11institutetext: 1Department of Physics, The Applied Math Program, and
Department of Astronomy, The University of Arizona, Tucson, Arizona 85721,
USA; 11email<EMAIL_ADDRESS>
2Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, and
School of Physics and Electronic Science, Guizhou Normal University, Guiyang
550001, China
3PMO, Nanjing 210023, China; Guangxi Key Laboratory for Relativistic
Astrophysics, Guangxi University, Nanning 530004, China; University of Chinese
Academy of Sciences, Beijing 100049, China
4PMO, Nanjing 210023, China

# Hint of a truncated primordial spectrum from the CMB large-scale anomalies

F. Melia1 John Woodruff Simpson Fellow Qingbo Ma2 Jun-Jie Wei3 and Bo Yu4

(Received May 4, 2021)

###### Abstract

Context. Several satellite missions have uncovered a series of potential
anomalies in the fluctuation spectrum of the cosmic microwave background
temperature, including: (1) an unexpectedly low level of correlation at large
angles, manifested via the angular correlation function, $C(\theta)$; and (2)
missing power in the low multipole moments of the angular power spectrum,
$C_{\ell}$.

Aims. Their origin is still debated, however, due to a persistent lack of
clarity concerning the seeding of quantum fluctuations in the early Universe.
A likely explanation for the first of these appears to be a cutoff, $k_{\rm
min}=$ $(3.14\pm 0.36)\times 10^{-4}$ Mpc-1, in the primordial power spectrum,
${\mathcal{P}}(k)$. Our goal in this paper is twofold: (1) we examine whether
the same $k_{\rm min}$ can also self-consistently explain the missing power at
large angles, and (2) we confirm that the introduction of this cutoff in
${\mathcal{P}}(k)$ does not adversely affect the remarkable consistency
between the prediction of Planck-$\Lambda$CDM and the Planck measurements at
$\ell>30$.

Methods. We use the publicly available code CAMB to calculate the angular
power spectrum, based on a line-of-sight approach. The code is modified
slightly to include the additional parameter (i.e., $k_{\rm min}$)
characterizing the primordial power spectrum. In addition to this cutoff, the
code optimizes all of the usual standard-model parameters.

Results. In fitting the angular power spectrum, we find an optimized cutoff,
$k_{\rm min}=(2.04^{+1.4}_{-0.79})\times 10^{-4}$ Mpc-1, when using the whole
range of $\ell$’s, and $k_{\rm min}=(3.3^{+1.7}_{-1.3})\times 10^{-4}$ Mpc-1,
when fitting only the range $\ell\leq 30$, where the Sachs-Wolfe effect is
dominant.

Conclusions. These are fully consistent with the value inferred from
$C(\theta)$, suggesting that both of these large-angle anomalies may be due to
the same truncation in ${\mathcal{P}}(k)$.

###### Key Words.:

cosmological parameters – cosmology: cosmic background radiation – cosmology:
observations – cosmology: theory – large-scale structure of the Universe

††offprints: F. Melia

##  1 Introduction

All three major satellite missions designed to study the cosmic microwave
background (CMB)—COBE (Hinshaw et al. 1996), WMAP (Bennett et al. 2003), and
Planck (Planck Collaboration VII 2020)—have uncovered several unexpected
features in its temperature fluctuations. These include missing correlations
at large angles and an unexpectedly low power in multipoles
$2\lesssim\ell\lesssim 30$, especially $\ell=2$ and $3$. In spite of these
apparent deficiencies, however, the agreement between the angular power
spectrum predicted by $\Lambda$CDM and the observations is quite remarkable
for $\ell>30$, forming the basis for the optimization of many cosmological
parameters.

The large-scale anomalies stand in sharp contrast to the general level of
success interpreting the CMB anisotropies, sustaining a simmering debate
concerning their origin, or possible misidentification due to unknown
systematic effects (see Bennett et al. 2011, Copi et al. 2010 for reviews). If
these two unexpected features are real, it is not even clear if they are due
to the same physical process, in spite of the fact that they both characterize
variations in temperature over very large angular scales.

A thorough analysis of these anomalies was carried out with the final release
of the Planck data by the Planck Collaboration VII (2020). They studied the
statistical isotropy and Gaussianity of the CMB using both the Planck 2018
temperature and polarization data. Previous work had focused solely on the
temperature fluctuations, due to only a limited ability to also probe the CMB
polarization. The large-angular-scale polarization measurements permit a
largely independent examination of the peculiar features seen in the
temperature which can, in principle, reduce or eliminate any subjectivity or
ambiguity in interpreting their statistical significance.

The possibility that the lack of large-angle correlations in the temperature
may be largely due to a statistical ‘fluke,’ as unlikely as that may appear to
be, was raised by the fact that the Planck Collaboration VII study found only
weaker evidence of an analogous lack of large-scale angular correlations in
the polarization data.

Nevertheless, in spite of the fact that no unambiguous detections of
cosmological non-Gaussianity, or of anomalies analogous to those seen in
temperature, could be claimed, efforts at reducing the systematic effects that
contaminated the earlier polarization maps on large angular scales could not
completely

exclude the presence of residual systematics that limit some tests of non-
Gaussianity and isotropy.

A comparison of both the temperature and polarization analyses thus tends to
mitigate our ability to conclude one way or the other whether the large-angle
anomalies are real. From a theoretical standpoint, the polarization
fluctuations also originated in the primordial gravitational potential, but
are largely sourced by different modes than the temperature anisotropies.
Thus, a measurement of large-scale anomalies in both the temperature and
polarization has the potential of providing a greater statistical
significance. It is important to see whether any anomalies seen in the
polarization maps are related to known features in the temperature. On the
other hand, if no such anomalies are seen in the polarization it might be the
case that the temperature anomalies, if not pure flukes, could be due to
secondary influences, such as the integrated Sachs-Wolfe effect (Planck
Collaboration XXI 2016), or more exotic scenarios in which they may be due to
physical processes that do not correlate with the polarization.

The situation with regard to the large-angle anomalies is thus far from clear,
and it is essential to continue broadening the study of possible causes of
such features in either the temperature, polarization or both. In earlier work
(Melia & López-Corredoira 2018), we showed that a likely theoretical
explanation for the missing large-angle correlations in temperature is the
presence of a cutoff, $k_{\rm min}$, in the primordial power spectrum,
${\mathcal{P}}(k)$. Indeed, our analysis of the Planck data revealed that a
zero cutoff is ruled out by these measurements at a confidence level $\gtrsim
8\sigma$. Instead, we found that the observed angular correlation function,
$C(\theta)$, could be reproduced satisfactorily at all angles with a cutoff
$k_{\rm min}=(4.34\pm 0.50)/r_{\rm dec}$, where $r_{\rm dec}$ is the comoving
distance between us and the last scattering surface at decoupling ($z_{\rm
dec}=1080$). A truncation such as this is not necessarily consistent with the
aims of slow-roll inflation to simultaneously fix the horizon problem and seed
a quantum fluctuation spectrum consistent with large-scale structure (Liu &
Melia 2020, Melia 2020), because $k_{\rm min}$ represents the time at which
inflation could have started, severely constraining the number of e-folds
available for the Universe to expand at an accelerated rate. It is therefore
imperative to pursue this line of inquiry further and see if additional
evidence may be found in favor of a non-zero $k_{\rm min}$.

The goal of this paper is to complete that analysis in two significant ways:
(1) to examine whether the same cutoff may be responsible for both large-angle
anomalies, and (2) to confirm that the introduction of $k_{\rm min}$ in
${\mathcal{P}}(k)$ does not adversely impact the remarkable agreement between
theory and observations at $\ell>30$. In § 2 we introduce the basic
theoretical background underlying this work, and then proceed to study the
effects of a truncated ${\mathcal{P}}(k)$ on the angular power spectrum in §
3. We discuss the results in § 4, and present our conclusion in § 5.

##  2 Theoretical background

The inflationary paradigm posits that an early accelerated expansion persisted
long enough to create a homogeneous and isotropic Universe within our current
Hubble volume. As such, the CMB temperature, $T$, measured in some direction
$\hat{e}$, is expected to be a Gaussian random field on the sky, expandable as
a series of spherical harmonics, $Y_{\ell m}(\hat{e})$, with independent
Gaussian random coefficients $a_{\ell m}$ of zero mean:

$\langle a_{\ell m}\rangle=0\;,$ (1)

for all $\ell>0$ and $m=-\ell,-\ell+1,...,+\ell$. If the Universe is indeed
statistically isotropic, this spherical harmonic decomposition, and the two-
point angular power spectrum derived from it, contain all of the physical
information needed to interpret the CMB anisotropies.

The less used (but equally important) angular correlation function $C(\theta)$
relates the temperature measured in two independent directions, $\hat{e}_{1}$
and $\hat{e}_{2}$, and depends solely on the dot product
$\cos\theta\equiv\hat{e}_{1}\cdot\hat{e}_{2}$. One may thus expand it in terms
of Legendre polynomials:

$\langle T(\hat{e}_{1})T(\hat{e}_{2})\rangle\equiv C(\theta)={1\over
4\pi}\sum_{\ell}(2\ell+1)\,C_{\ell}P_{\ell}(\cos\theta)\;,$ (2)

in which the variance $C_{\ell}$ is known as the angular power of multipole
$\ell$. Statistical independence implies that the expectation of a product of
$a_{\ell m}$’s with different $\ell$’s and $m$’s vanishes, while isotropy
requires the constant of proportionality for this product to be solely a
function of $\ell$, not $m$:

$\langle a_{\ell
m}^{*}\,a_{{\ell}^{\prime}{m}^{\prime}}\rangle=\delta_{\ell{\ell}^{\prime}}\,\delta_{mm^{\prime}}\,C_{\ell}\;.$
(3)

As one can see from figure 1 (discussed below), the standard model has been
strikingly successful in accounting for the angular power spectrum (i.e.,
$C_{\ell}$ versus $\ell$), at least on small angular scales ($\lesssim
1^{\circ}$), corresponding to $\ell\gtrsim 100$. The rather precise
determination of the various cosmological parameters (Planck Collaboration VI
2020) is based on this excellent agreement between theory and observation.

In principle, the angular correlation function $C(\theta)$ contains the same
information as the angular power spectrum (see Eq. 2), but there are good
reasons for analyzing the CMB anisotropies separately using these two
approaches. First, whereas the angular power spectrum highlights the relative
contributions of different spherical harmonics, the angular correlation
function describes the variations of $T$ in real space. Some features may
emerge more prominently in one description instead of the other. Second,
$C(\theta)$ highlights properties of the anisotropies at relatively large
angles (i.e., small $\ell$’s), while the power spectrum provides ample detail
in the opposite regime, namely $\ell\gg 10$. One can easily see the reason for
this with a quick inspection of figure 1. When using the full angular power
spectrum, the model fit is dominated by over 1,000 values of $\ell$ at
$\ell>30$, and while the optimized fit is striking in this regime, as we
discuss elsewhere in this paper, the error bars and variance at $\ell<30$ are
much larger and the fit is significantly less compelling. On the other hand,
the angular correlation function displays the temperature anisotropies with
equal weighting over all angles $0\leq\theta\leq 180^{\circ}$. When fitting a
model to these data, the dominant influence is therefore at
$\theta>10^{\circ}$, the opposite of what happens with the power spectrum.
Third, $C(\theta)$ is a direct pixel-based measure, not requiring any
reconstruction of the observed sky. Some of the systematics may therefore have
less impact in one approach relative to the other.

Figure 1: Angular power spectrum (blue dots) estimated with the NLIC method,
with $1\sigma$ Fisher errors. These error bars include cosmic variance,
approximated as Gaussian (see fig. 1 in Planck Collaboration VI 2020). The
$\Lambda$CDM best fit model ($C_{\ell}^{TT}$), with a power-law fluctuation
distribution and $k_{\rm min}=0$, is shown in black. The red curve (for
$k_{\rm min}=2.82/r_{\rm dec}$) is the best fit to the whole range of $\ell$’s
with an optimized $k_{\rm min}\not=0$. The two curves are indistinguishable at
$\ell\gtrsim 6$, but deviate significantly from each other for the low
multipole moments, corresponding to large angles ($\gtrsim 60^{\circ}$). The
Sachs-Wolfe contribution to $C_{\ell}^{TT}$ dominates at $\ell\lesssim 30$,
while local physical effects, such as acoustic oscillations, dominate at
$\ell\gtrsim 30$.

In spite of these differences, one may nonetheless reasonably expect that, of
the various large-angle anomalies seen in the CMB (Planck Collaboration VII
2020), the one associated with $C(\theta)$ could have the same physical origin
as a second emerging from the angular power spectrum. Many authors (see, e.g.,
Copi et al. 2010) have pointed out that the most striking feature of the
angular correlation function (as seen, say, in fig. 2 of Melia & López-
Corredoira 2018) is not just that it disagrees with $\Lambda$CDM at a very
high level of confidence ($\gtrsim 3\sigma$; Copi et al. 2015), but that it is
nearly zero at large angles ($\gtrsim 60^{\circ}$)—in significant tension with
inflation, which should have produced correlations throughout the visible
Universe.

Most of the statistical weight in the Planck analysis of the angular power
spectrum is provided by the high-$\ell$ multipoles, and one may see in figure
1 that the best-fitting portion of the curve traces the data at $\ell\gtrsim
60$ much better than those at low-$\ell$’s. Indeed, a general lack of angular
power appears within the range $2\lesssim\ell\lesssim 30$—especially with the
quadrupole $\ell=2$—at a confidence level exceeding $99\%$ (Bennett et al.
2011, Planck Collaboration VI 2020). Quite remarkably, none of the attempted
fixes invoking physically motivated inputs, such as neutrino properties, the
number of relativistic degrees of freedom, or a running of the spectral index
in the fluctuation distribution, have provided any significant improvement to
the fit (see, e.g., Planck Collaboration XXIV 2014; Planck Collaboration VI
2020). To be sure, the $C(\theta)$ anomaly is much more obvious than the low
harmonic-space quadrupole and octopole power (see also O’Dwyer et al. 2004)
but, given that they both reference CMB anisotropic structure at very large
angles ($\gtrsim 60^{\circ}$), it is tempting to consider the possibility that
these two anomalies may have a common physical origin (see discussion in
Planck Collaboration VII 2020).

This is still not generally accepted, however. The low power at large angles
may simply be due to cosmic variance (Efstathiou 2003b), in which our Universe
may represent a random extrapolation away from its most likely configuration.
Another possible explanation for the vanishing $C(\theta)$ at large angles is
that a range of low multipoles are ‘conspiring’ to cause the cancellation
(Copi et al. 2009). In this interpretation, the combined contribution of
$C_{2}$–$C_{5}$ is canceled by the contributions of $C_{\ell}$ with $\ell>5$,
an apparent conspiracy that seems to violate the independence of different
multipole powers expected in statistical isotropy. But all such attempts at
reconciling the missing angular correlations with low power at $\ell\lesssim
30$ have thus far been inconclusive. Sentiment has therefore shifted to the
idea that finding a theoretical explanation for the missing power at low
$\ell$’s may be best served by first explaining the vanishing $C(\theta)$ at
large angles.

Figure 2: Same as figure 1, except magnifying the view at $\ell\lesssim 30$,
where the dominant contribution to $C_{\ell}^{TT}$ is the Sachs-Wolfe effect.
The best-fit curves are indistinguishable for $\ell\gtrsim 6$, but the
comparison with the data is greatly improved for the low-$\ell$ multipoles.

##  3 Best-fit truncated spectrum

###  3.1 The angular correlation function

We recently addressed this issue (Melia & López-Corredoira 2018) by attempting
to show that the absence of large-angle correlations in the Planck data is
best explained by the introduction of a non-zero minimum wavenumber $k_{\rm
min}$ in the primordial fluctuation power spectrum,

${\mathcal{P}}(k)=\begin{cases}A_{s}\left({k\over
k_{0}}\right)^{n_{s}-1}&\mbox{if }k\geq k_{\rm min}\\\ 0&\mbox{if }k<k_{\rm
min}\end{cases}\;,$ (4)

where $n_{s}$ is the spectral index, measured to have the value $0.965\pm
0.004$ by Planck (Planck Collaboration VI 2020);
$A_{s}=2.196^{+0.051}_{-0.06}\times 10^{-9}$ is the spectral amplitude; and
$k_{0}=0.05$ Mpc-1 is the scalar pivot scale. Such a cutoff challenges
inflation’s ability to simultaneously fix the horizon problem and produce a
near scale-free distribution of quantum fluctuations matching the observed CMB
anisotropies (Liu & Melia 2020; Melia 2020), because the presence of a non-
zero $k_{\rm min}$ implies a specific time at which the accelerated expansion
could have started, thus restricting the number of e-folds available to expand
the Universe beyond the Hubble horizon. In contrast, standard inflationary
$\Lambda$CDM has no cutoff—and hence no limit (in principle)—to the degree of
inflated expansion. This is the reason, of course, why inflation predicts an
angular correlation at all scales which, however, has not been observed by any
of the CMB satellite missions (Wright et al. 1996; Bennett et al. 2003; Planck
Collaboration VI 2020).

Our earlier analysis of the angular correlation function, based on the notion
that quantum fluctuations were generated in the early Universe with this power
spectrum and $k_{\rm min}\not=0$, demonstrated that the Planck data rule out a
zero cutoff at a confidence level exceeding $8\sigma$. An optimized fit to
$C(\theta)$ showed that the correlations measured over the entire sky could be
reproduced remarkably well with a cutoff

$k_{\rm min}={4.34\pm 0.50\over r_{\rm dec}}\;,$ (5)

where $r_{\rm dec}$ is the comoving distance between us and redshift $z_{\rm
dec}=1080$, at which decoupling is thought to have occurred in standard
$\Lambda$CDM. Using the latest Planck parameters (Planck Collaboration VI
2020), one finds $r_{\rm dec}\approx 13,804$ Mpc, and therefore a
corresponding minimum wavenumber

$k_{\rm min}=(3.14\pm 0.36)\times 10^{-4}\;{\rm Mpc}^{-1}\;\;\;({\rm
ang.\;corr.\;function})\;,$ (6)

roughly 160 times smaller than the pivot scale $k_{0}$.

This previous study, however, did not complete the analysis by investigating
how such a cutoff would impact the angular power spectrum—both at low and high
$\ell$’s. Given how well inflationary $\Lambda$CDM reproduces the observed
$C_{\ell}$’s at $\ell\gtrsim 60$ (see fig. 1), the question in not merely
whether a non-zero $k_{\rm min}$ could also reduce the power of the low
multipoles, but whether it could do so while simultaneously preserving the
excellent agreement seen between theory and observation elsewhere in this
plot. Answering this question is the principal goal of this paper.

###  3.2 The angular power spectrum

The idea of reducing power at large angles by modifying the potential of
inflation is not new (for a comprehensive review, see Martin et al. 2013).
Other approaches have included: (1) altering inflation’s initial conditions
(Berera et al. 1998; Contaldi et al. 2003; Boyanovsky et al. 2006; Powell &
Kinney 2007; Wang & Ng 2008; Cicoli et al. 2014; Das et al. 2015; Broy et al.
2015; Liu & Melia 2020; Melia 2020); (2) a reconsideration of the Integrated
Sachs-Wolfe effect (Das & Souradeep 2014); (3) the introduction of spatial
curvature (Efstathiou 2003a); (4) the use of a non-trivial topology (Luminet
et al. 2003); (5) geometric effects (Campanelli et al. 2006; Campanelli et al.
2007); (6) a violation of statistical anisotropies (Hajian & Souradeep 2003);
the creation of primordial micro black-hole remnants (Scardigli et al. 2011);
and loop quantum cosmology (Barrau et al. 2014), among several others. Two
particular approaches stand out in terms of their overlap with our treatment
in this paper, notably those of Iqbal et al. (2015) and Santos et al. (2018),
to which we shall return in § 4 below, when we discuss our results in a
broader context.

The analysis we carry out here is novel for two principal reasons: (i) it is
complementary to—and required for the completion of—our previous study of the
angular correlation function (Melia & López-Corredoira 2018), and (ii) it is
very focused on the specific modification to the primordial fluctuation
spectrum exhibited in Equation (4), which is itself highly motivated by its
success in resolving the missing correlations at large angles. Can a truncated
spectrum ${\mathcal{P}}(k)$ with the same $k_{\rm min}$ resolve both the
$C(\theta)$ and low-$\ell$ anomalies?

We use the publicly available code CAMB111www.camb.info (Lewis et al. 2000) to
calculate the angular power spectrum, based on a line-of-sight approach
described in Seljak & Zaldarriaga (1996). The code is modified slightly to
include the additional parameter (i.e., $k_{\rm min}$) characterizing the
primordial power spectrum (Eq. 4). In addition to this cutoff, the code
optimizes all of the usual standard-model parameters but, as we shall see from
an inspection of figures 1–3 and Table 1, most of them remain unchanged from
their Planck-$\Lambda$CDM values. We therefore highlight here a comparison
only of the six basic parameters between the two cases in which $k_{\rm
min}=0$ and $k_{\rm min}\not=0$: baryon density, $\Omega_{\rm b}$, cold dark
matter density, $\Omega_{\rm c}$, the Thomson scattering optical depth,
$\tau$, due to reionization, the angular size of the acoustic horizon,
$\theta_{s}$, the spectral index, $n_{s}$, and the scalar amplitude, $A_{s}$.
Each of the quantities $\Omega_{i}$ represents the ratio of the energy density
$\rho_{i}$ of species “i” to the critical density, $\rho_{\rm c}\equiv
3c^{2}H_{0}^{2}/8\pi G$, in terms of the Hubble constant $H_{0}$. Our
optimization procedure is even less sensitive to the other parameters in
Planck-$\Lambda$CDM.

In optimizing the fit, we use the publicly available code cosmomc (Lewis &
Bridle 2002) with an initial cutoff range $0\leq k_{\rm min}\leq 10^{-3}$
Mpc-1, and a combined data set that includes: the low-$\ell$ TT and EE and
high-$\ell$ TT, TE and EE likelihoods from Planck 2018 (Planck Collaboration
VI 2020), the Dark Energy Survey (DES) Year 1 results (Abbott et al. 2018;
Troxel et al. 2018), the Baryon Acoustic Oscillation (BAO) compilation from
Alam et al. (2017), and the Pantheon Type Ia SN catalog (Scolnic et al. 2018).
Note that the error bars shown in figures 1 and 2 include cosmic variance,
approximated as Gaussian (see fig. 1 in Planck Collaboration VI 2020), and so
do the likelihoods also taken from Planck Collaboration VI (2020) used with
our cosmomc calculations.

The calculated angular power $C_{\ell}$ for Planck-$\Lambda$CDM, with $k_{\rm
min}=0$, is indicated by the black curve in figures 1 and 2, in comparison
with the Planck data (blue dots). Our corresponding optimized fit using
$k_{\rm min}$ as a free parameter is represented by the red solid curve. Based
on this fit, we find an optimized value

$k_{\rm min}={2.82^{+1.94}_{-1.09}\over r_{\rm dec}}\;\;\;({\rm
ang.\;power\;spectrum})\;,$ (7)

corresponding to a maximum fluctuation size $\theta_{\rm max}\approx
127^{\circ}$ in the plane of the sky. For a redshift $z_{\rm dec}=1080$ at
decoupling, this cutoff represents a maximum fluctuation size $\lambda_{\rm
max}\sim 31$ Mpc at that redshift. The six basic parameters for this fit, and
their confidence regions, are shown in figure 3 and listed in Table 1. Note
that, in spite of the fact that a $k_{\rm min}$ compatible with zero appears
to be possible within $1\sigma$ in the $\tau-k_{\rm min}$ panel of figure 3,
this is not the case because the $1\sigma$ and $2\sigma$ regions are different
in 1D and 2D. The distinction arises because the $1\sigma$ and $2\sigma$
levels in the 2D histograms are not the $68\%$ and $95\%$ values used for the
1D distributions. The relevant $1\sigma$ and $2\sigma$ levels for a 2D
histogram of samples are $1-\exp(-0.5)\rightarrow 39.3\%$ and
$1-\exp(-2)\rightarrow
86.5\%$.222https://corner.readthedocs.io/en/latest/pages/sigmas.html. Using
$39.3\%$ and $86.5\%$ of the region, the contour plots then do exclude a
$k_{\rm min}$ compatible with zero at the $1\sigma$ (1D) level.

The fact that a non-zero cutoff $k_{\rm min}$ has minimal, if any, impact on
the optimized fit at $\ell\gtrsim 30$, is consistent with the prevailing view
that the physical origin of the fluctuations is due to different physical
effects above and below $\sim 1^{\circ}$. It may therefore be useful to
compare this optimized value of $k_{\rm min}$ based on the whole range of
$\ell$’s with that found when $\ell$ is restricted to the range ($\lesssim
30$) thought to be dominated by the Sachs-Wolfe effect (Sachs & Wolfe 1967).
The optimized parameter values for this case, using the low-$\ell$ TT data
from Planck 2018, combined with the DES, BAO, and Type Ia SNe data sets, are
shown in figure 4. The corresponding best-fit cutoff $k_{\rm min}$ for this
restricted range is $(3.3^{+1.7}_{-1.3})\times 10^{-4}$ Mpc-1, consistent with
the full-$\ell$ range value, but even closer to the cutoff found using the
angular correlation function. In either case, the cutoff ‘fixing’ the low-
power anomaly at $\ell\lesssim 4$ is fully consistent with the value that
completely accounts for the missing correlations at angles $\gtrsim
60^{\circ}$, suggesting that both of these large-angle anomalies in the
standard model are due to the same truncation in ${\mathcal{P}}(k)$.

Table 1: Optimized Parameters in $\Lambda$CDM with and without $k_{\rm min}\not=0$ Parameters | $k_{\rm min}=0$ | $k_{\rm min}\not=0$
---|---|---
$H_{0}$ (${\rm km}\;{\rm s}^{-1}\;{\rm Mpc}^{-1}$) | $68.12\pm 0.37$ | $68.12\pm 0.38$
$\Omega_{\rm b}\,h^{2}$ | $0.02250\pm 0.00013$ | $0.02250\pm 0.00013$
$\Omega_{\rm c}\,h^{2}$ | $0.11830\pm 0.00082$ | $0.11832\pm 0.00082$
$\tau$ | $0.0586^{+0.0067}_{-0.0078}$ | $0.0587^{+0.0069}_{-0.0077}$
$n_{\rm s}$ | $0.9682\pm 0.0036$ | $0.9681\pm 0.0036$
$\ln(10^{10}A_{\rm s})$ | $3.050^{+0.013}_{-0.015}$ | $3.050\pm 0.015$
$k_{\rm min}$ (${\rm Mpc}^{-1}$) | 0 | $(2.04^{+1.4}_{-0.79})\times 10^{-4}$

Figure 3: The six basic parameters in $\Lambda$CDM optimized in our procedure,
based on the combined data sets SN+DES+BAO+Planck, together with the new
parameter ($k_{\rm min}$) we have introduced to represent a cutoff in the
primordial power spectrum, ${\mathcal{P}}(k)$.

##  4 Discussion

A side-by-side comparison of the best-fit parameters in $\Lambda$CDM with and
without a cutoff $k_{\rm min}$ (Table 1) reveals only negligible differences
between them, but the angular power spectrum for $\ell\lesssim 4$ is improved
significantly compared with the data. Thus, the excellent agreement between
standard $\Lambda$CDM and the CMB data for $\ell\gtrsim 60$ is completely
unaffected by our introduction of a cutoff to the primordial power spectrum.
Yet, based on the optimized $k_{\rm min}$ value using the whole $\ell$ range
(and even more so using the restricted range $\ell\lesssim 30$), we conclude
that a zero cutoff is ruled out at a confidence level of $\sim 2.6\sigma$. On
its own, this is already interesting enough for us to study its impact on
slow-roll inflation (Liu & Melia 2020; Melia 2020), but when combined with our
earlier measurement of $k_{\rm min}$ based on the angular correlation function
(Melia & López-Corredoira 2018), which showed that $k_{\rm min}=0$ was ruled
out at a confidence level exceeding $8\sigma$ for those data, these two
complementary indicators argue compellingly that the primordial power spectrum
${\mathcal{P}}(k)$ must have been truncated at $k_{\rm min}\sim 3\times
10^{-4}\;{\rm Mpc}^{-1}$.

Figure 4: Same as figure 3, except now using an optimization of the fit for
$\ell\leq 30$, and the low-$\ell$ TT data from Planck 2018 (Planck
Collaboration VI 2020), combined with the DES, BAO, and Type Ia SNe data sets.

These results are completely consistent with the prevailing view that the CMB
anisotropies are primarily due to two distinct physical influences: the Sachs-
Wolfe effect (Sachs & Wolfe 1967) at small $\ell$’s and acoustic oscillations
for $\ell\gtrsim 30$. The excellent fit to the data for large $\ell$’s (see
fig. 1) reinforces the perception that we understand fairly well the physical
process associated with small-angle anisotropies, in spite of the fact that
the angular correlation function has been problematic since the beginning
(Hinshaw et al. 1996; Bennett et al. 2003; Planck Collaboration VI 2020).

The emerging evidence for a non-zero $k_{\rm min}$, on the other hand, speaks
directly to the cosmological expansion itself. At $\ell\lesssim 30$, we see
the effects due to the metric perturbations associated with the cosmological
dynamics, and therefore the direct influence of the cosmological model itself.
In this range, we are probing ever closer to the onset of the hypothesized
inflationary expansion, culminating with the cutoff $k_{\rm min}$, that
signals the very first mode crossing the horizon at the beginning of the
quasi-de Sitter phase produced by the inflaton field (Liu & Melia 2020; Melia
2020).

The idea that at least one of the large-angle anomalies may be due to features
in the primordial power spectrum, ${\mathcal{P}}(k)$, has been with us for
several decades (see discussion in Planck Collaboration VII 2020). For
example, Shafieloo & Souradeep (2004) assumed an exponential cutoff at low
$k$’s, and found an optimized fit of the attenuation generally consistent with
the cutoff we have measured in this paper. Their ansatz for ${\mathcal{P}}(k)$
did not include an actual truncation, however, which would be required for
self-consistency with our previous treatment of $k_{\rm min}$ in Melia &
López-Corredoira (2018). The impact on inflation is different in these two
cases, since a sharp termination signals a precise time at which inflation
would have started. An exponential cutoff is not fully consistent with the
notion of horizon-crossing for the freezing of modes. In addition, this early
treatment was based on the use of WMAP data (Bennett et al. 2003), whereas our
analysis incorporates the higher precision Planck observations (Planck
Collaboration VI 2020). In figure 4, we have used only the Planck TT data at
$\ell\leq 30$, plus other low-redshift data sets, as described in the caption.
For figure 3, however, we used all of the Planck data, including TT, TE, EE
and also both the high-$\ell$ and other low-redshift data sets. A comparison
of these results shows that the optimized value of $k_{\rm min}$ is very
similar in these two cases, suggesting that the other low-redshift data sets,
such as DES, BAO and Type Ia SNe, do not significantly affect the limits on
the cutoff.

In related work, Nicholson & Contaldi (2009) and Hazra et al. (2014) modeled a
‘dip’ in ${\mathcal{P}}(k)$ on a scale $k\sim 0.002$ Mpc-1 for the WMAP data.
Their results confirmed those of an alternative approach (Ichiki et al. 2010),
in which an oscillatory modulation was identified around $k\sim 0.009$ Mpc-1.
These are not a truncation, however, nor are they consistent with our analysis
(Melia & López-Corredoira 2018) of a cutoff $k_{\rm min}$ in the angular
correlation function based on the Planck measurements.

Tocchini-Valentini et al. (2005) modeled both a dip at $k\sim 0.035$ Mpc-1 and
a ‘bump’ at $k\sim 0.05$ Mpc-1, also using WMAP data. As with the others,
however, these features are not the same as an actual truncation $k_{\rm
min}$, and our analysis utilizes the latest Planck data, rather than the less
precise WMAP measurements. Subsequent work by these authors, notably Tocchini-
Valentini et al. (2006), improved on this analysis considerably, though they
were still fully reliant on WMAP. These authors found evidence for three
features in ${\mathcal{P}}(k)$, one of which is analogous to our $k_{\rm
min}$, including a cutoff at $\sim 0.0001-0.001$ Mpc-1, with a confidence
level of about $2\sigma$. Our analysis based on the improved Planck data
produces a tighter constraint on this cutoff, more in line with the $k_{\rm
min}$ found from the angular correlation function. For example, their possible
cutoff range exceeds the actual value we have measured for $k_{\rm min}$ in
this paper.

More recently, Hunt & Sarkar (2014, 2015) found evidence in the WMAP data of a
cutoff $k<5\times 10^{-4}$ Mpc-1 to ${\mathcal{P}}(k)$, but concluded that
more accurate data, such as those provided by Planck, would need to be used to
confirm these results more robustly. The work most closely aligned with our
analysis in this paper is that of Iqbal et al. (2015) and Santos et al.
(2018). These two groups arrived at opposite conclusions to each other,
however. Though each carried out a thorough analysis of the possible cause for
the low CMB power at small $\ell$’s, and actually agreed on approximately what
value a cutoff (which they called $k_{c}$, analogous to our $k_{\rm min}$)
would be needed to address this anomaly, they adopted different model-
selection criteria and produced opposing outcomes.

This state of uncertainty has been with us for many years. The approach of
Iqbal et al. (2015) and Santos et al. (2018) was to focus solely on the
low-$\ell$ anomaly and to compare 7 or 8 different models, using a range of
model-selection tools. Our approach in this paper has been diametrically
opposite to this. We have only one goal in mind, motivated by the physics of
inflation. Most importantly, our previous analysis demonstrated that a zero
value for $k_{\rm min}$ is ruled out at over $8\sigma$. Notice, e.g., how
different this outcome is compared to that of Santos et al. (2018), who
concluded that standard inflationary cosmology, without any adjustment to
${\mathcal{P}}(k)$, is preferred by the CMB data and their choice of model
selection criteria.

The work we have carried out in this paper is based on the conclusion of our
previous work suggesting that only a simple cutoff $k_{\rm min}$ in
${\mathcal{P}}(k)$ is well motivated by the CMB angular correlation function.
We have not compared different models, nor have we based our analysis on a
subjective choice of model selection criteria, which seem to produce the
divergent outcomes seen, e.g., in Iqbal et al. (2015) and Santos et al.
(2018).

We showed in Liu & Melia (2020) that the cutoff $k_{\rm min}$ measured via the
CMB angular correlation function apparently rules out all inflationary models
based on slow-roll potentials, including those that may have been preceded by
radiation-dominated and/or kinetic-dominated phases (see also Handley et al.
2014). With such a cutoff, inflation could not simultaneously have solved the
horizon problem and accounted for the observed CMB fluctuation spectrum. This
is the reason why a confirmation of $k_{\rm min}$ based on the angular power
spectrum is so essential. The two large-angle anomalies are not necessarily
related (see also Planck Collaboration VII 2020), so our previous result with
$k_{\rm min}$ does not by itself ensure that it is also consistent with the
much better studied angular power spectrum, nor that it can also mitigate the
missing power at low-$\ell$’s. This paper has therefore been highly focused on
these two issues alone. Nevertheless, the fact that our measured value of
$k_{\rm min}$ is consistent with that found by Iqbal et al. (2015) and Santos
et al. (2018) is crucial because it demonstrates a consistent outcome by three
different groups.

We mention in passing that an extension of the work reported by Iqbal et al.
(2015) and Santos et al. (2018) to include the angular correlation function
would be very beneficial. Our previous assessment was that only a simple
cutoff in ${\mathcal{P}}(k)$ could completely account for the missing
correlations at large angles. It would be interesting to see if an independent
examination of these data, using a range of other models with various model
selection tools, such as those explored by these two groups, could alter this
conclusion.

In this paper, we have met our primary goal of showing that the introduction
of a non-zero cutoff in ${\mathcal{P}}(k)$ does not at all affect the
excellent agreement between the predictions of $\Lambda$CDM and the CMB
angular power spectrum at $\ell\gtrsim 30$. This is critical because it is
widely believed that the origin of the CMB anisotropies in this domain is well
understood, given that it relies on well-established astrophysical principles.

We have also demonstrated that the origin of the missing power in the
low-$\ell$ multipoles, particularly at $\ell=2$–$5$, is very likely due to the
same physical influence impacting the missing correlations at large angles.
The two optimized values of $k_{\rm min}$, i.e., $(3.14\pm 0.36)\times
10^{-4}$ Mpc-1 for the latter, and $(2.04^{+1.4}_{-0.79})\times 10^{-4}$ Mpc-1
for the former, are fully consistent with each other. These complementary
results reinforce the view that we are beginning to see direct evidence of the
initiation of inflation, if this process actually did occur. Nevertheless, the
disparity between the primordial fluctuation spectrum expected under these
conditions and what is actually required to produce the CMB anisotropies,
increases the tension between the concordance model and the observations.

##  5 Conclusion

The existence of large-angle anomalies in the temperature is uncontested
though, given the modest significance with which they disagree with the
standard model, and the fact that they were detected a posteriori, makes it
unclear how much evidence they actually provide for a true, physical origin
(Planck Collaboration VII 2020). They may simply be statistical fluctuations,
even though such an outcome appears to be highly unlikely. Nevertheless, if
any of them do indeed correspond to a real physical effect, it would be
extremely important for us to confirm this, justifying the continued attention
paid to these unusual features.

As noted earlier, however, we must still temper our conclusions regarding the
reality of these large-angle anomalies, given that the Planck Collaboration
VII (2020) study found only weaker evidence for them in the polarization data.
This is why upcoming missions designed specifically to measure the CMB
polarization will be so critical to the continuation of this work. For
example, LiteBIRD has been selected for development and launch in 2028 by the
Japan Aerospace Exploration Agency, with the goal of mapping the CMB
polarization over the whole sky with unprecedented precision (Hazumi et al.
2019). Its required angular coverage will correspond to $2\leq\ell\leq 200$,
perfectly attuned to the requirements for the analysis in this paper.

Perhaps even more impressively, the Probe of Inflation and Cosmic Origins
(PICO) mission (Hanany et al. 2019) is still in the study phase, but is
projected to be an imaging polarimeter scanning the sky for 5 years in 21
frequency bands spread between 21 and 799 GHz. If selected, PICO will produce
a full-sky survey of the intensity and polarization of the CMB with a final
combined-map noise level equivalent to an amazing 3,300 Planck missions.

And as a prominent third example, the Cosmic Origins Explorer (CORE) mission
(Delabrouille et al. 2018) proposed to the European Space Agency is projected
to have 19 frequency channels spanning the $60-600$ GHz range, and an angular
resolution from $2^{\prime}$ to $18^{\prime}$. It will observe the entire sky
repeatedly over four years of continuous scanning. Its design will be
optimized for complementarity with ground-based observations, performing
observations—such as that required by the analysis carried out in this
paper—essential to CMB polarization science not achievable without a dedicated
space mission.

Proposed, or in-progress, experiments to map the CMB polarization from the
ground include the Simons Observatory (Ade et al. 2019), located in the high
Atacama Desert in Northern Chile inside the Chajnator Science Preserve, at an
altitude of 5,200 meters. Though its goals are currently not as lofty as those
of the space-based missions, its principal aim is to produce a polarization
map of the sky with an order of magnitude better sensitivity than Planck.

The principal aim of all these future missions is, of course, to detect the
B-mode polarization in the tell-tale signature of gravity waves generated
during inflation. Nevertheless, a more precise measurement of the E-mode
polarization anisotropies at a much higher sensitivity than that available to
Planck should answer the question of whether or not the large-angle anomalies
seen in the temperature are also confirmed in the polarization maps. This
would then provide compelling evidence that they are not merely due to
secondary, background or instrumental effects, further motivating a study of
the impact on inflationary theory of a cutoff, $k_{\rm min}$, in the
primordial power spectrum, ${\mathcal{P}}(k)$.

For example, as shown by Liu & Melia (2020), a $k_{\min}$ like that measured
earlier in our analysis of the angular correlation function (Melia & López-
Corredoira 2018), and reinforced by the work reported in this paper, which is
based exclusively on the angular power spectrum, rules out the majority—if not
all—of the slow-roll inflaton potentials proposed thus far. If one insists on
inflation simultaneously fixing the horizon problem and accounting for the
observed primordial power spectrum, ${\mathcal{P}}(k)$, then the accelerated
expansion resulting from these hypothesized fields misses the required
comoving distance by a factor $\sim 10$. Moreover, neither a radiation-
dominated, nor a kinetic-dominated, phase preceding inflation can alleviate
this disparity (Liu & Melia 2020; see also Handley et al. 2014).

Our results reinforce the growing view that, at a minimum, inflation probably
needs to be modified or, at worst, needs to be replaced, in order to conform
with these observations.

###### Acknowledgements.

We are grateful to the anonymous referee for their helpful and thoughtful
review of this manuscript. FM is also grateful to Amherst College for its
support through a John Woodruff Simpson Lectureship. This work is partially
supported by the National Natural Science Foundation of China (grant No.
U1831122), the Youth Innovation Promotion Association (2017366), and the Key
Research Program of Frontier Sciences (grant No. ZDBS-LY-7014) of Chinese
Academy of Sciences, at Purple Mountain Observatory, and by the Innovation and
Entrepreneurial Project of Guizhou Province for High-level Overseas Talents
(grant No. 2019-02), the National Natural Science Foundation of China (grant
No. 11903010), and the Science and Technology Fund of Guizhou Province (grant
No. 2020-1Y020), at Guizhou University.

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