# Intimate Relationship Between Sterile Neutrino Dark Matter and $\Delta
N_{\rm eff}$

Kevin J. Kelly<EMAIL_ADDRESS>ORCID: 0000-0002-4892-2093 Theory
Department, Fermi National Accelerator Laboratory, Batavia, IL 60510, USA
Manibrata Sen<EMAIL_ADDRESS>ORCID: 0000-0001-7948-4332  Department
of Physics, University of California Berkeley, Berkeley, CA 94720, USA
Department of Physics & Astronomy, Northwestern University, Evanston, IL
60208, USA  Yue Zhang<EMAIL_ADDRESS>ORCID: 0000-0002-1984-7450
Department of Physics, Carleton University, Ottawa, ON K1S 5B6, Canada

###### Abstract

The self-interacting neutrino hypothesis is well motivated for addressing the
tension between the origin of sterile neutrino dark matter and indirect
detection constraints. It can also result in a number of testable signals from
the laboratories to the cosmos. We show that, in a broad class of models where
the sterile neutrino dark matter relic density is generated by a light
neutrinophilic mediator, there must be a lower bound on the amount of extra
radiation in early universe, in particular $\Delta N_{\rm eff}>0.12$ at the
CMB epoch. This lower bound will be further strengthened with an improved
$X$-ray search at the Athena observatory. Such an intimate relationship will
be unambiguously tested by the upcoming CMB-S4 project.

###### pacs:

Valid PACS appear here

††preprint: FERMILAB-PUB-20-576-T, N3AS-20-005, NUHEP-TH/20-12

Introduction – The origin of dark matter (DM) poses a fascinating puzzle. A
sterile neutrino with a keV-scale mass and small mixing with active neutrinos
is a simple, longstanding DM candidate. Along with other traditional
candidates (e.g., WIMPs/axions), sterile neutrinos are highly testable. In the
minimal setup, the active-sterile neutrino mixing that can account for the
observed DM relic density Dodelson and Widrow (1994) is already in tension
with searches for DM decaying into monochromatic $X$-rays Abazajian (2017). An
elegant solution for alleviating this tension is to introduce a new self
interaction among active neutrinos. This keeps neutrinos in thermal
equilibrium with themselves longer in the early Universe and facilitates
efficient sterile neutrino DM (S$\nu$DM) production De Gouvêa  _et al._
(2020); Kelly  _et al._ (2020). This mechanism is attractive because it
predicts a number of novel signatures for laboratory tests Berryman  _et al._
(2018); de Gouvêa  _et al._ (2020a); Brdar  _et al._ (2020). Interestingly,
the self-interacting neutrino scenario emerges from a broad class of well-
motivated beyond-Standard-Model frameworks Chikashige  _et al._ (1980);
Gelmini and Roncadelli (1981); Das  _et al._ (2017); Kelly and Zhang (2019);
Ng and Beacom (2014); Ioka and Murase (2014); Forastieri  _et al._ (2015,
2019); Kreisch  _et al._ (2019); Blinov  _et al._ (2019); Shalgar  _et al._
(2019); Bustamante  _et al._ (2020); Berryman  _et al._ (2015); Coloma and
Peres (2017); de Gouvêa  _et al._ (2020b); Chacko  _et al._ (2020a, b).

We will soon enter a new precision era of cosmology. The Cosmic Microwave
Background - Stage 4 (CMB-S4) project Abazajian  _et al._ (2016) is expected
to measure cosmological parameters to an unprecedented level. In particular,
it could restrict the amount of extra radiation (in units of extra neutrinos,
$\Delta N_{\rm eff}$) during the CMB epoch to a few-percent uncertainty, while
it is still allowed (perhaps even favored) to be order one by the current data
Ade  _et al._ (2016); Bernal  _et al._ (2016). This could shed light on
existing tensions resulting from various ways of measuring the Hubble
parameter Freedman (2017); Wong  _et al._ (2019); Riess  _et al._ (2019);
Aghanim  _et al._ (2020), as well as fundamental theories that can accommodate
deviations from the standard cosmology. Moreover, $\Delta N_{\rm eff}$
measurement during the big-bang nucleosynthesis (BBN) epoch puts further
restrictions on room for new physics.

In this letter, we point out an intriguing relationship between the two
important ingredients of our Universe – the origin of S$\nu$DM and $\Delta
N_{\rm eff}$ for both CMB and BBN. The connection is established by a light
mediator with a feeble coupling to active neutrinos such that it enters the
thermal bath after BBN but decays away before CMB. S$\nu$DM then is dominantly
produced from active neutrino self scattering through the exchange of on-shell
mediator. The post-BBN thermalization of the mediator inevitably modifies the
standard cosmology predictions by making new contributions to $\Delta N_{\rm
eff}^{\text{\sc cmb}}$ and $\Delta N_{\rm eff}^{\text{\sc bbn}}$. As a result,
the viable parameter space for S$\nu$DM relic density is highly correlated
with the predictions of $\Delta N_{\rm eff}$. We further point out a new
implication of the upcoming $X$-ray search by the Athena experiment for the
next-generation precision measurements in cosmology– this scenario connects
different frontiers of cosmological and astrophysical probes.

Models – S$\nu$DM is a linear combination of a gauge singlet fermion and the
active neutrino states from the Standard Model Dodelson and Widrow (1994),
$\nu_{4}=\nu_{s}\cos\theta+\nu_{a}\sin\theta$, where $\nu_{4}$ denotes a mass
eigenstate and $\theta$ is the active-sterile mixing angle in vacuum. The
sterile neutrino is a decaying DM candidate whose longevity is attributed to
its small mass/mixing. We introduce a new force among active neutrinos,

$\mathcal{L}_{\rm
int.}=\frac{\lambda_{\alpha\beta}}{2}\nu_{\alpha}\nu_{\beta}\phi+\mathrm{h.c.}\
,\quad(\alpha,\beta=e,\mu,\tau),$ (1)

which can arise from dimension-six gauge invariant operators at energy scales
well below the weak scale. The scalar field $\phi$ could be either real or
complex. If real, $\phi$ could be identified as the Majoron Chikashige  _et
al._ (1980); Gelmini and Roncadelli (1981), where $\lambda=iM_{\nu}/f$ and $f$
is the spontaneous lepton number breaking scale. If complex, $\phi$ also
contains the radial mode and restores $B-L$ symmetry at high scales Berryman
_et al._ (2018). The impact of active neutrino self-interactions on the relic
density of S$\nu$DM was first explored in De Gouvêa  _et al._ (2020). The new
force enables more frequent active neutrino scattering than normal weak
interactions, thereby, S$\nu$DM can be produced with a smaller mixing angle
than required by Dodelson-Widrow Dodelson and Widrow (1994). Ref. De Gouvêa
_et al._ (2020) focuses on $m_{\phi}$ between MeV–GeV, where the new
interaction is sufficiently strong to fully thermalize $\phi$ before BBN.

Figure 1: The Big Picture: full parameter space that can account for the relic
density of S$\nu$DM (thick black curve). We choose $m_{4}=4$ keV and
$\sin^{2}2\theta=5\times 10^{-9}$. The colored regions show the experimental
constraints on neutrino self-interaction mediated by a real scalar $\phi$,
including leptonic meson decays Pasquini and Peres (2016); Berryman  _et al._
(2018); Blinov  _et al._ (2019); de Gouvêa  _et al._ (2020a) (purple),
invisible $Z$ width Berryman  _et al._ (2018); Brdar  _et al._ (2020)
(orange), double beta decay Agostini  _et al._ (2015); Blum  _et al._ (2018);
Brune and Päs (2019) (green), and $\Delta N_{\rm eff}$ for successful BBN
Pitrou  _et al._ (2018) (blue). We will explore the S$\nu$DM production
mechanism in the the sub-MeV $m_{\phi}$ region marked by “This Work” which
could be fully tested by the upcoming CMB-S4 and Athena experiments.

The present work broadens the focus and opens up the possibility of a sub-MeV
mediator $\phi$. The constraint from BBN is lifted if $\phi$ does not
thermalize before/during the BBN epoch and has no primordial population from
other high-scale physics Shi and Fuller (1999); Shaposhnikov and Tkachev
(2006); Asaka  _et al._ (2006); Roland  _et al._ (2015); Hansen and Vogl
(2017); Johns and Fuller (2019); Bezrukov  _et al._ (2010); Nemevsek  _et al._
(2012); Dror  _et al._ (2020). This is achieved with a feeble coupling to
neutrinos and a phase space suppression for $\nu\nu\to\phi$ to occur at
temperatures much higher than $m_{\phi}$. The condition for not thermalizing
$\phi$ before BBN sets an upper bound on $\lambda$ that scales as
$1/m_{\phi}$, whereas correct DM relic density requires $\lambda$ to scale as
$\sqrt{m_{\phi}}$ in the small coupling regime. As a result, viable parameter
space opens up for $m_{\phi}$ between keV and MeV scales.

Remarkably, our result, in tandem with that from Ref. De Gouvêa  _et al._
(2020), nicely forms the “Big Picture” of Fig. 1. The thick black curve
depicts the entire range of the theory space where the relic density of
S$\nu$DM is explained, aided by neutrino self-interaction. Viable mediator
mass range divides into two windows, above and below MeV scale. They
correspond to different histories of the neutrino and S$\nu$DM sector in the
early universe. In the heavier mediator region, the strongest probes are from
terrestrial experiments. In contrast, the sub-MeV mediator regime will be best
explored by cosmological observations of $\Delta N_{\rm eff}$, to be detailed
below.

Cosmology – S$\nu$DM production is governed by

$\frac{df_{4}(E,z)}{dz}=\frac{\Gamma(E,z)\sin^{2}2\theta}{4Hz}f_{a}(E,z)\Theta(E-m_{4})\
,$ (2)

where and $f_{4}$/$f_{a}$ are phase space distribution functions of
sterile/active neutrinos, respectively. We introduce $z\equiv
m_{\phi}/T_{\nu}^{\text{\sc sc}}$ to label time, where $T_{\nu}^{\text{\sc
sc}}$ is defined as the neutrino temperature in the standard cosmology without
$\phi$; $z$ expands linearly with the Universe’s radius. The actual neutrino
temperature $T_{\nu}$ is affected by the interaction in Eq. (1). After weak-
interaction decoupling, $T_{\nu}$ deviates from $T_{\nu}^{\text{\sc sc}}$ as
the $\phi$ population builds. $\Gamma$ is the thermally-averaged scattering
rate for an active neutrino, and $H$ is the Hubble parameter. Since we are
interested in small $m_{\phi}$ and $\lambda$, the effective active-sterile
mixing angle is approximately equal to the vacuum one throughout the dominant
S$\nu$DM production window.

The distribution function $f_{a}$, after active neutrinos thermalize with
$\phi$, is characterized by $T_{\nu}$ and a chemical potential $\mu_{\nu}$,

$f_{a}(E,z)=\frac{1}{1+\exp\left[\left(E-\mu_{\nu}(z)\right)/T_{\nu}(z)\right]}\
.$ (3)

To derive their $z$ dependence, we introduce three time scales,

  * •

$z_{A}$: the onset of BBN – neutrinos have just decoupled from weak
interaction. We assume $\phi$ has not yet equilibrated with neutrinos;

  * •

$z_{B}$: the $\nu$-$\phi$ system has just equilibrated. In general, $\phi$
remains ultra-relativistic;

  * •

$z_{C}$: all $\phi$ have decayed away ($T\ll m_{\phi}$), mostly into
neutrinos, with a small fraction decaying into S$\nu$DM.

Before $z_{A}$, neutrinos and photons obey $T_{\nu}=T_{\nu}^{\text{\sc
sc}}=T_{\gamma}$. The $\nu$-$\phi$ interaction freezes in a sub-thermal
population of $\phi$ via the Boltzmann equation

$\frac{dY_{\phi}}{dz}=-\frac{\gamma_{\phi\leftrightarrow\nu\nu}}{sHz}\left(\frac{Y_{\phi}}{Y_{\phi}^{\text{\sc
eq}}}-1\right)\ ,$ (4)

where $Y_{\phi}=n_{\phi}/s$ is the yield, $n_{\phi}$ is the number density of
$\phi$, and $s$ is the entropy density of the Universe. The thermally-averaged
rate is defined as $\gamma_{\phi\rightarrow\nu\nu}\equiv
n_{\nu}^{eq}\langle(m_{\phi}/E)\Gamma_{\phi\rightarrow\nu\nu}\rangle$ where
$\Gamma_{\phi}=3\lambda^{2}m_{\phi}/(32\pi)$ is the decay rate at rest. Here
we assume $\phi$ couples universally to all neutrinos,
$\lambda_{ee,\mu\mu,\tau\tau}=\lambda$. In the small $m_{\phi}$ limit,
$\gamma_{\phi\rightarrow\nu\nu}\simeq
m_{\phi}^{3}\Gamma_{\phi}/(2\pi^{2}z^{2})$.111The
$\nu\bar{\nu}\to\phi\phi^{(*)}$ process is only important for the coupling
$\lambda$ much larger than those of interest to this work Huang  _et al._
(2018); Barenboim and Nierste (2020). Assuming no $\phi$ population at early
times, integrating Eq. (4) up to $z_{A}$ determines the population of $\phi$
and its contribution to $\Delta N_{\rm eff}^{\text{\sc bbn}}$.

Figure 2: Sterile neutrino DM relic density (black contours) and $\Delta
N_{\rm eff}$ (orange and blue contours for CMB and BBN, respectively), as a
function of $\lambda$ and $m_{\phi}$ for real (left) and complex (right)
scalar $\phi$. We present three choices of the S$\nu$DM mass with the largest
experimentally allowed mixing angles (see main text).

From $z_{A}$–$z_{B}$, the $\nu$-$\phi$ interaction induces chemical
equilibrium. At $z_{B}$, $T_{\nu}$ deviates from $T_{\nu}^{\text{\sc sc}}$; a
chemical potential is also generated, $\mu_{\phi}=2\mu_{\nu}=2\mu_{\bar{\nu}}$
for real scalar $\phi$, or $\mu_{\phi}=2\mu_{\nu}$ and
$\mu_{\phi^{*}}=2\mu_{\bar{\nu}}$ for complex $\phi$. Without CP violation,
$\mu_{\phi}=\mu_{\phi}^{*}$. At $z_{B}$, $T_{\nu}$ and $\mu_{\nu}$ are
dictated by energy/lepton-number conservation Chacko  _et al._ (2004);
Escudero and Witte (2020); Escudero Abenza (2020); Berlin and Blinov (2018).
Entropy is not preserved here; the $\phi$ thermalization process is
irreversible. In our supplemental material, we provide details of this
matching.

The most important epoch for S$\nu$DM production is from $z_{B}$–$z_{C}$, by
the $\nu\nu\to\nu\nu_{4}$ process with on-shell, $s$-channel $\phi$ exchange.
This is terminated after $\phi$ becomes non-relativistic and freezes out.
Entropy/lepton-number conservation allow us to derive $T_{\nu}(z)$ and
$\mu_{\nu}(z)$. The total energy of the Universe increases because some $\phi$
decay while turning non-relativistic. The neutrino self-interaction rate also
depends on $T_{\nu}(z)$ and $\mu_{\nu}(z)$,

$\begin{split}\Gamma(E,z)&=\frac{3\lambda^{2}m_{\phi}^{2}T_{\nu}}{8\pi
E^{2}}\left[\ln\left(e^{\mu_{\nu}/T_{\nu}}+e^{\omega}\right)-\omega\rule{0.0pt}{11.38109pt}\right]\sqrt{1-\frac{m_{4}^{2}}{m_{\phi}^{2}}}\
,\end{split}$ (5)

where $\omega={m_{\phi}^{2}}/({4ET_{\nu}})$. With the above, we integrate Eq.
(2) up to $z_{C}$ to obtain the relic abundance of S$\nu$DM,

$\Omega_{4}=\frac{m_{4}s_{0}}{2\pi^{2}s(z_{C})\rho_{0}}\int_{m_{4}}^{\infty}EdE\sqrt{E^{2}-m_{4}^{2}}f_{4}(E,z_{C})\
,$ (6)

where $s_{0}=2891.2\,{\rm cm^{-3}}$ and $\rho_{0}=1.05\times
10^{-5}\,h^{-2}\,{\rm GeV/cm^{-3}}$ are entropy and critical energy densities
today Zyla  _et al._ (2020). The total entropy density at time $z_{C}$ (we
take $z_{C}=30$) is

$s(z_{C})=\frac{2\pi^{2}}{45}\left[2T_{\gamma}^{3}+2s_{\nu}\left(T_{\nu}(z_{C}),\mu_{\nu}(z_{C})\rule{0.0pt}{11.38109pt}\right)\rule{0.0pt}{14.22636pt}\right]\
,$ (7)

where $T_{\gamma}\simeq 1.39T_{\nu}^{\text{\sc sc}}=1.39m_{\phi}/z_{C}$, and

$s_{\nu}(T_{\nu},\mu_{\nu})=-\frac{3}{\pi^{2}}\left[4{\rm
Li}_{4}(-e^{\mu_{\nu}/T_{\nu}})-\frac{\mu_{\nu}}{T_{\nu}}{\rm
Li}_{3}(-e^{\mu_{\nu}/T_{\nu}})\right]\ .$ (8)

The $\nu$-$\phi$ dynamics also allow one to calculate $\Delta N_{\rm
eff}^{\text{\sc bbn}}$ and $\Delta N_{\rm eff}^{\text{\sc cmb}}$. A positive
$\Delta N_{\rm eff}^{\text{\sc bbn}}$ is generated by the non-thermal
production of $\phi$ at temperatures above the MeV scale,

$\Delta N_{\rm eff}^{\text{\sc bbn}}\simeq
3.046\left(\frac{\rho_{\phi}(z_{A})}{\rho_{\nu}(z_{A})}\right)\ ,$ (9)

where $\rho_{\nu}(z_{A})={9\zeta(3)}T_{\nu}(z_{A})^{4}/({4\pi^{2}})$ and
$\rho_{\phi}(z_{A})$ can be calculated by solving Eq. (4). The thermal
reaction rate $\gamma_{\phi\leftrightarrow\nu\nu}$ is proportional to
$(\lambda m_{\phi})^{2}$, implying weaker BBN constraints for light $\phi$.

By CMB times, $\phi$ particles have decayed away. The contribution to $\Delta
N_{\rm eff}^{\text{\sc cmb}}$ is due to the non-standard distribution function
of active neutrinos, 222S$\nu$DM production via active-sterile mixing also
leads to a slight reduction in the active neutrino population. However, given
the lower S$\nu$DM number density compared to that of neutrinos, this modifies
$\Delta N_{\rm eff}^{\text{\sc cmb}}$ by less than 1%.

$\Delta N_{\rm eff}^{\text{\sc
cmb}}=3.046\left(\frac{\rho_{\nu}}{\rho_{\nu}^{\text{\sc sc}}}-1\right)\ .$
(10)

where

$\frac{\rho_{\nu}}{\rho_{\nu}^{\text{\sc
sc}}}=-\frac{720}{7\pi^{4}}\left(\frac{T_{\nu}(z_{C})}{T_{\nu}^{\text{\sc
sc}}(z_{C})}\right)^{4}{\rm Li_{4}}(-e^{\mu_{\nu}(z_{C})/T_{\nu}(z_{C})})\ .$
(11)

In short, neutrino self interactions in the early universe via a light
mediator $\phi$ imply a tight correlation between the origin of S$\nu$DM and
the amount of extra radiation. The post-BBN thermalization of $\phi$ provides
a novel mechanism that accounts for the correct relic density of S$\nu$DM.
Meanwhile, $\phi$ thermalization distorts the active neutrino phase-space
distribution, causing $\Delta N_{\rm eff}>0$ for both CMB and BBN. It is then
of great interest to quantify this interplay, especially the prediction for
$\Delta N_{\rm eff}^{\text{\sc cmb}}$, which will serve as a well-motivated
target for the upcoming CMB-S4.

Results and Discussions – We select three benchmark S$\nu$DM masses,
$m_{4}=\left\\{2.5,\,4,\,7.1\right\\}\,$keV, with the correspondingly largest
active-sterile mixing parameter, $\sin^{2}2\theta\simeq\left\\{5\times
10^{-9},\,10^{-9},\,7\times 10^{-11}\right\\}$, consistent with current
$X$-ray search limits Horiuchi  _et al._ (2014); Malyshev  _et al._ (2014);
Abazajian (2017); Dessert  _et al._ (2020). In Fig. 2, we scan over
($m_{\phi}$, $\lambda$), focusing on $m_{\phi}$ between keV and MeV, and
derive the black solid curves that can account for the observed DM relic
abundance, $\Omega_{\nu_{4}}h^{2}=0.1186$. For $m_{\phi}\gg m_{4}$, all the
curves follow $\lambda\sim m_{\phi}^{1/2}$, matching the result for
$m_{\phi}\gtrsim 1$ MeV De Gouvêa  _et al._ (2020). As $m_{\phi}\to m_{4}$,
the curves bend up, where larger coupling $\lambda$ is required, compensating
for the phase-space suppression of the $\phi\to\nu\nu_{4}$ decay. We also
present contours of constant $\Delta N_{\rm eff}^{\text{\sc bbn}}$ and $\Delta
N_{\rm eff}^{\text{\sc cmb}}$ generated by the $\phi$-$\nu$ interaction. Their
intersections with the black contours dictate the correlation between
$\Omega_{4}$ and $\Delta N_{\rm eff}$. For real (complex) scalar $\phi$, we
find the following predictions $0<\Delta N_{\rm eff}^{\text{\sc bbn}}<0.57\
(1.14)$ and $0.12\ (0.19)<\Delta N_{\rm eff}^{\text{\sc cmb}}<0.9\ (1.95)$.
The maximal values correspond to strong couplings with $\phi$ fully
thermalized by BBN, whereas the minimal values correspond to sufficiently
small $\lambda$ and/or $m_{\phi}$ and negligible pre-BBN population of $\phi$.
In the lower-left half of each $\lambda$-$m_{\phi}$ plane, both $\Delta N_{\rm
eff}^{\text{\sc bbn}}$ and $\Delta N_{\rm eff}^{\text{\sc cmb}}$ follow the
same parametrical dependence as the $\phi$ production rate
$\gamma_{\phi\leftrightarrow\nu\nu}$ in Eq. (4) and depend only on the product
$(\lambda m_{\phi})$.

Thanks to the late thermalization of $\phi$, the predicted $\Delta N_{\rm
eff}^{\text{\sc cmb}}$ values are always higher than $\Delta N_{\rm
eff}^{\text{\sc bbn}}$. However, we find $\Delta N_{\rm eff}^{\text{\sc bbn}}$
is not always negligible and it serves as an independent test of non-standard
neutrino cosmology. The prior-to-BBN production of $\phi$ particles can
increase the expansion rate of the universe, leading to a higher neutron-to-
proton ratio when BBN begins, and thus higher helium and deuterium abundances
Blinov  _et al._ (2019). The state-of-the-art limit, $\Delta N_{\rm
eff}^{\text{\sc bbn}}\lesssim 0.5$ Pitrou  _et al._ (2018), marginally
excludes the case of fully-thermalized real scalar $\phi$ prior to BBN and
sets a stronger constraint for complex $\phi$. Note that the scalar can only
mediate neutrino self-interactions, ensuring that weak interactions are
unaffected, and hence heat transfer from the plasma to the neutrino sea is
only indirectly impacted. The changes produced in the primoridial abundances
of helium and deuterium are well within observational uncertainties, and hence
the state-of-the-art limit applies Grohs  _et al._ (2020). Precision
calculations of neutrino decoupling taking into account of entropy transfer
from the plasma to $\nu$ (and $\phi$) could improve the accuracy of the
$\Delta N_{\rm eff}^{\text{\sc bbn}}$ prediction to $\mathcal{O}(1\%)$ level
Grohs  _et al._ (2015, 2016).

From Fig. 2, for each S$\nu$DM mass one can derive the lowest predicted values
of $\Delta N_{\rm eff}^{\text{\sc bbn}}$ and $\Delta N_{\rm eff}^{\text{\sc
cmb}}$. Reducing $\theta$ would require larger $\lambda$ to maintain
$\Omega_{\nu_{4}}$, resulting in larger $\Delta N_{\rm eff}^{\text{\sc bbn}}$
and $\Delta N_{\rm eff}^{\text{\sc cmb}}$ until they saturate to their maxima.
Remarkably, this predicts a target $\Delta N_{\rm eff}$ for precision BBN and
CMB measurements. Moreover, because $\theta$ is more strongly constrained for
higher S$\nu$DM mass, an upper bound on $\Delta N_{\rm eff}$ could set an
upper bound on $m_{4}$. This forms a novel interplay with DM indirect
detection experiments. Inspired by this, we present Fig. 3. For each $m_{4}$,
we combine the current $X$-ray constraint and the $\Omega_{\nu_{4}}$
requirement to derive minimal values of $\Delta N_{\rm eff}^{\text{\sc bbn}}$
and $\Delta N_{\rm eff}^{\text{\sc cmb}}$, as shown in the left panel. We
refer to supplemental material B for more details. We find $\Delta N_{\rm
eff}^{\text{\sc cmb}}>0.12$ for all regions of interest, a favorable target
CMB-S4. For $m_{4}\gtrsim 6\,$keV, $\Delta N_{\rm eff}^{\text{\sc bbn}}$ and
$\Delta N_{\rm eff}^{\text{\sc cmb}}$ have already reached their maxima. Fig.
3(right) shows the impact of projected $\sin^{2}2\theta$ constraints by the
Athena $X$-ray observatory Neronov and Malyshev (2016) on $\Delta N_{\rm
eff}$. With smaller $\sin^{2}2\theta$, higher minimal values of $\Delta N_{\rm
eff}^{\text{\sc bbn}}$ and $\Delta N_{\rm eff}^{\text{\sc cmb}}$, in
particular, $\Delta N_{\rm eff}^{\text{\sc cmb}}>0.3$, are necessary. These
predictions will be robustly tested by CMB-S4, with $\Delta N_{\rm
eff}^{\text{\sc cmb}}$ precision at the few-percent level. The two classes of
probes are highly complementary and will further restrict or discover
S$\nu$DM.

Figure 3: Minimal value of $\Delta N_{\rm eff}$ at the time of CMB (orange)
and BBN (blue) as a function of S$\nu$DM mass $m_{4}$. All contours correspond
to the largest allowed active-sterile mixing parameter
$\sin^{2}\left(2\theta\right)$ by the current (left) and upcoming (right)
searches of $X$-ray line from $\nu_{4}\to\nu\gamma$ decay.

The range of S$\nu$DM mass shown in Fig. 3 is $2\,{\rm keV}<m_{4}<10\,$keV.
Lower values of $m_{4}$ are inconsistent with the phase space density
constraint from dwarf galaxies Tremaine and Gunn (1979); Boyarsky  _et al._
(2009); Abazajian (2017). A stronger lower bound on $m_{4}$ may be set using
Lyman-$\alpha$ forest Yèche  _et al._ (2017) and Milky Way satellite dwarf
galaxy counts Nadler  _et al._ (2020). In our scenario, the DM free-streaming
length can be substantially shorter than regular warm DM. In numerical
calculation, we note that S$\nu$DM is dominantly produced around $z\sim 3-4$
where $\phi$ already starts to turn non-relativistic (see Fig. 2 in
supplemental material A). The initial DM velocity is thus suppressed if
$m_{\phi}\sim m_{4}$. Interestingly, such a near-degeneracy exactly occurs
where $\Delta N_{\rm eff}$ is minimized (see Fig. 2). As a result, small scale
constraints are relaxed.

So far we have assumed that $\phi$ couples equally to all active neutrinos.
Our conclusions qualitatively hold if this assumption is relaxed to generic
flavor non-universal and/or off-diagonal couplings. By the time of $\phi$
thermalization, there is sufficient time for neutrino flavor conversions to
occur, which necessarily equilibrates $\phi$ with all the active neutrinos.

Before closing, we comment on other relevant constraints. A sub-MeV $\phi$
that couples to the electron neutrino of the form Eq. (1) can affect the two-
neutrino double-beta decay rate. The current limit, $\lambda\lesssim 10^{-4}$
Agostini  _et al._ (2015); Blum  _et al._ (2018); Brune and Päs (2019), lies
above Fig. 2. Another slightly weaker constraint on the $\phi$-$\nu$ coupling
comes from the cooling of core-collapsing supernovae Raffelt (1996); Heurtier
and Zhang (2017); Brune and Päs (2019); Escudero  _et al._ (2020).

In summary, we explored the impact of active neutrino self-interaction,
mediated by an ultralight scalar $\phi$, on the relic density of S$\nu$DM and
the amount of extra radiation ($\Delta N_{\rm eff}$) in the early Universe.
Strong neutrino self-interactions can facilitate efficient production of DM
and generate a net $\Delta N_{\rm eff}$ at the same time. Our study reveals an
intimate relationship between these two important quantities and provides a
well-motivated target for the future cosmological and astrophysical
experiments, including CMB-S4 and Athena. These results, together with earlier
work, depicts the complete parameter space for S$\nu$DM, with neutrino self-
interactions mediated by a scalar whose mass spans over more than six orders
of magnitude. They serve as well a motivated target for future experimental
tests, from laboratories to the cosmos. Observations made in this work could
extend to other neutrino self-interaction models mediated by vector bosons,
such as gauged $U(1)_{L_{\mu}-L_{\tau}}$ Kelly  _et al._ (2020); Shuve and
Yavin (2014); Escudero  _et al._ (2019) and other anomaly free gauge theories
of baryon and lepton number.

###### Acknowledgements.

Acknowledgements: We thank Nikita Blinov for discussions and comments on the
draft. Fermilab is operated by the Fermi Research Alliance, LLC under contract
No. DE-AC02-07CH11359 with the United States Department of Energy. M.S.
acknowledges support from the National Science Foundation, Grant PHY-1630782,
and to the Heising-Simons Foundation, Grant 2017-228. Y.Z. is supported by the
Arthur B. McDonald Canadian Astroparticle Physics Research Institute.

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##  I Supplemental Material

###  I.1 A. Matching Conditions

In this section, we provide the details on the matching conditions among the
three time scales defined in the main text, $z_{A},z_{B},z_{C}$. This allows
us to derive the time $(z)$ dependence of the active neutrino temperature
$T_{\nu}(z)$ and chemical potential $\mu_{\nu}(z)$.

At time $z_{A}$, the active neutrinos have just decoupled from the thermal
plasma of other SM particles, thus $T_{\nu}(z_{A})=T_{\nu}^{\text{\sc
sc}}(z_{A})\simeq 1\,$MeV and $\mu_{\nu}(z_{A})=0$. The corresponding energy
and number densities of active neutrinos are

$\rho_{\nu}(z_{A})=\frac{7\pi^{2}}{80}T_{\nu}(z_{A})^{4},\quad
n_{\nu}(z_{A})=\frac{9\zeta(3)}{4\pi^{2}}T_{\nu}(z_{A})^{3}\ .$ (12)

Here we count neutrinos and antineutrinos separately, i.e.,
$\rho_{\bar{\nu}}=\rho_{\nu}$, $n_{\bar{\nu}}=n_{\nu}$. Meanwhile, there also
exist a sub-thermal population of $\phi$ that was frozen in via its
interaction with active neutrinos. The corresponding values, denoted by
$\rho_{\phi}(z_{A})$ and $n_{\phi}(z_{A})$, can be obtained by numerically
solving Eq. (5) of the main text.

Figure 4: Evolution of ratios $T_{\nu}(z)/T_{\nu}^{\text{\sc sc}}(z)$ and
$\mu_{\nu}(z)/T_{\nu}(z)$ as functions of $z$ for three values of
$\lambda_{\phi}$ and holding $m_{\phi}=5\,{\rm keV}$ fixed. Solid (dashed)
curves correspond to real (complex) scalar $\phi$ case.

The time $z_{B}$ is defined when $\phi$ has just entered chemical equilibrium
with active neutrinos. At this moment, the $\phi$-$\nu$ system shares the same
temperature $T_{\nu}(z_{B})$, and also develops chemical potential where
$\mu_{\phi}(z_{B})=2\mu_{\nu}(z_{B})$. The corresponding energy, number, and
entropy densities are

$\begin{split}\rho_{\nu}(z_{B})&=-\frac{9T_{\nu}(z_{B})^{4}}{\pi^{2}}{\rm
Li}_{4}(-e^{\mu_{\nu}(z_{B})/T_{\mu}(z_{B})})\ ,\\\
n_{\nu}(z_{B})&=-\frac{3T_{\nu}(z_{B})^{3}}{\pi^{2}}{\rm
Li}_{3}(-e^{\mu_{\nu}(z_{B})/T_{\mu}(z_{B})})\ ,\\\
s_{\nu}(z_{B})&=-\frac{12T_{\nu}(z_{B})^{3}}{\pi^{2}}{\rm
Li}_{4}(-e^{\mu_{\nu}(z_{B})/T_{\mu}(z_{B})})\\\ &\hskip
11.38092pt+\frac{3\mu_{\nu}(z_{B})T_{\nu}(z_{B})^{2}}{\pi^{2}}{\rm
Li}_{3}(-e^{\mu_{\nu}(z_{B})/T_{\mu}(z_{B})})\ ,\\\
\rho_{\phi}(z_{B})&=\frac{3T_{\nu}(z_{B})^{4}}{\pi^{2}}{\rm
Li}_{4}(e^{2\mu_{\nu}(z_{B})/T_{\mu}(z_{B})})\ ,\\\
n_{\phi}(z_{B})&=\frac{T_{\nu}(z_{B})^{3}}{\pi^{2}}{\rm
Li}_{3}(e^{2\mu_{\nu}(z_{B})/T_{\mu}(z_{B})})\ ,\\\
s_{\phi}(z_{B})&=\frac{4T_{\nu}(z_{B})^{3}}{\pi^{2}}{\rm
Li}_{4}(e^{2\mu_{\nu}(z_{B})/T_{\mu}(z_{B})})\\\ &\hskip
11.38092pt-\frac{2\mu_{\nu}(z_{B})T_{\nu}(z_{B})^{2}}{\pi^{2}}{\rm
Li}_{3}(e^{2\mu_{\nu}(z_{B})/T_{\mu}(z_{B})})\ ,\\\ \end{split}$ (13)

where we treat $\phi$ as an ultra-relativistic species ($m_{\phi}\ll
T_{\nu}$), which is a good approximation throughout the parameter space we
explore in this work. Again, in the case where $\phi$ is a complex scalar, the
densities of the $\phi^{*}$ degree of freedom are counted separately. In the
absence of CP violation, $\rho_{\phi^{*}}=\rho_{\phi}$,
$n_{\phi^{*}}=n_{\phi}$, $s_{\phi^{*}}=s_{\phi}$.

The matching conditions between $z_{A}$ and $z_{B}$ are dictated by energy and
lepton number conservation Chacko  _et al._ (2004). If $\phi$ is a real
scalar, it comes to chemical equilibrium with both $\nu$ and $\bar{\nu}$, thus

$\begin{split}\frac{\rho_{\nu}(z_{A})+\rho_{\bar{\nu}}(z_{A})+\rho_{\phi}(z_{A})}{\rho_{\nu}(z_{B})+\rho_{\bar{\nu}}(z_{B})+\rho_{\phi}(z_{B})}&=\frac{z_{B}^{4}}{z_{A}^{4}}\
,\\\
\frac{n_{\nu}(z_{A})+n_{\bar{\nu}}(z_{A})+2n_{\phi}(z_{A})}{n_{\nu}(z_{B})+n_{\bar{\nu}}(z_{B})+2n_{\phi}(z_{B})}&=\frac{z_{B}^{3}}{z_{A}^{3}}\
.\\\ \end{split}$ (14)

Alternatively, if $\phi$ is a complex scalar, it only comes into equilibrium
with neutrinos ($\phi^{*}$ equilibrates with $\bar{\nu}$), thus

$\begin{split}\frac{\rho_{\nu}(z_{A})+\rho_{\phi}(z_{A})}{\rho_{\nu}(z_{B})+\rho_{\phi}(z_{B})}&=\frac{z_{B}^{4}}{z_{A}^{4}}\
,\\\
\frac{n_{\nu}(z_{A})+2n_{\phi}(z_{A})}{n_{\nu}(z_{B})+2n_{\phi}(z_{B})}&=\frac{z_{B}^{3}}{z_{A}^{3}}\
.\\\ \end{split}$ (15)

By definition $z_{A}/z_{B}=T_{\nu}^{\text{\sc sc}}(z_{B})/T_{\nu}^{\text{\sc
sc}}(z_{A})$, and noting that $T_{\nu}(z_{A})=T_{\nu}^{\text{\sc sc}}(z_{A})$,
the above set of equations (14) or (15) allows us to derive the ratios

$\frac{T_{\nu}(z_{B})}{T_{\nu}^{\text{\sc
sc}}(z_{B})},\quad\frac{\mu_{\nu}(z_{B})}{T_{\nu}(z_{B})}\ .$ (16)

In particular, in the case of real $\phi$ and with a negligible population of
$\phi$ at time $z_{A}$ (corresponds to a tiny $\lambda$), we can derive

$T_{\nu}(z_{B})/T_{\nu}^{\text{\sc sc}}(z_{B})\simeq
1.12,\quad\mu_{\nu}(z_{B})/T_{\nu}(z_{B})\simeq-0.57\ ,$ (17)

consistent with the findings of Ref. Escudero Abenza (2020).

As the next step, we consider the evolution of $T_{\nu}$ and $\mu_{\nu}$
between $z_{B}$ and $z_{C}$, where $z_{C}$ is defined as a temperature much
lower than $m_{\phi}$. During this epoch, the $\nu$-$\phi$ system remains in
chemical equilibrium. As a result, we can describe their thermal distributions
at any time $z$ where $z_{B}<z<z_{C}$. Because neutrino masses are still
negligible, their energy, number and entropy densities are described by the
same functions as those in Eq. (13), by simply replacing $z_{B}\to z$. On the
other hand, the mass of $\phi$ becomes important and the corresponding
distribution functions are evaluated using

$\begin{split}\rho_{\phi}(z,m_{\phi})&=\frac{T_{\nu}(z)^{4}}{2\pi^{2}}\int_{\frac{m_{\phi}}{T_{\nu}(z)}}^{\infty}\frac{\left[x^{2}-(m_{\phi}/T_{\nu}(z))^{2}\right]^{1/2}x^{2}dx}{e^{x-\mu_{\phi}(z)/T_{\nu}(z)}-1}\
,\\\
n_{\phi}(z,m_{\phi})&=\frac{T_{\nu}(z)^{3}}{2\pi^{2}}\int_{\frac{m_{\phi}}{T_{\nu}(z)}}^{\infty}\frac{\left[x^{2}-(m_{\phi}/T_{\nu}(z))^{2}\right]^{1/2}xdx}{e^{x-\mu_{\phi}(z)/T_{\nu}(z)}-1}\
,\\\
p_{\phi}(z,m_{\phi})&=\frac{T_{\nu}(z)^{4}}{2\pi^{2}}\int_{\frac{m_{\phi}}{T_{\nu}(z)}}^{\infty}\frac{\left[x^{2}-(m_{\phi}/T_{\nu}(z))^{2}\right]^{3/2}dx}{e^{x-\mu_{\phi}(z)/T_{\nu}(z)}-1}\
,\\\
s_{\phi}(z,m_{\phi})&=\frac{\rho_{\phi}(z,m_{\phi})+p_{\phi}(z,m_{\phi})}{T_{\nu}(z)}-\mu_{\phi}(z)n_{\phi}(z,m_{\phi})\
,\end{split}$ (18)

where $\mu_{\phi}=2\mu_{\nu}$ still holds.

Applying the matching conditions between $z_{B}$ and $z$ ($z_{B}<z<z_{C}$),
which are entropy and lepton number conservation

$\begin{split}\frac{s_{\nu}(z_{B})+s_{\bar{\nu}}(z_{B})+s_{\phi}(z_{B})}{s_{\nu}(z)+s_{\bar{\nu}}(z)+s_{\phi}(z,m_{\phi})}&=\frac{z^{3}}{z_{B}^{3}}\
,\\\
\frac{n_{\nu}(z_{B})+n_{\bar{\nu}}(z_{B})+2n_{\phi}(z_{B})}{n_{\nu}(z)+n_{\bar{\nu}}(z)+2n_{\phi}(z,m_{\phi})}&=\frac{z^{3}}{z_{B}^{3}}\
,\\\ \end{split}$ (19)

for the case of a real $\phi$, or

$\begin{split}\frac{s_{\nu}(z_{B})+s_{\phi}(z_{B})}{s_{\nu}(z)+s_{\phi}(z,m_{\phi})}&=\frac{z^{3}}{z_{B}^{3}}\
,\\\
\frac{n_{\nu}(z_{B})+2n_{\phi}(z_{B})}{n_{\nu}(z)+2n_{\phi}(z,m_{\phi})}&=\frac{z^{3}}{z_{B}^{3}}\
,\\\ \end{split}$ (20)

for the complex $\phi$, and using the relation $z_{B}/z=T_{\nu}^{\text{\sc
sc}}(z)/T_{\nu}^{\text{\sc sc}}(z_{B})$, we will able to derive the ratio
$\mu_{\nu}(z)/T_{\nu}(z)$, and $r(z)$ which is defined as

$\frac{T_{\nu}(z)}{T_{\nu}^{\text{\sc
sc}}(z)}=r(z)\frac{T_{\nu}(z_{B})}{T_{\nu}^{\text{\sc sc}}(z_{B})}\ .$ (21)

Combining this result with the findings in Eq. (16), we can derive the ratio
of neutrino temperatures in this model to that in standard cosmology,
$T_{\nu}(z)/T_{\nu}^{\text{\sc sc}}(z)$.

Figure 5: Time dependence of S$\nu$DM, for three values of $m_{4}$ as
labelled. The other parameters are chosen for producing the observed DM relic
density. The solid (dashed) curves correspond to real (complex) scalar $\phi$
case.

For example, in the limit where $z=z_{C}\gg 1$ such that
$\rho_{\phi},n_{\phi},s_{\phi}\to 0$ are all Boltzmann suppressed, for real
scalar $\phi$ and negligible population of $\phi$ at $z_{A}$, we obtain

$T_{\nu}(z_{C})/T_{\nu}^{\text{\sc sc}}(z_{C})\simeq
1.04,\quad\mu_{\nu}(z_{C})/T_{\nu}(z_{C})\simeq-0.14\ ,$ (22)

also consistent with Ref. Escudero Abenza (2020).

In Fig. 4, we show the evolution of $T_{\nu}$ and $\mu_{\nu}$ as functions of
$z$ for three sets of model parameters. For very small $\lambda$, the
asymptotic values are consistent with the results in Eq. (17) and (22). For
sufficiently large $\lambda$, the freeze in population of $\phi$ (via Eq. (5)
of the main text) before BBN is non-negligible which affects the matching
results from $z_{A}$ to $z_{C}$. The trend is also consistent with the limit
where $\phi$ already thermalizes before BBN. In that case, we would have
$z_{A}=z_{B}$ and $\mu_{\nu}(z_{B})=0$.

Figure 6: Parametrical dependence of $\Delta N_{\rm eff}^{\text{\sc cmb}}$
(orange) and $\Delta N_{\rm eff}^{\text{\sc bbn}}$ (blue) on the product
$\lambda m_{\phi}$, for real (solid) and complex (dashed) scalar cases. This
result is valid as long as $m_{\phi}\ll 1\,$MeV.

With the above results, we calculate the S$\nu$DM relic density using Eq. (7)
in the main text. In Fig. 5, we plot the time dependence of
$d\Omega_{\nu_{4}}/d\ln z$ for a few sets of model parameters. In all cases,
we find that S$\nu$DM is dominantly produced around $z\simeq 3-4$ where $\phi$
already starts to become non-relativistic. If $\phi$ and $\nu_{4}$ is nearly
degenerate ($m_{\phi}\gtrsim m_{4}$), S$\nu$DM will be produced with a
velocity $v\ll c$.

Furthermore, the above results also allow us to calculate $\Delta N_{\rm eff}$
at CMB time. After $\phi$ decays away at $z_{C}\gg 1$, the ratios
$T_{\nu}(z_{C})/T_{\nu}^{\text{\sc sc}}(z_{C})$ and
$\mu_{\nu}(z_{C})/T_{\nu}(z_{C})$ approach to constant values throughout the
later evolution of the Universe, as shown in Fig. 4. We plug these functions
into Eq. (11) of the main text. In the case of negligible $\phi$ population by
the time of BBN, the value of $\Delta N_{\rm eff}^{\text{\sc cmb}}$ is 0.12
and 0.19 for real and complex $\phi$ case, respectively. The result for the
real scalar case is consistent with the findings in Refs. Chacko  _et al._
(2004).

We observe that $\Delta N_{\rm eff}^{\text{\sc bbn}}$ and $\Delta N_{\rm
eff}^{\text{\sc cmb}}$ depend on model parameter space $\lambda$ and
$m_{\phi}$ only through the combination $\lambda m_{\phi}$ as long as the
product is small (corresponding to the low-left half of Fig. 2 of the main
text). This dependence and the parametrical correlation are shown in Fig. 6
for real and complex scalars. We see that for sufficiently small $\lambda
m_{\phi}$, there is no impact on $\Delta N_{\rm eff}^{\text{\sc bbn}}$ (blue
curves) – no abundance of $\phi$ has developed by the time of BBN. On the
contrary, even for $\lambda m_{\phi}\approx 10^{-9}$ keV, $\Delta N_{\rm
eff}^{\text{\sc cmb}}$ has a lowest value, 0.12 or 0.19, depending on whether
$\phi$ is real or complex due to distortions of the active neutrino
distributions. For both BBN and CMB, large enough $\lambda m_{\phi}$ results
in a complete saturation of $\Delta N_{\rm eff}$ and these curves level off to
their maxima. This approximation is only valid for $m_{\phi}\ll 1$ MeV, where
$\phi$ cannot decay away before BBN.

###  I.2 B. Minimal $\Delta N_{\rm eff}$

Figure 7: Similar to Fig. 2 of the main text but with $m_{4}=2.5\,$keV,
$\sin\theta=5\times 10^{-9}$ and $10^{-10}$ for DM relic density calculation
(black curves). The corresponding lowest values of $\Delta N_{\rm
eff}^{\text{\sc cmb}}$ and $\Delta N_{\rm eff}^{\text{\sc bbn}}$ are shown by
the orange and blue curves, respectively. Here we consider the real scalar
$\phi$ case.

Here we provide some further context for the procedure used in generating Fig.
3 of the main text. For a given $m_{4}$, we determine the largest allowed
$\sin^{2}\left(2\theta\right)$ assuming current limits from $X$-ray searches
Horiuchi  _et al._ (2014); Malyshev  _et al._ (2014); Abazajian (2017);
Dessert  _et al._ (2020), as well as the projected upper limit by Athena
Neronov and Malyshev (2016). For a concrete choice, let us consider
$m_{4}=2.5$ keV. The current limit on this mass is
$\sin^{2}\left(2\theta\right)<5\times 10^{-9}$, whereas Athena will have
sensitivity at the level of $\sin^{2}\left(2\theta\right)=10^{-10}$. We
determine, for those two values of $\sin^{2}\left(2\theta\right)$, the
preferred region of $\lambda$ vs. $m_{\phi}$ parameter space for which
$\nu_{4}$ can account for all of the DM observed today – this is shown as the
two solid black lines in Fig. 7. Each of those curves, near the minimum in
$\lambda$ realized, attains some minimum value of both $\Delta N_{\rm
eff}^{\text{\sc cmb}}$ and $\Delta N_{\rm eff}^{\text{\sc bbn}}$. For the case
we have considered here, $m_{4}=2.5$ keV, we find that with current limits on
$\sin^{2}\left(2\theta\right)$, $\Delta N_{\rm eff}^{\text{\sc bbn}}$ ($\Delta
N_{\rm eff}^{\text{\sc cmb}}$) must be at least 0.02 (0.14), and with future
constraints (assuming Athena does not observe a signal compatible with
$m_{4}=2.5$ keV) will require $\Delta N_{\rm eff}^{\text{\sc bbn}}>0.4$
($\Delta N_{\rm eff}^{\text{\sc cmb}}>0.6$). In Fig. 7 we have assumed $\phi$
to be a real scalar, but the procedure is the same for complex $\phi$. These
minimum values are what is shown for that choice of $m_{4}$ in the left
(current constraint on $\sin^{2}\left(2\theta\right)$) and right (post-Athena
constraint) panels of Fig. 3.


irreducibles in the decomposition of
$\bigwedge(\mathbb{C}^{nm})\cong\bigwedge(\mathbb{C}^{n}\otimes\mathbb{C}^{m})$
as a $\mathfrak{gl}_{n}\otimes\mathfrak{gl}_{m}$-module. Moreover, our
decomposition de-quantizes appropriately, which means that the endomorphisms
induced by the quantum group generators tend to those induced by their
classical counterparts in the limit $q\to 1$. For instance, compare the action
of $E_{i}^{(n)}$ and $E_{j}^{(m)}$ on an arbitrary basis vector in the quantum
case as computed in Equation 15 and Equation 16 to the action of their
classical counterparts in the classical case, as computed in Equation 21 and
Equation 22.

Our proof of the first statement in 4.17 mimics the argument establishing
Theorem $6.16$ in [LZZ10]. First we construct a joint highest weight vector
for each isotypic component in the decomposition of Equation 17 in the next
lemma, and then we use a dimension count and the classical skew $GL_{n}\times
GL_{m}$ duality 3.10 to conclude.

Recall 3.14, which computes highest weight vectors in the classical case. In
the quantum case, the highest weight vectors are essentially the same: for
each partition $\mu$ fitting in an $n\times m$ rectangle we simply take the
product of all basis vectors in the boxes occupied by the Young diagram
corresponding to $\mu$. Figure 5 illustrates an example.

######  Lemma 4.19.

For each partition $\mu=(\mu_{1},\ldots,\mu_{n})$ satisfying $\mu_{1}\leq m$,
let

$v_{\mu}=\left(v_{1}\cdots v_{1+(\mu_{1}-1)n}\right)\left(v_{2}\cdots
v_{2+(\mu_{2}-1)n}\right)\cdots\left(v_{n}\cdots v_{2+(\mu_{2}-1)n}\right).$

Then

  1. (1)

The element $v_{\mu}$ is a highest weight vector with respect to the actions
of $U_{q}(\mathfrak{gl}_{n})$ and of $U_{q}(\mathfrak{gl}_{m})$.

  2. (2)

The $U_{q}(\mathfrak{gl}_{n})$-weight of $v_{\mu}$ is $\mu$, and its
$U_{q}(\mathfrak{gl}_{m})$-weight is $\mu^{\prime}$.

###### Proof.

It is understood that if $\mu_{i}=0$, then $v_{\mu}$ contains no
$v_{i+(b-1)n}$ as a factor. Since $\mu$ is a partition, this also implies that
there is no $v_{j+(b-1)n}$ factor for every $j\geq i$.

Much like in the classical case, Equations 15 and 16 show that positive root
vectors in $U_{q}(\mathfrak{gl}_{n})\otimes U_{q}(\mathfrak{gl}_{m})$ shift
occupied positions in any given state vector above and to the left, as
illustrated by Figures 2 and 3. Since $\mu$ is a left-justified partition,
every position above and to the left of an occupied position in $v_{\mu}$ is
occupied already, so

$\lambda_{q}(E_{i}^{(n)})\,v_{\mu}=\rho_{q}(E_{j}^{(m)})\,v_{\mu}=0,$

for $i=1,\ldots,n-1$ and $j=1,\ldots,m-1$. Hence, each $v_{\mu}$ is a joint
$U_{q}(\mathfrak{gl}_{n})\otimes U_{q}(\mathfrak{gl}_{m})$-highest weight
vector.

We can readily prove $(2)$ using 4.10 and 4.15. For instance, we see that

$\displaystyle\lambda_{q}(K_{i}^{(n)})\,v(\ell)$
$\displaystyle=q^{\sum_{j=1}^{m}\ell_{i+(j-1)n}-\ell_{i+1+(j-1)n}}\,v(\ell),\quad\text{and}$
$\displaystyle\rho_{q}(K_{j}^{(m)})\,v(\ell)$
$\displaystyle=q^{\sum_{i=1}^{n}\ell_{i+(j-1)n}-\ell_{i+jn}}\,v(\ell),$

for any $v(\ell)$, so

$\displaystyle\lambda_{q}(K_{i}^{(n)})\,v_{\mu}$
$\displaystyle=q^{\mu_{i}-\mu_{i+1}}\,v_{\mu}=q^{\langle\alpha_{i}^{(n)},\,\mu\rangle}\,v_{\mu},\quad\text{and}$
$\displaystyle\rho_{q}(K_{j}^{(m)})\,v_{\mu}$
$\displaystyle=q^{\mu_{j}^{\prime}-\mu_{j+1}^{\prime}}\,v_{\mu}=q^{\langle\alpha_{j}^{(m)},\,\mu^{\prime}\rangle}\,v_{\mu}.$

Here we used $\alpha_{i}^{(p)}=\epsilon_{i}-\epsilon_{i+1}$ to denote the
$i$th simple positive root of $\mathfrak{gl}_{p}$. ∎

###### Proof of 4.17..

For any $\mathbb{Z}_{\geq 0}$-graded module $M$, let $M_{=k}$ denote the
graded component of degree $k$. It follows from 4.19 that
$\bigoplus_{\mu}V_{\mu}^{(n)}\otimes V_{\mu^{\prime}}^{(m)}$ is a
$U_{q}(\mathfrak{gl}_{n})\otimes U_{q}(\mathfrak{gl}_{m})$-submodule of
$\bigwedge_{q}(V^{(nm)})$. Let $|\mu|$ denote the size of the partition $\mu$,
that is, the sum of its parts. Equations 15 and 16 prove that the
$U_{q}(\mathfrak{gl}_{n})$\- and the $U_{q}(\mathfrak{gl}_{m})$-actions
preserve subspaces of homogeneous degree, so that in fact

$\displaystyle\bigoplus_{|\mu|=k}V_{\mu}^{(n)}\otimes
V_{\mu^{\prime}}^{(m)}\subseteq\textstyle\left(\bigwedge_{q}(V^{(nm)})\right)_{=k}$

for each $k\leq nm$. The classical skew duality 3.10 then implies that

$\displaystyle\sum_{|\mu|=k}\dim_{\mathbb{C}(q)}\bigg{(}V_{\mu}^{(n)}\otimes
V_{\mu^{\prime}}^{(m)}\bigg{)}=\dim_{\mathbb{C}}\bigwedge(\mathbb{C}^{n}\otimes\mathbb{C}^{m}),$

since irreducible modules of $\mathfrak{gl}_{p}$ and of
$U_{q}(\mathfrak{gl}_{p})$ parametrized by the same dominant weight have the
same dimension. The $U_{q}(\mathfrak{gl}_{nm})$-module $V^{(nm)}$ is flat in
the sense of Lemma 2.32 in [BZ05], so the $v(\ell)$ form a basis of
$\bigwedge_{q}(V^{(nm)})$, which implies that
$\dim_{\mathbb{C}(q)}\bigwedge_{q}(V^{(nm)})=\dim_{\mathbb{C}}\bigwedge(\mathbb{C}^{n}\otimes\mathbb{C}^{m})$
and concludes our proof. ∎

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