# Ultrafast Carrier Dynamics in Ag-CdTe Hybrid Nanostructure: Non-radiative
and Radiative Relaxations

Sabina Gurungab<EMAIL_ADDRESS>Durga Prasad Khatuaab Asha Singha J.
Jayabalanab a Nano Science Laboratory, Materials Science Section, Raja Ramanna
Centre for Advanced Technology, Indore, India - 452013.
b Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai, India - 400094.

###### Abstract

In this article, we study non-radiative and radiative relaxation processes in
a hybrid formed by combining Ag nanoparticle (NP) and CdTe quantum dots (QD)
using transient transmission spectroscopy. The ultrafast transient
transmission of hybrid, when excited at 400 nm, shows a faster recovery of hot
electrons at a shorter time scale (few picoseconds) while it shows a slower
recovery at longer time scale (few tens of picoseconds). Further it is found
that the contribution of CdTe QD to the transient transmission is increased in
the presence of Ag NP. However, the radiative relaxation in CdTe QDs get
quenched in the presence of Ag NP. This work provides significant insight into
the various relaxation processes that leads to the charge transport and PL
quenching mechanisms in metal-semiconductor hybrids.

Hot-electrons and Transient and Metal Nanoparticles and Semiconductor Quantum
dots and Hybrid and Electron-Phonon coupling

###### pacs:

Valid PACS appear here

††preprint: Journal of Physics: Condensed Matter

##  I Introduction

Metal-semiconductor hybrid nanostructure (HNS) shows unique properties that
can supersede the combined function of individual material because of its
synergistic behavior. A metal nanoparticle (NP) when excited at its localized
surface plasmon resonance (LSPR), can modify the local field distribution
around it. The light-matter interaction in a semiconductor quantum dots (QD)
can be enhanced by placing it near to the metal NP. Such enhancement can
increase the efficiency of semiconductor devices like photodetector, solar
cells, LED etcSalamin _et al._ (2018); Atwater and Polman ; Jiang _et al._
(2014). It is also possible that charges excited in the metal NP can move to
the semiconductor QD in ultrafast time scaleMondal and Samanta (2015); Singh
_et al._ (2019). Such charge transfer can further increase the efficiency of
light harvesting devicesBae _et al._ (2013); Wu _et al._ (2015). Conversely,
metal NP can act as an electron sink that can disturb the electron-hole
recombination process in the semiconductor QD resulting in quenching of
PLNaldoni _et al._ (2013); Hai _et al._ (2013). The carrier dynamics that
takes place between metal NP and semiconductor QD plays a vital role in
determining the final optical response of HNS. The ultrafast response of
hybrid can have both the plasmonic and excitonic contributions. Both of these
contributions have different physical origin and are usually described by
different physical theories. In the case of isolated metal NPs, ultrafast
optical response can be explained by the increase in temperature of free
electronsJayabalan (2011). On the other hand, in the case of isolated
semiconductor QDs, it is explained by band to band absorption, band filling,
inter and intra band relaxation, etc Carey _et al._ (2009). In semiconductor
QDs, both radiative and non-radiative processes contribute to the total
carrier dynamics, while in metal NP only non-radiative processes contribute.
Thus the various dynamical processes that occurs in individual materials in
HNS as well as their synergistic interaction complicates the understanding of
its complete optical response. Nonetheless, it is important to comprehend the
basic interactions between these individual constituents for designing an HNS
focused toward a particular application.

In this article, we study the origin of the ultrafast optical response of
metal-semiconductor HNS using a colloidal mixture of Ag NP and CdTe QD,
labeled as Ag-nCdTe, using ultrafast transient spectroscopy. Ag-nCdTe HNS
formed by Ag NPs and CdTe QDs is an excellent candidate to study the optical
response under a strong exciton-plasmon coupling regime because the plasmon
band of Ag NP is well separated from the band edge emission of CdTe QD.
Besides, CdTe QDs has a fast electron injection time constant compared to
other II-VI semiconductor like CdSeBang and Kamat (2009). To reveal the origin
of optical process in HNS, we excite the carriers in Ag-nCdTe hybrid system
and probe the carrier dynamics near Ag NP plasmon peak and exciton peak in
CdTe QD. We study the changes in optical properties with respect to the
plasmonic response by keeping the concentration of CdTe QDs in colloid low
enough such that we do not get any measurable response from the bare CdTe QDs
colloid. The ultrafast dynamics of Ag-nCdTe when probed at plasmon peak of Ag
NP and deep conduction band in CdTe QD is different from that when probed near
to the band edge of the CdTe QD and away from the plasmon band. We have also
studied the PL spectra and time-resolved PL of Ag-nCdTe hybrid colloid and
compared it with that of bare CdTe QD colloid. Lastly the origin of the
observed optical response is explained using electromagnetic theory, two
temperature model and hot-electron transport process.

##  II Results and Discussions

Figure 1: Extinction spectra of the colloidal solutions of Ag NP, CdTe QD and
Ag-nCdTe hybrid.

The individual colloidal solutions of Ag NPs and CdTe QDs dispersed in water
were prepared and characterized separately. The volume fraction of CdTe QDs
and Ag NPs in water in the individual colloids were of the order of $10^{-6}$
and $10^{-7}$, respectively. Fig.1 shows the extinction spectra of Ag NP and
CdTe QD colloids. The extinction spectrum of Ag NP colloid shows a well-known
single LSPR peak at 413 nm, which is typical for small Ag spherical particles
in waterBohren and Huffman (2008). The average size of Ag NPs corresponding to
the LSPR is found to be 17 nm using Atomic Force Microscopy (AFM) topography
measurement. The extinction spectrum of CdTe QD colloid shows the lowest
energy excitonic peak, the 1s-1s transition, at 560 nm. The average size of
the CdTe QDs estimated using Peng’s formula is $\sim$ 3.4 nm and also
confirmed by AFM measurement Yu _et al._ (2003). In the final colloidal
solution, CdTe QDs are capped with thiol-glycolic acid (TGA) and Ag NPs are
capped by trisodium citrate. Due to the low volume fractions and in presence
of capping agents, the individual particles are expected to be well separated
in their respective colloidal solutions.

A mixing ratio of $\gamma$ = 0.56, where $\gamma$ is defined as the ratio of
volume of as-prepared Ag NP colloid to that of CdTe QD colloid ($\gamma$ =
VP/VD) was used to prepare HNS colloidal sample. Although the capping agents
TGA and trisodium citrate develops negative charges on CdTe QD and Ag NP; once
mixed, the CdTe QDs get attached to the Ag NPs due to the higher affinity of
TGA to the metal surfaceChang _et al._ (2017); Gallagher _et al._ (2013).
Thus, self-assembled Ag-nCdTe hybrid nanostructures of negatively charged CdTe
QDs and Ag NPs are expected to form in the mixed colloid. Based on our
previous study, at the mixing ratio of $\gamma$ = 0.56, in the Ag-nCdTe
hybrid, each of the Ag NP is completely surrounded by 32 CdTe QDsGurung _et
al._ (2018).

Once these CdTe QDs surrounds the Ag NP surface, further attachment of CdTe
QDs to the same Ag NP is prevented thus making a stable HNS in the mixed
solution. Ideally in the Ag-nCdTe HNS sample it is expected that there are
only HNS with no individual particles. Fig.1 also shows the measured
extinction spectrum of Ag-nCdTe hybrid colloids and is different from that of
both of its constituent colloidsGurung _et al._ (2018). The Ag-nCdTe hybrid
has an increasing absorption strength towards UV regime similar to that of a
bare CdTe QD colloid. Additionally, the Ag-nCdTe hybrid has an increased
absorption in the longer wavelength regime (starting from $\sim$ 600 nm). The
excitonic peak in case of Ag-nCdTe colloid also got bleached compared to the
bare CdTe QD colloid. Various groups have reported similar changes in the
extinction spectrum in metal-semiconductor HNS when compared to that of
individualsWang _et al._ (2006); Nahar _et al._ (2015); Wuister _et al._
(2003); Xia _et al._ (2008); An _et al._ (2010). The increase in the
absorption cross-section in the longer wavelength of Ag-nCdTe is attributed to
the defect states created by the attachment of Ag NP to CdTe QD. On the other
hand, the presence of MNP also dissociates the excitons in the semiconductor
QD resulting in the reduction of exciton peak strength.

Figure 2: Schematic of the pump-probe setup used for the measurement of
transient transmission from the samples.

To study the ultrafast transient phenomena in Ag-nCdTe hybrid nanostructure,
transient transmission measurement were carried out in two-color pump-probe
geometry (Fig.2) using a 1 kHz Ti-Sapphire laser of pulse duration $\sim$ 35
fs operated at 800 nm wavelengthKhatua _et al._ (2020). In this setup, the
output of the femtosecond oscillator-amplifier was split into two beams using
a beam splitter. One of these beams was converted to 400 nm by second-harmonic
generation in a Beta-barium Borate crystal and was used as the pump beam. The
other beam from the beam splitter was fed to an optical parametric amplifier
(OPA). The OPA output, centered either at 408 nm or 550 nm, was used as the
probe beam. The wavelength of the pump beam was chosen such that it excites
the sample near the LSPR of the Ag NP in water. Similarly, the probe was
chosen such that it can measure the changes in the optical properties near the
LSPR of Ag NP or near the band edge of the CdTe QD. The pump fluence at the
sample location was 2.1 $\mu$Jmm-2. Throughout the measurement, the colloidal
samples were circulated in a 1 mm flow cell to avoid any thermal damage.

Figure 3: Transient transmission of Ag NPs and Ag-nCdTe hybrid colloidal
samples when excited at 400 nm and probe at 408 nm. Inset shows the normalized
curve of the same data.

Fig.3 shows the transient transmission ($\Delta T/T$) measured for the
colloidal samples of bare Ag NP and Ag-nCdTe hybrid colloid when probed at 408
nm. Both Ag NP and Ag-nCdTe hybrid colloids showed measurable change in the
transmission of the sample at intensity used in the measurement. The magnitude
of $\Delta T/T$ in both the colloids start increasing with the arrival of the
pump pulse, reaching a maximum by about 450 fs. At later time delay, $\Delta
T/T$ reduces, leaving a residue even up to 100 ps. However, the peak of
$\Delta T/T$ of Ag-nCdTe colloid when probed at 408 nm is only 10% of that of
Ag NP colloid. Further, the reduction in the amplitude of fast component is
much higher than that of the longer decay component. This difference in the
reduction in the magnitudes can be clearly seen in the normalized plot of the
$\Delta T/T$ (inset of Fig.3). Relatively, the long decay time component is
much more in the case of Ag-nCdTe hybrid compared to that of bare Ag NP
colloid. For the number density of CdTe QD same as that in the Ag-nCdTe
hybrid, CdTe QD colloid did not show any measurable $\Delta T/T$ signal. Thus,
the finite $\Delta T/T$ measured for the Ag-nCdTe hybrid should be attributed
to the contribution by the Ag NP and the synergistic interaction between Ag NP
and CdTe QDs.

Let us first look at the origin of transient transmission signal from Ag NP
colloid. Over the decades, the origin of ultrafast optical response of the
bare metal NP colloid has been studied by several groupsGuillon _et al._
(2004); Singh _et al._ (2019); Del Fatti _et al._ (2000); Bigot _et al._
(2000); Jayabalan (2011). When an ultrashort pulse excites an Ag NP at or near
its LSPR, the absorbed energy creates a non-thermal energy distribution among
the electrons which changes to thermal distribution in about 300 fs to 500
fsDel Fatti _et al._ (2000); Guillon _et al._ (2004). At the end of the
thermalization, the temperature of the free electrons is much higher than that
of lattice. As the temperature of the free electrons increases, the real part
of the dielectric constant of metal also increasesJayabalan (2011). It is well
known that the peak of LSPR of metal nanosphere occur at a wavelength where
$\varepsilon_{m}^{r}$ = - 2 $\varepsilon_{h}$. Here, $\varepsilon_{m}^{r}$ is
the real part of dielectric constant of the metal and $\varepsilon_{h}$ is the
dielectric constant of the host medium (water in the present case). Once hit
by an ultrashort pulse, an increase in temperature of electrons will cause the
LSPR peak to shift towards a longer wavelength causing a change in the
transmission of the Ag NP colloid Jayabalan (2011); Del Fatti _et al._ (2000);
Bigot _et al._ (2000). Therefore the maximum in $\Delta T/T$ occur after the
thermalization of electrons which is about $\sim$ 450 fs after the excitation
by the pump pulse for the present Ag NP colloid. Further, when probed on the
blue side (408 nm), a red-shift of LSPR will cause an increase in transmission
(positive $\Delta T/T$). At later times the hot electrons heat the lattice
through electron-phonon (e-ph) interaction, together reaching a much lower
temperature than the initial hot electronsBigot _et al._ (2000). The cooling
of electron can be observed as a nearly exponential decay of the magnitude of
$\Delta T/T$ in picosecond timescales Bigot _et al._ (2000); Jayabalan (2011).
An exponential fit to the measured data in the case of Ag NP colloid yields
the e-ph thermalization time to be 2.3 ps at the pump fluence of 2.1
$\mu$Jmm-2. At the end of e-ph thermalization, the temperature of the particle
is higher than that of the surrounding host. At later times, the particle
cools by delivering heat to the host which takes few hundreds of
picosecondsLink _et al._ (2002); Pustovalov (2005). This leaves a small
residual, $\Delta T/T$, even after 100 ps delay between the pump and probe
pulses.

Let us now look at the origin of the ultrafast optical response of the Ag-
nCdTe hybrid colloid. In a metal-semiconductor hybrid, when a metal NP
interacts with semiconductor QD, the optical response of the combined
nanostructure is expected to be very different from that of the individual
components. When excited at the LSPR, the local field near the metal NP can be
much higher than the applied; such a large field can increase the absorption
in semiconductor QDCohen-Hoshen _et al._ (2012); Luo and Zhao (2019). It is
also possible that excited carriers may get transported from metal NP to
semiconductor QD or vice versaSingh _et al._ (2019). Additionally, energy
exchange processes like foster resonance energy transfer (FRET) and plasmon-
induced resonance energy transfer (PIRET) are also possible in the hybrid
systemWargnier _et al._ (2004); Cushing _et al._ (2015). Coupling between the
metal NP and semiconductor QD is also known to introduce additional defect
states in the hybrid nanostructuresNahar _et al._ (2015); Wuister _et al._
(2003). The individual contribution from each of these above mentioned
processes to the optical response of the hybrid depends on the shape, size and
material of the individual components, the properties of linking molecules and
their spatial distribution. To understand the origin of the optical response
of the hybrid arising due to their synergistic interaction, it is necessary to
understand changes in the LSPR of Ag NP due to the presence of CdTe QDs.

It is well known that the LSPR response of a metal NP depends strongly on the
dielectric constant of the surrounding medium. To understand the effect of the
presence of CdTe QD on the optical response of Ag NP, we have calculated the
optical response of the combined hybrid nanostructure using the T-matrix
technique. T-matrix is a numerical technique to calculate the optical response
of a collection of small spherical particles distributed in space using the
dielectric constant of the individual particles. It should be noted that the
optical response calculated using the T-matrix technique considers only
electromagnetic interaction between the Ag NP and CdTe QD. However in a real
metal-semiconductor hybrid, there is not only electromagnetic interaction but
several other charge and energy exchange processes mentioned earlier. Thus a
calculation which assumes only the electromagnetic interaction between the
metal and semiconductor would not mimic the true properties of the hybrid.
Nevertheless, such calculation would give some insight into the contribution
from electromagnetic interaction to the final optical response of the HNS. In
the T-matrix calculations, we used the experimentally measured sizes of the Ag
NP and CdTe QD. In the case of Ag, the size of NP is sufficiently large enough
to neglect the quantum confinement effects, and the bulk dielectric constants
($\varepsilon$) reported by Johnson and ChristyJohnson and Christy (1972) were
directly used for the T-matrix calculation. On the other hand, quantum
confinement effect plays a vital role in modifying the dielectric constant of
the CdTe QDs. Therefore to account for the quantum confinement effect, we
estimate the modified dielectric constant of CdTe QD using the method reported
by Marcelo Alves-Santos et.al. Alves-Santos _et al._ (2010). This method uses
a trial and error procedure to estimate the dielectric constants of CdTe QD
using its bulk counterpart with the experimentally measured absorption
spectrum. Figure.4 shows the calculated extinction cross-section of a Ag NP
surrounded by 32 CdTe QDs using T-matrix technique. For comparison, we have
also shown the calculated extinction spectra of a single Ag NP in water and 32
non-interacting CdTe QDs. As expected, the optical response of Ag-nCdTe
colloid is quite different with a strong red-shifted LSPR peak of about 75 nm
compared to that of Ag NP colloid Further, the peak strength of LSPR peak for
Ag NP also reduces to about 65% of that of Ag NP. Such red-shift of LSPR peak
and reduction in peak strength has been observed earlier in the hybrid formed
by Ag nanoplate and CdTe QDsSingh _et al._ (2019). Thus, an excitation at 400
nm would only weakly excite the LSPR in case of Ag-nCdTe hybrid.

Figure 4: The extinction cross-section calculated using T-Matrix technique for
a single Ag NP in water (orange), 32 non-interacting CdTe QDs and the hybrid
nanostructure Ag-nCdTe (1 Ag surrounded in all directions by 32 CdTe QDs).

It is well established that the peak change in $\Delta T/T$ directly depends
on the amount of energy absorbed by the metal NPJayabalan (2011); Singh _et
al._ (2019). Any change in the amount of absorbed energy in presence of CdTe
QDs should change the transient response of Ag NP in Ag-nCdTe hybrid. A
comparison between the calculated extinction spectra of Ag NP and Ag-nCdTe
hybrid can give an idea about the change in the absorbed energy when excited
at 400 nm. The numerically calculated LSPR peak of Ag NP in water is at 393 nm
while in the experiment, it is at 413 nm. Taking into account of such shift,
the absorption in Ag NP in the hybrid should be only 4% of that of bare Ag NP
at 400 nm. The measured peak change in $\Delta T/T$ of Ag-nCdTe hybrid colloid
is about 10% of that of Ag NP colloid (Fig.3). Further, an exponential decay
fit to the measured $\Delta T/T$ in case of Ag-nCdTe gives 1.3 ps while it is
2.3 ps in case Ag NP colloid. The e-ph thermalization time depends on the
absorbed energy in Ag NP. Thus a reduction in absorbed energy should also
reduce the decay time of $\Delta T/T$ as observed in the case of Ag-nCdTe
hybrid. Therefore, it looks tempting to attribute the observed changes in
$\Delta T/T$ to the reduction in absorbed energy in Ag NP when surrounded by
CdTe QDs in the hybrid colloid.

If the observed changes in $\Delta T/T$ is only due to such reduction in
absorption as predicted by electromagnetic theory, the measured dynamics in
Ag-nCdTe colloid can be explained using a two-temperature model (TTM), which
works well for pure Ag NP colloidJayabalan (2011); Del Fatti _et al._ (2000);
Bigot _et al._ (2000). Under low excitation conditions, change in dielectric
constant of Ag NP is proportional to the temperature of the electronsJayabalan
(2011). The relation between the electron temperature (Te) and lattice
temperature (Tl) in TTM is given by following differential equations:

$\displaystyle\frac{\partial T_{e}}{\partial t}$ $\displaystyle=$
$\displaystyle-\frac{g}{C_{e}}\left(T_{e}-T_{l}\right)+\frac{Q(t)}{C_{e}}$ (1)
$\displaystyle\frac{\partial T_{l}}{\partial t}$ $\displaystyle=$
$\displaystyle\frac{g}{C_{1}}\left(T_{e}-T_{l}\right)$ (2)

where, $g$ represents e-ph coupling constant, $Q(t)$ is the absorbed power
density, which is proportional to the pump intensity and $C_{e}$ (=$\gamma
T_{e}$) and $C_{l}$ is the specific heat capacities of the free electrons and
lattice respectively. Using the typical values reported for $g$, $C_{l}$, and
$\gamma$ the Eq.1 and Eq.2 were solved numerically to obtain the time
dependence of $T_{e}$. This estimated $T_{e}$ multiplied with a
proportionality constant $C$ was then fit to the decay part of the measured
$\Delta T/T$ with $Q$ and $C$ as fitting parameters. Figure.5 shows the best
fit obtained for the decay part of the measured $\Delta T/T$ of Ag NP colloid.
To fit the $\Delta T/T$ of Ag-nCdTe colloid, we keep all the parameters same
as that used in the fitting of Ag NP colloid including the constant $C$ and
reduce only $Q(t)$ until the peak $\Delta T/T$ matches that of the Ag-nCdTe
colloid. The transient curve thus obtained is also shown in Fig.5. Clearly,
for the corresponding peak change, the fast decay time should have been much
smaller than that observed in the experiment. Further, the magnitude of the
long decay component for Ag-nCdTe should also be much lower than that observed
in the experiment (inset of Fig.3). Thus, the measured transient signal in Ag-
nCdTe hybrid could not be just attributed to the transient plasmonic response
of Ag NP in the hybrid colloid. This suggests that the presence of CdTe QDs
coupled with Ag NP in the Ag-nCdTe hybrid do play a role in dictating the
final optical response. Several groups have measured the optical response of
pure CdTe QDs Kaniyankandy _et al._ (2010); Rawalekar _et al._ (2010); Mondal
and Samanta (2015). When CdTe QD colloid is pumped at 400 nm and probed on the
red-side close to 400 nm, it shows a small increase in transmission (positive
$\Delta T/T$). This increase in the transmission is attributed to the band-
filling effect when carriers are excited into the conduction band of CdTe QD
from its valance band Carey _et al._ (2009); Kaniyankandy _et al._ (2010).
Note that the bare CdTe QD colloid having same number density as that of Ag-
nCdTe hybrid colloid, did not show any measurable signal. Nonetheless, an
increased absorption in CdTe QD in presence of Ag NP near to it can contribute
to the increased signal. Further, hot carriers from Ag NP can also get
transported to CdTe QD in femtosecond time scale which can further increasing
the transient response of CdTe QD in Ag-nCdTe hybrid. However, since all these
proceseses contributes positively to the measured $\Delta T/T$, it is
difficult to distinguish between the various contributions.

Figure 5: The TTM fit to the experimental $\Delta T/T$ measured for Ag NP
colloid and Ag-nCdTe colloid. The data for Ag-nCdTe is increased by 10 times
to make it comparable with the Ag NP. Figure 6: Transient transmission of Ag
NPs and Ag-nCdTe hybrid colloidal samples when excited at 400 nm and probe at
550 nm. Inset shows the normalized curve of the same data.

When excited at 400 nm and probed close to the exciton peak, the CdTe QD
colloid show a much larger positive $\Delta T/T$ signal when compared to that
of probing near 400 nmKaniyankandy _et al._ (2010); Rawalekar _et al._ (2010);
Mondal and Samanta (2015). Further, when probed on the red-side of LSPR and
well away from the LSPR, Ag NP shows a weak negative $\Delta T/T$. Hence it
should be possible to distinguish various processes if probed close to the
1s-1s exciton transition of the CdTe QD, i.e. at 550 nm. Figure.6 shows the
transient transmission signal measured for the colloidal samples of bare Ag NP
and Ag-nCdTe hybrid colloid when probed at 550 nm. The transmission of both
the samples reduces with the arrival of the pump pulse ($\Delta T/T$ is
negative). Once again a maximum of $|\Delta T/T|$ occurs by about 450 fs after
the arrival of the pump pulse. Both the negative $\Delta T/T$ and the delayed
maximum are the signature of plasmonic contribution from Ag NP. In the next
few tens of picoseconds, the $\Delta T/T$ of Ag NP colloid recovers with a
time constant of 2.4 ps which is nearly same as that of e-ph thermalization
time observed at 408 nm probing. On the other hand, in case of Ag-nCdTe
colloid, the $\Delta T/T$ changes sign near about 3 ps, becoming positive at
later times. This positive signal which could be a signature of the CdTe QD
response, remains more or less constant up to 100 ps. The negative
contribution of plasmonic response of Ag NP to the $\Delta T/T$ of Ag-nCdTe
recovers within few picoseconds and is much faster than the positive
contribution of CdTe QD. Thus at longer delays, the sign of $\Delta T/T$
becomes positive explaining the observed signal in $\Delta T/T$ at delays more
than 3 ps in Fig.6. To estimate the time constants the measured $\Delta T/T$
of Ag-nCdTe colloid at 550 nm was fit to the equation:

$F(t)=\frac{1}{2}\mathcal{E}\left[t/\tau_{r}\right)\left[-Ae^{t/\tau_{1}}+B\left(1-e^{t/\tau_{1}}\right)+C\right],$
(3)

where, $\mathcal{E}$ is the error function. The first decay term corresponds
to the plasmonic response of Ag NP and has a negative amplitude while the
second term can be attributed to the positive contribution of band filling
effect in CdTe QDs. We find that Eq.3 fits well to the measured $\Delta T/T$
with a time constant $\tau_{1}$ = 1.3 ps. This time constant is same as that
of the decay constant of Ag-nCdTe hybrid when probed at 408 nm. Thus a model
with enhanced absorption in CdTe QDs along with hot carrier dynamics in Ag NP
can explain the observed changes in the measured transient $\Delta T/T$ at
these probe wavelengths.

Figure 7: (a) Photoluminescence spectra and (b) time-resolved
photoluminescence of CdTe QD and Ag-nCdTe hybrid colloids. The inset shows the
amplitudes of the best fit parameters obtained by fitting Eq.4 to the
experimental data.

Now, let us look at the effect of presence of Ag NP on the photoluminescence
(PL) properties of the CdTe QDs. The PL measurement was carried out by
exciting the sample at 405 nm. The PL emitted by the sample was collected
using lens combination and detected by using a spectrograph(for spectrum
measurement) or a fast photodetector-oscilloscope system(for dynamics). The
photodetector has a rise time of 300 ps and the oscilloscope has a bandwidth
of 1 GHz. The decay time estimated for the later part of the instrument
response function (IRF) is 350 ps. If the field enhancement due to Ag NP
increases the absorption in CdTe QD, after oscillation more number of
electrons and holes should be present in the CdTe QD coupled to an Ag NP when
compared to that of a bare CdTe QD. If all these electrons and holes recombine
radiatively, there should be an increase in the PL emission from Ag-nCdTe
hybrid compared to that of bare CdTe QD colloid. Figure.7(a) shows the
measured PL spectrum of CdTe QD colloid and Ag-nCdTe hybrid colloids. We find
that the integrated area under the PL curve of Ag-nCdTe hybrid did not
increase rather it got quenched to nearly 70% of that of CdTe QD colloid.
Figure.7(b) shows the time-resolved PL measured for CdTe QD and Ag-nCdTe
hybrid colloids. The time-resolved PL is fitted to tri-exponential function
convoluted with the experimental IRF ($I_{IRF}$), given by

$\displaystyle
F(t)=\left(I_{IRF}(\otimes)\sum_{i=1,2,3}{A_{i}e^{\frac{-t}{\tau_{i}}}}\right)$
(4)

where Ai is the signal amplitude corresponding to the decay time, $\tau_{i}$.
Figure.7(b) also shows the best fit obtained by fitting Eq.4 to the
experimental data. The fast decay time, $\tau_{1}$ = 0.23 ns, obtained for
CdTe QD colloid could be attributed to Auger like non-radiative relaxations
while the other decay times 7 ns and 65 ns could be attributed to the raditive
relaxation to the band-to-band recombination and defect state emission,
respectivelySagarzazu _et al._ (2011); Espinobarro-Velazquez _et al._ (2015);
Wang _et al._ (2003, 2003). The amplitude of each of these components are
shown in inset of Fig.7(b). The Ag-nCdTe hybrid also shows similar three
component PL decay and the corresponding amplitudes are also shown in inset of
Fig.7(b). When compared with the CdTe QD colloid, the Ag-nCdTe hybrid shows an
increase in the non-radiative component ($A_{1}$) while the radiative
components $A_{2}$ and $A_{3}$ decreases. Although carrier density in CdTe QD
is increased due to the presence of the Ag NP, such increase in carrier
density can also increase Auger like recombinationsLeBlanc _et al._ (2013);
Naiki _et al._ (2011). In Auger recombination the electron and hole
recombination energy is transferred non-radiatively to carriers deep into
their corresponding bands. These deeply excited carriers then relax non-
radiatively. Consequently, a fast quenching of radiative relaxation is
observed in Ag-nCdTe hybrid colloid (Fig.7(a).

##  III Conclusion

A hybrid formation between a metal and semiconductor is known to significantly
alter the static optical responses like absorption and PL spectra. We find
that both non-radiative and radiative components of the individual entities
also gets significantly altered when Ag NP and CdTe QD are brought in close
contact. Transient transmission measurements performed by exciting at 400 nm
and probing at two different wavelengths shows that the contribution of CdTe
QD to the transient response got increased in presence of Ag NP. However, such
increase in the CdTe QD response did not result in increasing the radiative
emission from the CdTe QDs. This work provides significant insight into the
various relaxation processes that leads to the charge transport and PL
quenching mechanisms in metal-semiconductor hybrids.

##  IV Acknowledgment

The authors are thankful to Tarun Kumar Sharma for fruitful suggestions and
encouragement. The authors, Sabina Gurung and Durga Prasad Khatua, are
thankful to RRCAT, Indore, under HBNI program, Mumbai, for the financial
support.

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# The Universe is not statistically isotropic

Joann Jones<EMAIL_ADDRESS>CERCA/ISO, Department of Physics, Case
Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA
Craig J. Copi<EMAIL_ADDRESS>CERCA/ISO, Department of Physics, Case
Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA
Glenn D. Starkman<EMAIL_ADDRESS>CERCA/ISO, Department of Physics,
Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA
Yashar Akrami<EMAIL_ADDRESS>Instituto de Física Teórica (IFT) UAM-
CSIC, C/ Nicolás Cabrera 13-15, Campus de Cantoblanco UAM, 28049 Madrid, Spain
CERCA/ISO, Department of Physics, Case Western Reserve University, 10900
Euclid Avenue, Cleveland, OH 44106, USA

###### Abstract

The standard cosmological model predicts statistically isotropic cosmic
microwave background (CMB) fluctuations. However, several summary statistics
of CMB isotropy have anomalous values, including: the low level of large-angle
temperature correlations, $S_{1/2}$; the excess power in odd versus even
low-$\ell$ multipoles, $R^{TT}$; the (low) variance of large-scale temperature
anisotropies in the ecliptic north, but not the south, $\sigma^{2}_{16}$; and
the alignment and planarity of the quadrupole and octopole of temperature,
$S_{QO}$. Individually, their low $p$-values are weak evidence for violation
of statistical isotropy. The correlations of the tail values of these
statistics have not to this point been studied. We show that the joint
probability of all four of these happening by chance in $\Lambda$CDM is likely
$\leq 3\times 10^{-8}$. This constitutes more than $5\sigma$ evidence for
violation of statistical isotropy.

††preprint: IFT-UAM/CSIC-23-132

Introduction. — Observations of the cosmic microwave background (CMB)
temperature and polarization anisotropies can rightfully be regarded as the
strongest evidence in support of a nearly spatially flat, statistically
isotropic $\Lambda$ Cold Dark Matter ($\Lambda$CDM) model of the Universe.
However, there are several large-angle features of the CMB, known as the
large-angle anomalies, that suggest violations of statistical isotropy (see,
e.g., [1, 2, 3, 4, 5]). Each feature is characterized by an anomaly-specific
statistic $S_{a}$ having a $p$-value (the probability of making an observation
at least as extreme as that actually observed) low enough to excite notice.
Typically $p(S_{a})<0.01$ in an appropriate ensemble of $\Lambda$CDM
realizations. Each anomaly was first identified in either the Cosmic
Background Explorer (COBE) [6] or Wilkinson Microwave Anisotropy Probe (WMAP)
data (e.g., Ref. [7]), and has subsequently been confirmed in more recent
Planck data (starting from Ref. [2]). (Within $\Lambda$CDM these anomalies are
separate from the parameter “tensions” that have received much attention,
especially the $H_{0}$ tension, the statistically significant difference
between the values of the Hubble constant determined from high and low
redshift probes; see, e.g., Ref. [5].)

Though each anomaly has a low $p$-value, they are often excused as statistical
flukes, especially as they were each identified a posteori. While this may be
convincing if the anomalies are all highly correlated, their joint $p$-value
may actually be far smaller than their individual ones, and far harder to
excuse. In this Letter, we consider four of these anomalies: the absence of
large-angle correlations, characterized by $S_{1/2}$; the parity asymmetry
($R^{TT}$); the low northern variance ($\sigma^{2}_{16}$, as a way of
characterizing hemispherical asymmetry); and the quadrupole-octopole alignment
($S_{QO}$). We explore whether or not these anomalies are correlated, i.e.,
how their joint probability distribution compares to the products of their
individual probability distributions.

While the correlations between anomaly statistics have previously been studied
[8], because each observation has a low $p$-value, we are less interested in
how the peaks of the distributions correlate than in how the tails do. This is
not captured by standard correlation measures. We show here that the joint
$p$-values of all pairs of these four anomalies in an ensemble of realizations
of best-fit $\Lambda$CDM are much less than the minimum of the pair,

$p(S_{A},S_{B})\ll\min[p(S_{A}),p(S_{B})]\,,$ (1)

for any pair of anomaly statistics $S_{A}$ and $S_{B}$, i.e., the statistics
are only weakly correlated in their tails.

We find that for three of the four Planck foreground-subtracted maps –
Commander, NILC, SMICA – the joint $p$-value of all four anomalies

$p(S_{1/2},R^{TT},\sigma^{2}_{16},S_{QO})\leq 3\times 10^{-8}\,.$ (2)

The outlier SEVEM map still has $p=1.8\times 10^{-7}$.

In those three maps, the $p$-value of the large-angle anomalies is less than
that of a $5.4\sigma$ Gaussian fluctuation. While there are undoubtedly look-
elsewhere penalties to be paid for this collection of mostly a posteori
statistical anomalies, it is clear that there is very strong evidence in the
CMB for the violation of statistical isotropy.

It is conventional to express both the predicted and observed CMB temperature
on the sky in terms of an expansion in spherical harmonics:

$T(\theta,\phi)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}a_{\ell m}Y_{\ell
m}(\theta,\phi)\,.$ (3)

The $\ell=0$ term represents the average temperature, while $\ell\geq 1$ are
the temperature anisotropies, $\Delta T(\theta,\phi)$. The $\ell=1$ term is
dominated by a large “Doppler” contribution due to our motion relative to some
appropriate average frame of rest, thus we conventionally focus our attention
on the cosmologically relevant terms $\ell\geq 2$.

$\Lambda$CDM as conventionally constructed predicts that the $a_{\ell m}$ are
statistically independent Gaussian (or nearly so) random variables of zero
mean that reflect the rotation and translation invariance of both the
underlying theory and the background geometry, so that their expected
correlations are purely diagonal (they are uncorrelated) and their variance
depends only on $\ell$,

$\langle a_{\ell
m}a_{\ell^{\prime}m^{\prime}}^{*}\rangle=C_{\ell}\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}.$
(4)

Here $\langle\cdots\rangle$ denotes an average over a hypothetical ensemble of
realizations of the underlying theory. The anomalies we discuss in this paper
call into question the assumptions and expectations encoded in Eq. 4.

The $C_{\ell}$ in Eq. 4 are the theoretical angular power spectrum. In
$\Lambda$CDM, they encode (nearly) all relevant cosmological information. They
depend on the 7 cosmological parameters that characterize the homogeneous
background geometry and its evolution, the stress-energy content of the
Universe, and the phenomenological power spectrum of primordial fluctuations
in the stress-energy and geometry. Similar descriptions apply to polarization,
but this work is confined to temperature fluctuations.

Enormous effort has been invested in building and operating CMB experiments so
as (primarily) to extract estimates of the underlying $C_{\ell}$.
Simplistically, this is done by obtaining a full-sky map $T(\theta,\phi)$,
extracting the $a_{\ell m}$, and computing the angular power spectrum

$C_{\ell}\equiv\frac{1}{2\ell+1}\sum^{\ell}_{m=-\ell}|a_{\ell m}|^{2}\,.$ (5)

In practice, there are many subtleties [9, 10] including the presence of
foregrounds and the consequent need to mask out parts of the sky where
foreground removal is unreliable.

The real-space angular correlation function is an alternative to the angular
power spectrum:

$C(\theta)=\langle
T(\bm{\hat{n}}_{1})T(\bm{\hat{n}}_{2})\rangle=\sum_{\ell=2}\frac{2\ell+1}{4\pi}C_{\ell}P_{\ell}(\cos\theta)\,,$
(6)

where $\bm{\hat{n}}_{1}$ and $\bm{\hat{n}}_{2}$ are two directions on the sky,
with $\cos\theta=\bm{\hat{n}}_{1}\cdot\bm{\hat{n}}_{2}$, and $P_{\ell}$ are
the Legendre polynomials. Strictly, the sum should start from $\ell=0$, but
for the reasons given above we consider the monopole-and-dipole-subtracted
angular correlation function. In a statistically isotropic universe or on a
full sky the $C_{\ell}$ in Eq. 6 are the same as the $C_{\ell}$ in Eq. 5 .

Maps and masks. — We make use of the data maps, masks, and best-fit
theoretical power spectrum provided by the Planck 2018 data release found on
the Planck Legacy Archive 111Available from https://pla.esac.esa.int/#maps..
We use the component-separated Commander, NILC, SEVEM, and SMICA maps and
apply the ‘common mask’ to all maps. Since the calculation of one of the
anomalies discussed below, $\sigma^{2}_{16}$, is based on the northern
ecliptic sky, all maps and the mask are rotated to ecliptic coordinates before
calculating any statistic. Since we are interested in large-scale features, we
work with maps downgraded to HEALPix 222See http://healpix.sourceforge.net.
resolution $N_{\mathrm{side}}=64$ or to $N_{\mathrm{side}}=16$. The rotating
and downgrading process is as follows. The Planck maps are provided at
$N_{\mathrm{side}}=2048$. We first rotate each map to ecliptic coordinates
using the healpy routine healpy.rotator.Rotator. We then downgrade the map to
$N_{\mathrm{side}}=64$ (for $S_{1/2}$ and $R^{TT}$) or to
$N_{\mathrm{side}}=16$ (for $\sigma^{2}_{16}$) including smoothing with the
appropriate Gaussian beams and applying the corresponding pixel window
functions as described in Ref. [8], i.e., we employ Gaussian beams with FWHM
of $0.0465421$ arcseconds and $0.186168$ arcseconds for $N_{\mathrm{side}}=64$
and $N_{\mathrm{side}}=16$, respectively. We follow the same procedure for the
masks, and choose a cutoff value of $0.9$ for resetting pixel values to 0 or
1.

Creating realizations. — We create $10^{8}$ $\Lambda$CDM realizations of the
CMB temperature using the Planck best-fit cosmological parameters [13]
333Available from https://pla.esac.esa.int/#cosmology. up to
$\ell_{\mathrm{max}}=200$. To remain consistent with the data maps, we
generate our simulated maps in ecliptic coordinates at $N_{\mathrm{side}}=64$
or $N_{\mathrm{side}}=16$, as required, including the pixel window function
and Gaussian beam smoothing appropriate for each $N_{\mathrm{side}}$. Here we
work with CMB-only simulated sky maps, i.e., we do not add simulated noise nor
foreground to our realizations of the CMB sky. (See below for detailed
discussions.)

The anomalies. — Broadly, the CMB anomalies are statistics other than the
$C_{\ell}$ that have values far into the tails of their distribution in the
ensemble of $\Lambda$CDM realizations. Many statistics have been studied and
many have been found to be anomalous, i.e., have a small $p$-value. Typically
the anomalous statistics that get reported have $p$-values less than a few
percent, though none have individual $p$-values smaller than approximately
$10^{-4}$.

Each of these anomalies is evidence against the validity of Eq. 4, i.e.,
against statistical isotropy of the Universe. However, they are all a posteori
– invented after looking at the data set in which they were first found to be
anomalous. This situation has led to something of a standoff – there are now
many examples of anomalies that cast statistical isotropy into doubt; however,
there is not yet a comprehensive fundamental model that post-dicts these
anomalies and makes other testable predictions. (Cosmic topology may be an
exception [15].) Further, some of us have argued that physical models for
anomalies are likely to make certain generic testable predictions for CMB
polarization [16, 17] and other observables [18, 19].

Meanwhile, scant attention has been given to the collective statistical
strength of the anomalies. The authors of Ref. [8] did study the correlations
of certain CMB anomaly statistics, but their attention was focused on the
peaks of those distributions. Below, we first recall a selection of four CMB
anomalies and then determine the correlation in the tails of their
distributions. The four statistics we have chosen are not exhaustive. We have
selected one representative statistic for each of four apparently unrelated
phenomena – lack of large-angle correlations, preference for odd parity,
difference between northern and southern hemispheres, and alignment of the
quadrupole and octopole. We follow closely the choices of Ref. [8] so as to
ameliorate any suspicion that we have cherry-picked our results, and we refer
the reader to Ref. [8] for in-depth discussions of many observational and
statistical issues associated with these statistics and the maps on which they
are calculated. Other statistics might certainly be added to this suite (e.g.,
the ones in Refs. [20, 21]) and might also be found to be largely uncorrelated
in $\Lambda$CDM with our four choices, strengthening the case for statistical
anisotropy.

## Lack of large-angle correlations, $S_{1/2}$:

The lack of large-angle correlations in the CMB temperature anisotropies was
first observed by COBE [22], quantified by WMAP [7, 23, 24], and most
precisely measured and characterized with Planck [2, 19, 3, 1, 4]. This
deficit is quantified by the $S_{1/2}$ statistic: the squared distance of the
two point angular correlation function $C(\theta)$ from zero for points on the
sky with angular separations from $60^{\circ}$ to $180^{\circ}$,

$S_{1/2}\equiv\int_{-1}^{1/2}[C(\theta)]^{2}\,\mathrm{d}(\cos\theta)\,.$ (7)

Though, $S_{1/2}$ can be calculated on full or masked skies, we only consider
the masked sky statistic in this work. In practice, $S_{1/2}$ is calculated
based on pseudo-$C_{\ell}$ extracted from $N_{\mathrm{side}}=64$ maps in
ecliptic coordinates with the common mask applied using NaMaster444Code
available from https://github.com/LSSTDESC/NaMaster.[26] up to
$\ell_{\mathrm{max}}=40$. See Ref. [24] for a discussion of its computation.

## Parity asymmetry, $R^{TT}$:

The parity asymmetry can be characterized in a number of ways. Here we choose
$R^{TT}$, a statistic characterizing the odd-parity preference at low-$\ell$
in the CMB [27],

$R^{TT}\equiv\frac{D_{+}(\ell_{\mathrm{max}})}{D_{-}(\ell_{\mathrm{max}})}\,,$
(8)

where (choosing $\ell_{\mathrm{max}}$ to be odd),

$D_{\pm}\equiv\frac{2}{\ell_{\mathrm{max}}-1}\sum_{\ell=2}^{\ell_{\mathrm{max}}}\frac{\ell(\ell+1)}{2\pi}\frac{(1\pm(-1)^{\ell})}{2}C_{\ell}\,.$
(9)

Following Ref. [8], we choose $\ell_{\mathrm{max}}=27$ and calculate $R^{TT}$
on unmasked $N_{\mathrm{side}}=64$ skies.

## Low northern variance, $\sigma_{16}^{2}$:

There are several indications that there is a statistically significant
difference between the northern and southern hemispheres of the CMB sky. This
was possibly first noted as an asymmetry in power [28] that was maximized when
the plane separating the hemispheres was approximately the ecliptic, and
independently using genus statistics in Ref. [29]. The anomaly was also
characterized as “dipole modulation” of the power [30] and as “variance
asymmetry” [31]. Planck 2013 [2] showed, using both full-resolution
($N_{\mathrm{side}}=2048$) and low-resolution ($N_{\mathrm{side}}=16$) maps
that the low variance was localized in the northern ecliptic hemisphere, with
a $p$-value of $\sim 0.1\%$, while the $p$-value for the southern hemisphere
was $\sim 45\%$. We adopt the $N_{\mathrm{side}}=16$ northern variance measure

$\sigma^{2}_{16}=\overline{(T-\bar{T})^{2}},$ (10)

where the overbar denotes an average over the pixels of an
$N_{\mathrm{side}}=16$ masked, northern ecliptic sky.

## Quadrupole-octopole alignment, $S_{QO}$:

There have been at least three ways presented to describe the alignment of the
quadrupole and octopole [32, 33, 34]. Following Ref. [8], we adopt the
approach of Ref. [33], which uses the multipole vector representation of CMB
temperature [35, 36]. The quadrupole is represented by two multipole vectors
$\bm{v}^{(2;j)}$, with $j=1,2$, and the octopole by three multipole vectors
$\bm{v}^{(3;j)}$, with $j=1,2,3$. This allows the quadrupole to be associated
with a single plane and the octopole to be associated with three planes. These
planes have oriented areas

$\bm{w}^{(\ell;i,j)}\equiv\bm{v}^{(\ell;i)}\times\bm{v}^{(\ell;j)}\,.$ (11)

It has been known since Ref. [32] that the octopole is unusually planar – its
three planes are unexpectedly aligned with one another – and that those planes
are unusually aligned with that of the quadrupole. A statistic used to
quantify the alignment of these four planes is

$S_{QO}\equiv\frac{1}{3}\sum_{i=1}^{2}\sum_{j=i+1}^{3}|\bm{w}^{(2;1,2)}\cdot\bm{w}^{(3;i,j)}|\,.$
(12)

Results. — The value of each of $S_{1/2}$, $R^{TT}$, $\sigma_{16}^{2}$, and
$S_{QO}$ for the four Planck maps is given in the second column of Table 1.
Along the diagonal of columns 3–6 and for rows 1–4 of each component-separated
map we present the $p$-values of those statistics from an ensemble of $10^{8}$
theoretical Planck-best-fit $\Lambda$CDM CMB skies. Each of those $p$-values
is $\lesssim 5\%$. In the off-diagonal entries below the diagonal we present
the $p$-values of pairs of statistics; in the off-diagonal entries above the
diagonal (blue) we present the factors by which these exceed the products of
individual $p$-values. For example, in SMICA $p(S_{1/2},R^{TT})=2.8\times
10^{-5}$, while $p(S_{1/2},R^{TT})/(p(S_{1/2})p(R^{TT}))=0.6$. Note that
consistently, only
$p(S_{1/2},\sigma_{16}^{2})/(p(S_{1/2})p(\sigma_{16}^{2}))\simeq 30\gg 1$; all
other pairs have factors which are relatively close to $1$. $R^{TT}$, in
particular, is very weakly correlated, or even anti-correlated, with the other
three. This is our first indication that the joint probability of all four
statistics, is much less than the smallest of them.

Stat. | Value | $S_{1/2}$ | $R^{TT}$ | $\sigma_{16}^{2}$ | $S_{QO}$
---|---|---|---|---|---
Commander
$S_{1/2}$ | 1272 | $1.5\times 10^{-3}$ | $\times{0.6}$ | $\times{27}$ | $\times{1.3}$
$R^{TT}$ | 0.7896 | $2.8\times 10^{-5}$ | $3.0\times 10^{-2}$ | $\times{1.1}$ | $\times{1.0}$
$\sigma_{16}^{2}$ | 617.6 | $1.2\times 10^{-4}$ | $1.0\times 10^{-4}$ | $3.1\times 10^{-3}$ | $\times{1.7}$
$S_{QO}$ | 0.7630 | $8.3\times 10^{-6}$ | $1.3\times 10^{-4}$ | $2.3\times 10^{-5}$ | $4.4\times 10^{-3}$
NILC
$S_{1/2}$ | 1218 | $1.3\times 10^{-3}$ | $\times{0.4}$ | $\times{29}$ | $\times{1.3}$
$R^{TT}$ | 0.7448 | $4.8\times 10^{-6}$ | $1.0\times 10^{-2}$ | $\times{1.0}$ | $\times{1.0}$
$\sigma_{16}^{2}$ | 605.9 | $9.2\times 10^{-5}$ | $2.4\times 10^{-5}$ | $2.5\times 10^{-3}$ | $\times{1.9}$
$S_{QO}$ | 0.8203 | $6.3\times 10^{-7}$ | $3.8\times 10^{-6}$ | $1.8\times 10^{-6}$ | $3.9\times 10^{-4}$
SEVEM
$S_{1/2}$ | 1215 | $1.3\times 10^{-3}$ | $\times{0.8}$ | $\times{33}$ | $\times{1.2}$
$R^{TT}$ | 0.8194 | $5.6\times 10^{-5}$ | $5.4\times 10^{-2}$ | $\times{1.2}$ | $\times{1.0}$
$\sigma_{16}^{2}$ | 583.4 | $6.5\times 10^{-5}$ | $1.0\times 10^{-4}$ | $1.6\times 10^{-3}$ | $\times{1.5}$
$S_{QO}$ | 0.6547 | $6.3\times 10^{-5}$ | $2.2\times 10^{-3}$ | $9.8\times 10^{-5}$ | $4.1\times 10^{-2}$
SMICA
$S_{1/2}$ | 1257 | $1.4\times 10^{-3}$ | $\times{0.6}$ | $\times{25}$ | $\times{1.3}$
$R^{TT}$ | 0.7906 | $2.8\times 10^{-5}$ | $3.0\times 10^{-2}$ | $\times{1.1}$ | $\times{1.0}$
$\sigma_{16}^{2}$ | 631.0 | $1.4\times 10^{-4}$ | $1.3\times 10^{-4}$ | $3.9\times 10^{-3}$ | $\times{1.8}$
$S_{QO}$ | 0.8048 | $1.7\times 10^{-6}$ | $2.9\times 10^{-5}$ | $6.6\times 10^{-6}$ | $9.2\times 10^{-4}$
Table 1:  $p$-values of the four anomaly statistics ($S_{1/2}$, $R^{TT}$,
$\sigma^{2}_{16}$, $S_{QO}$) for the four Planck component-separated CMB
temperature maps (Commander, NILC, SEVEM, and SMICA). For each map: the second
column shows the values of the statistics calculated with $S_{1/2}$ given in
$\mu\mathrm{K}^{4}$, $\sigma_{16}^{2}$ given in $\mu\mathrm{K}^{2}$, and
$R^{TT}$ and $S_{QO}$ both dimensionless; the “diagonal” entries of the four
rows and columns 3–6 are the individual $p$-values of the anomaly statistics
in an ensemble of $10^{8}$ realizations of best-fit $\Lambda$CDM; each off-
diagonal entry below the diagonal is the joint $p$-value of the statistic in
that row and column; the off-diagonal entries above the diagonal (blue) are
the factors by which the joint $p$-values exceed the products of the
individual $p$-values, i.e., $p(S_{A},S_{B})/(p(S_{A})p(S_{B}))$.

In Table 2, we examine the correlations between the most correlated pair of
statistics, $S_{1/2}$ and $\sigma_{16}^{2}$, and the next-most correlated
statistic $S_{QO}$. We find that there is little additional correlation among
them, in particular $0.6\leq
p(S_{1/2},\sigma_{16}^{2},S_{QO})/(p(S_{1/2},\sigma_{16}^{2})p(S_{QO}))\leq
2.1$.

|  $S_{1/2}$ and $\sigma_{16}^{2}$ | $S_{QO}$
---|---|---
Commander
$S_{1/2}$ and $\sigma_{16}^{2}$ | 1.2$\times 10^{-4}$ | $\times{1.7}$
$S_{QO}$ | 9.1$\times 10^{-7}$ | $4.4\times 10^{-3}$
NILC
$S_{1/2}$ and $\sigma_{16}^{2}$ | 9.1$\times 10^{-5}$ | $\times{0.6}$
$S_{QO}$ | 2.0$\times 10^{-8}$ | $3.9\times 10^{-4}$
SEVEM
$S_{1/2}$ and $\sigma_{16}^{2}$ | 6.5$\times 10^{-5}$ | $\times{1.3}$
$S_{QO}$ | 3.6$\times 10^{-6}$ | $4.1\times 10^{-2}$
SMICA
$S_{1/2}$ and $\sigma_{16}^{2}$ | 1.4$\times 10^{-4}$ | $\times{2.1}$
$S_{QO}$ | 2.7$\times 10^{-7}$ | $9.2\times 10^{-4}$
Table 2: For each of the four Planck component-separated CMB temperature maps
(Commander, NILC, SEVEM, and SMICA), along the diagonal are the joint
$p$-value of $S_{1/2}$ and $\sigma_{16}^{2}$, $p(S_{1/2},\sigma_{16}^{2})$,
and the $p$-value of $S_{QO}$, $p(S_{QO})$. In the off-diagonal below the
diagonal is the joint $p$-value of all three statistics
$p(S_{1/2},\sigma_{16}^{2},S_{QO})$, while above the diagonal (blue) is the
ratio
$p(S_{1/2},\sigma_{16}^{2},S_{QO})/(p(S_{1/2},\sigma_{16}^{2})p(S_{QO}))$.

Finally, Table 3 gives the joint $p$-values of all four anomaly statistics

$p_{4}\equiv p(S_{1/2},R^{TT},\sigma_{16}^{2},S_{QO})$ (13)

for the four component-separated maps and the “correlation factors” by which
they exceed the products of the four individual $p$-values,

$\frac{p(S_{1/2},R^{TT},\sigma_{16}^{2},S_{QO})}{p(S_{1/2})p(R^{TT})p(\sigma_{16}^{2})p(S_{QO})}\,.$
(14)

Map | $p_{4}$ | Correlation Factor
---|---|---
Commander | $3\times 10^{-8}$ | 51
NILC | $<1\times 10^{-8}$ | N/A
SEVEM | $18\times 10^{-8}$ | 40
SMICA | $1\times 10^{-8}$ | 64
Table 3: The joint $p$-values, $p_{4}$ of $S_{1/2}$, $R^{TT}$,
$\sigma_{16}^{2}$, and $S_{QO}$, and the correlation factors by which the
joint $p$-values exceed the products of the four individual $p$-values,
estimated on ensembles of $10^{8}$ realizations of Planck best-fit
$\Lambda$CDM for the four Planck component-separated CMB temperature maps
(Commander, NILC, SEVEM, and SMICA).

We see in Table 3 that $p_{4}$ for Commander, NILC, and SMICA are all $\leq
3\times 10^{-8}$, corresponding to $>5.4\sigma$ significance! Even the one
outlier, SEVEM, has a $p_{4}$ corresponding to $\sim 5.1\sigma$. There is no
particular reason to trust the SEVEM result over the others, and if it were
the correct value the other three would be extremely unlikely ($<1/10^{18}$)
to have happened by chance – there would have to be some reason why all three
of them come out so much smaller.

The correlation factor – the factor by which $p_{4}$ exceeds the product of
the four individual $p$-values – is thus consistently less than $100$, so that
$p_{4}$ is vastly smaller than the minimum of the individual $p$-values. Most
of that correlation factor is accounted for by the correlation between
$S_{1/2}$ and $\sigma^{2}_{16}$, but recall that even
$p(S_{1/2},\sigma^{2}_{16})<\frac{1}{10}\mathrm{min}(p(S_{1/2}),p(\sigma^{2}_{16}))$.

As remarked above, here we have worked with CMB-only simulated sky maps, i.e.,
we did not add simulated noise to our realizations of the CMB sky. This is
motivated by the fact that the Planck (and WMAP) measurements of large-scale
CMB temperature fluctuations are almost completely cosmic-variance-limited so
that we do not expect the noise to significantly affect our results and
conclusions. This is further supported by the fact that when we calculated
$p$-values of anomaly statistics and compared them to the corresponding Planck
values they agreed.

Working with noise-free simulations makes it possible to create ensembles of
$10^{8}$ realizations, which would not have been possible if we had added
noise to the simulations, as not more than $1000$ noise simulations are
available for the Planck sky maps. Additionally, WMAP and Planck, with very
different noise structures and properties, have both detected the large-scale
anomalies with very similar individual $p$-values, showing that the anomalies
are very unlikely to be due to experimental noise. Given that, it is very hard
to see how the noise would be responsible for all the anomalies together or
how the noise would significantly affect the joint $p$-values, i.e., the
correlations between anomalies.

We have also not attempted to simulate residual foregrounds. The Commander,
NILC, SEVEM, and SMICA teams have done that better than we can hope to. It
seems extraordinarily unlikely that all four of these foreground removal
methodologies would leave behind the identical anomalies, with nearly the same
values on all of the Planck maps (and similarly for all the foreground-removed
WMAP maps).

Conclusions. — The joint significance of the four statistics we chose to
consider much exceeds $5\sigma$, and for three of four maps exceeds
$5.5\sigma$. We should certainly mitigate this by the effect of some look-
elsewhere penalty. At the same time a more comprehensive study of all of the
many statistics that point toward the violation of statistical isotropy would
very likely increase the significance of this result. We conclude that it is
very unlikely that we just live in a rare realization of a statistically
isotropic universe. It is much more likely that the Universe is statistically
anisotropic.

###### Acknowledgements.

This work made use of the High Performance Computing Resource in the Core
Facility for Advanced Research Computing at Case Western Reserve University.
Some of the results in this paper have been derived using the healpy and
HEALPix packages [37, 38]. J.J., C.J.C., and G.D.S. acknowledge partial
support from NASA ATP grant RES240737; G.D.S. from DOE grant DESC0009946. Y.A.
acknowledges support by the Spanish Research Agency (Agencia Estatal de
Investigación)’s grant RYC2020-030193-I/AEI/10.13039/501100011033, by the
European Social Fund (Fondo Social Europeo) through the Ramón y Cajal program
within the State Plan for Scientific and Technical Research and Innovation
(Plan Estatal de Investigación Científica y Técnica y de Innovación)
2017-2020, and by the Spanish Research Agency through the grant IFT Centro de
Excelencia Severo Ochoa No CEX2020-001007-S funded by
MCIN/AEI/10.13039/501100011033.

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