# A Measurement of the Cosmic Expansion
Within our Lifetime

Fulvio Melia  Department of Physics, the Applied Math Program, and Department
of Astronomy, The University of Arizona, Tucson, AZ 85721
<EMAIL_ADDRESS>
###### Abstract

The most exciting future observation in cosmology will feature a monitoring of
the cosmic expansion in real time, unlike anything that has ever been
attempted before. This campaign will uncover crucial physical properties of
the various constituents in the Universe, and perhaps answer a simpler
question concerning whether or not the cosmic expansion is even accelerating.
An unambiguous yes/no response to this query will significantly impact
cosmology, of course, but also the standard model of particle physics. Here,
we discuss—in a straightforward way—how to understand the so-called ‘redshift
drift’ sought by this campaign, and why its measurement will help us refine
the standard-model parameters if the answer is ‘yes.’ A ‘no’ answer, on the
other hand, could be more revolutionary, in the sense that it might provide a
resolution of several long-standing problems and inconsistencies in our
current cosmological models. An outcome of zero redshift drift, for example,
would obviate the need for a cosmological constant and render inflation
completely redundant.

††: Eur. J. Phys.

  * July 16, 2021

Keywords: redshift drift, cosmology: theory, gravitation, ELT, SKA

##  1 Introduction

Objects receding from us with the general Universal expansion become fainter
with time, and their spectra are redshifted according to their distance. The
rate at which these quantities change is characterized by the expansion speed
and acceleration, but is scaled to the age of the Universe ($t_{0}\sim 13.5$
Gyr), which is considerably longer than a human lifetime. It would therefore
be farfetched to even consider ‘watching the Universe expand in real time.’

And yet, there is increasing excitement and anticipation today at the prospect
of actually measuring the evolving redshift of distant sources via a campaign
lasting several decades. The idea of measuring the cosmic expansion in real
time was first envisaged by Sandage (1962) over half a century ago, who
estimated that galaxies would exhibit spectroscopic velocity shifts (see Eq.
14 below) of order $1$ cm s-1 yr-1. But he concluded that with optical and
radio techniques available back then, there would be no hope of measuring such
relatively small changes, except over a very long time (certainly longer than
many decades). This type of cosmological experiment is quite unique, however,
because one does not need to pre-assume any particular model to acquire the
data and, more to the point, because the signal continues to grow larger with
time.

These are among the reasons why such a program has never been completely
abandoned, and today we find ourselves at the threshold of initiating several
monitoring campaigns expected to deliver compelling results by 2040. A
principal motivation for this type of observation is that it completely avoids
the inherent difficulties in measuring apparent magnitudes (Christiansen &
Siver 2012) which are subject to signal-to-noise limitations and intrinsic
source variability and evolution. The so-called ‘redshift drift’ expected from
the Sandage Test is free of all such constraints.

Many therefore expect these monitoring campaigns to deliver an unprecedented
improvement in our ability to measure the cosmological parameters. There isn’t
much doubt about this, but it is not widely recognized that there is an even
simpler question to be answered by these observations, whose outcome may
resolve several inconsistencies in our current models. Our goal in this paper
is to present a straightforward explanation for the relevance of redshift
drift to our comprehensive understanding in cosmology, and why a more precise
measurement of the acceleration of the Universe may lead to a paradigm shift
in our theoretical understanding by the time these monitoring campaigns
release their initial results a few decades from now.

We shall begin with a pedagogical explanation of redshift drift and then
convey the basic idea of how its measurement may be used to refine our
cosmological parameters in § 2. Beginning with §§ 3 and 4, however, we shall
focus more on the basic question we are addressing in this paper, i.e., will
the measured redshift drift confirm whether or not the Universe is actually
accelerating. We shall discuss the significance of a zero redshift drift on
our cosmological modeling in § 5, and present our conclusions in § 6.

##  2 Redshift Drift

Cosmology today is based on the Friedmann-Lemaître-Robertson-Walker (FLRW)
metric (see, e.g., Cook & Burns 2009; Melia 2020) for a spatially homogeneous
and isotropic three-dimensional space, expanding or contracting according to a
time-dependent expansion factor, $a(t)$:

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

This form of the metric is written using the coordinates of a comoving
observer, for whom $t$ is the cosmic time (and is the same everywhere), $r$ is
the comoving radius, which remains fixed for any source lacking so-called
peculiar motion, and $\theta$ and $\phi$ are the usual poloidal and azimuthal
angles. The geometric factor $k$, also known as the spatial curvature
constant, is $+1$ for a closed universe, $0$ for a flat universe, and $-1$ for
an open universe. All of the data today suggest that $k$ is very likely zero
(Planck Collaboration 2018), so we shall assume $k=0$ throughout this paper.
It is beyond the scope of this work to discuss the physical implication of
this measurement, but it is interesting to note that spatial curvature is
related to the local kinetic plus gravitational energy densities in the
Universe (Melia 2020), implying an important (and interesting!) initial
condition at the time of the Big Bang.

Figure 1: Schematic diagram showing the changing proper distance to a source,
$R_{\rm e}$, as the observer’s time advances from $t_{0}$ to $t_{0}+\delta
t_{0}$.

The coordinates $(ct,r,\theta,\phi)$ represent the perspective of a free-
falling observer. It is often also helpful to introduce the proper coordinates
that one may use to trace the physical location of a source as a function of
time. For example, the proper radius,

$R(t)\equiv a(t)r\;,$ (2)

is often used to express—not the co-moving (unchanging) distance $r$ between
two points but, instead—the increasing distance as the Universe expands, as
illustrated in Figure 1. This definition of $R$ is a consequence of Weyl’s
postulate (Weyl 1923), which states that no two worldlines should ever cross
(aside from any local peculiar motion, if it exists). According to Weyl, every
physical distance in an FLRW cosmology should thus be expressible as the
product of a fixed comoving radius $r$, and the aforementioned position-
independent function of time, $a(t)$. One sometimes sees $R$ referred to as
the areal radius—the radius of a two-sphere of symmetry—defined as
$R\equiv\sqrt{A/4\pi}$, independently of the coordinates, in terms of the area
$A$ of the two-sphere (Nielsen & Visser 2006; Abreu  Visser 2010).

The most reliable information concerning the expansion factor $a(t)$ arrives
to us in the form of electromagnetic waves, shifted in frequency due to the
combined influence of kinematic and gravitationally induced redshift effects
(Melia 2020). The null geodesic equation describing the propagation of such
waves along the $-\hat{r}$ direction, with fixed $\theta$ and $\phi$, is
easily obtained from Equation (1):

$c\,dt=-a(t)\,dr\;.$ (3)

Thus, an electromagnetic signal emitted at $r_{\rm e}$, at time $t_{\rm e}$,
will reach the observer at time $t_{0}$ given by

$\int_{t_{\rm e}}^{t_{0}}{dt\over a(t)}=r_{\rm e}\;.$ (4)

The comoving distance $r_{\rm e}$ is fixed in time, but the proper distance
shown in Figure 1 increases according to $R_{\rm e}(t_{0})=a(t_{0})r_{\rm e}$.

Information concerning the redshift of light emitted by the source is encoded
in Equation (4), which tells us how $t_{0}$ changes as a function of $t_{\rm
e}$ due to the evolution of $a(t)$ between these two times. For example, if we
consider the emission and detection of two crests of the wave, one at $t_{\rm
e}$ and $t_{0}$, and the second at $t_{\rm e}+\delta t_{\rm e}$ and
$t_{0}+\delta t_{0}$, then we see from Equation (4) that

$\int_{t_{\rm e}+\delta t_{\rm e}}^{t_{0}+\delta t_{0}}{dt\over
a(t)}=\int_{t_{\rm e}}^{t_{0}}{dt\over a(t)}\;.$ (5)

And given that $a(t)$ changes only by a miniscule amount during one period of
a typical wave, we may carry out these integrations trivially and find that

${\delta t_{\rm e}\over a(t_{\rm e})}={\delta t_{0}\over a(t_{0})}\;.$ (6)

The conventional definition of redshift, $z$, is based on the fractional
increase in wavelength,

$z\equiv{\lambda_{0}-\lambda_{\rm e}\over\lambda_{\rm e}}\;.$ (7)

Therefore, since the frequency of the wave is $\nu=c/\lambda$, and

${\nu_{\rm e}\over\nu_{0}}={\delta t_{0}\over\delta t_{\rm e}}\;,$ (8)

we arrive at one of the most fundamental equations in cosmology (Weinberg
1972):

$1+z={a(t_{0})\over a(t_{\rm e})}\;.$ (9)

Figure 2: Spectroscopic velocity shift $\Delta v/\Delta t$ associated with the
redshift drift predicted by the Planck-$\Lambda$CDM model ($k=0$, $\Omega_{\rm
m}=0.315$, $H_{0}=67.4$ km s-1 Mpc-1; thick black line), and several
variations with alternative values of $\Omega_{\rm m}$ (indicated in the
plot). In every case, dark energy is assumed to be a cosmological constant,
$\Lambda$, with $w_{\rm de}=-1$ and $\Omega_{\Lambda}=1-\Omega_{\rm m}$.

Of course, this relation gives us the redshift corresponding to cosmic
evolution over millions and billions of years. It is hardly useful as a probe
of the change occurring over a mere human lifetime. It is necessary for us to
derive from Equation (9) an expression yielding the incremental changes in $z$
expected during a much shorter time interval $\delta t_{0}$. Differentiating
Equation (9) with respect to the observer’s time $t_{0}$, we find that

${dz\over dt_{0}}=[1+z(t_{0})]H(t_{0})-{a(t_{0})\over a(t_{\rm
e})^{2}}{da(t_{\rm e})\over dt_{\rm e}}{dt_{\rm e}\over dt_{0}}$ (10)

where, by definition, the Hubble parameter is given as

$H(t)={1\over a(t)}{da(t)\over dt}\;.$ (11)

And since

$dt_{0}=[1+z(t_{0})]\,dt_{\rm e}\;,$ (12)

we arrive at an equally elegant and simple expression for the redshift drift
expected over an observable interval of time:

${dz\over dt_{0}}=(1+z)H_{0}-H(z)\;.$ (13)

To simplify the notation somewhat, we have used the definitions $z\equiv
z(t_{0})$ and $H_{0}\equiv H(t_{0})$.

During a monitoring campaign, the surveys will measure the spectroscopic
velocity shift, $\Delta v$, defined in terms of the redshift drift $\Delta z$
over an observation time $\Delta t$:

$\Delta v\equiv{c\Delta z\over 1+z}={c\Delta t\over 1+z}\,{dz\over dt_{0}}\;.$
(14)

The expectation within the cosmology community in regard to the measurement of
the redshift drift (i.e., $\Delta z$ over an interval of time $\Delta t$) is
that this equation may be used together with the new data to optimize the
standard model’s fit to the velocity shift as a function of redshift, thereby
‘measuring’ the cosmological parameters to very high precision. The model
dependence in Equation (14) enters exclusively through the Hubble parameter
$H(z)$ in Equation (13) which, for the standard model ($\Lambda$CDM), is
written as follows:

$H(z)\equiv H_{0}E(z)\;,$ (15)

where

$E^{2}(z)\equiv\Omega_{\rm m}(1+z)^{3}+\Omega_{\rm r}(1+z)^{4}+\Omega_{\rm
de}(1+z)^{3(1+w_{\rm de})}\;.$ (16)

This expression uses the standard ratios $\Omega_{i}\equiv\rho_{i}/\rho_{\rm
c}$ of the energy density for species $i={\rm m}$ (matter), $i={\rm r}$
(radiation) and $i={\rm de}$ (dark energy), in terms of the current critical
density $\rho_{\rm c}\equiv 3c^{2}H_{0}^{2}/8\pi G$, and the dark-energy
equation-of-state parameter, $w_{\rm de}\equiv p_{\rm de}/\rho_{\rm de}$, with
$p_{\rm de}$ the pressure.

For example, in the simplified approach of assuming a spatially flat Universe
(i.e., $k=0$) and dark energy in the form of a cosmological constant (Melia
2020) $\Lambda$ (with $w_{\rm de}=-1$), the monitoring of $\Delta v$ should
provide a direct measurement of $\Omega_{\rm m}$, and therefore of
$\Omega_{\rm de}\equiv\Omega_{\Lambda}=1-\Omega_{\rm m}$. To illustrate the
potential for carrying out this groundbreaking work, we show in Figure 2 the
variation of $\Delta v/\Delta t$ (in units of cm s-1 yr-1) with redshift and
the matter density $\Omega_{\rm m}$. The version of $\Lambda$CDM with the
parameters optimized by Planck (Planck Collaboration 2018) (conventionally
referred to as Planck-$\Lambda$CDM) is highlighted on this plot with a thick
black curve. Notice, e.g., that the redshift drift with time is positive at
low redshifts, and then turns negative for the more distant sources. This
unambiguous prediction by the standard model is simply based on the temporal
evolution of the matter ($\rho_{\rm m}$) and dark-energy ($\rho_{\rm de}$)
densities, which sees the Universe dominated by the former at $z>0.7$, giving
way to the latter towards the present. In $\Lambda$CDM, dark energy functions
as an agent of acceleration, whereas a matter-dominated cosmos is always
decelerating.

Measuring a spectroscopic velocity shift of $<1$ cm s-1 yr-1 sounds very
challenging, but as we shall discuss in § 4, the idea of observing redshift
drift has come a long way since the initial proposal by Sandage (1962) six
decades ago. Many groups have made projections for the next generation of
telescope facilities, and have concluded that a few thousand hours of
observing time spread over a baseline of several decades should provide a very
compelling signal. Indeed, as we shall see shortly, the most optimistic
predictions show that the European Extremely Large Telescope (ELT) (Liske et
al. 2014) (but also the Square Kilometer Array, SKA; Kloeckner et al. 2015)
will provide a $3\sigma$ result in under two decades. Before we consider the
details of this project, however, let us first direct our attention to what
may be the simplest question to be answered by a measurement of the redshift
drift.

##  3 Is the Universe Accelerating?

Recent trends have started to indicate the possibility that a simple yes/no
answer to this question may have an impact on several areas in cosmology and
the standard model of particle physics, so let us now focus on this particular
issue.

Figure 3: Spectroscopic velocity shift $\Delta v/\Delta t$ associated with
three of the $\Lambda$CDM models shown in Figure 2, i.e., Planck-$\Lambda$CDM,
$\Omega_{\rm m}=0.2$ and $\Omega_{\rm m}=0.5$, compared with an FLRW Universe
expanding at a constant rate (blue long-dashed line), which would exhibit
strictly zero redshift drift at all redshifts. The velocity shifts are
calculated over a twenty-year baseline, consistent with the expected errors
(red bars) after 20 years of monitoring with the ELT-HIRES (Liske et al.
2014).

The conventional wisdom developed over the past three decades would suggest
that the Universe is currently accelerating. Certainly, the analysis of Type
Ia supernovae (Perlmutter et al. 1998; Riess et al. 1998; Schmidt et al. 1998)
has provided a convincing, if not unchallenged (Melia 2020), argument that the
Universe has undergone several phases of deceleration and acceleration
consistent with Equations (15) and (16). But the growing tension between the
standard model’s predictions and our ever improving observations is beginning
to cast a big shadow over its formerly perceived infallibility. For example,
the Hubble constant, $H_{0}$, characterizing the expansion rate and cosmic
distance scale (Cadmus 1999), cannot find a self-consistent value from the
combined measurement of the cosmic microwave background (CMB) (Planck
Collaboration 2018), which gives $67.4\pm 0.5$ km $\rm s^{-1}$ $\rm Mpc^{-1}$,
and local Type Ia supernovae calibrated with the Cepheid distance ladder
(Riess et al. 2019), which instead yield $74.03\pm 1.42$ km $\rm s^{-1}$ $\rm
Mpc^{-1}$. Part of the problem is that this $4$–$5\sigma$ disparity seems to
get worse as the sensitivity of the instruments gets better.

But there are many indications, besides the problem with the Hubble constant,
that the observations may not be fully consistent with an accelerating
Universe. Indeed, the standard model suffers from several seemingly insoluble
hurdles and inconsistencies at a very fundamental level, in spite of the
general success it has enjoyed accounting for most of the observations over an
extended period of time. We shall return to this shortly, after first
describing how the upcoming measurement of the redshift drift will hopefully
bring this decades-long debate to a close.

The distinction between an FLRW Universe expanding at a constant rate and one
undergoing some form of acceleration or deceleration is quite simple to
understand (Melia 2016). If $a(t)\propto t$, meaning no acceleration, Equation
(11) gives $H(t)=1/t$, while Equation (9) shows that $1+z=t_{0}/t_{\rm e}$.
Thus,

$H(t_{\rm e})=H(t_{0})[1+z(t_{0})]\;,$ (17)

and

${dz\over dt_{0}}=0\;,$ (18)

so that $\Delta v=0$. In other words, a Universe without any acceleration has
strictly zero redshift drift at all redshifts. A source with a measured
redshift at the time of its discovery will retain that same redshift forever
in a Universe expanding at a constant rate. This is indicated by the blue
long-dashed line in Figure 3, along with the anticipated measurement precision
(red bars; see § 4 below) and a comparison of this outcome with three of the
curves in Figure 2.

##  4 The Next Generation of Telescopes

Today’s most powerful telescopes can attain a precision in measuring redshifts
better than $\sim 10^{-7}$, as used in searching for planets or measuring
time-varying fundamental constants (Lovis & Pepe 2007). Typical large-redshift
surveys, such as the Sloan Digital Sky Survey at Apache Point Observatory, do
worse than this, which means that the best one can hope for in measuring
$\Delta v$ is no better than a few meters s-1—nowhere near the level of
precision required in Figure 3. Instead of discerning between an accelerating
and non-accelerating Universe in $\sim 20$ years, one would need to wait many
centuries to even begin seeing the cosmic expansion in ‘real’ time.

This situation is changing dramatically with the development of new
telescopes, however, such as the ELT, with an order of magnitude improvement
in light-gathering power (Fig. 4), advances in adaptive optics to mitigate
atmospheric distortions, and new technologies in quantum optics to facilitate
the imprinting of ultra-stable spectra. Such devices are expected to measure
individual redshifts with an order of magnitude improvement in the precision
available today. With the enhanced statistics from the many thousands of new,
faint quasars that will be detected along the way, their overall capability to
measure $\Delta v$ should reach $\sim 10$ cm s-1, right in the ball park of
Figure 3. Integrating over a baseline of one to two decades should therefore
yield a signal-to-noise ratio sufficient to disentangle the various profiles
shown on this plot—in less than a human lifetime.

Figure 4: Artist’s impression of the European Extremely Large Telescope, a
39-meter aperture optical and infrared facility under construction on Cerro
Armazones in the Chilean Atacama Desert. (Courtesy ESO/L. Calçada)

According to Liske et al. (2008), the High-Resolution Spectrograph (HIRES) on
the ELT is expected to observe a spectroscopic velocity shift, $\Delta v$,
with an uncertainty of

$\sigma_{\Delta v}=1.35\,{2370\over{\rm S/N}}\sqrt{{30\over N_{\rm
QSO}}}\left({5\over 1+z_{\rm QSO}}\right)^{\alpha}\;{\rm cm}\;{\rm s}^{-1}\;,$
(19)

where $\alpha=1.7$ for $z\leq 4$, and $\alpha=0.9$ for $z>4$, with ${\rm
S/N}\approx 1,500$ after 5 years of monitoring $N_{\rm QSO}=10$ quasars in
each of three redshift bins at $z=2.5$, $3.5$ and $5.0$. With a baseline of 20
years, these uncertainties (shown as red bars in Fig. 3) are approximately
$6$, $4$ and $3$ cm s-1, respectively. Thus, if the Universe is expanding at a
constant rate (blue long-dashed line), we should see a $\sim 3\sigma$
difference between a measurement of zero and the velocity shift predicted by
Planck-$\Lambda$CDM over a span of just two decades. The difference continues
to increase with integration time beyond this limit.

##  5 The Impact of a Measured Acceleration

The standard cosmological model we use to interpret the data today has grown
and evolved over several decades, though always based firmly on the FLRW
metric shown in Equation (1). No other attempt at solving Einstein’s field
equations in general relativity has come close to supplanting this metric as
the most likely description of the cosmic spacetime.

But look carefully again at the coefficients in this metric, notably the
significant impact of the expansion factor $a(t)$ on yielding changes in time
as a function of position. Whereas Einstein’s equations provide us with this
formal expression, they do not tell us much about $a(t)$ itself, unless we
introduce specific information about the source of gravity, in the form of a
stress-energy tensor. And there is no fundamental theory we can rely on to do
this unambiguously.

Much of the effort in observational cosmology since the 1960’s has thus been
directed at identifying the various constituents that make up the fabric of
the Universe—the stars, galaxies, and gas between them, the background
radiation and whatever else is suggested by each new generation of higher
precision instrumentation. As noted earlier, today we believe the ‘cosmic
fluid’ contains baryonic matter, an unidentified dark matter, radiation, and a
very mysterious ‘dark energy.’ But we cannot escape the fact that all of this
description, which is essential in determining the behaviour of $a(t)$, is
largely empirical. And this limitation continues to thwart our attempts at
producing a complete theoretical understanding of the origin and evolution of
the Universe.

We can see this via Equation (16), for example, which characterizes the Hubble
parameter—and thus the expansion rate of the Universe—in terms of several
unknown factors, such as the density ratios of the various constituents and
the equation-of-state of dark energy. Our cosmological measurements help us
optimize their values, but how do we make sense of $\Omega_{\rm de}$ being,
say, $0.69$ or $0.72$? How does this differentiation inform the underlying
theory? To this point, everything we know about the evolution of the Universe
is based on measurements of integrated distances using standard candles, or
the assumed size of large-scale structures, such as galaxy clusters, or the
modeled growth of temperature variations in the CMB. But We’re not even sure
that this empirical approach makes sense.

What we mean by this is that a completely unavoidable prediction of the
formulation in Equation (16) is that the Universe must undergo various phases
of accelerated and decelerated expansion. The reason for this is quite simple
to understand. The various constituents represented by $\Omega_{\rm m}$,
$\Omega_{\rm r}$ and $\Omega_{\rm de}$ evolve over time at different rates
from each other. For example, if dark energy were a cosmological constant, the
quantity $\Omega_{\rm de}(1+z)^{3(1+w_{\rm de})}=\Omega_{\rm de}$ would never
change, while both $\Omega_{\rm m}(1+z)^{3}$ and $\Omega_{\rm r}(z+z)^{4}$
decrease as the Universe ages (i.e., as $z$ decreases). And since their
respective contributions to the equation-of-state in the cosmic fluid must
therefore also evolve with time, the expansion rate of the Universe cannot
itself remain constant, which therefore also means that $dz/dt_{0}$ in
Equation (13) cannot be zero. But we have never confirmed this fundamental
property of the standard model.

How wonderful it would be to finally prove this feature with the actual
measurement of a redshift drift! At the very least, we would then be certain
that a formulation like that expressed in Equation (16) does indeed make
sense, and that our current inability to relate the measured values of the
unknown parameters to some overarching theory will eventually give way to a
much deeper understanding. In concert with such a confirmation, we could then
set about actually measuring quantities such as $\Omega_{\rm de}$ via
Equations (13), (15) and (16) with the ever improving redshift-drift signal
over time.

As thrilling as such an outcome would be, an argument can be made that the
result would be even more surprising—and very perplexing—if it turns out that
$dz/dt_{0}=0$. That would be completely incomprehensible in the context of
Equation (16). It would imply that such a formulation of the Hubble parameter
is, at best, merely an approximation, and that in reality $H(z)$ is given by
the much simpler expression in Equation (17), which has only one unknown
factor, i.e., the value of the Hubble constant today.

This possibility has been explored for over a decade now, largely focusing on
the mitigation of several growing tensions between the standard model and
other types of observation, as we shall describe shortly. It turns out that a
constant expansion rate may actually fit the data better than the variable one
predicted by Equation (16) (see, e.g., Melia 2020), so there are good reasons
to believe that a measurement of zero redshift drift is not entirely out of
the question.

This is what we have been highlighting throughout this paper, i.e., that an
important question to be answered by the redshift-drift measurements is not
just which of the curves in Figure 2 best represents the data, but simply
whether the drift is different from zero (Fig. 3). The answer to this question
is a straightforward yes or no. And zero redshift drift could alleviate
several otherwise insurmountable problems with the standard model.

To begin with, the interplay between $\Omega_{\rm m}$ and $\Omega_{\rm de}$ in
Equation (16) that we have been discussing produces several alternating phases
of deceleration and acceleration throughout cosmic history, but with the
highly unlikely coincidence that we still end up with $H_{0}=1/t_{0}$ today,
within the measurement errors (Melia 2020). To achieve this endpoint within
the context of Planck-$\Lambda$CDM, dark energy needs to be a cosmological
constant, with $w_{\rm de}=-1$ and $\Omega_{\rm de}=\Omega_{\Lambda}\approx
0.7$. As is well known, however, the required value of $\rho_{\rm
de}=\rho_{\Lambda}$ is strikingly inconsistent with the much larger zero-point
energy suggested by quantum field theory. Depending on how one handles all the
energy contributions, the disagreement can be as high as 120 orders of
magnitude, a situation sometimes described as ‘the largest discrepancy between
theory and experiment in all of science’ (Adler et al. 1995; Harvey &
Schucking 2000).

In a constantly expanding cosmos, on the other hand, dark energy could not be
a cosmological constant. It would be a dynamic entity, akin to other particles
and fields present in the cosmic fluid. There is even a hint that some of it
may have decayed in the early Universe to produce the baryonic matter we see
today (Melia 2020). Regardless, in the event that the measured redshift drift
turns out to be zero, dark energy’s equation-of-state (different from $p_{\rm
de}=-\rho_{\rm de}$) and its coupling to other known particles, would provide
significant justification for its inclusion in an extended model of particle
physics.

Dark energy impacts the cosmic dynamics towards the end of the Universe’s
timeline, as seen from our perspective today. But there is an equally
troubling and persistent issue plaguing the standard model in the early
Universe, typically referred to as the ‘horizon’ problem. In fact, there are
several such problems. The first is associated with the temperature ($T_{\rm
cmb}$) of the CMB, which appears to be uniform across the sky (Planck
Collaboration 2018), save for tiny fluctuations on the order of 1 part in
$10^{5}$. The standard model, with the Hubble parameter in Equation (15),
would have experienced very rapid deceleration prior to the last scattering
surface at $z_{\rm cmb}\sim 1080$, preventing the Universe from expanding
sufficiently by that time for us to see this level of uniformity in $T_{\rm
cmb}$ today.

A very brief period of inflation prior to $z_{\rm cmb}$ was proposed as a
means of providing the missing expansion (Guth 1981). But in the intervening
four decades, attempts at identifying the inflaton field responsible for this
effect still have not determined its potential, and we cannot claim to
understand exactly how inflation works—or whether it is even consistent with
all of the observations. A careful analysis of the latest Planck data release
(Melia & López-Corredoira 2018) shows that several-large angle anomalies
associated with the aforementioned temperature fluctuations are probably due
to a cutoff in the primordial power spectrum. But these anisotropies are
attributed to quantum fluctuations in the inflaton field, and such a cutoff
signals the time at which inflation could have started. Unfortunately, the
data suggest that this initiation would have occurred too far beyond the Big
Bang to have allowed inflation to simultaneously solve the temperature horizon
problem and produce the fluctuations observed in the CMB (Liu & Melia 2020).

A second horizon problem, emerging with the recent discovery (Aad et al. 2012)
of the Standard Model Higgs boson, may be even worse. According to standard
particle physics, the Universe may have undergone several phase transitions,
in addition to the inflationary event we have just described, presumably
driven by the separation of the strong and electroweak forces in grand unified
theories. Now that we know the Higgs particle exists, it is very likely that
the Universe underwent another (electroweak) phase transition at $T=159.5\pm
1.5$ GeV. In Planck-$\Lambda$CDM, this would have occurred $\sim 10^{-11}$
seconds after the Big Bang. During this event, the electric and weak forces
separated, and the fermions gained mass (Englert & Brout 1964; Higgs 1964).
The latter, however, depends on the vacuum expectation value of the Higgs
field. But this too appears to be uniform across the sky, since we do not see
any evidence of fermion masses varying from one location to another. And
regardless of whether or not inflation happened, the Universe would have
resumed its deceleration afterwards, creating another horizon problem at
$t\sim 10^{-11}$ seconds—this time with respect to the Higgs vacuum
expectation value. This problem has been known since Zel’dovic et al. (1975)
and Kibble (Kibble 1976) suggested the possibility that domain walls might
have been created as a result of such scale transitions across the Universe,
constituting topological defects with significant observational consequences
(Vilenkin & Shellard 1994; Lazanu et al. 2015; Sousa & Avelino 2015) in
$T_{\rm cmb}$. But no evidence of such artifacts has ever been found, nor has
any viable explanation been offered within the context of the standard model
to account for the uniformity of the Higgs field vacuum expectation value
across the Universe.

But all of these horizon problems will disappear should the search for
redshift drift produce a null result. These problems emerge only in
cosmological models, such as Planck-$\Lambda$CDM, that predict an early period
of decelerated expansion. But if the Universe has been expanding at a constant
rate, it would always have had sufficient time to expose proper distances
beyond the most distant sources we see today. And this feature is independent
of the redshift of these sources or structures, including the last-scattering
surface at $z_{\rm cmb}$, and the apparent horizon at the time of the
electroweak phase transition (Melia 2020).

Between these two limits—where dark energy and $H_{0}$ appear to be
problematic in the local Universe, and where cosmic deceleration would
necessarily have led to a series of horizon inconsistencies at high
redshifts—the standard model also suffers from a long-standing time-
compression problem associated with the premature formation of structure.
Supermassive black holes are seen (Banados et al. 2018) at redshifts exceeding
$\sim 7.5$ which, in Planck-$\Lambda$CDM corresponds to about 690 Myr after
the Big Bang, barely a few hundred Myr after the formation of Pop III and II
stars. Conventional astrophysics, however, would indicate that a
$10\;M_{\odot}$ seed, created by a Pop II/III supernova, should have taken at
least 820 Myr to grow via Eddington-limited accretion (Melia 2020). The
timeline thus appears to have been overly compressed by a factor $\sim 2$ in
the standard model, at least beyond $z\sim 6$. Attempts at addressing this
issue invariably invoke exotic ‘fixes,’ such as imposing anomalously high
accretion rates (Volonteri & Riess 2005; Inayoshi et al. 2016), or the
creation of very massive ($\sim 10^{5}\;M_{\odot}$) seeds (Alexander et al.
2014). But no direct evidence of such unconventional processes and events has
ever been seen. A similar problem has also been observed via the premature
formation of galaxies, some of which appeared beyond $z\sim 10-12$, much
earlier than expected within the standard model (Steinhardt et al. 2016).

Contrasting with these inconsistencies, the formation of high-redshift quasars
and galaxies fits extremely well within the timeline associated with an FLRW
Universe expanding at a constant rate. In this case, a redshift of $\sim
15$—the start of the so-called Epoch of Reionization, when Pop III and II
stars began replenishing the interstellar medium with ionizing radiation—
corresponds to an age of $\sim 878$ Myr, as opposed to $\sim 400$ Myr in the
standard model. Most importantly, a $10\;M_{\odot}$ seed created at that time
would have easily grown into a $\sim 10^{9}\;M_{\odot}$ black hole at $z=7.5$
(i.e., $t\sim 1.65$ Gyr) via conventional Eddington-limited accretion (Melia
2020). A measurement of zero redshift drift could thus also cure the time-
compression problem in Planck-$\Lambda$CDM.

These principal illustrations already highlight what is at stake with a
precise measurement of the cosmic acceleration, but the derived benefits
extend beyond this brief survey. For example, if it turns out that the
expansion rate is constant, then (i) the concept of cosmic entropy would be
simplified, eliminating the so-called initial entropy problem, in which one
must scramble to explain how and why the entropy in the standard model began
with an anomalously low value and increased rapidly by the time the CMB was
produced (Melia 2021a); (ii) it would also help to explain how quantum
fluctuations in an incipient scalar field could have classicalized and turned
into density fluctuations growing under their own self-gravity (Melia 2021b);
and (iii) it would avoid the so-called trans-Planckian anomaly (Martin &
Brandenberger 2001; Brandenberger & Martin 2013), in which quantum
fluctuations would otherwise have mysteriously emerged out of the Planck
domain into the semi-classical Universe without an explanation from quantum
mechanics and without a viable theory of quantum gravity.

##  6 Conclusion

Planck-$\Lambda$CDM has been quite successful in accounting for a broad range
of cosmological observations, but careful scrutiny reveals several major
fundamental problems with its theoretical foundation. Some of these issues
have been with us for over half a century. Moreover, as the precision of our
instruments continues to improve, the tension between theory and experiment
gets bigger, not smaller. There are therefore many good reasons for continuing
to search for a more reliable (and complete) cosmological model.

This could simply be facilitated by the actual measurement of a redshift
drift, which over time would help us identify the values of the unknown
parameters in Equation (16), and thereby eventually helping us identify the
theory underlying their existence.

It is also possible that the upcoming measurement of redshift drift may
indicate to us that the expansion of the Universe is not accelerating after
all. If the measured redshift drift turns out to be non-zero, no matter the
value, the proposed solutions described in § 5 above become moot. But if the
outcome is unambiguously zero, the major problems and inconsistencies plaguing
the standard model simply go away. There are few instances in science when the
anticipated impact of an experiment carries this much weight.

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