# Highlights from PHENIX-I: Initial State and Early Times

Michael Leitcha (for the PHENIX collaboration)  Los Alamos National
Laboratory, P-25 MS H-846, Los Alamos, NM, 87545, USA

###### Abstract

We will review the latest physics developments from PHENIX concentrating on
cold nuclear matter effects, the initial state for heavy-ion collisions, and
probes of the earliest stages of the hot-dense medium created in those
collisions. Recent physics results from $p+p$ and $d+Au$ collisions; and from
direct photons, quarkonia and low-mass vector mesons in A+A collisions will be
highlighted. Insights from these measurements into the characteristics of the
initial state and about the earliest times in heavy-ion collisions will be
discussed.

††journal: Nuclear Physics A

##  1 Introduction

In this overview, we will discuss selected highlights from PHENIX in the areas
relating to the initial state and early times, focusing only on those which we
believe to be the most significant new results. These will include 1) the
suppression of rapidity-separated hadron pairs in $d+Au$ collisions, 2) the
contributions of quarkonia and Drell-Yan to the non-photonic single electrons
used to detect heavy quarks, 3) the continued suppression of the $J/\psi$ in
$Cu+Cu$ collisions to high-$p_{T}$, 4) the suppression of $\Upsilon$s in
$Au+Au$ collisions, and 5) estimates of the initial temperature from direct
photons in $Au+Au$ collisions.

##  2 Cold Nuclear Matter (CNM) and Gluon Saturation

The physics that modifies hard processes in nuclei relative to those on a free
nucleon, often called cold nuclear matter (CNM) effects, includes 1)
traditional shadowing either from global fits or from coherence models, 2)
gluon saturation at small momentum fraction ($x$) which is amplified in the
nuclear environment, and 3) initial-state energy loss and multiple scattering.
For hadron pairs with a rapidity separation between the two hadrons in the
pair, where one ”triggers” on a mid-rapidity ($|\eta|<0.35$) hadron and
studies correlations with a forward-rapidity ($3.1<\eta<3.9$) hadron, there
are two pictures which attempt to describe the characteristics of the process.
QCD based pictures, such as those used by Vitev [1] that include non-leading-
twist shadowing, give suppression of the pairs compared to the mid-rapidity
trigger particle. An alternative approach which represents gluon saturation in
the color-glass-condensate (CGC) model [2] also gives suppression and gives
broadening of the angular correlation peak between the two particles in the
pair. In the CGC picture, a mono-jet mechanism becomes important, where a
single jet has its momentum balanced by multiple gluons coupling to the
saturated gluon field.

Figure 1: (Color online) Centrality dependence of $I_{dAu}$ for rapidity-
separated hadron pairs (left). Correlation width, $\Delta\phi$, vs $p_{T}$ of
the associated mid-rapidity ($|\eta|<0.35$) $\pi^{0}$ (filled points) for
$p+p$ and for different centrality $d+Au$ collisions, showing no broadening
within the substantial uncertainties of the data points (right). Also shown
are similar results for higher energy clusters (open symbols) where
$\pi^{0}$’s and photons are not resolved.

Using the new Muon Piston Calorimeters (MPC) in PHENIX we are able to study
correlations of rapidity-separated hadron (h± or $\pi^{0}$) pairs, where one
triggers on a hadron at mid-rapidity and studies correlations with hadrons at
forward rapidity in the MPC. For these studies we use the ratio $I_{dAu}$,
which is the pair efficiency relative to the mid-rapidity ”trigger” hadron for
$d+Au$ divided by that for $p+p$,

$I_{dAu}={{N^{pair}_{d+Au}[(\eta=3.5)+(\eta=0)]/N^{trig}_{d+Au}(\eta=0)}\over{N^{pair}_{p+p}[(\eta=3.5)+(\eta=0)]/N^{trig}_{p+p}(\eta=0)}}$
(1)

Preliminary results [3] for the centrality dependence (in terms of number of
collisions, $N_{coll}$) of $I_{dAu}$ in Fig. 1 (left) show increasing
suppression for more central collisions. The angular correlations of the pairs
were also studied, but showed no broadening in the relative angle $\Delta\Phi$
outside the substantial uncertainties in the present preliminary result, Fig.
1 (right).

We have also studied hadron pairs in $d+Au$ collisions where both hadrons are
at mid-rapidity, this time in terms of $J_{dAu}$ which is basically the same
as $R_{dAu}$ for a single particle, but in this case for pairs,

$J_{dAu}={{(PairYield)_{dAu}}\over{<N_{coll}>(PairYield)_{pp}}}$ (2)

These pairs exhibit a very large Cronin-like enhancement, i.e. they scale
faster than $N_{coll}$ ($J_{dAu}>1$) and both $J_{dAu}$ and the angular
correlation width decrease for larger $p_{T}$ [4].

##  3 Open Heavy Quarks

Recent studies of the contribution of quarkonia and Drell-Yan to the spectrum
of single electrons from heavy quarks have determined that for transverse
momenta above about 5 GeV/$c$ these contributions can amount to up to 16% of
the total non-photonic electron yield [5]. The contributions of $J/\psi$,
$\Upsilon$, and Drell-Yan are shown in Fig. 2 (left) and one can see that the
$J/\psi$ gives the dominant contribution. After subtracting off these
estimates of the contributions to the $p+p$ collision data, with careful
attention to their uncertainty, shown in Fig. 2 (right), the net yield of
electrons from heavy quark decay has moved from a little above, to slightly
below, the FONLL model’s upper uncertainty limit. Similar corrections, but
with larger uncertainties have been applied for $Au+Au$ collisions. However,
because both $p+p$ and $Au+Au$ are lowered by about the same amount, the
resulting nuclear depencence in $Au+Au$ collisions, $R_{AA}$, is not
significantly changed.

Figure 2: (Color online) Contributions of quarkonia and Drell-Yan to the
background for single electrons from heavy meson decays and the change
relative to the background before their inclusion (left). Single electron
spectrum from heavy quarks corrected for these contributions compared to a
FONLL calculation [6] (right).

Most open-heavy flavor meson measurements at RHIC to date are not able to
separate contributions to the single electrons from charm and beauty, while
theoretical predictions of energy loss and flow in the hot-dense medium
created in high-energy heavy ion collisions are generally quite different for
charm and beauty. Recently a new method has been employed, where one studies
the correlations of hadrons near the observed electron and exploits the fact
that the decay of a beauty meson into an electron and hadron produces a
broader correlation and lower efficiency for observing the pair than that of a
charm meson. Using this technique, the fraction of $(b\rightarrow
e)/(b+c\rightarrow e)$ has been determined [7] and is shown vs $p_{T}$ in Fig.
3 (left).

PHENIX has also measured open-heavy flavor mesons at forward rapidity via
their decay to single muons, but so far not with enough precision to define
the shape of the cross section vs rapidity. However, three different methods
in PHENIX now yield consistent cross sections in $p+p$ collisions at mid
rapidity: single electrons via cocktail subtraction, a converter method, and
with di-electrons. Using the beauty fraction determined above, a beauty cross
section of $\sigma_{b{\bar{b}}}=3.2^{+1.2}_{-1.1}(stat)^{+1.4}_{-1.3}(sys)\
{\mu}b$ has also been determined [7].

Finally, the first proof-of-princible measurement of charm pairs via electron-
muon correlations in $p+p$ collisions has been made [8] and is shown in Fig. 3
(right). The peak at $\pi$ radians in $\Delta\Phi$ is from these correlated
pairs. This method promises to provide another independent measurement of
charm in the near future, as luminosities increase and allow substantial
yields for this rare signal.

##  4 Quarkonia Production and Suppression

The simultaneous theoretical description of both the cross section and the
polarization of the $J/\psi$ in hadron production has long been a challenge. A
new analysis of the 2006 PHENIX $p+p$ data agrees well with the previous
results, has significantly higher precision, and agrees well with the Lansberg
s-channel cut color-singlet model [9]. The decay polarization of the $J/\psi$
measured by PHENIX at mid and forward rapidity is shown in Fig. 4 (left),
where the Lansberg model reproduces the small polarization falling with
$p_{T}$ at mid rapidity (red points), but predicts a larger polization than
the null polarization seen at forward rapidity (by 2-3 sigma) [10]. Improved
polarization measurements at forward rapidity in several bins in $p_{T}$ are
expected soon, and may help clarify the situation.

Figure 3: (Color online) Fraction of beauty, $(b\rightarrow e)/(b+c\rightarrow
e)$, vs single-electron transverse momentum compared to FONLL calculations [6]
(left). Early electron-muon pair charm signal for $p+p$ collisions (right).

Figure 4: (Color online) Polarization vs $p_{T}$ in the helicity frame for
$J/\psi$ production in 200 GeV $p+p$ collisions with mid-rapidity points as
red circles and forward rapidity points as blue squares (left). $R_{CP}$ vs
rapidity for $J/\psi$ production in 200 GeV $d+Au$ collisions for three
different centrality bins, with the most central collisions (0-20%) on the
bottom (right).

New results for the $J/\psi$ from the 2008 $d+Au$ run with approximately
thirty times larger integrated luminosity than that of the previous (2003)
$d+Au$ results are beginning to emerge, with the first preliminary result in
terms of $R_{CP}$,

$R_{CP}^{0-20\%}={{N_{inv}^{0-20\%}/<N_{coll}^{0-20\%}>}\over{N_{inv}^{60-88\%}/<N_{coll}^{60-88\%}>}},$
(3)

shown in Fig. 4 (right) vs rapidity for three different centrality bins [10,
11]. One sees essentially no nuclear dependence at backward rapidity, a little
at mid rapidity, and increasing suppression with centrality at forward
rapidity in the nuclear shadowing region (large rapidity corresponds to small
momentum fraction down to about $x=2\times 10^{-3}$ and is in the shadowing
region). PHENIX is working on more comprehensive results for the near future
in terms of $R_{dAu}$, the nuclear dependence relative to $p+p$ \- the much
higher statistical precision of this new data requires precision systematics
and more careful analysis.

New preliminary results for $J/\psi$ $R_{AA}$ in $Cu+Cu$ collisions show
continuing suppression up to at least 7 GeV/$c$ in $p_{T}$. In Fig. 5 (left)
this suppression is compared to several theoretical models, including the
”hot-wind” AdS/CFT inspired model [12] which is inconsistent with the data.
Eventually, due to the Cronin effect seen in $d+Au$ collisons, which causes a
change from suppression to enhancement at high-$p_{T}$, one would expect
$R_{AA}$ to return to unity at large $p_{T}$, but there is no evidence of that
yet from these results.

Figure 5: (Color online) $R_{CuCu}$ vs $p_{T}$ for mid-rapidity $J/\psi$s out
to 9 GeV/$c$ compared to several theoretical models [12, 17] (left). Invariant
mass spectrum at mid rapidity for $p+p$ collisions at high mass showing the
$\Upsilon$ family and other components of the spectrum.

Figure 6: (Color online) Di-electron pair mass at mid rapidity for 200 GeV
Au+Au collisions in the $\Upsilon$ mass region, where $e^{+}e^{-}$ pairs are
shown as black points, like-sign pairs as red points, and mixed $e^{+}e^{-}$
background pairs in green (left). The probability distribution vs $R_{AuAu}$
determined from the $Au+Au$ and $p+p$ data (right).

With the increasing luminosities provided by the RHIC machine, PHENIX is now
beginning to accumulate useful number of $\Upsilon$s for various kinds of
collisions. From the 2006 $p+p$ run, as shown in Fig. 5 (right), we now have a
preliminary cross section for dielectron events in the $\Upsilon(1S+2S+3S)$
mass region [8.5,11.5 GeV/$c^{2}$] of $BR*d\sigma/dy\
(|y|<0.35)=114^{+46}_{-45}\ pb$ . A small number of dielectron pairs from
Drell-Yan and from open beauty pairs may also contribute in that mass region,
but this contribution is estimated to be less than 15% and is included in the
systematic uncertaintiy. Using a similar signal for $Au+Au$ collisions, shown
in Fig. 6 (left), and doing a very careful statistical analysis which takes
into account the small numbers of counts in both the $Au+Au$ and $p+p$
$\Upsilon$ mass regions, we have obtained the probability distribution for
$R_{AuAu}$ in this mass region shown in Fig. 6 (right). From this an upper
limit of $R_{AuAu}<0.64$ at 90% C.L. is determined [13].

Although $\Upsilon$s have long been touted as the standard candle for the
melting of quarkonia in the Quark Gluon Plasma (QGP), i.e. that they would not
be screened up to very high temperatures, it is clear that there are a number
of simple non-QGP effects that could easily cause a suppression at or below
the upper limit determined above. These include 1) the suppression of
$\Upsilon$ states seen in fixed target experiments [14] which would give about
$0.81^{2}$ in $R_{AuAu}$, 2) the fact that only about 52% of the
$\Upsilon_{1S}$ do not come from feeddown from the higher mass (2S, 3S)
$\Upsilon$ states and B decays [15], and 3) that we do not resolve the three
$\Upsilon$ states (1S+2S+3S) and the 1S is only about 73% of the total [16].

##  5 Initial State and Temperature

Figure 7: (Color online) Direct photon spectra vs $p_{T}$ in $p+p$ and $Au+Au$
collisions at 200 GeV, with the scaled $p+p$ reference compared to the data
from various centrality $Au+Au$ data as a black dashed line. The enhancement
over the $p+p$ reference at small $p_{T}$ for $Au+Au$ collisions is fit to an
exponential (solid black lines) to extract an inverse slope and an effective
temperature (left). Estimates of the initial temperature of the hot-dense
medium from several different models for the expansion of that medium, given
the average temperature of 221 MeV determined from the central $Au+Au$ spectra
(right).

Direct photon production in nucleus-nucleus collisions, although a difficult
measurement, is a clean probe of the initial-state gluon distributions in the
colliding nuclei. The latest measurements in PHENIX show no modification
relative to $p+p$ collisions except for transverse momenta above about 12
GeV/$c$. These modifications are likely due to CNM effects (Cronin) and isopin
(neutrons vs protons) [18].

A new method has been used recently to extract the yield of photons in the
low-$p_{T}$ thermal region where the production of these photons is inferred
from the low-mass ($M_{ee}<\ $300 GeV/$c^{2}$) $1<p_{T}<\ $5 GeV/$c$
$e^{+}e^{-}$ spectrum [19]. These low-mass photons show an enhancement over
the scaled $p+p$ reference, as shown in Fig. 7 (left). If interpreted as
thermal photons from the hot-dense medium and fit to an exponential slope, an
average temperature of the medium (for central collisions) of $T_{avg}=221\pm
23\pm$18 MeV is obtained. Since this is the average over the expansion, one
can ask within various theoretical descriptions for that expansion, what the
initial temperature is. Fig. 7 (right) shows various initial temperatures vs
the formation time assumed in each theoretical picture. All models indicate an
initial temperature of at least 300 MeV, well above the predicted QGP phase
transition at 170 MeV.

##  6 Summary

We highlight here some of the recent PHENIX results that we believe to be most
interesting in areas that relate to the initial state and early times,
including:

  1. 1.

Quarkonia contribute substantially to the electrons from heavy flavor for
$p_{T}>\ $5 GeV/$c$ and should be taken into account when comparing to
theoretical predictions.

  2. 2.

Beauty decays give 50% or more of the single electrons for $p_{T}>\ $4
GeV/$c$, so any differences between beauty and charm for energy loss and flow
may become apparent at these $p_{T}$ values.

  3. 3.

$J/\psi$ polarization measurements at mid rapidity agree with the Lansberg
color singlet model, but at forward rapidity the $p_{T}$-integrated value does
not.

  4. 4.

For $Cu+Cu$ collisions, $J/\psi$s continue to be strongly supressed up to
$p_{T}\simeq\ $8 GeV/$c$.

  5. 5.

Events in the ${\Upsilon}(1S+2S+3S)$ mass region at mid rapidity are
suppressed in $Au+Au$ collisions by at least 36%, but this is not unexpected
given cold nuclear matter effects and the likely strong suppression for
central $Au+Au$ collisions of the higher mass $\Upsilon$ states.

  6. 6.

Direct photons measured in the thermal region for central $Au+Au$ collisions
indicate intial temperatures of at least 300 MeV, well above the expected QGP
phase transition.

## References

  * [1] J. Qiu and I. Vitev, Phys.Lett. B632, 507 (2006) [hep-ph/0405068].
  * [2] D. Kharzeev et al., Nucl. Phys. A 748, 727 (2005).
  * [3] B. Meredith (for the PHENIX Collaboration), these proceedings.
  * [4] J. Jia (for the PHENIX Collaboration), QM09 poster [arXiv:0906.3776].
  * [5] A. Dion (for the PHENIX Collaboration), these proceedings.
  * [6] M. Cacciari et al., Phys. Rev. Lett 95, 122001 (2005); and private communication.
  * [7] A. Adare et al. (PHENIX Collaboration), [arXiv:0903.4851].
  * [8] T. Engelmore (for the PHENIX Collaboration), these proceedings.
  * [9] H. Haberzettl, J.P. Lansberg, Phys. Rev. Lett. 100, 032006 (2008) [arXiv:0709.3471].
  * [10] C. da Silva (for the PHENIX Collaboration), these proceedings.
  * [11] D. McGlinchey (for the PHENIX Collaboration), QM09 poster.
  * [12] H. Liu, K. Rajogopal, U. Wiedemann, Phys. Rev. Lett. 98, 182301 (2007) [arXiv:hep-ph/0607062].
  * [13] E. Atomssa (for the PHENIX Collaboration), these proceedings.
  * [14] D. Alde, et al. (E772 Collaboration), Phys. Rev. Lett. 64, 2479 (1990).
  * [15] S. Digal, et al. Phys. Rev. D 64, 094015 (2001).
  * [16] The CDF Collaboration, Published Proceedings Les Rencontres de Physique de la Vallee d’Aoste, La Thuile, Italy, March 3-9, 1996. FERMILAB-CONF-96/110-E; and G. Moreno et al. (E605) Phys. Rev. D43, 2815 (1991).
  * [17] X. Zhao, R. Rapp Phys. Lett. B664, 253 (2008) [hep-ph/07122407]; X.M. Xu, D. Kharzeev, H. Satz, X.N. Wang, Phys. Rev. C53 3051 (1996) [hep-ph/9511331]; B.K. Patra, V.J. Menon, Eur. Phys. J C44, 567 (2005) [nucl-th/0503034].
  * [18] B. Zhang, I. Vitev, [nucl-th/0810.3194].
  * [19] Y. Yamaguchi (for the PHENIX Collaboration), these proceedings, and [nucl-ex/0804.4168].


# Primordial black hole collision with neutron stars and astrophysical black
holes and the observational signatures

Sohrab Rahvar  Physics Department, Sharif University of Technology, Tehran
11365-9161, Iran
and
Research Center for High Energy Physics, Sharif University of Technology,
Tehran, Iran<EMAIL_ADDRESS>
(Day Month Year; Day Month Year; Day Month Year; Day Month Year)

###### Abstract

In this paper, we examine whether low-mass Primordial Black Holes (PBHs) can
be considered a plausible dark matter candidate in galactic halos. We derive
the relativistic dynamics of PBHs around the heavy compact objects and
evaluate their collision rate, as well as the likelihood of PBH capture in
neutron stars and black holes. Although the rate of these collisions in the
Milky Way is lower than our lifetime (i.e. almost one collision per hundred
years), it may still be observable on cosmological scales. Additionally, we
investigate the gravitational wave emission as an important observable window
for PBH-astrophysical black hole merging. For the allowed range of PBH mass,
gravitational wave signal is smaller than the sensitivity of present
gravitational wave detectors. We provide observational prospect for detection
of these events in future.

###### keywords:

Dynamics, Primordial Black Holes, Neutron Stars

##  1  Introduction

The large-scale structures of the universe are dominantly composed of dark
matter and Primordial black holes (PBHs) are one of the potential candidates
for dark matter, believed to have formed in the early universe due to quantum
fluctuations [1, 2]. While the theory of PBH formation predicts a wide range
of masses for PBHs [3], observations constrain their mass to two narrow
windows: low mass ($M<10^{-7}M_{\odot}$) and high mass ($M>1000M_{\odot}$) [4,
5]. It is believed that galaxies are surrounded by a halo of dark matter,
which we assume to be comprised of PBHs.

In this work, our aim is to investigate the interaction of PBHs with
astrophysical compact objects. This study has been done in numerous works such
as PBH merging with the astrophysical structures through the adiabatic process
[6, 7, 8, 9, 10, 11]. Also, the collision of PBH with stars and compact
objects are modeled as GRB source [12]. The consequences of this collision as
the Gamma-ray emission were also examined with the X-ray observatories [13].

Here, we focus on investigating the rate of PBH collisions with neutron stars
and astrophysical black holes (hereafter referred to as compact objects) and
studying the consequences of this interaction. While the PBH collision with
Earth has been studied in the context of Newtonian gravity [14], we extend
this argument to PBH collisions with compact objects with relativistic
dynamics, considering astrophysical consequences of collision of the PBHs with
the neutron stars and the astrophysical blackholes and possible observational
signatures from this collisions.

In Section (2), we introduce the phase space of PBHs and calculate the
relativistic path of PBHs around the compact objects and the rate of
collisions that occur with compact objects. In Section (3), we study the
consequences of these collisions with neutron stars, and in Section (4), we
investigate the collision of PBHs with astrophysical black holes. Finally,
Section (5) provides a summary and conclusion of our study.

##  2 Collision of PBHs with compact objects

In this section, we study the phase space of PBHs in the Galactic halo and the
rate of collisions with compact objects.

###  2.1 Dynamics of PBHs in halo

Let us assume the distribution of point mass PBHs in the halo of the Milky Way
galaxy is almost isothermal and follows the Maxwell-Boltzmann distribution as

$f(x,v)d^{3}v=n_{0}(x)(\frac{3}{2\pi\sigma^{2}})^{3/2}\exp{(\frac{-3v^{2}}{2\sigma^{2}})}v^{2}dvd\Omega,$
(1)

where $"x"$ represents the spatial part of phase space and $n_{0}(x)$ is the
number density of PBHs at any point of space. $d\Omega=d\phi\sin\theta
d\theta$ is the solid angle element and $\sigma$ is the dispersion velocity of
PBHs. Let us assume a coordinate system where a compact object is set at the
center of it.

This object gravitationally interacts with the PBHs in the halo. The flux of
particles in this coordinate system (at far distance of ”$r$” from the center)
entering the sphere with the radial distance of $"r"$ is [15]

${\cal F}(r)=\int f(r,v){\bf\hat{n}}\cdot{\bf v}v^{2}dvd\Omega,$ (2)

where ${\hat{n}}$ is the inwarding unit vector perpendicular to the surface
and $0<\theta<\pi/2$. We note that the overall current in the steady state
condition for the incoming and outgoing particles is zero. Here, our aim is to
consider just the incoming PBHs inside the sphere with the radius of $r$.
Multiplying the flux in equation (2) to the area at far distances from the
center, results in the rate of particles crossing the sphere with the radius
of $r$ (i.e. $dN/dt=4\pi r^{2}{\cal F}$).

We define the kinetic energy and the angular momentum of PBHs at
asymptotically flat spacetime with respect to the center of coordinate as
$E=mv^{2}/2$ and $J=rmv\sin\theta$. We note that the compact object at the
center of this coordinate system has a non-zero velocity with an arbitrary
direction with respect to the halo. For an ensemble of compact objects in the
Galaxy we can ignore the peculiar velocity of compact object while a detailed
calculation would provide a dipole interaction-rate of PBHs with the compact
object at the center of coordinate.

We can calculate the rate of incoming particles inside a sphere with a radius
of $r$ in terms of energy and angular momentum as

$\frac{dN}{dt}=2\pi^{2}n_{0}(\frac{3}{2\pi
m^{2}\sigma^{2}})^{3/2}\int\exp(-\frac{3E}{m\sigma^{2}})dEdJ^{2}.$ (3)

In the next section we derive the trajectory of PBHs around the gravitational
potential of the central compact object in terms of the two conserved
quantities of $E$ and $J$. Then our aim is to obtain the rate of interactions
of PBHs with the central compact object as a function of these two conserved
quantities.

###  2.2 Trajectory of PBHs around compact objects: Relativistic approach

Let us assume the Schwarzschild metric for the spacetime around a compact
object as

$ds^{2}=-(1-\frac{r_{s}}{r})c^{2}dt^{2}+(1-\frac{r_{s}}{r})^{-1}dr^{2}+r^{2}d\Omega.$
(4)

The Schwarzschild radius is $r_{s}=2GM/c^{2}$.

The action for a test particle of mass $m$ [16] (here a point mass PBH around
the compact object) is

$S=-mc^{2}\int d\tau=\int{\cal L}dt$ (5)

where

${\cal
L}=-mc^{2}\sqrt{(1-\frac{r_{s}}{r})-(1-\frac{r_{s}}{r})^{-1}\dot{r}^{2}-r^{2}\sin^{2}\theta\dot{\phi}^{2}-r^{2}\dot{\theta}^{2}}.$
(6)

For simplicity in the calculation, we can reduce space dimension into two,
taking into account the initial condition of the particle by setting
$\dot{\theta}=0$ and $\theta=\pi/2$. The corresponding canonical momentum for
each coordinate is given by $p_{i}=\partial{\cal L}/\partial\dot{q}_{i}$ where
from the Lagrangian the momentum is

$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle-\frac{m^{2}c^{2}\dot{r}}{{\cal L}}(1-\frac{r_{s}}{r})^{-1},$
(7) $\displaystyle J$ $\displaystyle=$
$\displaystyle-\frac{m^{2}c^{2}r^{2}\dot{\phi}}{\cal{L}},$ (8)

where $J$ is a conserved quantity. The corresponding Hamiltonian is ${\cal
H}=\sum p_{i}\dot{q}_{i}-\cal{L}$ where substituting the momentum and
coordinate, the Hamiltonian simplifies to

${\cal H}=-\frac{m^{2}c^{4}}{\cal{L}}(1-\frac{r_{s}}{r}).$ (9)

Dividing $J$ by the Hamiltonian we can find the angular velocity in terms of
these two quantities,

$\dot{\phi}=\frac{Jc^{2}}{\cal H}\frac{(r-r_{s})}{r^{3}}.$ (10)

Since Hamiltonian is time-independent, that is a conserved quantity and we
define it at the asymptotic flat space-time

$\displaystyle{\cal H}$ $\displaystyle=$
$\displaystyle\frac{mc^{2}}{\sqrt{1-v^{2}}}\simeq mc^{2}+E_{k},$
$\displaystyle\lim$ $\displaystyle~{}~{}r\rightarrow\infty$

where $E_{k}=\frac{1}{2}mv_{0}^{2}$ is the kinetic energy of PBH in the
Galactic halo.

On the other hand, let us take the closest distance of the test particle to a
compact object that happens at $r=r_{min}$ where at this position the radial
velocity is zero. Then the Hamiltonian at this point is given by

${\cal
H}^{2}=(1-\frac{r_{s}}{r_{min}})m^{2}c^{4}+\frac{J^{2}c^{2}}{r_{min}^{2}}.$
(12)

Equating the left and right-hand sides of equations (2.2) and (12), then the
angular momentum in terms of the kinetic energy of PBH at far distance and
$r_{min}$ is

$J^{2}=m^{2}c^{2}r_{min}^{2}\left((1+\frac{E_{k}}{mc^{2}})^{2}+\frac{r_{s}}{r_{min}}-1\right),$
(13)

where for the Newtonian limit (i.e. $r_{s}\rightarrow 0$) we recover the
conventional angular momentum as $J=r_{min}mv(r_{min})$. This equation
represents a constraint between the angular momentum and kinetic energy of a
particle. If the radius of a compact object, denoted as $r_{c}$ is greater
than the minimum distance between the compact object and a PBH, denoted as
$r_{min}$ (i.e. $r_{c}>r_{min}$), then the collision between the two objects
occur at $r_{c}$.

###  2.3 The rate of collisions

For calculating the rate of collisions, we substitute equation (13) as the
constrain between the energy and angular momentum of PBHs in (3), and after
integration the result is

$\frac{dN}{dt}=2\pi^{2}n_{0}(\frac{3}{2\pi})^{3/2}\sigma
r_{min}^{2}\left(\frac{2}{9}+\frac{2}{3}\frac{r_{s}}{r_{min}}\frac{c^{2}}{\sigma^{2}}+\frac{2}{27}\frac{\sigma}{c}\right),$
(14)

where the first and second terms are Newtonian terms; the third term is the
relativistic correction to the collision rate. Taking the local density of the
dark halo around the disk of the Milky Way [17] as $\rho_{D}\simeq 8\times
10^{-3}M_{\odot}pc^{-3}$ and the mass range of PBHs as $[10^{14},10^{23}]$gr,
the number density of PBHs in the halo obtain

$n_{0}=6\times
10^{-33}f\times(\frac{\rho_{h}}{0.008M_{\odot}/pc^{3}})(\frac{m_{pbh}}{10^{23}g})^{-1}\text{km}^{-3}$

where $f$ is the fraction of dark matter made of PBHs. Then the numerical
value of equation (14) is

$\displaystyle\frac{dN}{dt}$ $\displaystyle=$ $\displaystyle 2.7\times
10^{-11}\text{Gyr}^{-1}f\times(\frac{\rho_{h}}{0.008M_{\odot}/pc^{3}})(\frac{m_{pbh}}{10^{23}g})^{-1}$
$\displaystyle\times\left(0.22+2.25\times
10^{5}(\frac{rs}{km})(\frac{r_{m}}{10km})^{-1}\sigma_{200}^{-2}+5\times
10^{-5}\sigma_{200}\right),$

where $\sigma_{200}$ is the dispersion velocity of PBHs in the halo normalized
to $200$km/s. Ignoring the relativistic terms, the result agrees with the
Newtonian calculation [13]. For the collision of PBHs with compact objects in
the halo of Galaxy, we can ignore the first and third terms compared to the
second term; however for the dense regions as the center of Galaxy where the
PBHs move with relativistic velocities, we have to keep all the terms. For the
non-relativistic velocities the rate simplifies to

$\frac{dN}{dt}=6.1\times
10^{-6}\text{Gyr}^{-1}f\times(\frac{m_{pbh}}{10^{23}g})^{-1}(\frac{rs}{km})(\frac{r_{m}}{10km})^{-1}\sigma_{200}^{-2}.$
(16)

Taking into account the evaporation of PBHs from the early universe up to the
present time, we expect PBHs with the masses of $m>10^{14}$gr could survive at
the present time [14]. For the PBHs with Dirac-Delta mass function within the
mass range of $m\in[10^{14},10^{23}]$g [18, 4] (which is marginally larger
than the experimental constrain) the rate of collisions from equation (16) is
obtained as $dN/dt\in[6~{},6\times 10^{-9}]\text{Myr}^{-1}$. While the
collision rate of PBH with a compact object is high, however, the rate of
capturing (by means of trapping inside the compact object or making a
gravitationally bound state) is low. In the next sections, we investigate the
capture rate and the astrophysical signatures from the PBH collisions with
compact objects.

##  3 Physics of PBH collision with Neutron stars

In this section, we investigate the physical consequences of the PBH collision
with compact objects such as neutron stars. Let us image a small mass PBH
colliding with a neutron star where we can ignore the momentum transfer to the
neutron star as the mass of neutron star is much larger than the mass of PBHs.
Once PBH enters the neutron star two physical processes can happen (i)
decelerating due to the dynamical friction. (ii) accretion of mass of neutron
star on PBHs. For a PBH colliding with a neutron star, PBH interacts
gravitationally with the condensed material inside the neutron star. The
result is a drag force so-called dynamical friction [19] which is given by

$\frac{d\mathbf{v}_{pbh}}{dt}=-\frac{4\pi\ln(\Lambda)G^{2}\rho_{n}m_{pbh}}{v_{pbh}^{3}}\left[\mathrm{erf}(X)-\frac{2X}{\sqrt{\pi}}e^{-X^{2}}\right]\mathbf{v}_{pbh},$
(17)

where $\rho_{n}$ is the average density of a neutron star, $m_{pbh}$ is the
mass of PBH as a projectile, $v_{pbh}$ is the velocity of PBH inside the
neutron star that is derived from equation (12),
$X=v_{pbh}/(\sqrt{2}\sigma_{n})$ and $\sigma_{n}$ is the dispersion velocity
of particles inside the neutron star. $\Lambda$ is given by

$\Lambda=\frac{r_{n}\sigma_{n}^{2}}{Gm_{pbh}}=2(\frac{r_{n}}{r_{s(bh)}})(\frac{\sigma_{n}}{c})^{2},$
(18)

where $r_{n}$ is the radius of the neutron star, $r_{s(pbh)}$ is the
Schwartzchild radius of PBH. Substituting the numerical values:

$\Lambda=6\times
10^{10}(\frac{r_{n}}{10\text{km}})(\frac{m_{pbh}}{10^{23}\text{g}})^{-1}(\frac{\sigma_{n}}{c^{2}})^{2},$
(19)

where for the mass of PBHs in the range of $[10^{14},10^{23}]$g results in
$\ln\Lambda\in[24,45]$.

We note that the dispersion velocity of matter in the neutron star also is
given by the uncertainty principle and exclusion principle for the neutrons
(i.e. $pd=\hbar$) where $p$ is the momentum and $d$ is the distance between
the neutrons [20]. For the ultra-relativistic regime for the neutrons where
$E\simeq pc$ and we can set the dispersion velocity of neutrons as
$\sigma_{n}\sim c$. Substituting in definition of $X$ ,

$X=0.22(\frac{r_{s}}{1\text{km}})^{1/2}(\frac{r_{n}}{10\text{km}})^{-1/2}.$

and using equation (17), we define the time scale for dissipation energy of
PBH from the dynamical friction as $t_{df}=v_{pbh}/\dot{v}_{pbh}$ where the
numerical value is

$t_{df}=1.4\times
10^{5}\text{s}(\frac{m_{bh}}{10^{23}\text{g}})^{-1}(\frac{r_{n}}{10\text{km}})^{3/2}(\frac{M_{n}}{M_{\odot}})^{1/2}.$
(20)

For the lower and upper bands of PBH mass, the dynamical friction time scale
ranges from $10^{5}$ to $10^{14}$ second.

We can compare this time scale with the crossing time scale of PBH across the
neutron star by $t_{c}=r_{n}/v_{pbh}$ which is

$t_{c}\simeq
10^{-4}\text{s}(\frac{M_{n}}{M_{\odot}})^{-1/2}(\frac{r_{n}}{10\text{km}})^{3/2},$
(21)

this time scale is very small compared to the dynamical friction time scale
and PBHs can not lose a significant amount of their kinetic energy during the
crossing of the neutron stars.

Another important parameter in the collision of PBH-neutron star is the energy
release as a result of the collision. From the dynamical friction, we expect
the energy release would be $Q_{df}=-m_{pbh}\dot{v}_{pbh}r_{n}$. Substituting
deceleration from equation (17), the energy release is:

$Q_{df}=-\frac{3}{4}\ln\Lambda\left(erf(X)-\frac{2X}{\sqrt{\pi}}e^{-X}\right)(\frac{r_{s(pbh)}}{r_{n}})m_{pbh}c^{2}$
(22)

where using the numerical values the energy released as a result of dynamical
friction is

$Q_{df}=-5.4\times
10^{32}\text{erg}(\frac{m_{pbh}}{10^{23}\text{g}})^{2}(\frac{r_{n}}{10\text{km}})^{-1}.$
(23)

We divide this energy by the crossing time scale in equation (21) which
results in the power released from the dynamical friction inside the neutron
star

$P_{df}=5.4\times
10^{36}\text{erg/s}(\frac{m_{pbh}}{10^{23}\text{g}})^{2}(\frac{r_{n}}{10\text{km}})^{-5/2}(\frac{M_{n}}{M_{\odot}})^{1/2}.$
(24)

The other source of dissipation is the accretion of the neutron star’s
material by a PBH during the crossing of the interior of the neutron star. The
maximum flux from the accretion is given by the Eddington limit

$L_{edd}=\frac{4\pi Gm_{pbh}m_{p}}{\sigma_{TH}}=6.5\times
10^{27}\text{erg/s}(\frac{m_{pbh}}{10^{23}\text{g}}),$ (25)

where $\sigma_{TH}$ is the Thompson cross section for photon-electron
scattering and $m_{p}$ is the mass of proton. We note that in both two
mechanisms of energy dissipation by the dynamical friction and the accretion,
the majority of energy release happens at the interior of the neutron star.
This burst of energy can propagate to the surface of the star with the sound
speed and the result would be perturbation of the inertial tensor which may
cause a kind of glitch or temporary change in the spin of the neutron star.
The anomalies in the spin of the pulsar and burst of energy have already been
observed [21] and to interpret this observation with a PBH collision, detailed
modeling is needed.

Finally, we investigate the effect of energy dissipation of a PBH after
crossing a neutron star. Let us divide $Q_{df}$ from equation (23) (which
plays the major role in the dissipation) by the total kinetic energy of a PBH
(i.e. $E_{k}=m_{pbh}v^{2}/2$), $\epsilon=Q_{df}/E_{k}$. The result is

$\epsilon\simeq-(\frac{v}{1~{}\text{km/s}})^{-2}(\frac{m_{pbh}}{10^{23}\text{g}})(\frac{r_{n}}{10\text{km}})^{-1}$
(26)

In order to have confined PBH around the neutron star, $|\epsilon|\geq 1$
should be satisfied (let us define this velocity as $v_{th}=v$ with the
condition of $|\epsilon|\geq 1$). This means that PBHs with the initial
velocity less than $v_{th}$ in the halo can trap around the neutron star,
where

$v_{th}=1~{}\text{km/s}(\frac{m_{pbh}}{10^{23}\text{g}})^{1/2}(\frac{r_{n}}{10\text{km}})^{-1/2}$
(27)

Since the Galactic structures have different dispersion velocities, we expect
that the fraction of the PBHs can be captured by the neutron stars depends on
the corresponding dispersion velocity of the structures. Here we assume the
Maxwell-Boltzmann distribution for velocity of PBHs and the capturing fraction
can be evaluated from the integral

$P(\sigma,m_{pbh}|v<v_{th})=1-\sqrt{\frac{2}{\pi}}\int_{v_{th}}^{\infty}\frac{1}{\sigma^{3}}\exp(-\frac{v^{2}}{2\sigma^{2}})v^{2}dv,$
(28)

which depends on $\sigma$ as the dispersion velocity of PBHs inside the
Galactic structure and the mass of PBHs.

The dispersion velocity of particles inside the Galactic halo is
$\sigma_{\simeq}200\text{km/s}$. Also there is thin to thick dark disk models
with the dispersion velocities in the range of $\sigma=40-80\text{km/s}$ [22].
Figure (1) represents the result of the integration of equation (28) for
various dark components of the Galaxy. Now, we multiply this probability
function by the rate of PBH impact with a neutron star from equation (16) to
obtain the rate of a PBH capture for different Milky Way dark structures. The
estimation for the total number of neutron stars in the Milky Way [23] is
$N_{n}\simeq 10^{8}$. By multiplying the numerical values for the probability,
rate and number of stars (i.e. $\frac{dN}{dt}\times P(v<v_{th})\times N_{n}$),
we obtain the PBH capturing rate by the neutron stars for different dark
components of the Milky Way as shown in Figure (2).

The capturing rate, measured in Gyr-1, makes it unlikely for us to witness
these events in the Milky Way (except for the low mass PBHs, Figure 2) .
However, on a cosmological scale, along our past light-cone there is potential
for these events to be detected by observing large number of galaxies at
higher redshifts.

file=fig1.pdf,width=8.7cm

Figure 1:  This graph shows the likelihood of a PBH (Primordial Black Hole)
being trapped in a neutron star. The graph plots the probability against the
normalized mass of $m_{23}=(m/10^{23}\text{g})$ and includes three models: a
halo model (dashed line), a dark disk model with $\sigma=80\text{km/s}$
(dotted line), and a dark disk model with $\sigma=40\text{km/s}$ (solid line).

file=fig2.pdf,width=8.7cm

Figure 2: The rate of PBHs of Milky Way captured in a neutron star as a
function of normalized mass of $m_{23}=(m/10^{23}\text{g})$ for three
different models of the halo (dashed line), dark disk with
$\sigma=80\text{km/s}$ (dotted line) and dark disk with $\sigma=40\text{km/s}$
(solid line) in the Milky Way. The estimated total number of neutron stars in
the Milky Way is $10^{8}$ stars [23].

When a PBH is captured by a neutron star, it can either merge with the star
resulting in energetic outbursts, or be put in an orbital motion around the
star. The outcome depends on the impact parameter of the collision between the
PBH and the neutron star.

##  4 Physical consequence of PBH collision with the Astrophysical black holes

In this section, we study the gravitational collision between PBHs and
astrophysical black holes, which can be detected through the propagation of
gravitational waves. We note that the PBHs have been made in the early
Universe, while the astrophysical black holes are the final stage of the heavy
stars evolve to a black hole. Previous research [24, 25, 26, 27], has
extensively studied the signals produced by the interaction of PBH-PBH and
PBH-Astrophysical black holes.

We provide a basic estimation of the energy released from this collision and
study the observability of this event by estimating the energy release
resulting from the merger of two spin-less black holes with masses $M$ and
$m_{pbh}$ under head-on collision [28] and forming a black hole with mass
$M_{t}$ . Using the definition of the ADM-mass [29], we can express
$(m_{pbh}+M)c^{2}=E_{gw}+M_{t}c^{2}$ where $E_{gw}$ is the energy released by
the gravitational waves. On the other hand, the Entropy of a black hole is
proportional to the area of a black hole (i.e. $S\propto A\propto M^{2}$).
Thus, according to the second law of thermodynamics
$S(M_{t})>S(m_{bh})+S(M_{n})$, or in another word,
$M_{t}^{2}>m_{pbh}^{2}+M^{2}$. Combining the conservation of energy-momentum
with the second law of thermodynamics constrains the energy of gravitational
wave from this collision as

$E_{gw}<(m_{pbh}+M-\sqrt{m_{pbh}^{2}+M^{2}})c^{2},$ (29)

where in our case $m_{pbh}\ll M$ and equation (29) simplifies to

$E_{gw}<m_{pbh}c^{2}-m_{pbh}c^{2}\frac{m_{pbh}}{M}.$ (30)

Gravitational wave emission from the collision of an object with a black hole
has been analyzed in detail [30]. The energy of gravitational wave emission is
given by

$E_{gw}=5\times
10^{31}\text{erg}(\frac{m_{pbh}}{10^{23}\text{g}})^{2}(\frac{M}{10M_{\odot}})^{-1},$
(31)

where the total spectrum of gravitational wave peaks at the angular frequency
of

$\omega=0.32\frac{c^{3}}{GM}=0.6\times
10^{4}\text{Hz}(\frac{M}{10M_{\odot}})^{-1}.$ (32)

Multiplying equation (31) to (32) results in an estimation for the power of
the gravitational wave,

$P_{gw}\simeq 0.3\times
10^{36}\text{erg/s}(\frac{M}{10M_{\odot}})^{-2}(\frac{m_{pbh}}{10^{23}\text{g}})^{2}.$
(33)

Now, we estimate the amplitude of gravitational waves from this system at a
given distance from the Earth. The energy-momentum of gravitational waves
averaged over the several wavelengths [31] is given by

$T^{\mu\nu}=\frac{c^{4}}{32\pi
G}<\partial^{\mu}h^{\alpha\beta}\partial^{\nu}h_{\alpha\beta}>,$ (34)

where energy-momentum satisfies the conservation laws in Minkowski background
(i.e. $T^{\mu\nu}{}_{,\nu}=0$). Integrating over the spatial volume for
$\mu=0$ component is $\int T^{00}{}_{,0}d^{3}x=-\int T^{0i}ds_{i}$. Here the
left-hand side of this equation is the power of gravitational wave as in
equation (33) and the right-hand side can be substituted from equation (34).
For a plane wave at far distances from the source
$h^{\alpha\beta}\propto\exp(k\cdot r-\omega t)$, we obtain the amplitude of
the gravitational wave in terms of power and distance of the source from the
observer as

$|h|\simeq\frac{3GM}{rc^{4}}(\frac{8GP}{c})^{1/2}.$ (35)

We substitute the gravitational wave power from equation (33). The final
result for the amplitude of $h$ is

$|h|\simeq 4\times
10^{-27}(\frac{m_{pbh}}{10^{23}\text{g}})(\frac{r}{1\text{kpc}})^{-1}.$ (36)

The interesting feature of this equation is that the amplitude of
gravitational wave emission from PBH-astrophysical black hole collision is
independent of the mass of the astrophysical black hole. The physical reason
is that the gravitational wave power is proportional to $1/M^{2}$ from
equation (33); substituting in (35) results in a gravitational wave amplitude
independent from the mass of the astrophysical black hole. We note that in all
the calculation we assumed that $M\gg m_{pbh}$.

From equation (36), the amplitude of the gravitational wave at the observer’s
position is three orders of magnitude smaller than the sensitivity of the
Advanced LIGO detector [32]. Therefore, detecting PBH-astrophysical black hole
collisions remains a target for future gravitational wave detectors [33].

##  5 Conclusion

Summarizing this work, we investigated the probability and the physical
consequences of collision of the Primordial Black Holes (PBHs) as the
candidate for dark matter with compact objects. The compact object in our
study could be either a neutron star or an astrophysical black hole. Using a
Schwarzschild metric for the compact object as neutron star or an
astrophysical black hole, we obtained relativistic trajectories of PBHs around
these objects. The collision rate of PBHs with compact objects depends on the
mass function of PBHs and the mass of compact objects. We estimate the amount
of dissipation during the collision of PBHs with the neutron stars. The time
scale of dissipation is much larger than the PBH crossing time scale of the
neutron stars. However, having a Maxwell-Boltzmann distribution for the
velocity of PBHs, a small fraction of them can be captured in the neutron
stars, either by merging the neutron star or staying in an orbit around the
neutron star. The total rate of this capturing is very low and it is unlike to
be detected in the Milky Way during our lifetime. However, signals for this
event on the cosmological scale are expected. Also PBHs orbiting around the
neutron stars and black holes might be a target for the gravitational
microlensing observations [34] through binary microlensing channel. However,
we note that since the rate of binary formation according to Figure (2) is
low, we would expect that for a Milky Way type galaxy, there will be almost
$10$ PBH orbiting around the compact objects. However if the primordial black
holes theories let existence of smaller mass PBHs, then the number of neutron
star-PBH binaries would be higher and one could reveal the existence of PBHs
though the microlensing observations.

We also investigated the consequence of the collision of PBH with the
astrophysical black holes. We showed that the amplitude of the gravitational
wave signal for a PBH-astrophysical black hole collision located at kilo
parsec distance is three orders of magnitude smaller than the sensitivity of
the present-day detectors. Future detectors may observe these waves as a
signature for the existence of PBHs in the halo of our Galaxy.

## Acknowledgements

We would like to thank anonymous referee for his/her useful comments improving
this work.

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# Decoding Human Activities: Analyzing Wearable Accelerometer and Gyroscope
Data for Activity Recognition

###### Abstract

A person’s movement or relative positioning effectively generates raw
electrical signals that can be read by computing machines to apply various
manipulative techniques for the classification of different human activities.
In this paper, a stratified multi-structural approach based on a Residual
network ensembled with Residual MobileNet is proposed, termed as FusionActNet.
The proposed method involves using carefully designed Residual blocks for
classifying the static and dynamic activities separately because they have
clear and distinct characteristics that set them apart. These networks are
trained independently, resulting in two specialized and highly accurate
models. These models excel at recognizing activities within a specific
superclass by taking advantage of the unique algorithmic benefits of
architectural adjustments. Afterward, these two ResNets are passed through a
weighted ensemble-based Residual MobileNet. Subsequently, this ensemble
proficiently discriminates between a specific static and a specific dynamic
activity, which were previously identified based on their distinct feature
characteristics in the earlier stage. The proposed model is evaluated using
two publicly accessible datasets; namely, UCI HAR and Motion-Sense. Therein,
it successfully handled the highly confusing cases of data overlap. Therefore,
the proposed approach achieves a state-of-the-art accuracy of 96.71% and
95.35% in the UCI HAR and Motion-Sense datasets respectively.

Index Terms—  Activity recognition, Gyroscope Data, Accelerometer Data,
Activity Grouping

##  1 Introduction

Activity recognition using wearable sensors has been a trending topic of
research for its widespread applicability in diverse sectors ranging from
health care services to military applications[1]. Modern mobile devices
provide abundant sensor data, which are valuable for applications like
activity recognition. Various types of sensor data along with image and video
data have been employed for recognizing human activity[2]. In this paper, the
time series wearable sensor data (e.g. 3-axis accelerometer and gyroscope) are
mainly focused on, as they are pretty easy to obtain and can be used to
recognize human activity from distant data which are very small in volume and
easy to share through internet. In accelerometer-based activity recognition
research, benchmark evaluations compare commonly used features: FFT
coefficients, DCT coefficients, time domain features (including statistical
measures and spectral analysis), and advanced techniques like GAF, Recurrence
Plotting, Scattering Wavelet Transformation, and inter-axis correlations.

In numerous literature, support vector machine (SVM) [3],[4] and kth nearest
neighbor (KNN) classifiers are the most popular ML-based training method for
Body Activity Recognition. Furthermore, multiple-stage training through LSTM-
CNN-LSTM[5], ANN[6], CNN-LSTM[7] are also suggested by some literature.
However, augmented-signal features and a hierarchical recognizer are combined
to get a highly functional trained structure reported in [8].

In 1-D time domain data from accelerometers and gyroscopes, achieving high
training accuracy is challenging with raw data. Shallow networks struggle to
extract meaningful patterns, especially for activities like walking, lying,
and sitting, leading to irregular patterns. Feature extraction by shallow
networks often fails to provide the desired predictability. To illustrate,
walking and lying have different mean acceleration values of 10 and 2.5 as
shown in Fig. 1. However, for the case of walking and standing mean
accelerations are of the same value with distinction in their RMS values as
shown in Fig. 2. To tackle this issue, the activities are categorized into two
broad superclasses: Static (activities where the human body is stationary,
e.g. sitting, laying, etc.) and Dynamic (activities where the human body is in
motion, e.g. walking, jogging, etc.). This classification into super-classes
is deemed essential as it allows for the creation of a distinct and effective
differentiation among the various classes of activities. In this paper, a
multi-structural approach that combines two Residual Block with a Residual
MobileNet is proposed to improve the overall performance.

The main contributions of the paper are-

  * •

The paper introduces a novel multi-structural network architecture,
FusionActNet which leverages the strengths of residual network and mobileNet
components. This architecture is customized to excel in recognizing both
static and dynamic activities, addressing their distinctive features
effectively.

  * •

The proposed approach achieves a remarkable 96.71% accuracy by training
specialized Residual networks for static and dynamic activities and then
combining them through a Residual MobileNet. This represents a significant
advancement in classifying human activities using 1-D time domain data from
accelerometers and gyroscopes.

The remaining sections of the paper are organized in the following order. In
Section 2, a brief description of the dataset is presented. In Section 3, the
problems addressed in this research work and the detailed architecture of the
proposed neural network are presented. Section 4 depicts the experiment and
result whereas section 5 comprises the conclusion of the paper.

Fig. 1: Mean of z-axis acceleration between Walking and Lying Fig. 2: RMS
feature of z-axis acceleration between Walking and Standing

##  2 Dataset Description

In this work, a publicly accessible dataset named UCI HAR is used [9]. It
involved 30 volunteers aged 19 to 48, wearing a Samsung Galaxy S II smartphone
on their waist while performing six activities which are as follows: Walking,
Walking upstairs, Walking downstairs, Sitting, Standing, and Lying. The
smartphone’s accelerometers and gyroscopes recorded 3-axial linear
acceleration and angular velocity at 50Hz. The dataset was split into 70%
training and 30% testing data. Preprocessing included noise filtering, fixed-
width sliding windowing and separating gravitational and body motion
components. The dataset comprised 7352 training observations from 30 subjects
and 2947 test observations for model evaluation.

Fig. 3: Detailed architecture of FusionActNet

Another dataset used in this work is the Motion-Sense dataset, which consists
of 15 trials involving 24 individuals using an iPhone 6s placed in their
trouser pockets to record accelerometer and gyroscope data [10]. It includes
12 features, such as attitude, gravity, rotation rate, and user acceleration,
and covers activities like Walking Downstairs, Walking Upstairs, Sitting,
Standing, Walking, and Jogging.

##  3 Proposed Method

###  3.1 Network Architecture

Methods entailing extracted features of mixed-class activities with extensive
overlap passed through a deep neural net sometimes fail to generalize on a
large variety of datasets due to over-parameterization, low bias, and high
variance. It was evident through extensive experimentation that a single deep
network sometimes struggles to distinguish closely related features of various
activities. To address this, the concept of categorizing human activities into
two superclasses: Static and Dynamic, is introduced here. These two classes
are processed by separate models, allowing each model to specialize in a
specific type of data. Instead of relying on a single network, this approach
leverages two highly specialized models, making it more effective in capturing
subtle patterns within the data from these two superclasses. In this paper, a
novel deep neural network architecture has been proposed, that utilizes the
weighted concatenation of outputs from two expert models to recognize static
and dynamic human activities respectively. Because of this weighted fusion of
outputs from different expert models to recognize human activities from
different sensor data, this network is named FusionActNet. The proposed model
leverages the power of 1D Residual blocks as well as Residual MobileNet
building blocks, offering a dynamic and static pathway for information
processing.

####  3.1.1 Static Pathway (model static)

The model static component of FusionActNet is an expert for identifying the
static human activities (e.g. sitting, laying and standing) that comprises a
series of Residual blocks [11] in order to mitigate the vanishing gradients
problem, as shown in Fig. 3, each of which consists of a convolutional layer,
batch normalization, and rectified linear unit (ReLU) activation functions as
well as a skip connection. Each Residual Block $RB(d_{in},d_{out},d_{pool})$
where $d_{in}$ and $d_{out}$ refers to the input and output dimensions of 1D
convolution layers and $d_{pool}$ refers to the kernel size of the 1D
MaxPooling layer. The Residual Net Block begins with a convolutional layer
with a 3x3 kernel size and output channels $d_{out}$, followed by batch
normalization and ReLU activation. Subsequently, another 3x3 convolutional
layer with $d_{out}$ output channels is applied, followed by batch
normalization. An identity convolutional layer with a 1x1 kernel size is used
to match dimensions for residual connections. A max-pooling layer with a
kernel size of $d_{pool}$ is applied for down-sampling. The final layer of the
model static pathway includes adaptive average pooling to ensure a fixed-size
output irrespective of the input size. A fully connected layer reduces the
output of the static pathway to three output features, which are used for
classification or regression tasks.

####  3.1.2 Dynamic Pathway (model dynamic)

The model dynamic component is structurally identical to the model static but
is expected to capture dynamic patterns or temporal information in the input
dynamic data. It follows the same Residual block structure but is trained with
dynamic human activities such as Walking, Walking Upstairs and Walking
Downstairs.

####  3.1.3 Weight Prediction Submodule

A neural sub-module at the center of 3 acts as a feature fusion point,
combining information from both the static and dynamic pathways. This
combination is done through weighted concatenation of outputs of static and
dynamic models, where the weight $X_{weight}$ is obtained through the
mentioned neural sub-module, which will prioritize the output of the
appropriate expert model. An important element of this sub-module is the
Residual MobileNet Block $RMB(d_{in},d_{out})$ that consists of 1D convolution
layer, residual skip connection, the Batch normalization and ReLU activation.
Subsequently, two sets of MobileNet blocks [12] are introduced to further
refine the features: Each Residual MobileNet Block consists of depth-wise
separable convolutions for efficient feature extraction. A 3x3 depth-wise
convolution is applied for model size reduction and improving the efficiency,
followed by batch normalization and ReLU activation. A 1x1 point-wise
convolution is used for dimensionality reduction, followed by batch
normalization. A key innovation of this model lies in its ability to fuse
static and dynamic information seamlessly. This fusion process is achieved
through a weighted combination mechanism as shown in Fig. 3. By learning
appropriate weights, the model adapts to the characteristics of the input
data, effectively blending the outputs of the static and dynamic networks.
This fusion enhances the model’s adaptability and robustness, making it
capable of recognizing activities with varying characteristics.

##  4 Experiment and Result

Table 1: Evaluation results of different methods on UCI-HAR [9] dataset

Method | Accuracy | Precision | Recall | F1-score
---|---|---|---|---
MC-HF-SVM [4] | 0.8935 | 0.8993 | 0.8968 | 0.8980
SSAL [13] | 0.8124 | 0.8166 | 0.8125 | 0.8145
ResNet [14] | 0.9073 | 0.9095 | 0.8998 | 0.9046
CNN [15] | 0.9271 | 0.9321 | 0.9321 | 0.9293
LSTM [15] | 0.8901 | 0.8901 | 0.8899 | 0.8899
BLSTM [15] | 0.8940 | 0.8941 | 0.8941 | 0.8935
MLP [15] | 0.8683 | 0.8683 | 0.8658 | 0.8658
SVM [15] | 0.9050 | 0.9050 | 0.8986 | 0.8985
HAR-CT: CNN [16] | 0.9406 | 0.9358 | 0.9359 | 0.9359
HAR-CT: TWN [16] | 0.9342 | 0.9250 | 0.9250 | 0.9250
FusionActNet |  |  |  |
(Proposed) | 0.9671 | 0.97 | 0.96 | 0.96

Table 2: Evaluation results of different methods on MotionSense [10] dataset

Method | Accuracy | Precision | Recall | F1-score
---|---|---|---|---
MFCC + SVM [17] | 0.8935 | 0.8993 | 0.8968 | 0.8980
SVM + PSO [18] | 0.8124 | 0.8166 | 0.8125 | 0.8145
GA + ACO [18] | 0.9073 | 0.9095 | 0.8998 | 0.9046
DT + BGWO [18] | 0.9271 | 0.9321 | 0.9321 | 0.9293
Self-supervised TPN [19] | 0.8901 | 0.8901 | 0.8899 | 0.8899
FusionActNet |  |  |  |
(Proposed) | 0.9535 | 0.95 | 0.95 | 0.95

The results section presents an evaluation of various methods applied to the
UCI-HAR dataset, assessing their performance based on some major evaluation
criteria, such as accuracy, precision, recall, and F1-score. Each method is
listed along with its corresponding evaluation metrics in Table 1. The trend
seems to favor multilayer perceptron or neural network-based approaches as an
obvious choice for model prediction of activity recognition. To be precise,
HAR-CT: CNN and DT + BGWO with respective accuracy of 94.6% and 92.71% called
the closest shot with our proposed FusionActNet with an accurate
predictability of 96.71%. The performance of the proposed method is also
tested on another publicly available dataset called MotionSense [10] and the
results are shown in Table 2

From Fig. 4 it is observed in the confusion matrix that activity class
’walking’ was misclassified as ’walking upstairs’ and ’walking downstairs’
which are very logical. The misclassified cases were investigated closely and
it was found that in these cases, the nature of the data was quite similar.
Another point of limitation was found in ’sitting’ and ’standing’ data because
the x-axis and y-axis components of the gyro and accelerometer data points
were not in effect for both cases and the z-axis component data overlap was
too confusing and densely blended for the FusionActNet model to differentiate
the 10% (48 misclassified cases out of 566 trials) overlap cases as shown in
Fig. 4. However, Tables 1 and 2 serve as compelling evidence of the superior
performance of our proposed method, excelling in both accuracy and precision.

Fig. 4: Confusion Matrix

##  5 Conclusion

In conclusion, the proliferation of smartphones and the advent of multi-sensor
systems have ushered in an era where researchers can focus on monitoring daily
activities through the human physiological system. This has made the analysis
of human motion an integral part of our daily lives. Smartphones, equipped
with various sensors like accelerometers and gyroscopes, have become valuable
tools for recognizing human activities. In this paper, a comparatively newer
approach: a stratified multi-structural network combining residual and
mobileNet architectures, is proposed, referred to as ’FusionActNet’. This
proposed network is evaluated using the UCI-HAR and MotionSense datasets,
assessing its performance through key metrics including accuracy, precision,
recall, and F1-score. This research represents a significant step forward in
harnessing sensor-rich devices for precise human activity recognition,
offering promising applications in fields such as healthcare and smart
technology integration.

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