What is the Standard Cosmological Model?

Abstract

Reports of "cosmology in crisis" are in vogue, but as Mark Twain said, "the report of my death was an exaggeration". We explore what we might actually mean by the standard cosmological model, how tensions - or their apparent resolutions - might arise from too narrow a view, and why looking at the big picture is so essential. This is based on the seminar "All Cosmology, All the Time".

Full text

What is the Standard Cosmological Model?
Eric V. Linder1,2
1Berkeley Center for Cosmological Physics & Berkeley Lab,
University of California, Berkeley, CA 94720, USA
2Energetic Cosmos Laboratory, Nazarbayev University, Nur-Sultan 010000, Kazakhstan
(Dated: May 10, 2021)
Reports of “cosmology in crisis” are in vogue, but as Mark Twain said, “the report of my death
was an exaggeration”. We explore what we might actually mean by the standard cosmological
model, how tensions – or their apparent resolutions – might arise from too narrow a view, and why
looking at the big picture is so essential. This is based on the seminar “All Cosmology, All the
Time”.
I. INTRODUCTION
The origin of the word “crisis” comes from “decision”,
implying a framework – a standard model – and paths
forward that are being decided between. So to assess
whether some crisis exists and how severe it might be,
we must first explore what we mean by the standard cos-
mological model.
We discuss various definitions in Section II, then in-
vestigate a couple of oft claimed tensions and their
(un)resolutions in Section III. The primacy of using all
robust observations, at all redshifts – All Cosmology, All
The Time – is motivated in Section IV, and the conclud-
ing discussion is in Section V.
II. WHAT IS STANDARD COSMOLOGY?
What is the standard cosmological model depends very
much on where one draws the line on what is cosmology
or the universe.
A. Level 1: Global Properties
One could define Level 1 as saying cosmology is the
global properties of the universe:
•Connected – a signal can get from there to here and
from then to now. There is no discreteness (at least
at this level).
•Metric – we can figure out how far it is from there
to here and from then to now.
This gives the foundation and few would dispute that
they are part of the standard cosmological model; vio-
lation of either on the scale of the observable universe
would be revolutionary (and pretty exciting on smaller
scales as well!).
A third property comes from a plethora of observa-
tions, and is also generally accepted as part of our stan-
dard cosmological model:
•Homogeneity and Isotropy – on observable universe
scales.
This then implies the Robertson-Walker metric and we
are on familiar territory! In particular, the Robertson-
Walker metric has two characteristics (independent of the
theory of gravity):
•Evolution – a scale factor a(t).
•Spatial curvature – a constant kproportional to
the Gaussian curvature of space. It is worth em-
phasizing that a Robertson-Walker spacetime can
have spatial curvature but no spacetime curva-
ture (though this does not describe our universe),
and spacetime curvature but no spatial curvature
(which fits observations pretty well).
Unlike, say, discreteness, we don’t have to go down to
laboratory scales to test for a breakdown in homogeneity
and isotropy, and hence the Robertson-Walker metric.
However, one can show that these smaller scale devia-
tions do not significantly affect cosmological scales. This
is worth emphasizing: for the expansion and curvature,
and for observations involving propagation of light rays
along a line of sight, and for growth within three dimen-
sional volumes, the effect from small scales generally is
calculable and found to be small [1–4].
B. Level 2: History of the Universe
If one delves into the time structure, one could say that
a Level 2 answer to what is the standard cosmological
model is to say cosmology is the history of the universe.
From observations we could list five key stages:
•Early hot dense state – popularly, if nebulously,
known as the “Big Bang”. Here we consider it as a
generic description and source of initial conditions,
and whether it occurs at the Planck energy, 1015
GeV, or 103GeV is a (fascinating) detail, not a
source of crisis.
•Matter/antimatter asymmetry – I have absolutely
nothing to say! While we know the basic ingredi-
ents to deliver this (if not its magnitude) [5] one
could well argue this is a greater crisis than any
other raised, and yet it is rarely mentioned.
arXiv:2105.02903v1 [astro-ph.CO] 6 May 2021

2
•Radiation dominated era – primordial nucleosyn-
thesis, degrees of freedom g?(neutrino decoupling,
electron/positron annihilation), CMB thermaliza-
tion. Lots of good stuff!
•Matter dominated era – CMB scattering, growth of
structure (us!).
•Cosmic acceleration – “dark energy”, fate of the
universe?
As an aside, one of the most fascinating aspects of this
is that the argument could be made that the study of
dark energy actually grew out of investigation of the ra-
diation dominated era. The radiation era spans such a
huge range of e-folds of expansion (nominally some 55,
compared to the matter era’s 7, or dark energy’s 0.5),
and yet we know little of the details – did it stay ra-
diation dominated the entire time? We tend to simply
assume this but we have accurate windows on only tiny
slivers of this era, around primordial nucleosynthesis [6–
8] and toward the end, just before matter domination
and recombination [9–11]. In 1979, Robert Wagoner [12]
highlighted this by considering what freedom there was
of changing the equation of state of the dominant energy
density. This program of exploring the equation of state
at various epochs was picked up by two of his students
and in the 1980s was applied to the late universe and
became dark energy cosmology, developing both the cos-
mological model and observational probes, first for par-
ticular values of the equations of state [13,14] and then
for a general equation of state [15,16].
C. Level 3: Stuff In the Universe
A more specific view is that cosmology is the stuff in
the universe. After all, this is what is actually observed.
This would include:
•Cosmic microwave background radiation (CMB) –
CMB structure (anisotropies, polarization, spectral
distortions) is a rich probe of both history (Level 2,
including initial conditions such as adiabatic per-
turbations) and the other contents (Level 3, e.g.
matter stuff through scattering, gravitational po-
tentials). And of course it provides strong evi-
dence through isotropy (and homogeneity through
the CMB felt by distance objects) for Level 1 cos-
mology.
•Large scale structure – The continuous fields of
matter: the density field, velocity field, accelera-
tion (gravity) field – generally as probed by indi-
vidual sources. While these fields are related, the
relations do test the framework and each has par-
ticular incisive elements so they can be considered
distinct; plus, for cosmological purposes they are at
very different stages of observational development.
•Other – As probes of the standard cosmological
model, observations of other stuff such as neutri-
nos, gravitational waves, exotica (e.g. topological
defects) have not yet reached the same stage of
having a major impact, though this would be an
exciting development.
D. Turtles All the Way Down
Some researchers extend the standard model of cosmol-
ogy further and further down in detail, to “the stuff in
the stuff in the universe”, e.g. aspects of galaxies, galaxy
clusters, generation of various particles and fields (e.g.
neutrinos, gravitational waves, etc.). Others will stretch
to “the properties of the stuff in the stuff in the universe”,
e.g. cuspy cores of galaxies, tidal streams, Cepheid pul-
sations, etc.
Where one draws the line between the standard cos-
mological model and all else – call it astrophysics for sim-
plicity – is a personal choice. But it is rare that rain next
Tuesday in some place one had not predicted it throws
the standard meteorological model into crisis. It is not
impossible, but it is useful to remember that some par-
ticular tension has to work its way from Level Nto the
fundamental foundations.
E. Cosmologing Is Hard
But. . . cosmologing is hard. The properties of the stuff
in the stuff affect how and what we learn about the more
fundamental stuff. We are then faced with the puzzle of
whether we have fully understood the stuff in the stuff (or
mismeasured its properties) or whether indeed it prop-
agates cleanly to impacting the standard cosmological
model. Let us take two examples.
Example 1. Suppose one measured the CMB temper-
ature at scale factor a(redshift z=a−1−1) and found
that
TCMB(z)6=TCMB (0) ×(1 + z) ? (1)
What should one conclude: that the universe is not adi-
abatically expanding, or that there is some systematic
error (e.g. molecular collisional excitations)? How much
effort, and what proportion of the literature, should be
dedicated to investigating systematics before declaring a
crisis in the standard cosmological model?
Example 2. Suppose one measured the distance to an
object (or set of objects) at redshift zin terms of both its
luminosity distance and angular diameter distance and
found that the reciprocity relation is broken,
dL(z)6=da(z)×(1 + z)2? (2)
If one wishes to conclude that this places the standard
cosmological model in crisis (rather than some system-
atic error in the data), one must give up some founda-
tional element, since this relation arises from a) metricity,

3
b) geodesic completeness, c) photons propagate on null
geodesics, and d) adiabatic expansion. (Note conserva-
tion of photon phase space density is a big part of this,
but not all.)
What level of systematics investigation should one
(and the research community) carry out before declar-
ing an upending of the standard cosmological model? To
what extent should the proportion of systematics vs new
physics papers depend on the degree of new physics re-
quired? As the saying goes, if you tell me you saw out
the window a deer on the lawn, I might be willing to con-
sider how the deer got there, its effect on the shrubbery,
etc., but if you tell me you saw out the window a unicorn
on the lawn, I might want more evidence before devoting
time to the puzzle.
We have in place solutions for how to handle conflicting
expectations, observations, and theories:
•Rigorous data
•Multiple, disparate probes
•Crosschecks
•Consistency at all cosmic times
•Check the cosmic expansion history, cosmic growth
history, and light propagation (and soon gravita-
tional wave propagation)
We explore how these might be applied to an example
tension in the next section.
III. PAST TENSE, PRESENT TENSE?
A well known tension lies in current Hubble constant
H0values deduced from certain probes, taking all the
data at face value. In addition to the main puzzle, there
are some beyond the surface:
•Local measurements differ by some ∼2σdepending
on method, i.e. Cepheids vs tip of the red giant
branch [17,18].
•The tension is emphatically not “early vs late” cos-
mology since baryon acoustic oscillations (BAO)
distances (together with primordial element abun-
dances [19–21] or marginalizing over or sidestep-
ping the sound horizon at the baryon drag epoch
[22,23]), i.e. without use of the primordial CMB,
gives the same answer as from the CMB.
•Strong lensing time delays show a sharp transition
between low and high H0values around z∼0.4
[24,25], albeit with a small sample.
While CMB data alone constrains H0tightly only
within a ΛCDM cosmological model, while allowing a
considerable range of H0when the dark energy equation
of state wdiffers from −1, it is extraordinarily difficult
from a combination of cosmic probes such as CMB+BAO
or CMB+SN (supernovae distances) to obtain H0>70
[26] (we always write H0in units of km/s/Mpc). Basi-
cally, H0>70 requires a phantom dark energy (w < −1),
which is disfavored by the above combinations of probes;
see Figure 1.
There are two basic loopholes one might try by chang-
ing the cosmic expansion history – relax the tension in
the present or the past.
•Present tense (late time transition): arrange a
very sharp phantom excursion very close to the
present so that higher redshift distances are not
too strongly affected.
•Past tense (early time transition): arrange a lower
sound horizon scale rdrag with the Hubble param-
eter Hgoing up. Again, one must make it a sharp
transition – a spurt of extra early energy density
to raise H, then removing the early dark energy to
preserve the agreement with CMB data.
We begin with the early time transition. The covari-
ance between rdrag and H0has been known for a long
time [11,27–30]. Using CMB data, in 2013 Ref. [11] ac-
tually found evidence for an early time transition and its
effect on H0! – see Figure 2. This has been resuscitated
in many many articles in the last couple years. However,
early time transitions do not really work in removing the
Hubble constant tension (see, e.g., [31–34]). On the one
hand they cannot viably raise H0as far as the high values
favored by Cepheids, and on the other hand they gener-
ically violate other aspects of the cosmological model as
we discuss in Section IV.
The late time transition has the advantage that there
is much less effect on the CMB, and so more freedom
for change. If one raises H(z), distances will change.
To preserve distances (e.g. the distance to the CMB last
scattering surface, as well as BAO and SN distances),
with a higher H0one needs a smaller H(z > 0). This
means less energy density. This could be from a smaller
matter density Ωm(the matter density today as a frac-
tion of the critical density) but this is insufficient and one
requires a smaller dark energy density at some z > 0. To
give enough dark energy density today (especially if the
matter density is low and the total density is the criti-
cal density) requires dark energy density to appear quite
suddenly at low redshifts. This is again the phantom
regime w < −1. And again, such late time phase transi-
tions have been known for a long time (see, e.g., vacuum
metamorphosis [35–37]).
Since such late time transitions all have basically the
same physical effect, regardless of specific origin, let’s ex-
amine whether vacuum metamorphosis removes the H0
tension. Yes! but. . . . As the opening lines of the ab-
stract of [38] say, “We do obtain H0≈74 km/s/Mpc
from CMB+BAO+SN data in our model, but that is not
the point.” This – and essentially any late time transi-
tion – model fails because there is more to the standard

4
3.0 2.4 1.8 1.2 0.6
w
2
1
0
1
2
wa
Planck + R16
Planck + JLA + R16
3.0 2.4 1.8 1.2 0.6
w0
2
1
0
1
2
wa
Planck + R16
Planck + BAO
3.0 2.4 1.8 1.2 0.6
w0
2
1
0
1
2
wa
Planck + R16
Planck + H072
Planck + H070
Planck + H068
Planck + H066
FIG. 1. 68.3% and 95.4% constraints on the w0–waplane
in an 11 parameter extended space, using Planck CMB data
plus the R16 H0(∼74) prior. The top panel includes as
well JLA supernova data, the middle panel includes baryon
acoustic oscillations data: both strongly prefer w0≥ −1. The
bottom panel shows that w0≥ −1 is most cleanly achieved
by shifting the H0prior to H0≤70. (Adapted from [26].)
FIG. 2. Reconstruction of the expansion history deviations
δ(a) = δH2/H2
ΛCDM from ΛCDM is shown, with the mean
value (solid line) and 68% uncertainty band (shaded area).
Note the data prefers early dark energy δ6= 0. The in-
set demonstrates that the model independent reconstruction
then prefers a higher H0than the ΛCDM standard analysis.
(Adapted from [11].)
cosmological model than H0; it does not satisfy other
probes. We discuss the details in Section IV. For other
possible issues with late time transitions see [31,39–41].
IV. ALL COSMOLOGY, ALL THE TIME
So far we have considered only the expansion history.
However, one must take into account all cosmological
probes, e.g. how deviations from a standard cosmolog-
ical model affect the growth of large scale structure.
In the vacuum metamorphosis case of the previous sec-
tion, the combination of probes CMB+BAO+SN pro-
duced H0≈74. For a good fit to the CMB, preserving
Ωmh2means a low Ωm≈0.27. That can be ok. How-
ever, it also gives a high amplitude for mass fluctuations,
σ8≈0.88, which is quite high. This is due to the re-
duced dark energy density needed to get the distances
right, quite generally implying greater matter domina-
tion and growth at higher redshifts. We can start to see
that we do indeed need “all cosmology, all the time” –
use of all probes, over the full cosmic history.
One might brush high σ8under the rug and say that
with the lower Ωm, one has S8≡σ8(Ωm/0.3)0.5≈0.83,
which might be workable for some probes, i.e. roughly as
good as ΛCDM. So we could say that vacuum metamor-
phosis gives H0≈74 while not making any S8tension
worse, as apparently seen in Figure 3.
However, S8and H0are focusing on a single time (the
present) in cosmic history. This is a Bad Idea. In the
words of Lewis Carroll, “the rule is, jam tomorrow and
jam yesterday – but never jam today”. Figure 4shows

5
that the apparent removal of the H0tension is moot,
since Figure 3is only a tiny part of cosmic history, and
when one does all cosmology, all the time – taking into
account both the cosmic expansion and cosmic growth
over the span of cosmic history as in Figure 4– then
generically late time transitions do not work in giving a
viable cosmological model.
FIG. 3. Expansion and growth histories are here plotted si-
multaneously. At z= 0, hence these give H0and fσ8(0)
(≈0.52 S8for most viable cosmologies in general relativity;
see [38]), shown by the points. Excellent agreement with the
Cepheid value of H0(green dashed line), as opposed to the
ΛCDM value (cyan dashed line) is obtained by the flat vac-
uum metamorphosis (VM) late transition models with param-
eters fit to CMB+BAO+SN, and the VM VEV model agrees
well on S8as well. (Adapted from [38].)
This is general because the late time transition, an-
chored by the present, requires a lower dark energy den-
sity at higher redshifts and hence greater growth. If you
squeeze the model in one place, it will bulge out else-
where and fail to fit the array of probes. A similar gen-
eral “no-go” reasoning holds for early time transitions.
The higher expansion rate damps the CMB perturba-
tions, so to preserve the fit to CMB data one requires a
higher primordial curvature perturbation amplitude, and
this strengthens the later growth of the matter perturba-
tions, leading to high σ8. (See also [42].)
V. DISCUSSION
Very generally, neither late time nor early time at-
tempts to remove the H0tension survive all cosmology,
all the time. One needs to take into account all the
FIG. 4. A very different situation occurs if one looks be-
yond H0at the full conjoint evolutionary track of the various
models, not just the z= 0 (lower) endpoints as shown in Fig-
ure 3(which covers the small range between the dashed lines).
Over the histories the VM models diverge considerably from
ΛCDM. Curves extend from z= 0 at the bottom to z= 3
at the top, and the points with error bars show the trajec-
tory location at z= 0.6, 1, 2, where the error bars mimic 1%
constraints on each axis quantity to give a sense of separation
between the curves. Solid curves are for flat space, dotted
curves include Ωk. (Adapted from [38].)
probes, at all redshifts. It is not just H0, it is H(z).
It is not just Ωm, it is Ωm(z), and growth of structure,
light propagation, etc.
The standard cosmological model, whether one views
it at Level 1, 2, or whatever, is strong: I come to praise
it, not to bury it.
But, arguendo, suppose we ignored the questions of
Section II E and believed all data blissfully free of sys-
tematics, so that the order (two orders?) of magnitude
difference in number of papers squeezing the standard
cosmological model vs investigating the data is right and
proper. What could we do to remove the tension? From
the preceding section it is clear: we must break the con-
nection between the cosmic expansion history and growth
history. We can do this by a breakdown in general rela-
tivity, making gravity weaker so that σ8becomes lower;
we can do this by introducing new particle interactions,
again to suppress growth. Such changes will in turn al-
ter other cosmological probes, which must be considered,
and we must be wary of a spiral of epicycles. (See also
[43].)
Basically we would need to break the standard expan-
sion history to address the H0tension, and need to break
the standard growth history to keep the consequences of

6
the first break consistent with other probes. An interest-
ing possibility is to probe the connection of the growth of
structure to the cosmic expansion in a wholly new way.
Gravitational wave standard sirens offer this possibility,
in part. A direct connection between siren distances and
matter growth was developed by [44].
The propagation of gravitational waves is an excellent
probe of “spacetime friction” – a combination of the Hub-
ble friction from expansion and any time variation of the
gravitational strength; the growth of structure probes
both these quantities as well, in a different way. By com-
paring these two probes, possibly plus light propagation
which depends only on the expansion, we have the po-
tential to get a clear view of whether the connections
between them in the standard cosmological model hold
– and if they do not, is there consistency between the
deviations in one probe and in another.
A bit more technically: while distances found through
light propagation dEM (z) = dL(z) depend on the ex-
pansion H(z), those determined through gravitational
wave propagation depend on both H(z) and αM(z)≡
dln M2
P l(a)/d ln a, the running of the Planck mass or
inverse gravitational strength. Thus, a measurement
dGW (z)6=dEM (z) would signal a deviation from general
relativity (or systematics). Meanwhile, growth of cos-
mic structure depends on H(z) and the strength of grav-
ity Geff (z). In theories where the gravitational strength
is determined purely by the Planck mass, Geff (z) =
1/M2
P l(z), then the circle is complete and there is a tight
relation between the three probes. (Some theories of
gravity have a further factor, the braiding that mixes the
tensor and scalar parts, loosening the relation; Geff (z)
can also affect light propagation distances, and hence H0
from strong lensing, giving a redshift dependence to the
derived H0[25].) Thus we can crosscheck against sys-
tematics by seeing if the relation holds: a deviation in
one probe predicts a specific redshift dependence for a
deviation in another probe.
The consistency check can be quantified with a new
statistic combining the probe measurements [45],
DG(a)≡dMG
L,GW /dGR
L(a)
fσMG
8/fσGR
8(a).(3)
For general relativity, this equals one for all redshifts.
For a given modified gravity (MG) model, it has a spe-
cific redshift dependence predicted. Several examples are
shown in Figure 5.
In summary, the standard cosmological model has ex-
tremely deep foundations and multiple layers, and ap-
parent surface blemishes may have very little to do with
the fundamental basis. Especially if tensions cannot be
solved by one “tooth fairy”, such as an early or late time
transition, when confronted with application of the prin-
ciple of using all cosmology, all the time, but we are in-
stead led to a series of epicycles, we might recall the
unicorn vs the deer and pause.
New probes and data covering more cosmic history
will be essential in exploring the standard cosmological
FIG. 5. The new DGstatistic, using the complementarity
of the gravitational wave luminosity distance dL,GW and the
cosmic matter growth rate fσ8, can clearly distinguish dif-
ferent classes of gravity. Each class has a distinct shape in
its redshift dependence DG(a), though the curve amplitudes
will scale with Geff (z= 0). General relativity has constant
DG= 1. (Adapted from [45].)
model, at whatever level we define it, and whether it will
stand firm, need some patchwork, or be overturned. Each
of the levels has a wonderful array of areas for students
to research, and make significant and lasting contribu-
tions to what their students will learn as the standard
cosmological model.
ACKNOWLEDGMENTS
I thank the Asia-Pacific Center for Theoretical Physics
for inviting me to give this inaugural lecture in June 2020
for the series “Dark Energy in a Dark Age”. This work is
supported in part by the Energetic Cosmos Laboratory
and by the U.S. Department of Energy, Office of Science,
Office of High Energy Physics, under contract no. DE-
AC02-05CH11231.
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