of a
real ball bearing is extremely close to being a perfect
sphere---despite significant imperfections that can be seen under a
microscope---and it is therefore easily represented by a perfect
sphere. Similarly, the actual metric, $g_{ab}$, of our universe is
extremely close to a metric, $g_{ab}^{(0)}$, with FLRW
symmetry---except in the immediate vicinity of very strong field
objects, such as neutron stars and black
holes\footnote{\label{footnote:bh}The large deviation of $g_{ab}$ from
  $g_{ab}^{(0)}$ in the immediate vicinity of strong field objects is
  irrelevant to cosmology.  This is most easily seen by considering a
  new metric, $g_{ab}'$, corresponding to a universe in which each of
  the strong field objects is replaced by an object of the same mass
  but having a radius of $\sim 10^4$ Schwarzschild radii.  Then
  $g_{ab}'$ would be extremely close to $g_{ab}^{(0)}$ everywhere, but
  the cosmological properties of $g_{ab}$ and $g_{ab}'$ would be
  indistinguishable.}---and is easily represented as such. Thus, the
``fitting problem'' for $g_{ab}$ is trivially solved. On the other
hand, as noted above, the actual stress-energy tensor $T_{ab}$ of our
universe is very far from possessing FLRW symmetry. Nevertheless,
since $g_{ab}$ is so close to $g_{ab}^{(0)}$, there is no difficulty
in defining a suitable large scale average of $T_{ab}$, using either
$g_{ab}$ or $g_{ab}^{(0)}$. So, the first question above poses no
difficulty.

However, the second question is less trivial. Although order of
magnitude estimates clearly indicate that $t_{ab}^{(0)}$ should be
negligible for our universe \cite{Ishibashi:2005sj}, one would like to
have a controlled approximation scheme, wherein the dominant
corrections to $g_{ab}^{(0)}$ arising from $t_{ab}^{(0)}$ can be
computed. We have recently developed such an approximation
scheme~\cite{Green:2010qy}, wherein the actual metric $g_{ab}$ is
assumed to be close to a smooth metric $g_{ab}^{(0)}$, but derivatives
of $g_{ab}$ need not be close to derivatives of $g_{ab}^{(0)}$, and
second derivatives (i.e., the curvature) of $g_{ab}$ may have
unbounded fluctuations relative to $g_{ab}^{(0)}$. {\em A priori}, our
approach allows for significant backreaction effects, since the
dominant contribution to $t_{ab}^{(0)}$ comes from quadratic products
of first spacetime derivatives of metric perturbations, which are not
assumed to be small.  Our main result was the proof that---provided
the matter stress-energy tensor satisfies the weak energy
condition---$t_{ab}^{(0)}$ must be traceless and must itself satisfy
the weak energy condition.  Essentially, $t_{ab}^{(0)}$ can be large
only in the presence of significant amounts of gravitational
radiation, in accord with standard intuition. There cannot be any
large backreaction effects arising from matter inhomogeneities. We
will describe our approach and summarize the main results
of~\cite{Green:2010qy} in \sref{sec:framework}.

Even though no large backreaction effects can occur due to deviations
from an FLRW model on small scales, it is still of interest to know
exactly what the corrections due to backreaction are. In particular,
simulations of structure formation are done using Newtonian
cosmological models. Exactly what ``dictionary'' should be used to
translate Newtonian simulations into general relativistic cosmological
models? Exactly how accurate is this dictionary, particularly on large
scales? We have answered these questions in~\cite{Green:2011wc} using
a ``counting scheme'' based upon our approximation method, and we will
summarize our main results in \sref{sec:newtoniancosmology}. We also
include in that section a brief discussion of the degree to which
observable quantities in our universe can differ from an FLRW model.

The above results enable us to provide the following answer to the
question posed in the title of our paper: {\em The metric of our
  universe is extremely well approximated by an FLRW metric that
  satisfies~\eref{avgein} with negligible $t_{ab}^{(0)}$.}
Certain observable quantities involving geodesics (e.g., gravitational
lensing) and curvature (e.g., density contrasts) can differ very
significantly from an FLRW model, but these can be accurately computed
from Newtonian cosmological models using a dictionary of appropriate
accuracy, as described in \sref{sec:newtoniancosmology}.

Before describing our results in the following sections, we now
elucidate the nature of the ``fitting problem'' for our universe by
exploring more fully the ball bearing analogy alluded to
above. Consider a convex polyhedron with $30,000$ faces, where the
faces are taken to be tangent to randomly chosen points on a unit
sphere. We cannot easily draw such a figure with the required
resolution, so to illustrate the kind of surface we have in mind, we
have drawn such a polyhedron with 300 faces in
\fref{fig:polyhedronsphere}, but it should be kept in mind that the
polyhedron we are considering has a hundred times as many faces as
the one drawn in \fref{fig:polyhedronsphere}. For a polyhedron with
$30,000$ faces of the sort that we are considering, the typical
linear size of each face is $\sim 1/50$ of the radius of the sphere,
and the typical deviation of the polyhedral surface from the unit
sphere is no more than about 1 part in $10^4$. Thus, a solid steel
body with this polyhedral surface would pass inspection as a
semi-precision ball bearing (see~\cite{abbottballs}).
\begin{figure}
  \centering
  \includegraphics[width=\textwidth]{figure2.png}
  \caption{The ``fitting problem'' is analogous to the problem of
    approximating the surface of a ball bearing by a sphere.  Left: A
    300-face convex polyhedron constructed by intersecting the planes
    tangent to 300 randomly chosen points on the surface of a sphere.
    Our ``Sphereland'' is constructed by starting with a similar
    polyhedron with 100 times as many faces, and slightly rounding the
    vertices and edges.  Right: A sphere that solves the ``fitting
    problem'' for the polyhedron (compare with \fref{fig:ellis}).
    Note that we consider a 2-dimensional surface for illustrative
    purposes only.  In higher dimensions, the polygons must be
    replaced by higher dimensional polyhedra and the number of polyhedra
    would have to be corrected, but the construction would be
    otherwise identical.}
  \label{fig:polyhedronsphere}
\end{figure}

The curvature of our polyhedral surface is concentrated as
$\delta$-functions at the vertices, and the metric of the polyhedron also
fails to be smooth at the edges. We therefore slightly round the
vertices and edges, to produce a smooth surface, with large curvature
near the (former) vertices and nearly flat geometry elsewhere.

In the spirit of George Gamow's discussion of ``Flatland'' in his book
{\it One, Two, Three \ldots Infinity}~\cite{gamow:1947}, we now
consider how creatures making observations on our above constructed
``Sphereland'' would interpret what they experience. Our observers are
assumed to be tiny---much smaller than the typical distance between
the vertices of the original polyhedron. They are blind, and they cannot
move off the surface. Nevertheless, they have ropes that they can pull
tightly between any two points on the surface, and they can measure
rope lengths and angles between ropes extremely accurately. This
allows them to accurately map out the curvature of Sphereland by
measuring the deviation from $180^{\circ}$ of the sum of the angles of
a triangle constructed from taut ropes. When they plot the
curvature, they obtain results that depend upon the size of the
triangle they use, in a manner that qualitatively looks very much like
\fref{fig:ellis}. In particular, when they use triangles that are much
smaller than the typical distance between the vertices, the curvature
fluctuates enormously, usually being very small but occasionally being
extremely large. On the other hand, when they use triangles that are
large compared with the distance between vertices, they obtain results
that are reasonably consistent with spherical geometry. Nevertheless,
even when looking at phenomena on large scales, the observers find
some disturbing anomalies when attempting to model Sphereland by a
perfect sphere. For example, when they measure the angle subtended by
a distant object (by stretching ropes to the ends of the object and
measuring the angle between the ropes), they usually find it to be
somewhat smaller than predicted by the sphere model, but occasionally
they find it to be much larger. They also find that they can sometimes
span more than one taut rope between the same two points, even
though these points are not close to being antipodal in the sphere
model.

As a result of these observations, the observers might be tempted to
conclude that the perfect sphere model provides a reasonably good
description of Sphereland on large scales---although with some
significant deviations---but an extremely poor description of
Sphereland on small scales. However, the actual situation, of course,
is that the metric, $q_{ab}$, of Sphereland is everywhere extremely
close to the metric, $q_{ab}^{(0)}$, of a perfect sphere. This is true
on all scales and does not require any averaging of the Sphereland
metric. Nevertheless, despite the fact that $q_{ab}$ and
$q_{ab}^{(0)}$ are extremely close, their derivatives are not very
close, so, in particular, geodesics determined by $q_{ab}$ can differ
significantly from geodesics determined by $q_{ab}^{(0)}$. This
accounts for the angular size and other anomalies they find in their
large scale measurements. Finally, second derivatives of $q_{ab}$
differ enormously from second derivatives of $q_{ab}^{(0)}$, so the
curvature of $q_{ab}$ differs enormously from that of $q_{ab}^{(0)}$. In
summary, the geometry of Sphereland is described by a metric of the
form $q_{ab}^{(0)} + s_{ab}$, where $q_{ab}^{(0)}$ is the metric of a
perfect sphere and $|s_{ab}| \ll |q_{ab}^{(0)}|$, but first
derivatives of $s_{ab}$ are not small, and second derivatives of
$s_{ab}$ may be enormous.

In an exactly similar manner, the {\em spacetime} metric of our
universe takes the form\footnote{It is fair to ask how we ``know'' the
  facts asserted in this paragraph. As with all scientific
  ``knowledge,'' our beliefs are based on having a small set of simple
  assumptions that account for a vast amount of disparate data in a
  mathematically consistent manner. The $\Lambda$CDM model is based
  upon a simple set of assumptions and successfully accounts for a
  vast amount of disparate data. The results summarized in this
  article confirm that it is mathematically consistent.  Our figure of
  ``1 part in $10^4$'' comes from Newtonian cosmological simulations,
  which yield values of the Newtonian potential in the present
  universe no larger than $\sim 10^{-4}$ (as occurs near the center of
  the richest galaxy clusters); our dictionary (see section
  \ref{sec:newtoniancosmology}) implies that the spacetime metric
  deviations from FLRW are of the same size as the Newtonian
  potential.} $g_{ab} = g_{ab}^{(0)} + \gamma_{ab}$, where
$g_{ab}^{(0)}$ has FLRW symmetry and the components of $\gamma_{ab}$
are extremely small relative to $g_{ab}^{(0)}$---at the level of at
most about 1 part in $10^4$. This is true on all scales---except in
the immediate vicinity of black holes and neutron
stars\footnote{Korzy\'nski~\cite{Korzynski:2013tea} has recently
  extended the Lindquist-Wheeler universe consisting of a lattice of
  black holes~\cite{Lindquist:1957} to irregular black hole
  distributions, and has shown that an FLRW metric arises as a
  continuum limit as the number of black holes goes to infinity.  This
  construction illustrates that even in a universe containing only
  black holes, the metric can be very close to an FLRW metric except
  in the immediate vicinity of the black holes.}---and does not
require any averaging. However, first derivatives of $\gamma_{ab}$ are
not negligibly small, and second derivatives of $\gamma_{ab}$ can be
enormous, in accord with the large density inhomogeneities that we
observe on scales much less than the Hubble radius.

Nevertheless, the fact that $g_{ab}$ is very close to $g_{ab}^{(0)}$
does not imply that backreaction effects of inhomogeneities must be
negligible; as already discussed above, {\it a priori} the quantity
$t_{ab}^{(0)}$ appearing in \eref{avgein} could be large. In the next
section, we will summarize our results showing that, in fact, matter
inhomogeneities cannot produce large backreaction effects. In
\sref{sec:averaging} we will discuss the main flaw of some other
approaches that have suggested that large backreaction effects may
occur. In \sref{sec:newtoniancosmology}, we will describe how to very
accurately translate Newtonian cosmological models into general
relativistic models.

Our notation follows~\cite{Wald:1984}.  Roman letters from the early alphabet,
$\{a,b,c,\dots\}$, denote abstract spacetime indices. Roman letters
from mid-alphabet, $\{i,j,k,\ldots\}$, denote spatial
indices.

\section{Summary of our framework and results}\label{sec:framework}

In this section we briefly review the framework of~\cite{Green:2010qy}
and summarize our main results. Our framework is a generalization to
the non-vacuum case of a framework proposed by
Burnett~\cite{Burnett:1989gp}, which itself is a rigorous version of
Isaacson's approach~\cite{Isaacson:1967zz,Isaacson:1968zza}. Related
approaches that make Newtonian assumptions from the outset are given
in \cite{Futamase:1996fk,Carbone:2004iv,Baumann:2010tm}; see also
\cite{Zalaletdinov:1996aj}.

As indicated in the Introduction, we are interested in describing a
situation where the actual spacetime metric is of the form $g_{ab} =
g_{ab}^{(0)} + \gamma_{ab}$, where $g_{ab}^{(0)}$ has FLRW symmetry
and $\gamma_{ab}$ is small, but where first derivatives of
$\gamma_{ab}$ are not small and second derivatives of $\gamma_{ab}$
may be unboundedly large. The FLRW symmetry of $g_{ab}^{(0)}$ plays no
role in our framework or results, so in this section, $g_{ab}^{(0)}$
may be any smooth metric.

We would like to determine what restrictions Einstein's equation places
on $g_{ab}^{(0)}$. To do so, we could try to proceed by plugging the
metric form $g_{ab} = g_{ab}^{(0)} + \gamma_{ab}$ into Einstein's
equation and collecting terms of the various ``sizes.'' The largest
terms are those that are linear in the second derivatives of
$\gamma_{ab}$ and do not have any other factors of
$\gamma_{ab}$. However, these terms become negligibly small when
averaged. The next largest terms are of two distinct types: (a) terms
made from $g_{ab}^{(0)}$ alone and (b) terms quadratic in
$\gamma_{ab}$ that have a total of two derivatives acting on
$\gamma_{ab}$. When averaged over a spacetime region that is small
compared with the radius of curvature of $g_{ab}^{(0)}$ but large
compared with the variation scale of $\gamma_{ab}$, we will get an
equation of the form \eref{avgein}, where $t_{ab}^{(0)}$ is the
average of the quadratic terms (b). Since all terms in Einstein's
equation contain at most a total of only two derivatives acting on the
metric, all of the cubic and higher order terms in $\gamma_{ab}$
appearing in Einstein's equation become negligible when averaged.

However, if one tries to work with a {\em finite} perturbation $\gamma_{ab}$ in the above manner, it would be difficult to precisely define the ``averaging'' that must be done to obtain to obtain \eref{avgein}, as well as to justify various manipulations of terms. It would also be extremely difficult to obtain any precise, general results on the properties of $t_{ab}^{(0)}$. In order to have precise mathematical rules and obtain precise results, we would like to work with ``infinitesimal'' $\gamma_{ab}$. As in ordinary perturbation theory (see, e.g., section~7.5 of \cite{Wald:1984}), this means working with suitable one-parameter families of metrics $g_{ab} (\lambda)$ such that $[g_{ab} (\lambda) - g_{ab}^{(0)}] \to 0$ as $\lambda \to 0$. However, in order to represent the sort of perturbations that we are interested in, we do {\em not} want to require that spacetime derivatives of $[g_{ab} (\lambda) - g_{ab}^{(0)}]$ go to zero as $\lambda \to 0$. Our precise assumptions on $g_{ab}(\lambda)$ are as follows \cite{Green:2010qy}:
\begin{enumerate}
\item{\label{assumption1} Einstein's equation holds for all $\lambda>0$, i.e., we have
\begin{equation}
G_{ab}(g(\lambda)) + \Lambda g_{ab}(\lambda) = 8 \pi T_{ab}(\lambda),
\label{Ee}
\end{equation}
where $T_{ab}(\lambda)$ satisfies the weak energy condition,
i.e., for all $\lambda >0$ we have \mbox{$T_{ab}(\lambda) t^a(\lambda)
  t^b (\lambda) \geq 0$} for all vectors $t^a(\lambda)$ that are
timelike with respect to $g_{ab}(\lambda)$.}
\item{\label{assumption2}There exists a smooth positive function $C_1(x)$ on $M$ such that
\begin{equation}
|\gamma_{ab}(\lambda,x)| \leq \lambda C_1(x),
\end{equation}}
where $\gamma_{ab} (\lambda,x) \equiv g_{ab}(\lambda,x) - g_{ab}(0,x)$.
\item{\label{assumption3}There exists a smooth positive function $C_2(x)$ on $M$ such that
\begin{equation}
|\nabla_c \gamma_{ab}(\lambda,x)| \leq C_2(x).
\end{equation}}
\item{\label{assumption4}There exists a smooth tensor field $\mu_{abcdef}$ on $M$ such that
\begin{equation}
\wlim_{\lambda\to0}
\left[\nabla_a\gamma_{cd}(\lambda)\nabla_b\gamma_{ef}(\lambda) \right] =
\mu_{abcdef},
\label{mu}
\end{equation}
where ``$\wlim$'' denotes the weak limit.}
\end{enumerate}

Here the norms appearing in assumptions (\ref{assumption2}) and (\ref{assumption3}) are taken with respect to any fixed (i.e., $\lambda$-independent) Riemannian metric on spacetime and $\nabla_a$ denotes any fixed derivative operator.
The notion of ``weak limit'', which appears in assumption (\ref{assumption4}),
corresponds roughly to taking a local spacetime average, and then
taking the limit as $\lambda\to0$.  More precisely, $A_{a_1\dots
  a_n}(\lambda)$ is said to {\em converge weakly} to $A_{a_1\dots a_n}^{(0)}$ as
$\lambda\to0$ if and only if, for all smooth tensor fields
$f^{a_1\dots a_n}$ of compact support,
\begin{equation}
  \lim_{\lambda\to0}\int f^{a_1\dots a_n}A_{a_1\dots a_n}(\lambda)=\int f^{a_1\dots a_n}A_{a_1\dots a_n}^{(0)}.
\end{equation}
Assumptions (\ref{assumption2}) and (\ref{assumption3}) capture the notion that $\gamma_{ab}$ is small but its first spacetime derivatives need not be small. Assumption (iv) captures the notion that local spacetime averages of quadratic products of first derivatives of $\gamma_{ab}$ are well behaved.
An explicit example of a one parameter family satisfying assumptions (\ref{assumption1})--(\ref{assumption4}) is given in~\cite{Green:2013yua}.

Assumptions (\ref{assumption1})--(\ref{assumption4}) allow us to
rigorously derive an equation satisfied by $g_{ab}^{(0)}$ of the form
\eref{avgein}, where $t_{ab}^{(0)}$ is given by an explicit formula in
terms of $\mu_{abcdef}$. In \cite{Green:2010qy}, we then proved two
theorems constraining $t_{ab}^{(0)}$:
\begin{theorem}
Given a one-parameter family $g_{ab}(\lambda)$
  satisfying assumptions (\ref{assumption1})--(\ref{assumption4})
  above, the effective stress-energy tensor $t^{(0)}_{ab}$ appearing
  in equation~\eref{avgein} for the background metric $g^{(0)}_{ab}$ is
  traceless,
\begin{equation}
{t^{(0)a}}_a = 0.
\end{equation}
\end{theorem}
\begin{theorem}
  Given a one-parameter family $g_{ab}(\lambda)$ satisfying
  assumptions (\ref{assumption1})--(\ref{assumption4}) above, the
  effective stress-energy tensor $t^{(0)}_{ab}$ appearing in equation~\eref{avgein} for the background metric $g^{(0)}_{ab}$
  satisfies the weak energy condition, i.e.,
\begin{equation}
t^{(0)}_{ab} t^a t^b \geq 0
\end{equation}
for all $t^a$ that are timelike with respect to $g^{(0)}_{ab}$.
\end{theorem}

In essence, these theorems show that only those small scale metric
inhomogeneities corresponding to gravitational radiation can have a
significant backreaction effect; see \cite{Green:2010qy} for further discussion. In particular, it should be noted that in the case where $g^{(0)}_{ab}$ has FLRW
symmetry, even when short wavelength gravitational radiation is present and backreaction effects are large, the effective stress-energy tensor must be of the form of a
$P=\frac{1}{3}\rho$ fluid, and therefore cannot mimic dark energy.

In~\sref{sec:newtoniancosmology}, we will return to the issue of how
best to describe $g_{ab}$ in cosmology. Specifically, we will consider
how to translate Newtonian cosmologies into general relativistic
cosmologies in a way that yields accurate solutions to Einstein's
equation. However, we first give a brief discussion of some other
approaches that have suggested that backreaction might be large.

\section{Other approaches to backreaction}\label{sec:averaging}

As the results summarized in the previous section have shown, small
scale density inhomogeneities cannot produce large backreaction
effects in cosmology. However, a number of other approaches have come
to the opposite conclusion---or, at least, have suggested the
possibility that the cosmological backreaction effects due to matter
inhomogeneities on small and/or large scales could be large. Some of
these approaches are based upon questionable approximation methods
(see, e.g.,~\cite{Kolb:2005da}) and some are based upon questionable
interpretations of gauge-dependent quantities (see,
e.g.,~\cite{Mukhanov:1996ak,Abramo:1997hu,Unruh:1998ic,Geshnizjani:2002wp,Geshnizjani:2003cn}
with regard to long wavelength perturbations); we will not attempt to
review such approaches here. However, other approaches proceed by
assigning an FLRW metric, $g^{(0)}_{ab}$, to $g_{ab}$ via some
averaging procedure or the matching of some observables. As we shall
now illustrate, the main flaw with such approaches is that the FLRW
metric $g^{(0)}_{ab}$ that one thereby obtains may not be close to
$g_{ab}$---even in cases where $g_{ab}$ is well approximated by some
FLRW metric. Since $g^{(0)}_{ab}$ may be a poor approximation to
$g_{ab}$, there is no reason why $g^{(0)}_{ab}$ need be close to a
solution to Einstein's equation, so one may obtain large backreaction
terms in equations satisfied by $g^{(0)}_{ab}$. Of course, these
apparent backreaction effects are entirely spurious, being generated
by using a poor approximation to $g_{ab}$.

The approach of Buchert~\cite{Buchert:1999er,Buchert:2001sa} provides a good illustration of this point.
In Buchert's approach the
metric is written in a synchronous slicing comoving with the dust
particles,
\begin{equation}\label{eq:synchronousgauge}
  ds^2 = -dt^2 + {}^{(3)}g_{ij} (t) dx^idx^j.
\end{equation}
This is possible provided the dust 4-velocity $u^a$ is irrotational,
which we assume here. The volume, $V_{\mathcal{D}}$, of a comoving spatial region $\mathcal{D}$ on a constant-$t$ slice $\Sigma_t$ is given by
\begin{equation}
V_{\mathcal{D}}(t) = \int_{\mathcal{D}} \,\mathrm{d} \Sigma_t,
\end{equation}
where ${\mathrm{d}}\Sigma_t = \sqrt{{}^{(3)}g(t)} \, \mathrm{d}^3 x$ is the proper volume element on $\Sigma_t$. The averaged ``scale factor,'' $a_{\mathcal{D}}(t)$, is defined as proportional to
$V_{\mathcal{D}}^{1/3}$.
Spatial averages of scalars $\psi$ over $\mathcal{D}$ can then be defined by
\begin{equation}
  \langle \psi \rangle_{\mathcal{D}}(t) \equiv \frac{1}{V_{\mathcal{D}}}\int_{\mathcal{D}} \psi \,\mathrm{d}\Sigma_t  .
\end{equation}
In this way, one can define an averaged density, $\langle\rho\rangle_{\mathcal{D}}(t)$, and averaged scalar curvature, $\langle{}^{(3)}{\mathcal{R}}\rangle_{\mathcal{D}}(t)$. One may then assign an FLRW model, $g^{(0)}_{ab}$, to $g_{ab}$ based upon the values of these averaged quantities.

Equations governing the dynamics of $g^{(0)}_{ab}$ can be derived
by taking averages of the scalar components of Einstein's equation for $g_{ab}$. One obtains~\cite{Buchert:1999er,Buchert:2001sa}
\begin{eqnarray}
  \label{eq:buchert1}3\left(\frac{\dot{a}_{\mathcal{D}}}{a_{\mathcal{D}}}\right)^2=8\pi\langle\rho\rangle_{\mathcal{D}}-\frac{1}{2}\langle{}^{(3)}{\mathcal{R}}\rangle_{\mathcal{D}}- \frac{1}{2} \mathcal{Q}_{\mathcal{D}},\\
  \label{eq:buchert2}3\frac{\ddot{a}_{\mathcal{D}}}{a_{\mathcal{D}}} = -4\pi\langle{\rho}\rangle_{\mathcal{D}} + \mathcal{Q}_{\mathcal{D}},
\end{eqnarray}
where the quantity
\begin{equation}
  {\mathcal{Q}}_{\mathcal{D}} \equiv \frac{2}{3}\left( \langle \theta^2 \rangle_{\mathcal{D}} - \langle \theta\rangle_{\mathcal{D}}^2\right) - 2 \langle \sigma^2 \rangle_{\mathcal{D}}  .
\end{equation}
characterizes the ``backreaction,'' i.e., the failure of the FLRW spacetime $g^{(0)}_{ab}$, to satisfy Einstein's equation. There is no reason why $\mathcal{Q}_{\mathcal D}$ need be small\footnote{A further
  difficulty with the Buchert approach is that the equations are
  underdetermined, as there is no evolution equation given for
  $\mathcal{Q}_{\mathcal D}$.  However, one is not free to specify
  ${\mathcal{Q}}_{\mathcal D}(t)$ arbitrarily, as the exact (i.e., not averaged)
  metric must satisfy the Einstein equation with dust stress energy, but only two components of this equation have been used to get the Buchert equations.
  See~\cite{Green:2013yua} for further discussion.}, so there can be large backreaction within the Buchert framework.

The large backreaction that can occur in the Buchert framework is a direct consequence of the fact that $g^{(0)}_{ab}$ can be a poor approximation to $g_{ab}$. This is most easily seen in the case where $g_{ab}$ is not globally close to a single FLRW metric\footnote{If the actual metric, $g_{ab}$, of the universe were not globally close to a single FLRW metric, we do not believe that it would be useful to introduce any concept whatsoever of an ``averaged'' FLRW metric $g^{(0)}_{ab}$.}. A nice example of this is provided by a universe that consists of two disconnected dust FLRW universes in different stages of expansion. The Buchert prescription would represent this disconnected universe as a single FLRW universe, which provides a bad approximation to the actual metric everywhere. One can give an example of this sort~\cite{Ishibashi:2005sj} wherein the ``backreaction'' is so large that one obtains acceleration of the representative FLRW universe, even though each of the disconnected components of the actual universe is decelerating. Obviously, this acceleration is not a physical effect but is a spurious artifact of the poor approximation that $g^{(0)}_{ab}$ provides to the actual spacetime metric.

However, the Buchert prescription can also give a poor approximation to the actual spacetime metric $g_{ab}$---and, correspondingly, yield a large backreaction---even when $g_{ab}$ is well approximated by (or even equal to!) {\it some} FLRW metric. An example of this is provided by Minkowski spacetime, which has exact FLRW symmetry. Let $(T,X,Y,Z)$ be global inertial coordinates and consider the hypersurface defined by
\begin{equation}
  T=f(X,Y,Z),
\end{equation}
with
\begin{equation}
  f(X,Y,Z)=\frac{1}{3N}\sin(NX)\sin(NY)\sin(NZ)  ,
\end{equation}
with $N$ large. This hypersurface is extremely close to the hypersurface $T=0$, but is extremely ``wiggly.'' (However, note that $|\nabla_i f|<1$ everywhere, so the hypersurface is everywhere spacelike.) We choose this hypersurface as the initial hypersurface of a synchronous coordinate system \eref{eq:synchronousgauge}. We take the domain, $\mathcal{D}$, in the initial hypersurface to be a coordinate cube of side $[0,2\pi]$. Since the matter stress-energy vanishes, we obviously have $\langle\rho\rangle_{\mathcal{D}} = 0$. However, we have verified numerically that $\langle{}^{(3)}{\mathcal{R}}\rangle_{\mathcal{D}}$ is enormous for sufficiently large $N$. Thus, although the actual metric, $g_{ab}$, of Minkowski spacetime has exact FLRW symmetry, the Buchert procedure with the above choice of hypersurface instructs us to represent the Minkowski metric with an ``averaged'' FLRW metric $g^{(0)}_{ab}$ that is an extremely poor approximation to $g_{ab}$ on all scales. The backreaction term $\mathcal{Q}_{\mathcal D}$ is correspondingly large. Again, this backreaction is entirely spurious.

One might object to the above example on the grounds that there is no matter present in Minkowski spacetime and we have arbitrarily chosen some artificial hypersurface on which to perform the Buchert construction. However, we could modify this example by adding a tiny sprinkling of dust following geodesics orthogonal to our hypersurface. If the mass density of the dust is sufficiently small, the metric will be essentially unaffected by the dust, but the Buchert procedure will require us to use the above hypersurface and will provide us with an effective FLRW metric that is a very poor approximation to the actual spacetime metric. Of course, the velocity distribution of the dust in this example is not representative of any physically realistic situation. But the example is useful for showing that the Buchert construction depends upon the velocity distribution of the dust even when the dust makes a negligible contribution to the stress-energy. In realistic cosmological situations, the perturbations to the dust velocity will be much larger than the perturbations to the metric (see subsection~\ref{sec:mapping} below), and the comoving synchronous hypersurfaces of the Buchert construction will provide a poor choice for approximating the hypersurfaces with nearly FLRW symmetry. This will lead to a correspondingly large backreaction. We will comment further on this point at the end of subsection~\ref{sec:mapping}.

For similar reasons, large backreaction also appears in the approach
of Clarkson and Umeh~\cite{clarkson2011backreaction}. These authors
obtain expressions for an effective $H$ and $q_0$---and, thereby, a
representative FLRW metric $g^{(0)}_{ab}$---by writing down an
approximate formula for the angular diameter distance $d_{\mathrm{A}}$
as a function of redshift $z$ and performing an average. However,
lensing effects are sensitive to small scale structure since they
depend on derivatives of the metric; as discussed further in
subsection~\ref{sec:observations} below, lensing effects can be very
large even when the deviations of the metric from an FLRW metric are
very small. Consequently, for the power spectra they consider, they
obtain UV divergent expressions for their effective $H$ and $q_0$. The
expressions for backreaction are similarly divergent. Again, these
large backreaction effects arise from the fact that their choice of
FLRW metric, $g^{(0)}_{ab}$, is an extremely poor
approximation\footnote{It may sound odd that an FLRW metric
  $g^{(0)}_{ab}$ that is chosen to be the {\em best fit} to some
  observation made in the nearly FLRW metric, $g_{ab}$, should be an
  extremely poor approximation to $g_{ab}$. However, one might note
  that if our Sphereland observers chose the sphere metric that gave
  the best fit to their observation of, e.g., the average of the {\em
    square} of the curvature, they would get an extremely poor
  approximation to the metric of Sphereland.} to the actual spacetime
metric, $g_{ab}$.

\section{Cosmology with a Newtonianly-perturbed FLRW metric}\label{sec:newtoniancosmology}

As we have argued above, the metric of the universe is well described
by a metric of the form $g_{ab} = g^{(0)}_{ab} + \gamma_{ab}$, where
$g^{(0)}_{ab}$ is an FLRW metric satisfying the usual Einstein
equation with the averaged matter stress-energy as source, and
$\gamma_{ab}$ is ``small''---although its derivatives are not
small. We would like to determine $\gamma_{ab}$ more precisely. It is
clear that $\gamma_{ab}$ does not merely satisfy the linearized
Einstein equation about $g^{(0)}_{ab}$, since nonlinear terms in
derivatives of $\gamma_{ab}$ cannot be ignored. In particular, we
would like to know the leading order effects that these nonlinear
corrections have on large scale behavior, including the expansion rate
of the universe.

We analyzed this issue in~\cite{Green:2010qy} in the context of the completely general framework described in \sref{sec:framework} above. In that reference, we derived an equation satisfied by the ``long wavelength'' part of $\gamma_{ab}$ that takes into account the leading order nonlinear corrections due to the small scale inhomogeneities. However, this equation is extremely complicated, and it is not easy to see what effects it predicts in a general context. To make further progress, it is necessary to put in more assumptions about the nature of our universe. A key observational fact about our universe is that it is locally Newtonian, i.e., as seen by observers ``at rest'' with respect to $g^{(0)}_{ab}$, velocities of nearby objects are nonrelativistic and their dynamics is well described by Newtonian gravity---except, of course, in the immediate vicinity of black holes and neutron stars, which are not relevant for this discussion. Indeed, cosmologists normally simulate structure formation in our universe using only Newtonian gravity---even on scales comparable to the Hubble radius!

These considerations allow us to reformulate the issue of determining the leading order corrections to $g^{(0)}_{ab}$ as follows~\cite{Green:2011wc}: Consider a cosmological solution to the equations of {\em Newtonian gravity} of the kind produced in numerical simulations by cosmologists. We may ask the following questions: (1) What ``dictionary'' should be used to transcribe this Newtonian cosmology into a general relativistic cosmology? (2) How accurate is this dictionary, i.e., how close does the resulting general relativistic cosmology come to satisfying Einstein's equation? Of course, these questions are not independent: If the dictionary did not produce accurate results, this would indicate that we need a better dictionary. As we shall describe below, there is a straightforward dictionary suggested by the correspondence between linearized Newtonian gravity and linearized general relativity. This dictionary is quite satisfactory, but it can be further improved~\cite{Green:2011wc} to take into account post-Newtonian effects at small scales as well as the leading order large scale effects produced by the nonlinear effects of inhomogeneities. The resulting improved dictionary should yield an extremely accurate general relativistic description of our universe, and, in particular, it provides the leading order backreaction effects at large scales produced by small scale inhomogeneities.

We will now review Newtonian cosmology. We will then present a dictionary that
maps Newtonian solutions into general relativistic cosmologies and discuss its accuracy.  Finally we discuss some effects that these
inhomogeneities have on light propagation and observables.

\subsection{Newtonian cosmology}

It is a remarkable fact that the evolution equations for a homogeneous and isotropic cosmology with dust matter and a cosmological constant are precisely the same
in Newtonian gravity and general relativity.  To see this, begin with
the equations of Newtonian gravity with a cosmological constant,
\begin{eqnarray}
  \label{eq:poisson}\partial^i\partial_i\phi+\Lambda=4\pi\rho,\\
  \label{eq:masscons}\partial_t\rho+\partial_i(\rho v^i)=0,\\
  \label{eq:euler}\partial_t(\rho v^i)+\partial_j(\rho v^i v^j)=-\rho\partial^i\phi.
\end{eqnarray}
Plugging in the cosmological ansatz, $\rho=\rho_0(t)$, $v^i=H(t)x^i$,
and requiring solutions to be non-singular, it can be
seen~\cite{Peebles1980} that \eref{eq:poisson}--\eref{eq:euler}
reduce to the Friedmann equations for a dust cosmology,
\begin{eqnarray}
  \label{eq:friedmann1}\frac{dH}{dt}+H^2=-\frac{4\pi}{3}\rho_0+\frac{\Lambda}{3},\\
  \label{eq:friedmann2}H^2=\frac{8\pi}{3}\rho_0+\frac{\Lambda}{3}-\frac{k}{a^2}.
\end{eqnarray}
Here $a$ is the radius of any comoving ball, so that
\begin{equation}
  H=\frac{1}{a}\frac{da}{dt}.
\end{equation}
The constant $k$ is a constant of integration, and the density $\rho_0$ varies as $1/a^3$.  Thus, there is an obvious correspondence
between Newtonian and relativistic dust cosmologies in the homogeneous and isotropic case, which takes exact solutions to exact solutions. In the following, we restrict consideration to the $k=0$ case.

It is useful to rewrite the exact Newtonian equations relative to a background
solution and to transform to a comoving coordinate system of the
background. We obtain~\cite{Green:2011wc}
\begin{eqnarray}
  \label{eq:poisson1}\partial^i\partial_i\psi_{\mathrm{N}}=4\pi a^2\rho_0\delta_{\mathrm{N}},\\
  \label{eq:masscons1}\dot{\delta}_{\mathrm{N}}+\partial_i((1+\delta_{\mathrm{N}})v_{\mathrm{N}}^i)=0,\\
  \label{eq:euler1}\dot{v}_{\mathrm{N}}^i+v_{\mathrm{N}}^j\partial_jv_{\mathrm{N}}^i+\frac{\dot{a}}{a}v_{\mathrm{N}}^i=-\partial^i\psi_{\mathrm{N}},
\end{eqnarray}
where $v_{\mathrm{N}}^i$ is the velocity relative to the ``Hubble flow,'' $\psi_{\mathrm{N}}$ is the Newtonian potential relative to the background, and $\delta_{\mathrm{N}}$ is the fractional density perturbation. We have inserted the subscript ``N'' on
these expressions in order to be able to clearly distinguish them from the corresponding general relativistic quantities that we will consider below. Equations \eref{eq:poisson1}--\eref{eq:euler1} (or variants describing
discrete matter sources) form the basis of numerical simulations
exploring structure formation in cosmology.

\subsection{Mapping to general relativity}\label{sec:mapping}

It is perhaps an even more remarkable fact that the exact
correspondence between homogeneous, isotropic Newtonian and general
relativistic dust cosmologies also extends to linearized solutions in the
$k=0$ case, i.e., linearized solutions of
\eref{eq:poisson1}--\eref{eq:euler1} correspond exactly to (scalar and
vector sector) solutions of the $k=0$ general relativistic
perturbation equations~\cite{Bardeen:1980kt} when an appropriate
``dictionary'' is used. For scalar perturbations, this is most easily
seen by noting that, when written in terms of Bardeen's gauge
invariant variables, the Einstein equation with dust matter,
linearized off of FLRW, is identical to the linearized form of
\eref{eq:poisson1}--\eref{eq:euler1}, with the identifications
\begin{eqnarray}
  \label{eq:Bardeen1}\psi_{\mathrm{N}} \longleftrightarrow \Phi_{\mathrm{A}}=-\Phi_{\mathrm{H}},\\
  \label{eq:Bardeen2}v_{\mathrm{N}}^i \longleftrightarrow v_{\mathrm{s}}^i,\\
  \label{eq:Bardeen3}\delta_{\mathrm{N}} \longleftrightarrow \epsilon_{\mathrm{m}}  ,
\end{eqnarray}
where we have used Bardeen's notation for the gauge invariant variables $\Phi_{\mathrm{A}}$, $\Phi_{\mathrm{H}}$, $v_{\mathrm{s}}^i$, and $\epsilon_{\mathrm{m}}$.
A similar correspondence holds in the vector sector, although there it is necessary to also solve a Poisson-type
equation to obtain the vector-type metric perturbation~\cite{Green:2011wc}. The
spacetime metric and stress-energy tensor may be straightforwardly
reconstructed in any gauge from the gauge invariant variables.

While the gauge invariant Bardeen variables are extremely convenient
to use in linear perturbation theory, there are no analogs of them
beyond linear order. Thus, in order to attempt to extend the above
exact Newtonian-relativistic correspondence for linearized solutions
to an approximate Newtonian-relativistic correspondence for exact
solutions, we must make a choice of gauge. We choose to work in
``longitudinal gauge,'' where the metric takes the form
\begin{equation}
  \label{eq:metric-longitudinal}ds^2=a^2(\tau)\left\{-(1+2A)d\tau^2 - 2B_i dx^id\tau + [(1+2H_{\mathrm{L}})\delta_{ij}+h_{ij}]dx^idx^j\right\},
\end{equation}
where $\partial^i B_i = \partial^j h_{ij} = h^i_{\phantom{i}i} =
0$. As argued in~\cite{Green:2011wc}, it should be possible to make
this gauge choice for metrics that are sufficiently close to FLRW
metrics. We will comment further on this gauge choice below.

We could obtain a dictionary by simply transcribing the linearized
dictionary to the nonlinear regime, replacing the Bardeen variables in
the linearized correspondence with their metric component counterparts
in longitudinal gauge.  However, we can easily improve this dictionary
by slightly modifying the definitions of $v^i$ and $B^i$ so as to
obtain consistency with the nonlinear momentum constraint at small
scales.  We obtain~\cite{Green:2011wc}
\begin{eqnarray}
  \label{eq:dict1}A=-H_L=\psi_{\mathrm{N}},\\
  \label{eq:dict2}(1+\delta_{\mathrm{N}})v^i=(1+\delta_{\mathrm{N}})(v_{\mathrm{N}}^i+B^i)-\overline{(1+\delta_{\mathrm{N}})v_{\mathrm{N}}^i},\\
  \label{eq:dict3}\delta=\delta_{\mathrm{N}}-\frac{3}{4\pi\rho_0a^2}\left[\left(\frac{\dot{a}}{a}\right)^2\psi_{\mathrm{N}}+\frac{\dot{a}}{a}\dot{\psi}_{\mathrm{N}}\right],\\
\label{eq:dict4}\partial^j\partial_j B^i = -16\pi\rho_0a^2\left.\left((1+\delta_{\mathrm{N}})v_{\mathrm{N}}^i-\overline{(1+\delta_{\mathrm{N}})v_{\mathrm{N}}^i}\right)\right|_{\mathrm{v}},\\
  \label{eq:dict5}h_{ij}=0.
\end{eqnarray}
In these expressions, the overline denotes an average over all of
space, while the subscript ``v'' denotes the ``vector part''.  Note
that it is necessary to solve the Poisson equation~\eref{eq:dict4} to
obtain $B^i$, but this is a straightforward procedure that could be
incorporated into numerical simulations.  The quantity $B^i$ and the
corrections to $v^i$ will normally be small.  If we neglect these
terms, we obtain a continuum version of the dictionary developed
in~\cite{Chisari:2011iq}.  We note, however, that very recent efforts
to incorporate $B^i$ into Newtonian simulations have suggested that it
may be possible to measure the frame dragging effects of
$B^i$~\cite{Bruni:2013mua,Adamek:2013wja}.

If we start with an exact Newtonian cosmology and translate it into a
general relativistic cosmology using the above dictionary, we wish to
know how well Einstein's equation is satisfied as well as the
corrections to the dictionary that would be needed to make Einstein's
equation hold to greater accuracy. In order to proceed, we need to
have an {\it a priori} notion of the ``size'' of the Newtonian
quantities. To assign such orders, we break up each quantity into its
(``long wavelength'') averaged part and its (``short wavelength'')
remainder. We introduce a ``small parameter'' $\epsilon$, and take the
long wavelength parts of $\psi_{\mathrm{N}}$, $v_{\mathrm{N}}$, and
$\delta_{\mathrm{N}}$ to be $\Or(\epsilon)$. We also take all space
and time derivatives of the long wavelength parts of these quantities
to be $\Or(\epsilon)$. On small scales, where nonlinear motion is
important, our assignment of ``sizes'' is based on usual
post-Newtonian counting.  We take $\psi_{\mathrm{N}}=\Or(\epsilon)$
and $v_{\mathrm{N}}=\Or(\epsilon^{1/2})$. Each spatial derivative is
assumed to contribute an inverse power of $\epsilon$, $\partial_i \sim
\Or(\epsilon^{-1})$, and each time derivative is assumed to contribute
an inverse half-power of $\epsilon$, $\partial_0\sim
\Or(\epsilon^{-1/2})$. Consistent with the Newtonian equation
\eref{eq:poisson1}, we then have
$\delta_{\mathrm{N}}=\Or(\epsilon^{-1})$.  Oliynyk has proven
existence of a wide class of {\em one-parameter families} of
inhomogeneous solutions to the Einstein equation with this
behavior~\cite{Oliynyk:2013jqa,Oliynyk:2014ufa}.

With this counting scheme in hand, we can check how well the Einstein
equation is satisfied by the relativistic cosmology given by our
dictionary. {\it A priori}, the long wavelength part of Einstein's
equation contains terms at most $\Or(\epsilon)$, whereas the short
wavelength part of Einstein's equation contains terms that are
$\Or(\epsilon^{-1})$. We find that, with the above dictionary, the
long wavelength part fails to hold at $\Or(\epsilon)$---due to the
appearance of such terms as the average of quadratic products of
derivatives of the short wavelength part of
$\psi_{\mathrm{N}}$---whereas the short wavelength part of Einstein's
equation fails at $\Or(1)$. However, we showed in~\cite{Green:2011wc}
how to improve our dictionary so that the long wavelength part of
Einstein's equation holds at $\Or(\epsilon)$, and the short wavelength
part of Einstein's equation holds at $\Or(1)$. The short wavelength
corrections are post-Newtonian corrections and are negligible for any
present-day observations.  The long wavelength corrections essentially
correspond to including as source terms in Einstein's equation the
effects of the energy and stresses of the Newtonian potential and the
kinetic motions. {\em These provide the leading order backreaction
  effects on large scales of the small scale inhomogeneities.}
Although these corrections are formally $\Or(\epsilon)$ in our
counting scheme, they are, in fact, negligible\footnote{In fact, the
  next order correction to the pressure vanishes completely when
  averaging virialized systems~\cite{Baumann:2010tm}; this was
  recently demonstrated numerically~\cite{Adamek:2014gva}.  The
  correction to the energy density corresponds to the inclusion of
  kinetic energy and Newtonian binding energy.}. Thus, while it is
very comforting to know that the improved ``Oxford dictionary''
of~\cite{Green:2011wc} exists, the
dictionary~\eref{eq:dict1}--\eref{eq:dict5} should provide an
excellent description of our universe. The smallness of Newtonian
perturbations guarantees that the universe is well-described by an
FLRW model, even though the stress-energy perturbations are very large
on small scales.

Finally, we make a comment on our choice of metric
form~\eref{eq:metric-longitudinal}. It is extremely important that we
did {\em not} choose a metric form---such as the comoving synchronous
form~\eref{eq:synchronousgauge}---that is tied to the matter motion,
i.e., geodesics of the actual spacetime. This is because the geodesics
of the actual spacetime differ significantly from the geodesics of the
background spacetime, so the metric components in, e.g., synchronous
coordinates can be very different even when the metrics are actually
very close. In the present instance, this phenomenon is manifested as
follows: Even at the linear level, when the
dictionary~\eref{eq:Bardeen1}--\eref{eq:Bardeen3} is expressed in a
particular gauge, there will be some ``mixing'' between the velocity,
the potential, and the density.  For example, in the longitudinal
gauge, the relativistic density perturbation \eref{eq:dict3} involves
$\psi_{\mathrm{N}}$. However, in the longitudinal gauge, the
dictionary provides metric perturbation variables that consistently
remain $\Or(\epsilon)$ in our counting scheme. By contrast, in the
comoving synchronous gauge, metric components pick up contributions
from $v_{\mathrm{N}}$. These cause the metric perturbation to become
$\Or(1)$ on a timescale of order $\epsilon^{1/2}$, thereby potentially
leading to large, spurious backreaction effects. This is closely
related to the example given near the end of \sref{sec:averaging}.

\subsection{Effects on observables}\label{sec:observations}

As we have emphasized throughout this article, the spacetime metric, $g_{ab}$, of our universe is very close to an FLRW metric, $g_{ab}^{(0)}$. However, most things that we {\em observe} about the universe involve electromagnetic radiation, which propagates on null geodesics. However, the
geodesics of the actual metric $g_{ab}$ are {\em not} necessarily very close to corresponding geodesics of  $g_{ab}^{(0)}$.
Moreover, the time evolution of the shear and convergence of a bundle of
geodesics depends on the Riemann curvature~\cite{Wald:1984} (i.e., second
derivatives of the metric), which can be very large.  As a result,
observables in the actual universe described by $g_{ab}$ can differ
considerably from observables computed in the FLRW background
spacetime $g_{ab}^{(0)}$.

The redshift--luminosity relation provides a good example of the kinds
of differences that can occur for observables, even when $g_{ab}$ and
$g_{ab}^{(0)}$ are very close. In an FLRW spacetime, the Weyl
curvature vanishes and the Ricci curvature is uniform. However, in the
actual universe, most of the matter is clumped into structures such as
galaxies. If a beam of light passes through a clump of matter, it will
encounter high Ricci curvature and experience large convergence. If it
passes near---but not through---a clump of matter, it will encounter
only Weyl curvature, producing shear and, eventually, convergence. If
it does not pass near any clumps of matter, it will not experience any
significant curvature at all. Thus, if the actual universe is
sufficiently lumpy, then the light beams from most sources will
encounter little curvature and will be dimmed compared with the FLRW
spacetime~\cite{Dyer:1972distance}. However, occasionally, the beams
will pass through strong curvature, and the sources will be greatly
magnified and/or multiply imaged. Thus, large and highly non-Gaussian
deviations in the apparent luminosity of standard candle high redshift
sources will occur~\cite{Holz:1997ic,holz1998lensing}.

By flux conservation, the {\em average} of the apparent luminosity of
sources (including multiple images) must match the FLRW value to a
high degree of accuracy. Thus, if one were to plot the {\em mean
  apparent luminosity} of standard candle sources versus redshift, one
should obtain good agreement with the underlying FLRW model. However,
if one were to plot the mean of a nonlinear function of
luminosity---such as the luminosity distance---versus redshift, one
could get significant departures from the underlying FLRW values at
high redshift, on account of the large, non-Gaussian
fluctuations. This illustrates the
point that, as a general matter of principle, one should {\em not}
start at the level of observables and then try to design some
effective FLRW metric that captures the behavior of these
observables. Rather, one must proceed as follows:

\begin{enumerate}
\item Start with a background FLRW metric that is believed to be close
  to the actual metric of the universe (except, of course, in
  the immediate vicinity of strong-field objects such as black holes).
\item Determine the corrections to the FLRW metric that describe
  inhomogeneities.  To an excellent approximation, these are the
  (small) Newtonian corrections described in the previous subsection.
\item Calculate observables in the full metric (background plus
  perturbations).
\end{enumerate}

The $\Lambda$CDM model has been remarkably successful in carrying out
this strategy and thereby making predictions in excellent agreement
with all cosmological observations. As we have reviewed in this
article, this model is mathematically consistent and fully in accord
with general relativity. Unless some observational discrepancy arises
in the future, the $\Lambda$CDM model deserves to be ranked among the
great achievements of modern science.

\ack

This research was supported in part by NSF
grant PHY 12-02718 to the University of Chicago, and by NSERC. SRG is
supported by a CITA National Fellowship at the University of Guelph,
and he thanks the Perimeter Institute for hospitality. Research at
Perimeter Institute is supported through Industry Canada and by the
Province of Ontario through the Ministry of Research \& Innovation.

redpajama_set_name: RedPajamaArXiv


This short film portrays a teenager who in a certain way illustrates this cliché expression of the "lost generation" who no longer hopes, no longer dreams, does not delude himself. Bia refuses to make choices, to comply with the expectations of her teachers and her parents.Perhaps it is also by pure provocation, by a spirit of contradiction. In any case, this idea of contrasts, oppositions and counter-senses seems to guide the film in its development, in the composition of the shots and the staging.
With his short film Nada , the Brazilian Gabriel Martins offers us a kind of postmodern teen movie where adolescence is no longer a kind of lost paradise but a time when the individual in crisis with the world discovers the rules of the game which defines the world of adults, a world apprehended with suspicion. Far from being a beginner, Gabriel Martins has a dozen short films to his credit, and worked as a cinematographer on two films by André Novais Oliveira, previously selected for the Directors' Fortnight. Nada is a small introspective pause on life choices. Choose a job, study, work, start a family: the film evokes the expectations of a society that exerts invisible pressure on individuals.
Odé is a black man. His brother, Luiz, was beaten to death during a conflict for the rights of a community occupation in a big city. Political, racial, ancestry and urbanization relationships in the most recent Brazilian scenario.


text:
Q: Can not change lay-out of dotplot() (lattice package) based on ranef(lmerMod) We're trying to use dotplot to create a caterpillarplot based on the random effect structure of a lmerMod object created by lmer() in the lme4 package.
The plot itself is created perfectly fine, but when trying to change the lay-out errors occur.
Sample of code without lay-out instructions:
all_bv_C <- lmer(RQ_EvT_A ~ SD_Lft_M_Cat4 + SD_Opl_M_Cat3 + OV_Gez_M_4 +   (1|VSVnr), data=BV2, REML=TRUE)

random <- ranef(all_bv_C, condVar = TRUE)

dotplot(randoms, scales = list(x = list(relation = 'free')))

This creates the plot:

We wish to change the title, axis labels and the color palette. For example, to change the title, the usual syntax would be
dotplot(randoms, scales = list(x = list(relation = 'free')), main="Title")

This throws the error:

Error in if (main) nx : argument is not interpretable as logical

We've been unable to get around this error. Everywhere we looked, this should work for any dotplot usage. Can anyone shed some light?
P.S.: We're using dotplot() over ggplot() due to some irregularities in extracting the random effect structure into a data frame as would be suggested here: ggCaterpillar. The function specified throws a NULL due to:
pv   <- attr(x, "postVar")

We've also tried other routes for extracting the variance/covariance matrix to adapt the function, but felt that dotplot was the easier route after fumbling for a day.

A: You can not here without changing the code. Looking at the source code of the S3 method dotplot for ranef.mer class:
getS3method("dotplot","ranef.mer")

You can not set the titles suing arguments. If you look in the function in some line it is written explicitly:
 mtit <- if (main) nx

where nx is the names(x)( your ranef object).
So if you do  somthing like :
names(randoms) <- "Title"
dotplot(randoms)

the plot title will change. But this is a hack. Better here to change the code of the function and customize it as you like.

A: If anyone else sees this, a simpler solution would be to create the dotplot object first, and then change the title using the $ operator:
dp <- dotplot(randoms, scales = list(x = list(relation = 'free')))
dp$VSVnr$main <- "Title"

redpajama_set_name: RedPajamaStackExchange


Facts (as representations) do not exist independent of humans. If science worked by just picking up facts that existed independent of us, it would not work.
Just a moment. I'm asking Straggler to close his eyes for a moment.
Whether facts can be said to exist independent of humans, depends on whether you are a Platonist or a nominalist.
It is okay to reopen those eyes now.
But, I don't think Straggler was saying this: I think he was using anthropomorphic language to say, "The scientific method requires one to make predictions."
We use the word "prediction" very loosely, and apply it to what are not really predictions.
I still have no idea what 'it' was that was not about facts but about concepts or why it is relevant. Or why you decided to start talking about 'facts' and 'concepts' in the first place. I just pointed out that Newton used pendulums as evidential support for his laws - could you explain what your point here is?
Assuming it isn't really relevant (just to type something else as a response), do you accept that Newton's Laws have empirical support? That this support relies on a sequence of observations that are in accord with the laws? Do you agree that to suggest that "All the swans I've ever seen are white" supports (with some degree of uncertainty) "All swans are white"? Do you agree that 'all the pendulum experiments Newton had ever seen" supports (with some degree of uncertainty) "Every reaction has an equal and opposite reaction"'?
Great. But we're talking about using pendulums. What relevance does the nature of facts or concepts have to do with anything?
Newtons third law tells us that a force is not merely something defined as being that which causes a change of velocity. It is a statement of the much more fundamental idea of force as an interaction between bodies. This underpins the logic of the other two laws.
Forces always arrive from interactions. That is what Newton's third law tells us.
We have never observed an acceleration that is not the result of an interaction.
We have never observed an acceleration without a corresponding equal and opposite force acting on an interacting body.
If we did observe an isolated body undergoing an acceleration Newtons laws would have been violated in a fundamental way.
We never have and inductively we conclude that we never will.
If you were to start with people who were unfamiliar with Newton's science, and ask them about Newton's first law, they would tell you that it is obviously wrong. Nothing is more obvious than that moving things just slow down all by themselves.
So in depth evidence based investgation sometimes refutes common sense notions of how the world works.
Newton's science, partly built on Galileo's science, involved a profound change in how we conceptualize the world. We had to start calling things forces (things such as friction), which were previously seen as just the ordinary slowing down of moving things.
Yes we had to start thinking about the world and how it works in a way that transends everyday common sense notions.
If gravity suddenly stops tomorrow, that just means that predictions for anything beyond that time will be wrong.
ALL of our scientific theories are based on the constancy of natural phenomeon.
When we say we are not saying that this relationship applies only in the past. We are saying that this relationship applies now and into the future.
Hence our ability to scientifically make conclusions about the timing of eclipses and such like. Albeit tentatively.
The possibility does not stop us from making predictions.
It's possible that gravity and friction etc. will suddenly stop behaving as observed so far. But all scientific theories are based on the inductive conclusion that they won't. The fact that we can and do apply our theories to future events is based on the inductive conclusion natural phenomen will continue to behave as observed thus far.
You have invented a form of "science" that cannot say anything about any future event because any conclusion based on natural phenomenon behaving in the future as they have been observed to behave thus far is a "guess" or an "opinion" by the terms of your silly silly argument.
As I see it, evolution is a pragmatic system - selecting what works. If you are going to insist that pragmatism involves making inductive predictions about the future, then you are implicitly saying that evolution depends on inductive predictions of the future. That agrees with the ID position that it would require intelligence so is not explained by natural selection.
It should be noted hat the key difference between genuinely scientific theories and pseudoscience like ID is the ability to successfully predict new observations.
Any fool can construct an internally consistent explanatory theory (look at omphalism) but one that predicts and discovers can be considered genuinely scientific. Ask any IDist what they have doscovered recently if you doubt that.
So don't you dare dismiss the relevance of prediction and then accuse others of advocting creationist style reasoning.
The reason for your intense confusion is that you are looking at this in simplistic black and white terms of science either being wholly inductive or not inductive at all.
Science is neither wholly inductive nor wholly deductive. It comprises elements of both. An inductively reasoned speculative conclusion is drawn. The logical consequences of this are deductively derived (i.e. predictions are made). These predictions are tested. The original inductively derived conclusion is thus either verified or refuted as being in accordance with reality. There is more to it than that regarding tentativity and such matters.
But you hopefully get the gist.
Actually, they are not. It gets hotter in the summer and colder in the winter, which is not constant.
I just had a cup of coffee. It quenches my thirst and I like the taste. I guess it will still quench my thirst in a weeks time, and I will still like the taste. But why should that guess matter to whether I drink coffee now?
We use scientific laws because they work and are useful. Why should we worry about whether they will still be useful next week, when we are concerned with using them today?
It seems to me that you are projecting an implicit prediction, but most people are not actually making that prediction.
Where have I said that we cannot say anything about future events. I only say that we cannot know. That doesn't prevent us from following what seem to be best practices.
You are changing the subject.
The point is that evolution is a natural pragmatic system that works quite well without making predictions of the future. And sure, occasionally there are failures, as when a species goes extinct. But, for the most part, it works.
If tomorrow turns out to be hotter than today, I'll skip wearing a sweater. If tomorrow turns out to be colder, I put on a coat. We take the world as it comes, and adapt to change. Sure, some things seem to change very little, so we don't have to do much about those. But it doesn't require that we are making predictions about everything.
If you think I am suggesting that science need only produce internally consistent theories, then you are hopelessly confused.
So...the unchanging pattern of summers being hotter and winters being colder (in the northern hemisphere) has no constancy?
Varying between hot and cold is still change, and change is not the same as constant.
If you had been paying attention to the news on global warming, you would know that even the pattern is changing.
Please answer the question I asked and not the question you made up in your head.
The question asks whether a variable has some constancy.
The question does not make a lot of sense.
Here on planet Earth, summers tend to be warmer than winters, very consistently. The actual day to day and year to year high and lows may vary, as expected, but still, baring a major volcanic eruption or two, summers are warmer than winters.
Here on planet Earth, summers tend to be warmer than winters, very consistently.
Sure. But what does "consistently" actually mean.
If I tell somebody that the weather is consistent, they will probably agree with me.
If I point out that some summers are a lot hotter than other summers, and that the weather is inconsistent, they will probably agree with me.
Used that way, I suggest that "consistent" is just a "feel good" word that doesn't mean much.
If our world were totally homogeneous, then that would make it perfectly consistent (in some sense). From a mathematical point of view, it would be highly patterned. It would look the same no matter how you looked at it. That's a case of multiple symmetries, about as ideally patterned as you can get.
In such a world our science would be useless. It would be as if we were in a very dense fog all the time, unable to see anything. What we need is contrast. Science needs a lumpy world, not a highly patterned world.
Climate scientists routinely assume that physical laws will continue to operate in the future as they have in the past to predict what climate trends are likely to be in the future.
I'd also like to restate my question from an earlier post: What do you make of the fact the Newton said he used inductive reasoning to develop his laws of motion?
If you have to bet the rent money, that's the way to bet.
If you are betting that physical laws are going to change radically, ask for huge odds.


Alicia and I took a weeklong trip to Houston where we had a lot of fun. You can read more about it in the news section. We also got to see our friends Shaun and Kelli Stracener, however we didn't get any pictures with them. It was a great week, and it was very nice to get away for a while. We were gone July 5th through July 11th.
This was the place that we stayed for the week that we were down here. It was really nice and cozy. It was great to get away for a while.
You can see our dining room and our kitchen in this shot. I am really skinny to not get in the shot.
Here is Alicia with the gift I got her for her birthday, she couldn't wait to open it up, so she got to open it early.
For her birthday, I got Alicia some new scriptures. She was excited that they had her married name on them.
Here is a picture of Alicia in the green of Texas. We found this nice bench on a walk through the town.
Alicia is so dreamy in a swimsuit that the camera just took the picture this way, i didn't have to do anything to have it come out this way.
Here we are in our last day at the Condo. We were really relaxed on our last day in Houston.
Alicia continutes to look more pregnant, but still not showing that she is almost seven months along.
Here is the space ship that we took to travel around for the day.
Here is where it all took place, from landing on the moon to hooking up with the MIR Space Station.
Here we are at the place where those words were first heard.
This grove of trees are dedicated to those astronauts who lost their lives in the Challenger and Columbia spacecrafts.
Here I am in the space shuttle holding on as we encounter some space debris.
Alicia trying to hold it as she realizes what she has to do to go in a space shuttle.
Alicia are making our final approach after a week long trek.
Alica is about to be attacked by a giant spider.
Here is the front of the showroom for Denton Jones Inc.
Here is a picture of Alicia in the showroom.
Here we all are in the showroom in Houston with Denton.


text:
A Plaza Full of Mirrors
The latest addition to the Improbability Vault is "A Plaza Full of Mirrors" which reviews four different men some have prior claimed were present in Dealey Plaza during the Kennedy assassination. Each claimed person is subjected to verifiable evidence placing them elsewhere.
Enduring rumors blame various famous and infamous people for the assassination of President John F. Kennedy. Prior claims based often upon speculative photographic interpretation attempts to place many alleged conspirators in Dealey Plaza. Facts, possibilities, and myths cast varied reflections within the Kennedy case, but ultimately some choose to see what they desire. Thus was born the opportunity for people to claim they and others were at the crime scene. If they are present, to some it means they "must" be responsible. In following decades four men are repeatedly accused of being present in Dealey Plaza, they are Joseph Milteer, Frank Sturgis, E. Howard Hunt, and George H. W. Bush. Some have attempted to insert these men into the Kennedy assassination but were they present in Dallas?
It is highly improbable any conspirator or related well-known figure would be at the crime scene, the entire goal of a clandestine operation is to avoid identification and capture, and those who plan it do not often attend the criminal act. They likely have reasonable alibis in distant locations to avoid the slightest inference of complicity. A conspirator attending would subject themselves to public and press photography, film recording, and legal statements. It would be complex enough for a small group to undertake a plot without having to worry about conspirators being possibly apprehended at the scene. Some demand certain men were present and responsible;
the first of them is Joseph Milteer.
An Unidentified SPectator (AT LEFT) some Call Milteer is NOted To be heavily Balding in 1963, MiLteer (AT RIGHT) in a Photograph Years After the Assassination has a Full Head of Hair. This is Just One of the Multiple Differences Between the Men
The House Select Committee on Assassinations (HSCA) offers, "Joseph Milteer was a "militant right-wing organizer who is alleged to have been a possible coconspirator in the assassination" by members of the public based upon his "strong resemblance" to a motorcade witness. Milteer did have a conversation less than two weeks prior to the assassination with a police informant discussing the possible assassination of President Kennedy. Yet no direct evidence supports that Milteer had definite foreknowledge, his commentary is never specific to the exact events in Dallas. Those seeking to place him in Dallas refer to unproven photographic claims to cement their ideas. This is despite no verified witnesses placing Milteer at the scene in their statements.
House Select Committee on Assassinations COmparison and Analysis of the Unknown Spectator and Milteer
The HSCA conducted a photographic study using photographs of Milteer from before and following November 22, 1963. After extensive comparison and analysis of the photography by consultants, they concluded the spectator called Milteer by some was in fact another unidentified person. Officials did note some similarities "...age and general facial configuration...eyeglasses similar in general style to those favored by Milteer." However, investigators additionally state this spectator "...does not resemble Milteer in upper lip thickness; he is also partially bald, whereas Milteer apparently had a full head of hair in the photograph that was taken several years after the assassination. Most significantly, Milteer's reported stature of 64 inches places him about 6 inches under the spectator's estimated stature...the motorcade spectator could not have been Joseph Milteer."i Yet not just photographic evidence corroborates Joseph Milteer's presence feasibly elsewhere during the time in question.
The Federal Bureau of Investigation had informants who observed Joseph Milteer during November of 1963 who reported these sightings to the Bureau. Milteer traveled extensively to cities across the southern United States. Officials learn Milteer is in Florida a day after President Kennedy's assassination, the distance between his location in Jacksonville and Dallas is roughly 1,000 miles. This requires nearly fifteen hours of driving without stops for sleep and food based on current highway travel. Consider Milteer would have to stop for gas multiple times on less developed roadways and likely required food and sleep. We should also include the time used for the purpose of his journeys, meetings to recruit, fund raise, and discuss issues with militants who shared his fringe ideas. All these things add time and diminish his chances of being in Dallas a day prior.
Milteer was reportedly in route to Columbia, South Carolina from Jacksonville, Florida during the evening of November 23, 1963. It takes roughly just under four hours to reach South Carolina from Jacksonville, an informant observes Milteer at the Jacksonville train station.ii Milteer spent the night reaching a hotel room after travelling from Florida to South Carolina. This roughly four-hour trip is less than one-third the distance from Dallas to Jacksonville. Simply adding the usual eight hours most people sleep to the fifteen hours of time for travel, a few hours of meetings per city, a couple meals per day, and Milteer is likely out of time if he was racing from Dallas.iii
Most evidence supports he did not have the necessary time to make a trip from Dallas, photographic analysis supports Milteer was not the spectator claimed by some in Dealey Plaza, and yet further evidence supports he actually was at home. Three official documents confirm after investigation by agents of the FBI's Atlanta Field Office that Milteer was at home in Quitman, Georgia the day President Kennedy was shot.iv v vi Despite his private claims of knowledge and influencing the assassination, when he offers public blame for the assassination in his notes to fellow militants he claimed that Zionists were responsible. Milteer was a fringe right-wing proponent who supported violent extremist groups and did speak of the President's death preceding the event. Yet based on repeated evidence he was not in Dealey Plaza.
The DEALEY PLAZA TrAMP Some CLaimed was Sturgis (AT Left) and Frank Sturgis (AT Right) Differing Facial Traits, Hairline, and Physique support they are two different Men
Another often-repeated story alleges former member of the US and Cuban military, actor, and soldier of fortune Frank Sturgis was present in Dealey Plaza. Some have claimed Frank Sturgis was one of the Three Tramps photographed in Dealey Plaza.vii He additionally would publicly fund raise for his anti-Communist groups and faced multiple arrests for attempts to violate United States neutrality laws, aviation rules, and smuggling. It is improbable conspirators would select a publicly known figure of failed operations for an important clandestine mission.
HSCA COmparison and analysis of TRAMP A and Sturgis
The Federal Bureau of Investigation undertook a photographic comparison of Frank Sturgis during the President's (Rockefeller) Commission, Sturgis himself provided materials and testimony pursue this end. Officials collected dozens of different black and white photographs of varying quality that presented a range of different photographs for study. Based on the facial indices, hairline differences, and varying physique Sturgis was ruled not one of the Tramps, yet officials could not fully rule him out just based on metric traits alone.viii
According to further legal evidence, Frank Sturgis and multiple witnesses affirmed he was at home in Miami on November 22, 1963.ix x Consider when Sturgis was testifying before officials and subject to legal prosecution for perjury, he repeatedly affirms being in Miami. During subsequent interviews, he makes allusions of being in Dallas while not subject to any legal consequences. Similar to Lee Harvey Oswald, without sufficient evidence, Sturgis too enjoys the presumption of legal innocence despite his later unproven assertions and those seeking to implicate him. Without tangible proof to support the Dealey Plaza claim, there is no reason to regard it.
Sturgis additionally states in the press he is not one of the men arrested and photographed in Dallas that day. He additionally offered to undertake a polygraph on public assertions of him being involved in the assassination of President Kennedy.xi Sturgis did attempt to benefit using CIA funded programs to aid his often-illegal financial schemes. The facts indicate his activities included selling arms shipments, supporting anti-Castro militants, using fraudulent pretenses to collect sponsorship funds, Cuban air raids, and military vehicle sales. It remains unlikely intelligent conspirators would directly utilize such a well-known and outspoken figure with an extensive recent arrest record to undertake a covert plot. No substantial evidence suggests Sturgis was directly part of the Kennedy Assassination.
THE DEALEY PLAZA TRAMP SOME CLAIMED was Hunt (At LEft) E. Howard Hunt (At Right) Investigators Determined They were Two Different MEn BASED ON AGE AND FACIAL TRAITS
Central Intelligence Agency Officer E. Howard Hunt is the third person some assert was present in Dealey Plaza and part of the Kennedy assassination. Public speculation is the basis for these assertions similar to prior ones.xii Agency Inspector General S.D. Breckenridge contacted the Office of Security and reviewed related files on Hunt. He then contacted the Agency Office of Finance asking for Hunt's information on the four-week pay period ending on November 23. Officials note he took only 11 hours of sick leave over multiple days during the time and no annual leave.
Further review of Hunt's travel records state he undertook no travel during November 1963.xiii Accordingly, Hunt did not have the time, nor do formerly classified documents support he was in Dallas November 1963. The Agency did not announce the results publicly; they were content in the fact that sufficient verifiable evidence did not support the
public claims about Hunt.
HSCA COMPARISON and Analysis of Tramp C and E. Howard Hunt.
Official photographic assessment panels also dismissed Hunt as a potential tramp after extended review of comparison photographs. Hunt further made statements in the press denying any connection to allegations linking him to the Kennedy assassination. He also filed suit against publications claiming he was part of a plot.xiv During a court battle with Mark Lane, Hunt would not honestly state where he was publicly on November 22, 1963. Some presumed this meant he was in Dallas, this assumption has proven incorrect.
Hunt's possible reasoning for being dishonest was the potential legal consequences of his covert activities. No substantial evidence links him to official assassination plots and verifiable evidence supports he was in Washington DC. Based on House Select Committee files Hunt was in Washington DC with CIA Deputy Director for Plans Richard Helms, CIA Inspector General Lyman Kirkpatrick, and Cuban exile Enrique Harry Ruiz-Williams on November 22, 1963.xv No verified evidence places him in Dallas.
Long before Hunt's later claims of involvement in Rolling Stone magazine during a serious illness, he originally stated he was in Washington DC at work and then went shopping with his wife. Hunt claimed to have been on annual leave, yet Agency records disprove this. The later cover story is likely to hide his meeting with high-ranking Agency
leaders plotting illegal activities that day. It may also be possible Hunt did go shopping in
Washington D.C. after his meeting.
Hunt's later claim that he was a "bench warmer" involved in the JFK assassination has no proven basis in evidence. Similar to claims made by Frank Sturgis, the other alleged suspects Hunt named without evidence of complicity are his speculation at best. He then omits these claims in later interviews due to improving health. Hunt was feasibly muddying the already blackened waters of the case for his own purposes.
A SPECTATOR SOME CLAIM IS Bush (AT LEFT) GEorge H.W. Bush (AT RIGHT) Differences Between the two MEn are observable in THE Nose, CHIN, and Hairline
Modern speculative media has attempted to link a final person to Dealey Plaza and the Kennedy assassination. Former President George H.W. Bush according to some has a link to the Kennedy assassination. His later one-year term overseeing the Central Intelligence Agency is suggested as confirmation of these claims, yet that just infers connections to the CIA not the Kennedy assassination. Additionally, unverified photographs have been prior offered as proof that Bush was in Dealey Plaza. However, some evidence contends such claims.
AN EXCERpt From One Reported WITNESS Account
Former Kiwanis Club vice president Aubrey Irby was attending a meeting of businesspersons at the Blackstone Hotel in Tyler, Texas to support George H. W. Bush's political candidacy. Irby states "George had just started to give his speech when Smitty, the head bellhop, tapped me on the shoulder to say that President Kennedy had been shot." A room full of people observed George H.W. Bush in Tyler when the President was shot in Dallas. George Bush then reportedly quickly ended his speech.xvi
Bush subsequently contacts Federal Bureau of Investigation officials, "Houston on November 22, 1963 advised that George H.W. Bush, a reputable businessman, furnished information to the effect that James Parrot has been talking of killing the President when he comes to Houston. A check with Secret Service at Houston, Texas revealed that agency had a report that Parrot stated in 1961 he would kill President Kennedy if he got near him."xvii James Parrot was a fringe Republican operative who despised President Kennedy and denied ever making such a statement. This information ties Bush to the Kennedy assassination, but it is merely the role of an informant providing an unproven lead. Bush also sent a prior letter to author J.E. Hansen describing his location at the time
he learned President Kennedy was shot.xviii
If Bush had any relation to a plot, why would he make any statement to officials to connect him to the plot? Any reasonable person would remain silent, take no memorable action, and thus provide no association with the crime. Additionally, George Bush signed the JFK Records Collection Act into law.xix Bush also was among those within the Central Intelligence Agency who enforced a prior moratorium on the destruction of CIA records with possible relation to the Kennedy assassination.xx It was not in his interest to aid these official inquiries if he was complicit in related illegality.
George Bush did have a long-term association with his adviser and former Agency employee Thomas Devine.xxi He did notify the FBI of a perceived threat to President Kennedy the day of the assassination. It is possible he even served as more than a business contact for the Agency prior to his subsequent role as its Director for a year. Yet none of those facts verifies the claims often made about him regarding the Kennedy assassination. This is not a proven nefarious conspirator and certainly not someone who is verifiably present in Dealey Plaza.
Milteer, Sturgis, Hunt, and Bush, each have reasons for their detractors to despise them. Multiple books and videos assert they were in Dealey Plaza and future examples may unfortunately follow. The facts support that any story that relies on these men being present
in Dealey Plaza is questionable and not based on verified evidence. Some people likely shall further claim one or all of these four men were in Dealey Plaza. However, based upon the verifiable documents, they never were.
i. Report of the House Select Committee on Assassinations, Appendix Volume VI, Comparison of photographs of Joseph Milteer with that of an unidentified Dallas motorcade spectator, pp. 242-248
ii. President's Commission Document 1347, FBI Gemberling Report of 16 Jul 1964, Re: Allegations against persons other than Oswald, November 26, 1963, p. 1
iii. HSCA, Federal Bureau of Investigation Subject Files, C-D, Church Committee, No Title, January 22, 1964, pp. 13-14
iv. Ibid, No Title, February 6, 1967, p. 2
v. FBI, JFK Headquarters File, Section 16, Memo from Belmont to Rosen, November 27, 1963, p. 2
vi. HSCA, Numbered Photographs, United States Secret Service Investigative Report, No Title, November 27, 1963, p. 1
vii. FBI, JFK HQ File, Section 177, Memo from the Director, September 25, 1974
viii. Report of the HSCA, Appendix VI, The Three Tramps, pp. 259-263
ix. Senate Select Committee on CIA Activities, Boxed Files, Deposition of Frank Sturgis, 4 Apr 1975, pp. 11-15
x. Ibid, Oswald in New Orleans, Allegations and Sources Draft I, June 23, 1975, p. 7
xi. Jack Anderson and Les Whitten, (April 12, 1975),Watergate Burglar Quizzed on CIA, The Washington Post, p. E-55
xii. Central Intelligence Agency, Russ Holmes Work File, Transcript of Proceedings, Statement of Victor Marchetti, July 9, 1984, pp. 72-77 xiii. HSCA, Segregated Central Intelligence Agency file, Jaffe's Inquiry into Hunt and Sturgis Whereabouts, Box 43, September 20, 1974
xiv. Federal Bureau of Investigation, JFK Headquarters File, Section 177, Teletype from Baltimore to the Director, pp. 1-3
xv. Senate Select Comm., Boxed Files, Oswald in New Orleans, Allegations and Sources Draft I, June 23, 1975, p. 7
xvi. Kitty Kelley, "The Family: The Real Story of the Bush Dynasty, Anchor Books, May 2005, p. 212
xvii. President's Commission Document 301, FBI Gemberling Report of 18 December 1963 re: Assassination of President Kennedy, Re: James Milton Parrot, November 22, 1963
xviii. Jodie Elliott Hansen, November 22, 1963: Ordinary and Extraordinary People Recall Their Reactions When They Heard The News, Thomas Dunn Books, rear cover
xix. Background on the Collection, The John F. Kennedy Assassination Records Collection Act, The United States National Archives, archives.gov
xx. CIA, Miscellaneous Series, Breckenridge Files (HSCA), Status of Senate Select Committee on Intelligence Document Control Review, March 15, 1977
xxi. CIA, Deputy Director for Plans file, Memorandum: Messrs. George Bush and Thomas J. Devine, Subject: Thomas J. Devine, January 30, 1968
Hunt -N- Sturgis
Research by: C.A.A. Savastano
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EZGO Parts, Club Car Parts, Yamaha Golf Cart Parts > Club Car Golf Cart - Parts > Brakes Cables, Shoes, Drum & Other Parts-Club Car > Brake Linkage Clevis Pin Club Car Models DS and Precedent 1981 & UP.
Club Car Golf Cart Brake Clevis Pin Fits Models DS and Precedent 1981 & UP. Measures 5/16" x 1".


Donald Glover created and stars in FX's comedy "Atlanta."
Donald Glover's fans have waited about three years between the news that he was creating a new comedy, "Atlanta," and its much anticipated premiere Tuesday at 10 p.m. on FX.
While it has felt like a long time to wait for a show, the former "Community" star and rapper feels it was just the right amount of time. But the public's knowledge of things that used to be more hidden has increased with the internet.
"First of all, when I was a kid, you wouldn't hear, 'Adam Sandler has a TV show coming.' That wasn't news," Glover told Business Insider during the recent Television Critics Association press tour. "There was no internet really for that. You'd just start seeing the ads a month or two before, and you're like, 'Oh, I guess I'm going to check this out.' So I think it's like people are like, 'What took so long?' I had to make the show. That was it ... It's just that people know more about the process, so they're like, 'It's been three years.' And I'm like, 'That's how long it takes to get a show done usually.'"
Glover acknowledges that the show hasn't been his sole focus since leaving the NBC comedy "Community" in 2013. Under his rap pseudonym, Childish Gambino, he released a couple of albums and toured. He has also appeared in some big projects such as the films "The Martian" and "Magic Mike XXL."
"I guess it could have happened faster, maybe if I wasn't doing so many other things," Glover said. "But in general, I think it's just like people are more aware of the process, so it's different. But also, I like taking my time. I don't like to be rushed at all."
From left, Donald Glover, Brian Tyree Henry, and Keith Stanfield on FX's "Atlanta."
For viewers and TV critics, the time Glover spent making "Atlanta" will be well worth it. The offbeat comedy follows Earn (Glover) and his rapper cousin Alfred (Brian Tyree Henry), who are trying to get the music business to pay attention to them.
Their efforts lead to plenty of discouraging and hilarious roadblocks while they knock around the tough streets and colorful characters of black Atlanta.
Glover knows the show's portrayal of Georgia's capital will be unique for TV.
"It's the part of Atlanta I don't think that people show because it's not 'interesting' to people," Glover said, making air quotes. "I put that in quotes because it's not the part of Atlanta that's been proven to sell. And I felt like I just knew there was a hunger for Atlanta s--- like that, but nobody had really figured out how to sell it. And I was like, I know how to sell it. It needs to just feel like you're black, you know?"
And as for the Atlanta music scene, viewers will get some of that, but Glover isn't promising an education in the music industry on the show. That wouldn't be authentic for him.
"Most people who live in everyday lives — white people, Chinese people, Mexican people — they're living their lives, and they're trying to eat. I was more interested in that."


A brilliant result for those accessing this page yesterday but, for sure, by not having subscribed membership, you missed some excellent stuff elsewhere.
Today looks tricky as we have rain falling, which will scupper my numbers if we have a going change of any significance.
I'm offering you what looks a very trappy one and if we get something here then for sure, this season is going to be spellbinding!
NOTES: As tight as a mallards backside. Fahey again has one securing all three icons and I expect EASTERN IMPACT to run a solid race. His personal stats give cause for optimism….1-2 at Newmarket, 2-6 at the trip and a 40% strike rate on the current going. He is in my dutch and will be my 20/80.
My top rated has previous winning form here (1-3), has a 20% strike rate at the trip but that one win from six attempts at winning on this ground concerns me.
HUNTSMANS CLOSE will be the third in my dutch. Again the ground is an issue, just one win from eight on it and he;s never visited Newmarket before. Three wins from 16 shots at the trip also has me rubbing my chin but, he's got a decent draw to work with if I have read this correctly.
Of the rest ZANETTO is very much worth keeping an eye on. Track, trip and ground are right and he'll be a player if fit enough on debut. In his 3-y-o season his early form was very good and it will be down to Balding'S skill at getting one handicapped again with this one.


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text:
Q: PHP extension that uses memcached I am thinking of writing a PHP extension library that will use the memcached library. It is trivial to simply link my library to the memcache shlib.
However, I am not sure what will happen if my (extension library) user already uses memcache on his/her website. My questions are:

*

*Is it possible to have (possibly different versions) of memcache on the machine?

*Is it best to statically link or dynamically link to memcache when building the extension library? (to cater for version incompatibilities - assuming memcache is backward compatible, otherwise all bets are off)

The questions basically degenerate to how may one safeguard an extension library they have written if it has a dependence on a third party file which may already be being used on the website that the extension library is going to be used on?
The question may probably be slightly ill-posed, but I hope you understand the gist of what I am asking.

A: Mind that there are two memcache extensions for PHP, one is called  memcache, the other memcached, the first uses it's own implementation of the memcache protocol, the later uses the library.
If you're using the first you shouldn't have a conflcit but have to take care of memcache on your own. I'd suggest building an extension which depends on the memcached and re uses the library it found.


Selling my 6120-DSW in all it's Western glory! Great guitar in excellent condition, but I need to thin the herd and already have my Dynasonic and hollowbody needs covered.
Playability is fantastic, and the tone is incredible. Dynas in a hollow body are such an excellent combo - Sure, it'll do the Duane Eddy twang thang all day, but I also love the neck and middle positions, which provide a nice, round assortment of sounds with excellent note separation.
JT03 serial, which means it's a 2003 from the Terada factory. Fit and finish are excellent. Full western appointments, with the G-brand and fretboard inlays to the gold hardware. From a cosmetic perspective, it has very little playwear, just some light swirling in the finish and, given it's age, a bit of tarnish on the gold hardware. No smoke odors, dents or buckle rash whatsoever, just a beautiful guitar overall.
Sure is beautiful and a good price. Hope it sells for you.
Is that one of the early ones with an ebony fretboard?
This has a Rosewood fretboard, which looks great against the inlays.
Shame. Rosewood imports are a problem now. Wrong side of the pond.
Also... PRICE DROP! Asking $1675 w/ shipping to the lower 48!