# Emergence of smooth distance and apparent magnitude in a lumpy Universe

Obinna Umeh  Institute of Cosmology & Gravitation, University of Portsmouth,
Portsmouth PO1 3FX, United Kingdom
Department of Physics, University of the Western Cape, Cape Town 7535, South
Africa
Department of Mathematics and Applied Mathematics, University of Cape Town,
Rondebosch 7701, South Africa<EMAIL_ADDRESS>
###### Abstract

The standard interpretation of observations such as the peak apparent
magnitude of Type Ia supernova made from one location in a lumpy Universe is
based on the idealised Friedmann-Lemaître Robertson-Walker spacetime. All
possible corrections to this model due to inhomogeneities are usually
neglected. Here, we use the result from the recent concise derivation of the
area distance in an inhomogeneous universe to study the monopole and Hubble
residual of the apparent magnitude of Type Ia supernovae. We find that at low
redshifts, the background FLRW spacetime model of the apparent magnitude
receives corrections due to relative velocity perturbation in the observed
redshift. We show how this velocity perturbation could contribute to a
variance in the Hubble residual and how it could impact the calibration of the
absolute magnitude of the Type Ia supernova in the Hubble flow. We also show
that it could resolve the tension in the determination of the Hubble rate from
the baryon acoustic oscillation and local measurements.

  *

Keywords: Cosmological parameters–cosmology: observations–dark energy

##  1 Introduction

The Universe we observe today from a single location on Earth is not
homogeneous and isotropic. This is obvious from the map of the distribution of
structures in the nearby Universe by the Sloan Digital Sky Survey (SDSS) [1].
The observed distribution is clearly inhomogeneous [2, 3]. On very large
scales, that is more than 200 Mpc away from the observer, the features of
isotropy in the distribution of structures begin to emerge [4]. Geometric
homogeneity on the one hand cannot be established independent of the
Copernican principle assumption [5, 6, 7]. Yet, when we interpret distances to
or the apparent magnitudes of sources for cosmological inference, we model the
underlying null geodesics using the Friedmann-Lemaître Robertson-Walker (FLRW)
metric without considering the fact that null geodesics that terminate at the
observer could be impacted by structures in the neighbourhood of the observer
[8].

This practice was motivated by the 1976 paper by Steven Weinberg [9]. Weinberg
showed that if conservation of photon number holds, then it is consistent to
use the background FLRW spacetime on all scales and at all times. Weinberg’s
argument/proof relies on several assumptions which include: (1) the
cosmological principle holds(i.e., that the average number density of photons
is described in the FLRW spacetime) [10], (2) the impact of gravity on rulers
and clocks is not important [11, 12], (3) the impact of strong redial lensing
effects or caustics on the luminosity distance is negligible [13, 14]. Also,
the proof focused on the area of the screen space and not on the
area/luminosity distance. Recent works have shown that the weak gravitational
lensing contributes to the area/luminosity distance measurement at high
redshift [15, 16, 17].

The impact of caustics on the area distance was first studied in [13] within
the Swiss-cheese model of the Universe. The authors showed that the presence
of radial lensing caustics leads to an increase in the area distance on
average when compared to the FLRW prediction. Similarly, it is well-known that
rulers and clocks behave differently in the neighbourhood of caustics and
these effects were neglected in Weinberg’s photon number conservation argument
[14]. Our focus here is to study the impact of radial lensing contribution to
the area distance within the standard cosmology(i.e., perturbed FLRW
spacetime). We work in the Conformal Newtonian gauge which is the most viable
gauge for studying the dynamics of structures in the immediate neighbourhood
of the observer [18]. The details of the derivation of the area distance used
in this work were presented in [19]. Our target here is simply to focus on the
impact of inhomogeneities on the calibration of the apparent magnitude of the
Type Ia supernova (Sn Ia). We study how the inhomogeneities could contribute
to the Hubble residuals which are usually attributed to the host galaxy
properties [20, 21, 22, 23].

Essentially, this paper shows that the calibration of the absolute magnitude
of the Sn Ia in the Hubble flow with a set of local distance anchors is
impacted by inhomogeneities along the line of sight. Furthermore, it shows
that the contribution of the inhomogeneities to the apparent magnitude of the
Sn Ia could resolve the supernova absolute magnitude tension [24]. The
supernova absolute magnitude tension is another way of expressing the Hubble
tension, i.e. the tension between the determinations of the Hubble rate by the
early and late Universe experiments [25]. The correction to the apparent
magnitude is due to the effect of radial lensing or Doppler lensing [26]. How
the Doppler lensing effects could impact or contribute to the Hubble residual
is also discussed. Finally, the estimate of the Alcock-Paczyński parameters
[27] in a perturbed FLRW spacetime is provided. It is compared to the
background FLRW spacetime prediction using the cosmological fitting
prescription proposed by Ellis and Stoeger in 1987 [28, 29].

This paper is organised as follows: In section 2, the impact inhomogeneities
along the line of sight to the area distance at low redshift is provided.
Section 3 contains discussions on how inhomogeneities impact the apparent
magnitude of Sn Ia. For instance, in sub-section 3.1, details on how it
impacts the calibration of the absolute magnitude of Sn Ia is provided. How
the impact of small scale inhomogeneities could provide a possible resolution
of the supernova absolute magnitude tension is presented in sub-section 3.3.
The impact of the small scale inhomogeneities on the inferred Hubble rate from
the Alcock-Paczyński parameters is discussed in section 3.4. The argument on
existence of a fundamental limit of the expanding global coordinate system is
provided in section 4 and conclusion is provided in section 5. For all the
numerical analyses, the cosmological parameters determined from the analysis
of Cosmic Microwave Background (CMB) anisotropies by Planck collaboration were
used [4].

##  2 Impact of inhomogeneities on the area distance in cosmology

We describe the bundle of light rays coming from a source as null geodesics
and define a deviation vector as a difference between a central ray,
$\bar{x}^{i}$, in a bundle and the nearby null geodesic, $x^{i}$, at the same
affine parameter, $\lambda$, as
$\xi^{i}(\lambda)=x^{i}(\lambda)-\bar{x}^{i}(\lambda)$. With respect to the
observer’s line of sight direction, ${{\bf\hat{n}}}$, $\xi^{i}$ maybe
decomposed into two components:
$\xi^{i}=\xi_{{}_{\parallel}}n^{i}+\xi_{\perp}^{i}$, where
$\xi_{{}_{\parallel}}=\xi_{i}\,n^{i}$ and ${{\xi_{\perp}^{i}}}$ are components
parallel and orthogonal to ${{\bf\hat{n}}}$ respectively. The distortions
experienced by the nearby geodesics due to the impart of curvature modify the
area distance on an FLRW spacetime according to [19]

$\displaystyle{{{d}_{A}(z,{{\bf\hat{n}}})}}$
$\displaystyle\simeq\bar{d}_{A}(z)\bigg{[}1+\frac{\xi{{}_{\parallel}}}{\bar{d}_{A}}+\kappa-\frac{1}{4}\bigg{(}\gamma_{ij}\gamma^{ij}-\omega_{ij}\omega^{ij}\bigg{)}\bigg{]}\,,$
(1)

where $\bar{d}_{A}$ is the area distance on the background FLRW spacetime,
$\kappa=-\nabla_{\perp i}\xi_{\perp}^{i}/2$ describes the isotropic
magnification of the source image due to the weak gravitational lensing,
$\gamma_{\langle ij\rangle}=\nabla_{\perp\langle i}\xi_{\perp j\rangle}$ is
the shear distortion, it describes the stretching of the image tangentially
around the foreground mass and
$\omega=\varepsilon^{ij}\nabla_{\perp[i}\xi_{\perp j]}$ is the twist.
${\xi{{}_{\parallel}}}/{\bar{d}_{A}}$ is the radial convergence [13, 30]. It
is central to our work. The impact of the heliocentric peculiar motion will be
discussed elsewhere, in the meantime see [31] for the estimate of the expected
impact of heliocentric peculiar motion on the area distance.

We work in Conformal Newtonian gauge on a flat FLRW background spacetime:

$\displaystyle{\rm{d}}s^{2}$
$\displaystyle=a^{2}\big{[}-\big{(}1+2\Phi\big{)}{\rm{d}}\eta^{2}+\big{(}\big{(}1-2\Phi\big{)}\delta_{ij}{{{\rm{d}}x^{i}{\rm{d}}x^{j}}}\big{]}\,.$
(2)

Here $\delta_{ij}$ is the flat metric on Minkowski spacetime, $a$ is the scale
factor of the Universe and $\Phi$ is the Newtonian potential. The all-sky
average of equation (1) on the surface of constant redshift is given by

$\displaystyle\left\langle{d}_{A}(z_{s})\right\rangle_{\Omega}$
$\displaystyle=\bar{d}_{A}\bigg{[}1+\left\langle\frac{\xi{{}_{\parallel}}}{\bar{d}_{A}}\right\rangle_{\Omega}-\frac{1}{2}\left\langle\gamma_{ij}{\gamma}^{ij}\right\rangle_{\Omega}\bigg{]}\,.$
(3)

We made use of Gauss’s theorem to set the all-sky average of $\kappa$ to zero
$\left\langle\kappa\right\rangle_{\Omega}=0$. $\omega_{ij}\omega^{ij}$
vanishes at second order in perturbation theory and
$\left\langle\gamma_{ij}{\gamma}^{ij}\right\rangle_{\Omega}$ becomes important
only at high redshift and its total contribution is about a few percent at
$z\geq 1$ [17]. Hence, we do not consider the weak gravitational lensing
correction in the discuss that follows.

The leading order correction to the area distance at low redshift is given by
the second term in equation (3), i.e.
$\left\langle{\xi{{}_{\parallel}}}/{\bar{d}_{A}}\right\rangle_{\Omega}$. This
correction comes from the perturbation of the redshift, $\delta z$, due to
Doppler effect [26, 19]. When interpreting distances to sources on constant
redshift surface(i.e., area distance as a function of the cosmological
redshift), the monopole of the radial lensing correction,
$\left\langle{\xi{{}_{\parallel}}}/{\bar{d}_{A}}\right\rangle_{\Omega}$,
translates to an increase in distance to the source, $\delta r=\delta
z/\mathcal{H}\,$, the dominant term in $\delta z$ is the relative peculiar
velocity;

$\displaystyle{{\delta
z(z,{{\bf\hat{n}}})\approx\partial_{\|}{v_{s}}-\partial_{\|}v_{o}+\mathcal{O}\left(\Phi_{o}-\Phi_{s}\right)\,,}}$
(4)

where $\partial_{\|}v_{o}$ and $\partial_{\|}v_{s}$ are the peculiar
velocities of the observer and the source respectively, $\mathcal{H}\,$ is the
conformal Hubble rate. At the linear order in perturbation theory, this term
may be evaluated on the background spacetime, this is the well-known Born
approximation [32], however, at the second order, we need to include the post-
Born correction terms [33]:

$\displaystyle\delta z({\bar{r}},{\bar{\theta}^{i}})$ $\displaystyle\to$
$\displaystyle\delta z({r}_{s}-\delta
r_{s},{\theta}^{i}-\delta\theta^{i}_{s})\,,$ (5) $\displaystyle=$
$\displaystyle\delta z({{r}_{s}},{{\theta}^{i}_{s}})-\delta
r_{s}\partial_{{}_{\parallel}}\delta
z({\bar{r}},{\bar{\theta}^{i}})-\delta\theta^{i}_{s}\nabla_{\perp}\delta
z({\bar{r}},{\bar{\theta}^{i}})\,,$ (6)

where, $\delta\theta^{i}_{s}$ is the angular perturbation.
$\delta\theta^{i}_{s}\nabla_{\perp}\delta z$ captures the effect of
gravitational deflection, it is not important to our discussion because it is
sub-dominant at low redshift. Our focus is on the radial correction, $\delta
r_{s}\partial_{{}_{\parallel}}\delta z$, it dominates at very small redshifts
[19]

$\displaystyle\left\langle\frac{\xi{{}_{\parallel}}(z_{s})}{\bar{d}_{A}}\right\rangle_{\Omega}$
$\displaystyle\approx$
$\displaystyle\frac{1}{r_{s}\mathcal{H}\,^{2}_{s}}\bigg{[}\left\langle\partial_{||}v_{s}\partial^{2}_{||}v_{s}\right\rangle_{\Omega}-\left\langle\partial_{\|}{v_{o}}\partial^{2}_{\parallel}{v_{s}}\right\rangle_{\Omega}\bigg{]}\,,$
(7)

where $r_{s}$ is the comoving distance to the source. The all-sky average of
$\left\langle\partial_{||}v_{s}\partial^{2}_{||}v_{s}\right\rangle_{\Omega}$
vanishes, this implies that the key leading order contribution to
$\left\langle{\xi{{}_{\parallel}}}/{\bar{d}_{A}}\right\rangle_{\Omega}$ is
given by [19]

$\displaystyle\left\langle\frac{\xi{{}_{\parallel}}(z_{s})}{\bar{d}_{A}}\right\rangle_{\Omega}$
$\displaystyle\approx-\frac{1}{r_{s}\mathcal{H}\,^{2}_{s}}\left\langle\partial_{\|}v_{o}\partial^{2}_{\parallel}{v_{s}}\right\rangle_{\Omega}\,.$
(8)

Equation (8) is the leading order term in the expression for the area distance
given in equation (3). The coupling between the peculiar velocity term
evaluated at the observer position and the Kaiser redshift space distortion
term evaluated at the observer suggests that this term describes the "parallax
effect". That is the apparent displacement in the background source position
as the observer move relative to the local over-density. Astronomers utilise
this effect as a tool to calculate distances to both galactic and
extragalactic sources [34, 35].

It is straightforward to expand equation (8) in spherical harmonics and
estimate the ensemble average if the peculiar velocity term (i.e.,
$\partial_{\parallel}v_{o}$) is known. In our case, it corresponds to the
peculiar velocity of the heliocentric observer. Our perturbation theory
formalism is not valid on the solar system’s scale [36]. Therefore, to handle
this correctly, we borrow insights from the analysis of the temperature
anisotropies of the Cosmic Microwave Background radiation and expand equation
(7) in full-sky spherical harmonics and drop the first two harmonics $\ell=0,$
$\ell=1$ which are coordinate dependent(see [37, 19] for detail on how to do
this). The impact of $\ell=0$, $\ell=1$ harmonics are then included by Lorentz
boosting to the Heliocentric frame of reference [38, 39]. The monopole of
coordinate independent part of equation (7) is given by [19]

$\displaystyle\left\langle\frac{\xi{{}_{\parallel}}(z_{s})}{\bar{d}_{A}}\right\rangle_{\Omega}$
$\displaystyle\approx$
$\displaystyle\frac{1}{r_{s}}(f(z_{s})D(z_{s}))^{2}\sum_{\ell=2}^{\ell_{\rm{max}}}(2\ell+1)\int\frac{dk_{1}}{2\pi^{2}}k_{1}P_{m}(k_{1})j^{\prime}_{\ell}(k_{1}r_{s})j^{\prime\prime}_{\ell}(k_{1}r_{s})\,,$
(9)

where $j_{\ell}$ is the spherical Bessel function of order $\ell$, ′ on
$j_{\ell}(kr)$ denotes derivative wrt the argument, i.e. $(kr)$, $f$ is the
rate of growth of structure, $P_{m}$ is the matter density power spectrum, $k$
is amplitude of the wave vector $k=|{\bm{k}}|$. For the results we discuss
here, we consider the linear perturbation theory prediction of $P_{m}$ and the
halo model prediction [40, 41]. When we quote percentage correction, we refer
to only the results obtained using the linear matter perturbation theory
prediction for $P_{m}$.

Figure 1: We show the monopole of the radial lensing contribution
$\left\langle{\xi{{}_{\parallel}}}/{\bar{d}_{A}}\right\rangle_{\Omega}$ to the
area distance. At low redshifts, this is proportional to the fractional
difference between the monopole of the area distance and the background FLRW
expectation:
$\left\langle{\xi{{}_{\parallel}}}/{\bar{d}_{A}}\right\rangle_{\Omega}\approx(\left\langle{d}_{A}\right\rangle_{\Omega}-\bar{d}_{A})/\bar{d}_{A}$.
The legend shows the dependence of the size of the correction on the model
used to estimate the matter power spectrum(linear perturbation theory (dashed
line) and halo model (thick line)). For the inset, we zoomed in on the very
small redshifts $(z<0.01)$. We identified two redshift positions on the inset:
$z_{\rm{cut}}$ and $z_{\rm{ngc}}$. The $z_{\rm{cut}}$ corresponds to the
minimum redshift below which the spacetime is not expanding (we describe how
to determine $z_{\rm{cut}}$ in detail in section 4.). From the analysis, we
find that $z_{\rm{cut}}=2.4\times 10^{-4}$. At $z_{\rm{cut}}$, we find about
12% correction to the area distance based on the background FLRW spacetime.
Furthermore, $z_{\rm{ngc}}$ is the cosmological redshift for the NGC 4258. We
find about 2% correction to the area distance at $z_{\rm{ngc}}=0.0018$. NGC
4258 is one of the key local distance anchors used to calibrate the co-located
cepheids(cepheids in the same host as the Sn Ia). The co-located cepheids are
then used to calibrate the Sn Ia in the Hubble flow. In the numerical
computation, we find that it is enough to set $\ell_{\rm{max}}$ to
$\ell_{\rm{max}}=100$.

The correction to the area distance is shown in figure 1. It is clear that at
$z\geq 0.023$, the corrections to the background FLRW model are less than
$0.1\%$, hence the background FLRW spacetime model is valid. However, for
$z\leq 0.01$, the correction to the background could exceed 12% at
$z_{\rm{cut}}$. Here, the $z_{\rm{cut}}$ is the minimum value the cosmological
redshift can take in an expanding universe. It corresponds to the minimum
length scale below which the spacetime is not expanding. The $z_{\rm{cut}}\leq
z\leq 0.023$ redshift range corresponds to the first and second rungs of the
cosmic distance ladder according to the SH0ES collaboration classification
scheme [42]. As we shall see in the subsequent section, the apparent optical
properties of sources in $z_{\rm{cut}}\leq z\leq 0.023$ is very important,
they play a crucial role in the calibration of the absolute magnitude of the
Sn Ia in the Hubble flow.

The most essential point from the section is that the effective area distance
model in the low redshift limit in the presence of inhomogeneities is given by

$\displaystyle\left\langle{d}_{A}(z_{s})\right\rangle_{\Omega}$
$\displaystyle=\frac{{d}^{\rm{eff}}_{H}(z_{s})}{(1+z_{s})}\int_{z_{\rm{cut}}}^{z_{s}}\frac{dz^{\prime}}{{\sqrt{\Omega_{m}(1+z^{\prime})^{3}+1-\Omega_{m0}}}}\,,$
(10)

where $\Omega_{m0}$ is the matter density parameter,
${d}^{\rm{eff}}_{H}=d_{H}\left[1+\left\langle{\xi{{}_{\parallel}}}/{\bar{d}_{A}}\right\rangle_{\Omega}\right]$
is the effective Hubble distance and $d_{H}=c/{H}_{0}$ is the global Hubble
distance. Using this model for the distance to NGC 4258 gives about 2%
correction to the prediction of the background FLRW model. In the next
section, this point is investigated in greater detail in order to understand
how it impacts the apparent and absolute magnitude of the Sn Ia in the Hubble
flow.

##  3 Cosmology in the presence of inhomogeneities

###  3.1 Apparent magnitude and calibration of the cosmic distance ladder

The apparent magnitude, $m_{b}$, of the Sn Ia is a function of the bandpass
(filter), the spectral flux density and the fundamental standard spectral flux
density. Our interest is on the effect of the small scale inhomogeneities on
the luminosity distance, $d_{L}$. Assuming photon number conservation, we can
obtain $d_{L}$ from $d_{A}$ which we have already calculated using the
distance duality relation: $d_{L}=(1+z)^{2}d_{A}$. Within a given filter,
$m_{b}$ is related to the flux density $F_{d_{L}}$ according to

$\displaystyle m_{b}=-2.5\log_{10}\left[F_{d_{L}}\right]\,.$ (11)

For cosmological inference, the apparent magnitude is useless without
calibration. The peak apparent magnitude of the Sn Ia in the Hubble flow is
calibrated using secondaries, for example, the Tip of the Red Giant Branch
(TRGB) [43], cepheids [44], etc. The essence of calibration is to determine
the absolute magnitude ($M_{b})$ of the Sn Ia. The absolute magnitude is
defined as the apparent magnitude measured at a distance $D_{F}$:
$M_{b}=-2.5\log_{10}\left[{F_{D_{F}}}\right],$ where $F_{D_{F}}$ is called the
reference flux or the zero-point of the filter [45, 46, 47]. The observed flux
density at both distances obeys the inverse square law with the luminosity
distance: ${F_{d_{L}}}/{F_{D_{F}}}=\left[{D_{F}}/{d_{L}}\right]^{2}$. The
cosmological magnitude-redshift relation or the distance modulus is defined by
setting $D_{F}=1$[Mpc]

$\displaystyle\mu=m_{b}-M_{b}=5\log_{10}\left[\frac{d_{L}}{[\rm{Mpc}]}\right]+25\,,$
(12)

where $\mu$ is the distance modulus. The SH0ES collaboration calibrates the
absolute magnitude of the Sn Ia in the Hubble flow using a sub-sample of
nearby Sn Ia co-located with cepheids in the second rung of distance ladder:

$\displaystyle\ {{M_{b}={{m_{0,\rm{SnIa}}}}-\mu_{0,\rm{ceph}}\,,}}$ (13)

where $m_{0,\rm{SnIa}}$ is corrected-peak apparent luminosity of Sn Ia in the
same host as cepheid, $\mu_{0,\rm{ceph}}$ is the distance modulus of the co-
located cepheid. $\mu_{0,\rm{ceph}}$ is obtained from the calibrated Leavitt
law [48]

$\displaystyle
m_{0,{\rm{ceph}}}=\mu_{0,{\rm{ceph}}}+M_{H,{\rm{ceph}}}+b_{w}\left({{\log_{10}}}P-1)+Z_{W}\left[O/H\right]\right]\,,$
(14)

where $m_{o,{\rm{ceph}}}$ is the apparent magnitude of Cepheids,
$\left[O/H\right]$ is the metallicity correction, $P$ is the period of the
host measured in days, $M_{H,\rm{ceph}}$ is the fiducial absolute magnitude of
cepheid evaluated when $\log_{10}P=1$ or $P=10$ days and the metallicity
correction set to the solar metallicity. $b_{W}$ and $Z_{W}$ define the
empirical relation between the cepheid period, metallicity and luminosity.

The recent SH0ES analysis made use of an independent geometric distance
estimate to NGC 4258 as an anchor to calibrate $M_{H,\rm{ceph}}$ for cepheids
in NGC 4258 [25]

$\displaystyle{{M_{H,\rm{ceph}}=m_{\rm{NGC}}-\mu_{\rm{NGC}}\,,}}$ (15)

where, $m_{\rm{NGC}}$ is the corrected apparent magnitude of cepheids in NGC
4258 and $\mu_{\rm{NGC}}=5\log_{10}[d_{\rm{NGC}}/{\rm{[Mpc]}}]$ is the
independent distance modulus to the NGC 4258. The NGC 4258 is an intermediate
mass spiral galaxy with a central supermassive black hole. It is about 7.5
[Mpc] away from earth with a cosmological redshift of about $z\approx 0.0018$.
The distance estimate to NGC 4258 (i.e $d_{\rm{NGC}}$) is obtained from the
shape of the MASER luminosity profile in the accretion disk of the
supermassive black hole at the centre of NGC 4258. From the width of the
luminosity profile, $d_{\rm{NGC}}$ is given by

$\displaystyle{{d_{\rm{NGC}}=\frac{\partial_{\theta}v_{\rm{LoS}}}{a_{\rm{LOS}}}\,,}}$
(16)

where $v_{\rm{LoS}}$ is the Line of Sight (LoS) velocity to the localised
MASER emission around the supermassive black hole,
$\partial_{\theta}v_{\rm{LoS}}$ is the gradient of $v_{\rm{LoS}}$ on the sky,
$a_{\rm{LOS}}$ is the LoS acceleration and $\theta$ is the angle on the sky as
measured by the observer [49].

Other geometric distance anchors include the Milky-Way Cepheids [34]. In this
case, the parallax method is used to estimate the distance to the Milky-Way
Cepheids and the Large Magellanic Cloud (LMC), where the distance is obtained
from the dynamics of the detached eclipsing binary systems [50]. Both of the
distances to LMC and the milky-way cepheids are less than 1[Mpc] away. The
cosmological perturbation theory technique is inadequate for estimating
distances to objects that are less than 1 [Mpc] away from the observer [19].
That is $d_{A}\approx d_{L}<1$[Mpc] is non-perturbative. As a result, this
study focuses on NGC 4258 which is about 7.5 [Mpc] away. In general, for
$d_{A}\approx d_{L}\geq 1$[Mpc], the cosmological perturbation theory
technique provides a valid description of the impact of inhomogeneities.

###  3.2 Hubble residuals and inhomogeneities along the line of sight

There has been a huge interest in understanding the observed Hubble residuals
in the samples of the Sn Ia [22, 23]. The Hubble residual is defined as the
difference between the distance modulus of the corrected peak apparent
magnitude of the Sn Ia inferred from the light curves as a function of the Sn
Ia host cosmological redshift and the apparent magnitude calculated assuming
the background FLRW spacetime predication for the luminosity distance [20,
51]. Correlations have been found between the Hubble residuals and host galaxy
properties such as the size, stellar mass, ${\rm{H}}\beta$ content etc [52,
23, 22].

Using the cosmological perturbation theory technique, we can calculate the
Hubble residuals provided that the calibration of the absolute magnitude is
held fixed. For example, keeping the calibration of $M$ fixed, we find that
the Hubble residuals is given by

$\displaystyle\left\langle\Delta\mu(z_{s})\right\rangle_{\Omega}\equiv\left\langle\mu\right\rangle_{\Omega}-\bar{\mu}$
$\displaystyle=$ $\displaystyle
5\log_{10}\left[1+\left\langle\frac{\xi{{}_{\parallel}}}{\bar{d}_{A}}\right\rangle_{\Omega}-\frac{1}{2}\left\langle\frac{\xi^{2}{{}_{\parallel}}}{\bar{d}_{A}^{2}}\right\rangle_{\Omega}\right]\,,$
(17) $\displaystyle=$
$\displaystyle{{\frac{5}{\ln({10})}\left[\left\langle\frac{\xi{{}_{\parallel}}}{\bar{d}_{A}}\right\rangle_{\Omega}-\frac{1}{2}\left\langle\frac{\xi^{2}{{}_{\parallel}}}{\bar{d}_{A}^{2}}\right\rangle_{\Omega}\right]}}\,,$
(18)

where the spherical harmonic expansion of
$\left\langle{\xi^{2}{{}_{\parallel}}}/{\bar{d}_{A}^{2}}\right\rangle_{\Omega}$
is given by

$\displaystyle{\left\langle\frac{\xi^{2}{{}_{\parallel}}}{\bar{d}_{A}^{2}}\right\rangle_{\Omega}}$
$\displaystyle\approx$
$\displaystyle\frac{f^{2}(z_{s})D^{2}(z_{s})}{r_{s}^{2}}\sum_{\ell=2}^{\ell_{\rm{max}}}(2\ell+1)\int\frac{{\rm{d}}k}{2\pi^{2}}P_{m}(k)j^{\prime}_{\ell}(kr_{s})j^{\prime}_{\ell}(kr_{s})\,.$
(19)

The variance(error) in $\Delta\mu$ due to the radial lensing effect maybe
estimated using

$\displaystyle{{\sigma^{2}_{\Delta\mu}=\left\langle(\Delta\mu)^{2}\right\rangle_{\Omega}-\left\langle\Delta\mu\right\rangle_{\Omega}^{2},}}$
(20)

The standard deviation is obtained from
${{\sigma^{2}_{\Delta\mu}:\sigma_{\Delta\mu}=\sqrt{\sigma^{2}_{\Delta\mu}}}}$
[53]. The full sky harmonic expansion of
$\left\langle(\Delta\mu)^{2}\right\rangle_{\Omega}$ is given by

$\displaystyle{{\left\langle(\Delta\mu)^{2}\right\rangle_{\Omega}\approx\left[\frac{5}{\ln({10})}\right]^{2}\left[\left\langle\frac{\xi^{2}{{}_{\parallel}}}{\bar{d}_{A}^{2}}\right\rangle_{\Omega}\right]\,.}}$
(21)

Figure 2: This is the contribution to the Hubble residual due to the small
scale inhomogeneities along the line of sight as a function of the
cosmological redshift. For the dashed lines, we made use of the linear power
spectrum and for the thick lines, we used the halo model estimate of the
matter power spectrum. The shaded area corresponds to the range of expected
standard deviation due to small scale inhomogeneities.

The plot of the Hubble residual plus or minus the expected error from the
small scale inhomogenieties as a function of the cosmological redshift is
shown in figure 2. As expected, the effect of the small scale inhomogeneities
contribute most significantly to the monopole of the Hubble residuals in the
regions where the distance anchors live (i.e., very low redshifts). However,
the impact of the variance extends to intermediate redshifts. Figure 2 also
shows that the impact of effect of small scale inhomogeneities along the line
of sight could account for a part or majority of the observed Hubble
residuals. More detailed study would be required to account for all the
expected variance in the Hubble residual especially at high redshifts. Note
that the effects of tangential weak gravitational lensing was neglected in
this analysis, it will become important at high redshifts. Also, the improved
modelling of the peculiar velocity would be required in order to account for
the effect of the bulk flow.

###  3.3 Resolution of the supernova absolute magnitude tension

It is possible to calibrate the absolute magnitude of the Sn Ia in the Hubble
flow using the sound horizon scale at the surface of the last scattering,
$r_{\star}$, as an anchor [54]. This approach is known as the inverse distance
ladder approach. The approach assumes that the distance duality relation holds
and that the background FLRW spacetime predication for the luminosity distance
is valid on all scales [55].

Using the inverse distance ladder technique and the CMB-Planck’s constraint on
$r_{\star}$, the authors of [54] found that the absolute magnitude for
pantheon Sn Ia samples in the Hubble flow to be $M^{\rm{P18}}=-19.387\pm
0.021\,\,{\rm{mag}}$. However, using the distance estimate to NGC 4258 as
described in sub-section 3.1, the SH0ES collaboration found the absolute
magnitude for the same sample to be $M^{\rm{SH0ES}}=-19.269\pm 0.029$ [25].
Using different local distance anchors lead to a value which is consistent
with this. The difference between the use of early Universe sound horizon as
an anchor and the late Universe local distance anchors is given by
$M^{\rm{SH0ES}}-M^{\rm{P18}}=0.118\pm 0.035\,[{\rm{mag}}]$. This difference is
known as the supernova absolute magnitude tension [24].

Using the flux inverse square law $F_{d_{L}}=L/4\pi d_{L}^{2}$ which holds in
any spacetime and keeping the intrinsic luminosity, $L$ fixed, equation (11)
can be written as

$\displaystyle\left\langle\Delta m(z_{s})\right\rangle_{\Omega}$
$\displaystyle\equiv$ $\displaystyle\left\langle
m_{b}\right\rangle_{\Omega}-\bar{m}_{b}\,,$ (22) $\displaystyle=$
$\displaystyle
5\log_{10}\left[1+\left\langle\frac{\xi{{}_{\parallel}}}{\bar{d}_{A}}\right\rangle_{\Omega}-\frac{1}{2}\left\langle\frac{\xi^{2}{{}_{\parallel}}}{\bar{d}_{A}^{2}}\right\rangle_{\Omega}\right]\,,$
(23)

where $\bar{m}_{b}=-2.5\log_{10}L+5\log_{10}\bar{d}_{L}$ is the apparent
magnitude with the luminosity distance given by the background FLRW spacetime.
This is in line with the early Universe approach.

Figure 3:  The plot above shows the difference between the monopole of the
apparent magnitude and the apparent magnitude predicted by the background FLRW
spacetime as function of the luminosity distance. The dashed vertical line at
$d_{L}=1$[Mpc], indicates a difference of about $0.10$ [mag] at the
corresponding horizontal intersection. This is where the absolute magnitude of
the Sne Ia in the Hubble flow are determined. The position where the curve
goes through zero will likely disappear when the next to leading order terms
are added. The dashed and thick lines indicate the cosmological perturbation
theory and halo model predication for the matter power spectrum respectively.

Now, recall that the local determination of the supernova absolute magnitude
does not assume a model for the luminosity distance when this calibration is
done. Within our perturbation theory framework, we can calculate the monopole
of the apparent magnitude down to $1$ [Mpc]. Therefore, our claim is that our
calculation of the monopole of the apparent magnitude i.e. $\left\langle
m_{b}\right\rangle_{\Omega}$ in a perturbed FLRW spacetime gives a better
representation of what is measured than the background FLRW model on scales
where the absolute magnitude is determined. Evaluating the difference in
apparent magnitudes at 1 [Mpc] gives the measured difference in supernova
absolute magnitude between the late and early Universe approaches.

It is clear from figure 3 that by including the effect of inhomogeneities on
the apparent magnitude, the difference can easily be accounted for. In other
words, if the early Universe calibration of the supernova absolute magnitude
uses equation (10) instead of the background FLRW spacetime equivalent, the
supernova absolute magnitude tension could be resolved. We can see this by
calculating this difference explicitly using the monopole of the generalised
distance modulus

$\displaystyle\left\langle m_{b}\right\rangle_{\Omega}-\left\langle
M\right\rangle_{\Omega}^{R}$ $\displaystyle=$ $\displaystyle
5\log_{10}\bar{d}_{L}+25\,.$ (24)

We made use of the general expression for the luminosity distance
$d_{L}=\bar{d}_{L}\left[1+\xi_{{}_{\parallel}}/\bar{d}_{A}\right]$ and have
absorbed the correction due to inhomogeneities into what we call the
renormalised absolute magnitude(this corresponds to the supernova absolute
magnitude measured by the SH0ES collaboration)

$\displaystyle\left\langle M\right\rangle_{\Omega}^{R}-M_{b}$ $\displaystyle=$
$\displaystyle\frac{5}{\ln({10})}\left[\left\langle\frac{\xi{{}_{\parallel}}}{\bar{d}_{A}}\right\rangle_{\Omega}-\frac{1}{2}\left\langle\frac{\xi_{{}_{\parallel}}^{2}}{\bar{d}_{A}^{2}}\right\rangle_{\Omega}\right]_{d_{L}=1[\rm{Mpc}]}\,,$
(25) $\displaystyle\approx$
$\displaystyle{{\frac{1}{{3\ln(10)}}\frac{\sigma_{ij}\sigma^{ij}}{{H}^{2}_{0}}\bigg{|}_{d_{L}=1[\rm{Mpc}]}\approx
0.10\,[{\rm{mag}}].}}$ (26)

where $\sigma_{ij}=\partial_{i}v_{j}-\delta^{ij}\partial_{i}v_{j}/3$ is the
rate of shear deformation tensor experienced by the matter field. In order to
obtain equation (26) in terms of the shear tensor, we have analytically
continued the redshift dependence of the terms in equation (25) to zero. This
helps to appreciate the physical phenomenon at play here. That is the
spacetime around the observer is tidally deformed due to the presence of
inhomogeneous structures. Evaluating the ensemble average of the terms in
equation (26) with a smoothing scale set at 1[Mpc] gives a result consistent
with figure 3. This is also consistent with the result presented in [29] where
a non-perturbative approach was used.

###  3.4 Fitting the Alcock-Paczyński parameters

Similar to the CMB, the plasma physics of acoustic density waves before
decoupling imprints a characteristic length scale in the matter distribution.
Alcock and Paczyński [27] introduced a clever way to recover the true
separation between galaxies which then allows to use the Baryon Acoustic
Oscillation (BAO) as a standard ruler. The observed separation could be
decomposed into the radial ${\alpha}_{{}_{\parallel}}$ and the orthogonal
${\alpha}_{\perp}$ components [56]. These are the observables that are
measured by the large scale structures [57]. ${\alpha}_{{}_{\parallel}}$ and
${\alpha}_{\perp}$ are proportional to the Hubble rate and the area distance
at the mean redshift of the observation respectively. When
${\alpha}_{{}_{\parallel}}$ and ${\alpha}_{\perp}$ are interpreted in terms of
the background FLRW model

$\displaystyle{{\bar{\alpha}_{{}_{\parallel}}(z_{s})=\frac{{H}^{\rm{fid}}}{{H}}\,,\qquad{\rm{and}}\qquad\bar{\alpha}_{\perp}(z_{s})=\frac{\bar{d}_{A}}{\bar{d}^{\rm{fid}}_{A}}}}$
(27)

and evolved to today, it leads to a discrepant value for the Hubble rate. Here
${H}^{\rm{fid}}$ and $d^{\rm{fid}}_{A}$ are the fiducial Hubble rate and area
distance.

In a perturbed Universe, the radial and orthogonal distances can be written
as: $r{{}_{\parallel}}=\bar{r}{{}_{\parallel}}+\delta
r_{{}_{\parallel}}=\bar{r}{{}_{\parallel}}+\delta z/H$ and
${r}^{A}_{\perp}=\bar{r}^{A}_{\perp}+\delta r^{A}_{\perp}$, where $\delta
r_{{}_{\parallel}}$ and $\delta r^{A}_{\perp}$ are radial and tangential
perturbations due to inhomogeneities respectively. Using these, the monopole
of the generalised Alcock-Paczyński parameters in the presence of
inhomogeneities become [19]

$\displaystyle\left\langle\alpha_{{}_{\parallel}}(z_{s})\right\rangle_{\Omega}$
$\displaystyle\simeq\frac{{H}^{\rm{fid}}}{{H}}\bigg{[}1+\left\langle\frac{1}{H}\frac{\partial\delta
z}{\partial r}\right\rangle_{\Omega}+\mathcal{O}(\delta z)\bigg{]}\,,$ (28)
$\displaystyle\left\langle{\alpha}_{\perp}(z_{s})\right\rangle_{\Omega}$
$\displaystyle=\frac{\left\langle{d}_{A}\right\rangle_{\Omega}}{d^{\rm{fid}}_{A}}\,.$
(29)

Comparing equations (28) and (29) to their corresponding background FLRW limit
(equation (27)) and matching the corresponding terms
($\left\langle\alpha_{{}_{\parallel}}\right\rangle_{\Omega}=\bar{\alpha}_{{}_{\parallel}}$
and $\left\langle\alpha_{\perp}\right\rangle_{\Omega}=\bar{\alpha}_{\perp})$,
we find that the inferred Hubble rate from $\alpha_{{}_{\parallel}}$

$\displaystyle\mathcal{H}\,^{\alpha_{\parallel}}(z_{s})$
$\displaystyle\approx\mathcal{H}\,\left[1-\frac{1}{\mathcal{H}\,^{2}_{s}}\left[\left\langle\partial^{2}_{\|}{v_{s}}\partial^{2}_{\parallel}{v_{s}}\right\rangle_{\Omega}+\left\langle\partial_{\|}{v_{s}}\partial^{3}_{\parallel}{v_{s}}\right\rangle_{\Omega}-\left\langle\partial_{\|}v_{o}\partial^{3}_{\parallel}{v_{s}}\right\rangle_{\Omega}\right]\right]\,.$
(30)

Expanding the correction in spherical harmonics, the leading order
approximation at low redshift is given by

$\displaystyle\frac{\mathcal{H}\,^{\alpha_{\parallel}}(z_{s})}{\mathcal{H}\,}-1$
$\displaystyle\approx-\sum_{\ell=2}^{\ell_{\rm{max}}}(2\ell+1)\int\frac{{\rm{d}}k}{2\pi^{2}}k^{2}P_{m}(k)(f(z_{s})D(z_{s}))^{2}$
(31)
$\displaystyle\times\bigg{[}j^{\prime\prime}_{\ell}(kr_{s})j^{\prime\prime}_{\ell}(kr_{s})+\bigg{(}j^{\prime}_{\ell}(kr_{s})-\frac{\mathcal{H}\,_{o}}{\mathcal{H}\,_{s}}\frac{D_{o}f_{o}}{f(z_{s})D(z_{s})}j^{\prime}_{\ell}(kr_{o})\bigg{)}j^{\prime\prime\prime}_{\ell}(kr_{s})\bigg{]}\,.$

We show a plot of the fractional difference between the Hubble rate that would
be inferred from the analysis of radial component of the BAO and the global
Hubble rate as function of the cosmological redshift in figure 4. The
difference between the effective Hubble rate and the global Hubble rate begin
to emerge at about the redshift of $z\leq 0.01$.

Figure 4: We show the fractional difference between the Hubble rate inferred
from the radial component of BAO and the background FLRW spacetime
prediction.The vertical dashed line indicates the position of $z_{\rm{cut}}$.
The dashed and thick lines indicate the cosmological perturbation theory and
halo model predication for the matter power spectrum respectively.

When the expression in equation (30) is analytical continued to $z=0$, i.e.,
zero redshift limit both $H^{\alpha_{\parallel}}$ and ${H_{0}^{{d}_{A}}}$ give
the same result (see [19, 29] for further details).

$\displaystyle\frac{({H_{0}^{{d}_{A}}}-{H}_{0})}{{H}_{0}}=-\lim_{z\to
0}\left\langle\frac{\xi{{}_{\parallel}}}{\bar{d}_{A}}\right\rangle_{\Omega}\approx\frac{2}{15}\left\langle\frac{\sigma_{ij}\sigma^{ij}}{H^{2}_{0}}\right\rangle_{\Omega}=\frac{H^{\alpha_{\parallel}}_{0}-H_{0}}{H_{0}}\,.$
(32)

Evaluating the ensemble average of equation (32) at 1[Mpc] gives about 10%
difference. This corresponds to the amount required to solve the Hubble
tension. Equation (32) shows that the neighbourhood of the observer is curved
and photons travel longer distance to get the observer.

##  4 Breakdown of the expanding coordinate system

On the background FLRW spacetime one can easily extrapolate the magnitude-
redshift relation down to zero redshift. In the presence of virialised
structures (our local group), this is no longer possible [29]. This
immediately constrains the map between the cosmological redshift and the
monopole of the area distance shown in figure 1. It is possible to estimate
the length scale where the transition occurs given a halo model. There are
also pieces of evidence that the transition length scale have been detected in
the analysis of the large scale peculiar velocity survey, in this case, it is
called the radius of the zero-velocity surface [58, 59].

Given a geodesic deviation equation for time-like geodesics with the initial
conditions at $\tau_{\rm{ini}}=0$, zero-velocity surface is a conjugate point
or caustic [10, 29]. The appearance of a conjugate point indicates a breakdown
of the global expansion coordinate chart [60]. Within the halo model, we can
estimate this scale by calculating the comoving radius where the trace of the
covariant derivative of the observer 4-velocity vanishes:
$R_{0}=-({c}/{3H_{0}})({\rm{d}\ln\rho}/{\rm{d}\ln r})$, where $c$ is the speed
of light and $\rho$ is the mass-density. It is straightforward to calculate
${\rm{d}\ln\rho}/{\rm{d}\ln r}$ using any of the halo profiles [61]. There are
observational constraints for our local group, it is given by
$R_{0}\sim(0.95-1.05)[{\rm{Mpc}}]$ [58, 62, 59]. In terms of the cosmological
redshift, it corresponds to about $z_{\rm{cut}}=2.4\times 10^{-4}$.

The existence of $z_{\rm{cut}}$ implies that when the distance between two
points in the universe is less than the radius of the zero-velocity surface
($|{\bm{x}}_{2}-{\bm{x}}_{1}|<R_{0}$), the global FLRW model or the
cosmological perturbation theory on an FLRW background cannot give the best
fit model. The discussion on how to calculate the distance to the Milky-way
cepheids or the Large Magellanic cloud and the impact of the heliocentric
proper motion on the area distance would be presented elsewhere.

##  5 Conclusions

This paper shows that the area distance at very low redshift limit includes a
non-negligible correction due to nonlinear structure formation to the
background FLRW spacetime predication. The amplitude of the correction is
about 10% at $z=z_{\rm{cut}}=2.4\times 10^{-4}$. The nonlinear correction is
due to the effect of radial lensing or the Doppler lensing, i.e. a small scale
lensing effects by galaxies orbiting about the centre of the massive clusters.
The leading order part of this effect is due to a correlation between observer
peculiar velocity and redshift space distortion term at the source. This
effect manifests as a displacement in the apparent position of a source with
respect to the peculiar velocity of the observer. The implications of this
effects on the apparent magnitude of the Sn Ia was studied in detail in sub-
section 3.3. The result show that the Doppler lensing effects impact the
calibration of the absolute magnitude of the Sn Ia in the Hubble flow and that
it could explain the supernova absolute magnitude tension without the need for
any exotic early or late dark energy component.

Although the impact of the heliocentric observer relative velocity was
neglected because the cosmological perturbation theory used is not valid in
the multi-stream region. This is the major reason the analysis reported here
focused on the coordinate independent contribution of the effects of small
scale inhomogeneities on the apparent magnitude of the Sn Ia. However, it is
important to note that the impact or correction to the supernova absolute
magnitude will remain unchanged when the impact of the heliocentric observer
relative velocity is included. In cosmological analysis, the impact of the
heliocentric observer relative velocity could be calculated and subtracted off
from the observed signal.

Using the expression for the area distance that includes the radial/Doppler
lensing effects, the generalised Alcock-Paczyński parameters could be derived,
in the paper it was shown that it is possible to reconcile the inferred Hubble
rate from the BAO measurements with the Hubble rate from the analysis of the
apparent magnitude of the SN Ia by the SH0ES collaboration without any need
for exotic late/early dark energy model. Including the effect of small scale
non-linearity does not lead to any difference between the TRGB and the SH0ES
calibration of the absolute magnitude of the Sn Ia in the Hubble flow.

Finally, there are several other proposed solutions to Hubble or Sn Ia
absolute magnitude tension. These solutions usually invoke exotic early or
late dark energy model [63, 64] (see [65, 66] for a list of all possible
models within this framework). There are also frame-dependent dark energy
model approach [67] (this approach explains the SH0ES result but gives a large
value of the Hubble constant from the BAO analysis). There is also proposals
that explains the Hubble tension as a manifestation of the quantum measurement
uncertainties [68], or the impact of the evolving gravitational constant
leading to a modification of gravity at late-time [69, 70, 71]. All these
approaches except the frame dependent dark energy(frame dependent dark energy
model breaks 4D diffeomorphism invariance) assume that the FLRW background
spacetime is valid on all scales and at all times. One unique about our
approach is that it explains the Hubble tension within the standard model by
simply including the effect of small scale perturbations. It will be
interesting to see whether there is a projection of the hyperconical universe
that explains both the dark energy and the Hubble tension better than the
$\Lambda$CDM universe [72, 73].

## Acknowledgement

I benefited immensely from the works of G.F.R. Ellis on this topic. OU is
supported by the UK Science & Technology Facilities Council (STFC)
Consolidated Grants Grant ST/S000550/1 and the South African Square Kilometre
Array Project. For the purpose of open access, the author(s) has applied a
Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript
version arising. Most of the tensor algebraic computations in this paper were
done with the tensor algebra software xPand [74] and xTensor [75].

## Data Availability

Supporting research data are available on reasonable request from the author.

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# The runaway nature and origin of $\alpha$ Crucis system

M. Torosyan<EMAIL_ADDRESS>Corresponding author Astronomy
Department, Yerevan State University, 1 Alex Manoogian Street, 0025 Yerevan,
Armenia N. Azatyan<EMAIL_ADDRESS>Byurakan Astrophysical Observatory,
0213 Aragatsotn Province, Armenia  E. Nikoghosyan<EMAIL_ADDRESS>Byurakan
Astrophysical Observatory, 0213 Aragatsotn Province, Armenia  A. Samsonyan
<EMAIL_ADDRESS>Byurakan Astrophysical Observatory, 0213 Aragatsotn
Province, Armenia  D. Andreasyan<EMAIL_ADDRESS>Byurakan
Astrophysical Observatory, 0213 Aragatsotn Province, Armenia

###### Abstract

Massive stars are always the focus of astronomical research and a significant
part of them (10–20%) moves in space at a high (supersonic) velocity. This
paper presents the results of a study of the $\alpha$ Crucis system, located
at $\sim$114 pc distance from the Sun, with an observed bow shock around it.
We used data and images from the Gaia and WISE space telescopes. The
coordinates, distance, and proper motion of the $\alpha$ Crucis system were
used to determine its space velocity. We managed to find a stellar cluster to
which the $\alpha$ Crucis system belongs, that is, it has not been ejected
from its parent cluster, but is moving in space together with other members of
the cluster. The $\alpha$ Crucis system has a velocity of $\sim$1.3 km/s
relative to the star cluster. The geometric parameters of the bow shock are
compatible with other known bow shocks. The bow shock is unaligned, i.e., most
likely interstellar medium (ISM) large-scale motions are responsible for the
resulting bow shock, which is further evidence that the $\alpha$ Crucis system
is not runaway in nature.

_Keywords: Infrared: ISM – Open clusters and associations: Scorpius-Centaurus
– Stars: individual: HD 108248, HD 108249, and HD 108250_

##  1 Introduction

The massive stars within the Milky Way are not randomly dispersed;
approximately 80% of the O-type stars are part of OB associations located in
the galaxy’s spiral arms (Brown et al., 1999). The kinematic characteristics
of the remaining 20%, which belong to the field population, indicate that
these O stars are runaways—originating from OB associations but subsequently
escaping (Blaauw, 1993). Two primary scenarios are widely accepted to explain
the presence and high velocities of runaway stars: (1) the supernova explosion
of a companion star within a massive binary system (Blaauw, 1961), and (2) the
dynamic ejection from a young star cluster during its early developmental
stages (Poveda et al., 1967). Identifying the original OB association of a
runaway star is crucial, as it provides insights into the star’s evolutionary
history. By knowing the proper motions of both the runaway star and its parent
OB association, the kinematic age of the runaway star can be determined.

Another method of searching for high-velocity runaway star candidates is to
bow-shocks associated with these objects, which are generated by the
interaction of the star’s wind with the interstellar medium (ISM) (e.g., van
Buren & McCray, 1988, Peri et al., 2012, Azatyan et al. submited). Therefore,
OB runaways can be used also as probes of the ISM. As rule, bow shocks emit
strong radiation in the mid-infrared range (e.g., Gvaramadze et al., 2011).

In our paper, we have chosen the $\alpha$ Crucis system as the target of
study. The $\alpha$ Crucis system consists of three stars: HD 108248, HD
108249, and HD 108250. HD 108248 and HD 108249 are known as a1 Cru and a2 Cru
with spectral types B0.5IV and B1V, respectively (Houk & Cowley, 1975, Reed,
2005). HD 108250 is also known as $\alpha$ Cru C with spectral type B4V
(Tokovinin et al., 1999, Reed, 2005). Figure 1 shows a combination of 22
$\mu$m (red), 12 $\mu$m (green), and 3.4 $\mu$m (blue) images of the $\alpha$
Crucis system and the bow shock detected around it. The three stars of the
system are indicated by white arrows. The HD 108248 and HD 108249 pair is
significantly brighter compared to HD 108250. HD 108250 is 90 arcseconds away
from the pair HD 108248 and HD 108249 and has the same space motion,
suggesting they may be gravitationally bound (Shatsky & Tokovinin, 2002).
Rizzuto et al. (2011) calculated that this system, including HD 108250, is a
member of the Lower Centaurus–Crux subgroup in the Scorpius–Centaurus
association with a 66% probability. The radial velocities of stars HD 108248,
HD 108249, and HD 108250 are 11.2$\pm$2, 0.6$\pm$2, and 27$\pm$5 km/s,
respectively (Wilson, 1953). The parallaxes and proper motions of the three
stars in the system are almost the same. The parallaxes and proper motions of
HD 108248 and HD 108249 are 10.13$\pm$0.50 mas, -35.83$\pm$0.47 mas/yr, and
-14.86$\pm$0.43 mas/yr, respectively (van Leeuwen, 2007). The third data
release of Gaia observatory (Gaia DR3) provides reliable information only on
the star HD 108250. The parallax is 9.3964$\pm$0.1289 mas (Gaia Collaboration,
2022), giving a distance of $\sim$114 pc from the Sun. The fractional parallax
uncertainty (parallax/parallax_error) is $\sim$73\. The proper motion in RA is
-39.548$\pm$0.117 mas/yr and in Dec is -13.760$\pm$0.132 mas/yr (Gaia
Collaboration, 2022). The Renormalized UnitWeight Error (ruwe) coefficient is
1.316, indicating good quality astrometric solutions for this star.

Although the star HD 108250 is at the center of symmetry of the observed bow
shock, this bow shock has previously been associated with the HD 108248 and HD
108249 pair (Bodensteiner et al., 2018). Since these three stars are
physically connected, the observed bow shock is most likely associated with
all three. The aim of this work is to elucidate the runaway nature of the
$\alpha$ Crucis system, find the parent star cluster, calculate the relative
velocity of the system with respect to the parent star cluster, and determine
the kinematic age. Another goal is to study the properties of the observed bow
shock.

We have organised the paper as follows: Section 2 describes the data used;
Section 3 details the results along with discussions, covering the search for
a parent star cluster, the determination of the relative velocity of the
$\alpha$ Crucis system, and the characteristics of the observed bow shock; the
summarising of the results is in Section 4.

Figure 1: Colour-composite image of $\alpha$ Crucis system and the bow shock
around it: 3.4 $\mu$m (blue), 12 $\mu$m (green), and 22 $\mu$m (red) bands.
The three stars of the system are indicated by white arrows.

##  2 Data used

Since the bow shock observed around the $\alpha$ Crucis system radiates in the
mid-infrared range, we used the Wide-field Infrared Survey Explorer (WISE;
Wright et al., 2010) images at 3.4, 4.6, 12, and 22 $\mu$m wavelengths. For
kinematic parameters, we used the Gaia Data Release 3 (Gaia DR3; Gaia
Collaboration, 2022) catalog data.

##  3 Results and discussion

###  3.1 Searching for a parent star cluster

We searched for members of candidate parent stellar cluster/association of HD
108250 in the Gaia DR3 catalog in the range $(l,b)=(291.0^{\circ}\text{ to
}320.0^{\circ},-3.5^{\circ}\text{ to }+3.5^{\circ})$. Since Gaia DR3 only
provides reliable data for the star HD 108250, we assume that the obtained
results are correct for the entire system. We apply a series of filters on the
data, which are briefly described below:

  * •

We excluded sources with ruwe greater than 1.4, which probably indicates poor
astrometric solutions for these sources.

  * •

We excluded sources for which the ratio of parallax to parallax uncertainty
(parallax/parallax_error) was less than 5.

  * •

We excluded sources for which the parallax is not in the range of 8.3 to 10.3
mas.

As a result, 3959 sources were obtained. The obtained sample were reflected by
Tool for OPerations on Catalogs And Table (TOPCAT; Pössel, 2020) through the
distribution of proper motions. The left panel of Figure 2 shows the
distribution of the proper motions of objects obtained based on the search in
the direction of the star HD 108250. The blue arrow indicates a star cluster
with its proper motions clearly separated from the field. The star HD 108250,
with its measured proper motions and distance, most likely belongs to that
cluster. On the right panel of Figure 2 is the histogram of object distances.
The red one is the histogram of all objects, and the blue one – the cluster
members, which again stands out clearly from the field stars and has a maximum
at the distance where the $\alpha$ Crucis system is located. There are 93
members of the cluster. The average parallax of the cluster members is
$9.33\pm 0.25$ mas, which means that the cluster is located at $107\pm 3$ pc
distance. The average proper motion of members in the direction of RA is
$-37.91\pm 0.89$ mas/yr, and in the direction of Dec is $-10.62\pm 1.41$
mas/yr. Performing a comparison between the parameters of the star HD 108250
and the members of the cluster, namely the parallax (and hence the distance)
and proper motions, it is clear that the star HD 108250 or the $\alpha$ Crucis
system has not escaped from its parent cluster and is gravitationally bound to
the other members, i.e., together they are moving in space with almost the
same space velocity. The cluster is one of the subgroups in the
Scorpius–Centaurus association, and these subgroups are located quite far from
each other, but move as a whole (Blaauw, 1964, Goldman et al., 2018).

Figure 3 shows the distribution of cluster members in the field. Members are
marked with blue circles. The green arrow shows the position of the bow shock
observed around the $\alpha$ Crucis system. As can be seen, the members have a
large dispersion in the field and the $\alpha$ Crucis system is located in the
peripheral part of the star cluster.

Figure 2: Left: Distribution of proper motions of the objects obtained based
on the search in the direction of the star HD 108250. Red circles are objects
obtained based on the search. Blue arrow shows the star cluster clearly
separated from the field. Right: Histogram of object distances. The red
histogram is for all objects, and the blue one is for the cluster members.
Figure 3: Distribution of the cluster members in the field on WISE W4 (22
$\mu$m) image. The members are marked with blue circles. The green arrow shows
the position of the bow shock observed around the $\alpha$ Crucis system.

###  3.2 Relative velocity of the $\alpha$ Crucis system

One of the conditions for the runaway nature of the $\alpha$ Crucis system is
its high space velocity relative to its parent star cluster. To determine the
space or peculiar velocity relative to the parent cluster, the coordinates,
distance, proper motions, and apparent radial velocities of HD 108250 and the
parent cluster members are required. Since the radial velocities in Gaia DR3
are known for only a part of the cluster members, i.e., the data are
incomplete, we therefore set the radial velocity component equal to zero and
calculated the relative transverse velocity of HD 108250. Therefore, the
peculiar velocity of the star HD 108250 or the $\alpha$ Crucis system relative
to the parent cluster is approximately 1.3 km/s. This finding supports the
conclusion that the $\alpha$ Crucis system, despite the bow shock observed
around it, is not a runaway system, since the relative velocity of runaway
stars exceeds 30 km/s (Blaauw, 1961).

###  3.3 Bow shocks

Runaway stars move in the ISM at velocities that exceed the speed of sound.
Their stellar wind hits the ISM and produces so-called bow shocks (Wilkin,
1996). Due to the stellar wind, the dust is heated and re-radiates at infrared
wavelengths. Bow shocks have also been observed in the optical range (e.g.,
Brown & Bomans, 2005) and in a few cases at radio wavelengths (Benaglia et
al., 2010). The $\alpha$ Crucis system and the surrounding bow shock is well
resolved in the mid-infrared wavelength range. Figure 4 shows the bow shock
around the system in the WISE 22 $\mu$m channel. The open red circle shows the
position of HD 108250, and the white arrow shows the direction of motion of
the $\alpha$ Crucis system. As we see, the observed bow shock is unaligned,
that is, the apparent motion of the system does not coincide with the angular
opening of the nebula (bow shock). This can be explained by large-scale
motions in the ISM, which may also be responsible for the high relative
velocity between the star and the ISM (Bodensteiner et al., 2018). This is
probably why the bright pair HD 108248 and HD 108249 in the $\alpha$ Crucis
system, with which the observed bow shock was previously associated, is offset
from the center of symmetry of the bow shock, which is usually observed in
unaligned bow shocks (Wilkin, 2000). The fact that the bow shock is unaligned
probably proves that the $\alpha$ Crucis system is not runaway.

To compare the properties of the observed bow shock with other known bow
shocks, we measured its geometrical parameters, such as the spatial length
$l=17.2\pm 0.57$ pc, the width $w=4.2\pm 0.14$ pc, and the distance from the
star to the center of the bow shock (standoff distance) $R=5.5\pm 0.18$ pc.
Table 4 presented in Peri et al. (2012) paper contains the same parameters for
a number of other bow shocks. We compared them with our results and concluded
that the parameters of the bow shock surrounding the $\alpha$ Crucis system
are compatible with the parameters of those already known bow shocks.

Figure 4: The bow shock observed around the $\alpha$ Crucis system on WISE W4
(22 $\mu$m) image. Open red circle shows the location of HD 108250 star. White
arrow indicates direction of motion of the $\alpha$ Crucis system.

##  4 Conclusion

Within the framework of this paper, the results of the study of the $\alpha$
Crucis system at a distance of $\sim$114 pc and the bow shock observed around
it are presented. We have discovered a star cluster that is clearly separated
from the stars of the field by their proper motions and distances, the number
of members of which is 93. The cluster is one of the subgroups in the
Scorpius-Centaurus association. The $\alpha$ Crucis system, with its proper
motion and distance, most likely belongs to that cluster. We concluded that
the $\alpha$ Crucis system has not escaped from its parent star cluster and is
gravitationally bound to the other members, that is, they move together in
space, having almost the same space velocity. The peculiar velocity of the
$\alpha$ Crucis system with respect to the star cluster is $\sim$1.3 km/s.

We also studied the parameters of the bow shock observed in the mid-infrared
wavelength around the $\alpha$ Crucis system. By measuring the geometrical
parameters of this bow shock, we concluded that its properties are similar to
other known bow shocks. The apparent motion of the $\alpha$ Crucis system does
not match the angular opening of the nebula (bow shock), i.e., the observed
bow shock is unaligned. This means that the bow shock was formed because of
the large-scale motions in the ISM, not the velocity of the $\alpha$ Crucis
system.

Taking the above results into account, we note as the main conclusion that the
$\alpha$ Crucis system, despite the bow shock observed around it, is not
runaway.

## Acknowledgements

This work was partially supported by a research grant number No 21AG-1C044
from Higher Education and Science Committee of Ministry of Education, Science,
Culture and Sport RA. This work presents results from the European Space
Agency (ESA) space mission Gaia. Gaia data are being processed by the Gaia
Data Processing and Analysis Consortium (DPAC). Funding for the DPAC is
provided by national institutions, in particular the institutions
participating in the Gaia MultiLateral Agreement (MLA). This publication also
makes use of data products from the Wide-field Infrared Survey Explorer, which
is a joint project of the University of California, Los Angeles, and the Jet
Propulsion Laboratory/California Institute of Technology, funded by the
National Aeronautics and Space Administration.

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