# Direct Detection of Light Dark Matter from Evaporating Primordial Black
Holes

Roberta Calabrese  Dipartimento di Fisica “Ettore Pancini”, Università degli
studi di Napoli “Federico II”, Complesso Univ. Monte S. Angelo, I-80126
Napoli, Italy  INFN - Sezione di Napoli, Complesso Univ. Monte S. Angelo,
I-80126 Napoli, Italy  Marco Chianese<EMAIL_ADDRESS>Dipartimento di
Fisica “Ettore Pancini”, Università degli studi di Napoli “Federico II”,
Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy  INFN - Sezione di
Napoli, Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy  Damiano F.G.
Fiorillo  Dipartimento di Fisica “Ettore Pancini”, Università degli studi di
Napoli “Federico II”, Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy
INFN - Sezione di Napoli, Complesso Univ. Monte S. Angelo, I-80126 Napoli,
Italy  Ninetta Saviano  INFN - Sezione di Napoli, Complesso Univ. Monte S.
Angelo, I-80126 Napoli, Italy  Scuola Superiore Meridionale, Università degli
studi di Napoli “Federico II”, Largo San Marcellino 10, 80138 Napoli, Italy

###### Abstract

The direct detection of sub-GeV dark matter interacting with nucleons is
hampered by the low recoil energies induced by scatterings in the detectors.
This experimental difficulty is avoided in the scenario of boosted dark matter
where a component of dark matter particles is endowed with large kinetic
energies. In this Letter, we point out that the current evaporation of
primordial black holes with masses from $10^{14}$ to $10^{16}$ g is a source
of boosted light dark matter with energies of tens to hundreds of MeV.
Focusing on the XENON1T experiment, we show that these relativistic dark
matter particles could give rise to a signal orders of magnitude larger than
the present upper bounds. Therefore, we are able to significantly constrain
the combined parameter space of primordial black holes and sub-GeV dark
matter. In the presence of primordial black holes with a mass of
$10^{15}~{}\mathrm{g}$ and an abundance compatible with present bounds, the
limits on DM-nucleon cross-section are improved by four orders of magnitude.

Introduction.— Though Dark Matter (DM) is one of the backbones of the standard
cosmological model, its nature is still unknown Bertone and Tait (2018). So
far, all the evidences of its existence are related only to its gravitational
imprints, while we have no clue about its non-gravitational interactions
Kahlhoefer (2017); Pérez de los Heros (2020). Among the different ways to
probe DM properties, direct detection experiments are achieving very stringent
constraints on DM-nucleon Angloher  _et al._ (2005); Aprile  _et al._ (2012);
Agnes  _et al._ (2015); Angloher  _et al._ (2016); Aalbers  _et al._ (2016);
Agnese  _et al._ (2017); Aalseth  _et al._ (2018); Aprile  _et al._ (2018);
Akerib  _et al._ (2020); Aprile  _et al._ (2019a, b); Angloher  _et al._
(2017); Abdelhameed  _et al._ (2019); Aprile  _et al._ (2020); Adhikari  _et
al._ (2021); Meng  _et al._ (2021) and DM-electron Essig  _et al._ (2012a, b,
2017); Crisler  _et al._ (2018); Agnes  _et al._ (2018); Aguilar-Arevalo  _et
al._ (2019); Andersson  _et al._ (2020) interactions (see Ref. Schumann (2019)
for a recent review). These experiments search for the nuclear and electron
recoil energy caused by the possible scatterings with DM particles that
surround us. Due to a rapidly decreasing sensitivity at low recoil energies,
the constraints on DM interactions dramatically weaken for DM masses smaller
than about 1 GeV (1 MeV) for DM interactions with nuclei (electrons), thus
leaving light DM candidates poorly explored by direct searches. There are
complementary approaches to the exploration of light DM. Fermionic DM
particles lighter than 0.1 keV are highly constrained by phase-space arguments
Boyarsky  _et al._ (2009); Di Paolo  _et al._ (2018) and structure formation
Tremaine and Gunn (1979). Other model-dependent constraints are placed by
astrophysical and cosmological observations, as well as by colliders Cohen
_et al._ (2015); Daci  _et al._ (2015); Ali-Haïmoud  _et al._ (2015);
Baryakhtar  _et al._ (2017); Bondarenko  _et al._ (2020); Coogan  _et al._
(2021a); Gluscevic and Boddy (2018); Xu  _et al._ (2018); Slatyer and Wu
(2018); Nadler  _et al._ (2019).

The experimental limitation of direct detection can be circumvented in the
framework of boosted dark matter, where a fraction of DM particles gains a
velocity higher that the virial one due to a number of different mechanisms
Agashe  _et al._ (2014); Giudice  _et al._ (2018); Fornal  _et al._ (2020). In
recent years, it has been pointed out that light DM particles can be
upscattered to (semi-)relativistic velocities through collisions with cosmic-
rays Cappiello  _et al._ (2019); Bringmann and Pospelov (2019); Ema  _et al._
(2019); Cappiello and Beacom (2019); Ema  _et al._ (2021); Andriamirado  _et
al._ (2021). The unavoidable presence of such a subdominant boosted DM
component has improved the model-independent constraints on the DM-nucleon
cross-section at the level of $\sim 10^{-31}~{}{\rm cm^{2}}$.

In this Letter, we propose the evaporation of Primordial Black Holes (PBHs) at
present times as a source of boosted light DM particles (hereafter denoted as
$\chi$). Throughout this work, we refer to this scenario as ePBH-DM. PBHs are
hypothetical black holes formed soon after the inflationary epoch through the
gravitational collapse of density fluctuations in the early universe
Zel’dovich (1967); Harrison (1970). As a result of combining quantum field
theory and general relativity, PBHs emit Hawking radiation in the form of
graybody spectrum with a temperature $T_{\rm PBH}$ inversely proportional to
the PBH mass Hawking (1974, 1975); Zeldovich (1976); Carr (1976); Page (1976,
1977); MacGibbon and Webber (1990). PBHs with a mass
$M_{\mathrm{PBH}}=\mathcal{O}(10^{15}~{}\mathrm{g})$ are evaporating at the
present times, process that has been employed to set stringent upper bounds on
the PBHs abundance Boudaud and Cirelli (2019); Laha (2019); Ballesteros  _et
al._ (2020); Dasgupta  _et al._ (2020); Coogan  _et al._ (2021b); Calabrese
_et al._ (2021); De Romeri  _et al._ (2021); Baker and Thamm (2021) (see Ref.
Carr  _et al._ (2020) for a comprehensive review). DM particles with a mass
$m_{\chi}<T_{\mathrm{PBH}}$ are also emitted with (semi)-relativistic momenta
by PBHs. This mechanism has been vastly studied as a way to produce DM
particles in the early Universe Morrison  _et al._ (2019); Baldes  _et al._
(2020); Bernal and Zapata (2021a, b, c); Auffinger  _et al._ (2021); Gondolo
_et al._ (2020); Masina (2021); Cheek  _et al._ (2021a, b). However, the
production of DM particles in the present Universe, to our knowledge, has
never been investigated.

Here we explore for the first time the phenomenological implications of the
ePBH-DM scenario in direct detection experiments. Remarkably, we find that
even a tiny fraction of evaporating PBHs is enough to give rise to a sizeable
flux of boosted light DM particles (see Fig. 1). This translates into a
detectable event rate in current experiments such as XENON1T in case DM
particles interact with nucleons (see Fig. 2). Hence, the existence of PBHs
would imply incredibly strong constraints on the DM parameter space, and
viceversa, as shown in Fig. 3. We find that, assuming PBHs abundances
compatible with current bounds, the limits on the spin-independent (SI) DM-
nucleon cross-section are improved up to four orders of magnitude. Conversely,
the non-observation of the ePBH-DM signal allows us to deduce upper bound on
the PBHs abundance a few orders of magnitude stronger than current
constraints, depending on the strength of DM-nucleon interactions. We remark
that our constraints do not require the $\chi$ particles to be the dominant DM
component.

DM flux from evaporating PBHs.—

Figure 1: Diffuse DM flux from evaporating PBHs. We show the diffuse flux of
DM particles incident at the Earth as a function of the kinetic energy. The
galactic (solid) and extragalactic (dashed) lines are shown separately.
Different colors are used to identify different PBH masses
($M_{\text{PBH},15}=M_{\text{PBH}}/10^{15}$ g). The PBH abundance is chosen as
the maximum value allowed by the present constraints Carr  _et al._ (2020):
$f_{\text{PBH}}$ is equal to $2.9\times 10^{-10}$, $3.9\times 10^{-7}$,
$3.7\times 10^{-4}$ for a PBH mass of $0.5$, $1.0$, and $8.0$ in units of
$10^{15}~{}{\rm g}$, respectively. Effects of energy losses in Earth and
atmosphere are not included.

The first step of our work consists in obtaining the DM emission rate from an
evaporating PBH. We here make a conservative choice by considering chargeless
and spinless PBHs. Spinning PBHs would evaporate faster, thus enhancing the DM
emission rate. For the sake of concreteness, we examine the case of DM Dirac
fermions with four degrees of freedom, $g_{\chi}=4$. However, this framework
can be easily extended to scalar and vector DM particles. Differently from
Ref. Baker and Thamm (2021), the addition of only one new species to the
Standard Model particle content does not significantly alter the standard
emission rate of the Hawking radiation. The Hawking temperature of an
evaporating PBH with mass $M_{\mathrm{PBH}}$ is given by Hawking (1975); Page
(1976, 1977); MacGibbon and Webber (1990)

$k_{\mathrm{B}}T_{\mathrm{PBH}}=10.6\left[\frac{10^{15}~{}{\rm
g}}{M_{\mathrm{PBH}}}\right]~{}\mathrm{MeV}\,,$ (1)

where $k_{\mathrm{B}}$ is the Boltzmann constant. The differential spectrum
per unit time is

$\frac{{\rm d}N_{\chi}}{{\rm d}T{\rm
d}t}=\frac{g_{\chi}}{2\pi}\frac{\Gamma(T,M_{\mathrm{PBH}})\,}{\exp\left[(T+m_{\chi})/k_{\rm
B}T_{\rm PBH}\right]+1}\,,$ (2)

where $T$ is the kinetic energy of the particle, and $\Gamma$ is the graybody
factor, which we compute by means of the BlackHawk code Arbey and Auffinger
(2019).

From the differential spectrum we can compute the flux of DM particles
reaching the Earth for a monochromatic PBH mass distribution. It consists of a
galactic ($\rm gal.$) and an extragalactic ($\rm egal.$) component:

$\frac{{\rm d}\phi_{\chi}}{{\rm d}T{\rm d}\Omega}=\frac{{\rm
d}\phi^{\mathrm{gal.}}_{\chi}}{{\rm d}T{\rm d}\Omega}+\frac{{\rm
d}\phi^{\mathrm{egal.}}_{\chi}}{{\rm d}T{\rm d}\Omega}\,.$ (3)

The first component can be written as

$\frac{{\rm d}\phi^{\mathrm{gal.}}_{\chi}}{{\rm d}T{\rm
d}\Omega}=\frac{f_{\mathrm{PBH}}}{4\pi\,M_{\mathrm{PBH}}}\frac{{\rm
d}N_{\chi}}{{\rm d}T{\rm d}t}\int_{0}^{+\infty}{\rm d}s\,\rho^{\rm NFW}_{\rm
DM}(r)\,.$ (4)

Here, the quantity $f_{\mathrm{PBH}}=\rho_{\mathrm{PBH}}/\rho_{\rm DM}$ is the
fraction of PBHs with respect to the average DM density $\rho_{\rm DM}$ of the
Universe, which is determined by Planck Aghanim  _et al._ (2020). The galactic
flux is proportional to the integral over the line-of-sight distance $s$ of
the galactic DM density, denoted as $\rho^{\rm NFW}_{\rm DM}$, for which we
assume a Navarro-Frank-White profile Navarro  _et al._ (1997). This is a
function of the galactocentric distance
$r=(r_{\odot}^{2}+s^{2}-2\,s\,r_{\odot}\cos\ell\cos b)^{1/2}$ with
$r_{\odot}=8.5$ kpc and $(b,\ell)$ the galactic coordinates.

For the extragalactic component in Eq. (3), the differential flux takes the
expression

$\frac{{\rm d}\phi^{\mathrm{egal.}}_{\chi}}{{\rm d}T{\rm
d}\Omega}=\frac{f_{\mathrm{PBH}}\,\rho_{\rm
DM}}{4\pi\,M_{\mathrm{PBH}}}\int_{t_{\mathrm{min}}}^{t_{\mathrm{max}}}{\rm
d}t\,\left[1+z(t)\right]\frac{{\rm d}N_{\chi}}{{\rm d}T{\rm d}t}\,,$ (5)

where we take into account the effect of redshift $z(t)$ on the energy in the
DM emission rate, and we integrate from the time of matter-radiation equality
($t_{\mathrm{min}}$) to the age of the Universe ($t_{\mathrm{max}}$).
Differently from the galactic component, which is enhanced towards the
galactic center, the extragalactic DM flux is isotropic.

Given the evaporation temperatures of PBHs from Eq. (1), DM particles are
mainly produced at energies between $1$ MeV and $100$ MeV, for PBHs masses
from $10^{14}$ and to $10^{16}$ g. Hence, DM particles lighter than about $1$
MeV are emitted by PBHs with ultra-relativistic velocities. In a sense, the
evaporation of PBHs is a new mechanism for boosted DM. In Fig. 1, we show the
galactic and extragalactic diffuse flux of DM particles at the Earth, averaged
over the whole solid angle, for a reference DM mass $m_{\chi}=1$ MeV and for
different PBH masses. The kinetic energy distribution of particles peaks at
different energies depending on the PBH temperature, which can be much larger
than the DM mass. For kinetic energies much smaller than the PBH mass, the
galactic spectrum flattens since the particles are produced practically at
rest. The normalization of the flux is mainly determined by the PBH mass and
the concentration of PBHs $f_{\text{PBH}}$. Larger PBH masses lead to slower
evaporation rate and lower fluxes. However, at larger PBH masses the
constraints on $f_{\text{PBH}}$ are also weaker (see the right plot in Fig.
3), leading to larger fluxes in Fig. 1.

Propagation through Earth and atmosphere.—  Direct detection experiments probe
DM particles through their potential interactions with nucleons. These
processes, however, might also cause an attenuation of the DM flux at the
detector position due to the propagation in the Earth and in the atmosphere
Starkman  _et al._ (1990); Mack  _et al._ (2007); Kavanagh  _et al._ (2017);
Emken and Kouvaris (2018). For the sake of simplicity, we here model this
phenomenon by following the analytical approach described in Ref. Bringmann
and Pospelov (2019). In particular, we employ a ballistic-trajectory
approximation and compute the energy loss of DM particles reaching the
detector. Since the kinetic energy is typically $T\lesssim
100~{}\mathrm{MeV}$, we can neglect quasi-elastic and inelastic scatterings as
well as the nuclear form-factors. Under this assumption, the interaction
lengths in the Earth ($\ell^{\oplus}_{\mathrm{int}}$) and in the atmosphere
($\ell^{\rm atm}_{\mathrm{int}}$) read as

$\ell^{i}_{\mathrm{int}}=\left[\sum_{N}n^{i}_{\mathrm{N}}\sigma_{\chi\mathrm{N}}\frac{2m_{\mathrm{N}}m_{\chi}}{(m_{\mathrm{N}}+m_{\chi})^{2}}\right]^{-1}\,,$
(6)

where the sum respectively runs over the most abundant elements in the Earth
(from iron to aluminium) and the atmosphere (nitrogen and oxygen) with number
density $n^{i}_{\mathrm{N}}$, whereas the cross-section is

$\sigma_{\chi\mathrm{N}}=\sigma_{\chi}^{\mathrm{SI}}\mathrm{A}^{2}\left(\frac{m_{\mathrm{N}}(m_{\chi}+m_{\mathrm{p}})}{m_{\mathrm{p}}(m_{\chi}+m_{\mathrm{N}})}\right)^{2}\,.$
(7)

Here we focus on the spin-independent scatterings which feature the standard
coherent enhancement with the mass number $\mathrm{A}$. Moreover, we limit
ourselves to the case of isospin-singlet structure assuming equal the DM
interaction with protons and neutrons. In this framework, DM particles with an
initial kinetic energy $T_{0}$ reach the detector with a smaller kinetic
energy $T_{d}$ after travelling a total geometrical distance $d=d_{\rm
atm}+d_{\oplus}$. We take into account that the XENON1T detector is located in
the Gran Sasso National Laboratory at a depth of about $1.4~{}{\rm km}$
Capuano  _et al._ (1998). Hence, the actual DM flux reaching the XENON1T
detector can be obtained as

$\frac{{\rm d}\phi_{\chi}^{d}}{{\rm d}T_{d}{\rm
d}\Omega}\approx\frac{4m_{\chi}^{2}e^{\tau}}{\left(2m_{\chi}+T_{d}-T_{d}e^{\tau}\right)^{2}}\left(\frac{{\rm
d}\phi_{\chi}}{{\rm d}T{\rm d}\Omega}\bigg{|}_{T_{0}}\right)\,,$ (8)

where $\tau=d_{\rm atm}/\ell^{\rm atm}_{\rm int}+d_{\oplus}/\ell^{\oplus}_{\rm
int}$, and the incoming DM flux is evaluated at

$T_{0}\left(T_{d}\right)=\frac{2m_{\chi}T_{d}\,e^{\tau}}{2m_{\chi}+T_{d}-T_{d}e^{\tau}}\,.$
(9)

For a given incoming direction defined by the galactic coordinates $(b,\ell)$,
the distance $d$ depends on the time-dependent longitude of the experiment.
Thus, in the following, we consider the average of Eq. (8) over a day.

In closing this section, we may stress that the approach of Ref. Bringmann and
Pospelov (2019) is based on the approximation of collinear, or ballistic,
propagation of DM in matter. As pointed out in Ref. Cappiello and Beacom
(2019), this approximation does not apply to the collisions of light DM with
heavy nuclei; rather, the propagation of DM happens in a diffusive way,
leading to a larger energy loss and a possible reflection of DM from the Earth
and the atmosphere. Such a more complicated propagation might lower the ePBH-
DM flux detectable at the experiment. In order to understand the impact of the
ballistic approximation, we have computed the flux under the most conservative
assumption that particles are completely depleted by the scattering (in other
words, particles which have undergone even a single scattering are not
considered). Even under this extreme assumption, we still find large DM fluxes
observable in underground experiments. Therefore, we expect that a more
realistic treatment of the DM propagation should not disrupt the scenario
presented here. As will be shown later, the ePBH-DM scenario allows us to
probe DM-nucleon cross-section at the level of $10^{-35}$ cm2, for which the
effects of propagation are marginal, since the atmosphere is completely
transparent to DM.

Figure 2: Ratio of XENON1T event rate over DM-nucleon cross-section. We show
the ratio of the event rate induced by DM scatterings inside the XENON1T
detector over the DM-nucleon cross-section. Here, we account for the effects
of propagation through the Earth and the atmosphere. The lines correspond to
different values of the DM-nucleon cross-section. The remaining parameters are
chosen as $M_{\text{PBH}}=10^{15}$ g, $f_{\text{PBH}}=3.9\times 10^{-7}$, and
$m_{\chi}=1$ MeV.

Figure 3: Constraints on the ePBH-DM parameter space. We show the regions
excluded by XENON1T experiment in the planes
$m_{\chi}$–$\sigma^{\text{SI}}_{\chi}$ (left) and
$M_{\text{PBH}}$–$f_{\text{PBH}}$ (right). In the left panel, the excluded
regions are obtained for different values of $M_{\text{PBH}}$: for each case,
$f_{\text{PBH}}$ is set to the maximum value allowed by present constraints
Carr  _et al._ (2020) (also shown with the thin black line in the right plot).
For comparison, we also show previous DM constraints from cosmic-ray boosted
DM particles Bringmann and Pospelov (2019); Cappiello and Beacom (2019) (1),
from CRESST experiment Angloher  _et al._ (2017); Abdelhameed  _et al._ (2019)
(2), and from cosmology Gluscevic and Boddy (2018); Xu  _et al._ (2018);
Slatyer and Wu (2018); Nadler  _et al._ (2019) (3). In the right panel, the
different thick colored lines correspond to the upper bound on
$f_{\mathrm{PBH}}$ obtained for different DM-nucleon cross-section and
choosing $m_{\chi}$ smaller than 1 MeV.

XENON1T event rate.—  We can now compute the event rate expected in the
XENON1T detector in the ePBH-DM scenario. In this experiment, the main target
for DM scatterings is provided by Xenon nuclei. The differential event rate
(number of events per ton year) per Xenon recoil energy $E_{r}$ can be
obtained as

$\frac{{\rm d}R}{{\rm
d}E_{r}}=\sigma_{\chi\mathrm{Xe}}\,\mathcal{N}_{\mathrm{Xe}}\int{\rm
d}T_{d}{\rm d}\Omega\,\frac{{\rm d}\phi_{\chi}^{d}}{{\rm d}T_{d}{\rm
d}\Omega}\frac{\Theta(E_{r}^{\mathrm{max}}-E_{r})}{E_{r}^{\mathrm{max}}}\,,$
(10)

where $\mathcal{N}_{\mathrm{Xe}}$ is the total number of targets, and

$E_{r}^{\mathrm{max}}=\frac{T_{d}^{2}+2m_{\chi}T_{d}}{T_{d}+(m_{\chi}+m_{\mathrm{Xe}})^{2}/(2m_{\mathrm{Xe}})}$
(11)

is the maximum allowed value for the recoil energy, for which we assume a flat
distribution. In this estimate, the cross-section $\sigma_{\chi\mathrm{Xe}}$
includes the corresponding nuclear form-factor as provided in Ref.s Helm
(1956); Lewin and Smith (1996). Moreover, we consider an exposure of one ton
year.

The event rate at XENON1T detector is shown in Fig. 2 as a function of the
recoil energy. In order to highlight the effect of propagation through the
Earth and the atmosphere, we report the ratio of the event rate and the DM-
nucleon cross-section $\sigma_{\chi}^{\mathrm{SI}}$. In the absence of
propagation effects, this ratio would be independent of the cross-section.
This can be seen in the case of the solid yellow and dotted blue curve
(corresponding to low cross-sections), which are practically identical. For
$\sigma^{\text{SI}}_{\chi}=10^{-31}$ cm2 and
$\sigma^{\text{SI}}_{\chi}=10^{-28}$ cm2, the effects of propagation are more
evident, pushing the events to lower recoil energies and weakening the
sensitivity to the ePBH-DM signal.

Results.—  In order to probe the ePBH-DM scenario, we need to compare the
expected event rate with the current data from XENON1T Aprile  _et al._
(2018). This experiment has observed no excess of recoil events in the energy
window from $E_{r,1}=4.9~{}\mathrm{keV}$ to $E_{r,2}=40.9~{}\mathrm{keV}$.
Following Ref. Bringmann and Pospelov (2019), this can be translated into an
upper limit on the total event rate as

$\int_{E_{r,1}}^{E_{r,2}}{\rm d}E_{r}\frac{{\rm d}R}{{\rm
d}E_{r}}\lesssim\kappa\,\overline{v}_{\rm DM}\,\rho^{\odot}_{\rm
DM}\,\mathcal{N}_{\rm Xe}\left(\frac{\sigma_{\chi{\rm
Xe}}}{m_{\chi}}\right)_{m_{\chi}\gtrsim 100~{}{\rm GeV}}\,,$ (12)

where $\kappa\simeq 0.23$, $\overline{v}_{\rm DM}\simeq 235$ km/s is the mean
DM velocity in the usual scenario, $\rho^{\odot}_{\rm DM}=0.3$ GeV/cm3 is the
local DM density as assumed by XENON1T Collaboration, and the term in
parentheses is $2.5\times 10^{-40}{\rm cm^{2}/GeV}$ according to the current
XENON1T limit for large DM masses Aprile  _et al._ (2018).

Equation (12) translates into the limits on the combined parameter space of DM
and PBHs shown in Fig. 3. In particular, the left plot shows the regions of
the DM parameter space that are constrained by XENON1T, in the case of three
different PBH masses. For each PBH mass, we consider the maximum allowed value
for $f_{\mathrm{PBH}}$ in agreement with current limits Carr  _et al._ (2020).
The grey region is excluded by previous constraints on sub-GeV DM particles
from cosmic-ray upscatterings Bringmann and Pospelov (2019); Cappiello and
Beacom (2019), CRESST experiment Angloher  _et al._ (2017); Abdelhameed  _et
al._ (2019) and cosmology Gluscevic and Boddy (2018); Xu  _et al._ (2018);
Slatyer and Wu (2018); Nadler  _et al._ (2019). Here, we only focus on the
region $m_{\chi}\geq 10^{-3}~{}\mathrm{MeV}$ since lighter fermionic DM is
highly disfavored. However, our constraints virtually extend to lower DM
masses. For $m_{\chi}=1~{}{\rm MeV}$, we have checked that the DM constraints
are unchanged for complex scalar DM particles, while they weaken by about
$80\%$ for vector DM particles.

In the right plot, we report the upper bounds on the PBH abundance as a
function of the PBH mass, assuming four benchmark values for the DM-nucleon
cross-section. Here, we consider DM masses smaller than 1 MeV. For low DM-
nucleon cross-sections, the constraints on $f_{\rm PBH}$ are almost inversely
proportional to $\sigma_{\chi}^{\rm SI}$. For
$\sigma^{\mathrm{SI}}_{\chi}=10^{-28}\mathrm{cm}^{-2}$, instead, the peculiar
behaviour of the constraint is due to the considerable effects of the
propagation in the Earth and the atmosphere. We emphasize that our constraints
are valid for $0.5\lesssim M_{\mathrm{PBH}}/(10^{15}~{}\mathrm{g})\lesssim
8.0$. Lighter PBHs are almost completely evaporated, whereas heavier ones emit
DM particles with too low kinetic energies.

Discussion.—  The ePBH-DM scenario with PBH masses from $10^{14}$ to
$10^{16}~{}{\rm g}$ leads to signatures that are already testable with direct
detection experiments. The constraints presented in this work apply to any
model with the simultaneous presence of PBHs and DM coupled to nucleons. For
low DM-nucleon cross-sections, our results mainly depend on the spectrum of DM
from PBHs, and therefore are expected to be robust. For large cross-sections,
the different approaches proposed in the literature to describe the DM
propagation through Earth and atmosphere might lead to slightly different
results. However, this does not influence the possibility of constraining the
scenario under consideration.

Since PBHs constitute only a very small fraction of the total energy of the
non-relativistic DM in the Universe, the emitted DM particles are expected to
not change significantly the properties of the galactic DM distribution and
the extragalactic DM density. Indeed, the magnitude of the signal is not
connected with a particularly large DM flux from PBHs, but rather with the
poorly constrained DM-nucleon cross-sections for sub-GeV DM. Our work shows
that cross-sections from $10^{-35}$ to $10^{-31}$ cm2, while allowed by
present constraints, may be inconsistent with the existence of PBHs
evaporating at present times.

We emphasize that there is a strong degeneracy between the amount of PBHs and
the strength of the DM interaction with nucleons. For low cross-sections,
where the effects of propagation are negligible, the ePBH-DM signal is indeed
proportional to the product $f_{\text{PBH}}\cdot\sigma^{\text{SI}}_{\chi}$.
Future observations of evaporating PBHs, for example through the gamma-ray and
neutrino burst produced in a PBH evaporation Tesic (2012); Glicenstein  _et
al._ (2013); Abdo  _et al._ (2015); Archambault (2018); Ackermann  _et al._
(2018); Albert  _et al._ (2020); Dave and Taboada (2021); López-Coto  _et al._
(2021), might confirm their existence and remove this degeneracy, allowing for
definite constraints on the DM-nucleon interaction.

## Acknowledgements

We thank Gennaro Miele and Stefano Morisi for useful comments and discussions.
This work was supported by the research grant number 2017W4HA7S “NAT-NET:
Neutrino and Astroparticle Theory Network” under the program PRIN 2017 funded
by the Italian Ministero dell’Università e della Ricerca (MUR) and by the
research project TAsP (Theoretical Astroparticle Physics) funded by the
Istituto Nazionale di Fisica Nucleare (INFN).

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In this note, we extend the analysis for the residual-based a posteriori error estimators in the energy norm defined for the algebraic flux correction (AFC) schemes [6] to the newly proposed algebraic stabilization schemes [9, 10]. Numerical simulations on adaptively refined grids are performed in two dimensions showing the higher efficiency of an algebraic stabilization with similar accuracy compared with an AFC scheme.

keyword steady-state convection diffusion reaction equations; algebraically stabilized finite element methods; a posteriori estimator; adaptive grid refinement

§ INTRODUCTION

§ ALGEBRAIC STABILIZATION SCHEMES

§ A POSTERIORI ERROR ESTIMATORS

§ NUMERICAL STUDIES

§ SUMMARY

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# Mining DEV for social and technical insights about software development
††thanks: To appear in the Proceedings of the 18th International Conference on
Mining Software Repositories (MSR 2021).

Maria Papoutsoglou12, Johannes Wachs34, Georgia M. Kapitsaki2  1Aristotle
University of Thessaloniki, Greece  2University of Cyprus, Cyprus  3Vienna
University of Economics and Business, Austria  4Complexity Science Hub Vienna,
Austria
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Software developers are social creatures: they communicate, collaborate, and
promote their work in a variety of channels. Twitter, GitHub, Stack Overflow,
and other platforms offer developers opportunities to network and exchange
ideas. Researchers analyze content on these sites to learn about trends and
topics in software engineering. However, insight mined from the text of Stack
Overflow questions or GitHub issues is highly focused on detailed and
technical aspects of software development. In this paper, we present a
relatively new online community for software developers called DEV. On DEV
users write long-form posts about their experiences, preferences, and working
life in software, zooming out from specific issues and files to reflect on
broader topics. About 50,000 users have posted over 140,000 articles related
to software development. In this work, we describe the content of posts on DEV
using a topic model, showing that developers discuss a rich variety and
mixture of social and technical aspects of software development. We show that
developers use DEV to promote themselves and their work: 83% link their
profiles to their GitHub profiles and 56% to their Twitter profiles. 14% of
users pin specific GitHub repos in their profiles. We argue that DEV is
emerging as an important hub for software developers, and a valuable source of
insight for researchers to complement data from platforms like GitHub and
Stack Overflow.

###### Index Terms:

Developers, social networks, human factors, topic modeling, GitHub, DEV

##  I Introduction

Online communities and platforms play an important role in the professional
lives of software developers [1]. Developers find ideas, collaborators, and
even jobs through connections on software oriented platforms like GitHub and
Stack Overflow, and on traditional social media including Facebook and Twitter
[2]. Their interactions and the content posted on these platforms are a
valuable source of information about the practice of software engineering. The
software engineering research community has long been interested in predicting
current practice and future trends using digital trace data [3, 4].

Despite the richness of these data sources, researchers using them in this way
often make broad inferences about software engineering from highly
contextualized content. GitHub issues or commit messages generally refer to
specific pieces of code. Stack Overflow questions also tend to be highly
specialized. On more socially oriented platforms like Twitter, which limits
post lengths to 280 characters, discussions about software mix with an endless
variety of other content.

To address this gap we present a novel source of long-form text data created
by people working in software called DEV (https://dev.to). DEV is “ _a
community of software developers getting together to help one another out_ ,”
focused especially on facilitating cooperation and learning. Content on DEV
resembles blog and Medium posts and, at a glance, covers everything from
programming language choice to technical specifications of systems to social
factors in software. Other users can interact with posts by liking them,
commenting on them, or sharing them on other platforms. Contributions to DEV
are longer than tweets: the nearly 140,000 texts in our dataset contain on
average 710 words (median 461, stdev. 990). Users can fill out an extensive
profile, including links to their other accounts on the web and to their
personal projects on GitHub. This suggests that DEV has significant potential
for developers to promote themselves and to drive attention to their software
projects.

Text posted on DEV has a broader perspective than content that can be
collected from GitHub or Stack Overflow, while remaining focused on software.
Potential applications include predicting language and framework popularity,
measuring the social attitudes of developers, and understanding the general
trends in software. As we will show, many DEV users share links to their
profiles on multiple platforms including GitHub, Twitter and Linkedin,
highlighting the platform’s potential for the study of cross-platform behavior
[5]. About 85% of the users in our dataset listing both a GitHub and Twitter
account are not in the state of the art dataset linking GitHub and Twitter
from MSR 2020 [6]. A significant number of users link to platforms of emerging
interest to the software engineering community such as Youtube and Twitch.

The aim of our work is to give the empirical software engineering community a
first look at DEV and to promote its potential for research. We collect data
from the platform and apply a topic modeling approach to map the topics of
conversation in the community. We find that users discuss a rich mixture of
social and technical aspects of software development, often in the same post.
We also describe the information users share about themselves, noting that
many users link to their accounts on multiple other sites, and to personal
projects. We highlight the potential of this new platform to complement data
sources traditionally used by the software engineering community to study
social aspects of software and trends in development.

##  II Related work

Open source software developers use a variety of channels to promote
themselves and their projects [7]. As part of these channels social media
plays an important role in software engineering [8], offering communication
(sharing content, meeting new people), coordination (about events, releases,
calls to action) and promotion (launches, new opportunities). Social media and
blogs also play an important role in the labor market for job-seekers and
recruiters alike, especially when combined with content from online coding
platforms. Companies and projects expend considerable effort and resources to
attract the best developers, including examining the social web [9]. This
online presence of developers can be utilized directly, for example matching
GitHub profiles to job ads [10].

Text analysis has been applied to many kinds of data generated by software
developers, including data coming from the above sources. Though typically
unstructured, text contains important information about how people think and
feel about software development. Common sources for text about software
include GitHub [11], Bugzilla [12], Stack Overflow and mailing lists [13]. A
common approach is to fit a topic model to a corpus. A topic model is an
unsupervised method which assigns topics to documents. This approach has been
applied to data from technical contexts including GitHub pull requests [14],
Stack Overflow Q&A and other Stack Exchange sites [15, 16, 17], and Gitter
[18]. These sources focus on specific aspects of the software development
process, and the resulting corpora are oriented towards technical themes.

But software engineering researchers are not only interested in how developers
talk about specific technical issues. Seeking to find data that better
captures social aspects of software development, including information about
preferences, teamwork, collaboration, and potential trends, researchers study
not just technical platforms like GitHub [19] but also broader social
communities like Twitter [20]. Within such communities, researchers can to
some extent identify conversations which are likely about software
engineering. Researchers have not had access to data from an online community
dedicated to social aspects of software development and computing. We present
this analysis of DEV to fill this gap in the literature. Our work, to our
knowledge the first to analyze DEV, seeks to promote the use of this new
dataset in software engineering research.

##  III Data collection and analysis setup

Data collection: On DEV, users typically tag their articles to improve the
article’s visibility and attract readers. Users browsing DEV can search for
articles by tag. We wrote and deployed a customized web crawler that searches
for all tags on the platform and collects all articles using that tag. We
stored metadata and author information for every article. For each unique user
we crawled the information available on their public profile, including links
to other platforms. Using the official DEV API (https://docs.forem.com/api/),
we identified more than 53,000 posting users, which we used to verify the
completeness of our collection of articles. Roughly 33% of users wrote 80% of
articles in our corpus. As an article can have more than one tag, we removed
duplicate articles based on their URL and title. We extracted the full text of
147,030 articles. Using a language detection method we filtered out non-
English articles leaving a corpus of 138,925 articles.

Preprocessing articles: In the data collection step we retrieved the HTML
format of the text; so we were able to find specific HTML elements which we
used to process the data. In DEV posts, as in Stack Overflow questions and
answer, code snippets are enclosed in the $<$code$>$ $<$/code$>$ HTML element.
As we are interested for the natural language part of the articles, we removed
these code snippets, other HTML elements (e.g. $<$li$>$ or $<$p$>$), and URLs.
Using the Quanteda [21] R package we ran the texts through a standard
preprocessing pipeline, removing punctuation, number symbols, and stopwords,
and casting to lowercase. We stemmed the remaining words in each text and
created unigrams and bigrams. After evaluating preliminary results from
various topic models, we discarded the unigrams as topics derived from bigrams
were found to have significantly richer meaning [22].

Topic modeling setup: Before proceeding with the topic modeling, we trimmed
bigrams appearing in fewer than 90 texts and less than 200 times in total to
reduce noise in the corpus. An important step before finalizing a topic
modeling approach is to determine the best $K$ number of topics. A popular
metric to find the best fitted $K$ value is the use of $C_{v}$ coherence score
[23, 16]. We performed a range of different experiments by varying $K$ from 5
to 50 in steps of five recording each run’s coherence score. 30 topics had a
high coherence score and was also evaluated by two authors for
interpretability. Having chosen $K=30$ topics, we fit the model using the STM
package in the R programming language [24]. This package provides different
implementations of structured models including two of the most common
initialization methods: “spectral” and “LDA”. LDA (Latent Dirichlet
Allocation) is the most frequently used topic modeling method, but the
spectral model recovers similarly high quality topics from large (greater than
50,000 documents) corpora at lower computational cost [25, 26]. As our corpus
is large in this sense, we proceed using spectral topic modeling.

##  IV Results

# | Description | Category | Rel. Freq. | Key Bigrams
---|---|---|---|---
T1 | ESS EPICS Environment (E3) | Technical | .016 | e3_e3, host_github, nil_return, overrid_fun, error_messag
T2 | NPM | Technical | .040 | project_post, npm_instal, npm_run, file_call, config_file
T3 | Learning to program | Mixed | .046 | start_learn, learn_python, learn_javascript, front_end, game_develop
T4 | Supporting software and tools | Mixed | .046 | week_back, make_life, life_easier, make_sens, find_work
T5 | Text editors | Technical | .026 | visual_studio, text_editor, oper_system, command_prompt, code_extens
T6 | Databases | Technical | .031 | primari_key, graphql_api, sql_server, graphql_queri, app_express
T7 | User activity logging | Technical | .007 | logrocket_record, pixel-perfect_video, instrument_dom, stacktrac_network, replay_problem
T8 | Asynchrony | Technical | .029 | email_list, promis_resolv, async_function, callback_function, event_handler
T9 | React | Technical | .041 | subscrib_email, react_import, react_compon, button_onclick, compon_render
T10 | Team management | Social | .041 | great_product, softwar_develop, project_manag, technic_debt, team_work
T11 | ASP.NET | Technical | .020 | asp_net, net_core, public_class, spring_boot, async_task
T12 | Web Development/Learning | Mixed | .035 | blog_post, web_monet, rubi_rail, node_js, learn_node
T13 | Work-life balance | Social | .080 | week_time, remot_work, work_home, social_media, spend_time
T14 | Automatic testing | Technical | .021 | open_full, unit_test, write_test, autom_test, test_suit
T15 | Foundations | Technical | .037 | prime_number, data_structur, time_complex, sort_algorithm, hash_tabl
T16 | Open Source | Mixed | .040 | open_sourc, sourc_contribut, pull_request, view_github, version_control
T17 | Data Science | Technical | .028 | machin_learn, data_scienc, neural_network, deep_learn, artifici_intellig
T18 | REST APIs and the Web | Technical | .028 | rest_api, http_request, status_code, web_server, api_call
T19 | APIs | Technical | .036 | api_key, send_messag, access_token, send_email, twitter_api
T20 | Code Review, Challenges | Mixed | .045 | code_review, code_challeng, solv_problem, clean_code, make_code
T21 | Function Handling | Technical | .043 | function_call, return_function, function_program, data_type, declar_variabl
T22 | Containers and versioning | Technical | .051 | docker_ps, git_commit, git_add, docker_run, git_push
T23 | Mobile | Technical | .037 | subscrib_youtub, react_nativ, mobil_app, io_android, build_app
T24 | Site accessibility/UX | Mixed | .038 | static_site, site_generat, load_lazi, screen_reader, user_experi
T25 | Skills and learning | Social | .048 | program_languag, soft_skill, junior_develop, appli_job, find_job
T26 | Cloud computing | Technical | .029 | api_gateway, googl_cloud, ec2_instanc, s3_bucket, kubernet_cluster
T27 | Systems | Technical | .032 | smart_contract, distribut_system, web_applic, microservic_architectur, applic_secur
T28 | Webforms | Technical | .010 | input_type, type_text, form_field, usest_const, form_onsubmit
T29 | CSS | Technical | .031 | flex_contain, display_flex, posit_absolut, css_grid, media_queri
T30 | Rust | Technical | .011 | amp_mut, sourc_code, code_generat, compil_time, syntax_tree

TABLE I: Topics of the DEV corpus. We derive both the listed description and
category from the key bigrams and an inspection of characteristic articles and
their tags. The relative frequency of a topic refers to how often documents
draw from that topic.

###  IV-A What do developers write about?

We present and summarize the topics found by our analysis in Table I. The
table includes a description of each topic, our categorization of the topic as
either social, technical or mixed, the relative frequency of the topic, and
the top five distinguishing bigrams. The topics we found reflect the rich
variety of content on DEV. Though we categorize most topics as relating to
technical topics, several of the most frequently mentioned topics can be
classified as social.

The assignment of posts to topics in a topic model is not one to one. Indeed,
a strength of the topic modeling approach is that each document is modeled as
a mixture of topics. The notion that a text can address multiple topics is
referred to as heteroglossia [27]. As topics can appear together, we can
visualize the overall structure of the DEV corpus using a topic network [28].
In a topic network, nodes represent topics and two nodes are connected by an
edge weighted by the correlation between those topics across the corpus.

As all pairs of topics have some correlation between them, this network needs
to be filtered. We filter the topic-topic correlation network by calculating
its planar maximally filtered graph (PMFG) [29]. The PMFG extends the maximal
spanning tree (MST) approach to filtering weighted networks. While the MST
returns a tree, the PMFG returns a planar subgraph of the original graph with
maximum edge weights (correlations). Planar graphs can include triangles and
4-cliques, making them significantly more dense than trees while still
filtering out most edges. We plot the resulting network in Figure 1,
distinguishing nodes we classified as social, mixed, or technical by their
color. Node sizes increase with topic frequency in the corpus.

Figure 1: Topic correlation network. Nodes are topics, connected by edges if
they frequently appear in the same documents. Edges were selected using the
planar maximally filtered graph method. Node sizes increase with topic
frequency in the corpus. Numbers refer to topics described in Table I. Red
nodes are topics we classified as social, green nodes as technical, and yellow
nodes as mixed.

The network reveals that topics we labeled as social (T10, T13, T25) often
occur in the same document and are among the most frequent topics. We also
note that general programming topics like T8 (Asynchrony), T15 (Foundations),
and T21 (Function Handling) play a bridging role between web development-
related topics (T9, T28, T29) and the rest of the network. T22 (Containers and
versioning) is also a bridging topic. Our interpretation is that these topics
are describing aspects of software dealt with by a wide variety of developers
and so it is natural that they connect topics that are more specialized. At a
glance, the network reveals the multi-faceted nature of the DEV community and
what its users write about.

###  IV-B How do developers present themselves?

To learn more about the users themselves, we now describe how they present
themselves in their profiles. As in most online communities, DEV users have
personal profile pages. On these pages users share information about
themselves, including a brief biography, their location, workplace, and
education. They can also link to their accounts on other platforms including
GitHub, GitLab, Stack Overflow, Medium, Twitter, and Linkedin. There is also a
part of the user profile reserved for GitHub repositories, allowing users to
directly promote their work [30]. Users can also list skills, interests, what
they are learning, and what they are available for. The profile reports user
statistics, like the number of posts and comments a user has made, and lists
badges, gamification elements on DEV, that the user has collected.

Here we focus on the extent to which users fill out their profiles and how
often they link to other platforms. We do this to emphasize the role of DEV as
a social hub for developers and as a source for identifying users across
platforms. We report the share and count of users sharing links to other
platforms in their profiles in Table II. Users most frequently share their
GitHub and/or GitLab profiles (84%). A majority of users (56%) share their
Twitter profiles. Over 12 thousand posting users share their Linkedin
accounts, with potential value for studies of the labor market in software. We
also note that some users on DEV also link their Youtube and Twitch accounts,
two platforms which are of growing interest to the software engineering
community [macleod2015code, alaboudi2019exploratory].

We also report several key intersections: about 40% of users share both their
GitHub profiles and their Twitter accounts. To what extent is this new data on
cross-site linkages? We downloaded the replication materials of a paper
published in MSR 2020 by Fang et al. which linked around 70 thousand
individuals across the two platforms[6] and checked to what extent our data
overlaps. Of the 15,316 users in our dataset sharing both a GitHub and Twitter
account, 13,027 (85%) were not in the Fang et al. dataset, representing a
substantial increase in the number of accounts that can be linked. As the DEV
community continues to grow (13,732 posters joined in 2019 vs 6,006 in 2018),
its value as a data source for cross-platform linkages will only increase.

Linked Platform | Share of Users | Count
---|---|---
GitHub/GitLab | 84% | 32,136
— Has pinned Repo(s) on DEV | 14% | 5,346
Twitter | 56% | 21,492
Linkedin | 32% | 12,320
Medium | 12% | 4,595
Stack Overflow | 11% | 4,083
Instagram | 8% | 3,144
Facebook | 8% | 2,864
Youtube | 3% | 1,190
Twitch | 2% | 778
GitHub/GitLab & Twitter | 40% | 15,316
GitHub/GitLab & Linkedin | 29% | 11,117
GitHub/GitLab & Twitter & Linkedin | 19% | 7,207
TABLE II: Links to other platforms on DEV user profiles.

##  V Discussion

In this paper we presented DEV, an online platform for software developers
that serves as a unique platform for social developers. We shared a first look
at the community using a topic modeling approach to understand what its users
are talking about. We found that they discuss a variety of technical and
social aspects of software development. A significant number of posts in our
corpus deal with purely social aspects of software development, underscoring
that DEV offers information quite different from what researchers might find
on Stack Overflow or GitHub. We also found that DEV users are highly networked
in the sense that many of them link to their accounts on other platforms in
their profiles.

In future work we plan to explore the dynamic evolution of topics, to better
assess DEV’s potential for forecasting trends. We also plan to integrate user
feedback on posts into our analysis, aiming to describe which posts and topics
are more popular or controversial. Different users, characterized by how they
fill out their profiles, likely post different kinds of content. We also plan
to relate such user differences to differences in content. For example, users
describing themselves as managers or leads might be more likely to post about
coordination and collaboration than about technical subjects.

DEV also has significant potential for research on the labor market for
software, complementing data from GitHub or Stack Overflow [31]. On DEV users
describe, in plain text, their skills and preferences on their user profiles
and indirectly via their posts. Data from DEV could be used to revisit
important social factors in software engineering including gender gaps and
disparities [32, 33, 34], the relationship between geography and collaboration
[35, 36], and the effects of gamification on user behavior [37]. Of particular
interest is the site’s social network, the structure and dynamics of which
could be contrasted with developer networks on GitHub [38]. Before these
projects can proceed, DEV data needs to be suitably anonymized to respect user
privacy, especially when linking user identities across platforms [6].

In the broad and growing ecosystem of online communities relating to software
development, DEV plays an important role. On DEV, users post about software
from a broad perspective, integrating social and technical ideas in one place.
For the software engineering research community, DEV presents an opportunity
to better understand the social and networked software developer.

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