Lifshitz cosmology: quantum vacuum and Hubble tension
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MNRAS 507, 3473–3485 (2021) https://doi.org/10.1093/mnras/stab2345
Advance Access publication 2021 August 6
Lifshitz cosmology: quantum vacuum and Hubble tension
‹
Dror Berechya and Ulf Leonhardt‹
Weizmann Institute of Science, Rehovot 7610001, Israel
Accepted 2021 August 9. Received 2021 June 15; in original form 2020 November 24
ABSTRACT
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Dark energy is one of the greatest scientific mysteries of today. The idea that dark energy originates from quantum vacuum
fluctuations has circulated since the late ’60s, but theoretical estimations of vacuum energy have disagreed with the measured
value by many orders of magnitude, until recently. Lifshitz theory applied to cosmology has produced the correct order of
magnitude for dark energy. Furthermore, the theory is based on well-established and experimentally well-tested grounds in
atomic, molecular and optical physics. In this paper, we confront Lifshitz cosmology with astronomical data. We find that the
dark–energy dynamics predicted by the theory is able to resolve the Hubble tension, the discrepancy between the observed
and predicted Hubble constant within the standard cosmological model. The theory is consistent with supernovae data, Baryon
Acoustic Oscillations and the Cosmic Microwave Background. Our findings indicate that Lifshitz cosmology is a serious
candidate for explaining dark energy.
Key words: dark energy.
(Plebanski 1960; Leonhardt 2010) which is also the foundation of the
1 I N T RO D U C T I O N
well-developed field of transformation optics (Service & Cho 2010).
The cosmological standard model, the Cold Dark Matter (CDM) A homogeneous and isotropic, expanding universe with scale factor
model, has been spectacularly successful. With a few basic principles, a(t) is perceived by the electromagnetic field as a medium with a
it explains a vast range of phenomena over an enormous range of homogeneous and isotropic but evolving refractive index n(t) ∝ a(t).
time scales. With only six free parameters, it fits the complex and Then, calculating the vacuum energy in the universe should be done
detailed fluctuation spectra of the cosmic microwave background as if it were a dielectric medium with an evolving refractive index in
(CMB). Nevertheless, the CDM model lacks an explanation of the what is known as Lifshitz theory (Lifshitz 1954; Landau, Lifshitz &
underlying nature of three of its pillars, known as the dark sector – Pitaevskii 1980). Applied to cosmology, the Lifshitz vacuum energy
inflation, dark matter, and dark energy. turns out to have the same order of magnitude as the measured
In recent years, the cosmology community has been actively cosmological constant (Leonhardt 2019).
looking for cracks in the CDM model in the form of tensions In this paper, we compare the predictions of Lifshitz theory with
between several independent phenomena (Verde, Treu & Riess astronomical data. We also formulate the theory such that it can be
2019). Presently, the most severe such tension is known as the Hubble taken up by astronomers. Lifshitz theory in cosmology has not been
tension: the discrepancy between the Hubble constant (the present– designed to alleviate the Hubble tension, but we show that the most
day expansion rate) inferred from early–universe phenomena and naive choice of its coupling parameter fits the SH0ES value (Riess
the value obtained by local probes of cosmic expansion (Verde et al. et al. 2021) with perfect precision. We also find that the theory is
2019; Riess 2020). Not everyone agrees that these tensions are real consistent with the Pantheon type Ia supernova (SN Ia) data at the
(Efstathiou 2020) but by revealing cracks in the CDM model they same level or slightly better than the CDM model, that it agrees with
may shed light on the dark sector. the measured baryon acoustic oscillations (BAO) and does not lead
There have been numerous attempts to explain the Hubble tension to deviations from the measured CMB spectra within the accuracy of
(Di Valentino et al. 2021). Without exception, they either require the cosmic parameters. There are still many opportunities for further
significant changes to general relativity, the cosmological principle, analysis, but the findings reported here already show that Lifshitz
or modifications to the standard model of particle physics that have cosmology is a serious contender for a realistic explanation of dark
not been experimentally tested elsewhere. energy, rooted in established physics.
Here enters the Lifshitz theory in cosmology (Leonhardt 2019).
This theory is based on solid foundations in atomic, molecular, and
2 L I F S H I T Z T H E O RY I N C O S M O L O G Y
optical (AMO) physics that have been experimentally tested with
percent-level precision (Decca 2014). The connection to cosmology
2.1 Background
is the analogy between curved space-times and dielectric media
Most of our universe is empty space. Yet, this ‘emptiness’ is far from
being ‘nothingness.’ According to the modern view of quantum field
E-mail: dror.berechya@weizmann.ac.il (DB); ulf.leonhardt@ theory (QFT), the universe is filled with quantum fields in at least their
weizmann.ac.il (UL) ground state – also known as the vacuum state. Since the early days
C The Author(s) 2021.
Published by Oxford University Press on behalf of Royal Astronomical Society. This is an Open Access article distributed under the terms of the Creative
Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium,
provided the original work is properly cited.3474 D. Berechya and U. Leonhardt
of QFT, it is known that the vacuum state of a quantum field contains The theory that agrees with modern measurements (Decca 2014) of
non-vanishing energy density, and due to Casimir in the late ’40s, Casimir forces is the Lifshitz theory (Lifshitz 1954; Dzyaloshinskii
we know that this energy density may even exert measurable forces et al. 1961; Landau et al. 1980; Rodriguez et al. 2011; Scheel
(Casimir 1948; Casimir & Polder 1948). The physics of the quantum 2014). Due to the analogy mentioned above between space-times
vacuum has been well tested (Munday, Capasso & Parsegian 2009; and media, Lifshitz theory can be applied to cosmology; in that
Rodriguez, Capasso & Johnson 2011; Decca 2014; Zhao et al. 2019) case, the electromagnetic field and its fluctuations perceive the
and explains a vast set of phenomena, from the adhesion of geckos (spatially-flat, homogeneous, and isotropic) expanding universe as
to walls (Autumn & Gravish 2008) to the limit trees can grow (Koch a spatially-uniform but time-dependent medium with a refractive
et al. 2004). index that is proportional to the scale factor a (Leonhardt 2019,
So, the state of affairs is as follows. We know the universe is filled 2020). Admittedly, when applying Lifshitz theory to that specific
with quantum fields at their ground state, we know that this ground kind of medium, we extrapolate the theory outside its well-tested
state exhibits non-vanishing energy density and may exert forces, zone and introduce some new ideas. Nevertheless, the application of
and finally, we know that the universe is also filled with a mysterious Lifshitz theory to the expanding universe was shown to produce the
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energy density we call dark energy. It is therefore tempting to correct order of magnitude for the dark energy density (Leonhardt
combine the physics of the quantum vacuum and dark energy. 2019).
Zel’dovich was the first to suggest, in 1968, that the cosmolog- In time-dependent media, the vacuum energy turns out to be time-
ical constant comes from the physics of the quantum vacuum dependent and responding to the evolution of the refractive index, or
(Zel’dovich 1968). By calculating the bare energy density of the in the case of cosmology – to the evolution of the universe. Then,
vacuum, with a cut-off at the Planck scale where presumably GR by the Friedmann equations, the universe is reacting to the vacuum
breaks, one gets the correct structure of the cosmological constant. energy. In the following, we present the resulting self–consistent
So, have we found an explanation of dark energy? Not quite yet. dynamics.
The problem is that the quantitative prediction of the vacuum energy
density is off by about 120 order of magnitude (Weinberg 1989).
2.2 Equations of motion
Furthermore, if the theory is made to agree with the observed value
of the vacuum energy density by choosing a sufficiently low cut-off In the framework of the flat–CDM model, the background (homo-
for the vacuum fluctuations, it severely disagrees with measurements geneous and isotropic) universe evolves by the Friedmann equations,
of vacuum forces (Mahajan, Sarkar & Padmanabhan 2006). which can be summarized into one equation as
This situation does not seem very encouraging. However, the
case for a Casimir cosmology is not closed yet (Leonhardt 2019, H 2 (a) = H02 (r a −4 + m a −3 + ) (CDM) (1)
2020); the idea that dark energy stems from vacuum fluctuations where H(a) is the Hubble parameter, H0 the Hubble constant (Hubble
(Sakharov 1967; Zel’dovich 1968; Weinberg 1989) may still be parameter at the present–day), a is the scale factor, and x with x =
valid. The one encouraging insight is that curved space-times are r, m, are the density parameters for radiation, matter, and the
the same as dielectric media in the eyes of the electromagnetic cosmological constant .
field: Maxwell’s equations in curved space-time are equivalent to Let us now see how this equation changes for the Lifshitz theory in
Maxwell’s equations in dielectric media (Plebanski 1960; Leonhardt cosmology (hereafter, ‘Lifshitz cosmology,’ LC). For a given cosmic
2010), and our spatially flat, expanding universe is just another curved expansion, i.e. for a given a(t), Lifshitz theory predicts for a medium
space-time. It would be a far more unreasonable assumption that the with n(t)∝a(t) the energy-momentum tensor of the quantum vacuum
universe is one particular space-time with different rules or that in that medium (Leonhardt 2019), in our case, in the universe.
vacuum physics is different in the lab and the universe. Therefore, In turn, the vacuum energy and stress react back on the cosmic
we assume that we can calculate vacuum energy in the universe as if evolution through the Friedmann equation, influencing a(t). This
it were the corresponding dielectric medium. mutual interaction between the vacuum energy and the background
Now, since Zel’dovich, substantial progress has been made in universe results in self-consistent dynamics (Leonhardt 2019), which
understanding the quantum vacuum forces such that formal ar- we express here as1
guments can be replaced by empirically tested theory (Rodriguez 2
et al. 2011; Scheel 2014; Simpson & Leonhardt 2015). Without H (a) = H02 (r a −4 + m a −3 + LC )
(2)
exception, the empirical evidence for forces of the quantum vacuum ˙ LC = 8α H ∂t3 H −1
H02 (Lifshitz cosmology)
and the comparison with theory comes from AMO physics. There
where LC is the density parameter for dark energy in the Lifshitz
the quantum vacuum produces attractive or repulsive forces (Munday
cosmology. α is a dimensionless coupling parameter that depends
et al. 2009; Zhao et al. 2019) between dielectric objects and inside
on the cut–off, assumed near the Planck scale, and on the possible
inhomogeneous media. For example, in the Casimir effect (Casimir
contributions of other fields in the standard model of particle physics
1948), vacuum fluctuations cause two dielectric plates to attract each
(Leonhardt 2019). As these influences are not known within the
other. Here the spatial variation of the refractive index from free
present theory, α is a free parameter that must be fitted against
space to the material of the plates generates a vacuum force on the
observations. Taking only the electromagnetic field into account
surface of each plate. This effect is a general phenomenon: variations
and assuming a sharp cut-off at exactly the Planck scale, we get
of the refractive index create variations in the electromagnetic energy
(Leonhardt 2019) α TH
= (9π )−1 (the ‘TH’ superscript indicates that
density and stress σ in media (Lifshitz 1954; Dzyaloshinskii, Lifshitz
this value is a theoretical prediction under the above-mentioned
& Pitaevskii 1961; Landau et al. 1980; Scheel 2014), which gives
the force density ∇ · σ . This fact means that Casimir forces do not
only act between dielectric bodies such as mirrors but also inside 1 The dynamics that would result from the original calculations in Leonhardt
inhomogeneous bodies. Inhomogeneous dielectric media do exert (2019) are somewhat different from the dynamics we bring here; the reason
local vacuum forces (Landau, Lifshitz & Pitaevskii 1995; Griniasty for this difference is a different definition of the vacuum state. See Appendix A
& Leonhardt 2017). for more details.
MNRAS 507, 3473–3485 (2021)Lifshitz cosmology 3475
conditions). A dot above a character denotes differentiation with in equations 2). Thus, we get
respect to cosmological time t.
H 2 = H02 (m a −3 + LC ),
The second equation in equations (2) describes the response of the
vacuum energy to changes of the scale factor a (or, in the language m −3 3
LC = ∞ 1 + 18 α ln ( a + 1) −
of the Lifshitz theory, the refractive index). This equation hides an ∞ ∞ 3
a +1
m
integration constant, which remains a free parameter that must also
(for aeq a). (3)
be fitted against observations. Thus, Lifshitz cosmology replaces one
of the CDM parameters, namely , with two new parameters: The next step is to fit the theory’s parameters with cosmological
α and the integration constant (giving us a total of only seven data sets, such as CMB power spectra, SN Ia, and BAO. The complete
parameters). way of fitting the parameters is to include the modified equation
for the background dynamics, i.e. the new equation for H(a), in
the relevant computer codes and perform statistical analysis (such
3 A P P ROX I M AT E S O L U T I O N as likelihood–based MCMC or Fisher information). In this paper,
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however, we aim to explore the Lifshitz dynamics for the first
In Section 2.1, we have presented the ideas behind the description of time to test whether this theory can plausibly resolve the Hubble
dark energy as the vacuum energy produced in a time-dependent tension at all, which would then justify further research. For this,
dielectric medium using Lifshitz theory (for further details, see we take the CDM’s value for the sound horizon, r∗ , as we assume
Leonhardt (2019) and Leonhardt (2020)). In Section 2.2, we saw that the deviations of Lifshitz cosmology from the CDM model
that this theory also gives a testable prediction: a modified expansion are negligible in the early universe. The following section shows
history, embodied in equations (2). In this section, we will find that the resulting dynamics are consistent with BAO and SN Ia
an approximate solution for the dynamics predicted by Lifshitz measurements. To preserve the acoustic angular scale of the CMB
cosmology. Later, we will use this approximate solution to analyse fluctuations (θ ∗ ≡ r∗ / DM ), we must demand that the angular diameter
the dynamics and demonstrate the theory’s plausibility. Here we distance to the surface of last–scattering, DM , is unchanged as well:
present the approximate solution along general lines; for further z∗
details, see Appendix B. dz (CDM)
DM = c = DM (4)
The contribution of the cosmological constant in the CDM model 0 H (z )
is negligible at last-scattering as / m (1 + z∗ )3 ≈ 1.7 · 10−9 with (CDM)
where DM is calculated with the CDM model. In effect, this
values provided by Planck Collaboration (2020), and it is even
demand gives us a relationship between α and the combination
smaller before that time. We assume that in Lifshitz cosmology
H02 ∞ for the following reason. Since H02 m is proportional to
the vacuum contribution is negligible before last–scattering as well
the physical matter density, it should be a model–independent
and verify this later. The right-hand side of the second equation
quantity; therefore, we may use CDM’s value for this combination.
in equations (2) is zero for linear H−1 ; this means that LC is
The Planck collaboration2 determined ωm P
≡ [m h2 ]P = 0.1430 ±
constant in both radiation- and matter-domination eras where H−1 is
0.0011 (Planck Collaboration 2020) via the relative heights of the
linear. Lifshitz cosmology may intervene only in the transition period
CMB acoustic peaks (approximately) model-independently (Planck
around aeq (as we will see in detail in Section 5). Hence, if we start
Collaboration et al. 2014). Here and throughout this paper h ≡ H0 /
with a negligible vacuum contribution during radiation domination,
100[km s−1 Mpc−1 ] and the ‘P’ superscript, hereafter, denotes that the
then the vacuum contribution will remain negligible at the beginning
value is determined by Planck. Fig. 1 shows the resulting relationship.
of matter-domination if the effects of Lifshitz cosmology around aeq
The ∞ h2 errors presented in this figure are estimates based solely
are small. Here, we assume that this is the case; in Section 5 we check (CDM)
on propagating the CDM errors in determining DM ; that is,
the validity of this assumption (see Fig. 6). Thus, we adopt CDM’s
for any other CDM’s quantity, we take the mean value given by
dynamics at the early universe and focus our attention on the late
Planck’s TT,TE,EE+LowE+lensing analysis (Planck Collaboration
universe. Hence we drop the radiation term in our calculations (as
et al. 2020) without errors (see Appendix B1 for more details).
it turns out, see Appendix B, this is a crucial simplification for our
Thus, we are left only with α as a free parameter. The value of
calculations).
α will determine H(a), and hence, will fix the value of H0 , Fig. 2,
Even without the radiation term, finding a closed analytical
as well as the values of the other parameters in Table 1.
solution for equation (2) remains a real challenge. Moreover, finding
Fig. 2 shows that whatever the actual value of H0 may be, Lifshitz
a numerical solution is no less challenging, mainly for the following
cosmology may reproduce it (at least nominally, see the discussion
two reasons. First, equations (2) are ‘stiff equations,’ causing havoc
˙ LC in Section 4.1). The theoretical prediction under the assumptions of
with step size and accuracy, and second, the equation for
only electromagnetic contributions and a sharp cut-off at exactly the
depends on high derivatives of a (up to fourth–order), which is
Planck length is more or less at the middle of the local measurements,
problematic since the highest derivatives take the lead in differential
and remarkably, it is right on the latest measurement by the SH0ES
equation solvers. In reality, the dynamics of LC are a mere correction
team (Riess et al. 2021).
to the dynamics of the universe.
To study the influence of different values of α and hence of
Therefore, we solve for the dynamics after last–scattering pertur-
different sets of parameters, we will explore the resulting dynamics
batively. α is presumably small (recall that the theoretical prediction
of two representative realizations of Lifshitz cosmology. The first
is αTH
= (9π )−1 ≈ 0.035), so we calculate LC up to first–order
one, which we call ‘M1,’ is the theoretical prediction, for which we
in α . We plug the zeroth–order Hubble parameter (the CDM’s
have α M1
= αTH
= (9π )−1 . For the second realization, which we
Hubble parameter, equation (1) without the radiation term) into
the equation for ˙ LC (the second equation in equations 2), and we
integrate it (analytically, see Appendix B) with ∞ ≡ lim LC as 2 Throughout
a→∞ the paper, we will use italic letters to designate the Planck
the integration constant. In this way, we get the first–order correction collaboration and distinguish it from Planck the person or other contexts in
to LC , that we substitute into the equation for H(a) (the first equation which this name might appear.
MNRAS 507, 3473–3485 (2021)3476 D. Berechya and U. Leonhardt
call ‘M2,’ we take α M2
= 0.0225. Here we choose two values for
α as examples, and then ∞ h2 is determined by equation (4). The
obtained parameter values for the two realizations M1 and M2 are
presented in Table 1. We regard these two realizations as examples
for the viability of the theory.
4 L OW R E D S H I F T P RO B E S
At this point, we have in our hands a theory explaining the physical
origin of dark energy, one that stems from well-known physics, an
approximate solution for the theory’s dynamics assuming unmodified
early evolution, and two sets of parameters (in Table 1) for two real-
izations of the theory: M1 (α M1
= (9π )−1 , a theoretical prediction)
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and M2 (α = 0.0225).
M2
Now, we are ready to compare the resulting dynamics with low
Figure 1. ∞ h2 ≡ lim LC h2 as a function of α in the range of interest. redshift probes of cosmic expansion, viz. SNe Ia and BAOs. We
a→∞
This relationship results when demanding that Lifshitz cosmology preserves shall see that M1 fits better the SN data with SH0ES calibration of
DM (equation 4), and it is calculated according to equation (B17) with ωm P =
the absolute magnitude MB , which might be crucial regarding the
0.1430 and ω P = 0.3107 from Planck Collaboration et al. (2020). The bands
Hubble tension (Efstathiou 2021; Benevento, Hu & Raveri 2020);
show the ±1σ (dark grey) and ±2σ (light grey) errors in ∞ h2 as estimated
(CDM) however, M1’s fit to BAO data is somewhat lesser than CDM’s. On
by propagating the errors in DM (see Appendix B1). The two realizations
the other hand, M2 fits BAO data better than M1 and seemingly falls
of Lifshitz cosmology considered in this paper are also shown: M1 (black and
yellow point), the theoretical prediction for the electromagnetic contribution from CDM’s fit only by a small margin; yet, M2 fits SN data with
M1 = α T H = (9π )−1
alone with a sharp cut–off at exactly the Planck length, α a lower value (more negative) of MB and can only relieve the tension
and M2 (black point), α M2 = 0.0225. (see the discussion in the following subsection).
We will compare the resulted dynamics of M1 and M2 with
observational findings, refraining from a more complex statistical
analysis for the time being. Our analysis already indicates the
viability of Lifshitz cosmology. However, only a complete statistical
analysis will determine the actual set of values for the theory’s
parameters, instead of M1 and M2, which are demonstrations
obtained by choosing α and imposing equation (4), and will enable
us to decide which is the better theory. This further analysis poses an
opportunity for future research.
4.1 The Hubble diagram and distance ladders
We start with SNe Ia observations. Ultimately, each SN Ia measures
the luminosity distance via the relation
DL (z)
μ ≡ mB − MB + δμ = 5 log10 + 25, (5)
Mpc
where
z
dz
DL (z) = c (1 + z) , (6)
0 H (z )
Figure 2. H0 in units of km s−1 Mpc−1 as a function of α in the range of μ is the distance modulus, MB is the absolute magnitude (in the B–
interest. The bands show the ±1σ (dark grey) and ±2σ (light grey) errors in band), mB is the apparent magnitude, and δμ summarizes corrections
H0 obtained by propagating the errors in ∞ h2 . The theory points M1 and due to effects such as colour, light-curve’s shape, and host-galaxy
M2 are as in Fig. 1. We also show several local measurements of H0 , done mass; these effects can be either measured or fitted using SN Ia data
by several independent groups using several independent methods: Cepheids alone, independently of cosmology (Riess et al. 2018a). Roughly
- SN Ia [73.2 ± 1.3 by Riess et al. (2021), SH0ES team], TRGB - SN
speaking, in each measurement, we measure mB and z, and we wish to
Ia [72.1 ± 2.0 by Soltis, Casertano & Riess (2021) and 69.6 ± 1.88 by
infer DL (z). MB is thus a nuisance parameter that must be determined
Freedman et al. (2020)], Tully Fisher [76.00 ± 2.55 by Kourkchi et al.
(2020)], Surface Brightness Fluctuations (SBF) [73.3 ± 2.5 by Blakeslee or marginalized over. This nuisance parameter is degenerate with
et al. (2021) and 70.5 ± 4.1 by Khetan et al. (2021)], SN II [75.8+5.2 −4.9 by
H0 in the SN Ia data: as a prefactor in H(z), H0 would shift MB by
de Jaeger et al. (2020)], and Time-delay Lensing [74.5+5.6 and 67.4 +4.2 5log10 H0 in equation (5); thus, both MB and H0 get swallowed into
−6.1 −3.2 by
−1 −1
Birrer et al. (2020)]. All the values are in units of km s Mpc and quoted the intercept of the magnitude–redshift relation.
from the compilation in Di Valentino (2021). As can be seen, whatever the One way to break that degeneracy is to use a distance ladder to
actual value of H0 is, Lifshitz cosmology may reproduce the correct value (at infer MB by calibrating SN Ia. Generally, there are two approaches
least nominally, see the discussion in Section 4.1). The theoretical prediction to measuring H0 using distance ladders. One is to use geometrical
(M1) is more or less at the middle of the local measurements, and remarkably, measurements to anchor local probes of distance (first rung), such
it is right on the latest measurement by the SH0ES team (red point). as Cepheids (e.g. SH0ES team, Riess et al. (2021)) or TRGB (e.g.
Soltis et al. 2021 and Freedman et al. (2020)), then use these probes
MNRAS 507, 3473–3485 (2021)Lifshitz cosmology 3477
Table 1. Two realizations of Lifshitz cosmology. For each realization, we choose α . Then ∞ h2 is fixed by equation (4) (taking
P = 0.1430 and ωP = 0.3107 (Planck Collaboration 2020), see Appendix B for details of the calculations), and the rest of the
ωm
parameters of Lifshitz cosmology follow. The errors in the parameters result from propagating the errors in ∞ h2 that are estimated
(CDM)
by propagating the errors in DM while calculating ∞ h2 .
km s−1
α H0 Mpc ∞ h2 LC (z = 0) h2 m h2 LC (z = 0) m
M1 (9π )−1 73.2 ± 0.8 0.527 ± 0.013 0.393 ± 0.012 0.143 ± 0.017 0.733 ± 0.028 0.267 ± 0.028
M2 0.0225 69.9 ± 0.7 0.426 ± 0.011 0.345 ± 0.010 0.143 ± 0.015 0.707 ± 0.026 0.293 ± 0.026
and go farther to calibrate the absolute magnitude of SN Ia in the measure, for each model, we calculate the root–mean–square devia-
same host galaxies (second rung), and finally, go farther still and tion (RMSD) given by
use the calibrated absolute magnitudes to infer H0 from SNe Ia in
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N
i=1 (μ(zi ) − μi )
the Hubble flow (third rung). This approach is known as the local 2
RMSD = (8)
distance ladder. The other approach starts instead with BAO standard N
rulers (assuming a value for rd , see Section 4.3, which makes this
where zi and μi are, respectively, the measured redshift and distance
approach model–dependent) to calibrate the absolute magnitude of
modulus (with a given MB ) of measurement i, μ(zi ) is the theoretical
far–away supernovae; then, it uses the calibrated absolute magnitudes
prediction, and N is the number of data points (for the Pantheon
to infer H0 from lower redshift SNe Ia. This approach is known as
sample, N = 1, 048). For each model, we have found MB that gives the
the inverse distance ladder (Efstathiou 2021).
lowest RMSD using the (unbinned) Pantheon data. For comparison,
The SH0ES team takes the approach of the local distance ladder
we also calculated the RMSD with the reproduced SH0ES absolute
and uses Cepheids to calibrate the absolute magnitude of SN Ia. To
magnitude (MB = −19.244) (Efstathiou 2021) and with absolute
reproduce SH0ES MB , Efstathiou (2021) combined the geometrical
magnitude obtained by fitting a late dark energy model with Planck,
distance estimates of the maser galaxy NGC 4258 (Reid, Pesce &
BAO, and Pantheon data (retaining MB in the Pantheon likelihood
Riess 2019), detached eclipsing binaries in the Large Magellanic
without the SH0ES constraint), MB = −19.415 (Benevento et al.
Cloud (Pietrzyński et al. 2019), and parallax measurements for 20
2020). The results are presented in Table 2. The bottom panel of Fig. 3
Milky Way Cepheids (Benedict et al. 2007; Van Leeuwen et al. 2007;
shows μ for the binned Pantheon data with the best MB of M1 and
Riess et al. 2018b), the SH0ES Cepheid photometry and Pantheon
M2, together with the theoretical curves. We find MB∗ = −19.421
SN peak magnitudes, then Efstathiou (2021) finds (equation 6 there)
for CDM, MB∗ = −19.330 for M1, and MB∗ = −19.388 for M2
MB = −19.244 ± 0.042 mag. (7) (the ‘∗’ superscript indicates a value corresponding to the lowest
RMSD). The three RMSD values that correspond to the three MB∗ ’s
We have adopted this value and used it with the Pantheon data set are comparable to one another, so it seems that the three models fit
(which is given by Scolnic et al. (2018) and publicly available in the unbinned Pantheon data comparably well.
doi: 10.17909/T95Q4X) to extract the μ’s of observed SNe. We These results show that while M2 can only mitigate the MB tension
also calculated μ (equation 5) for CDM and the two Lifshitz by about 19 per cent at mean value, as MB∗ ≡ MB(SH0ES) − MB∗ =
cosmologies (M1 and M2). Fig. 3 shows μ ≡ μ − μCDM for 0.144, M1 can relieve it considerably by about 51 per cent at mean
the theories and the observed data points with MB = −19.244 (top value, as MB∗ = 0.086 (for CDM, one gets MB∗ = 0.177). On
panel). This figure also shows binned data from Scolnic et al. (2018). the other hand, M2 fits the shape of the binned Pantheon data
It has been noted (Efstathiou 2021; Benevento et al. 2020) that, exceedingly well, as shown in Fig. 3, while M1’s fit to the shape
in principle, SH0ES does not measures H0 directly but measures MB is only moderate. The shape of the binned data might depend on
instead; H0 is inferred from the low redshift (z < 0.15 (Efstathiou the model, e.g. via model dependence of the redshift weights of the
2021; Benevento et al. 2020)) SNe in the Pantheon sample with the surveys (Benevento et al. 2020); in addition, the RMSD(MB ) profile
measured MB . According to this view, the Hubble tension is really (for the unbinned data) is shallow around the minimum, so the two
an MB tension: a significant gap of about MB ≈ 0.2 between the Lifshitz cosmologies might do even better (this seem to be especially
SH0ES MB and the one inferred from the Pantheon data without true for M1). Only a rigorous statistical analysis would be able to
including the SH0ES constraint (retaining MB in the likelihood) tell. Nevertheless, based on our current analysis, we conclude that the
(Benevento et al. 2020) or the one obtained by inverse distance two Lifshitz cosmologies fit the Pantheon sample as well as CDM
ladder (Efstathiou 2021). For theories that modify the dynamics does, and they both reduce the MB tension (M1 might even, hopefully,
above z ≈ 0.15, these two viewpoints should be equivalent; however, resolve this tension). As we will see later, while both cosmologies
for theories that modify the dynamics below that redshift, only (M1 and M2) seem to fit the BAO data comparably to CDM, M2
the latter viewpoint (MB tension) should be considered, as in this is a better fit there.
case, H0 is not constrained by the Pantheon data (see figure 1 in
Benevento et al. (2020)) and SH0ES analysis would be oblivious to
4.2 Distance-ladder-independent analysis
this modification (Efstathiou 2021; Benevento et al. 2020). That is,
if our universe would evolve according to a theory that modifies the Before we turn to BAO data, let us compare our dynamics with
dynamics below z ≈ 0.15, it will not appear in the SH0ES analysis, measurements of E(z) ≡ H(z) / H0 that are independent of any
and they would approximately measure the same CDM value for distance ladder, going around the MB dilemma. The quantity E(z) is
H0 as inferred from the CMB. independent of H0 and thus avoids the H0 –MB degeneracy; therefore,
Even though Lifshitz cosmology starts to modify the dynamics it can be measured using SN Ia data alone (Riess et al. 2018a).
at z > 0.15 (see Fig. 5), we would like to estimate how the theory To extract E(z) from the SN Ia data, Riess et al. (2018a)
will perform regarding the MB tension. To give some quantitative parametrize it by its value at several specific redshifts and interpolate
MNRAS 507, 3473–3485 (2021)3478 D. Berechya and U. Leonhardt
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Figure 3. μ ≡ μ − μCDM as a function of z. Top panel: Pantheon data (Scolnic et al. 2018) (yellow (unbinned data) and blue (binned data) points) with the
reproduced SH0ES absolute magnitude MB = −19.244 from Efstathiou (2021). The theoretical predictions are also shown: unbroken black cure represents M1,
and dashed black curve represents M2. The grey band around each curve shows the ±1σ errors in μ obtained by propagating the errors in ∞ h2 . Bottom
panel: The binned Pantheon data are shown with MB = −19.330 (Green) and with MB = −19.388 (Purple), the best MB in terms of RMSD (equation 8) for M1
and M2, respectively.
Table 2. Root–mean–square deviation (RMSD)
calculated with equation (8) for the (unbinned)
Pantheon data (Scolnic et al. 2018). The top block
shows the best (in terms of RMSD) MB and the
corresponding RMSD for each model. The bottom
block shows results with the reproduced SH0ES
absolute magnitude (MB = −19.244) (Efstathiou
2021) and the absolute magnitude obtained by
fitting a late dark energy model with Planck, BAO,
and Pantheon data (MB = −19.415) (Benevento
et al. 2020).
MB CDM M1 M2
−19.421 0.1449 – –
−19.330 – 0.1511 –
−19.388 – – 0.1453 Figure 4. [E −1 ] ≡ E −1 − ECDM −1
as a function of z. The six grey points
are model–independent measurements of E−1 performed by Riess et al.
−19.244 0.2291 0.1736 0.2046
(2018a) based on SN Ia data alone. The theoretical predictions are also shown:
−19.415 0.1450 0.1736 0.1477
unbroken black cure represents M1, and dashed black curve represents M2.
The grey band around each curve shows the ±1σ errors in [E−1 ] obtained
by propagating the errors in ∞ h2 . Among the six data points, three (at z =
to define the complete E(z) function, which can then be used to 0.07, 0.35, and 0.9) are situated closer to M1’s curve, and the remaining three
compute the luminosity distance and compare to the data while (at z = 0.2, 0.55, and 1.5) are situated closer to the CDM baseline. M2
fully marginalizing over the absolute magnitude. This way, they lies in between, and it seems to agree with all data points. All in all, Lifshitz
constrained the value of E−1 (z) at six different redshifts model– cosmology appears to fit the data comparably to CDM.
independently (except for assuming a spatially–flat universe). Fig. 4
−1
shows [E −1 ] ≡ E −1 − ECDM for these six data points together
with the two Lifshitz cosmologies. It can be seen that among the six
MNRAS 507, 3473–3485 (2021)Lifshitz cosmology 3479
data points, three (at z = 0.07, 0.35, and 0.9) are situated closer to
M1’s curve, and the remaining three (at z = 0.2, 0.55, and 1.5) are
closer to the CDM’s baseline. M2 lies in between, and it seems
to agree with all data points. All in all, we conclude that Lifshitz
cosmology does fit the E−1 data to a degree comparable to CDM.
4.3 BAO measurements
Now we turn to BAOs, the second low redshift probe we consider
in this paper. BAO measurements can be used for measuring H(z),
as these measurements constrain the product H(z)rd (Di Valentino
et al. 2021; Arendse, Agnello & Wojtak 2019), where rd is the sound
horizon at the end of the baryon–drag epoch (zd = 1, 059.94 (Planck
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Collaboration et al. 2020)).
The BAO constraint (at z ≥ 0.38) on H(z)rd can be extrapolated
to z = 0 using a lower redshift probe, such as SN Ia, to obtain
a constraint on H0 rd (Arendse et al. 2019). This procedure of
extrapolating the BAO measurements is model–dependent (Arendse
et al. 2019). Nonetheless, the extrapolation can be performed us-
ing various cosmographic techniques, such as cosmology–agnostic Figure 5. H(z) / (1 + z) in units of km s−1 Mpc−1 as a function of z.
The theoretical predictions are shown: red curve – CDM and black curves
expansions of the Hubble parameter or distances, so that the final
– Lifshitz cosmology: unbroken – M1 and dashed – M2. The grey band
measurement might be considered as independent of a cosmological
around each black curve shows the ±1σ errors in H(z) / (1 + z) obtained
model (Arendse et al. 2019). Therefore, a point has been made that by propagating the errors in ∞ h2 . Also shown are BAO results with rd =
due to the extrapolated BAO constraint on H0 rd , one cannot rise 147.09 Mpc at several redshifts from: galaxy correlations in BOSS DR12
H0 without reducing rd since this would introduce tension with the (Alam et al. 2017), quasar correlation in eBOSS DR16 (Hou et al. 2021),
extrapolated BAO measurements (Arendse et al. 2019). the correlations of Lyα absorption in eBOSS DR14 (Agathe et al. 2019), and
However, while the cosmographic techniques are agnostic to cross-correlation of Lyα absorption and quasars in eBOSS DR14 (Blomqvist
cosmology, they are still models, and it is not clear how well they et al. 2019). The SH0ES measurement at z = 0 (Riess et al. 2021) is shown
may capture the Lifshitz cosmology. Di Valentino et al. (2021) has as well. While the overall fit of CDM to the BAO measurements seems to
noted that the BAO data are extracted under the assumption of a be somewhat better, the fit of the two Lifshitz cosmology realizations seems
to be reasonably acceptable. The point at z = 0.61 already disagrees with
CDM scenario, so one should be careful in excluding all the ‘Late
CDM, but more severely so with M2 and even more with M1. The point at
Time solutions’ only using this argument. Moreover, the use of SN Ia
z = 0.38 agrees with Lifshitz cosmology (M1 and M2) slightly better, and so
to extrapolate the BAO measurements in this inverse distance ladder does the point at z = 1.48.
procedure might be problematic, as also noted by Di Valentino et al.
(2021), which recommended not to use this approach. They write,
‘the fiducial absolute magnitude[’s] [...] value depends on the method M2 and even more with M1; the data point at z = 0.38 agrees with the
used to produce a light curve fit, which bands are included, the Lifshitz cosmologies (M1 and M2) slightly better than with CDM,
light curve age where it is defined, and the fiducial reference point and so does the point at z = 1.48. By observing Fig. 5, we conclude
chosen. Errors would arise from unintended mismatches between that while the overall fit of CDM to BAO measurements of H(z)
SN analyses and missing covariance data’ (Di Valentino et al. 2021). seems to be somewhat better, the fit of the Lifshitz cosmologies is
Lastly, Arendse et al. (2020) used strong gravitational lensing to acceptable.
break the degeneracy between rs and H0 in the extrapolated BAO
constraints; they found a small trend in the measured rd when using
5 E A R LY U N I V E R S E
each lens (at different redshift) separately (see fig. 5 there). While
statistically insignificant (1.6σ ) at the moment, this trend might Up to this point, we have ignored the early universe and solved the
signal residual systematics, either in the lenses themselves or in theory assuming only late-universe modifications. Now, we shall
the procedure used to extrapolate the BAO measurements (Arendse turn our attention to this point. We will estimate the expected
et al. 2020). Arendse et al. (2020) has noted that a recent (z ≈ 0.4) early-universe modifications due to Lifshitz cosmology to assess
change in dark energy might produce this behaviour and re–absorb the validity of the assumption that led us to drop the radiation term.
this trend. Specifically, we will verify that our Lifshitz cosmology’s dynamics
Only a more thorough analysis of Lifshitz cosmology would are consistent with negligible dark energy contribution at the early
be able to answer the question posed by extrapolating the BAO universe.
measurements; at the moment, we may take Planck’s value for rd We return to the two coupled equations, equation (2), that describe
(147.09 Mpc (Planck Collaboration et al. 2020)) to see whether the the mutual interaction between the expanding universe and dark
Lifshitz cosmologies (M1 and M2) are consistent with the BAO energy according to Lifshitz cosmology. This time, we do not drop
measurements of H(z) at z ≥ 0.38. We take CDM’s value for the the radiation term, and we need to find a new solution that includes
sound horizon at the end of the baryon–drag epoch to approximate the this term. In addition, we can no longer assume that the sound horizon
Lifshitz cosmology’s value since zd shortly follows last–scattering at last-scattering r∗ is unchanged. Therefore, if we wish to proceed
(z∗ = 1, 089.92 (Planck Collaboration 2020)). Fig. 5 shows H(z) / in the spirit of Section 3 and find a set of values for the theory’s
(1 + z) at z = 0.38, 0.51, 0.61, 1.48, 2.34, and 2.35 from BAO parameters by imposing a relationship between α and ∞ h2 ;
measurements (with rd = 147.09 Mpc). In this figure, the data point then, instead of demanding that DM is kept unchanged (equation
at z = 0.61 already disagrees with CDM, but more severely so with 4), we should demand that the CMB’s angular acoustic scale θ ∗ is
MNRAS 507, 3473–3485 (2021)3480 D. Berechya and U. Leonhardt
inset). The dark energy’s dynamics kick in again around the transition
period from matter to vacuum domination at zvm ≈ 0.29 (calculated
from fde (z = zvm ) = 0.5). There, it rises drastically and becomes at
the present fde (z = 0) ≈ 0.733 for M1 or ≈0.707 for M2. It finally
approaches 1, far in the future.
This solution, which considers the early features of dark energy, in-
volves a further approximation in addition to first–order perturbation
theory: the parameters are obtained from the late-universe dynamics.
Therefore, we should be more careful with drawing cosmological
conclusions, but only regard it as an indication of the validity of
our assumption for the late-universe dynamics. If this approximate
solution would have predicted more noticeable modifications around
last–scattering, then our assumption would be in question; the fact
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that this is not the case makes our assumption sensible. As a side
remark, we cautiously mention that Philcox et al. (2021) suggests that
early modification to CDM should treat matter–radiation equality
Figure 6. Relative dark energy contribution fde as a function of z, as calcu-
and last-scattering scales similarly to solve the Hubble tension; it
lated from the approximation of Lifshitz cosmology’s dynamics including the
seems that Lifshitz cosmology does just that.
early universe. The unbroken curve shows the case M1 and the dashed curve
M2. The ±1σ errors in fde due to errors in ∞ h2 are not shown here since
they are thinner than the curve’s width. Three special times are presented:
matter–radiation equality zeq = 3, 402 (Planck Collaboration et al. 2020) 6 DISCUSSION
(orange), last–scattering z∗ = 1, 089.92 (Planck Collaboration 2020) (red),
and vacuum-matter equality zvm ≈ 0.29 (calculated from fde (z = zvm ) = 0.5) Viewing the universe as one giant ‘dielectric medium’ with time-
(blue). The early-universe evolution of fde , according to Lifshitz cosmology, dependent refractive index and applying Lifshitz theory for calculat-
takes place roughly at the range 223 ≤ z ≤ 30, 350 for M1 or 388 ≤ z ≤ 17, ing the vacuum energy inside the medium, one can find a physical
950 for M2 where fde ≤ −0.005, and peaks around zeq with fde ≈ −0.019 at description of dark energy (Leonhardt 2019). This description is
the peak for M1 or ≈−0.012 for M2 (inset). At late times, fde rises drastically
based on well-established and well-tested physics (Landau et al.
and becomes at the present fde (z = 0) ≈ 0.733 for M1 or ≈0.707 for M2. Far
1980; Leonhardt 2020) which makes it unique among all other
in the future, it approaches 1.
models of dark energy. The theory comes with two free parameters,
α and ∞ h2 (replacing h2 of CDM, such we have a total of
unchanged. That is, we should, in principle, demand seven parameters). We call this theory Lifshitz cosmology.
r∗ In this paper, we have investigated two realizations of Lifshitz
θ∗ ≡ = θ∗(CDM) . (9) cosmology; for each realization, we choose a value for the coupling
DM
parameter α , and then, by demanding equation (4), ∞ h2 is fixed
As it turns out (see Appendix B2), it is not straightforward to together with the predicted dynamics.
generalize our solution (equation 3) to accommodate r a−4 and Our first considered realization (M1) is α M1
= αTH
= (9π )−1
then solve equation (9). Therefore, we will approximate by taking a based on the assumption of only electromagnetic contribution to
detour: we take α and ∞ h2 that we found for M1 and M2 when the vacuum energy with a sharp cut-off at exactly the Planck length.
considering only late modifications (Table 1), and then we plug (as Amazingly, this naive theoretical prediction gives the SH0ES value
the zeroth-order solution) H 2 = H02 (r a −4 + m a −3 + ∞ ) into for H0 (73.2 [km s−1 Mpc−1 ] at mean value). We may (and in some
the equation for H02 ˙ LC (the second equation in equations 2), instances, we should (Benevento et al. 2020; Efstathiou 2021)) view
and we integrate it (numerically, see Appendix B2) with ∞ h2 the Hubble tension as a tension between the SH0ES value for MB
as the integration constant. This way, we get an approximation for and the one obtained by calibrating the Pantheon data with CDM
the first-order (in α ) LC h2 that includes early–universe effects. or using inverse distance ladders (Benevento et al. 2020; Efstathiou
Now, we can use the so-obtained LC h2 to calculate the relative 2021). Table 2 shows that M1 can considerably relieve the tension
contribution of the vacuum energy, fde , throughout the entire cosmic by 51 per cent at the best MB value; the relatively small difference in
evolution RMSD between the best MB and the SH0ES value suggests that M1
ωLC might even resolve this MB tension completely. M1 also seems to fit
fde ≡ (10)
ωr a −4 + ωm a −3 + ωLC E−1 measurements based on distance-ladder-independent SN Ia data
(Fig. 4). On the other hand, while M1 appears to fit the shape of the
where ωx ≡ x h2 for x = r, m, LC.
binned SN Ia data (Fig. 3) at the lower redshift region (z ∼ 0.2), it
The results are shown in Fig. 6. As expected,3 the dark energy’s
does not fit the shape at the higher redshift region (to the extent that
dynamics kick in around the transition period from radiation to
this shape does not depend on the model). M1 also seems to fit BAO
matter domination at zeq = 3, 402 (Planck Collaboration et al. 2020).
measurements of H(z) only moderately (Fig. 5).
The early-universe evolution of fde , according to Lifshitz cosmology,
Our second considered realization (M2) is α M2
= 0.0225. This
takes place roughly at the range (that includes both last-scattering
model seems to be the middle ground between M1 and CDM; it
and matter-radiation equality) 223 ≤ z ≤ 30, 350 for M1 or 388 ≤
gives a nominal value of H0 = 69.9 ± 0.7 [km s−1 Mpc−1 ], and it
z ≤ 17, 950 for M2 where fde ≤ −0.005, and it peaks around zeq
shrinks MB by only ∼19 per cent. On the other hand, M2 fits all
with fde ≈ −0.019 at the peak for M1 or ≈−0.012 for M2 (Fig. 6’s
the E−1 data points (Fig. 4), it perfectly fits the shape of the binned
SN Ia data over the entire redshift range (Fig. 3), and its fit to BAO
3 Recall that Lifshitz cosmology predicts dark energy evolution only in measurements of H(z) is comparable to CDM’s and only slightly
transition periods worse (Fig. 5).
MNRAS 507, 3473–3485 (2021)Lifshitz cosmology 3481
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1976), where the quantum vacuum defined with respect to creation B1 Late universe
and annihilation operators in an inertial frame in Minkowski space
To obtain the late-universe evolution, we start from equation (2),
appears in an accelerated frame as thermal radiation – not as a
which we write again here:
vacuum.
2
This frame dependence means that a state defined as a vacuum in H (a) = H02 (r a −4 + m a −3 + LC ),
one frame is not necessarily the same state defined as a vacuum in (B1)
˙ LC = 8α H ∂t3 H −1 .
H02
another frame; these two states could be different from one another,
i.e. two different physical settings. Let us emphasize that the general These two coupled equations describe the mutual interaction
coordinate invariance does not break; instead, the ‘vacuumness’ between the cosmic expansion and the evolution of dark energy. By
of a quantum state is a frame-dependent quality. One may under- solving these equations, we can find the evolution of the background
stand this frame-dependence with an analogy to a point-mass at universe (homogeneous and isotropic) according to Lifshitz cosmol-
rest. ogy. The problem is that these equations are not easy to solve. As
The rest frame of a point-mass is one unique frame (up to mentioned in Section 3, even obtaining a reliable numerical solution
is considerably hard for the following two reasons. First, the equation
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translations), in which the point-mass appears to sit at rest; in other
frames, the same point-mass appears to move. Of course, the physics for the dark energy’s dynamics, the second equation in equation (B1),
describing the point-mass is independent of the frame in which we depends on high derivatives of the scale–factor a (up to fourth-order
choose to observe it. Nonetheless, two point–masses which appear derivative); this is a problem because, in differential equation solvers,
at rest in two different frames (not related by translations), do not the highest derivatives take the lead, whereas in reality, for most of
represent the same physical system but two different systems. In the period of interest, the dynamics of LC are a mere correction
this analogy, the ‘vacuumness’ of a quantum field is akin to the to the dynamics of the universe. Second, equation (B1) constitute
‘restness’ of a point-mass. To conclude this idea, two quantum states what is known as ‘stiff equations,’ causing havoc with step size and
defined as a vacuum in two different frames are two different physical accuracy. Therefore, we will approximate and solve perturbatively,
states; they are not one and the same state observed in two different where α will be our small parameter. As also discussed in Section 3,
coordinates. our first simplification will be dropping the radiation term H02 r a −4 .
After clarifying this point, one question is raised when considering Next, the combination H02 m that appears in the first equation of
Lifshitz cosmology: Which is the relevant frame for defining the equation (B1) is proportional to the present-day physical density of
universe’s vacuum state? matter ρ 0, m ; as this density is a physical entity, H02 m should be a
Leonhardt (2019) defined the vacuum state with respect to confor- model-independent combination. Indeed, H02 m can be determined
mal time τ , model–independently by the relative heights of the CMB acoustic
peaks (Planck Collaboration 2014). Therefore, we may replace ωm ≡
dt
τ= . (A3) m h2 (where h ≡ H0 / 100[km s−1 Mpc−1 ]) in equation (B1) by its
a(t) CDM’s equivalent
In conformal time, the FLRW metric becomes conformally flat. As P
ωm ≡ [m h2 ]P = 0.1430 ± 0.0011
Maxwell’s equations are conformally invariant (Birrell & Davies
(Planck, TT,TE,EE+lowE+lensing), (B2)
1982), the electromagnetic field and its fluctuations perceive the
conformally flat expanding universe as flat Minkowski space with 2 P
where [m h ] is obtained by Planck’s TT,TE,EE+lowE+lensing
constant Hamiltonian and hence, an exact ground state. For this CDM analysis (Planck Collaboration 2020) (the ‘P’ superscript
reason, Leonhardt (2019) thought to define the cosmological vacuum denotes that we use Planck’s CDM value). In the following, we
as a vacuum state with respect to conformal time, which leads to will also use Planck’s value (Planck Collaboration 2020) of
equation (A2). We performed the same analysis as in Section 3,
with equation (A2) replacing equation (A1) in equation (2). We
P
ω ≡ [ h2 ]P = 0.3107 ± 0.0082
found that Lifshitz cosmology with conformal vacuum state leads (Planck, TT,TE,EE+lowE+lensing). (B3)
to H0 < 67 [km s−1 Mpc−1 ] for any α in the range of interest. We
Now, for mathematical convenience, we re–scale and re–define
thus conclude that the original version of Lifshitz cosmology, with
the variables
a conformal vacuum state, is ruled out by the Hubble tension (that
ωLC
demands a higher H0 ). ν ≡ ln [(ωP
) a], ξ ≡ ω
P 1/3
/ωm P
t, η ≡ P , (B4)
In this paper, with hindsight, we have tried to define the cos- ω
mological vacuum as a vacuum state with respect to cosmological where ωLC ≡ LC h2 . We also regard ξ (time) as a function of ν
time t. This definition of a vacuum state seems more natural as (scale–factor) and define as the derivative of ξ with respect to ν.
it means that the cosmological vacuum is co-moving with the It is easy to show that = ω P
H −1 (expressed in terms of a and
expanding universe alongside anything else, matter and radiation t):
alike. By so defining the vacuum state, we obtain equation (A1). dν −1 da dν −1 −1
dξ P da 1
Our current work shows that this definition seems able to resolve (or ≡ = = = ω
considerably mitigate) the Hubble tension. We thus conclude that the dν dξ dξ da dt a
cosmologically relevant vacuum state should be defined as a vacuum = P
ω H −1 . (B5)
with respect to cosmological time t.
Then, equation (B1) become (dropping the radiation term and
P
replacing ωm with ωm )
A P P E N D I X B : C A L C U L AT I O N S θ = (e−3ν + η)−1/2 , (B6)
In this appendix, we will detail the calculations that were briefly
1 θ 1θ2
described in Sections 3 and 5. η = 8 α 2 ∂ν ∂ν − (B7)
θ θ 2 θ2
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