Missing large-angle correlations versus even-odd point-parity imbalance in the cosmic microwave background
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A&A 660, A121 (2022)
https://doi.org/10.1051/0004-6361/202142296 Astronomy
c ESO 2022 &
Astrophysics
Missing large-angle correlations versus even-odd point-parity
imbalance in the cosmic microwave background
M.-A. Sanchis-Lozano1 , F. Melia2,? , M. López-Corredoira3,4 , and N. Sanchis-Gual5
1
Instituto de Física Corpuscular (IFIC) and Departamento de Física Teórica, Centro Mixto Universitat de València-CSIC,
Dr. Moliner 50, 46100 Burjassot, Spain
e-mail: Miguel.Angel.Sanchis@ific.uv.es
2
Department of Physics, The Applied Math Program, and Department of Astronomy, The University of Arizona, Tucson, AZ 85721,
USA
e-mail: fmelia@email.arizona.edu
3
Instituto de Astrofísica de Canarias, 38205 La Laguna, Tenerife, Spain
e-mail: fuego.templado@gmail.com
4
Departamento de Astrofisica, Universidad de La Laguna, 38206 La Laguna, Tenerife, Spain
5
Departamento de Matemática da Universidade de Aveiro and Centre for Research and Development in Mathematics
and Applications (CIDMA), Campus de Santiago, 3810-183 Aveiro, Portugal
Received 24 September 2021 / Accepted 13 February 2022
ABSTRACT
Context. The existence of a maximum correlation angle (θmax & 60◦ ) in the two-point angular temperature correlations of cosmic
microwave background (CMB) radiation, measured by WMAP and Planck, stands in sharp contrast to the prediction of standard
inflationary cosmology, in which the correlations should extend across the full sky (i.e., 180◦ ). The introduction of a hard lower cutoff
(kmin ) in the primordial power spectrum, however, leads naturally to the existence of θmax . Among other cosmological anomalies
detected in these data, an apparent dominance of odd-over-even parity multipoles has been seen in the angular power spectrum of the
CMB. This feature, however, may simply be due to observational contamination in certain regions of the sky.
Aims. In attempting to provide a more detailed assessment of whether this odd-over-even asymmetry is intrinsic to the CMB, we
therefore proceed in this paper, first, to examine whether this odd-even parity imbalance also manifests itself in the angular correlation
function and, second, to examine in detail the interplay between the presence of θmax and this observed anomaly.
Methods. We employed several parity statistics and recalculated the angular correlation function for different values of the cutoff kmin
in order to optimize the fit to the different Planck 2018 data.
Results. We find a phenomenological connection between these features in the data, concluding that both must be considered together
in order to optimize the theoretical fit to the Planck 2018 data.
Conclusions. This outcome is independent of whether the parity imbalance is intrinsic to the CMB, but if it is, the odd-over-even
asymmetry would clearly point to the emergence of new physics.
Key words. cosmological parameters – cosmic background radiation – cosmology: observations – cosmology: theory –
inflation – large-scale structure of Universe
1. Introduction tially flat, if the initial spatial curvature was indeed set arbitrarily
(but see Melia 2014). If this initial condition were truly indeter-
As is well known, observations of the temperature fluctuations in minate, the Universe would have required an astonishing degree
the cosmic microwave background (CMB) radiation show that of fine-tuning at the time of the Big Bang to evolve into what we
our Universe is quite uniform on scales much larger than the see today without the effects of inflation.
apparent (or Hubble) horizon (Melia 2018) at the time of decou- The most popular inflation models tend to adopt the slow-
pling. According to standard cosmology, this so-called horizon roll condition, positing that the inflaton potential V(φ) changed
problem can be overcome by assuming an inflationary phase very slowly during the phase of exponentiated expansion. For
lasting a tiny fraction of a second almost immediately after the these scenarios, it is convenient to measure the inflation time as
Big Bang (Starobinsky 1979; Kazanas 1980; Guth 1981; Linde a function of the number of e-folds, Ne , in the expansion factor
1982). Meanwhile, quantum fluctuations in the underlying infla- a(t). The duration of inflation would then be roughly equal to
ton field φ (Mukhanov 2005) would have grown and (some- Ne Hφ−1 , where Hφ ≡ ȧ/a was the Hubble parameter at that time.
how) classicalized to produce density perturbations that were
In principle, the horizon and flatness problems might both be
also stretched enormously by the accelerated expansion, even-
solved by requiring an inflation corresponding to Ne & 60, one
tually forming the seeds of today’s large-scale structure, includ-
of the more notable successes of the inflation paradigm.
ing galaxies and clusters (Peebles 1980). Inflation has also been
As we show below, however, the existence of a
invoked to explain why the Universe today appears to be spa-
maximum correlation angle (θmax & 60◦ ) observed in
the CMB by all three major satellite missions, COBE
?
John Woodruff Simpson Fellow. (Hinshaw et al. 1996), WMAP (Bennett et al. 2003), and Planck
Article published by EDP Sciences
A121, page 1 of 10A&A 660, A121 (2022)
(Planck Collaboration VI 2018), implies a smaller number In this paper, we do not, however, include the potentially use-
of e-folds (Ne ≈ 55), contrasting with most inflationary sce- ful information concerning the polarization of the CMB radia-
narios (Melia & López-Corredoira 2018; Liu & Melia 2020; tion available from Planck for several practical reasons. The E-
Melia 2021). Moreover, this discrepancy does not stand in mode polarization arising from the Thomson scattering of pho-
isolation. Many other puzzles and anomalies contributing tons by free electrons is dominated by optically thin plasma on
to an increasing level of tension with standard cosmology small spatial scales. Therefore, these polarization effects can-
(ΛCDM) have emerged in recent years as the accuracy of the not extend over large angular scales. In addition, the subtraction
observations has improved (see, e.g., the recent review by of foreground contamination is more difficult to carry out for
Perivolaropoulos & Skara 2021). For example, Schwarz et al. polarized CMB light based on the current Planck data because
(2016) included in their list of CMB anomalies an apparent of the required complex multicomponent fitting. Future mis-
alignment of the lowest multipole moments with each other and sions such as Litebird (Errard et al. 2016), PICO (Hanany et al.
with the motion and geometry of the Solar System, a hemi- 2019) and COrE (The COrE Collaboration 2011) will achieve
spherical power asymmetry, and an unexpectedly large cold spot more precise measurements at much higher sensitivity than is
in the southern hemisphere. Di Valentino et al. (2021) argued currently available with Planck, and should be able to answer
against general concordance by demonstrating that a combined the question of whether the large-angle anomalies seen in
analysis of the CMB angular power spectrum obtained by the temperature fluctuations are confirmed by the polarization
Planck and the luminosity distance inferred simultaneously maps.
from type Ia supernovae (SNe) excludes a flat universe and a
cosmological constant at the 99% confidence level. A broader
view of the general tension between the predictions of ΛCDM
and the observations may be found in López-Corredoira (2017). 2. Angular correlations in the CMB
Whether some of these anomalies have a common origin One of the main goals of analyzing the angular correlation func-
becomes of paramount importance to the fundamental basis of tion of the CMB is to extract information regarding the differ-
our cosmological modeling. ent stages of the evolution of the Universe. Based on very gen-
In this paper we focus on two of the more recent discrepan- eral grounds, small (large) angles between CMB photon trajec-
cies. The first is the lack of large-angle correlations seen in the tories can be associated with small (large) length scales in the
CMB data, which seems to suggest that the primordial power source plane (the opposite of using energy scales). Furthermore,
spectrum, P(k), had a hard cutoff at a kmin distinctly different it offers us the possibility of analyzing an important assumption
from zero. This explanation for the angular-correlation anomaly in ΛCDM, that is, that the fluctuations are Gaussian and sta-
has recently been shown to also account self-consistently for tistically homogeneous and isotropic. As previously noted, the
the missing power at low `s in the angular power spectrum angular correlation function of the CMB is already known to
(Melia 2021). This feature in P(k) indicates the time at which exhibit several statistical anomalies. In this section, we focus on
inflation could have started (Liu & Melia 2020), hence setting the lack of large-angle correlations, but we first recall features in
an upper limit to the possible number of e-folds by the time the CMB that are of special interest to this study.
it ended. It is kmin that now appears to create an inconsistency It is customary to distinguish primary anisotropies arising
between the number of e-folds required to solve the horizon prior to decoupling, from secondary anisotropies developed as
problem and that corresponding to the measured fluctuation the CMB photons propagate from the last scattering surface to
spectrum (Melia & López-Corredoira 2018). The second is an the observer. We do not distinguish between recombination, last
apparent preference of the CMB data for an odd point-parity, first scattering surface, or freeze-out times, but approximate all of
inferred from the analysis of the angular power spectrum (see, them as equal to a cosmic time td ≈ 3.8 × 105 years (z ' 1100).
e.g., Kim et al. 2012; Schwarz et al. 2016). We seek to confirm For angles greater than a few degrees, the primary contributor
whether this odd-even imbalance is also present in the measured to the former is the Sachs–Wolfe (SW) effect (Sachs & Wolfe
angular correlation function, and if it is, we attempt to find a phe- 1967), representing fluctuations in the metric leading to temper-
nomenological connection between these two recently identified ature anisotropies via perturbations of the gravitational potential
features in the CMB fluctuation distribution. at the time of decoupling. There are two Sachs–Wolfe influ-
We note, however, that an interpretation of the odd-even ences: the above-mentioned, nonintegrated Sachs–Wolfe effect,
imbalance as being intrinsic to the CMB anisotropies is not uni- and the integrated Sachs–Wolfe effect (ISW). The latter is some-
versally accepted. For example, Creswell & Naselsky (2021b) times also further subdivided into an early ISW (taking place
suggested that this asymmetry may be due to a contamination just after decoupling; for convenience, this is often just included
from a few regions of the sky. We return to this viable possi- in the SW), and a late ISW, arising while the CMB photons
bility toward the end of our discussion in Sect. 4. The work we propagated through the expanding medium. The ISW contributes
carry out in this paper can therefore provide a more quantita- non-negligibly to the CMB anisotropies only if the universal
tive assessment of the idea that the an odd-even imbalance may expansion is at least partially driven by something other than
originate within the CMB anisotropies for a more detailed com- purely nonrelativistic matter. In the standard model, dark energy
parison with the alternative scenario in which it is primarily due – possibly in the form of a cosmological constant – started
to some observational contamination. In doing so, we attempt to influencing the expansion at z ∼ 0.5, corresponding to a cos-
answer the questions whether (i) either one or the other of these mic time ∼10 Gyr. The ISW has the effect of mainly raising the
characteristics is sufficient to account for the observed angular Sachs plateau at low multipoles. However, the detection of this
correlation function, or if are both required; and (ii) if the latter is ISW due to dark energy is not fully confirmed yet. The putative
true, whether the reoptimized value of kmin is different from that detections may simply be noise with underestimated error bars
reported earlier (Melia & López-Corredoira 2018; Melia 2021) (López-Corredoira et al. 2010; Dong et al. 2021).
and might mitigate the current level of tension between the lat- A way to address the angular dependence and anisotropies
est Planck release and the predictions of standard inflationary of the CMB is through its primary power spectrum, originally
cosmology. defined by the Fourier transform of the primordial fluctuation
A121, page 2 of 10M.-A. Sanchis-Lozano et al.: Large-angle anomalies in the CMB
by unitary vectors n1 and n2 ,
1200
C(θ) = hT (n1 )T (n2 )i. (2)
1000
The angle θ ∈ [0, π] is defined by the scalar product n1 · n2 .
One typically expands C(θ) in terms of Legendre polynomi-
800
als (assuming azimuthal symmetry)
C(θ)
600 ∞
1 X
C(θ) = (2` + 1)C` P` (cos θ), (3)
400 4π `=2
200 where the C` coefficients encode the information with cosmo-
logical significance from the sky. The sum starts at ` = 2 and
0 ends at a given `max , dictated by the resolution of the data. The
first two terms are excluded because (i) the monopole (` = 0) is
-200 simply the average temperature over the whole sky and plays no
0 20 40 60 80 100 120 140 160 180
θ (degrees) role in the correlations, other than a global scale shift; and (ii)
the dipole (` = 1) is greatly affected by Earth’s motion, creating
an anisotropy that dominates the intrinsic cosmological dipole
1200 signal.
1000
2.2. Maximum angle in two-point angular correlations
800
Large-angle correlations in the CMB provide information about
the earliest stages of the primitive Universe, well before recom-
C(θ)
600
bination and the subsequent formation of cosmic structure. In
400 this context, it may be useful to point out an interesting anal-
ogy with the angular correlations among the final-state par-
200 ticles in heavy-ion or proton-proton collisions at the Large
Hadron Collider, and the early formation of nonconventional
0 matter, such as a quark-gluon plasma or hidden valley particles
(Sanchis-Lozano et al. 2020). In Fig. 1a we plot the observed
-200 angular correlation function (black dots) measured by Planck
0 20 40 60 80 100 120 140 160 180
θ (degrees) (Planck Collaboration VI 2018), compared with fitted (blue and
red) curves to be discussed below. Above '60◦ , the correlations
Fig. 1. Two-point correlation function C(θ) (solid curves) optimized to drop to near zero, except for a downward tail at ∼180◦ , which
fit the Planck 2018 data (black points) (Planck Collaboration VI 2018). we also examine in more detail below. This shape of C(θ), par-
(a) Top panel: umin = 4.5 and odd-even parity balance of multipoles ticularly its suppression at large angles, was unexpected in stan-
(blue curve). (b) Bottom panel: umin = 4.5 and odd-even parity domi-
nance (red curve). See main text.
dard cosmology, given that inflation was supposed to begin early
enough (with kmin → 0) to provide the required number of e-
folds to solve the horizon and flatness problems, thereby provid-
spectrum, usually parameterized as ing coverage across the full sky (see, e.g., Melia 2014).
It is worth mentioning at this point that these expectations on
k ns −1 the value of kmin and correlations at all angles are primarily based
P(k) = A , (1) on the correctness of the standard model. Large-angle correla-
k0
tions are not necessarily expected in all cosmological models,
where ns is the scalar spectral index. The spectrum would be however. For example, in the alternative cosmology known as
perfectly scale free (with ns = 1) if the Hubble parameter Hφ the Rh = ct universe, which does not have an inflationary epoch,
were strictly constant during inflation. In typical slow-roll infla- the expansion factor is linear in time and the maximum angle
tionary models, however, this is only approximately true, and corresponds to the size of the apparent horizon (Melia 2018) at
Hφ evolves slowly, which produces a slight deviation of the decoupling,
spectral index from one. The observations show that in fact 2π
ns = 0.9649 ± 0.0042 (Planck Collaboration VI 2018), adding θmax ' (in radians), (4)
ln (t0 /td )
some observational support for a slow-roll potential, V(φ).
where td and t0 denote the decoupling and present cosmic times,
2.1. Two-point angular correlation function respectively. The Rh = ct universe is an FLRW cosmology in
which the equation of state is constrained by the zero active mass
The anisotropies in the CMB are very small, of about one part condition in general relativity, that is, ρ + 3p = 0. The Ray-
in 105 , but they carry a wealth of information pertaining to the chaudhuri equation clearly shows that ä = 0 in that case, which
possible influence of V(φ) and the subsequent evolution of the leads to a Universe expanding at a constant rate (Melia 2013a).
Universe after reheating. A very powerful probe of these fluctua- This Universe has no horizon problem, and spatial flatness is
tions is the two-point angular correlation function, defined as the ensured because the total energy density is zero. It therefore has
ensemble product of the temperature differences with respect to no need for inflation. In such a universe, the Hubble parameter
the average temperature, from two directions in the sky defined is always exactly equal to the inverse of the age of the Universe,
A121, page 3 of 10A&A 660, A121 (2022)
and the Hubble radius satisfies Rh = ct at all times, hence the
eponymous origin of its name. Setting td = 3.8 × 105 years and
t0 = 13.8 Gyr, we obtain θmax ∼ 40◦ . Thus, if inflation were
to fail to adequately explain the existence of such a maximum
correlation angle, it might be considered evidence supporting a
noninflationary model, such as Rh = ct.
We generalize the expression in Eq. (4) by writing it in terms
of the maximum fluctuation size λmax (td ) at decoupling time td
and the proper distance Rd ≡ a(td )rd (with rd the comoving dis-
tance) from us to the last scattering surface,
λ
max
θmax = 2 tan−1 . (5)
Rd
Because we study the formation of primordial physical quan-
tities, the apparent horizon (equal to the Hubble radius in this
case) should set the basic length scale at stake (Melia 2013b,
2020b). In particular, the maximum fluctuation size can be esti-
mated as λmax = α2πRh , where α . 1 denotes a coefficient
dependent on the cosmological model. For instance, α = 1 for de 180
Sitter space and α ' 0.5 for ΛCDM (Melia 2013b, 2018). This
scale changes as the expansion factor a(t) grows, starting with 160
the assumed slow-roll inflation, followed by radiation and then 140
matter-dominated evolution, before reaching decoupling, where
the CMB radiation was released. Nevertheless, knowledge of Rh 120
at decoupling is sufficient to estimate λmax associated with the
θmax
100
largest fluctuation we can see in the CMB anisotropies today.
On the other hand, to compute the proper distance Rd 80
between us and the last scattering surface, we must know
the expansion history from decoupling to today. An excellent 60
approximation for a(t) during this time may be written Ne = 60
40
Ne = 55
a(t) ∼ sinh2/3 (H̃ t), (6) 20 Ne = 0
√
where H̃ ≡ (3/2) ΩΛ H0 , with ΩΛ and H0 denoting the normal- 0
0.001 0.01 0.1 1 10
ized dark matter density and Hubble constant today, respectively, Cosmic Time (Gy)
and t the cosmic time since the Big Bang (see Appendix A).
In Fig. 2 we show the maximum correlation angle obtained Fig. 2. Maximum correlation angle due to inflation. Top panel (a): 3D
in our study under different assumptions concerning the infla- plot of the maximum correlation angle as a function of cosmic time and
the number of e-folds. Bottom panel (b): maximum correlation angle
tionary epoch. Figure 2a shows a 3D rendition of the maximum
evolution with cosmic time for different Ne values, corresponding to
correlation angle as a function of cosmic time and the number vertical sections in the 3D plot. We highlight that the plateau for Ne
of e-folds (Ne ). The correlation angle remains confined to just a is smaller than about 50, implying a large inhomogeneity of the CMB
few degrees for any number of e-folds Ne . 40. For Ne ≥ 62, on along cosmic time.
the other hand, the curve quickly reaches 180◦ along the whole
cosmic time until now. For the sake of clarity, the maximum cor-
relation angle evolution with cosmic time is shown in Fig. 2b for 2.3. Low cutoff in the CMB power spectrum
different Ne values, corresponding to vertical cuts in the 3D plot ΛCDM predicts an angular correlation curve that crosses the
(Fig. 2a) (see Appendix A for the mathematical details). zero-axis twice and extends over the whole 180◦ range of
The late-time ISW effect yielding additional secondary poloidal angles (see, e.g., Fig. 1 in Melia 2014). This result
anisotropies should tend to broaden the angular correlations as is manifestly inconsistent with observational evidence, which
lower-` multipoles are enhanced (due to a raising of the Sachs- shows a maximum correlation angle of ≥60◦ , as already dis-
plateau). Therefore, the number of e-folds required to com- cussed in the previous section.
ply with the observed maximum value of θmax should decrease In order to mitigate this tension, Melia & López-Corredoira
even further once the ISW is taken into account. The Planck (2018) introduced a cutoff to the primordial power spectrum,
2018 data therefore suggest that Ne . 55, creating even greater representing a lower limit to the integral
tension with the conventional slow-roll inflationary scenario.
Z ∞
Liu & Melia (2020) provided more details concerning the dif-
ficulties faced by the slow-roll paradigm to simultaneously solve C` = N dk kns −1 j2` (k rd ), (7)
kmin
the horizon problem and missing correlations at large angles.
Addressing these constraints from the latest CMB data where the normalization constant N and the minimum mode
would require a more complicated inflationary process than is wavenumber kmin are optimized using a global fit to the whole
usually conjectured. As we show below, this conclusion goes in observed angular correlation function. Although this procedure
the same direction as the need for a cutoff in the power spec- represents a phenomenological introduction of the cutoff kmin ,
trum in order to correctly reproduce the whole angular correla- this truncation has some theoretical justification in that it rep-
tion function of the CMB. resents the first quantum fluctuation to either (i) have crossed
A121, page 4 of 10M.-A. Sanchis-Lozano et al.: Large-angle anomalies in the CMB
the Hubble horizon once inflation started, or (ii) have emerged with the projectors defined as γ`+ = cos2 (`π/2) and γ`− =
out of the Planck domain if inflation never happened (see Melia sin2 (`π/2). Assuming that `(` + 1)C` is approximately constant
2019; Liu & Melia 2020). The need for a kmin may also be related at low `, P± can clearly be considered as a measurement of the
to the infrared regularization of the inflaton field commutator, degree of parity asymmetry: below unity, it implies odd-parity
although in standard cosmology, it should then likely be much dominance, and vice versa. Any deviation of this statistic from
smaller than its value (i.e., '3 × 10−4 Mpc−1 , corresponding to unity points to an odd-even parity imbalance.
umin = 4.34) optimized using the Planck 2018 data (Liu & Melia A second statistic useful for checking a point-parity imbal-
2020). ance may be defined as the average ratio of the power in adja-
In the computation of the C` coefficients, only the SW effect cent odd and even multipoles up to a given ` value (Aluri & Jain
is taken into account, ignoring other effects such as the baryon 2012; Panda et al. 2021),
acoustic oscillations (BAO), whose influence extends primarily
over smaller angles (θ . 5◦ ) and hence only the very large-` `max odd
2 X D` − 1
multipoles (certainly >100). Changing the integration variable odd
Q(`max ) = odd , (11)
from k to u ≡ krd in Eq. (7) and setting ns = 1 for simplicity, we `max − 1 `=3 D`
obtain
where `max
odd
is the maximum odd multipole up to which the statis-
j 2 (u)
Z ∞
C` = N du ` · (8) tic is computed, and D` ≡ `(` + 1)C` /π. In contrast to P(`max ),
odd
umin u the new statistic, Q(`max ), ensures that there are always the same
number of odd and even powers along the whole considered
We point out that only those C` coefficients with ` . 20 are
multipole range, so no sawtooth oscillations are present. As for
actually affected by the existence of the cutoff umin in the above
P(`max ), this statistic is also expected to fluctuate about the value
integral.
of one at low `s.
In a previous computation of these coefficients using the
In case of Gaussian fluctuations, the angular power spectrum
Planck 2103 dataset, Melia & López-Corredoira (2018) found
and the angular correlation function contain the same informa-
that the best fit to the angular correlation function is obtained
tion concerning the angular distribution of temperature in the
with umin = 4.34 ± 0.50, which translates into a minimum
CMB. Nevertheless, although the angular power spectrum cov-
wavenumber kmin = 4.34/r(td ) (see also the similar limit placed
ers all of the ` dependence, it emphasizes large-` values, such
on kmin by an analogous study of the angular power spectrum
that most of the information at θ & 10◦ is squeezed into a very
itself; Melia 2021). In the present paper, we repeated this analy-
narrow interval, making it difficult to pick out any disagreement
sis using the more recent Planck 2018 dataset, obtaining umin =
between theory and observation for the low multipoles. Con-
4.5 ± 0.5, and kmin = 4.5/r(td ), which is compatible with the pre-
versely, the angular correlation function covers the fluctuation
vious results from Planck 2013. We provide more details about
distribution more evenly over all angles, thereby making it rela-
the statistical analysis yielding this result below.
tively easier to study the large-angular region, corresponding to
Very importantly, an almost zero correlation plateau above
lower multipoles. Ultimately, both approaches should be mutu-
the maximum angle (θmax ≈ 60◦ ) can be obtained by setting
ally consistent as well as complementary for the extraction of
a lower cutoff to the integration variable u, corresponding to a
useful information.
lower cutoff in the power spectrum. Mathematically, this result odd
With this goal in mind, we require the statistic Q(`max )
can be understood as a delicate balance between even and odd
(shown in red) to match (up to a given accuracy) the data (shown
multipole contributions to C(θ). We return to this crucial point in
in black) in Fig. 3a. We do this by heuristically tuning the
the next section.
weights
3. Odd versus even point-parity in the CMB C`even
q(`even ) = (12)
C`even +1
Among the other anomalies observed in the CMB, an odd-even
parity violation may indicate a nontrivial topology of the Uni- to optimize the fit. Then, using the above ratios, we fit the two-
verse, unexpected physics at the pre- or inflationary epochs, or point angular correlation function keeping two parameters free,
some unsolved systematic errors in the data reduction. In the namely, the normalization, N, and the cutoff kmin (i.e., umin ). As
following, we focus on the apparent odd-dominance of the CMB noted earlier, the latter was already constrained to the interval
fluctuations, that is, the fact that the weight of odd multipoles in umin ∈ (4.34 ± 0.50) in Melia & López-Corredoira (2018) using
either the power spectrum or the two-point angular correlation the old Planck 2013 data and without any consideration of a pos-
function is larger than the corresponding weight of the even mul- sible odd-even parity imbalance. We have carried out this anal-
tipoles. This imbalance is commonly referred to as a point-parity ysis again for the latest Planck 2018 data at small, middle, and
asymmetry of the CMB, and we address two statistics below that now large angles, incorporating the odd-parity dominance.
are widely employed to analyze it. To incorporate these weights into the angular correlation
function, we modified Eq. (3) to read
∞
3.1. Odd-even parity statistics 1 X
C(θ) = (2` + 1) w` C` P` (cos θ), (13)
We employ the parity statistic (Panda et al. 2021) 4π `=2
P+ (`max ) with
P(`max ) = , (9)
P− (`max ) q(`even )
where w`even = , w`even +1 = 1 − w`even , (14)
q(`even ) + 1
`max
X `(` + 1) where `even denotes even values of ` within the selected inter-
P± (`max ) = γ`± C` , (10)
`=2
2π val. The weights w` for the coefficients C` , introduced ad hoc
A121, page 5 of 10A&A 660, A121 (2022)
Table 1. χ2d.o.f. values for different fits to the two-point correlation func-
tion C(θ) and the parity statistic P(`max ), inferred from the Planck data.
1
PB: parity balance; OD: odd dominance.
0.8 Curve/parameters: umin = 0+PB umin = 0+OD umin = 4.5+PB umin = 4.5+OD
C(θ) 6.1 3.8 0.80 0.70
Q(lmax)
P(`max ) 30 5.3 18.0 0.99
0.6
0.4 plots in Fig. 4a that show the odd and even multipole contribu-
tions to C(θ). Each (odd and even) contribution behaves quite
observed distinctly due to the intrinsic parity properties of the Legendre
0.2 umin = 4.50 weighted polynomials about θ = 90◦ . Consequently, once summed to pro-
duce the full C(θ), these two contributions produce constructive
0 10 20 30 40 50 60 70 80 interference at small angles (i.e., θ < 90◦ ) and destructive inter-
lmax ference at large angles (θ > 90◦ ).
One of our most important conclusions from this work is that
in order to produce the optimum fit of the Planck 2018 data with
1.4 the lowest χ2d.o.f. in Fig. 1b, we must include in our theoretical
C(θ) both the cutoff umin in Eq. (8) and an odd-parity dominance
1.2 via the weighting factors in Eq. (13). Otherwise, our fit to the
observed data points for either the angular correlation function or
1 the parity statistics worsens considerably, yielding a χ2d.o.f. much
larger than one. Using both the cutoff (kmin , 0) and an odd-
P+ / P-
even parity imbalance simultaneously produces the best fit for
0.8
C(θ), Q(`max ) and P(`max ) (see Table 1), however.
As noted earlier, we find from our study an optimized cutoff
0.6 umin ' 4.5 ± 0.5, in which the mean value and σ = 0.5 error were
obtained using a Monte Carlo analysis to sample the variation of
0.4 C(θ) within the measurement errors. Since the C(θ) points are
observed
umin = 4.50 weighted highly correlated, we circumvented this problem using a Monte
0.2 Carlo procedure as described in Melia & López-Corredoira
0 10 20 30 40 50 60 70 80
(2018). Starting from 100 mock CMB catalogs, we computed
lmax the two-point correlation function Ci (θ), (i = 1, 100) in each
case. From these, we obtained ∆Ci (θ) = Ci (θ) − C0 (θ), where
Fig. 3. Statistics used to describe the point-parity asymmetry of the C0 (θ) is the angular correlation function in standard cosmology.
CMB. (a) Top panel: Q(`max odd
) statistic as a function of `max for umin = Then we calculated umin,i for C(θ) = C Planck (θ) + ∆Ci (θ) for each
4.50 (red), compared to the Planck 2018 data. The overlap is visually realization i. Next, from the resulting roughly Gaussian distribu-
almost perfect. (b) Bottom panel: same for the P(`max ) statistic using the tion of umin,i , we determined its average value and rms, yielding
odd
values of C` determined from the matching of the Q(`max ) statistic with umin ' 4.5 ± 0.5, whose corresponding χ2d.o.f. distribution from
the data shown on the left. The reduced χ for this fit is χ2d.o.f. ≈ 0.99.
2
the C(θ) fits varies smoothly around its minimum at 4.5. This
umin range is slightly larger than but compatible with the interval
obtained from the Planck 2013 dataset: umin = 4.34 ± 0.50.
to model the odd-even imbalance, should not be confused with A quantitative comparison of the various fits we have
the window factors. Obviously, by requiring q(`even ) ≡ 1 ∀ `even , explored is provided in Table 1, which lists the χ2d.o.f. values for
we obtain w`even = w`even +1 = 1/2, thereby restoring the odd-even different choices of parameters, including the odd-even parity
parity balance. relative weights. A remark is in order: the improvement of the
In Fig. 3a we show the function Q(`max ) for the Planck 2018 C(θ) fit, once umin = 4.5 is fixed, is somewhat modest when
data (black) with the fit (red) corresponding to the set of param- passing from parity balance to parity breaking. This is easy to
eters that provides the minimum reduced χ2d.o.f. for the angu- understand because the main effect of imposing odd-dominance
lar two-point correlations, keeping the cutoff we introduced to takes place at large angles, where the observational uncertainties
improve the fit (see Table 1). The matching of the observed and are quite large and its impact on the total χ2d.o.f. is thus relatively
theoretical values achieved by tuning the ratios q(`) (and their small. On the other hand, the agreement between the theoreti-
respective weighting factors, w` ) that modify the C` coefficients cal prediction and the parity statistic P(`max ) improves dramati-
is excellent. In addition, the set of weights optimized to find this cally, as expected, given the observed odd-dominance, as seen in
match for Q(`max ) also produces an excellent fit in the P(`max ) Fig. 3a.
plot, corresponding to a reduced χ2d.o.f. near unity, as seen in We stress that the main benefit of using a kmin , 0 together
Fig. 3b and Table 1. with parity breaking is not so much the improvement of the
Figure 1b shows the plot of C(θ) over the whole range of fit to the two-point correlations (with a focus on the tail at
angles with umin = 4.5, although this time, it incorporates the large angles) at the cost of increasing the number of degrees
odd-dominance via the coefficients C` , modified as explained of freedom, but the fact that this approach can simultaneously
above. The fit now also matches the downward tail at large resolve several apparently disconnected anomalies in the CMB.
angles. To understand what is happening, consider the separate The end result is an excellent fit of C(θ) at small, medium,
A121, page 6 of 10M.-A. Sanchis-Lozano et al.: Large-angle anomalies in the CMB
2000 pieces:
1500 C(θ) = C(θ ; ` ≤ `0 ) + C(θ ; ` ≥ `0 + 1)
`0
X (2` + 1)
1000 = w` C` P` (θ)
`=2
4π
500 `up
X (2` + 1)
+
C(θ)
w` C` P` (θ). (15)
0 4π
`=` +1
0
-500 The first piece, C(θ ; ` ≤ `0 ), corresponds to the C` coefficients
that are significantly affected by umin in the integral of Eq. (8) and
-1000 the different odd-even weighting factors used to produce Fig. 3a.
The second summation runs from `0 + 1 up to `up = 401 (ideally,
-1500 infinity).
In contrast to conventional wisdom, the contribution of high-
-2000
0 20 40 60 80 100 120 140 160 180 ` multipoles, C(θ ; ` ≥ `0 + 1), can certainly influence C(θ) even
θ (degrees) at angles θ ∼ 180◦ . To see this, we separately plot in Fig. 4b the
contributions of even and odd multipoles for `0 > 40. Due to the
750
oscillatory behavior of the Legendre polynomials, both contri-
butions add to always produce a positive correlation at smaller
angles, while a delicate balance exists at θ ∼ 180◦ that yields
500 zero correlation when the parity symmetry is exact. A slight
imbalance between odd and even high-` multipoles, however,
250 has a small (but observable) influence on the shape of the tail.
We thus conclude that the value of C(θ ' 180◦ ) contains very
interesting information concerning a possible odd-even parity
C(θ)
0 imbalance even for high-` multipoles.
-250
4. Discussion
-500 In spite of the high degree of isotropy in the CMB, certain
anomalies have been found that create some tension with stan-
dard inflationary cosmology. The lack of long-range angular cor-
-750 relations beyond a maximum angle (θmax & 60◦ ) is difficult to
0 20 40 60 80 100 120 140 160 180 reconcile with the basic inflationary paradigm, which is founded
θ (degrees) on the principle of a slow-roll potential producing an acceler-
Fig. 4. Contributions from the even (blue) and odd (magenta) multipoles ated expansion of over 60 e-folds. This maximum correlation
to the two-point angular correlation function C(θ) in Eq. (13) for (a) ` ∈ angle instead allows only about 55 e-folds, well below the num-
[2, 401] (top panel) and (b) ` ∈ [40, 401] (bottom panel). The delicate ber required to solve the temperature horizon problem. More-
balance between odd and even polynomials is clear even at high ` for over, as shown in Melia & López-Corredoira (2018), the feature
large angles. Selecting a different weighting of each contribution has a (i.e., kmin ) that maps into such a maximum correlation angle also
strong influence on C(θ) at θ ' 180◦ . produces a zero-correlation plateau in the two-point angular cor-
relation function at all larger angles.
Meanwhile, it has been known for some time that the angular
and large angles (reproducing the existence of a θmax & 60◦ ) power spectrum of the CMB favours a weighting of odd multi-
with a p-value '0.95, and the implementation of an odd-parity poles over even (see, e.g., Kim et al. 2012). This anomaly could
dominance seen also in the angular power spectrum. This out- mean a breakdown of the odd-even parity expected in the cos-
come constitutes the principal result of our analysis in this mological principle, so its study carries great interest. In this
paper. paper, we therefore sought to determine whether these two fea-
tures, that is, kmin and an odd-even parity imbalance, are related,
3.2. Relevance of the angular correlations at θ ' 180◦ and/or whether both are required to produce the best fit to the
Planck 2018 data.
Large-angle correlations are commonly associated with low We have incorporated a possible odd-even parity imbalance
multipoles, typically ` ∈ [2, 20]. We have already established in our analysis by introducing nonuniform weighting factors that
that the introduction of a cutoff kmin mainly affects multi- modify the C` coefficients in accordance with the parity statistic
poles with ` . 20, and significantly alters the shape of the Q(`max ). The outcome of this analysis has resulted in an excel-
C(θ) curve at θ & 60◦ . In contrast, the net contribution of lent fit to the two-point correlation function, including the tail
multipoles with ` & 20 practically cancels out, producing associated with the odd-parity dominance at θ ∼ 180◦ . We have
a plateau of zero correlation if nonuniform weights w` are stressed that this angular region is influenced not only by the
excluded. expected low-` multipoles, but also to some degree by the high
Nevertheless, the tail of C(θ) at θ ∼ 180◦ is influenced by multipoles when the odd-even parity is broken.
the higher-order polynomials, even ` & 20. To demonstrate this At this stage, we can only speculate about a possible physical
effect, we have split the angular correlation function into two origin of kmin and/or an odd-even parity imbalance. Certainly, in
A121, page 7 of 10A&A 660, A121 (2022)
the context of slow-roll inflation, the cutoff signals the time at nates from the CMB itself. As we indicated in the introduc-
which inflation could have started. This is the principal reason tion, it may simply be due to an observational artifact. Recently,
why a nonzero value of this truncation to the primordial power Creswell & Naselsky (2021b) discussed a link between the par-
spectrum is so restrictive for the ability of inflation to solve the ity asymmetry and the low-` peak anomaly, established in
temperature horizon problem while simultaneously producing the presence of highly asymmetric regions in the sky, due
the distribution of anisotropies seen in the CMB. A natural ques- to some foreground contamination in four regions near the
tion that arises in this context is whether the presence of a cutoff, Galactic plane ([`, b] = [212◦ , 21◦ ], [32◦ , 21◦ ], [332◦ , 8◦ ], and
kmin , should also impact the angular power spectrum itself. The [152◦ , 8◦ ]). This asymmetric distribution increases the odd-
answer appears to be yes, and it does so in a very intriguing multipole power, while the deficit of symmetric regions leads
way. It appears that the same value of umin required to optimize to a corresponding deficit of even-multipole peaks. There-
the fit to the angular correlation data also completely accounts fore, the odd-even parity imbalance in the CMB could be
for the ‘missing’ power seen in the low-multipole components explained, to a large statistical significance, as a consequence
(Melia 2021). The fact that the same feature in P(k), that is, a of an anomalous density of antipodal peaks in the sky once
umin = 4.5 ± 0.5 (as we have found in this paper) can account for the dipole contribution from the motion of the Solar Sys-
both empirically derived anomalies adds weight to its possible tem is removed, without resorting to actual cosmological
reality. effects.
We here demonstrated that both kmin and an odd-even par- Having said that, we can safely conclude that our study in
ity imbalance are required to optimize the fit to the Planck this paper establishes the compatibility of an infrared cutoff kmin
2018 data, however. So what could be the origin of this par- in the power spectrum with an odd-over-even parity imbalance,
ity violation? There is no known mechanism that can pro- regardless of its origin, either cosmological or due to contami-
duce such an imbalance due to inflation on its own. Prior nation, or an incorrect foreground subtraction. Our analysis has
to classicalization, all of the quantum fluctuations seeded in shown that both kmin and the parity asymmetry are necessary in
the early Universe and expanded during inflation were spher- order to provide a best fit to the angular correlation function of
ically symmetric (Melia 2021). Suggestions have therefore the CMB.
tended to focus on possible nonstandard beginnings or trans-
Planckian issues. For example, topological models involving
multiconnected universes have been invoked to account for the
anomalous cold spot, or the aforementioned missing power 5. Conclusion
at low multipoles (Efstathiou 2003; Land & Magueijo 2006). This brief survey of possible causes of an odd-even parity imbal-
Although these models can lower the power of the small- ance is by no means exhaustive, but it is fair to conclude that
` multipoles, they apparently cannot create an asymmetry, a resolution of its origin will probably rely on new theoreti-
however. cal ideas. This stands in contrast to the meaning of kmin , which
At face value, an odd-even parity imbalance might be viewed can indeed in some way be attributed to the inflaton potential,
as a possible trans-Planckian effect (Brandenberger & Martin V(φ). At least theoretically, there would be no obvious con-
2013), given that this is the first instance following the Big Bang nection between this cutoff and the odd-even parity imbalance.
when features measurable today would have emerged into the Nevertheless, the observational evidence suggests that both are
semi-classical Universe (Melia 2020a). This topic touches on necessary to optimize the fit to the Planck 2018 data, as hinted
a broader issue related to the self-consistency of basic infla- also for the WMAP observations by the earlier work reported
tionary theory because the quantum fluctuations in the inflaton in Kim et al. (2012), although the WMAP measurements had a
field would have been seeded in the so-called Bunch–Davies lower precision than the Planck 2018 observations, and these
vacuum (Bunch & Davies 1978), well below the Planck scale. authors did not analyze in depth the two-point angular correla-
It is unclear, however, how or why quantum mechanics as we tion function over its whole angular range, as we have done here
know it and general relativity could be used meaningfully to for Planck.
describe the evolution of these fluctuations on scales smaller Our main conclusion is that neither the odd-even imbalance
than their Compton wavelength (see, e.g., Melia 2020a). In other nor the cutoff kmin = 4.5/rd on their own and separately are
words, an odd-even parity imbalance may turn out to be a signa- sufficient to minimize the χ2d.o.f. of the C(θ) fit to the Planck
ture of trans-Planckian physics once a viable theory of quantum 2018 data. Both are required, and while a nonzero kmin may be
gravity is devised, but there is no evidence of this right now. attributed to an as yet undiscovered inflaton potential, the odd-
Any oscillatory and sharp features in P(k) tend to become com- even imbalance would appear to signal entirely new physics, if
pletely smeared out by the time the CMB power spectrum is pro- it is not simply due to contamination.
duced (Bennett et al. 2011). Other possibilities may also include
bouncing cosmologies (Agullo et al. 2021).
Acknowledgements. This work has been partially supported by Agencia Estatal
If confirmed to be of cosmological origin, an odd-parity de Investigación del Ministerio de Ciencia e Innovación under grant PID2020-
dominance would violate the Cosmological Principle. Together 113334GB-I00/AEI/10.13039/501100011033, by Generalitat Valenciana under
with the missing large-angle correlations, these two features grant PROMETEO/2019/113 (EXPEDITE), by the Center for Research and
could be an indication that new physics is required to modify Development in Mathematics and Applications (CIDMA) through the Por-
the standard model accordingly, perhaps even leading to some tuguese Foundation for Science and Technology (FCT – Fundação para a Ciên-
cia e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020, by
exciting new discoveries about the origin and evolution of the national funds (OE), through FCT, I.P., in the scope of the framework con-
Universe. tract foreseen in the numbers 4, 5 and 6 of the article 23, of the Decree-
Regardless of its origin, however, the reality of an odd-over- Law 57/2016, of August 29, changed by Law 57/2017, of July 19 and
even parity imbalance is becoming more firmly established in by the projects PTDC/FIS-OUT/28407/2017, CERN/FIS-PAR/0027/2019 and
PTDC/FIS-AST/3041/2020. This work has further been supported by the
the analysis of the Planck data, making speculation such as European Union’s Horizon 2020 research and innovation (RISE) programme
this interesting to consider. Nevertheless, the existence of this H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740 and by FCT through
asymmetry is by no means a certain indication that it origi- Project No. UIDB/00099/2020.
A121, page 8 of 10M.-A. Sanchis-Lozano et al.: Large-angle anomalies in the CMB
References López-Corredoira, M., Sylos Labini, F., & Betancort-Rijo, J. 2010, A&A, 513,
A3
Abramowitz, M., & Stegun, I. A. 1970, Handbook of Mathematical Functions: Melia, F. 2013a, A&A, 553, A76
with Formulas, Graphs, and Mathematical Tables (New York: Dover Books Melia, F. 2013b, Class. Quant. Grav., 30, 155007
on Mathematics) Melia, F. 2014, A&A, 561, A80
Agullo, I., Kranas, D., & Sreenath, V. 2021, Class. Quant. Grav., 38, 065010 Melia, F. 2018, Am. J. Phys., 86, 585
Aluri, P. K., & Jain, P. 2012, MNRAS, 419, 3378 Melia, F. 2019, Eur. Phys. J. C, 79, 455
Bennett, C. L., Hill, R. S., Hinshaw, G., et al. 2003, ApJS, 148, 97 Melia, F. 2020a, Astron. Nachr., 341, 812
Bennett, C. L., Hill, R. S., Hinshaw, G., et al. 2011, ApJ, 192, 17 Melia, F. 2020b, The Cosmic Spacetime (Oxford: Taylor & Francis)
Brandenberger, R. H., & Martin, J. 2013, Class. Quant. Grav., 30, 113001 Melia, F. 2021, Phys. Lett. B, 818, 136632
Bunch, T. S., & Davies, P. C. W. 1978, Proc. R. Soc. A, 360, 117 Melia, F. 2022, Astron. Nachr., 343
Creswell, J., & Naselsky, P. 2021a, J. Cosmol. Astropart. Phys., 2021, 103 Melia, F., & López-Corredoira, M. 2018, A&A, 610, A87
Creswell, J., & Naselsky, P. 2021b, ArXiv e-prints [arXiv:2105.08658] Melia, F., Ma, Q., Wei, J.-J., & Yu, B. 2021, A&A, 655, A70
Di Valentino, E., Melchiorri, A., & Silk, J. 2021, ApJ, 908, L9 Mukhanov, V. F. 2005, Physical Foundations of Cosmology (Cambridge:
Dong, F., Yu, Y., Zhang, J., Yang, X., & Zhang, P. 2021, MNRAS, 500, Cambridge University Press)
3838 Panda, S., Aluri, P. K., Samal, P. K., & Rath, P. K. 2021, Astropart. Phys., 125,
Efstathiou, G. 2003, MNRAS, 346, L26 102582 [erratum: Astropart. Phys., 130, 102582]
Errard, J., Feeney, S. M., Peiris, H. V., & Jaffe, A. H. 2016, J. Cosmol. Astropart. Peebles, P. J. E. 1980, The Large-Scale Structure of the Universe (Princeton:
Phys., 2016, 052 Princeton University Press)
Guth, A. H. 1981, Phys. Rev. D, 23, 347 Perivolaropoulos, L., & Skara, F. 2021, ArXiv e-prints [arXiv:2105.05208]
Hanany, S., Alvarez, M., Artis, E., et al. 2019, ArXiv e-prints Planck Collaboration VI 2018, A&A, 641, A6
[arXiv:1902.10541] Sachs, R. K., & Wolfe, A. M. 1967, ApJ, 147, 73
Hinshaw, G., Branday, A. J., Bennett, C. L., et al. 1996, ApJ, 464, L25 Sanchis-Lozano, M. A., Sarkisyan-Grinbaum, E. K., Domenech-Garret, J. L., &
Kazanas, D. 1980, ApJ, 241, L59 Sanchis-Gual, N. 2020, Phys. Rev. D, 102, 035013
Kim, J., Naselsky, P., & Hansen, M. 2012, Adv. Astron., 2012, 960509 Starobinsky, A. A. 1979, J. Exp. Theor. Phys. Lett., 30, 682
Land, K., & Magueijo, J. 2006, MNRAS, 367, 1714 Schwarz, D. J., Copi, C. J., Huterer, D., & Starkman, G. D. 2016, Class. Quant.
Linde, A. D. 1982, Phys. Lett. B, 108, 389 Grav., 33, 184001
Liu, J., & Melia, F. 2020, Proc. R. Soc. A, 476, 20200364 The COrE Collaboration (Armitage-Caplan, C., et al.) 2011, ArXiv e-prints
López-Corredoira, M. 2017, Found. Phys., 47, 711 [arXiv:1102.2181]
A121, page 9 of 10A&A 660, A121 (2022)
Appendix A: Expansion factor following decoupling 2 F 1 [a, b; c; z] = (1 − z)−a 2 F1 [a, b − c; c; z/(z − 1)] →
The comoving distance to the last scattering surface may be 2 F 1 [1/2, 1/6; 7/6; −x
2
=]
written 1
2 F 1 [1/2, 1; 7/6; x /(1 + x )] ,
2 2
(A.6)
(1 + x2 )1/2
Z t 0
dt
rd = c 0
. (A.1)
td a(t ) and reverting back to the original variable, we find the proper dis-
The scale factor a(t) determined from Friedmann’s equations for tance required to determine the maximum angle in Equation (5)
an isotropic and homogeneous Universe made of matter (dust) to be
and dark energy reads 3 sinh2/3 [H̃td ] sinh[H̃t]1/3
Rd = × ×
ȧ 2 Ω H̃ cosh[H̃t]
m
= H02 + Ω Λ
a a(t)3
d ,
2
3 p 2/3 F
2 1 1/2, 1/6; 7/6; tanh [ H̃t] − (t → t ) (A.7)
→ a(t) ∼ sinh ΩΛ H0 t , (A.2)
2 where t here denotes the cosmic time of observation since the
Big Bang. If the observation is today, then t = t0 is to be identi-
where ΩΛ ' 0.7 and H0 denote the normalized dark matter
fied with the present age of the universe.
density and Hubble parameter today, √ respectively. To simplify
the notation, we define H̃ = (3/2) ΩΛ H0 , noting that in fact
H̃ ' H0 . Appendix B: Error due to the finite cutoff `up
The proper (or physical) distance is given by
In this appendix, we estimate the accuracy of the Legendre
Rd = a(td ) rd , (A.3) expansion of C(π) over `up polynomials instead of infinity. For
simplicity, we assume that the relation C` = 2`(` + 1) is satis-
so that fied for all `; we therefore obtain 4π C(π) = 1/4, as defined in
Z t
dt0 Equation (3) for all ` from 2 to ∞.
Rd = sinh2/3 [H̃td ] 2/3
. (A.4) Thus, we may write
td sinh [H̃t0 ]
`up
Changing to the variable x = sinh[H̃t], we obtain for the indefi- X (2` + 1) (−1)`up +1
4π C(π) = 1/4 = (−1)` + . (B.1)
nite integral
`=2
2`(` + 1) 2(`up + 1)
3 sinh1/3 x
Z
1 dx This equation allows us to estimate the relative error made by
= 2 F 1 [1/2, 1/6; 7/6; −x ] .
2
neglecting polynomials of order higher than `up . In this work we
H̃ x2/3 (1 + x2 )1/2 H̃
(A.5) set `up = 401, so that the relative error is about 0.5%. Other
effects such as BAO at large ` have been neglected throughout
Next, invoking the basic property of the hypergeometric series this analysis because they have very little or no influence on the
(Abramowitz & Stegun 1970), large angles we are focused on in this paper.
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