# Constraints on electromagnetic form factors of sub-GeV dark matter from the
Cosmic Microwave Background anisotropy

Gaetano Lambiase a, Subhendra Mohantyb, Akhilesh Nautiyalc and Soumya Raob a
Dipartimento di Fisica “E.R Caianiello”, Universit degli Studi di Salerno, Via
Giovanni Paolo II, 132 - 84084 Fisciano (SA), Italy
b Theory Division, Physical Research Laboratory, Navrangpura, Ahmedabad 380
009, India
c Department of Physics, Malaviya National Institute of Technology, JLN Marg,
Jaipur 302017,India

###### Abstract

We consider dark matter which have non-zero electromagnetic form factors like
electric/magnetic dipole moments and anapole moment for fermionic dark matter
and Rayleigh form factor for scalar dark matter. We consider dark matter mass
$m_{\chi}>\cal{O}({\rm MeV})$ and put constraints on their mass and
electromagnetic couplings from CMB and LSS observations. Fermionic dark matter
with non-zero electromagnetic form factors can annihilate to $e^{+}e^{-}$ and
scalar dark matter can annihilate to $2\gamma$ at the time of recombination
and distort the CMB. We analyze dark matter with multipole moments with Planck
and BAO observations. We find upper bounds on anapole moment
$g_{A}<7.163\times 10^{3}\text{GeV\textsuperscript{-2}}$, electric dipole
moment ${\cal D}<7.978\times 10^{-9}\text{e-cm}$, magnetic dipole moment
${\mu}<2.959\times 10^{-7}\mu_{B}$, and the bound on Rayleigh form factor of
dark matter is $g_{4}/\Lambda_{4}^{2}<1.085\times
10^{-2}\text{GeV\textsuperscript{-2}}$ with $95\%$C.L.

###### pacs:

98.80.Cq, 98.80.Ft, 26.35.+c, 95.35.+d

##  I Introduction

It is well accepted that formation of large scale structures and the rotation
curves of galaxies require an extra dark matter component beyond than the
known particles of the standard model. The particle properties of this dark
matter are, however, completely unknown. Direct detection experiments, which
rely on nuclear scattering, have ruled out a large parameter space. But, these
techniques are not efficient in measuring dark matter of sub-GeV mass
Undagoitia:2015gya ; Mohanty:2020pfa . To measure sub-GeV mass dark matter a
suitable method is scattering electrons from heavy atoms Essig:2011nj ;
Essig:2012yx ; Agnes:2018oej ; Aprile:2019xxb ; Catena:2019gfa ;
Catena:2020tbv . Dark matter with non-zero electric or magnetic dipole moments
Pospelov:2000bq ; Sigurdson:2004zp ; Masso:2009mu  or anapole moment Ho:2012bg
; DelNobile:2014eta  can be very effective in scattering electrons. The
electromagnetic form factors can be viewed as effective operators
Primulando:2015lfa ; DeSimone:2016fbz ; Chu:2018qrm ; kavanagh , which arise
by integrating out the heavy particles in a ultraviolet complete theory
Kopp:2014tsa .

The electromagnetic couplings of dark matter can be constrained from cosmic
microwave background and large scale structures observations. The electric and
magnetic dipole moment vertex can give rise to dark matter-baryon coupling.
For heavy dark matter ($\sim 100\,{\rm GeV}$) the baryon drag on the dark
matter will show up in structure formation and CMB Sigurdson:2004zp . Light
dark matter ($\cal{O}({\rm MeV})$ will annihilate to radiation and lower the
effective neutrino number ($N_{\rm eff}$) Ho:2012br ; Boehm:2013jpa ;
Brust:2013ova .

In this paper we will analyze the effect of light dark matter with
electromagnetic form factors on CMB anisotropy and polarization from dark
matter annihilation to $e^{+}e^{-}$ or photons close to recombination era,
$z\sim 1100$. The effects of annihilating dark matter on CMB are studied in
Chen:2003gz ; Padmanabhan:2005es ; Galli:2009zc ; Slatyer:2009yq ;
Finkbeiner:2011dx . Production of relativistic $e^{+},e^{-}$ heats up the
thermal gas and ionizes the neutral Hydrogen, which increases the free
electron fraction. Due to this increased free electron fraction there is a
broadening of the last scattering surface and suppression of CMB temperature
anisotropy. The low-$l$ correlations between polarization fluctuations are
also enhanced due to increased freeze-out value of the ionization fraction of
the universe after recombination. These effects on CMB are significant and can
be used to put constraints on thermal averaged annihilation cross-section
$\langle\sigma v\rangle$. Planck-2018 reports $\langle\sigma
v\rangle<\left(3.2\times
10^{-28}/f\right)\times\left(M_{DM}/\left(\text{GeV}/\text{c\textsuperscript{2}}\right)\right)\text{cm\textsuperscript{3}/\text{s}}$
for velocity independent thermal average cross-section Aghanim:2018eyx . Here
$f$ is the fraction of energy injected to the intergalactic medium (IGM) by
annihilating dark matter. Forecasts for upcoming CMB experiments such as
AdvACTPol, AliCPT, CLASS, Simons Array, Simons Observatory, and SPT-3G has
been studied by Cang:2020exa  in detecting decaying/annihilating dark matter,
and it is found that $\langle\sigma
v\rangle\sim\left(10^{-29}/f\right)\times\left(M_{DM}/\left(\text{GeV}/\text{c\textsuperscript{2}}\right)\right)\text{cm\textsuperscript{3}/\text{s}}$.

The annihilation $\chi\chi\rightarrow e^{+}e^{-}$ occur with one dipole or
anapole vertex and the annihilation cross sections are quadratic in dipole or
anapole moments. For fermionic dark matter the
$\chi\chi\rightarrow\gamma\gamma$ annihilation cross section are quartic in
dipole moments and the bounds from this process are much weaker Arellano-
Celiz:2019pax  than the ones we derive in this paper from CMB. For anapole
dark matter the cross section for the process
$\chi\chi\rightarrow\gamma\gamma$ is zero Ho:2012bg . Scalar dark matter can
have dimension-6 Rayleigh operator vertex with two-photons. For such Rayleigh
dark matter the leading order contribution to annihilation will be from the
$\phi\phi\rightarrow\gamma\gamma$ process which can distort the CMB near
recombination and from this we put bounds on the dark-matter photon Rayleigh
coupling.

This paper is organized as follows. In Section II we list the electromagnetic
form factors of dark matter which we shall constrain from CMB data. In Section
III we discuss the physics of recombination and the effect of dark matter
annihilation on the CMB. In Section IV we compare the CMB analysis with data
from Planck and BAO and using COSMOMC we put constraints on the dark matter
form factors. In Section V we compare our bounds with earlier results and from
other experiments, and in Section VI we summarize our results and give our
conclusions.

##  II Electromagnetic form factors of dark matter

Spin-1/2 dark matter can have the following electromagnetic form factors.
These are the magnetic moment described by the dimension-5 operators,

${\cal L}_{\rm
magnetic}=\frac{g_{1}}{\Lambda_{1}}\bar{\chi}\sigma^{\mu\nu}\chi F_{\mu\nu}$
(1)

where $g_{1}$ is a dimensionless coupling and $\Lambda_{1}$ is the mass scale
of the particles in the loop which generate the dipole moment. The magnetic
moment of Dirac fermions is $\mu=2g_{1}/\Lambda_{1}$ and the operator (1) is
zero for Majorana fermions.

Similarly electric dipole operator is of dimension-5,

${\cal L}_{\rm
electric}=\frac{g_{2}}{\Lambda_{2}}\,i\,\bar{\chi}\sigma^{\mu\nu}\gamma_{5}\chi
F_{\mu\nu}$ (2)

where the electric dipole moment of Dirac fermions is ${\cal
D}=2g_{2}/\Lambda_{2}$ and the operator (2) is zero for Majorana fermions.

Finally the anapole moment is a dimension-6 operator

${\cal L}_{\rm
anapole}=\frac{g_{3}}{\Lambda_{3}^{2}}\,i\,\bar{\chi}\gamma^{\mu}\gamma_{5}\chi\partial^{\nu}\,F_{\mu\nu}$
(3)

This operator is non-zero for Dirac as well as Majorana fermions and the
coefficient $g_{A}=g_{3}/(\Lambda_{3}^{2})$ is called the anapole moment of
$\chi$.

Stringent bounds on sub-GeV mass dark matter are put from the observation of
$\chi e^{-}\rightarrow\chi e^{-}$ scattering Essig:2011nj  in direct detection
experiments like Xenon-10 Essig:2012yx , DarkSide Agnes:2018oej  and Xenon-1T
Aprile:2019xxb .

Using the experimental limits on dark matter electron scattering from
Xenon-10, Xenon-1T and DarkSide bound on the electric dipole, magnetic dipole
and anapole form factors of dark matter have been put in ref.Catena:2019gfa ;
Catena:2020tbv .

Real and complex scalar dark matter can have interaction with 2-photons by
dimension-6 Rayleigh operator

${\cal L}_{2\phi\,2\gamma}=\frac{g_{4}}{\Lambda_{4}^{2}}\phi^{*}\phi
F_{\mu\nu}F^{\mu\nu}$ (4)

These will contribute to $\phi\phi\rightarrow\gamma\gamma$ annihilations which
can be constrained from CMB kavanagh . In the absence of CP violation the
$\tilde{F}F$ operator does not arise. The annihilation
$\phi\phi\rightarrow\gamma\gamma$ takes place via s-wave in the leading order
and cross section
$\sigma(\phi\phi\rightarrow\gamma\gamma)v\simeq(g_{4}^{2})m_{\phi}^{2}/\Lambda_{4}^{4}$
kavanagh . Bounds on the operator (4) from Xenon1T Aprile:2019xxb  and gamma
ray searches from dwarf spheroidal satellites (dSphs) Fermi-LAT:2016uux  and
halo of the Milky way Ackermann:2015lka  by Fermi-LAT are obtained in ref.
kavanagh  for dark matter with mass larger than $\cal{O}(\rm GeV)$.

##  III Thermal history of the universe with annihilating dark matter

Recombination occurs around $z=1100$ when electrons and protons combine
together to form neutral hydrogen. If the annihilation cross-section of dark
matter particles is sufficiently large, it can modify the history of
recombination and hence can leave a clear imprint on CMB power spectrum. The
shower of particles produced due to annihilation can interact with the thermal
gas in three different ways. (i) The annihilation products can ionize the
thermal gas, (ii) can induce induce Lyman-$\alpha$ excitation of the hydrogen
that will cause more electrons in $n=2$ state and hence increase the
ionization rate and (iii) can heat the plasma. Due to the first two effects
the evolution of free electron fraction $\chi_{e}$ changes and the last effect
changes the temperature of baryons. The equation governing the evolution of
ionization fraction in the presence of annihilating particles is given as

$\frac{d\chi_{e}}{dt}=\frac{1}{\left(1+z\right)H(z)}\left[R_{s}(z)-I_{s}(z)-I_{X}(z)\right].$
(5)

Here $R_{s}$ is the standard recombination rate, $I_{s}$ is the ionization
rate due to standard sources and $I_{X}$ is the ionization rate due to
annihilating dark matter particles. The computation of standard recombination
rate was done in Peebles:1968ja ; Zeldovich:1969en ; Seager:1999km  and it is
described as

$\left[R_{s}(z)-I_{s}(z)\right]=C\times\left[\chi_{e}^{2}n_{H}\alpha_{B}-\beta_{B}\left(1-\chi_{e}\right)e^{-h_{p}\nu_{2s}/k_{B}T_{b}}\right].$
(6)

Here $n_{H}$ is the number density of hydrogen nuclei, $\alpha_{B}$ and
$\beta_{B}$ are the effective recombination and photo-ionization rates for
principle quantum numbers $\geq 2$ in Case B recombination,$\nu_{2s}$ is the
frequency of the $2s$ level from the ground state and $T_{b}$ is the
temperature of the baryon gas. The factor $C$ appearing in Eqn. (6) is given
by:

$C=\frac{\left[1+K\Lambda_{2s1s}n_{H}\left(1-\chi_{e}\right)\right]}{\left[1+K\Lambda_{2s1s}n_{H}\left(1-\chi_{e}\right)+K\beta_{B}n_{H}\left(1-\chi_{e}\right)\right]}.$
(7)

Here $\Lambda_{1s2s}$ is the decay rate of the metastable $2s$ level,
$n_{H}\left(1-\chi_{e}\right)$ is the number of neutral ground state $H$ atoms
and $K=\frac{\lambda_{\alpha}^{3}}{8\pi H(z)}$, where $H(z)$ is the Hubble
expansion rate at redshift $z$ and $\lambda_{\alpha}$ is the wavelength of the
$Ly-\alpha$ transition from the $2p$ level to the $1s$ level.

The term $I_{X}$ appearing in Eq. (5) represents the evolution of free
electron density due to nonstandard sources. In our case it is due to
annihilation of dark matter during recombination. which increases the
ionization rate in two ways. (i) By direction ionization from the ground state
and (ii) by additional $Ly-\alpha$ photons, which boosts the population at
$n=2$ increasing the the rate of photoionization by CMB. Hence, the ionization
rate $I_{X}$ due to dark matter annihilation is expressed as

$I_{X}(z)=I_{Xi}(z)+I_{X\alpha}(z).$ (8)

Here $I_{Xi}(z)$ represents the ionization rate due to ionizing photons and
$I_{X\alpha}$ represents the ionization rate due to $Ly-\alpha$ photons.

The rate of energy release $\frac{dE}{dt}$ per unit volume by a relic self-
annihilating dark matter particle can be expressed in terms of its thermally
averaged annihilation cross-section $\langle\sigma v\rangle$ and mass
$m_{\chi}$ as

$\frac{dE}{dt}=2g\rho_{c}^{2}c^{2}\Omega_{DM}^{2}\left(1+z\right)^{6}f(z)\frac{\langle\sigma
v\rangle}{m_{\chi}},$ (9)

where $\Omega_{DM}$ is the dark matter density parameter, $\rho_{c}$ is the
critical density today, $g$ is degeneracy factor $1/2$ for Majorana fermions
and $1/4$ for Dirac fermions, and $f(z)$ is the fraction of energy absorbed by
the CMB plasma, which is $O(1)$ factor and depends on redshift. A detailed
calculation of redshift dependence of $f(z)$ for various annihilation channels
is done in Slatyer:2009yq ; Huetsi:2009ex ; Evoli:2012qh ; Galli:2013dna ;
Madhavacheril:2013cna  using generalized parameterizations or principle
components. It is shown in Galli:2011rz ; Giesen:2012rp ; Finkbeiner:2011dx
that the redshift dependence of $f(z)$ can be ignored up to a first
approximation, since current CMB data are sensitive to energy injection over a
relatively narrow range of redshift, typically $z\sim 1000-600$. Hence $f(z)$
can be replace with a constant $f$, which we take as $1$ for our analysis.
Here we use ’on-the-spot’ approximation, which assumes that the energy
released due to dark matter annihilation is absorbed by IGM locally
Zhang:2006fr ; Zhang:2007zzh ; Galli:2009zc .

The terms appearing on the right hand side of Eq. (8) are related to the rate
of energy release as

$\displaystyle I_{Xi}$ $\displaystyle=$ $\displaystyle
C\chi_{i}\frac{\left[dE/dt\right]}{n_{H}(z)E_{i}}$ (10) $\displaystyle
I_{X\alpha}$ $\displaystyle=$
$\displaystyle\left(1-C\right)\chi_{\alpha}\frac{\left[dE/dt\right]}{n_{H}(z)E_{\alpha}}.$
(11)

Here $E_{i}$ is the average ionization energy per baryon, $E_{\alpha}$ is the
difference in binding energy between the $1s$ and $2p$ energy levels of a
hydrogen atom, $n_{H}$ is the number density of Hydrogen Nuclei, and
$\chi_{i}$ and $\chi_{\alpha}$ represent the fraction of energy going
ionization and $Ly-\alpha$ photons respectively; which can be expressed in
terms of free electron fraction as
$\chi_{i}=\chi_{\alpha}=\left(1-\chi_{e}\right)/3$ Chen:2003gz .

A fraction of energy released by annihilating dark matter particles also goes
into heating the baryon gas, which modifies the evolution equation for the
matter temperature $T_{b}$ by contributing one extra term $K_{h}$ as

$\left(1+z\right)\frac{dT_{b}}{dz}=\frac{8\sigma_{T}a_{R}T_{CMB}^{4}}{3m_{e}cH(z)}\frac{\chi_{e}}{1+f_{He}+\chi_{e}}\left(T_{b}-T_{CMB}\right)-\frac{2}{3k_{B}H(z)}\frac{k_{h}}{1+f_{He}+\chi_{e}}+2T_{b}.$
(12)

Here the nonstandard term $K_{h}$ arising due to annihilating dark matter is
given in terms of rate of energy release as

$K_{h}=\chi_{h}\frac{\left(dE/dt\right)}{n_{H}(z)},$ (13)

with $\chi_{h}=\left(1+2\chi_{e}\right)/3$ being the fraction of energy going
into heat.

In this work we consider annihilating dark matter with electromagnetic form
factors. We will now obtain the energy deposition rate for dark matter with
anapole moment, electric dipole moment and magnetic dipole moment. One can
define a quantity $p_{ann}$ that depends on the properties of dark matter
particles as

$p_{ann}=f\frac{\langle\sigma v\rangle}{m_{\chi}}.$ (14)

The current constraint on $p_{ann}$ with velocity independent $\langle\sigma
v\rangle$ is $1.795\times 10^{-7}$ m3s-1kg-1 $95\%\;C.L$ from Planck-2018
Aghanim:2018eyx . In our analysis we will use various electromagnetic form
factors and mass of the dark matter as our model parameters rather than
$p_{ann}$. Hence we will express energy deposition rate in terms of these
parameters for annihilating dark matter with anapole and dipole moments.

###  III.1  Dark matter with anapole moment

The annihilation cross-section for dark matter with anapole moment is given as
Ho:2012bg ,

$\langle\sigma v\rangle_{\chi\chi\rightarrow
e^{+}e^{-}}=\frac{2g_{A}^{2}\alpha m_{\chi}^{2}}{3}v_{rel}^{2}.$ (15)

where $\alpha=e^{2}/(4\pi)$ and $v_{rel}$ is average relative velocity between
the annihilating dark matter particles in the centre of mass frame. The
thermally averaged velocity can be expressed in terms of temperature by
$\frac{1}{2}\left(\frac{1}{2}m_{\chi}\right)\langle
v^{2}_{rel}\rangle=\frac{3}{2}T$. Hence the cross-section (15) can be
expressed in terms of temperature as

$\langle\sigma v\rangle_{\chi\chi\rightarrow e^{+}e^{-}}=4g_{A}^{2}\alpha
m_{\chi}^{2}\left(\frac{T}{m_{\chi}}\right).$ (16)

After decoupling the temperature of the dark matter behaves as
$T\propto\left(1+z\right)^{2}$. Assuming the decoupling temperature of the
dark matter $T_{d}$ of the order of $\frac{m_{\chi}}{10}$ we get

$\displaystyle T$ $\displaystyle=$ $\displaystyle
T_{d}\frac{\left(1+z\right)^{2}}{\left(1+z_{d}\right)^{2}}=T_{d}\frac{T_{0}^{2}}{T_{\gamma
d}^{2}}\left(1+z\right)^{2}$ (17) $\displaystyle=$
$\displaystyle\frac{10T_{0}^{2}}{m_{\chi}}\left(1+z\right)^{2}.$

Here $z_{d}$ and $T_{\gamma d}$ are the redshift of dark matter decoupling and
the temperature of photons at that redshift respectively, which is same as
$T_{d}$. $T_{0}$ is the current temperature of CMB. Using Eq. (17) the
annihilation cross-section (16) becomes

$\langle\sigma v\rangle_{\chi\chi\rightarrow e^{+}e^{-}}=40g_{A}^{2}\alpha
T_{0}^{2}\left(1+z\right)^{2}.$ (18)

Hence using Eqs. (18) and (14) we can obtain the expression for $p_{ann}$ as

$p_{ann}=\frac{40g_{A}^{2}\alpha T_{0}^{2}}{m_{\chi}}\left(1+z\right)^{2}.\\\
$ (19)

As mentioned earlier we will choose $f\sim 1$ here. Since $p_{ann}$ is
velocity dependent, the rate of energy release given by Eq. (9) will be

$\frac{dE}{dt}=\rho_{c}^{2}c^{2}\Omega_{DM}^{2}\frac{40g_{A}^{2}\alpha
T_{0}^{2}}{m_{\chi}}\left(1+z\right)^{8}.$ (20)

Here the redshift dependence of the energy deposition rate is modified as
compared to (9).

###  III.2 Dark matter with electric dipole moment

For DM with electric dipole moment the annihilation cross-section is given by
Masso:2009mu ,

$\langle\sigma v\rangle_{\chi\chi\rightarrow
e^{+}e^{-}}=\frac{\alpha\mathcal{D}^{2}}{12}v^{2}_{rel}$ (21)

where $v_{rel}$ is the relative velocity of two annihilating WIMPS. For
thermal averaged cross-section $T=m_{\chi}\langle v^{2}_{rel}\rangle/3$. So
the annihilation cross-section for dark matter can be expressed in terms of
temperature as

$\langle\sigma v\rangle_{\chi\chi\rightarrow
e^{+}e^{-}}=\frac{\alpha\mathcal{D}^{2}}{4}\left(\frac{T}{m_{\chi}}\right).$
(22)

Assuming $T_{d}\sim\frac{m_{\chi}}{10}$ and using Eq. (17) for temperature of
the dark matter the annihilation cross-section for dark matter with electric
dipole moment becomes

$\langle\sigma v\rangle_{\chi\chi\rightarrow
e^{+}e^{-}}=\frac{5\alpha\mathcal{D}^{2}T_{0}^{2}}{2m_{\chi}^{2}}\left(1+z\right)^{2}.$
(23)

Hence using (14) we get

$p_{ann}=\frac{5\alpha\mathcal{D}^{2}T_{0}^{2}}{2m_{\chi}^{3}}\left(1+z\right)^{2}.$
(24)

In this case the energy deposition rate will be

$\frac{dE}{dt}=\frac{1}{2}\rho_{c}^{2}c^{2}\Omega_{DM}^{2}\frac{5\alpha\mathcal{D}^{2}T_{0}^{2}}{2m_{\chi}^{3}}\left(1+z\right)^{8}.$
(25)

Here also the redshift dependence is modified as compared to (9), since the
thermally averaged cross-section is velocity dependent.

###  III.3 Dark matter with magnetic dipole moment

For dark matter with magnetic dipole moment the annihilation cross-section is
given as Masso:2009mu ,

$\langle\sigma v\rangle_{\chi\chi\rightarrow e^{+}e^{-}}=\alpha\mu^{2},$ (26)

and hence

$p_{ann}=\frac{\alpha\mu^{2}}{m_{\chi}}.$ (27)

Here the annihilation cross-section does not depend on the velocity of dark
matter so the energy deposition rate will be

$\frac{dE}{dt}=\frac{1}{2}\rho_{c}^{2}c^{2}\Omega_{DM}^{2}\frac{\alpha\mu^{2}}{m_{\chi}}\left(1+z\right)^{6},.$
(28)

which has the same redshift dependence as in Eq. (9).

###  III.4 Rayleigh dark matter

For scalar dark matter with Rayleigh coupling (4) the annihilation cross-
section is given by kavanagh ,

$\langle\sigma
v\rangle_{\phi\phi\rightarrow\gamma\gamma}=\frac{(g_{4}^{2})m_{\phi}^{2}}{\Lambda_{4}^{4}},$
(29)

and hence

$p_{ann}=\frac{(g_{4}^{2})m_{\phi}}{\Lambda_{4}^{4}}.$ (30)

Here again the annihilation cross-section is independent of the velocity of
dark matter, so the energy deposition rate will be same as (9).

$\frac{dE}{dt}=\frac{1}{2}\rho_{c}^{2}c^{2}\Omega_{DM}^{2}\frac{(g_{4}^{2})m_{\phi}}{\Lambda_{4}^{4}}\left(1+z\right)^{6}.$
(31)

##  IV CMB constraints on various multipole moments of dark matter

As mentioned earlier annihilating dark matter increases the ionization
fraction during recombination and heats the plasma. Hence the evolution
equations of free electron fraction and matter temperature get modified as
given by Eq. (5) and Eq. (12) respectively. The non-standard ionization rate
$I_{X}$ to compute free electron fraction can be obtained using Eq. (8) along
with Eqs. (10) and (11). We use these equations along with energy deposition
rates (20), (25), (28) and (31) for dark matter with anapole moment, electric
dipole moment and magnetic dipole moment, and Rayleigh coupling to modify
RECFAST routine Seager:1999km  in CAMB Lewis:1999bs . We have also checked our
analysis using CosmoRec Chluba:2010ca  and HyRec AliHaimoud:2010dx ;
Giesen:2012rp  code instead of RECFAST, and we found similar results. With
this we obtain modified theoretical angular power spectra, which can be used
to compute the bounds on various electromagnetic form factors and mass of the
dark matter from Planck-2018 data using COSMOMC Lewis:2002ah . The priors for
the multipole moments and mass of the dark matter are given in Table 1. All
these priors are sampled logarithmically to cover a larger range for the new
parameters. We also vary the other six parameters of $\Lambda$CDM model with
priors given in Ade:2013zuv . We have imposed flat priors for all parameters.

Type of dark matter coupling | Priors
---|---
Anapole |  |  $5.0<$$\ln\left(10^{9}\left(g_{A}/GeV^{-2}\right)\right)$$<40$ |  $\;\;-3.0<$$\log_{10}\left(m_{\chi}/GeV\right)$ $<2.0$
---|---
Electric dipole |  |  $-5.0<$ $\ln(10^{18}({\cal D}/(e-cm))$ $<40$ |  $\;\;\;-3.0<$ $\log_{10}\left(m_{\chi}/GeV\right)$ $<2.0$
---|---
Magnetic dipole |  |  $-10.0<$ $\ln(10^{9}(\mu/\mu_{B}))$ $<15.0$ $\;\;\;\;-3.0<$ $\log_{10}\left(m_{\chi}/GeV\right)$ $<2.0$
---
Rayleigh coupling |  |  $-10.0<$ $\ln(10^{9}g_{4}/(\Lambda_{4}^{2}GeV^{-2}))$ $<20.0$ $\;\;\;\;-12.0$ $<\log_{10}\left(m_{\chi}/GeV\right)$ $<2.0$
---
Table 1: Priors on input parameters for annihilating dark matter.

We use the lower bound for the mass of dark matter with anapole moment, and
electric and magnetic dipole moment as $1$ MeV since the annihilation channel
for this case is $\chi\;\chi\rightarrow\;e^{+}\,e^{-}$. However, in case of
scalar dark matter with Rayleigh coupling, kavanagh , the dark matter
annihilates to photons having energy around $1$ eV during recombination, which
is used as the lower bound for mass of Rayleigh dark matter. We also use BAO
and Pantheon data along with Planck-2018 observations for our analysis. We
perform MCMC convergence diagnostic tests on 4 chains using the Gelman and
Rubin ”variance of mean”/”mean of chain variance” R-1 statistics for each
parameter.

The constraints obtained for anapole moment and mass of the dark matter along
with other six parameters of $\Lambda$CDM model are shown in Table 2. Fig. 1
represents the marginalized constraints on anapole moment and mass of the dark
matter along with joint 68% CL and 95% CL constraints on both the parameters
from Planck-2018 and BAO data.

Parameter | 68% limits | 95% limits | 99% limits
---|---|---|---
$\Omega_{b}h^{2}$ | $0.02243\pm 0.00013$ | $0.02243^{+0.00026}_{-0.00026}$ | $0.02243^{+0.00034}_{-0.00034}$
$\Omega_{c}h^{2}$ | $0.11917\pm 0.00090$ | $0.1192^{+0.0018}_{-0.0018}$ | $0.1192^{+0.0024}_{-0.0023}$
$\tau$ | $0.0567\pm 0.0072$ | $0.057^{+0.015}_{-0.014}$ | $0.057^{+0.020}_{-0.018}$
${\rm{ln}}(10^{9}(g_{A}/GeV^{-2}))$ | $<22.6$ | $<29.6$ | $<31.9$
${\rm{log_{10}}}(m_{\chi}/GeV)$ | $---$ | $---$ | $---$
${\rm{ln}}(10^{10}A_{s})$ | $3.048\pm 0.014$ | $3.048^{+0.029}_{-0.028}$ | $3.048^{+0.039}_{-0.036}$
$n_{s}$ | $0.9670\pm 0.0037$ | $0.9670^{+0.0073}_{-0.0073}$ | $0.9670^{+0.0097}_{-0.0096}$
$H_{0}$ | $67.73\pm 0.41$ | $67.73^{+0.81}_{-0.80}$ | $67.7^{+1.1}_{-1.1}$
Table 2: Planck-2018 and BAO constraints on anapole momentum and mass of the
dark matter with other 6 parameters of $\Lambda$CDM

(a) Marginalized constraints (b) Joint 68% CL and 95%CL constraints

Figure 1: Constraints for anapole moment and mass of the dark matter using
Planck-2018 and BAO data

It can be seen from Table 2 that

$g_{A}<7.163\times 10^{3}\text{GeV\textsuperscript{-2}}\;~{}95\%C.L.$ (32)

The constraints obtained using Planck-2018 and BAO data on electric dipole
moment and the mass of the dark matter along with the other six parameters of
$\Lambda$CDM model are listed in Table. 3. Fig. 2 depicts the marginalized
constraints on electric dipole moment and mass of the dark matter and with
joint 68% CL and 95%CL constraints on both the parameters from Planck-2018 and
BAO data.

Parameter | 68% limits | 95% limits | 99% limits
---|---|---|---
$\Omega_{b}h^{2}$ | $0.02244\pm 0.00014$ | $0.02244^{+0.00027}_{-0.00026}$ | $0.02244^{+0.00036}_{-0.00034}$
$\Omega_{c}h^{2}$ | $0.11921\pm 0.00091$ | $0.1192^{+0.0018}_{-0.0018}$ | $0.1192^{+0.0024}_{-0.0024}$
$\tau$ | $0.0565\pm 0.0073$ | $0.056^{+0.015}_{-0.014}$ | $0.056^{+0.020}_{-0.018}$
${\rm{ln}}(10^{18}({\cal D}/(e-cm))$ | $<12.9$ | $<22.8$ | $<26.7$
${\rm{log_{10}}}(m_{\chi}/GeV)$ | $>-0.963$ | $---$ | $---$
${\rm{ln}}(10^{10}A_{s})$ | $3.048\pm 0.014$ | $3.048^{+0.029}_{-0.028}$ | $3.048^{+0.039}_{-0.037}$
$n_{s}$ | $0.9670\pm 0.0037$ | $0.9670^{+0.0071}_{-0.0073}$ | $0.9670^{+0.0093}_{-0.0097}$
$H_{0}$ | $67.72\pm 0.41$ | $67.72^{+0.82}_{-0.79}$ | $67.7^{+1.1}_{-1.0}$
Table 3: Planck-2018 and BAO constraints on electric dipole momentum and mass
of the dark matter with other 6 parameters of $\Lambda$CDM

(a) Marginalized constraints (b) Joint 68% CL and 95%CL constraints

Figure 2:  Constraints for electric dipole moment and mass of the dark matter
using Planck-2018 and BAO data.

We can see from Table. 3 that

${\cal D}<7.978\times 10^{-9}\text{e-cm}\;~{}95\%C.L.$ (33)

Table. 4 represents the constraints on magnetic dipole moment and mass of the
dark matter obtained from Planck-2018 and BAO data. Here also we have quoted
the constraints on other six parameters of $\Lambda$CDM. Fig. 3 represents the
marginalized constraints on magnetic dipole moment and mass of the dark matter
along with joint 68% CL and 95%CL constraints on both the parameters.

Parameter | 68% limits | 95% limits | 99% limits
---|---|---|---
$\Omega_{b}h^{2}$ | $0.02244\pm 0.00013$ | $0.02244^{+0.00026}_{-0.00026}$ | $0.02244^{+0.00034}_{-0.00034}$
$\Omega_{c}h^{2}$ | $0.11920\pm 0.00091$ | $0.1192^{+0.0018}_{-0.0018}$ | $0.1192^{+0.0023}_{-0.0023}$
$\tau$ | $0.0564\pm 0.0073$ | $0.056^{+0.015}_{-0.014}$ | $0.056^{+0.020}_{-0.018}$
${\rm{ln}}(10^{9}(\mu/\mu_{B}))$ | $-2.0^{+4.0}_{-6.4}$ | $<5.69$ | $<7.36$
${\rm{log_{10}}}(m_{\chi}/GeV)$ | $---$ | $---$ | $....$
${\rm{ln}}(10^{10}A_{s})$ | $3.047\pm 0.014$ | $3.047^{+0.029}_{-0.028}$ | $3.047^{+0.039}_{-0.037}$
$n_{s}$ | $0.9671\pm 0.0037$ | $0.9671^{+0.0073}_{-0.0072}$ | $0.9671^{+0.0096}_{-0.0093}$
$H_{0}$ | $67.72\pm 0.41$ | $67.72^{+0.82}_{-0.79}$ | $67.7^{+1.1}_{-1.0}$
Table 4: Planck-2018 and BAO constraints on magnetic dipole momentum and mass
of Majorana fermion dark matter with other 6 parameters of $\Lambda$CDM

(a) Marginalized constraints

(b) Joint 68% CL and 95%CL constraints

Figure 3: Constraints for magnetic dipole moment and mass of the dark matter
using Planck-2018 and BAO data.

Again we can read from Table 4 that

${\mu}<2.959\times 10^{-7}\mu_{B}\;~{}95\%C.L.$ (34)

Similarly Table. 4 lists the constraints on Rayleigh coupling and mass of the
scalar dark matter. Fig. 3 represents the marginalized constraints on both of
these parameters along with joint 68% CL and 95%CL constraints from
Planck-2018 and BAO data. Again we have mentioned constraints on other six
parameters of $\Lambda$CDM.

Parameter | 68% limits | 95% limits | 99% limits
---|---|---|---
$\Omega_{b}h^{2}$ | $0.02244\pm 0.00013$ | $0.02244^{+0.00026}_{-0.00026}$ | $0.02244^{+0.00034}_{-0.00034}$
$\Omega_{c}h^{2}$ | $0.11919\pm 0.00092$ | $0.1192^{+0.0018}_{-0.0018}$ | $0.1192^{+0.0023}_{-0.0024}$
$\tau$ | $0.0567\pm 0.0074$ | $0.057^{+0.015}_{-0.014}$ | $0.057^{+0.020}_{-0.019}$
${\rm{ln}}(10^{9}g_{4}/(\Lambda_{4}^{2}GeV^{-2}))$ | $<6.43$ | $<16.2$ | $<19.7$
${\rm{log_{10}}}(m_{\chi}/GeV)$ | $<-3.55$ | — | —
${\rm{ln}}(10^{10}A_{s})$ | $3.048\pm 0.015$ | $3.048^{+0.030}_{-0.028}$ | $3.048^{+0.040}_{-0.037}$
$n_{s}$ | $0.9671\pm 0.0037$ | $0.9671^{+0.0073}_{-0.0073}$ | $0.9671^{+0.0097}_{-0.0096}$
$H_{0}$ | $67.73\pm 0.41$ | $67.73^{+0.81}_{-0.80}$ |
| $67.7^{+1.1}_{-1.1}$ |  |
Table 5: Planck-2018 and BAO constraints on Rayleigh coupling and mass of dark
matter with other 6 parameters of $\Lambda$CDM

(a) Marginalized constraints

(b) Joint 68% CL and 95%CL constraints

Figure 4: Constraints on Rayleigh coupling and mass of scalar dark matter
using Planck-2018 and BAO data

We can see from the Table 5 that

${\frac{g_{4}}{\Lambda_{4}^{2}}}<1.085\times
10^{-2}\;\text{GeV\textsuperscript{-2}}\;~{}95\%C.L.$ (35)

The upper bounds given by Eqns. (32), (33), (34) and (35) are obtained after
margenalizing over all other parameters.

##  V Comparison with constraints from other experiments

In order to obtain the correct relic density of sub-GeV dark matter we require
mediators in the sub-GeV mas range Mohanty:2020pfa . We can describe dark
matter interactions, where the energy transfer is so small, with effective
operators like anapole, dipole and Rayleigh form factors, which are effective
below the cutoff scale $\Lambda\sim{\rm GeV}$. These low energy effective form
factors cannot be constrained from colliders, and the best bounds are obtained
from the low energy processes like electron scattering in direct detection
experiments or in searches of sub-GeV scale gamma rays from galaxies. The
constraints which can be obtained on ${\rm TeV}$ scale effective theories from
colliders is studied in Arina:2020mxo . We compare the bounds from CMB
distortion by dark matter annihilation obtained in this paper with the bounds
on the electromagnetic form factors from other experiments and astrophysical
observations in this section.

###  V.1 Electric, Magnetic and Anapole moments

For sub-GeV mass fermionic dark matter the best bounds come from the
ionization of atoms with electron scattering process
$\chi+e^{-}\rightarrow\chi+e^{-}$ in direct detection experiments like
Xenon-10 Essig:2012yx , DarkSide Agnes:2018oej  and Xenon-1T Aprile:2019xxb .
The most stringent bounds come from Xenon-10 Essig:2012yx  and Xenon-1T
Aprile:2019xxb  which are dark matter mass dependent. The bound on anapole
moment of Majorana dark matter is $g_{A}<(10^{2}-0.5\times 10^{-1}){\rm
GeV^{-2}}$ in the mass range $m_{\chi}=(0.2\,{\rm MeV}-1\,{\rm GeV})$
Catena:2019gfa . This is more stringent than the CMB bound $g_{A}<7.163\times
10^{3}\,{\rm Gev^{-2}}$ (for $m_{\chi}\geq 0.5\,{\rm MeV}$) since the CMB
bound is based on annihilation process $\chi\chi\rightarrow e^{+}e^{-}$, whose
cross section is velocity suppressed (15).

The bound on electric dipole moment of Xenon-10 Essig:2012yx  and Xenon-1T
Aprile:2019xxb  of Dirac dark matter with mass in the range
$m_{\chi}=(0.2\,{\rm MeV}-1\,{\rm GeV})$ is ${\cal D}<(6.6\times
10^{-18}-3.9\times 10^{-20})\,{\rm e-cm}$ Catena:2019gfa . This is again more
stringent than the CMB bound ${\cal D}<7.978\times 10^{-9}\,\text{ e-cm}$ (for
$m_{\chi}\geq 0.5{\rm MeV}$) as the annihilation cross section for electric
dipole annihilation is velocity suppressed (21).

Finally bound on magnetic dipole moment from Xenon-10 Essig:2012yx  and
Xenon-1T Aprile:2019xxb  of Dirac dark matter with mass in the range
$m_{\chi}=(0.2\,{\rm MeV}-1\,{\rm GeV})$ is $\mu<(1.2\times 10^{-5}-5.9\times
10^{-8})\mu_{B}$ Catena:2019gfa . This is comparable to the CMB bound
$\mu<2.959\times 10^{-7}\mu_{B}$ (for $m_{\chi}\geq 0.5{\rm MeV}$). The CMB
bound is comparable with the direct detection bounds as the annihilation cross
section for magnetic dipole annihilation is not velocity suppressed (26).

###  V.2 Rayleigh form factor of scalar dark matter

Constraints on the Rayleigh form factor (4) are obtained from electron
ionisation by Xenon-1T Aprile:2019xxb  and by searches for $\gamma$ ray line
spectrum in the Milky Way center by Fermi-LAT Ackermann:2015lka . For dark
matter of mass $\sim{\rm GeV}$ the bound from Fermi-LAT search is
$g_{4}/\Lambda_{4}^{2}<10^{-3}\,{\rm GeV^{-2}}$, and from Xenon-1T the bound
is $g_{4}/\Lambda_{4}^{2}<0.25\,{\rm GeV^{-2}}$ kavanagh . The CMB bound
obtained in the paper $g_{4}^{2}/\Lambda_{4}^{2}<1.1\times 10^{-2}\,{\rm
GeV^{-2}}$ is weaker than the Fermi-LAT bound but is valid for larger range of
dark matter masses up to $m_{\phi}\geq{\rm eV}$.

##  VI Conclusions

Electromagnetic form factors are an important class of interactions in the
effective theories framework of classifying dark matter interactions. Dirac
fermions dark matter with non-zero electric and magnetic dipole moments can
give the correct relic density $\Omega_{m}h^{2}=0.11$ by the
$\chi\chi\leftrightarrow f\bar{f}$ freeze-out process if ${\cal D}=2.5\times
10^{-16}\,\text{ e-cm}$ and $\mu=8.2\times 10^{-7}\mu_{B}$ respectively
Masso:2009mu . Majorana fermions with anapole moment of mass $10$ MeV with
anapole moment $g_{A}=0.11{\rm Gev^{-2}}$ can be dark matter with the correct
freeze-out relic density Ho:2012bg .

Electromagnetic dark matter can be observed not only via electron scattering
direct detection experiments Essig:2011nj ; Essig:2012yx ; Agnes:2018oej ;
Aprile:2019xxb ; Catena:2019gfa ; Catena:2020tbv  but can also be constrained
from the CMB.

In this paper we have considered anapole and dipolar dark matter matter with
masses $m_{\chi}>\cal{O}({\rm MeV})$. We find that dark matter with
electromagnetic dipole or anapole form factors will distort the CMB during
recombination era by producing relativistic electron via the process
$\chi\chi\rightarrow e^{+}e^{-}$. We find that the Planck data gives the
bounds on electromagnetic form factors ${\cal D}<7.978\times 10^{-9}\,\text{
e-cm}$ and $\mu<2.959\times 10^{-7}\mu_{B}$, and $g_{A}<7.163\times 10^{3}{\rm
Gev^{-2}}$.

Dark matter with $\cal{O}({\rm MeV})$ mass in thermal equilibrium with
radiation will be ruled out by BBN constraints on $N_{\rm eff}$. These can
only have be created after the BBN era by the freeze-in mechanism. Freeze-in
requires very small couplings and our bounds on electric dipole and anapole
moments also rules out the freeze-out mechanism for relic density. These may
be produced by the freeze-in mechanism which requires smaller couplings
Hall:2009bx ; Mohanty:2020pfa .

For scalar dark matter there is the dimension-six Rayleigh operator coupling
with photons. We put the bound on the Rayleigh coupling as
$\frac{g_{4}}{\Lambda_{4}^{2}}<1.085\times
10^{-2}\text{GeV\textsuperscript{-2}}$ ( $95\%$C.L).This bound is valid for
dark matter mass as low as $\cal{O}({\rm eV})$. Such light dark matter can
only be produced by the freeze-in mechanism to evade bounds from BBN.

Spectral distortion of the CMB can also arise from radiatively decaying dark
matter Bolliet:2020ofj . The bounds derived on the radiative lifetime can be
used for deriving bounds on dipolar couplings of Majorana dark matter which
can have non-zero transition electric and magnetic moments dipole.

##  VII ACKNOWLEDGEMENTS

A. N. would like to thank ISRO Department of Space Govt. of India to provide
financial support via RESPOND programme Grant No.
DS_2B-13012(2)/47/2018-Sec.II.

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# Terrestrial-Satellite Spectrum Sharing in the
Upper Mid-Band with Interference Nulling

Seongjoon Kang† Giovanni Geraci♯ Marco Mezzavilla† Sundeep Rangan† S. Kang, M.
Mezzavilla, and S. Rangan were supported by NSF grants 1952180, 2133662,
2236097, 2148293, 1925079, the NTIA, and the industrial affiliates of NYU
WIRELESS. G. Geraci was supported by the Spanish Research Agency grants
PID2021-123999OB-I00 and CEX2021-001195-M, by the UPF-Fractus Chair, and by
the Spanish Ministry of Economic Affairs and Digital Transformation and the
European Union NextGenerationEU through the UNICO 5G I+D SORUS project. †NYU
Tandon School of Engineering, Brooklyn, NY, USA  ♯Telefónica Research and
Universitat Pompeu Fabra, Barcelona, Spain

###### Abstract

The growing demand for broader bandwidth in cellular networks has turned the
upper mid-band (7–24 GHz) into a focal point for expansion. However, the
integration of terrestrial cellular and incumbent satellite services,
particularly in the 12 GHz band, poses significant interference challenges.
This paper investigates the interference dynamics in terrestrial-satellite
coexistence scenarios and introduces a novel beamforming approach that
leverages available ephemeris data for dynamic interference mitigation. By
establishing spatial radiation nulls directed towards visible satellites, our
technique ensures the protection of satellite uplink communications without
markedly compromising terrestrial downlink quality. Through a practical case
study, we demonstrate that our approach maintains the satellite uplink signal-
to-noise ratio (SNR) degradation under 1 dB and incurs a median SNR penalty of
only 0.1 dB for the terrestrial downlink. Our findings offer a promising
pathway for efficient spectrum sharing in the upper mid-band, fostering a
concurrent enhancement in both terrestrial and satellite network capacity.

##  I Introduction

The upper mid-band from 7–24 $\mathrm{G}\mathrm{H}\mathrm{z}$ has attracted
considerable interest to expand cellular services [1]. These frequencies offer
much more abundant spectrum than that available in the congested sub-7
$\mathrm{G}\mathrm{H}\mathrm{z}$ bands with more favorable propagation and
coverage than the mmWave frequencies. With this balance of coverage and
spectrum, the upper mid-band has become the focus of significant work in
industry [2, 3], the Federal Communications Commission (FCC) [4, 5], and the
3rd Generation Partnership Project (3GPP) [6]. In 3GPP, the upper mid-band is
referred to as frequency range 3 (FR3).

A key challenge for expanding cellular service in the upper mid-band is to
coexist with incumbent services, particularly commercial communication
satellites. Mega-constellations of Low Earth Orbit (LEO) satellites can enable
new services and provide vital high-capacity coverage in rural areas where
fixed infrastructure is costly [7, 8, 9, 10]. Starlink, currently the largest
service provider, has deployed approximately $4700$ LEO satellites and is
planned to expand to over 25000 [11]. The upper mid-band is a key part of
Starlink’s service expansion as shown, for instance, by a recent FCC filing
[12].

Operating cellular networks in the upper mid-band may disrupt incumbent
satellite service links [13, 14]. In particular, as cellular base stations
(BSs) employ large antenna arrays and high transmit power, the sidelobes
generated by their beamformed transmissions toward the user equipment (UEs)
can result in strong interference to the satellite uplink. While early
attempts were made at mitigating cellular-to-satellite interference through
null-steering [15, 16], these approaches focus on one-dimensional beam spaces
and are not directly applicable to the rectangular array configurations of
current BSs. Morevoer, they may incur a significant beamforming gain loss when
the nulling directions are close to the main serving direction.

In this work, we consider practical cellular deployments in the upper mid-band
with realistic antenna configurations and present a novel approach to mitigate
cellular-to-satellite interference. In particular, we focus on the
$12\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ band, given the growing interest
it has attracted thanks to its abundant available spectrum and benign
propagation [17].111Note that the FCC released a proposed rulemaking notice
(NPRM) to discuss the spectrum sharing of the
$12\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ band with 5G systems [4]. Our
proposed approach leverages LEO satellite tracking information using
_ephemeris_ data, and it aims to mitigate interference generated by the
downlink of a cellular network to the uplink of the LEO satellite as shown in
Fig. 1, since this represents the worst case. Our main contributions are as
follows:

  * •

_Cellular-to-satellite interference analysis:_ We analytically illustrate the
relationship between the satellite elevation angle and the corresponding
cellular-to-satellite pathloss, identifying ranges susceptible to
interference.

  * •

_Proposed interference nulling_ : We propose an interference nulling method in
which terrestrial beamforming is carried out while creating spatial radiation
nulls onto certain directions based on line-of-sight (LOS) channels computed
from practically available ephemeris data.

  * •

_Case study_ : We track Starlink satellites from a sample rural area near
Boulder, Colorado, and evaluate terrestrial-satellite spectrum sharing in the
upper mid-band. We demonstrate that the proposed interference nulling method
can consistently keep the degradation in the satellite uplink signal-to-noise
ratio (SNR) below $1\text{\,}\mathrm{d}\mathrm{B}$ while only penalizing the
terrestrial downlink SNR by $0.1\text{\,}\mathrm{d}\mathrm{B}$ in median.

Figure 1: Illustration of the interference caused by the cellular downlink to
the satellite uplink through BS antenna sidelobes.

##  II Single Victim Satellite in a LOS Channel

To understand the nature of the interference from the terrestrial downlink to
the satellite uplink, we first consider the case of a single victim satellite
in a LOS scenario as illustrated in Fig. 1. We will analyze multiple
satellites and non-LOS channels later. The interference in this scenario is
influenced by two primary factors: the BS antenna element gain and the
satellite elevation angle. On the one hand, when satellites are positioned at
higher elevation angles, they are typically situated more directly above the
BSs. Consequently, the distance between satellites and BSs decreases and so
does the path loss, potentially intensifying interference to the detriment of
the satellites. On the other hand, since BS antennas are generally slightly
down-tilted, their antenna gains diminish at higher elevation angles, thereby
mitigating interference. As a result, the level of interference experienced by
any given satellite can vary significantly across different elevation angles
and this variability necessitates a suitable approach to managing
interference.

The separation between the target satellite and the BS, termed the _slant
distance_ , is given as follows

$\displaystyle
d(\theta)=\sqrt{R_{E}^{2}\sin^{2}(\theta)+h^{2}+2hR_{E}},-R_{E}\sin(\theta)$
(1)

where $R_{E}$ denotes the Earth’s radius, $h$ is the height of BS, and
$\theta$ is the elevation angle. It can be observed that the slant distance
increases as the elevation angle $\theta$ in the global coordinate system
decreases. Given $d(\theta)$, the free space path loss (FSPL) at a carrier
frequency $f$ can be expressed as

$\displaystyle{\rm
FSPL}(\theta)=20\log_{10}(d(\theta))+20\log_{10}(f)-147.55.$ (2)

The red dotted curve in Fig. 2 depicts the variation in path loss as a
function of the satellite elevation angle $\theta$, with the azimuth angle
held constant at $\phi=0^{\circ}$.

Figure 2: Change in the total BS-to-satellite propagation loss caused by
accounting for the antenna element gain as a function of the satellite
elevation angle $\theta$ for an azimuth angle fixed to $\phi=0^{\circ}$.

To characterize the sidelobes of BS antenna arrays, we consider solely the
antenna element gain, as the gains due to beamforming vary depending on the
specific transmission directions. We adopt the 3GPP antenna field pattern
defined in [18], described by

$\displaystyle G_{E}(\theta,\phi)$
$\displaystyle=G_{\max}\\!-\\!\min\left\\{\\!-\\!\left[G_{E,V}(\theta)\\!+\\!G_{E,H}(\phi)\right]\\!,A_{\mathrm{m}}\right\\}$
(3) $\displaystyle G_{E,V}(\theta)$
$\displaystyle=-\min\left\\{12\left[\left(\theta-90^{\circ}\right)/\theta_{3\mathrm{dB}}\right]^{2},SLA_{\mathrm{V}}\right\\}$
(4) $\displaystyle G_{E,H}(\phi)$
$\displaystyle=-\min\left\\{12\left[\phi/\phi_{3\mathrm{dB}}\right]^{2},A_{\mathrm{m}}\right\\},$
(5)

and where $G_{\max}=$ $8\text{\,}\mathrm{d}\mathrm{B}$ is the maximum gain,
$SLA_{\mathrm{V}}=$ $30\text{\,}\mathrm{d}\mathrm{B}$ denotes the side-lobe
level limit, $\phi_{\mathrm{3dB}}=65^{\circ}$ and
$\theta_{\mathrm{3dB}}=65^{\circ}$ are the horizontal and vertical
$3\text{\,}\mathrm{d}\mathrm{B}$ beamwidths, respectively, and
$A_{\mathrm{m}}=$ $30\text{\,}\mathrm{d}\mathrm{B}$ represents the front-to-
back ratio. Note that $\theta$ and $\phi$ are the elevation and azimuth angles
of the BS expressed in the local coordinate system.

For the purpose of this analysis, we focus on the impact of the BS antenna
sidelobes on the LOS channels between the BS and the satellite. We also assume
that the element gain of the satellite remains nearly at its maximum value,
representing a worst-case scenario. The solid black curve in Fig. 2 shows the
variation in propagation loss vs. satellite elevation angle when accounting
also for the antenna element gain in (3). We observe that, for elevation
angles below $40^{\circ}$, the BS antenna sidelobes further reduce the
propagation loss. Given the higher likelihood of encountering lower elevation
angles [19], Fig. 2 suggests that LOS channels between BSs and satellites may
pose a significant interference risk to uplink satellite communications.

##  III Proposed Interference Nulling Technique

We now consider the case where multiple victim satellites suffer interference
by potential _multi-path_ channels from a BS. As before, we focus on downlink
transmission (BS $\rightarrow$ UE). The BS has $N_{t}$ transmit antennas and
the UE has $N_{r}$ receive antennas, and we let $\boldsymbol{{H}}_{\rm
ter}\in\mathbb{C}^{N_{r}\times N_{t}}$ denote the multiple-input multiple-
output (MIMO) channel matrix from the BS to the UE, which can be frequency
selective due to multi-path fading. We are concerned with the sidelobe
interference to $N_{\rm sat}$ satellites, indexed $i=1,\ldots,N_{\rm sat}$.
Each potential victim satellite is assumed to have some beam focused on the
area covering the BS which is sensitive to the transmission from the BS. We
let $\boldsymbol{{h}}_{i}\in\mathbb{C}^{N_{t}\times 1}$ denote the multi-path
channel vector from the terrestrial BS to the $i$-th victim satellite, which
can be frequency dependent.

#### Multi-path interference nulling

The terrestrial BS and terrestrial UE must select, respectively, transmit and
receive beamforming vectors $\boldsymbol{{w}}_{t}$ and $\boldsymbol{{w}}_{r}$.
The first algorithm we consider is _multi-path interference nulling_ where the
beamforming vectors are selected by solving the following problem:

$\displaystyle{\boldsymbol{{w}}}_{r}^{\rm opt},{\boldsymbol{{w}}}_{t}^{\rm
opt}$
$\displaystyle=\arg\\!\max_{\boldsymbol{{w}}_{r},\boldsymbol{{w}}_{t}}\left[|\boldsymbol{{w}}_{r}^{\sf\scriptscriptstyle
H}\tilde{\boldsymbol{{H}}}_{\rm
ter}\boldsymbol{{w}}_{t}|^{2}-\lambda\sum_{i=1}^{N_{\rm
sat}}\lVert\tilde{\boldsymbol{{h}}}_{i}\boldsymbol{{w}}_{t}\lVert^{2}\right],$
(6)
$\displaystyle\textrm{s.t.}\quad\lVert\boldsymbol{{w}}_{t}\rVert^{2}=1\quad\text{and}\quad~{}\lVert\boldsymbol{{w}}_{r}\rVert^{2}=1,$

where $\tilde{\boldsymbol{{H}}}_{\text{ter}}$ and
$\tilde{\boldsymbol{{h}}}_{i}$ are the normalized versions of MIMO multi-path
channels $\boldsymbol{{H}}_{\text{ter}}$ and $\boldsymbol{{h}}_{i}$,
respectively, and $\lambda\geq 0$ denotes a regularization parameter trading
off beamforming gain at the terrestrial UE for nulling accuracy at the
satellites. Normalizing channel matrices ensures
$\lVert\tilde{\boldsymbol{{H}}}_{\rm ter}\rVert_{F}^{2}=N_{t}N_{r}$ and
$\lVert\tilde{\boldsymbol{{h}}}_{i}\rVert^{2}=N_{t}$ [20] and circumvents the
pronounced path loss disparity between terrestrial and satellite channels,
thus restricting the range of values of interest for $\lambda$. We
approximately solve the optimization problem (6) by selecting
$\boldsymbol{{w}}_{r}^{\rm opt}$ as the maximum singular vector on the left of
$\boldsymbol{{H}}_{\rm ter}$. Then, $\boldsymbol{{w}}_{t}^{\rm opt}$ is given
by maximum eigenvector of

$\displaystyle\tilde{\boldsymbol{{H}}}^{\sf\scriptscriptstyle H}_{\rm
ter}\boldsymbol{{w}}_{r}\boldsymbol{{w}}_{r}^{\sf\scriptscriptstyle
H}\tilde{\boldsymbol{{H}}}_{\rm
ter}-\lambda\sum_{i=1}^{N_{sat}}\tilde{\boldsymbol{{h}}}_{i}\tilde{\boldsymbol{{h}}_{i}}^{\sf\scriptscriptstyle
H}$ (7)

Due to a multi-path environment, the solutions of the problem (6), i.e.,
${\boldsymbol{{w}}}_{r}^{\rm opt}$ and ${\boldsymbol{{w}}}_{t}^{\rm opt}$,
depend on frequency.

#### Proposed LOS interference nulling

A practical challenge in implementing multi-path interference nulling is that
the BS must keep track of the entire wideband channel $\boldsymbol{{h}}_{i}$
to each satellite. To avoid the complexity of real-time multi-path channel
estimation between each BS and all satellites, we propose a simple _LOS
nulling_ method. In this method, we replace $\tilde{\boldsymbol{{h}}}_{i}$
with $\boldsymbol{{e}}(\theta_{i},\phi_{i})$ where $(\theta_{i},\phi_{i})$ are
the azimuth and elevation angle to the $i$-th victim satellite and
$\boldsymbol{{e}}(\theta_{i},\phi_{i})$ is the spatial signature of the BS
antenna array in that direction. This formulation is motivated by the power
predominance of LOS paths with respect to NLOS paths. Furthermore, all
ephemeris data from commercial satellites are publicly available. Therefore,
the channel vector $\boldsymbol{{e}}(\theta_{i},\phi_{i})$ can be obtained at
each BS using these public data for real-time satellite tracking, along with
the respective elevation and azimuth angles.

#### Example beamforming pattern

To illustrate the LOS beamforming method, Fig. 3 shows an example of the
transmit beamforming gains $|{\boldsymbol{{e}}^{\sf\scriptscriptstyle
H}(\theta,\phi)}\boldsymbol{{w}}_{t}|^{2}$ across the angular space of a BS
equipped with an $8\times 8$ uniform rectangular array (URA) with a vector
$\boldsymbol{{w}}_{t}$ calculated from the proposed LOS interference nulling
method. In this case, there are $N_{\rm sat}=10$ satellites with locations
shown in the red cross marks. We set $\lambda=10$ and we can see that the
beamforming pattern places strong nulls at all satellite locations. The
desired destination towards the terrestrial UE is shown in the black circle.
In addition, the beamforming gain towards the desired UE computed with
interference nulling is $17.96\text{\,}\mathrm{d}\mathrm{B}$. The theoretical
maximum beamforming gain is
$10\log_{10}(64)=$$18.06\text{\,}\mathrm{d}\mathrm{B}$, so the loss, at least
in this case, is negligible.

#### Effects of Motion

A potential issue with either full multi-path or LOS interference nulling is
that the interference channel can vary over the transmission time of a BS. To
assess this possibility, recall that, in 5G, the transmission time interval
(TTI) is as large as $T=$ $1\text{\,}\mathrm{m}\mathrm{s}$. Now, a LEO
satellites orbiting at an altitude of $h=$ $600\text{\,}\mathrm{k}\mathrm{m}$
would travel at a velocity of
$v=\,$$7.56\text{\,}\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{s}$. Hence, the
maximum change in the the angle is at most,
$\Delta\theta=\tan^{-1}\left({vT}/{h}\right)\approx 7.2\times 10^{-4}$
degrees. Hence, the angular change is minimal, and LOS interference channel
$\boldsymbol{{e}}(\theta_{i},\phi_{i})$ can considered to be approximately
constant over practical TTIs.

Figure 3: Beamforming gains in beam space for an $8\times 8$ BS URA.

##  IV Case Study and Performance Evaluation

We now present a case study to evaluate the performance of the proposed
interference nulling. The main assumptions are detailed in the sequel and
summarized in Table I.

###  IV-A System Model and Methodology

#### Terrestrial channel model

We focus on downlink transmission, since this represents the worst case in
terms of interference caused at satellites. We consider a
$24\text{\,}\mathrm{k}\mathrm{m}$ $\times$ $15\text{\,}\mathrm{k}\mathrm{m}$
rural area near Boulder, Colorado, illustrated in Fig. 4, where BSs are
deployed on a grid with $1732\text{\,}\mathrm{m}$ inter-site distance (ISD).
Terrestrial users (UEs) are deployed uniformly at random throughout the region
and associate to the BS providing the largest signal strength. We consider a
scenario with $20\%$ network load, resulting in 21 BSs (out of the total 104)
transmitting simultaneously, each with an $8\times 8$ URA. For this setup,
following the 3GPP specifications in [21], we simulate the propagation channel
using the commercial ray-tracing software Wireless InSite [22].

#### Satellite channel model

We assume that LEO satellites in a mega-constellation are employed to provide
service to satellite UEs within the same rural area, sharing the same spectrum
as the terrestrial deployment in the upper mid-band. To capture the real-time
positions of the satellites, we use ephemeris data in the two-line element set
(TLE) format [23], available from public databases [24], and process it
through the Python library _pycraf_ [25]. As shown in Fig. 4, we consider a
single Earth Station within the given rural area where satellites are tracked
at every minute. We consider a fixed number of $\rm N_{\rm sat}$ satellites
chosen randomly at each interval from the pool of visible satellites. For
those, we obtain the multi-path channel components via ray-tracing and add the
relevant attenuation effects from atmospheric conditions, rain, clouds, and
scintillation using the Python library ITU-Rpy [26].

#### Real-time channel extraction

While ray-tracing simulations allow us to obtain all propagation channels, the
BS-to-satellite channels vary dynamically over time due to satellite motion,
and it is impractical to perform real-time ray-tracing. To overcome this
issue, we pre-calculate these channels for a set of fixed positions
categorized by satellite elevation and azimuth angles, creating a look-up
table as shown in Fig. 5. Specifically, we include discrete points spaced by
$60^{\circ}$ in azimuth and $10^{\circ}$ in elevation. This look-up table
allows us to approximate real-time satellite channels by selecting entries in
the table closest to the current satellite position.

Figure 4: Rural area near Boulder, CO, used for our case study. Figure 5:
Computation of satellite multi-path channels: (i) satellite coordinates
$(\theta^{t},\phi^{t})$ at time $t$ are tracked via ephemeris data; (ii) ray-
tracing is employed to pre-compute the multi-path channels for each set of
coordinates in the look-up table; and (iii) over time the multi-path channel
instances are extracted from the look-up table.

###  IV-B Key Performance Indicators

#### Satellite interference-to-noise ratio

The interference-to-noise ratio (INR) serves as a key metric to quantify
interference in spectrum sharing scenarios. For a satellite $i$ and BS $j$
employing transmit beamforming vector $\boldsymbol{{w}}_{j}$, the INR is
calculated as follows:

${\mathsf{INR}}_{i}\\!=\\!P_{\rm tx}+10\log_{10}\sum_{j=1}^{\rm N_{\rm
BS}}{|\boldsymbol{{w}}_{j}^{\sf\scriptscriptstyle
H}\boldsymbol{{h}}_{ij}|^{2}}+\frac{G}{T}-L_{a}-10\log_{10}(B\cdot\kappa),$
(8)

where $P_{\rm tx}$ denotes the BS transmit power, ${\rm N_{\rm BS}}$ is the
number of concurrent BS transmissions, $\boldsymbol{{h}}_{ij}$ is the multi-
path interference channel between BS $j$ and satellite $i$, $G/T$ is the
satellite antenna gain-to-noise-temperature ratio, $L_{a}$ incorporates
additional losses such as atmospheric and scintillation attenuation [23], $B$
is the bandwidth, and $\kappa$ is Boltzmann constant. Note that the INR on the
satellite link can be directly mapped to an SNR degradation $\rho^{\rm S}$ as
follows

$\displaystyle\rho^{\rm
S}=10\log_{10}\left(1+10^{0.1\cdot\mathrm{INR}}\right),$ (9)

e.g., for $\rho^{\rm S}<$ $1\text{\,}\mathrm{d}\mathrm{B}$ the INR should be
kept below $-6\text{\,}\mathrm{d}\mathrm{B}$.

#### Terrestrial SNR degradation

In addition, to evaluate the loss of terrestrial SNR we define the following
quantity [1]

$\rho^{\rm T}=10\log_{10}\left(\frac{|{\boldsymbol{{w}}}_{r}^{\rm
opt}\boldsymbol{{H}}_{\rm
ter}{\boldsymbol{{w}}}_{t}|^{2}}{|{\boldsymbol{{w}}}_{r}^{\rm
opt}\boldsymbol{{H}}_{\rm
ter}\widehat{{\boldsymbol{{w}}}}_{t}^{\lambda}|^{2}}\right)$

where $\boldsymbol{{w}}_{r}^{\rm opt}$ is the receive beamformer obtained from
(6), whereas $\boldsymbol{{w}}_{t}$ and
$\widehat{{\boldsymbol{{w}}}}_{t}^{\lambda}$ are the transmit beamformers
obtained from (6) for $\lambda=0$ and for $\lambda>0$, respectively, i.e.,
without and with interference nulling.

###  IV-C Coexistence Evaluation Without Interference Nulling

To establish a baseline, we assess the interference experienced by satellites
in the absence of interference nulling, when terrestrial BSs apply singular
value decomposition (SVD)-based transmit beamforming. This is achieved by
resolving (6) with $\lambda=0$, thus ignoring any interference to satellites.
Fig. 6a illustrates the cumulative distribution function (CDF) of the INR
across different satellite elevation angle ranges $\theta$: specifically, for
$\theta\in[25^{\circ},45^{\circ}]$ (dotted red),
$\theta\in[45^{\circ},70^{\circ}]$ (dash-dot blue), and
$\theta\in[70^{\circ},90^{\circ}]$ (dashed gray). The CDF for the entire
spectrum of elevation angles, i.e., $\theta\in[25^{\circ},90^{\circ}]$, is
depicted as well (solid black).

As previously discussed in Section II, lower elevation angles may lead to
increased interference due to the gain from the BS antenna elements. Fig. 6a
specifically demonstrates that when $\theta\in[25^{\circ},45^{\circ}]$ or
$\theta\in[45^{\circ},70^{\circ}]$, over $70\%$ of terrestrial transmissions
result in an $\text{INR}>$$-6\text{\,}\mathrm{d}\mathrm{B}$ for the satellite
uplink. In contrast, an $\text{INR}>$$-6\text{\,}\mathrm{d}\mathrm{B}$ is
observed only $30\%$ of the time when $\theta>70^{\circ}$. However, the
likelihood of encountering satellites at lower elevation angles is greater.
The probability distribution function (PDF) of the elevation angle for the
satellites observed from the Starlink constellation is shown in Fig. 6b,
aligning with the analytical findings in [19]. Consequently, the aggregate CDF
of the INR for all elevation angles (solid black line in Fig. 6a) corroborates
the presence of considerable terrestrial-to-satellite interference,
underscoring the importance of employing interference mitigation strategies.

(a)

(b)

Figure 6: (a) CDF of the INR on the satellite uplink without interference
nulling vs. elevation angle range, and (b) PDF of the elevation angles.

###  IV-D Performance Evaluation of Interference Nulling

TABLE I: System-level simulation parameters. Terrestrial network |
---|---
Geographical area |  24 km $\times$ 15 km near Boulder, CO
BS deployment |  $1732$ m ISD, 3 sectors/site
UE deployment |  Uniform random
Carrier frequency and noise figure |  $12$ GHz and $7$ dB (UE)
BS transmit power and network load |  $33$ dBm, 20% of the BSs active
Antenna height |  $35$ m (BS) and $1.6$ m (UE)
Antenna element radiation pattern |  As per [18], $12^{\circ}$ downtilt (BS)
Antenna array configuration |  $8\times 8$ (BS) and $1\times 2$ (UE)
Satellite network |
Tracked satellite constellation |  Starlink, min. elevation $25^{\circ}$ [27]
Coordinates of the Earth station |  $None$N, $None$W
Satellite bandwidth and $G/T$ |  $30$ MHz, $13$ dB/k

We evaluate three different beamforming approaches:

  * •

_Multi-path interference nulling_ : Achieved by solving (6) with the full
multi-path interference channels $\tilde{\boldsymbol{{h}}}_{i}$. Note that in
this case all the evaluation quantities are averaged in linear scale over
available carrier frequencies, since the multi-path channels can be frequency
selective.

  * •

_LOS interference nulling_ : Achieved by solving (6) employing the LOS
interference channel
$\tilde{\boldsymbol{{h}}}_{i}=\boldsymbol{{e}}(\theta_{i},\phi_{i})$. Recall
that the BSs can obtain this matrix by tracking the elevation and azimuth
angles of satellites from ephemeris data.

  * •

_No nulling_ : A baseline obtained by setting $\lambda=0$ in
$(\ref{eq:bf_eq})$, corresponding to SVD-based beamforming, which does not
perform any interference nulling.

#### Satellite uplink INR

Fig. 7a shows the CDF of the INR on the satellite link, defined in (8), for
the above three beamforming schemes. When interference nulling is performed,
two values of $\lambda$ are considered, namely $1$ and $10$. In all cases,
interference nulling (blue and red curves) is highly beneficial to the
satellite uplink, which otherwise incurs $\text{INR}>$
$-6\text{\,}\mathrm{d}\mathrm{B}$ nearly two thirds of the time (black curve).
With a sufficiently large regularization parameter, i.e., $\lambda\geq 10$,
both nulling schemes consistently keep the INR below
$-6\text{\,}\mathrm{d}\mathrm{B}$, thus guaranteeing a satellite uplink SNR
degradation $\rho^{\rm S}<$ $1\text{\,}\mathrm{d}\mathrm{B}$. Importantly, the
proposed LOS interference nulling exhibits nearly the same performance as full
multi-path nulling, making the former highly preferable due to its greater
implementation feasibility. Indeed, non-LOS interference paths, such as those
caused by reflections or diffractions, seem to cause negligible interference
to the satellite uplink and do not justify tracking the multi-path channel for
nulling purposes.

(a)

(b)

Figure 7: CDF of (a) the INR on the satellite uplink, and (b) the
corresponding SNR loss on the terrestrial downlink.

#### Terrestrial downlink SNR degradation

Similarly, Fig. 7b shows the CDF of the SNR loss $\rho^{\rm T}$ on the
terrestrial dowlink for the three different nulling schemes and two values of
the regularization parameter $\lambda$. Observing Fig. 7b alongside Fig. 7a
confirms the role played by $\lambda$, whereby higher values (thicker dashed
blue/red curves) reduce the INR on the satellite uplink while increasing the
SNR degradation on the terrestrial downlink. Fig. 7b also shows that even for
$\lambda=10$ (which guarantees very low INR on the satellite uplink, as shown
in Fig. 7a) the proposed LOS interference nulling incurs an SNR loss on the
terrestrial downlink that amounts to only $0.1\text{\,}\mathrm{d}\mathrm{B}$
in median and remains below $3\text{\,}\mathrm{d}\mathrm{B}$ $95\%$ of time.

Altogether, the results in Fig. 7 suggest that, by choosing a sufficiently
large value of $\lambda$, nulling the LOS interference paths can successfully
eliminate terrestrial-to-satellite interference with a negligible terrestrial
performance loss.

##  V Conclusion

In this paper, we studied spectrum sharing between terrestrial network
downlink and LEO satellite uplink in the upper mid-band. We addressed the
critical challenge of interference from terrestrial BSs, particularly through
LOS sidelobe transmissions, by introducing a novel interference mitigation
technique. This technique optimizes terrestrial beamforming by creating
spatial nulls onto certain channel directions, which can be computed using
readily available satellite ephemeris data. Our system-level simulations,
which include tracking Starlink satellites over rural Colorado, confirmed that
our method can effectively prevent degradation of the satellite uplink while
maintaining terrestrial downlink performance. Possible extensions of our work
include considering multi-user MIMO terrestrial transmissions in the upper
mid-band and generalizing the proposed interference nulling technique
accordingly.

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