# What is flat $\Lambda$CDM, and may we choose it?

Stefano Anselmi 1,2 Matthew F. Carney3 John T. Giblin, Jr4,5 Saurabh Kumar4
James B. Mertens3 Marcio O’Dwyer4 Glenn D. Starkman4,6<EMAIL_ADDRESS>Chi Tian3,7 1INFN-Padova, Via Marzolo 8, I-35131 Padova – Italy  2 LUTH, UMR
8102 CNRS, Observatoire de Paris, PSL Research University, Université Paris
Diderot, 92190 Meudon – France  3Department of Physics and McDonnell Center
for the Space Sciences, Washington University, St. Louis, MO 63130, USA
4Department of Physics/CERCA/Institute for the Science of Origins, Case
Western Reserve University, Cleveland, OH 44106-7079 – USA  5Department of
Physics, Kenyon College, 201 N College Rd, Gambier, OH 43022, USA  6Department
of Physics, Imperial College, London, UK  7School of Physics and
Optoelectronics Engineering, Anhui University, Hefei,China

###### Abstract

The Universe is neither homogeneous nor isotropic, but it is close enough that
we can reasonably approximate it as such on suitably large scales. The
inflationary-$\Lambda$-Cold Dark Matter ($\Lambda$CDM) concordance cosmology
builds on these assumptions to describe the origin and evolution of
fluctuations. With standard assumptions about stress-energy sources, this
system is specified by just seven phenomenological parameters, whose precise
relations to underlying fundamental theories are complicated and may depend on
details of those fields. Nevertheless, it is common practice to set the
parameter that characterizes the spatial curvature, $\Omega_{K}$, exactly to
zero. This parameter-fixed $\Lambda$CDM is awarded distinguished status as
separate model, “flat $\Lambda$CDM.” Ipso facto this places the onus on
proponents of “curved $\Lambda$CDM” to present sufficient evidence that
$\Omega_{K}\neq 0$, and is needed as a parameter. While certain inflationary
model Lagrangians, with certain values of their parameters, and certain
initial conditions, will lead to a present-day universe well-described as
containing zero curvature, this does not justify distinguishing that subset of
Lagrangians, parameters and initial conditions into a separate model. Absent
any theoretical arguments, we cannot use observations that suggest small
$\Omega_{K}$ to enforce $\Omega_{K}=0$. Our track record in picking
inflationary models and their parameters a priori makes such a choice dubious,
and concerns about tensions in cosmological parameters and large-angle cosmic-
microwave-background anomalies strengthens arguments against this choice. We
argue that $\Omega_{K}$ must not be set to zero, and that $\Lambda$CDM remains
a phenomenological model with at least 7 parameters.

Concordance cosmology, a Universe that is dominated by dark energy of fixed
energy density and cold dark matter ($\Lambda$CDM) and that began, long ago,
with a nearly scale-free spectrum of adiabatic density fluctuations, is a
surprisingly simple description of a remarkably complex universe. In this
setup, just about everything we see on cosmological scales can be described in
terms of a nearly homogeneous and isotropic space-time containing homogeneous
and isotropic vacuum energy, and filled with ordinary baryonic and dark
matter, photons, and neutrinos that are nearly homogeneous on cosmological
scales, but have sizable fluctuations on astrophysical scales. The dynamics of
this energy density is governed by general relativity (GR) and the Standard
Model of particle physics (and possibly some other interactions if the dark
matter is not of the Standard Model), though the evolution of cosmological
perturbations can be tracked almost entirely with linearized GR. In addition,
one commonly postulates an early-universe “inflationary” origin for the
fluctuations, which explains why they are primordially adiabatic, with a
nearly scale-invariant power spectrum over some wide range of scales.

This cosmology can be parametrized in terms of just 7 real parameters:
$\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, $H_{0}$, $\tau$, $n_{s}$, $A_{s}$, and
$\Omega_{K}$.111 One might add $T_{\rm CMB}$, the current temperature of the
cosmic microwave background radiation (CMB), which fixes the cosmological
moment at which we are describing the universe. $T_{\rm CMB}$ also fixes the
current energy density in radiation $\Omega_{r}$ once one has specified
$N_{\rm eff}$, the effective number of relativistic neutrino species at the
time of big bang nucleosynthesis. $N_{\rm eff}$ is calculable in the Standard
Model. These parameters are those of what is often called “vanilla”
$\Lambda$CDM. There are also a number of additional parameters that
characterize detailed features of the Universe or departures from the simplest
model. These include the neutrino masses and mixings, and the amplitude and
spectral parameters of any tensor modes. There are many other such potential
parameters. In more detail, these are222Throughout, we set $c=1$.: the ratio
$\Omega_{b}$ of the energy density in baryons $\rho_{b}$ to the critical
energy density, $\rho_{crit}=3H_{0}^{2}/8\pi G_{N}$, for a homogeneous flat
Friedmann-Lemaître-Robertson-Walker (FLRW) universe, times $h^{2}$, the square
of the current Hubble parameter $H_{0}$ in units of $100$ km/s/Mpc; $h^{2}$
times $\Omega_{c}$, the ratio of the energy density in cold dark matter
$\rho_{c}$ to $\rho_{crit}$; the current Hubble parameter $H_{0}$; the
reionization optical depth $\tau$ (which should in principal be a calculable
function of the other 6, but in practice depends on complicated non-linear
astrophysics); the index $n_{s}$ of the power spectrum of scalar fluctuations,
and the amplitude $A_{s}$ of that power spectrum. Finally, there is
$\Omega_{K}$, a measure of spatial curvature, written to appear like the ratio
of the energy density in spatial curvature to the $\rho_{crit}$. Actually
there is no energy density associated with spatial curvature, which is part of
the geometric (Einstein tensor $G_{\mu\nu}$) term of the Einstein field
equations $G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi G_{N}T_{\mu\nu}$, not the source
(stress-energy tensor $T_{\mu\nu}$) term. However, in an FLRW universe,
$\Omega_{K}$ is $1$ minus the sum of the $\Omega_{i}$ associated with each of
the actual sources of energy density – once the cosmological constant
$\Lambda$ has been rewritten as a vacuum-energy-density contribution to
$T_{\mu\nu}$.

It is noteworthy that all these parameters are phenomenological, i.e. not one
can be readily understood as a parameter of the action of the fundamental
theory underlying cosmology. They are more closely analogous to the masses,
orbital parameters and similar properties of the Sun and planets in a “theory”
of the solar system, than to General Relativity’s only parameters, $G_{N}$ and
the value of the cosmological constant $\Lambda$, or to the 19-20 parameters
of the Standard Model of particle physics (plus a few more if one includes
neutrino masses and mixings). The $\Lambda$CDM parameters depend in
complicated, and in some cases not-yet-understood, ways not just on the
parameters of the fundamental theory–which is not yet fully known–but also, at
least potentially, on boundary conditions. A rough analogy is to the dynamics
of a plasma inside a cavity, which depend not just on the micro-physics of the
plasma constituents, but also on the shape of the cavity, the conductivity of
its surface, and even the roughness of that surface. It should therefore
perhaps be surprising that just 7 parameters seem to suffice to approximately
describe the Universe.

It has nevertheless become the widespread practice among cosmological
phenomenologists and observers to regard the last of these, $\Omega_{K}$, as
optional; to set $\Omega_{K}\equiv 0$ and analyze the “6-parameter
flat-$\Lambda$CDM” concordance cosmology. This argument rests on three legs:
First, the are the historical low-precision measurements of $\Omega_{K}$, from
which there was no evidence that $\Omega_{K}\neq 0$. Second, there are long-
standing arguments that $\Omega_{K}=0$ is a generic prediction of inflation,
though, during a period in the 1990s when $\Omega_{m}\simeq 0.3$ was the
preferred observational value, and it was believed that
$\Omega_{\Lambda}\equiv 0$, so $1-\Omega_{m}-\Omega_{\Lambda}=0.7=\Omega_{K}$,
some argued that $\Omega_{K}\neq 0$ was generic, or at least Linde (1995) that
there was no good reason to prefer any particular value of $\Omega_{K}$. These
arguments have largely not been revisited since the emergence of tensions in
the fits of $\Lambda$CDM parameters to cosmological data, and of statistical
anomalies in that data. Finally, third, there is a sense among cosmologists
that there are three choices – $k=0$, $k=1$, and $k=-1$ – and that these are
separate models that could be treated separately, even though this is a choice
not between distinct fundamental Lagrangians but between boundary conditions
(i.e. between three classes of possible manifolds, that don’t even exhaust the
full set of possible manifolds).

We set out here to challenge this practice of regarding
$(\Omega_{K}=0)$-$\Lambda$CDM as a distinct model with one fewer parameter
than $\Lambda$CDM, which can be evaluated separately. It has been suggested
that it is “premature” Leonard et al. (2016) to set $\Omega_{K}=0$ in
$\Lambda$CDM, and argued that this is a “very restrictive assumption that
needs to be tested empirically” Ade et al. (2016a), leading many others to
explore $\Lambda$CDM with $\Omega_{K}\neq 0$ (e.g., Aghanim et al. (2020); Di
Valentino et al. (2019); Handley (2021); Efstathiou and Gratton (2020)) with a
range of conclusions. Still others have worried about the possibility of
drawing erroneous conclusions about evolving dark energy Clarkson et al.
(2007); Wright (2006), the nature of inflation Handley (2019), or alternatives
to inflationary $\Lambda$CDM Melia et al. (2018). Here we argue that one
simply is not entitled set $\Omega_{K}=0$ in $\Lambda$CDM. By downgrading an
inescapable phenomenological parameter of the model universe to an optional
parameter, we introduce–through our priors–information that we simply don’t
have a priori. By insisting that the parameter should be restored only if the
need for it is justified by strong evidence, we potentially mislead ourselves
about tensions among data sets, how they might be solved, our confidence in
our concordance model, the utility of future and planned experiments, and
other key questions in the era of precision cosmology. The point of this paper
is not to take sides for or against particular tensions, but to argue that the
correct stage on which they should be examined cannot be assumed to be flat –
Without a theory of initial fluctuations, such as inflation, indentifying
$\Omega_{K}$ indistinguishable from $0$ as a “model” relies completely on
arbitrary assumptions about the properties of cosmic fluctuations to even make
sense of a background cosmology, much less select the specific Euclidean
background. Within the inflationary $\Lambda$CDM paradigm, it is a prediction
of certain, but not all, inflationary Lagrangians, with certain, but
definitely not all, values of those Lagrangian’s parameters, and certain, but
again not all, initial conditions – all in a context where there is no well-
understood measure on the models/parameter/initial-condition “space” from
which one is choosing.

Properly including $\Omega_{K}$ as a free parameter has practical
consequences: changing the best-fit values of other parameters and their
uncertainties, and exacerbating or moderating the strengths of parameter
tensions. In principle, one may also wish to replace $\Omega_{K}$ with one or
more underlying model parameters that can be more closely connected to
calculable predictions for observables.

##  I What does it mean to be flat?

The subtext of this question is that FLRW cosmology equates/conflates at least
three distinct meanings of the statement “the Universe is flat”:

  1. 1.

The Universe is a Euclidean three-manifold: All of spacetime may be foliated
by spacelike hypersurfaces each of which is a Riemannian three-manifold (an
appropriately smooth space) with the appropriate topology to “admit” a single
homogeneous metric, and that is the Euclidean metric. This property might
extend back to just after a spacelike “Big Bang” singularity, or to
$t=-\infty$ in a cyclic or bouncing cosmology; but most importantly, it
includes the time slice we inhabit.

  2. 2.

The observable Universe has zero average curvature: the “three-geometry” of
the constant-time spacelike hypersurface through us is characterized by a
3-Riemann-curvature tensor ${}^{3}R_{ijkl}$. The spatial average of this
3-geometric quantity (or the 3-Ricci tensor, or the 3-Ricci scalar) is $0$
over the region of space that we observe.

  3. 3.

The (first) Friedmann equation requires no term that scales like curvature:
cosmological observations are analyzed in the context of a phenomenological
model—background FLRW cosmology plus perturbations. The background has only
one dynamical degree-of-freedom, the scale factor $a(t)$. The Friedmann
equation, supplemented with an equation-of-state relationship between pressure
and energy density, relates the rate of change of $a(t)$, to the energy
density $\rho$. In “vanilla” $\Lambda$CDM, four contributions to $\rho$ are
included, which scale as333 When placed in the context of the Standard Model
of particle physics, or various extensions thereto, specific particles
contribute to either the matter or radiation component at different epochs. As
the Universe expands, particles with mass transition from radiation-like to
matter-like. This slightly complicates the simple four-component description,
and is of particular current relevance for neutrinos. $a^{-4}$ (“radiation”),
$a^{-3}$ (“matter”), $a^{-2}$ (“curvature”444 In General Relativity, the
presumed underlying theory of FLRW, an $a^{-2}$ term can have a geometric
origin, arising from an isotropic homogeneous curvature of space. It thus
enters the Einstein equations through the Einstein tensor, not the stress-
energy tensor. Certain forms of energy density, such as certain networks of
cosmic strings, would also scale as $a^{-2}$, but these are assumed not to be
present in vanilla $\Lambda$CDM. ), and $a^{0}$ (“vacuum energy”). The current
fractional contributions to the total energy density are given by
$\Omega_{i}$, with $i$ indexing these four sources. (Thus
$\sum_{i}\Omega_{i}\equiv 1$.) $\Omega_{K}$ is the symbol for the fractional
coefficient of the $a^{-2}$ contribution to the Friedmann equation.

The word flat is thus non-specific. Our not-precisely homogeneous or isotropic
universe may or may not be flat in each of the above three senses.

We examine each of these meanings of flatness below, the goal being to ask if,
and to what extent, we are justified in assuming that any or all of the above
flatness conditions are satisfied a priori, i.e. what are the implications of
placing a strict prior that perfectly enforces any or all of these conditions
on the cosmology, and which then requires strong evidence to overturn.

The question is not, for example, how good or bad an approximation it is to
expand around a homogeneous, isotropic, zero-spatial-curvature background, but
rather whether one can justify regarding perfectly flat FLRW as a separate
model, in which, in particular, $\Omega_{K}$ is simply not a parameter that
one must measure.

###  I.1 Topological flatness

An underlying assumption of FLRW is that $3+1$-dimensional classical (i.e.
ignoring any underlying quantum nature) spacetime is smooth555 We try to steer
clear of specific mathematical definitions when those are not necessary.—a
semi-Riemannian manifold—and can be described as a time-ordered sequence of
smooth $3$-d spatial slices, at least from soon after some possible initial
singularity to the present and into the foreseeable future. We know that this
is a problematic assumption given the existence of black holes, which are
thought to hide singularities, or quantum-mechanically resolved singularities,
behind horizons. Nevertheless, we have the sense, and assume, that we can
excise these short-distance defects in the geometry.

We also assume for concordance cosmology that this large-scale spatial
3-manifold has no boundary hypersurfaces. This is not an essential feature of
spacetime; in certain models of large extra dimensions in which interest
exploded in the 1990’s Arkani-Hamed et al. (1998); Randall and Sundrum (1999);
Kaloper et al. (2000), we (i.e. all or most Standard Model particles) are
confined to “the brane”–a 3+1-dimensional sub-spacetime of the full “bulk”
spacetime. This brane generically imposes boundary conditions on bulk fields.
Nevertheless, a no-boundary prior is defensible, if not essential.

Having assumed that space on large scales is a three-manifold, we confront
their vast diversity. Spatial three-manifolds can be infinite (non-compact) or
finite (compact). Compact manifolds without boundary are called closed666 This
meaning of a closed manifold should not be confused with the cosmologists’
description of a universe that eventually stops expanding and then collapses
toward a future big crunch as a “closed universe.” $S^{3}$ LFRW universes are
closed in both senses if $\Lambda$ is sufficiently small that curvature causes
the cosmic expansion to halt before $\Lambda$ can cause accelerated expansion.
and have been extensively studied. Many cosmologists are familiar with the
fact that in two dimensions every simply connected Riemann surface admits one
of just three homogeneous geometries – Euclidean ($E^{2}$), spherical
($S^{2}$), or hyperbolic ($H^{2}$). Effectively, it is possible to smooth out
any local ripples in the spacetime and leave behind a homogeneous and
isotropic geometry (i.e. metric).

The situation is much less simple in three dimensions. Thurston, in 1982, put
forward the eponymous Geometrization Conjecture Thurston (1982), since proved
by Perelman Perelman (2002, 2003). It states, approximately, that every closed
3-manifold is the connected sum of irreducible pieces called prime
3-manifolds. This decomposition is unique for orientable manifolds777 We may
or may not inhabit an orientable manifold. The Standard Model is chiral – it
includes different left-handed and right-handed fermionic particles/fields. On
a non-orientable manifold this is not possible. On the other hand, it seems
risky to infer global properties of spacetime from no-go theorems whose subtle
assumptions may prove consequential. . Each prime submanifold admits a
homogeneous geometry–however there are 8 of them, not 3. Non-compact three-
manifolds in general are much less well studied.

FLRW considers only the three homogeneous and isotropic 3-geometries—Euclidean
($E^{3}$), spherical ($S^{3}$), and hyperbolic ($H^{3}$). Because the
dynamical equations of General Relativity depend only on the metric, the
Einstein field equations for the evolution of the homogeneous metric (but not
their boundary conditions) are entirely agnostic about which of the many
manifolds of each type we might inhabit. Cosmologists therefore tend to equate
these geometries with the covering spaces of these geometries – for each
geometry, the manifold that has the full unbroken symmetry group of that
geometry – because the field equations of GR are independent of the topology.
Cosmologists thus refer to all $E^{3}$ manifolds as $k=0$, all $S^{3}$
manifolds as $k=1$, and all $H^{3}$ manifolds as $k=-1$. Two more of the 8
possible homogeneous geometries follow quite naturally from the two-
dimensional case: $S^{2}\times E^{1}$, and $H^{2}\times E^{1}$. They are
anisotropic—flat in one dimension and isotropically curved in the other two.
The final three, $\widetilde{SL}(2,R)$, Nil, and Sol, are less familiar, and
also anisotropic.

It is a challenge to place any measure on this rich space of possibilities.
There are only $10$ topologically distinct closed 3-manifolds with the $E^{3}$
geometry. On the other hand there are a countable infinity of distinct
topologies for manifolds admitting the $S^{3}$ geometry. Meanwhile “there are
enormous numbers of examples of … [$H^{3}$ manifolds], and their
classification is not completely understood.” wik (2022). This suggests that
flat 3-manifolds are a set of measure zero in the space of closed 3-manifolds.
However, we should be cautious to adopt the topologists’ counting for
cosmological purposes. For example, hyperbolic three-manifolds are rigid
Mostow (1968) – the only deformation of the fundamental domain permitted if
one wants to preserve the homogeneity of the metric is an overall scaling
(i.e. a change in the curvature scale). In contrast, the simplest Euclidean
three-torus is usually described as a cube with opposite faces identified;
however, each of the three side lengths can be independently and continuously
varied, as can the three independent angles between the sides. Thus the finite
number of flat three-manifolds each require up to 6 real parameters to specify
them, whereas the countable infinity of hyperbolic three-manifolds require
only one each. On the other hand, it is not clear why we should count the
parameters that specify the homogeneous geometry, and disregard the
overwhelmingly larger set of real parameters required to specify
inhomogeneities – represented, for example, by the amplitudes of eigenmodes of
the appropriate Laplacian.

Returning to and refining our initial assertion – it is a challenge to select
a single compelling measure for cosmology on the space of possible three-
manifolds.888We avoid even contemplating the complications if the topology of
the $3+1$d spacetime precludes a global decomposition into a spatial manifold
cross time.

In summary, the cosmologists’ classification of FLRW models as $k=0,\,\pm 1$
cannot be taken as a rational basis for assigning priors to three-manifolds,
as it is grossly incomplete, and arbitrarily implies something like equal
weight for each of the three special classes of manifolds that it happens to
include, and zero weight for all other three-manifolds, not to mention spaces
that aren’t manifolds, but that would/might be just as viable cosmologically.

It is distinctly possible that the entire question of cosmic topology is
obviated by a prolonged epoch of cosmic inflation. Suppose the pre-
inflationary universe was homogeneous but with a geometry other than $E^{3}$.
Or perhaps was nearly homogeneous, but well described by a background geometry
other than $E^{3}$ plus perturbations. During cosmic inflation, the length
scale(s) associated with that (background) geometry would increase compared
with the Hubble Horizon scale and any other scale determined by microphysics.
Local measurements of geometry–i.e. measurements made over fixed
distances–might be increasingly well-approximated by Euclidean geometry.999
Cosmologists would normally say that local measurements of geometry _would_ be
increasingly well-approximated by Euclidean geometry; however, that statement
hides the important assumption of a short-distance cut-off in the spectrum of
initial inhomogeneities – perhaps the physical Planck scale at the onset of
inflation.  After inflation ended, the physical scale over which such “local”
measurements could be made would begin to increase–this is often described as
scales coming (back) “into the horizon.” If inflation was sufficiently brief,
then we could today, once again, detect the non-Euclidean nature of the
background geometry. If inflation lasted long enough, then even today, the
deviation of the background geometry from Euclidean might be too small to
detect.

Detecting the background curvature and detecting the non-triviality of cosmic
topology are not one and the same thing, though they are certainly connected.
Moderately stringent limits have been placed on observables related to the
topology of the Universe. The shortest closed loops “around the Universe”
passing through us must, at 99.7% CL, be longer than at least 98.5% of the
diameter of the last scattering surface of the cosmic microwave background
(CMB) Cornish et al. (2004); Shapiro Key et al. (2007); Vaudrevange et al.
(2012) – a statement that can be made regardless of the specific manifold in
which we might live, and of the specific background geometry given the
inferred levels of curvature fluctuaions. This is known based on the absence
of “matched circle pairs” Cornish et al. (1998) in Wilkinson Microwave
Anisotropy Probe (WMAP) temperature maps WMA , slightly extended to special
cases using Planck data Ade et al. (2014, 2016b). The Planck team also
searched for evidence of non-trivial topology using a correlation-function
technique, but that search was limited to certain classes of manifolds, to
certain values of the parameters of the applicable fundamental domain, and,
for most topologies, to certain locations within that fundamental domain. The
possibility remains of non-trivial topology on length scales short enough that
it could be detected.

In summary, a spatial slice through us may have the appropriate topology to
allow us to put the homogeneous Euclidean 3-metric on its entirety (once we
somehow excise the black holes and “stitch up” those excisions). This could
reasonably be called a topologically flat space. It is hard (or impossible) to
put a meaningful measure on the space of 3-spaces to evaluate the probability
for this to be the case. We have not measured the topology (and may never do
so), so do not yet have any basis for settling the question observationally.
Neither do we have any basis from cosmic topology for assuming the Universe is
flat.

###  I.2 Zero average spatial curvature

Whether or not the spatial slice through us (or at least a judiciously chosen
slice) is a topologically flat 3-manifold (i.e. admits a homogeneous Euclidean
metric), the actual metric is not homogeneous – everywhere around us is
evidence of non-Euclidean geometry. Perhaps, however, the Universe is flat
because the average spatial curvature over the observable Universe is zero.
There are at least three questions pertaining to this suggestion:

  1. 1.

Which average of which quantity is zero?

  2. 2.

Which theoretical models predict that that average will be precisely zero?

  3. 3.

Is there justification for assigning a prior to one or more of those models
that plucks them out of the vast space of models and distinguishes them among
their alternatives?

####  I.2.1 Which average curvature?

While the geometry of spacetime is described by a metric $g_{\mu\nu}$, the
curvature is characterized by the Riemann tensor
$R_{\alpha\beta\gamma\delta}$, which, written this way with 4 covariant
indices, is a function of the second derivatives of the metric with respect to
the spacetime coordinates. $R_{\alpha\beta\gamma\delta}$ has 20 independent
degrees of freedom. The Einstein field equations, the dynamical equations of
GR, depend only on the Ricci tensor, a trace of the Riemann tensor
$R_{\mu\nu}=g^{\alpha\gamma}R_{\alpha\mu\gamma\nu}$ (and its trace, the Ricci
scalar, $R=g^{\mu\nu}R_{\mu\nu}$), through the Einstein tensor101010 The
remaining degrees of freedom of the Riemann tensor are contained in the Weyl
or conformal tensor.  $G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$.
$G_{\mu\nu}$ has just 10 independent components. The field equations state
that $G_{\mu\nu}$ is proportional to the stress-energy tensor $T_{\mu\nu}$,

$G_{\mu\nu}=8\pi G_{N}T_{\mu\nu}\,.$ (1)

For a universe that is close to homogeneous and isotropic, and which seems to
be reasonably well-modeled by some mixture of perfect fluids (characterized by
their density $\rho$ and isotropic pressure $p$), the diagonal elements of
$T_{\mu}^{\nu}$ in a Cartesian coordinate system – $\rho,p,p,p$ – are most
definitely not zero on average at most cosmological epochs – during the
radiation-dominated and matter dominated periods they are positive definite;
during the current accelerated-expansion epoch (and any past inflationary
epoch), $\rho$ is positive definite while $p$ is negative definite. Thus none
of $G_{\mu\nu}$, $R_{\mu\nu}$, and $R$ average to 0 over a cosmologically
interesting volume, at least for any reasonable choice of average.

This is not surprising to any cosmologist. “Flat” cannot mean that spacetime
has no curvature – it is a claim about space not spacetime, in other words
about the induced metric on spatial slices. We must first perform a “space-
time decomposition” – we must take the $3+1$-dimensional spacetime manifold
and represent it as a sequence of non-intersecting $3$-dimensional spatial
manifolds, a foliation. On any given spatial slice any two points are outside
one anothers’ light-cones. The coordinates within any given slice are spatial
coordinates, the label for the different slices is time. On each such slice,
we can then find111111${}^{3}g$ can be computed directly from $g$ and the
specification of the foliation. the metric ${}^{3}g_{ij}$ describing distances
between pairs of points (with $i,j$ now running over 3 spatial coordinates),
and we can compute the 3-d spatial Riemann tensor, ${}^{3}R_{ijkl}$ from
spatial derivatives of ${}^{3}g$. When cosmologists speak of a flat universe,
they are describing the properties of ${}^{3}R_{ijkl}$. ${}^{3}R_{ijkl}$ has 6
degrees of freedom, but in the Euclidean 3-geometry $E^{3}$,
${}^{3}R_{ijkl}=0$.

We can immedately identify several challenges. First, the separation of
spacetime into space and time is not unique. Even if there is a choice of
foliations that gives ${}^{3}R_{ijkl}=0$, this will not be true on other
foliations. We could adopt a very mathematical approach and merely require
that there exist such a foliation, but how would one verify this
observationally? And what precisely would one mean? After all, the geometry is
very far from Euclidean on small scales. More on that below.

The second and related challenge is that we do not observe spatial slices. The
only point on our spatial slice that we know we can observe is the one we
occupy. Mostly we observe photons (or lately gravitational waves and
neutrinos) that emanate from points on (or incredibly near, in the case of
neutrinos) our past light cone – approximately these are nested spheres of
increasing radius as one moves further into the past. (We do get information
from our past worldline – e.g. geological data; we also get photons that have
scattered in the past, offering in principle an opportunity to fill in the
light cone, but that is far less extensive. Cosmic rays do this to a limited
extent as well, travelling at all speeds $\leq c$.) But this simple
description relies on the Universe being nearly homogeneous. The presence of
mass concentrations, which create gravitational lenses, black holes, et
cetera, complicates this description. On which surface are we meant to average
the curvature and find zero?

Finally, what quantity are we meant to average and how? Is it ${}^{3}R_{ijkl}$
or ${}^{3}R^{i}_{jk}l$; the 3-Ricci tensor ${}^{3}R_{ij}$ or
${}^{3}R^{i}_{j}$; or the 3-Ricci scalar, ${}^{3}R$? Or is it some function of
these? The average of a function is not the function of the average; averaging
doesn’t commute with most mathematical operations. And how do we weight the
average?

And is it zero? One clear challenge in placing a delta-function prior on any
average of the curvature at zero is that no viable theory predicts that the
curvature will be precisely zero. This is absolutely clear on “small scales” –
the existence of structures from Cavendish experiments to superclusters
testifies to the curvature of space. Most of these structures, while non-
linear from the point of view of their stress-energy, source gravitational
fields that can be studied at low order in perturbation theory; however this
is certainly not the case for the mergers of stellar mass black hole binaries,
which we now detect regularly via their gravitational radiation. Thus any
attempt to assert that the curvature is zero must involve a scheme to remove
the contribution of these linear and non-linear deviations from an idealized
flat background. Many authors Zel’dovich (1964); Bertotti (1966); Gunn (1967a,
b); Kantowski (1969); Dyer and Roeder (1972, 1974); Kaiser and Peacock (2016);
Bonvin et al. (2015) have explored analytically the question of which averages
of which observables might be best suited for recovering an unbiased measure
of the background geometry, although the expected variance has not as far as
we can tell been studied. Recently, Adamek et al. Adamek et al. (2019)
explored the influence of small-scale non-linear structures on determinations
of certain curvature averages and provided numerical evidence supporting the
claim Kaiser and Peacock (2016) that well-designed measurements could at least
be unbiased. (See below.)

Meanwhile, current theories of the origins of structure ascribe the origin of
the small scale structures we inhabit and observe to a spectrum of
fluctuations in some primordial field, typically a scalar field termed the
inflaton, with associated fluctuations in the metric. This spectrum naturally
extends to wavelengths much larger than the size of currently known
structures, indeed larger than the Hubble scale. Kleban Kleban and Schillo
(2012) argued that if the fluctuations in the Newtonian potential are
$\Phi_{N}$, then they generate ${}^{3}R=\frac{4\nabla^{2}\Phi_{N}}{a^{2}}$.
(Of course there can be vector and tensor perturbations, anisotropic stress,
and other non-Newtonian contributions.)

There is no compelling reason to expect a sudden cutoff in the spectrum at
scales just larger than observed structures. For example, some of us Copi et
al. (2019) demonstrated that such a cutoff, somewhat unexpectedly, does not
naturally explain the absence of large angle correlations in the CMB
temperature fluctuations. Cosmic topology would provide a cutoff, but only on
scales larger than the fundamental domain, which, as described above, is
already known not to be much smaller than the sphere of last scattering – the
limits of our direct electromagnetic observations. Both the “subhorizon” and
“superhorizon” contributions to the metric fluctuations would contribute to
any average of the curvature on large scales. Kleban showed that for a
canonical inflationary power spectrum, the square root of the expectation
value of the spatial average of $(^{3}R)^{2}$ is non-zero. Interpreted as a
background curvature it would lead to $|\Omega_{K}|\simeq 10^{-5}$. The
magnitude of the contribution of large-scale modes to the average curvature
(as characterized by $|\Omega_{K}|$) was explored numerically in Adamek et al.
(2019) and by some of us in Tian et al. (2021), where we also considered
different averaging procedures. While long-wavelength fluctuations make a non-
vanishing contribution to inferred cosmological parameters, for that
contribution to be observationally significant would require that their
amplitude is larger than anticipated in vanilla inflationary $\Lambda$CDM.
Exactly this has been suggested as a possible solution to the $H_{0}$ tension
Adhikari and Huterer (2020).

This spatial average of $(^{3}R)^{2}$ may, anyway, not be an observationally
relevant quantity to explore. Certainly, while $(^{3}R)^{2}=0$ implies
$\Omega_{K}=0$, the converse is not true; it is not necessary to have
$(^{3}R)^{2}=0$ in order to get $\Omega_{K}=0$. We are interested in
observables, such as the coefficient of the $a^{-2}$ term in the Friedmann
equation, not an integral over a gauge-dependent spatial hypersurface.
Moreover, a similar computation to those in Adamek et al. (2019) or Tian et
al. (2021) would presumably yield a non-isotropic effective curvature,
probably of similar size to the average curvature, but which might be more
detectable.

###  I.3 $\mathbf{\Omega_{-2}=0}$ for scale-factor evolution

In a homogeneous and isotropic universe, the metric in spherical polar
coordinates can be written in the form

$ds^{2}=dt^{2}-a(t)^{2}\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)\right]$ (2)

where $k$ is the curvature parameter described above. This metric is
consistent with a homogeneous diagonal stress energy tensor of the form
$T^{\mu}_{\nu}(t)=\mathrm{diag}(\rho(t),p(t),p(t),p(t))$. The Einstein
equations for the dynamics of the metric then reduce to a pair of independent
equations for the first and second time derivatives of $a(t)$ in terms of the
density $\rho(t)$, the pressure $p(t)$, and two constants, the cosmological
constant $\Lambda$ and the curvature constant $k$. These are the Friedmann
equations:

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}$ $\displaystyle=\frac{8\pi
G\rho}{3}-\frac{1}{3}\Lambda-\frac{k}{a^{2}}$ (3)
$\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle=-\frac{4\pi
G\rho}{3}\left(\rho+3p\right)-\frac{1}{3}\Lambda\,.$ (4)

Applied to a homogeneous isotropic perfect fluid with a given equation of
state, $p=p(\rho)$, or to a collection of such fluids (i.e.
$\rho=\sum_{i}\rho_{i}$), the Friedmann equations yield the time evolution of
the one dynamical function in (2)–the scale factor $a(t)$.

We know that the contributions to the energy density in the Universe include
certain elements–relativistic particles such as photons, massive slow-moving
composite “particles” such as atoms, ions, and nuclei (and the still-more-
composite objects built out of them like stars and galaxies). The latter
category is widely understood to also include cold dark matter, non-
relativistic non-interacting “particles”. A gas of photons, which are massless
and non-self-interacting, is very well approximated as a perfect fluid with
$p=\rho c^{2}/3$. Because $p$ is the kinetic energy density of a perfect
fluid, for non-relativistic non-interacting species, $p\simeq\frac{1}{2}\rho
v^{2}\ll\rho c^{2}$, so $p=0$ is a good leading order approximation. This is
known as “dust.”

Conservation of stress-energy (or equivalently the Friedmann equations) imply
that a fluid with $p=\rho/3$ has $\rho\propto a^{-4}$, while one with $p=0$
has $\rho\propto a^{-3}$. This allows the first Friedmann equation in a
homogeneous isotropic universe containing only radiation
($p_{r}=\rho_{r}c^{2}/3$) and pressureless matter to be rewritten as a first-
order non-linear ordinary differential equation:

$\left(\frac{\dot{a}}{a}\right)^{2}=H_{0}^{2}\sum_{i=0,-2,-3,-4}\Omega_{i}\left(\frac{a}{a_{0}}\right)^{i}$
(5)

with

$\displaystyle\Omega_{0}$ $\displaystyle\equiv-\frac{\Lambda
c^{2}}{3H_{0}^{2}}\,,\quad\Omega_{-2}\equiv-\frac{kc^{2}}{a_{0}^{2}H_{0}^{2}}\,,\quad\Omega_{-3}\equiv\frac{8\pi
G}{3H_{0}^{2}}\rho_{m0}\,,$ (6)
$\displaystyle{\mathrm{and}}\quad\Omega_{-4}\equiv\frac{8\pi
G}{3H_{0}^{2}}\rho_{r0}\,.$

$H_{0}$ is $\dot{a}/a$ evaluated today, and $\rho_{m0}$ and $\rho_{r0}$ are
the dust and radiation energy densities when $a=a_{0}$. Conventionally,
cosmologists replace $\Omega_{0}\to\Omega_{\Lambda}$,
$\Omega_{-2}\to\Omega_{K}$, $\Omega_{-3}\to\Omega_{m}$,
$\Omega_{-4}\to\Omega_{r}$.

The framework described above relies implicitly on the confidence that the
metric that solves the full Einstein equations sourced by the real stress
energy tensor is well approximated by the homogeneous metric that is sourced
by the average stress-energy tensor

$\displaystyle\bar{g}_{\alpha\beta}\left[G_{\mu\nu}\\!=\\!8\pi
G_{N}{T}_{\mu\nu}\right]$ (7) $\displaystyle\qquad\qquad\simeq
g_{\alpha\beta}\left[G_{\mu\nu}\\!=\\!8\pi G_{N}\bar{T}_{\mu\nu}\right]\,,$

(where spatial averages over FLRW hypersurfaces are represented by an
overbar); and, moreover, that the average stress-energy tensor is well
described by the sum of homogeneous perfect fluids that have the same
equations of state as the constituent fluids would if they were actually
homogeneous

$\displaystyle\bar{T}_{\mu\nu}\simeq T_{\mu,\nu}[\bar{\rho},p(\bar{\rho})]\,.$
(8)

This casual commutation of averaging with the solving of systems of coupled
non-linear partial differential equations is not to be taken lightly, and has
long been the subject of concern under the name of “the cosmlogical
backreaction problem” and remains unsettled (see, for example, Schander and
Thiemann (2021) and references therein for a recent review). It raises several
questions:

  * •

Is the average stress energy tensor well described as the sum of a homogeneous
radiation fluid and a homogeneous dust fluid? Even though the matter has non-
linear inhomogeneities and complex physics such as magneto-hydrodynamics is
required to describe the evolution of at least the baryonic component of that
matter? Even though certain neutrino species may currently be transitioning
from the relativistic (radiation) regime to the non-relativistic weakly
interacting (dust) regime?

  * •

If so, is the average of the metric obtained from the full Einstein tensor
(which is proportional to the full stress-energy tensor) equal to the metric
obtained from the Einstein tensor that is proportional to the average stress
energy tensor?

  * •

If the answer is “yes, though not precisely”, how good is the approximation?

These questions themselves hide subtleties like – what is meant by spatial
average? Over which spatial hypersurfaces? The Einstein equations are local,
and respect causality – the spatial average of a quantity over some volume
cannot causally determine the change of another quantity at any point until
such time as the volume over which the average is taken is inside the past
light cone of that point. In other words averaging requires abandoning local
differential equations.

What can we imagine are the possible consequences of averaging on the
evolution of the spatially averaged metric $\bar{g}_{\alpha\beta}$:

  * •

The average metric over the finite Hubble volume may not be precisely
isotropic – local anisotropies may average away slowly, or not at all.

  * •

The Friedmann equation obeyed by the average metric may not be represented to
sufficient accuracy by a sum of terms that are proportional to $a^{0}$,
$a^{-2}$, $a^{-3}$, or $a^{-4}$. In other words, (5) may not be a good
representation of the evolution of the effective stress-energy, curvature, and
cosmological-constant contributions to the right-hand-side of the Friedmann
equation.

  * •

$\Omega_{i}$ may not be sourced exclusively by the homogeneous sources to
which they are ascribed – for example there may be contributions to
$\Omega_{-2}$ that are sourced by matter non-linearities.

There is another set of concerns relating to how we extract the values of
$\Omega_{i}$ in (5). In many cases this involves modeling the dynamics of the
perturbations in the metric and in the stress-energy in the background metric.
This is true for example in both CMB and large-scale structure inferences of
cosmological parameters. Errors in the assumed form of (5) will propagate into
the dynamical equations underlying those perturbations.

Thus to use (5) in the context of precision cosmology requires more than
having a prior that favors a model for the stress energy that includes only
radiation, baryonic matter, and non-interacting dark matter. It requires
demonstrating that none of the concerns raised by the above questions lead to
measurable deviations from the FLRW background model sourced by the average
fluids.

As remarked above, Adamek et al. Adamek et al. (2019) provided numerical
evidence supporting the claim (e.g., Kaiser and Peacock (2016)) that, despite
the presence of non-linear structure, well-designed averages of observations
could recover estimates of parameters in a background FLRW model that are
reasonably unbiased. Specifically they found that a redshift-binned average of
$1/d_{L}^{2}$ for Type 1a supernova luminosity distances yields $\Omega_{K}$
consistent with the fiducial $\Omega_{K}=0$ universe, with a $1\sigma$
uncertainty of $\simeq 0.001$. However they analyzed only one simulation from
which the precise value of neither the bias nor the theoretical variance in
$\Omega_{K}$ can be inferred. That study could resolve halos down to $5\times
10^{11}M_{\odot}$ – how finer resolution (and greater non-linearity) or
baryonic physics might affect the bias or variance remains uncertain. Also
uncertain is how inhomogeneities affect other cosmological observables from
which FLRW background parameters are estimated. This has been addressed in
some cases analytically in perturbation theory Bonvin et al. (2015), but not
in the context of non-linear simulations. Heinesen and Macphereson Heinesen
(2020, 2021a); Macpherson and Heinesen (2021); Heinesen (2021b) have worked to
build a framework for further explorations of these matter.

One way to understand these issues is that the meaning of $\Omega_{K}$ may
itself have an irreducible ambiguity at the level of $\sim 10^{-3}$, and so
its value may well vary at this level among even well-designed observables
that are used to infer “it.” As the precision with which experiments seek to
infer $\Omega_{K}$, one should be cautious that this ambiguity not be
misinterpreted as disagreement. On the other hand, this also suggests that
properly designed estimates that result in values larger than this ambiguity
cannot be dismissed. Truthfully, this caution extends to other potential
parameters of FLRW cosmology that we quietly ignore, e.g. $\Omega_{-1}$ in the
language of (6). $\Omega_{K}$ stands out principally because it is an inherent
parameter of FLRW, i.e. it belongs to the geometric (left-hand) side of (1),
not (or not only, if we mean $\Omega_{-2}$) to the stress-energy (right-hand)
side.

Finally, we must note that Adamek et al. (2019) studied the effect of
inhomogeneities only in a background $\Omega_{K}=0$ FLRW model. It is unclear
how the results generalize to $\Omega_{K}\neq 0$, or indeed to the wider set
of allowed homogeneous but anisotropic models, or the even wider set of
manifolds without, or with boundaries. A real program would seek to measure
the full metric everywhere and then, if appropriate, fit it to a homogeneous
metric plus perturbations. While this is perhaps an impossible program, we
should be aware of the true physical problem and its challenges.

At this point we might be tempted to ignore that to describe our Universe we
need a theory of the initial fluctuations. Within vanilla-$\Lambda$CDM, this
role is served by any one of a large set of possible inflation models, each
with all of its attendant parameters. However, proceeding boldly to ignore the
questions of inhomogeneities – whether as initial conditions or generated in
the very early universe — one could, instead of fitting for $\Omega_{K}$ (or
better for fitting for a more general homogeneous background metric), opt to
include a specification of the boundary conditions121212Even manifolds without
boundaries have boundary conditions – e.g. the periodic boundary conditions
that define a three-torus. for the homogeneous background geometry as part of
the model. One would then argue that a well-defined class of manifolds (with
their associated boundary conditions) identifies a model, and that we can do
this in such a way as to focus our attention on FLRW models, and in particular
flat-FLRW models.

To execute this boundary-condition approach, one must first assign a prior of
$0$ to all manifolds with boundaries, manifolds that do not admit a single
homogeneous geometry, and manifolds that admit only an anisotropic geometry.
One does this, despite the lack of any fundamental principle that excludes
them, and in fact in the full knowledge that today we appear to live in a
manifold with boundaries (black-hole singularities), or at least with regions
hidden behind horizons. One would be left with just the “space” of manifolds
that admit a homogeneous and isotropic geometry, i.e. the FLRW models. One
would then, despite lacking a measure on this space, separate it into three
classes – those that admit homogeneous positive, zero, or negative curvature.
We could then call each class a model, and assign it a prior. Homogeneous
geometries in each class would have their own set of parameters that describe
the metric131313 There will be others that characterize the different boundary
conditions within each class – e.g. the relative side lengths of the
fundamental domain of a three-torus. As well, one will have some indexing of
the different members of each model class, i.e. of the boundary conditions
that all allow the same homogeneous geometry. The Euclidean metric has one
fewer than the other two – i.e. $\Omega_{K}\equiv 0$. One could then test
whether the background curvature is flat, negatively or positively curved by
performing model selection among the three models. Usually this is done by
fixing the boundary conditions within each class to select the covering space
of that geometry.

This approach seems very hard to justify, especially given the absence of any
theory that dynamically selects the homogeneous isotropic boundary-free
manifolds from among all possible spaces, or that chooses one of the Euclidean
3-spaces from among these manifolds. Moreover, it is only through the
fluctuations, for which one has adopted no theory, that one has observables
through which to perform the model selection.

In the next section we explain why including inflation as the theory of the
fluctuations prevents us from pursuing this program by generating through a
random process non-primordial fluctuations that supplement (or replace) the
initial fluctuations.

##  II Inflationary Model Priors

How does one justify assuming an FLRW background metric and then setting
$\Omega_{K}=0$, i.e. removing $\Omega_{K}$ from the set of free parameters of
the theory, when no model yields $\Omega_{K}$ (or indeed the rest of
${}^{3}R_{ijkl}$) exactly zero? The standard answer is that inflation predicts
that the geometry is currently (perturbed) FLRW, with $|\Omega_{K}|$ smaller
than the uncertainty in any measurement we are likely to perform, justifying
setting a $\delta(\Omega_{K})$ priorJaffe (2022).

Certainly, there are inflationary models – by which we mean Lagrangians of
field theories coupled to GR141414 There must also be coupling to the Standard
Model in order to effect “reheating”–the transfer of a sufficient fraction of
the potential energy density driving inflation into ordinary particles as
inflation ends., as opposed to our previous use of model to denote a choice of
background manifold – that, for certain ranges of the Lagrangian parameters,
take many cosmological initial conditions and map them into late-time
universes in which certain spatial averages of the spatial Riemann tensor are
small. For those models, parameter choices and initial conditions, the flat-
FLRW metric, and thus $\Omega_{K}=0$, is an attractor of the classical theory
— given any patch of the early universe in which the initial conditions allow
inflation to begin and in which there is a short-wavelength cutoff to the
initial perturbation spectrum, then, ignoring quantum fluctuations, the
Riemann tensor in that patch will be driven toward that of the flat-FLRW
metric, and $\Omega_{K}$ will be driven toward zero.

This effective classical convergence to $\Omega_{K}$ indistinguishable from
$0$ is not universal in inflationary models Efstathiou and Gratton (2020).
Certainly, if the inflation lasts forever, then one is driven classically
toward a universe in which the only energy density is that of the inflation
plus (any other source of) vacuum energy, and in which the spatial metric is
precisely Euclidean. On the other hand, if one is not in the right part of
model parameter space or one has the wrong initial conditions, then inflation
may not begin, or the Riemann tensor may not reach that of flat-FRW. In “just-
so” inflationary models, this leads to $\Omega_{K}$ arriving near $0$, but to
a value that may be observationally distinguishable from $0$. Meanwhile, there
are also inflationary models that predict $\Omega_{K}\neq 0$ for appropriate
parameters Linde (1995); Hergt et al. (2022) – such models were particularly
popular when it was thought that $\Omega_{K}\neq 0$ and are once-again
becoming of interest.

Ironically, one should now realize that if inflation carries the scale of
topology far beyond the Hubble scale, then it removes the possibility for an
observer to use topology to determine which unique homogeneous geometry our
spatial 3-manifold admits. Thus if we wanted to fix $\Omega_{K}=0$, we would
have to rely on a local-geometry meaning of $\Omega_{K}$. Here the fact that
inflation is used not just as an explanation for the near-homogeneity, near-
isotropy and near-flatness of the universe, but also as an explanation for its
inhomogeneities is crucial, as we now discuss.

The great success of the inflationary scenario is widely regarded as the
prediction of a spectrum of scalar quantum fluctuations that (at least by the
time they are observed $14$ Gyr later): are predominantly or entirely
adiabatic; are nearly, but not precisely, scale free; extend at least to
scales that until recently were super-horizon; and are nearly Gaussian at
early times. These quantum fluctuations are produced at sub-horizon scales in
an ongoing process during inflation even as the initial fluctuations in the
stress-energy and geometry are being stretched to ever-longer and eventually
super-horizon physical scales. These quantum fluctuations carry components of
the Riemann tensor away from zero even as the accelerated expansion of the
Universe is redshifting the initial fluctuations and their contribution to the
Riemann tensor.

To summarize, the background initial conditions pre-inflation could be
$\Omega_{K}=0$ (in the topological sense), but long-enough inflation removes
the ability of the observer to use the topology to anchor a unique choice of
homogeneous geometry, and thereby set $\Omega_{K}$ exactly to $0$. Inflation
simultaneously generates fluctuations in the metric which create an
irreducible ambiguity for any observer about the precise value of $\Omega_{K}$
in their Hubble patch – different “nearby” choices of homogeneous background
geometry will yield “nearby” descriptions of the fluctuations. All of them
will be consistent with the theory– there is no unique way to perform the
background-plus-perturbations split. Hence we cannot assume $\Omega_{K}=0$,
with a theory that can only describe $|\Omega_{K}|\simeq 0$; but how “nearby”
does inflation predict these descriptions to be, i.e. does inflation have a
generic prediction of the amplitude and spectrum of post-inflationary
fluctuations?

When inflation was first proposed, it was expected that the amplitude of the
quantum-induced scalar fluctuations would naturally be ${\cal O}(1)$ – and
that in most inflationary models a fine-tuning of model parameters would be
necessary to suppress the fluctuations, so as not to spoil the inflationary
prediction of homogeneity and flatness. As the search for CMB fluctuations
continued, it resulted in increasingly stringent upper limits on their
amplitude, so the required fine tuning became increasingly stringent. The
scalar fluctuations were finally detected by the Cosmic Background Explorer
Smoot et al. (1992), but at a level that requires either stringent fine tuning
or clever models in which the necessary small parameters emerge “naturally.”
The original a priori space of parameter of inflationary models was slowly
shrunk by gradually improving observational limits, with the result that what
we confidently believed about inflationary models to begin with was not
matched by our posterior inferences.151515 Of course every improved
measurement of a fundamental parameter shrinks the allowed parameter space,
and one would not want to view that as generically problematic for a theory.
This makes the precise characterization of fine-tuning difficult; however, if
dimensionless parameters of models are forced to have very small values, or
very particular ratios without an explanation through symmetry, that is
generally viewed as concerning.  Moreover, this evolution to more restricted
parts of parameter space – in which FLRW would be a better approximation – was
driven by observations, not new understanding of theoretical principles.

There have been concerted efforts to demonstrate that there is a large set of
initial conditions in which inflation begins and persists for long enough to
homogenize, isotropize, and flatten the Universe. Arguments have been made
that the appropriate measure on the space of initial conditions favors those
that support inflation, and that the more inflation they support they more
heavily they should be weighted. There has also been a history of cosmologists
remaining or becoming unconvinced by the explanatory power of this mapping.
Priors on possible initial conditions are difficult to assess, and the on-
again-off-again interest in the “measure problem” for inflationary models is
probably a reflection of this difficulty. (See, e.g. Vanchurin et al. (2000);
Garriga et al. (2006); Bousso (2006); Guth (2007); Linde et al. (2009); Linde
and Noorbala (2010); Guth and Vanchurin (2011).)

The tensor fluctuations from inflation have not yet been detected as “B-modes”
in the cosmic microwave background. That non-detection has itself begun
driving not just a shrinking of the parameter spaces of inflationary models,
but a shrinking of the allowed space of inflationary models. Priors on the set
of possible inflationary models are even more difficult to assess than the
priors on model parameters. However, over the last decades many of the single-
field inflationary models that were regarded as heavily favored a priori are
now effectively ruled out. Yet, the claim that a very narrow prior on
$\Omega_{K}$ is justified rests on confidence in our ability to pre-judge
inflationary models.

Should we have such confidence in an unspecified and shrinking set of
inflationary models that we insist that an observable cosmological parameter
$\Omega_{K}$ should not be estimated because in some of those models, with
certain parameter choices (and some range of initial conditions) that have
been dictated by observations, its expected value cannot currently be
distinguished from zero?

More concisely, within the inflationary framework, one is not meant to include
a specification of the boundary conditions for the homogeneous and isotropic
background as part of the model, the FLRW background geometry is meant to
emerge dynamically. Meanwhile, the theory for the inhomogeneous part of the
metric, make it impossible to expect that the average curvature, and hence
$\Omega_{K}$, is exactly zero. Instead, the observed properties of
fluctuations is what drives limits on inflationary models and their
parameters. Perhaps one could try to argue that it is possible to test
$\Omega_{K}\simeq 0$ FLRW against the data. However this choice of priors is
not motivated by our theoretical understanding, i.e. there are no theoretical
reasons to select initial conditions patches and inflationary model parameter
values to fulfill the $\Omega_{K}\simeq 0$ FLRW condition and then call this a
“cosmological model”. Only data can tell us if our inflationary-$\Lambda$CDM
description works with FLRW geometry and $\Omega_{K}$ consistent with zero or
not. Hence, even within an FLRW assumption, the value of $\Omega_{K}$ needs to
be properly inferred from observational data.

##  III Some practical takeaways regarding spatial curvature

The universe is not precisely homogeneous and isotropic, it has fluctuations.
$\Omega_{K}$, and for that matter the other cosmological parameters of FLRW,
are phenomenological parameters, not fundamental parameters. They describe a
background universe that doesn’t actually exist, but which is meant to
approximate the Universe (or the piece of it in which we live) well enough
that we can treat deviations from FLRW as perturbations with statistics we can
relate simply to our model of the origin and evolution of those perturbations.
Certain inflationary models, with certain values of their parameters, map
certain parts of the space of initial conditions to regions of spacetime in
which spatial curvature is small but not exactly zero, but our track record in
reliably placing strong priors on this extremely complicated space of
inflationary models, parameters and initial conditions does not justify
placing strongly informative priors on cosmological parameters and then
claiming that this set of priors defines a new “cosmological model” named
flat-$\Lambda$CDM where the background is homogeneous, isotropic, and
Euclidean ($\Omega_{K}=0$).

One should not fix an FLRW parameter to a specific value without a theoretical
framework to enforce that value. It is not appropriate to regard such fixing
as “a restrictive assumption that must be tested empirically” Ade et al.
(2016a), if that means demanding strong Bayesian evidence to negate the
assumption and restore the parameter. In particular, there is no compelling
theoretical reason to set $\Omega_{K}=0$. It must be included in the set of
FLRW that are to be fit.

If one finds that $\Omega_{K}\neq 0$, then it will be important to explore
whether other degrees of freedom of the spatial curvature (${}^{3}R_{ijkl}$)
are non-zero. This is necessary to understand the cause of the non-zero value
of $\Omega_{K}$, in particular whether the deviation is consistent with the
universe being nearly FLRW.

Notice (e.g. Wright (2006); Clarkson et al. (2007); Aghanim et al. (2020); Di
Valentino et al. (2019); Handley (2021)) that fitting for $\Omega_{K}$ rather
than fixing it has consequences for best-fit values and errors of other
cosmological parameters, and for assessments of the consistency of
cosmological data sets (e.g. Gonzalez et al. (2021)).

Theorists should also consider predicting and observers then measuring
quantities that have clear meanings not just in the background metric but in
the actual metric (and that may reduce to simpler quantities in a homogeneous
and isotropic limit). For example, components of the ($3+1$-d) Riemann or
Ricci tensors, or the Ricci scalar.

##  IV Summary

We examined the common practice among cosmologists of omitting $\Omega_{K}$
from the set of 7 standard FLRW parameters to be fit to data, by setting it
precisely to $0$, and then arguing that to add it requires sufficient evidence
that it is not $0$. This practice is given the name “flat $\Lambda$CDM,”
suggesting that it is a well-defined separate and fundamental theory of the
Universe from curved $\Lambda$CDM. However, the 7 parameters of inflationary
$\Lambda$CDM cosmological theory are phenomenological parameters of an
approximate effective theory of the Universe on large, but not necessarily
superhorizon, scales. The map to them from the much larger number of
parameters of an ultimate complete underlying microscopic theory – General
Relativity, plus the Standard Model (SM) of particle physics, plus any Beyond
the SM fundamental physics ultimately needed to account for dark matter, dark
energy, inflation, baryogenesis or leptogenesis, the quantum nature of
gravity, and whichever current observational anomalies ultimately prove to be
physical – is unknown, but undoubtedly highly nonlinear, and probably chaotic
in many parts of the parameter space of the fundamental theory. It may well
also involve a specification of a wide variety of data on some Cauchy surface;
or perhaps it has no deterministic specification.

Since the injection of this information, $\Omega_{K}\equiv 0$ a priori may
alter the inferred values of the 6 other parameters and will reduce the
uncertainty on them, we need to understand the underlying justification for
this delta-function prior on a phenomenological parameter.

We began by exploring the meaning of the term flat and its connection to any
arguments underlying the prior. We noted that a spatial hypersurface on which
we find ourselves (“space”) could be a manifold that admits a unique
homogeneous geometry, and that could be the Euclidean geometry. It is
extremely challenging to justify any specific measure on the space of
manifolds, and to use this to argue convincingly for a strong preference for a
topologically flat manifold. A compelling theory of quantum cosmology might
help in this regard, but it is currently lacking. The topology of space thus
seems destined to be a question that can be answered only by observation, if
it is answerable at all. Cosmic topology can in principle be determined from
appropriate measurements if the spatial hypersurface is topologically non-
trivial and the relevant length scales are not too superhorizon; efforts to
search for such topology has so far yielded only lower limits on certain
associated observational length scales.

We therefore have no theoretical justification to choose, out of all possible
manifolds, those manifolds that admit a Euclidean metric and declare them to
be a separate privileged (high-prior) class.

A second, more relevant definition of flatness is that appropriate averages of
the Riemann tensor of the 3-geometry of space are zero. We pointed to some of
the many questions or challenges surrounding this definition. Perhaps salient
among them is that we have had very little observational access to any sizable
sub-volume of space at a fixed time – our information about the distant
reaches of the Universe comes to us at the speed of light, and so we observe
on our past light cone. 161616We will eventually get some information from
observables like the Sunyaev-Zeldovich (SZ) effect where light is scattered
into our past light cone, and also from messengers, like neutrinos and cosmic
rays, that are subluminal. We could in principle make observations of the
geometry and the stress energy on that past light cone and evolve it forward
to the current time, but even in principle that would only determine the
geometry here, since every other location in space has points on its past
light cone that are outside our past light cone. We are thus limited, perhaps
in principle, to checking consistency of observations with a theory that
predicts the statistical properties of the spatial Riemann tensor on our past
light cone.

We do have a theory that makes predictions about the behavior of the spatial
Riemann tensor – inflation. At least in regimes where deviations from a flat
FLRW spacetime are linear, the primary effect of inflation on a classical
level is to change the relation between comoving and physical scales, so that
short-wavelength fluctuations become long-wavelength. Under the assumption
that the initial conditions have no fluctuations below some cutoff scale, this
will cause the Riemann tensor measured over a fixed physical volume to
eventually approach that of a Euclidean cosmological spacetime. This
stretching out of the fluctuations is balanced by quantum mechanical
generation of new fluctuations on small scales (which are then also stretched)
which can be arranged to be of small amplitude. At the end of inflation, the
nearly homogeneous energy density driving inflation could be transformed
efficiently into ordinary matter and radiation, and the Riemann tensor could
be close to that of $k=0$ FLRW – spatially nearly homogeneous and nearly
isotropic, with nearly zero 3-Riemann tensor. Whether or not inflation begins,
and how closely this classical evolution would cause the patch we are in to
approach the Euclidean LFRW geometry, depends on the initial pre-inflationary
geometric inhomogeneities, on the parameters of the inflationary theory, on
the initial conditions of the inflaton field, and on other specifics of the
inflationary model.

For a very long time after that post-inflationary thermalization, or perhaps
forever – depending on the underlying topology, and the magnitude of vacuum
energy (or the value of $\Lambda$), and amplitude of the quantum fluctuations
– the Universe would be well described approximately by an FLRW metric with
$k=0$. Its dynamics would therefore be close, by some measure, to those of an
$\Omega_{K}=0$ FLRW universe.

Predictions of a small current value of $\Omega_{K}$ – i.e. below the
observational uncertainties of our current experiments – are thus
inflationary-model dependent, model-parameter dependent, and (to a widely
debated extent) initial-condition dependent. We have to be in an inflationary
model that has a flat-FLRW attractor; we have to be in a region of initial
condition “space” that is in the basin of that attractor; the quantum
fluctuations cannot drive us too far away from the FLRW Riemann tensor on
large scales; the non-linear fluctuations on small scales must average in some
precise way to recover a Hubble-scale flat-FLRW Riemann tensor with some
appropriate observational scheme.

Our ability to predict a priori which is the correct inflationary Lagrangian
has already been shown to be poor. We have even less theoretical justification
for choosing the values of the associated Lagrangian parameters, the initial
inflaton field value, the initial metric and inflaton inhomogeneities, and
whatever else determines the duration of inflation and the amplitude of
inflationary fluctuations. These are determined from observation, not
theoretical principles. Since $\Omega_{K}$ depends on these, we cannot justify
an $\Omega_{K}=0$ prior, or even a prior that enforces that $|\Omega_{K}|$ be
less than our observational error bars.

Setting $\Omega_{K}=0$ does not define a “cosmological model”, it is just an
arbitrary insertion of strong information into FLRW parameter determination.
Within FLRW, $\Omega_{K}$ must be estimated from observations.

##  V Acknowledgements

GDS is partly supported by a Department of Energy grant DESC0009946 to the
particle astrophysics theory group at CWRU. SA is party supported by the
project “Combining Cosmic Microwave Background and Large Scale Structure data:
An Integrated Approach for Addressing Fundamental Questions in Cosmology,”
funded by the MIUR Progetti di Rilevante Interesse Nazionale (PRIN), Grant
Number 2017YJYZAH, Bando 2017. JBM is partly supported by a Department of
Energy grant DE-SC0017987 to the particle astrophysics theory group at WUSTL.
J. T. G. is supported by the National Science Foundation, Grant No.
PHY-2013718. We thank Y. Akrami, C. Copi, A. Jaffe, and A. Kosowsky for
enlightening discussions.

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