# Average bolometric corrections and optical to X-ray flux measurements as a
function of accretion rate for X-ray binaries

K. Anastasopoulou,1,2,3,4 A. Zezas,2,3,4 J. F. Steiner,4 P. Reig2,3
1 INAF-Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate
(LC), Italy
2 Institute of Astrophysics, Foundation for Research and Technology-Hellas,
GR-71110 Heraklion,Greece
3 Department of Physics, University of Crete, Herakleio, Greece
4 Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, USA
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

In this paper we use an RXTE library of spectral models from 10 black-hole and
9 pulsar X-ray binaries, as well as model spectra available in the literature
from 13 extra-galactic Ultra-luminous X-ray sources (ULXs). We compute average
bolometric corrections (BC=$\mathrm{L_{band}/L_{bol}}$) for our sample as a
function of different accretion rates. We notice the same behaviour between
black-hole and pulsars BCs only when ULX pulsars are included. These
measurements provide a picture of the energetics of the accretion flow for an
X-ray binary based solely on its observed luminosity in a given band. Moreover
it can be a powerful tool in X-ray binary population synthesis models.
Furthermore we calculate the X-ray (2-10 keV) to optical (V-band) flux ratios
at different Eddington ratios for the black-hole X-ray binaries in our sample.
This provides a metric of the maximum contribution of the disk to the optical
emission of a binary system and better constraints on its nature (donor type
etc). We find that the optical to X-ray flux ratio shows very little variation
as a function of accretion rate, but testing for different disk geometries
scenarios we find that the optical contribution of the disk increases as the
$p$ value decreases ($T(r)\sim r^{-p}$). Moreover observational data are in
agreement with a thicker disk scenario ($p<0.65$), which could also possibly
explain the lack of observed high-inclination systems.

###### keywords:

X-rays: binaries – stars: black holes – pulsars: general

††pubyear: 2020††pagerange: Average bolometric corrections and optical to
X-ray flux measurements as a function of accretion rate for X-ray binaries–A

##  1 Introduction

Studies of X-ray binaries (XRBs) are generally conducted in the limited energy
pass-bands provided by the available X-ray observatories. The majority of
these observatories (e.g. Chandra, Swift, XMM-Newton, NICER, MAXI) cover the
$\sim 0.1$ – $\sim 10.0$ keV band, which encompasses a significant part of the
thermal emission of the accretion disk. Therefore these observations have
provided important information on the physics of the accretion disks in XRBs,
the absorption by local and/or intervening material, and of course their
demographics in different environments (e.g. Remillard & McClintock, 2006;
Fabbiano, 2006; Muno et al., 2009; Coleiro & Chaty, 2013; Haberl & Sturm,
2016; Fornasini et al., 2017). However, as demonstrated by X-ray observations
extending beyond 10.0 keV (e.g. with the RXTE, INTEGRAL, NuSTAR observatories)
the $0.1$–$10.0$ keV band probes a small fraction of the bolometric output of
XRBs, particularly in the intermediate and hard accretion states (e.g.
Remillard & McClintock, 2006; Done, Gierliński, & Kubota, 2007). In particular
it does not trace the low-energy tail of the accretion disk thermal emission
and the hard X-ray tail of the Comptonized hard spectral component.

The bolometric luminosity ($L_{bol}$; 0.02-200 keV) is a defining property of
XRBs since it provides a direct picture of the energetics of the accretion
flow which is inextricably linked to the accretion disk structure and its
impact in the surrounding medium (e.g. Done, Gierliński, & Kubota, 2007). The
spectral-energy distribution changes significantly with the accretion state of
the XRB, presumably in part owing to the associated changes in the geometry of
the accretion flow. Despite its importance, it is not easy to measure the
bolometric luminosity of an XRB since that would require simultaneous
observations from multiple observatories over a wide wavelength range. We aim
to overcome this limitation by determining bolometric corrections
(BCs=$\rm{L_{band}/L_{bol}}$) appropriate for XRBs of different compact object
types, accretion rates, or accretion states. Based on these BCs one could
estimate the bolometric luminosity of any XRB based on its observed luminosity
in a given band and its accretion rate. BCs are also crucial for comparing
predictions for the demographics or luminosity distribution of a given X-ray
binary population (e.g. from populations synthesis models Belczynski et al.,
2008; Fragos et al., 2013). These models predict the accretion rate for a
given X-ray binary which is then converted to an X-ray luminosity in a given
energy band. This conversion depends critically on the spectrum of the
accretion flow, which can be parameterized in terms of a bolometric
correction.

The characterization of the donor stars of XRBs is a key component for
understanding their nature and disentangling the contribution of the accretion
disk in the observed optical emission. The X-ray to optical flux ratio in
particular has been used for the identification of high-mass XRBs (HMXBs) in
nearby galaxies when optical spectroscopic observations are not available (e.g
Antoniou et al., 2009). Similarly, extrapolation of the accretion disk
emission to the optical or UV band has been used as a way to estimate its
contribution in the measured multi-wavelength emission of their counterparts
and the more accurate characterisation of the donor star (e.g. Kaaret, Ward, &
Zezas, 2004; Kaaret, Feng, & Roberts, 2017). Moreover van Paradijs &
McClintock (1994) found that the contribution of the accretion disk to the
optical emission of an X-ray binary can be quite important in the case of low-
mass XRBs (LMXBs). They studied 18 LMXBs and found that the absolute visual
magnitude ($M{{}_{V}}$) originating from their disks ranges from 5 to -5,
depending on the X-ray luminosity and the outer radius of the accretion disk,
affecting significantly the inferred spectral type of donor stars.

However, distinguishing between accretion disk and donor star optical emission
is rather complex. For example, Kaaret, Ward, & Zezas (2004) found a spectral
type of O4V-B3Ib for the ULX counterpart in Holmberg II but this result is
true only if there is no significant contribution of optical light from the
accretion disk. For ULX NGC 1313 X2 Pakull, Grisé, & Motch (2006) measured a B
type donor star (using the $\Sigma-M{{}_{V}}$ relation of van Paradijs &
McClintock (1994)), while Liu et al. (2007) found a spectral type of O7V
stating that the contribution of the accretion disk is insignificant, but
agreed with Pakull, Grisé, & Motch (2006) in scenarios where they consider the
contribution of disk important.

In HMXBs the relative contribution of the disk to the optical emission of the
system is expected to be small and mainly originating from the donor star
(e.g. van den Heuvel & Heise, 1972; Treves et al., 1980), In low-mass XRBs
(LMXBs) the relative contribution from the disk is considered to be the
dominant emission process in the soft state, while in the hard state the
coronal emission dominates the X-ray signal, and a small contribution from the
donor star and the jet is possible. For example, Russell et al. (2006) studied
quasi-simultaneous optical, near-infrared (NIR), and X-ray observations of 33
XRBs in order to estimate the contributions of various emission processes
(disk, donor star, jet), in those sources. They found that for black-hole XRBs
(BHXBs) X-ray reprocessing from the disk is the dominant emission process in
the optical, while the jet dominates ($\sim$90%) in the NIR and the hard
state. Moreover X-ray reprocessing from the disk dominates the optical and NIR
in neutron star XRBs with possible contribution from the jet only at high
luminosities. Therefore a reliable measurement of the optical luminosity (i.e
spectral type) of the donor star, unless the XRB is on quiescent, requires the
exclusion of any contamination from the accretion disk. Since observationally
this is very challenging, a theoretical calculation of the fraction of optical
to X-ray flux ($\rm{\xi=f_{opt}/f_{x}}$) in a given band will make possible
the calculation of the optical flux of the accretion disk only with the
knowledge of its X-ray flux which could be then used in any study of the
optical emission of the XRBs.

In this paper we use $\sim$10000 model spectra of young XRBs (pulsars; black-
holes; $<$100 Myr) and create prescriptions for calculating average BCs as
well as the average $\xi$ values for the accretion disk as a function of the
accretion rate. In Section 2 we present the data used in this study as well as
the analysis procedure. We discuss our results and their implications in other
studies in Section 3. Finally in Section 4 we summarise our findings.

##  2 Data

This paper focuses on the analysis of X-ray spectral models of 10 Galactic
BHXBs, 9 pulsar XRBs and 13 extra-galactic ultra-luminous X-ray sources
(ULXs). The scope is to understand the average emission properties (BCs,
$\xi$) of XRBs at different accretion rates, and for that reason we selected
XRBs with reliable distance and mass measurements. Detailed information of
known physical properties of the selected sample of BH, pulsar XRBs, and ULXs
and their corresponding references are reported in Tables 1, 2, and 3
respectively. For pulsar XRBs as well as for confirmed ULX pulsars (ULXps), we
assume a mass of 1.4 M⊙. For the rest of the ULXs we either assume that they
are all neutron stars with a mass of 1.4 M⊙, or BHs with a mass of
$9.1\,M_{\odot}$, which is the average mass of the Galactic BHs included in
our sample (see Table 1).

###  2.1 The spectral models

####  2.1.1 Black-hole X-ray binaries

In this work we used the BH XRB spectral models (Table 1), for 10 out of the
29 stellar-mass BH and BH candidates (total 15,000 spectra), reported in
Steiner et al. (2016), and collected over the 16 years of Rossi X-ray Timing
Explorer (RXTE) operations. The selection of the 10 systems was based on the
existence of reliable distance and mass measurements. The spectra were fitted
with a simplistic spectral model of phabs$\times$[smedge(simpl$\otimes$
diskbb)+Gauss] (hereafter Model 1), where phabs represents the absorption
along the line of sight, diskbb and simpl the disk and the Compton components
respectively, smedge the relativistically broadened Fe–K absorption edge, and
Gauss the reflection generated by illumination of the cold disk by the power-
law component” which is used to fit a relativistically smeared Fe K$\alpha$
line, or in some cases the absorption due to an outflow. Moreover, we used
three other spectral models just for comparison purposes in our analysis.
These spectral models are phabs$\times$[smedge(simplcut$\otimes$
diskbb)+Gauss] (hereafter Model 2), where simplcut is a Compton component with
a cut off at high energies, phabs$\times$[smedge(simpl$\otimes$ diskbb)+laor]
(hereafter Model 3), where laor represents a relativistically broadened Fe-
line fluoresced from the disk by coronal illumination and
phabs$\times$[(simpl$\otimes$ diskbb)+Gauss] (hereafter Model 4).

####  2.1.2 Pulsar X-ray binaries

Accreting pulsars is a dominant component of the HMXB populations (e.g. Reig,
2011). In particular, HMXBs with a main-sequence companion – the so-called
Be/X-ray binaries– represent the most numerous group of X-ray pulsars (Reig,
2011; Paul & Naik, 2011). They are hard X-ray transient systems that remain
most of their lives in quiescence. Occasionally they undergo giant X-ray
outbursts. When this occurs, the X-ray luminosity changes by 3 or 4 orders of
magnitude, reaching Eddington values. Because of these large amplitude
variations in luminosity, BeXBs represent excellent laboratories to test
accretion models. In this work, we analysed 9 accreting pulsars. These are all
the systems that have been through a giant outburst (type II) during the RXTE
life time, and also have reliable distance measurements. The data were
retrieved from the RXTE archive and covered an X-ray outburst. The broadband
spectra were fitted with a simple phenomenological model consisting of an
absorbed power-law model modified at high energies by an exponential cutoff,
phabs$\times$cutoffpl. The discrete components (spectral lines) to account for
Fe K fluorescence and cyclotron resonant scattering features when present were
fitted with a narrow Gaussian (Gauss), and Lorentzian (cyclabs) profiles,
respectively. The continuum fitted with a power-law leaves residuals in the
10–20 keV energy range in most cases. We added a broad Gaussian in emission to
account for these residuals (see e.g. Klochkov et al., 2008).

Table 1: Black-hole X-ray binaries

Name | Distance | Reference | Mass | Reference | Spectral Type | $\mathrm{N_{models}}$
---|---|---|---|---|---|---
| kpc |  | $M_{\odot}$ |  |  |
XTE J1118+480 | $4.16_{-2.12}^{+3.91}$ | G19 | 6.9-8.2 | K13a | K7-M1V | 124
GRO J1655-40 | $3.74\pm 0.50$ | G19 | $6.6\pm 0.5$ | SH03 | F6IV | 987
XTE J1550-564 | $4.40\pm 0.50$ | O02, O11 | $9.1\pm 0.6$ | O11 | K3III | 517
GRS 1915+105 | $8.60\pm 2.00$ | G19 | $12.5\pm 2.0$ | R14 | K1-5III | 2566
V4641 Sgr | $6.24_{-1.34}^{+2.04}$ | G19 | $6.4\pm 0.6$ | M14 | B9III | 94
4U 1543-475 | $8.60_{-2.46}^{+3.84}$ | G19 | $8.4-10.4$ | O98, O03 | A2V | 130
MAXI J1659-152 | $8.60\pm 3.70$ | K13b |  $6.0-8.0$111for a distance of 8.6 kpc | Y12 | M5V | 71
Cyg X-1 | $2.38_{-0.17}^{+0.20}$ | G19 | $14.8\pm 1.0$ | T14 | 09.7Iab | 2446
LMC X-1 | $48.10\pm 2.20$ | S14 | $10.9\pm 1.5$ | O09 | O8 | 704
LMC X-3 | $48.10\pm 2.20$ | S14 | $7.0\pm 0.6$ | O14 | B2.5V | 1598

References. G19—Gandhi et al. (2019), K13a–Khargharia et al. (2013),
K13b—Kuulkers et al. (2013), M14—MacDonald et al. (2014) ,O98—Orosz et al.
(1998), O02—Orosz et al. (2002), O03—Orosz (2003), O09—Orosz et al. (2009),
O11—Orosz et al. (2011), O14—Orosz et al. (2014), R14—Reid et al. (2014)
SH03—Shahbaz (2003), S14—Steiner et al. (2014b), T14—Tomsick et al. (2014),
Y12—Yamaoka et al. (2012), Spectral types have been adopted from BlackCAT
catalogue (Corral-Santana et al., 2016) and references therein.

Table 2: Pulsar X-ray binaries

Name | Distance | Spectral type | Reference | $\mathrm{N_{models}}$ | $E(B-V)$ | $\rm{N_{H}}$
---|---|---|---|---|---|---
| kpc |  |  |  |  | $10^{22}\ cm^{-2}$
4U 0115+63 | 7.19${}_{-1.09}^{+1.47}$ | B0.2Ve | PF15 | 26 | 1.33 | 0.91
1A 0535+262 | 2.12${}_{-0.20}^{+0.25}$ | O9.7IIIe | PF15 | 31 | 0.74 | 0.51
1A 1118-616 | 2.92${}_{-0.21}^{+0.25}$ | O9.5III-Ve | R11 | 32 | 0.92 (R11) | 0.63
EXO 2030+375 | 3.63${}_{-0.88}^{+1.34}$ | B0Ve | PF15 | 133 | 1.65 | 1.13
GX 304-1 | 2.01${}_{-0.13}^{+0.14}$ | B2Vne | R11 | 21 | 2.0 (R11) | 1.37
KS 1947+300 | 15.19${}_{-2.73}^{+3.66}$ | B0Ve | PF15 | 115 | 0.90 | 0.62
MXB 0656-072 | 5.11${}_{-0.93}^{+1.35}$ | O9.7Ve | B09 | 74 | 0.812 | 0.56
V0332+53 | 5.13${}_{-0.75}^{+1.01}$ | O8.5Ve | PF15 | 80 | 1.64 | 1.12
XTE J1946+274 | 12.64${}_{-2.87}^{+3.88}$ | B0-1IV-Ve | PF15 | 65 | 1.65 | 1.13

References. B09—Belczynski & Ziolkowski (2009), R11—Reig (2011), RF15—Reig &
Fabregat (2015) , for the distances GAIA measurements were used from Bailer-
Jones et al. (2018).

Table 3: Ultra-luminous X-ray sources

Name | Distance | References | Accretor | $\mathrm{N_{models}}$ | References
---|---|---|---|---|---
| Mpc |  |  |  |
NGC 7793 P13 | 3.50 | P10 | Pulsar | 1 | W18A
NGC 300 ULX1 | 1.88 | G05 | Pulsar | 1 | C18
NGC 5907 ULX1 | 17.06 | T16 | Pulsar | 2 | W15, F17
M82 X-1 | 3.30 | F14 | Unknown | 4 | B20
IC 342 X-1 | 3.40 | T10 | Unknown | 3 | R15
IC 342 X-2 | 3.40 | T10 | Unknown | 1 | R15, S17
NGC 1313 X-1 | 4.20 | M2, T16 | Unknown | 8 | W19
NGC 5204 X-1 | 4.90 | T16 | Unknown | 1 | M15
Circinus ULX5 | 4.20 | F77 | Unknown | 1 | W13
Holmberg IX X-1 | 3.55 | P02, T16 | Unknown | 6 | W17
Holmberg II X-1 | 3.40 | K02, T16 | Unknown | 1 | W18B
M33 X-8 | 0.83 | B16 | Unknown | 1 | WE19
NGC 5643 X-1 | 13.90 | S03 | Unknown | 2 | K16

References. B16—Bhardwaj, et al. (2016), B20—Brightman, et al. (2020),
C18—Carpano, et al. (2018), G05—Gieren, et al. (2005), F77—Freeman, et al.
(1977), F14—Foley, et al. (2014), F17—Fürst, et al. (2017), K02—Karachentsev,
et al. (2002), K16—Krivonos & Sazonov (2016), M2—Méndez, et al. (2002),
M15—Mukherjee, et al. (2015), P02—Paturel, et al. (2002), P10—Pietrzyński, et
al. (2010), R15—Rana, et al. (2015), S03—Sanders, et al. (2003), S17—Shidatsu,
Ueda & Fabrika (2017), T10—Tikhonov & Galazutdinova (2010), T16—Tully,
Courtois & Sorce (2016), W13—Walton, et al. (2013), W15—Walton, et al. (2015),
W17—Walton, et al. (2017), W18A—Walton, et al. (2018), W18B Walton, et al.
(2018), W19—Walton, et al. (2020), WE19—West, et al. (2018)

####  2.1.3 Ultra-luminous X-ray sources

In order to include in our analysis sources at very high accretion rates we
used all ULX spectral models, for which details on their best-fit parameters,
were available from the literature (see “references” column of Table 3). These
models are usually the result of simultaneous spectral fitting from NuSTAR
data, which extend to harder energies ($>$ 10 keV), and XMM-Newton, Chandra or
Suzaku data, which cover softer energies (up to $\sim$10 keV). For the
selection of the appropriate models for this work, we used wherever it was
possible physical models that describe the non-thermal component of the
spectra, like Compton scattering of seed photons from the disk etc. Those
models also describe well the soft part of the unabsorbed model (i.e show a
convergence at soft energies; comptt, simpl). On the contrary the power-law
component which is widely used to describe non-thermal emission from XRBs is a
good phenomenological model to describe the spectra of XRBs, however it
diverges unphysically at energies well below the thermal peak (e.g. Steiner et
al., 2009).

###  2.2 The hydrogen column density

Throughout this work, and for each source we have adopted different values for
the hydrogen column density ($\rm{N_{H}}$) in our spectral models. The values
used, depend on the class of the source (BHXRB, pulsar XRB or ULX) and the
limitations introduced by the spectral modelling. We explain in more detail
below, the reason behind this approach and the different values of the
$\rm{N_{H}}$ values adopted:

  1. 1.

The original $\rm{N_{H}}$ which is the $\rm{N_{H}}$ resulting from the
spectral fitting. For the BHs this value was fixed at a specific value for the
same object (see Table 1 of Steiner et al. (2016)) while for the pulsar XRBs
the value was a free parameter during the fitting process. For the ULXs, it
was the value provided in the corresponding papers (see References Table 3).

  2. 2.

A fiducial value of $0.1\times 10^{22}\ atoms\ cm^{-2}$, for a Galactic source
which is relatively unobscured, was adopted in order to have a uniform
absorption for all models and spectra, and for easier comparison.

  3. 3.

The $\rm{N_{H}}$ was set to the value of 0, in order to get the intrinsic
spectral shape. In this way, we calculated the physical properties of the
sources without the absorption hindering their luminosity and altering the
properties of the spectrum especially at soft energies ($<$2 keV). Since the
spectral models for the pulsar XRBs are based on power-law models, the
estimated unabsorbed flux diverges as the energy band extends to lower
energies. Therefore it was not possible to calculate the unabsorbed flux by
setting the $\rm{N_{H}}$ to 0. This was not possible as well, for 7 out of 13
ULXs (NGC 300 ULX1, NGC 5907 ULX1, Holmberg II X-1, IC 342 X-1, IC 342 X-2,
M33 X-8, and NGC 5643 X-1 ), where phenomenological models (e.g. power-law
etc.) have been used.

  4. 4.

In order to avoid the divergence of a power-law model, we have assumed for
pulsar XRBs absorption due to the circumstellar material with a line-of-sign
extinction calculated through the Bayestar3 model (Green et al., 2019) which
provides a 3D dust reddening map based on Gaia distances and photometric data
from Pan-STARRS 1 and 2MASS. We assumed an $\rm{R_{V}=A_{V}/E(B-V)=3.1}$
reddening law to derive the extinction $\rm{A_{V}}$ from the reddening, and
the Güver & Özel (2009) relation to calculate the equivalent line-of-sight
$\rm{N_{H}}$ column density from the extinction. It was not possible for the
reddening to be calculated for sources 1A 1118-616 and GX 304-1 due to the
lack of measurements from the Bayestar3 model in the vicinity of these
sources, and therefore the reddening values reported in Reig (2011) were used
instead. The extinction, as well as the intrinsic $\rm{N_{H}}$ are reported in
the last two columns of Table 2.

###  2.3 Energy bands

The energy bands we present along this paper are the bands reported in Table
4. They are the “standard” bands of most X-ray observatories like Chandra,
XMM-Newton, eROSITA, NuSTAR. We also use a bolometric band which we define at
$\mathrm{0.02-200\,keV}$.

Table 4: Energy bands | Name | Band (keV)
---|---|---
Bolometric | Bol | 0.02-200
Broad | B1 | 0.5-10
| B2 | 4-25
Soft | S1 | 0.5-2
| S2 | 4-12
| H1 | 2-10
Hard | H2 | 12-25

###  2.4 Analysis

In the following section we present the basic analysis performed in order to
further process the model spectra, which are described in Section 2.1. This
was done for the purposes of calculating BCs, and the contribution of the disk
to the optical flux of the binary system.

####  2.4.1 Calculations of Bolometric Corrections

In order to calculate the bolometric luminosity (band Bol) for all our
sources, we have used the XSPEC v12.11.1 software (Arnaud, 1996). In more
detail, we extrapolated the spectra using the command energies of XSPEC to
cover the energy range 0.02-200 keV, which corresponds to the band of the
bolometric luminosity. This extrapolation is being made under the assumption
that the physical mechanisms that produce the shape of the spectrum we
observe, remain unchanged, and no new mechanism is introduced at softer or
harder energies. However, caution is advised when using the pulsar XRBs BCs,
since a soft excess (modelled usually as a black-body with $kT_{BB}\sim 0.1$
keV) is typically present in the spectra of pulsar XRBs. This soft excess is
thought to originate from reprocessing of hard X-rays from the neutron star by
the inner region of the accretion disk (Hickox, Narayan, & Kallman, 2004).

As mentioned above, in order to calculate BCs, necessary was the measurement
of the bolometric luminosity for each one of the sources in our sample, and
therefore the extrapolation of all spectral models in order to cover the 0.02
to 200 keV energy range. This method was performed for the absorbed and
fiducial absorption model spectra for all classes of sources. For the
calculation of BCs free of the Galactic absorption introduced by each source,
the unabsorbed BH spectral models, as well as, the intrinsic $\rm{N_{H}}$
pulsar XRB spectral models were also extrapolated to cover the bolometric
energy range.

Then, we calculated the fluxes for all spectral models, using XSPEC command
flux for all energy bands provided in Table 4. Fluxes were converted to
luminosities, assuming spherical symmetry, and using the “distance” columns in
Tables 1, 2, and 3.

The Eddington luminosity of each source was calculated using the following:

$\rm{L_{edd}}=1.3\times 10^{38}\frac{M}{M_{\odot}}ergs^{-1}$ (1)

The Eddington ratio, which provides a metric of the accretion rate of material
originating form the disk onto the compact object, is defined as:

$Eddratio=\frac{L_{bol}}{L_{edd}}$ (2)

and was calculated for each model spectrum in our sample. Black hole systems
were aggregated into 10 Eddington ratio bins and pulsar XRBs into 5 Eddington
ratio bins, spaced logarithmically starting from the lowest to the highest
Eddington ratios. Each bin contains at least 15 spectra, and their range in
Eddington ratios can be seen in the second and third Columns of Tables 5, 10,
and 9.

We then computed the BCs for each spectral model as:

$BC=\frac{L_{band}}{L_{bol}}$ (3)

where “band” is one of the energy bands specified in Table 4.

Finally for each Eddington ratio bin we calculated the median value (50
percent percentile) of the BCs, as well as their 1$\sigma$ errors (68 percent
percentiles). We present our results for the average bolometric correction at
different accretion rates in Figs. 1 and 6, and Tables 5 to 8, for BHs and in
Figs. 2 and 3, and Tables 10, and 9 for the pulsar XRBs and ULXs. Examining
the behaviour of the BCs, for example at band 0.5-10 keV, and for the
different classes of XRBs, we notice that for BH systems (Fig. 1) the higher
values of BCs are present at lower and higher Eddington ratios, revealing that
the bulk of the bolometric X-ray emission in those ratios, is emitted at the
energy range of 0.5-10 keV. Regarding the pulsar XRBs and the 0.5-10 keV band,
higher BC values are present only at lower Eddington ratios (Fig. 2), but when
in our sample we include the ULXs (Fig. 3) we observe the higher BC values at
higher Eddington ratios, which is the same pattern observed in BH systems. We
discuss our results in more detail and for different energy bands in Section
3.

####  2.4.2 Calculation of disk’s optical contribution

In this section we present the analysis performed for the calculation of the
XRB optical luminosity originating from the accretion disk and the comparison
with its X-ray luminosity at different accretion rates. This test can only
work under the assumption that the same physical mechanisms responsible for
the observed spectrum in the X-rays are responsible also for the emission of
the disk in the optical. Therefore we use only the BH XRB unabsorbed model
spectra (Table 1) since these spectra are described by physical models (simpl)
and could in principle be extrapolated down to lower energies and the optical
regime. In addition none of these models contains jet emission or stellar
companion emission, and therefore are representative of the disk/corona
emission from the system. In our analysis we consider only Model 1 out of the
four available models for BHXBs, since no significant difference between the
different models (see discussion in Section 3.1.1) was observed.

In order to measure the optical flux, we extrapolated the model spectra, using
the command energies of XSPEC, down to $10^{-4}$ keV. We then measured the
optical flux of the disk using the tool
PYPHOT222https://mfouesneau.github.io/docs/pyphot/. The tool computes the
photometry of input stellar spectra using a library that contains a
significant amount of common filters. We calculated the flux in the V band,
using the filter ”GROUND_JOHNSON_V” from the PYPHOT filter library, with
effective wavelength 5438.689 Å.

We then calculated, the ratio of the optical luminosity $L_{V}$ and the X-ray
luminosity in the 2-10 keV band ($L_{X}$), at different accretions rates.
Throughout the text often we refer for simplicity to the X-ray luminosity
originating from the disk, corona and reflection components as ”disk X-ray
luminosity”. The methodology was the same as presented in Section 2.4.1, but
this time we used L(2-10 keV)/$L_{edd}$ instead of $\mathrm{L_{bol}/L_{edd}}$.
We selected the 2-10 keV energy band, because is one of the bands most
commonly used in X-ray observatories and in studies of individual XRBs, and
therefore our results can be compared easier with observations. In Table 14 we
present the values of the optical to X-ray emission from the disk, for 10 bins
at different Eddington ratios ($\mathrm{L_{X}/L_{edd}}$), and in Fig. 4 we
show how the ratio of optical to X-ray luminosity varies as a function of
different X-ray to $L_{edd}$ ratios. We discuss our results in more detail in
Section 3.2.

##  3 Results and discussion

###  3.1 Bolometric corrections

In this section we discuss the results of the average BCs for BH and pulsar
XRBs based on the unabsorbed model spectra for the BHs and the fiducial and
intrinsic absorption for pulsars and ULXs.

####  3.1.1 Black-hole XRBs

In Fig.1, we show for model 1 and unabsorbed spectra of BHXBs how the average
BC changes at different accretion rates. BCs of bin 1 ($L_{bol}/L_{edd}\sim
10^{-4}$), should be interpreted with caution, since they have significant
systematics related to the background subtraction. The different colours
correspond to BCs for different energy bands. For band 0.5-10 keV (solid black
line) we notice that BCs are higher at lower and higher accretion rates. This
is the result of a softer spectrum and a stronger accretion disk for higher
accretion rates and a decrease in the hard X-ray flux at lower accretion
rates. Therefore most of the 0.5-10 keV band emission dominates the spectrum
at high accretion rates above 0.1 $L_{edd}$. Similar behaviour show the sub-
bands 0.5-2 keV (solid red line) and 2-10 keV (solid blue line). However it is
clear that the 2-10 keV contributes more to the higher BCs at Eddington ratios
$>$0.1.

The rest of the bands which correspond to higher energy parts of the spectrum,
4-25 keV (dotted black line), 4-12 keV (red dotted line), and 12-25 keV (blue
dotted line) show similar behaviour. First, in all cases the BCs are quite low
($<0.3$ BC) with the lowest ($<0.1$ BC) for the harder band of the three,
revealing that the contribution of the emission above 4 keV to the bolometric
emission is rather low. The variation of the BC as a function of Eddington
ratio is also not significant ($<$0.1 BC). For band 4-25 keV, there is a small
increase for low ($\sim 10^{-4}$), high ($\sim 1$) and intermediate ($\sim
10^{-2}$) Eddington ratios. These small increases could be explained by the
different contributions in the 2 sub-bands. Namely, band 4-12 keV contributes
to this increase for low ($\sim 10^{-4}$) and high ($\sim 1$) Eddington
ratios, while band 12-25 keV contributes in the intermediate ($\sim 10^{-2}$)
Eddington ratios.

Moreover, we observe (Fig. 6) that the BCs for the different spectral models
of BHXBs (described in section 2.1.1), show the same behaviour for all bands.
Therefore, using a different model does not affect the BC calculation. In
Tables 5 to 8, we present the values of BCs for all available models of BHs
XRBs at different Eddington ratio bins.

Figure 1: Average BCs for our sample of BHXBs described in Table 1 as a
function of the Eddington ratio ($\mathrm{L_{bol}/L_{edd}}$). Here we present
the BCs for spectral model 1 (see Section 2.1.1) and all energy bands
described in Table 4. Namely the dash-dotted lines represent the BCs for 4-12
keV, 12-25 keV, and 4-25 keV in red, blue, and black respectively. The solid
lines represent the BCs for 0.5-2 keV, 2-10 keV, and 0.5-10 keV in red, blue,
and black respectively.

####  3.1.2 Pulsars XRBs

In Fig. 2, we show how the BC change as a function of the Eddington ratio for
pulsar XRBs and intrinsic absorption (see section 2.2). Very interesting is
the fact that the behaviour of BC as a function of Eddington ratio for
pulsars, looks very similar with those of BHs XRBs, however for the higher
accretion rates this is only true when we include the confirmed pulsar ULXs
(see following Section). Moreover for pulsars XRBs, we overall notice that at
harder bands (4-25 keV, 4-12 keV, 12-25 keV), the BCs have higher values than
the ones observed in BHs. This is expected since pulsar XRBs generally show
harder spectra comparison to those of BHXBs (e.g. Vrtilek & Boroson, 2013). We
report the BC values for pulsar XRBs and intrinsic $\rm{N_{H}}$ in Table 9.

####  3.1.3 ULXs

In Fig. 3, we present together with the data for the pulsar XRBs also how the
BC change at higher Eddington ratios ($>$1), if we add spectral models for all
ULXs of Table 3. We consider three cases, 1) where all confirmed pulsar ULXs
are added (left panel); 2) assuming that all ULXs are NS with masses of 1.4 M⊙
(middle panel) and 3) assuming that all non-confirmed pulsar ULXs are BHs with
a mass of 9.1 M⊙ (right panel). We see that no matter the assumption on the
nature of the ULXs, the behaviour is equivalent for the same Eddington ratios.
Under the assumption however that all ULXs are NS, the correlation expands to
higher Eddington ratios ($\sim 10^{3}$). For cases 1) and 2) where the ULXs
are treated as NS we decided to plot separately the source NGC 300 ULX1 (6th
bin; $\mathrm{L_{bol}/L_{edd}}\sim$ 10), due to its more complicated spectrum
compare to other ULXps (see Carpano, et al. (2018). For this particular reason
this bin has no errors in our plots, and also is the reason why the BC shows
an abrupt increase for the band 0.5-2 keV. We report the fiducial $\rm{N_{H}}$
BC values for pulsar, ULXps, ULXs assumed as NS, and ULXs assumed as BHs in
Tables 10, 11, 12, and 13 respectively.

Figure 2: Average BCs for our sample of pulsar XRBs described in Table 2 as a
function of the Eddington ratio ($\mathrm{L_{bol}/L_{edd}}$), and assuming
intrinsic $\rm{N_{H}}$. The colour coding is the same as described in Fig.1

Figure 3:  Average BCs for our sample of pulsar XRBs and ULXs as described in
Tables 2 and 3, and as a function of the Eddington ratio
($\mathrm{L_{bol}/L_{edd}}$) and assuming fiducial $\rm{N_{H}}$. From left to
right plots, we account for the contribution of confirmed pulsar ULXs, then
for ULXs assuming that they are all NS, and finally for ULXs assuming that
they all BHs (excluding the confirmed pulsar ULXs). The colour coding is the
same as described in Fig.1.

###  3.2 BHXBs disk optical to X-ray luminosity as a function of accretion
rate.

In XRB studies physical quantities regarding the system are usually inferred
by measuring the optical emission of the donor star in quiescence. In
transient systems such as these, during outburst the donor-star emission is
generally negligible compared to that of other components (disk, jet etc).

In BH HMXBs the donor star is so massive ($\gtrsim 10M_{\odot}$), that
dominates the optical X-ray output of the system (van den Heuvel & Heise,
1972; Treves et al., 1980), while in BH LMXBs ($M_{donor}\lesssim
3M_{\odot}$), optical emission is mainly originating from the outer accretion
disk as a result of reprocessing of X-rays produced in the inner disk
(Cunningham, 1976; Vrtilek et al., 1990). Other processes might be
contributing such as thermal emission from the companion star while the source
is in quiescence (e.g. Greene, Bailyn, & Orosz, 2001; Mikołajewska et al.,
2005), intrinsic thermal emission from a viscously heated outer disk (Shakura
& Sunyaev, 1973; Frank, King, & Raine, 2002), as well as possibly jet
contribution in the hard state although it is expected to be higher in the NIR
(Russell et al. (2006), for a review see Fender (2006)).

In this section we focus on the optical emission originating from X-ray
reprocessing in the outer accretion disk and we are measuring its expected
contribution at different accretion rates. In more detail, in Fig. 4 (red
line) we determine the optical (V-band; $L_{V}$) to X-ray (2-10 keV; $L_{X}$)
ratio originating from the disk/corona and we show how it changes as a
function of Eddington ratio. Here we use the extrapolated fluxes from the
models as described in Section 2.4.2. We see that accounting for the 1$\sigma$
errors the $L_{V}/L_{X}$ value shows no significant variation
($L_{V}/L_{X}\sim 4\times 10^{-5}$) as a function of accretion rate except of
a small increase at $L_{X}$/$L_{edd}$ $\sim 10^{-2}$, which reveals that
independent of the state a binary is in (soft, hard, etc), and under the
assumption that the contribution of the jet is negligible in the soft and not
significant during the hard state and the optical band, the amount of the
optical to X-ray emission remains constant as a function of Eddington ratio.

This result is not unexpected, since correlations between X-ray and optical
emission have been identified. For example, Russell et al. (2006) using quasi
simultaneous optical and X-ray observations from XRBs observed that the hard
state near-infrared/optical luminosity of BH LMXBs (originating from the disk
and the jet) increases with increasing 2-10 keV X-ray luminosity (see top
panel of their Fig.1). Regarding the soft state, the optical luminosity (see
bottom panel of their Fig.1) seems to increase as the X-ray luminosity is
increasing more or less at the same rate as was observed in the hard state.
Russell et al. (2006) conclude also that the contribution of the jet is less
important for the optical compared to the NIR band. This could give an
explanation of why the optical luminosity increases with increasing X-ray
luminosity almost at the same rate, for both the soft (where the jet is
quenched) and the hard state. This result is is in agreement with a more or
less constant X-ray to optical ratio luminosity we have calculated along
different accretion rates (Fig. 4).

Figure 4: Disk optical V-band to X-ray 2-10 keV band luminosity ratio of the
BHXRBs as a function of the Eddington ratio L(2-10 keV)/$L_{edd}$. Different
colours correspond to different dependencies of the local disk temperature
$T(r)$ to the radius $r$, i.e different disk geometries. Namely the red line
corresponds to $T(r)\sim r^{-0.75}$ (XSPEC model: diskbb), the green and blue
lines to $T(r)\sim r^{-0.65}$ and $T(r)\sim r^{-0.55}$ respectively (XSPEC
model: diskpbb).

####  3.2.1 Testing different disk geometries

The BH model spectra reported in this work, were fitted using an XPSEC diskbb
model to account for the X-ray emission originating from the accretion disk
(Steiner et al., 2016). The diskbb model represents the spectrum from an
accretion disk consisting of multiple black-body components where the local
disk temperature is described as function of disk radius $T(r)\sim r^{-p}$
where $p=3/4$. This represents the standard picture of an optically thick
geometrically thin accretion disk (Shakura & Sunyaev, 1973).

However studies have shown that radial advection could be important and then
$p$ can have smaller values (e.g. Mineshige et al., 1994; Watarai et al.,
2000; Kubota & Makishima, 2004). Another mechanism that could contribute to
this effect is X-ray irradiation of the outer accretion disk by X-rays
produced in the inner accretion flow (e.g. van Paradijs & McClintock, 1994;
van Paradijs, 1996), which is believed to have a large impact on the outer
disk during outburst. Accretion disks with $p<0.75$ are described by the so-
called slim accretion disk models which were proposed by Abramowicz et al.
(1988) and represent optically thick, geometrically not very thin accretion
disks. A different disk geometry could have significant implications on the
reprocessing of X-rays in the disk and therefore on the optical emission
originating from the disk.

We tested this effect by replacing in our spectral models for the BHXBs the
diskbb component with a diskpbb component using two different values of $p$,
namely $p=0.65$ and $p=0.55$. The substitution was made under the assumption
that the 2-10 keV flux remains unchanged between the diskbb and diskpbb
models, and then the V-band luminosity was calculated following the method in
Section 2.4.2. We show in Fig. 4 how the $L_{V}/L_{X}$ changes as a function
of accretion rate for different disk geometries. We observe that the optical
flux increases significantly, one and two orders of magnitude for $p=0.65$ and
$p=0.55$, respectively, therefore the thicker the disk geometry is, the higher
is expected to be its optical emission.

####  3.2.2 Comparison with observations

The optical to X-ray luminosity ratio of the disk as function of different
Eddington ratios and disk geometries as presented in Fig. 4 provides a
diagnostic tool, when compared with observational data, not only to determine
the disk’s optical contribution in a BH XRB system at different accretion
rates/spectral states, but also to indicate the geometry of the disk. For that
reason we used a number of simultaneous and quasi-simultaneous optical and
X-ray data of BHXBs available in the literature.

We used the same distance and mass measurements reported in Table 1 for the
sources that also exist in our sample, while for the rest, we used the
measurements reported in the corresponding papers or in the BlackCAT catalogue
(Corral-Santana et al., 2016). In Fig. 5 we present the same data as in Fig. 4
with the difference that on the y axis we show, the optical luminosity
$L_{V}$, for easier comparison with the observational data which are also
over-plotted.

In more detail, regarding simultaneous observations, we have used Swift XRT
and ultraviolet/optical (UVOT) data from the 2005 outburst of the BH LMXB GRO
J1655-40 (Brocksopp et al., 2006). We used the V magnitude, and the X-ray flux
at 2-10 keV computed from XRT spectral models of six of their observations
(Table 4 in Brocksopp et al., 2006), which correspond to the high/soft or very
high state. We corrected the optical V magnitudes for reddening using the
value of $A_{V}=3.7\pm 0.3$ reported in Jonker & Nelemans (2004). We then
calculated the V-band luminosity using the AB magnitude zeropoint, the
distance of the source reported in Table (1), and the flux as $\lambda\
F_{\lambda}$ assuming that the spectral range of the filter can be
approximated by its central wavelength. In Fig. 5 we show the data for GRO
J1655-40 with magenta circles, which are located at Eddington ratios close to
1.0, which is generally expected for systems in high/soft or very high states,
and between the green line for diskpbb model (p=0.65) and the blue line for
the diskpbb model (p=0.55). The optical observations are not expected to
include contribution from a jet, since it is considered to be quenched at the
high/soft and very high spectral states Fender (2006). We calculated (using
the spectral type absolute magnitude reported in Carroll & Ostlie (2006), and
solar properties from Prša et al. (2016)) that the the donor star, having a
spectral type of F6IV (see Table 1), is expected to have a contribution up to
14% to the systems optical emission when the system is at $\sim 0.05-0.1\
\mathrm{L_{edd}}$ (assuming the rest of the optical emission is originating
from the accretion disk since the jet is quenched in the soft state). For that
reason, we also corrected for the contribution of the donor star. We show in
Fig. 5, that the disk optical luminosity of GRO J1655-40, agrees more with a
what is expected from a thicker disk ($p\sim 0.60$).

We then included quasi-simultaneous observations of a number of the BHXBs
presented in Russell et al. (2006)(see references therein). In Fig. 5 we
included data for BH LMXBs in the soft state (blue colour), in the hard state
(red colour), as well as BH HMXBs (cyan colour), keeping always the same
symbol for the same source. We should note here that the hard-state data taken
from Russell et al. (2006) have been corrected for the fractional contribution
of the donor star, and uncertainties connected with the uncertain spectral
type of the donor star are propagated in the errors. Also they corrected for
the contribution of the donor star, during outburst in cases the donor is
comparatively bright, and as consequence all data points from Russell et al.
(2006), represent the optical flux originating from all other than the donor
emission processes in the system.

In more detail, regarding the data adopted from Russell et al. (2006), we show
the BH HMXBs Cyg-X1 (cyan circle) and V4641 Sgr (cyan star). We show the BH
LMXBs LMC X-3 (square), 3A 0620-003 (pentagon), 4U 1543-475 (down triangle),
V404 Cyg (big circle), GRO J1655-50 (x symbol), and XTE J1118+480 (plus
symbol).

All BH HMXBs are located above the blue line ($p=0.55$), and this is expected
as the donor star dominates the optical output of the system. However the
source V4641, could be characterised as an intermediate XRB being of spectral
type B9III, as the $L_{V}$ contribution from the donor star is calculated
(using absolute magnitude reported in Carroll & Ostlie (2006), and solar
properties from Prša et al. (2016)) to be of the order of $L_{V}\sim 5.0\times
10^{35}$ erg s-1 , which lies between 1% and 5% of the overall V-band
luminosity of the system accounting for different observations which lie
between 0.1 and 1.0 $\mathrm{L_{edd}}$ of the system. Therefore we consider
more representative of the BH HMXB class, Cyg X-1. Overall, BH HMXBs are
treated here as reference and an indication of a limit above which a source
could be characterised as a BH HMXB.

Regarding the BH LMXBs, we notice that all sources regardless of the state
they are in, they fall a bit below or above the blue line ($p=0.55$), which
corresponds to the geometrically thicker disk case. In the soft state (blue
points), the jet is quenched and therefore the optical emission we measure is
dominated by X-ray emission and/or reprocessing from the disk. V404 Cyg shows
much higher optical luminosity than other soft BH LMXBs. This could be
consequence of a very bright outburst it went through that completely alters
the geometry of the disk, or the fact that it has a sub-giant companion. In
the hard state (red points), where the contribution of the donor star has been
subtracted, we see only the contribution of the disk and the jet. However the
contribution of the jet in the optical is not expected to be significant
compared to that of the disk. In fact, Russell et al. (2006) showed that the
transition between an optical thick to an optical thin jet happens around the
K band, with the jet NIR contribution being $\sim 90\%$. They also measure
that the optical contribution of the jet could be between zero and 76% but the
optical spectral energy distributions of BHXBs show a thermal spectrum
indicating that X-ray reprocessing from the disk dominates the optical output.
Therefore significant contribution from the jet should be expected mainly in
the hard state and in the NIR emission, while in the optical the disk
dominates. In all cases, all hard state data follow or are a bit above the
blue line ($p=0.55$), in agreement with thicker disk geometries. Moreover we
see that none of the observed sources is below the red line ($p=0.75$), which
is expected since no source could physically have an optical luminosity lower
than the one produced by the disk at a given Eddington ratio.

####  3.2.3 Discussion

Overall with the calculation of the $L_{V}/L_{X}$ as a function of Eddington
ratio and the comparison with the observational data, we aim to provide a
limit of the disk optical contribution to the BH XRB system. This could be
useful for a better determination of donor star type, or for estimating the
disk geometry. In fact we see that all observational data shown in Fig. 5,
(under the assumption that the jet contribution in the hard state is not
significant) do not agree with the standard scenario of a geometrically thin
disk but rather with that of a geometrically thicker disk with $p\lesssim
0.60$.

However, there are certain assumptions included in our calculations that could
introduce some level of uncertainty. For example, the spectral models we are
using, assume an intrinsically steady state where the X-ray emission
originating from the inner disk is coupled with the optical emission
originating from the outer disk. However, two regimes of this behaviour are
considered to exist, and our assumption is true in the regime where X-ray disk
irradiation is dominant (e.g. van Paradijs & McClintock, 1994; van Paradijs,
1996). In this case, the emission at longer wavelengths corresponding to the
outer portion of the disk is coupled to the emission produced in the inner
disk. The second regime is dominated by viscous dissipation and therefore the
outer disk optical emission and the inner disk X-ray emission are correlated
with a time delay which depends on the viscous timescale, and in XRBs can be
of the order of weeks (e.g. Steiner et al., 2014a). In that latter case, our
assumption would not hold, and any variability or any time delay effectively
will add some noise on the relations we provide. However, since we do not use
a time dependent model this point of uncertainty is not addressed to this work
and could be let to some other kind of analysis that investigates the time
dependent behaviour of disk models.

Another source of uncertainty introduced in our calculations is the
contribution and effect of the corona into our results. On the one hand, the
contribution of the corona is expected to be significant in the X-ray regime,
and the subtraction of that contribution would result in somehow larger
optical emission and therefore steeper relations. On the other hand, the
corona could be in part responsible for the irradiation of the disk, resulting
in higher optical emission and shallower relation. Disentangling the corona
contribution is not possible in our calculations since it could potentially
have an effect on both the X-ray and optical regime. However, even though the
coronal contribution, might be important in particular accretion regimes (i.e.
very high state, low state), in the soft state where a source spends most of
its time during outburst (e.g. Tetarenko et al., 2016) it is not the dominant
mechanism. Therefore we do not expect that the coronal contribution would have
significant implications into our results.

Interestingly, the fact that all of the observational data agree with a
thicker disk scenario ($p\lesssim 0.60$), seems to be also in agreement with
the lack of high inclination systems in transient BHXBs with low-mass
companions (Narayan & McClintock, 2005; Corral-Santana et al., 2013). Although
high-mass BHXBs can exhibit eclipses even at relatively low inclinations (e.g.
M33 X-7; Orosz et al., 2007), for low-mass BHXBs a lack of eclipses has been
observed which can be attributed solely on inclinations effects. For example,
Narayan & McClintock (2005) after studying 11 sources with inclination
estimates, found that no source has $i>75^{\circ}$. They showed that the
probability that this would happen by chance is less than 4%, and the lack of
such systems is possibly the result of a flared accretion disk (corrugated or
warped) by about $\pm 15\ \mathrm{degrees}$. Moreover they found that BHXBs
with higher inclinations $i=70-75\ \mathrm{degrees}$ show more noisy light-
curves, which could be explained if they are close to be occulted by the disk.
Therefore a flared thicker disk could be an explanation, for not seeing
perfect eclipses in low-mass black-hole XRBs and as a consequence missing a
large number of high-inclination systems.

Figure 5: Optical V-band luminosity versus 2-10 keV band X-ray luminosity
normalised by the Eddington luminosity. The solid lines represent the expected
disk optical luminosity at different accretion rates and disk geometries of
$p=0.75$, $p=0.65$, and $p=0.55$ with red, green, and blue respectively. The
magenta circles correspond to simultaneous optical and X-ray observations of
GRO J1655-40 (Brocksopp et al., 2006). Blue and red points correspond to
quasi-simultaneous soft and hard state observations of XRBs while cyan to
observations of BH HMXBs taken from Russell et al. (2006) and references
therein. The same symbol is used for the same source.

##  4 Summary

In this paper we use a spectral model library of 10 BHXBs (9237 models), 9
pulsar XRBs (577 models) and 13 ULXs (32 models). Using this library we
calculated:

  1. 1.

Bolometric corrections in six energy bands (0.5-2 keV, 2-10 keV, 0.5-10 keV,
4-12 keV, 12-25 keV, and 4-25 keV) as a function of different Eddington
ratios. We cover Eddington ratios from $\sim 10^{-4}\ L_{edd}$ for the BH and
pulsar XRBs, and reaching up to $\sim 10^{3}\ L_{edd}$ when including the ULXs
to study the higher-end of Eddington ratios. We notice that the same pattern
is observed, shifted by $\sim 2$ orders in the Eddington ratio, between the
BCs calculated for BH and pulsar XRB, only when we include the pulsar ULXs in
the latter case. Moreover fitting with different spectral models for BHs does
not affect the BC calculations.

Average BCs as a function of the accretion rate, are useful tools for
calculating the bolometric luminosity of any XRB using its luminosity in the
narrow band provided by the existing X-ray observatories, and in calculations
for XRB population synthesis models as observable constraints.

  2. 2.

For BHXRBs, optical to X-ray luminosity ratio as a function of Eddington ratio
originating from the accretion disk and corona, and for different disk
geometries ($p=0.55$,$p=0.65$, and $p=0.75$). We observe that the optical to
X-ray contribution for the same disk geometry does not show any significant
variation, while as we move to thicker disks (smaller $p$ values), the optical
contribution of the disk to the binary system increases. Including quasi-
simultaneous optical and X-ray observational data, which are representative of
the disk emission, we find that they agree with a thicker disk scenario
instead of the standard thin disk. The existence of a disk thicker than the
standard one ($p<0.75$) could also explain the fact that we do not observe any
high inclination low-mass BH systems.

Overall, the knowledge of the average optical contribution from the disk as a
function of the accretion rate, and for different geometries, is useful for a
better determination of the spectral type of the donor star (after exclusion
of the disk luminosity), as well as for the estimation of the disk geometry.

## Acknowledgements

K. A. and A. Z. acknowledge funding from the European Research Council under
the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant
Agreement n. 617001 and from the European Union’s Horizon 2020 research and
innovation program me under the Marie Sklodowska-Curie RISE action, grant
agreement No 691164 (ASTROSTAT). K. A. acknowledges funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No 865637). We also made use of NASA’s
Astrophysics Data System Bibliographic Services.

## Data availability

Spectra models of BH and pulsar XRBs used in this work, resulted from spectra
extracted by public data of the RXTE observatory. Spectral models used for
ULXs are reported in the corresponding references.

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##  Appendix A Tables and plots

Figure 6: Average BCs for our sample of BHXBs described in Table 1 as a function of the Eddington ratio ($\mathrm{L_{bol}/L_{edd}}$). Here we present the BCs for spectral models 2, 3, and 4 (see Section 2.1.1), and all energy bands described in Table 4 : Model 2 in the top, model 3 in the middle, and model 4 in the bottom panel. Color coding is the same as in Fig. 1.  Table 5: Average BCs for model 1 of the BHXBs in 10 Eddington ratio bins. N | $\log_{min}\frac{L_{bol}}{L_{edd}}$ | $\log_{max}\frac{L_{bol}}{L_{edd}}$ | S1 | H1 | B1 | S2 | H2 | B2
---|---|---|---|---|---|---|---|---
1 | -4.3 | -3.86 | 0.3${}_{-0.11}^{+0.12}$ | 0.36$\pm 0.07$ | 0.66${}_{-0.21}^{+0.14}$ | 0.18$\pm 0.05$ | 0.06$\pm 0.04$ | 0.26${}_{-0.1}^{+0.06}$
2 | -3.86 | -3.42 | 0.3${}_{-0.11}^{+0.1}$ | 0.22${}_{-0.06}^{+0.09}$ | 0.57${}_{-0.19}^{+0.07}$ | 0.11${}_{-0.03}^{+0.05}$ | 0.07$\pm 0.02$ | 0.18${}_{-0.04}^{+0.08}$
3 | -3.42 | -2.98 | 0.16${}_{-0.08}^{+0.19}$ | 0.17${}_{-0.05}^{+0.06}$ | 0.35${}_{-0.11}^{+0.18}$ | 0.11${}_{-0.04}^{+0.05}$ | 0.09$\pm 0.03$ | 0.19${}_{-0.07}^{+0.08}$
4 | -2.98 | -2.54 | 0.16${}_{-0.11}^{+0.27}$ | 0.15${}_{-0.04}^{+0.1}$ | 0.33${}_{-0.16}^{+0.31}$ | 0.1${}_{-0.03}^{+0.06}$ | 0.08${}_{-0.05}^{+0.04}$ | 0.17${}_{-0.05}^{+0.1}$
5 | -2.54 | -2.1 | 0.08${}_{-0.04}^{+0.14}$ | 0.14${}_{-0.03}^{+0.07}$ | 0.25${}_{-0.09}^{+0.13}$ | 0.11${}_{-0.04}^{+0.05}$ | 0.11${}_{-0.06}^{+0.02}$ | 0.22${}_{-0.09}^{+0.06}$
6 | -2.1 | -1.66 | 0.09${}_{-0.04}^{+0.08}$ | 0.16$\pm 0.02$ | 0.26${}_{-0.05}^{+0.08}$ | 0.12$\pm 0.01$ | 0.11$\pm 0.01$ | 0.24${}_{-0.02}^{+0.01}$
7 | -1.66 | -1.22 | 0.36${}_{-0.25}^{+0.14}$ | 0.17${}_{-0.03}^{+0.05}$ | 0.55${}_{-0.28}^{+0.14}$ | 0.09$\pm 0.03$ | 0.06${}_{-0.05}^{+0.04}$ | 0.16${}_{-0.08}^{+0.07}$
8 | -1.22 | -0.78 | 0.38${}_{-0.1}^{+0.09}$ | 0.44${}_{-0.13}^{+0.16}$ | 0.85${}_{-0.21}^{+0.05}$ | 0.15${}_{-0.06}^{+0.15}$ | 0.01${}_{-0.01}^{+0.03}$ | 0.18${}_{-0.08}^{+0.14}$
9 | -0.78 | -0.34 | 0.32${}_{-0.09}^{+0.1}$ | 0.47$\pm 0.08$ | 0.82${}_{-0.12}^{+0.06}$ | 0.22${}_{-0.09}^{+0.1}$ | 0.05$\pm 0.05$ | 0.29${}_{-0.14}^{+0.12}$
10 | -0.34 | 0.1 | 0.3${}_{-0.08}^{+0.06}$ | 0.54$\pm 0.09$ | 0.84${}_{-0.05}^{+0.04}$ | 0.27${}_{-0.05}^{+0.12}$ | 0.05${}_{-0.04}^{+0.02}$ | 0.33${}_{-0.1}^{+0.11}$
Table 6: Average BCs for model 2 of the BHXBs in 10 Eddington ratio bins. N | S1 | H1 | B1 | S2 | H2 | B2
---|---|---|---|---|---|---
1 | 0.33${}_{-0.14}^{+0.09}$ | 0.34${}_{-0.05}^{+0.07}$ | 0.69${}_{-0.17}^{+0.08}$ | 0.19${}_{-0.06}^{+0.05}$ | 0.08$\pm 0.04$ | 0.26${}_{-0.1}^{+0.13}$
2 | 0.3${}_{-0.13}^{+0.1}$ | 0.3${}_{-0.09}^{+0.05}$ | 0.61${}_{-0.14}^{+0.09}$ | 0.16${}_{-0.05}^{+0.06}$ | 0.09${}_{-0.03}^{+0.05}$ | 0.25$\pm 0.09$
3 | 0.15${}_{-0.07}^{+0.13}$ | 0.2${}_{-0.06}^{+0.12}$ | 0.39${}_{-0.15}^{+0.17}$ | 0.13${}_{-0.06}^{+0.1}$ | 0.11${}_{-0.04}^{+0.07}$ | 0.24${}_{-0.1}^{+0.17}$
4 | 0.16${}_{-0.1}^{+0.23}$ | 0.17${}_{-0.07}^{+0.14}$ | 0.37${}_{-0.19}^{+0.3}$ | 0.11${}_{-0.05}^{+0.09}$ | 0.1${}_{-0.05}^{+0.07}$ | 0.22${}_{-0.1}^{+0.14}$
5 | 0.09${}_{-0.05}^{+0.21}$ | 0.15${}_{-0.04}^{+0.11}$ | 0.29${}_{-0.12}^{+0.19}$ | 0.12${}_{-0.06}^{+0.09}$ | 0.12${}_{-0.07}^{+0.06}$ | 0.23${}_{-0.12}^{+0.14}$
6 | 0.1${}_{-0.04}^{+0.07}$ | 0.16$\pm 0.02$ | 0.26${}_{-0.05}^{+0.09}$ | 0.12$\pm 0.01$ | 0.12$\pm 0.01$ | 0.24${}_{-0.02}^{+0.01}$
7 | 0.38${}_{-0.26}^{+0.13}$ | 0.17${}_{-0.04}^{+0.07}$ | 0.57${}_{-0.29}^{+0.14}$ | 0.09$\pm 0.03$ | 0.06${}_{-0.04}^{+0.05}$ | 0.15$\pm 0.08$
8 | 0.4${}_{-0.11}^{+0.07}$ | 0.44${}_{-0.11}^{+0.16}$ | 0.86${}_{-0.22}^{+0.04}$ | 0.15${}_{-0.05}^{+0.14}$ | 0.01${}_{-0.01}^{+0.03}$ | 0.18${}_{-0.07}^{+0.14}$
9 | 0.34${}_{-0.1}^{+0.08}$ | 0.46${}_{-0.08}^{+0.09}$ | 0.81${}_{-0.12}^{+0.07}$ | 0.21${}_{-0.07}^{+0.1}$ | 0.06${}_{-0.05}^{+0.04}$ | 0.28${}_{-0.12}^{+0.12}$
10 | 0.35${}_{-0.1}^{+0.1}$ | 0.48${}_{-0.19}^{+0.09}$ | 0.8${}_{-0.08}^{+0.07}$ | 0.23${}_{-0.1}^{+0.1}$ | 0.05${}_{-0.04}^{+0.02}$ | 0.28${}_{-0.11}^{+0.13}$
Table 7: Average BCs for model 3 of the BHXBs in 10 Eddington ratio bins. N | S1 | H1 | B1 | S2 | H2 | B2
---|---|---|---|---|---|---
1 | 0.24${}_{-0.12}^{+0.08}$ | 0.25${}_{-0.06}^{+0.13}$ | 0.48${}_{-0.11}^{+0.26}$ | 0.15${}_{-0.07}^{+0.06}$ | 0.07${}_{-0.03}^{+0.05}$ | 0.21${}_{-0.07}^{+0.1}$
2 | 0.28${}_{-0.08}^{+0.13}$ | 0.22${}_{-0.07}^{+0.09}$ | 0.54${}_{-0.16}^{+0.12}$ | 0.11${}_{-0.04}^{+0.05}$ | 0.06${}_{-0.02}^{+0.03}$ | 0.17${}_{-0.04}^{+0.09}$
3 | 0.18${}_{-0.09}^{+0.19}$ | 0.15${}_{-0.04}^{+0.08}$ | 0.35${}_{-0.12}^{+0.21}$ | 0.09${}_{-0.03}^{+0.05}$ | 0.08$\pm 0.03$ | 0.17${}_{-0.06}^{+0.08}$
4 | 0.17${}_{-0.1}^{+0.24}$ | 0.14${}_{-0.05}^{+0.1}$ | 0.33${}_{-0.16}^{+0.31}$ | 0.08${}_{-0.02}^{+0.07}$ | 0.07$\pm 0.04$ | 0.15${}_{-0.05}^{+0.1}$
5 | 0.08${}_{-0.04}^{+0.22}$ | 0.14${}_{-0.03}^{+0.07}$ | 0.26${}_{-0.1}^{+0.14}$ | 0.11${}_{-0.03}^{+0.05}$ | 0.11${}_{-0.05}^{+0.02}$ | 0.22${}_{-0.08}^{+0.06}$
6 | 0.08${}_{-0.03}^{+0.07}$ | 0.16${}_{-0.02}^{+0.03}$ | 0.25${}_{-0.05}^{+0.08}$ | 0.13$\pm 0.01$ | 0.12${}_{-0.01}^{+0.0}$ | 0.24$\pm 0.01$
7 | 0.41${}_{-0.3}^{+0.1}$ | 0.18${}_{-0.04}^{+0.05}$ | 0.6${}_{-0.33}^{+0.11}$ | 0.09$\pm 0.03$ | 0.05${}_{-0.04}^{+0.05}$ | 0.14${}_{-0.07}^{+0.08}$
8 | 0.38${}_{-0.11}^{+0.09}$ | 0.43${}_{-0.13}^{+0.17}$ | 0.85${}_{-0.21}^{+0.05}$ | 0.15${}_{-0.06}^{+0.15}$ | 0.01${}_{-0.01}^{+0.03}$ | 0.18${}_{-0.08}^{+0.14}$
9 | 0.32${}_{-0.08}^{+0.11}$ | 0.46${}_{-0.07}^{+0.09}$ | 0.82${}_{-0.12}^{+0.06}$ | 0.23$\pm 0.09$ | 0.05${}_{-0.05}^{+0.04}$ | 0.3${}_{-0.15}^{+0.11}$
10 | 0.29${}_{-0.07}^{+0.08}$ | 0.53${}_{-0.08}^{+0.1}$ | 0.84${}_{-0.05}^{+0.04}$ | 0.28${}_{-0.07}^{+0.12}$ | 0.05${}_{-0.04}^{+0.02}$ | 0.34${}_{-0.12}^{+0.11}$
Table 8: Average BCs for model 4 of the BHXBs in 10 Eddington ratio bins. N | S1 | H1 | B1 | S2 | H2 | B2
---|---|---|---|---|---|---
1 | 0.3${}_{-0.1}^{+0.13}$ | 0.33$\pm 0.08$ | 0.67${}_{-0.21}^{+0.08}$ | 0.19${}_{-0.09}^{+0.04}$ | 0.07$\pm 0.04$ | 0.25${}_{-0.13}^{+0.09}$
2 | 0.34${}_{-0.18}^{+0.09}$ | 0.24$\pm 0.06$ | 0.59${}_{-0.19}^{+0.08}$ | 0.13${}_{-0.04}^{+0.06}$ | 0.07$\pm 0.03$ | 0.2${}_{-0.05}^{+0.09}$
3 | 0.15${}_{-0.07}^{+0.23}$ | 0.19${}_{-0.04}^{+0.05}$ | 0.37${}_{-0.1}^{+0.2}$ | 0.13${}_{-0.05}^{+0.03}$ | 0.1${}_{-0.04}^{+0.03}$ | 0.23${}_{-0.08}^{+0.06}$
4 | 0.19${}_{-0.13}^{+0.29}$ | 0.18${}_{-0.05}^{+0.1}$ | 0.38${}_{-0.16}^{+0.3}$ | 0.12$\pm 0.05$ | 0.1${}_{-0.08}^{+0.03}$ | 0.22${}_{-0.12}^{+0.07}$
5 | 0.07${}_{-0.04}^{+0.25}$ | 0.15${}_{-0.04}^{+0.06}$ | 0.26${}_{-0.11}^{+0.17}$ | 0.11${}_{-0.04}^{+0.05}$ | 0.11${}_{-0.06}^{+0.02}$ | 0.22${}_{-0.08}^{+0.07}$
6 | 0.05${}_{-0.01}^{+0.07}$ | 0.15$\pm 0.03$ | 0.2${}_{-0.02}^{+0.07}$ | 0.12${}_{-0.02}^{+0.01}$ | 0.11$\pm 0.01$ | 0.23${}_{-0.04}^{+0.02}$
7 | 0.39${}_{-0.27}^{+0.11}$ | 0.16${}_{-0.04}^{+0.13}$ | 0.56${}_{-0.3}^{+0.22}$ | 0.09${}_{-0.03}^{+0.02}$ | 0.05${}_{-0.04}^{+0.05}$ | 0.15$\pm 0.06$
8 | 0.39${}_{-0.11}^{+0.08}$ | 0.43${}_{-0.11}^{+0.18}$ | 0.86${}_{-0.22}^{+0.05}$ | 0.15${}_{-0.05}^{+0.15}$ | 0.01${}_{-0.01}^{+0.03}$ | 0.18${}_{-0.08}^{+0.14}$
9 | 0.32$\pm 0.09$ | 0.48${}_{-0.08}^{+0.07}$ | 0.82${}_{-0.12}^{+0.06}$ | 0.23${}_{-0.08}^{+0.11}$ | 0.05$\pm 0.05$ | 0.3${}_{-0.14}^{+0.12}$
10 | 0.3${}_{-0.08}^{+0.05}$ | 0.54${}_{-0.09}^{+0.1}$ | 0.84${}_{-0.06}^{+0.04}$ | 0.26${}_{-0.04}^{+0.14}$ | 0.05${}_{-0.04}^{+0.02}$ | 0.33${}_{-0.1}^{+0.11}$
Table 9: Average BCs for pulsar X-ray binaries and intrinsic $\rm{N_{H}}$ in 5 Eddington ratio bins. N | $\log_{min}\frac{L_{bol}}{L_{edd}}$ | $\log_{max}\frac{L_{bol}}{L_{edd}}$ | S1 | H1 | B1 | S2 | H2 | B2
---|---|---|---|---|---|---|---|---
1 | -3.54 | -2.79 | 0.06${}_{-0.02}^{+0.04}$ | 0.57$\pm 0.03$ | 0.62${}_{-0.03}^{+0.05}$ | 0.44${}_{-0.09}^{+0.04}$ | 0.25${}_{-0.05}^{+0.04}$ | 0.7${}_{-0.07}^{+0.04}$
2 | -2.79 | -2.04 | 0.03${}_{-0.01}^{+0.03}$ | 0.49${}_{-0.06}^{+0.1}$ | 0.52${}_{-0.06}^{+0.14}$ | 0.46$\pm 0.05$ | 0.28${}_{-0.06}^{+0.05}$ | 0.75$\pm 0.05$
3 | -2.04 | -1.29 | 0.02$\pm 0.01$ | 0.42${}_{-0.13}^{+0.05}$ | 0.45${}_{-0.14}^{+0.05}$ | 0.4${}_{-0.11}^{+0.05}$ | 0.32${}_{-0.03}^{+0.02}$ | 0.73${}_{-0.13}^{+0.04}$
4 | -1.29 | -0.55 | 0.02$\pm 0.01$ | 0.35${}_{-0.1}^{+0.05}$ | 0.37${}_{-0.1}^{+0.06}$ | 0.34${}_{-0.07}^{+0.03}$ | 0.33${}_{-0.05}^{+0.04}$ | 0.67${}_{-0.08}^{+0.06}$
5 | -0.55 | 0.2 | 0.02${}_{-0.01}^{+0.02}$ | 0.32$\pm 0.07$ | 0.33${}_{-0.05}^{+0.1}$ | 0.32${}_{-0.08}^{+0.13}$ | 0.32${}_{-0.08}^{+0.1}$ | 0.58${}_{-0.09}^{+0.28}$
Table 10: Average BCs for pulsar X-ray binaries and fiducial $\rm{N_{H}}$ in 5 Eddington ratio bins.  N | $\log_{min}\frac{L_{bol}}{L_{edd}}$ | $\log_{max}\frac{L_{bol}}{L_{edd}}$ | S1 | H1 | B1 | S2 | H2 | B2
---|---|---|---|---|---|---|---|---
1 | -3.5 | -2.75 | 0.16${}_{-0.05}^{+0.06}$ | 0.51${}_{-0.04}^{+0.06}$ | 0.68${}_{-0.06}^{+0.02}$ | 0.38${}_{-0.07}^{+0.08}$ | 0.22${}_{-0.04}^{+0.05}$ | 0.59${}_{-0.08}^{+0.1}$
2 | -2.75 | -2.01 | 0.08$\pm 0.03$ | 0.48${}_{-0.07}^{+0.09}$ | 0.56${}_{-0.08}^{+0.12}$ | 0.43${}_{-0.05}^{+0.04}$ | 0.26${}_{-0.05}^{+0.06}$ | 0.71${}_{-0.07}^{+0.06}$
3 | -2.01 | -1.26 | 0.06$\pm 0.02$ | 0.42${}_{-0.13}^{+0.06}$ | 0.48${}_{-0.15}^{+0.05}$ | 0.38${}_{-0.09}^{+0.05}$ | 0.31${}_{-0.03}^{+0.02}$ | 0.68${}_{-0.1}^{+0.06}$
4 | -1.26 | -0.52 | 0.04${}_{-0.02}^{+0.05}$ | 0.35${}_{-0.1}^{+0.04}$ | 0.38${}_{-0.12}^{+0.1}$ | 0.33${}_{-0.06}^{+0.03}$ | 0.32${}_{-0.06}^{+0.04}$ | 0.65${}_{-0.09}^{+0.06}$
5 | -0.52 | 0.23 | 0.04${}_{-0.03}^{+0.09}$ | 0.33${}_{-0.07}^{+0.05}$ | 0.36${}_{-0.05}^{+0.14}$ | 0.28${}_{-0.05}^{+0.16}$ | 0.28${}_{-0.07}^{+0.12}$ | 0.54${}_{-0.08}^{+0.3}$
Table 11: Average BCs for ULXps for bins 6 and 7 shown in left panel of Fig. 3 and fiducial $\rm{N_{H}}$. N | $\log_{min}\frac{L_{bol}}{L_{edd}}$ | $\log_{max}\frac{L_{bol}}{L_{edd}}$ | S1 | H1 | B1 | S2 | H2 | B2
---|---|---|---|---|---|---|---|---
6 | 0.23 | 1.48 | 0.34 | 0.44 | 0.79 | 0.31 | 0.08 | 0.4
7 | 1.48 | 3.11 | 0.16${}_{-0.06}^{+0.08}$ | 0.58${}_{-0.22}^{+0.07}$ | 0.85${}_{-0.34}^{+0.01}$ | 0.37${}_{-0.12}^{+0.08}$ | 0.08${}_{-0.01}^{+0.0}$ | 0.44${}_{-0.11}^{+0.09}$
Table 12: Average BCs for ULXs assuming they are all NS for bins 6, 7, and 8 shown in the middle panel of Fig. 3 and fiducial $\rm{N_{H}}$. N | $\log_{min}\frac{L_{bol}}{L_{edd}}$ | $\log_{max}\frac{L_{bol}}{L_{edd}}$ | S1 | H1 | B1 | S2 | H2 | B2
---|---|---|---|---|---|---|---|---
6 | 0.23 | 1.66 | 0.34 | 0.44 | 0.79 | 0.31 | 0.08 | 0.4
7 | 1.66 | 1.87 | 0.26${}_{-0.1}^{+0.1}$ | 0.47${}_{-0.05}^{+0.11}$ | 0.78${}_{-0.08}^{+0.04}$ | 0.33${}_{-0.08}^{+0.06}$ | 0.09$\pm 0.03$ | 0.43${}_{-0.11}^{+0.08}$
8 | 1.87 | 3.11 | 0.22${}_{-0.03}^{+0.06}$ | 0.51${}_{-0.02}^{+0.14}$ | 0.75${}_{-0.04}^{+0.12}$ | 0.37${}_{-0.04}^{+0.05}$ | 0.1${}_{-0.03}^{+0.05}$ | 0.49${}_{-0.09}^{+0.03}$
Table 13: Average BCs for ULXs assuming they are all BHs for bin 6 shown in the right panel of Fig. 3 and fiducial $\rm{N_{H}}$. N | $\log_{min}\frac{L_{bol}}{L_{edd}}$ | $\log_{max}\frac{L_{bol}}{L_{edd}}$ | S1 | H1 | B1 | S2 | H2 | B2
---|---|---|---|---|---|---|---|---
6 | 0.23 | 1.9 | 0.24${}_{-0.05}^{+0.07}$ | 0.5${}_{-0.05}^{+0.11}$ | 0.76${}_{-0.06}^{+0.07}$ | 0.35$\pm 0.06$ | 0.11${}_{-0.04}^{+0.03}$ | 0.47${}_{-0.1}^{+0.05}$
Table 14: Average optical V-band to X-ray 2-10 keV band ratios as a function of Eddington ratio $L_{(2-10\ keV)}/L_{edd}$ for our sample of BHXBs. N | $\log_{min}\frac{L2-10}{L_{edd}}$ | $\log_{max}\frac{L2-10}{L_{edd}}$ |  $\frac{L_{V}}{L_{X}}$ (p=0.75) |  $\frac{L_{V}}{L_{X}}$ (p=0.65) |  $\frac{L_{V}}{L_{X}}$ (p=0.55)
---|---|---|---|---|---
|  |  | $\times 10^{-5}$ | $\times 10^{-4}$ | $\times 10^{-3}$
1 | -4.52 | -4.1 | 4.05${}_{-2.21}^{+5.88}$ | 4.05${}_{-2.07}^{+5.2}$ | 9.32${}_{-4.45}^{+8.64}$
2 | -4.1 | -3.67 | 2.41${}_{-1.42}^{+6.16}$ | 2.36 ${}_{-1.26}^{+4.5}$ | 4.61${}_{-2.15}^{+6.28}$
3 | -3.67 | -3.24 | 3.4${}_{-2.69}^{+12.28}$ | 2.87${}_{-1.98}^{+8.96}$ | 5.64${}_{-3.31}^{+21.66}$
4 | -3.24 | -2.81 | 0.92${}_{-0.4}^{+4.68}$ | 1.03${}_{-0.42}^{+3.85}$ | 2.78${}_{-1.06}^{+4.67}$
5 | -2.81 | -2.39 | 1.98${}_{-1.19}^{+3.2}$ | 1.77${}_{-0.87}^{+2.27}$ | 3.54${}_{-1.21}^{+2.91}$
6 | -2.39 | -1.96 | 12.36${}_{-7.99}^{+9.81}$ | 10.58${}_{-6.35}^{+10.28}$ | 20.96${}_{-12.88}^{+25.76}$
7 | -1.96 | -1.53 | 3.44${}_{-1.11}^{+6.94}$ | 4.28${}_{-1.12}^{+6.33}$ | 14.5${}_{-4.21}^{+11.33}$
8 | -1.53 | -1.1 | 2.3${}_{-1.21}^{+0.88}$ | 2.96${}_{-1.4}^{+1.1}$ | 11.22${}_{-4.23}^{+3.63}$
9 | -1.1 | -0.68 | 1.67${}_{-0.77}^{+0.66}$ | 2.19${}_{-0.83}^{+0.77}$ | 7.95${}_{-2.14}^{+3.56}$
10 | -0.68 | -0.25 | 0.96${}_{-0.35}^{+0.47}$ | 1.45${}_{-0.37}^{+0.62}$ | 6.49${}_{-0.71}^{+2.47}$


Dark Matter from freeze-in
and its inhomogeneities

Alessandro Strumia

Dipartimento di Fisica dell’Università di Pisa, Italia
Abstract

> We consider generic freeze-in processes for generation of Dark Matter,
> together with the consequent re-thermalization of the Standard Model fluid.
> We find that Dark Matter inherits the Standard Model adiabatic
> inhomogeneities on the cosmological scales probed by current observations,
> that were super-horizon during freeze-in. Thereby, freeze-in satisfies the
> bounds on iso-curvature perturbations.

###### Contents

  1. 1 Introduction
  2. 2 Intuitive argument based on ‘separate universes’
  3. 3 Iso-curvature perturbations during freeze-in?
    1. 3.1 Intrinsic non-adiabatic energy transfer
    2. 3.2 Intrinsic non-adiabatic pressure
  4. 4 Conclusions

##  1 Introduction

Freeze-in is a possible mechanism that could have generated the Dark Matter
(DM) cosmological abundance [1]. It assumes that the Standard Model (SM)
cosmological thermal plasma was not initially accompanied by any DM abundance.
Since all SM components self-interact thermalising to a common temperature,
cosmological inhomogeneities were initially adiabatic.

Next, ‘freeze-in’ particle physics processes produce DM particles with mass
$M$ out of the SM plasma. For example, one can have decays SM $\rightarrow$ DM
DM or scatterings SM SM $\rightarrow$ DM DM, dominated either at large
temperatures $T\gg M$ (‘UV-dominated freeze-in’) or at low temperatures $T\sim
M$ (‘IR-dominated freeze-in). In order to match the observed cosmological DM
density [2], the rate of freeze-in processes must be much smaller than the
Hubble rate $H$. Freeze-in automatically generates DM inhomogeneities out of
SM inhomogeneities.

Observations are consistent with dominant adiabatic inhomogeneities (namely,
the SM/DM fluid is the same everywhere), while iso-curvature inhomogeneities
(namely, DM inhomogeneities different from SM inhomogeneities) are
constrained, on cosmological scales, to be below a few $\%$ level [2].

We consider if freeze-in leads to acceptable DM inhomogeneities.

Weinberg answered positively this issue for thermal freeze-out: since freeze-
out dominantly happens in the non-relativistic regime, computing
inhomogeneities in the DM number density was enough [3]. On the other hand,
freeze-in can be relativistic, and the iso-curvature issue started being
considered discussed recently: [4] claims that a specific freeze-in model is
excluded because it generates too large scale-independent iso-curvature
perturbations. The authors of [4] argue that all freeze-in models are
similarly problematic. In the model considered in [4] DM has a small electric
charge and is thereby produced by IR-dominated scatterings of two SM
particles, such as $e^{-}e^{+}\rightarrow{\rm DM}\,{\rm DM}$. This generates,
at any given time, a contribution to the DM density $\rho_{\rm DM}$
proportional to the square of the SM density $\rho_{\rm SM}$, and DM
inhomogeneities might be not proportional to SM inhomogeneities. However, one
must consider the cumulative cosmological process taking into account that all
regions of the Universe undergo a similarly diluting $\rho_{\rm SM}$. As we
will see, this leads to negligible iso-curvature effects. A simple argument is
presented in section 2, and the general formalism is used in section 3.
Section 4 presents our conclusions.

##  2 Intuitive argument based on ‘separate universes’

We start presenting an intuitive argument. Working in the Newtonian gauge,

$ds^{2}=-[1+2\Phi(t,\vec{x})]dt^{2}+a^{2}(t)[1-2\Psi(t,\vec{x})]d\vec{x}^{2}$
(1)

the primordial adiabatic perturbations $\delta\rho_{\alpha}(t,\vec{x})$ in the
density $\rho_{\alpha}(t)$ of a fluid $\alpha$ can be characterised in a
simple geometric way as [5, 6, 7, 3]

$\delta\rho_{\alpha}=\frac{d\rho_{\alpha}}{dt}\,\delta t$ (2)

working at first-order in the small $\delta\rho_{\alpha}\ll\rho_{\alpha}$. In
eq. (2) $\delta t(t,\vec{x})$ is some universal function common to all fluids
that can be intuitively thought as a delay in the time evolution of the
different regions.

Observation constrain iso-curvature perturbations only on scales comparable to
the horizon today, while the freeze-in DM density was generated before
matter/radiation equality (much before in most freeze-in models). This means
that we only need to worry if freeze-in generated iso-curvature perturbations
on scales much larger than the small horizon at freeze-in.

We can thus apply the ‘separate universes’ picture (see e.g. [5]): the very
early Universe at freeze-in can be thought as many homogeneous regions without
causal contact, given that inhomogeneities on different scale evolve
independently in first-order approximation. Freeze-in dynamics produces DM
with adiabatic perturbations because all regions undergo the same dynamics, up
to the delay $\delta t$. So eq. (2) holds for the DM density, no matter how
complicated the freeze-in dynamics is. Explicitly, freeze-in processes that
produce one DM particle with space-time density rate $\gamma(T)$ lead to a
constant DM number density $n_{\rm DM}$ in units of the entropy density $s$
given by

$\frac{{n}_{\rm DM}}{s}=\int\frac{dT}{T}\frac{\gamma(T)}{Hs}.$ (3)

In regions where the SM plasma was denser, freeze-in initially produced more
DM by some amount that depends on the freeze-in model, but in these region the
DM average density changed more rapidly leading to adiabatic DM
inhomogeneities.

The next section substantiates the above intuitive reasoning by explicit
computations.

##  3 Iso-curvature perturbations during freeze-in?

A general formalism to compute the cosmological evolution of inhomogeneities
in interacting fluids was developed in [8, 9]. We adopt its presentation as
summarized in [10], that makes more explicit the sources of iso-curvature
inhomogeneities.

Simple first-order evolution equations for the various densities are obtained
by combining the Einstein gravity equations into the conservation of the
energy-momentum tensor $T^{\mu\nu}=\sum_{\alpha}T^{\mu\nu}_{(\alpha)}$. The
energy-momentum tensors $T^{\mu\nu}_{(\alpha)}$ of fluid $\alpha$ only is not
conserved because interactions transfer energy-momentum $Q^{\nu}_{(\alpha)}$
to other fluids. So one has

$\nabla_{\mu}T^{\mu\nu}_{(\alpha)}=Q^{\nu}_{(\alpha)}\qquad\hbox{with}\qquad\sum_{\alpha}Q^{\nu}_{(\alpha)}=0$
(4)

because of total energy conservation. In the homogeneous limit, this implies
that the average densities evolve as
$\dot{\rho}_{\alpha}+3H(\rho_{\alpha}+\wp_{\alpha})=Q^{0}_{(\alpha)}\equiv
Q_{\alpha}$, where $\wp_{\alpha}$ is the pressure of fluid $\alpha$. The
energy component of $Q_{(\alpha)}$ is expanded in small inhomogeneities as
$Q_{(\alpha)0}=-Q_{\alpha}(1+\Phi)-\delta Q_{\alpha}$ [10] so that
$\sum_{\alpha}\delta Q_{\alpha}=0$ by total energy conservation. The total
density is $\rho=\sum_{\alpha}\rho_{\alpha}$.

Under reparametrizations $t\rightarrow t+\delta t$ one has
$\Psi\rightarrow\Psi+H\,\delta t$ and
$\delta\rho\rightarrow\delta\rho-\dot{\rho}\,\delta t$, so
$\zeta=-H[\Psi/H+\delta\rho/\dot{\rho}]$ is the reparametrization-invariant
displacement between uniform-density and uniform-curvature surfaces. This
‘curvature perturbation’ can be defined for each fluid

$\zeta_{\alpha}=-\Psi-H\frac{\delta\rho_{\alpha}}{\dot{\rho}_{\alpha}}$ (5)

and it evolves as [10]

$\dot{\zeta}_{\alpha}=-\frac{H}{\dot{\rho}_{\alpha}}\delta Q_{{\rm
intr},\alpha}+3\frac{H^{2}}{\dot{\rho}_{\alpha}}\delta\wp_{{\rm
intr},\alpha}-\dot{H}\frac{Q_{\alpha}}{\dot{\rho}_{\alpha}}\left(\frac{\delta\rho_{\alpha}}{\dot{\rho}_{\alpha}}-\frac{\delta\rho}{\rho}\right)+{\cal
O}(k^{2})$ (6)

where $\delta Q_{{\rm intr},\alpha}$ and $\delta\wp_{{\rm intr},\alpha}$ will
be defined later. As usual, small perturbations are conveniently expanded in
comoving Fourier modes $k$, and the ‘separate universe’ argument amounts to
consider the limit $k\rightarrow 0$ of the full equations. We focus on large
super-horizon scales, thereby omitting the label $k$ and neglecting Laplacians
and other terms suppressed by $k^{2}/a^{2}H^{2}$. Such terms are indeed
negligible whenever freeze-in occurs way before matter/radiation equality, for
relevant cosmological scales $k$.

The equations (6) can be written in a slightly more convenient form by
avoiding using the total density $\rho$ and defining instead the iso-curvature
relative perturbations $S_{\alpha\beta}$ between two fluids $\alpha$ and
$\beta$

$S_{\alpha\beta}\equiv
3(\zeta_{\alpha}-\zeta_{\beta})=-3H\left(\frac{\delta\rho_{\alpha}}{\dot{\rho}_{\alpha}}-\frac{\delta\rho_{\beta}}{\dot{\rho}_{\beta}}\right)$
(7)

that evolve as

$\dot{S}_{\alpha\beta}=-3H\left(\frac{\delta Q_{{\rm
intr},\alpha}-3H\,\delta\wp_{{\rm
intr},\alpha}}{\dot{\rho}_{\alpha}}-\frac{\delta Q_{{\rm
intr},\beta}-3H\,\delta\wp_{{\rm
intr},\beta}}{\dot{\rho}_{\beta}}\right)+\hbox{multiplicative}+{\cal
O}(k^{2}).$ (8)

We again ignore the terms suppressed by $k^{2}$, and we also drop the
‘multiplicative’ terms (namely, proportional to combinations of
$S_{\alpha^{\prime}\beta^{\prime}}$ terms) because we are only concerned in
understanding if non-zero iso-curvature perturbations are generated by the
‘source’ terms explicitly shown in eq. (8). The formalism summarized in [10]
makes clear that, in the long-wavelength limit $k\rightarrow 0$, iso-curvature
perturbations are only sourced by the non-adiabatic energy transfer $\delta
Q_{{\rm intr},\alpha}$ and by the non-adiabatic pressure $\delta\wp_{{\rm
intr},\alpha}$ intrinsic in each fluid $\alpha$. These terms will be now be
defined and evaluated.

###  3.1 Intrinsic non-adiabatic energy transfer

One source of iso-curvature perturbations is the intrinsic non-adiabatic
energy transfer, the part of energy transfer $\delta Q_{\alpha}$ from fluid
$\alpha$ ‘biased’ with respect to its energy density $\rho_{\alpha}$ [10]:

$\delta Q_{{\rm intr},\alpha}\equiv\delta
Q_{\alpha}-\frac{\dot{Q}_{\alpha}}{\dot{\rho}_{\alpha}}\delta\rho_{\alpha}.$
(9)

We next consider its value during freeze-in, where the relevant fluids are
$\alpha=\\{{\rm SM},{\rm DM}\\}$.

The rate of freeze-in particle collisions can be computed, in any given
particle-physics model, as a function of the local temperature of the SM
fluid, that also controls its density. Thereby the energy transfer from the SM
fluid only depends on its local density, $Q_{\rm SM}(\rho_{\rm SM})$.
Consequently $\delta Q_{{\rm intr,SM}}=\delta Q_{\rm SM}-\delta\rho_{\rm
SM}dQ_{\rm SM}/d\rho_{\rm SM}=0$ vanishes in a generic freeze-in model.

Next, energy conservation demands $\delta Q_{\rm SM}+\delta Q_{\rm DM}=0$, so
that the intrinsic non-adiabatic energy transfer to the DM fluid can be
written as

$\delta Q_{\rm intr,DM}\equiv\delta Q_{\rm DM}-\frac{\dot{Q}_{\rm
DM}}{\dot{\rho}_{\rm DM}}\delta\rho_{\rm DM}=\dot{Q}_{\rm
DM}\left(\frac{\delta\rho_{\rm SM}}{\dot{\rho}_{\rm SM}}-\frac{\delta\rho_{\rm
DM}}{\dot{\rho}_{\rm DM}}\right).$ (10)

This potential ‘source’ terms thereby becomes a ‘multiplicative’ term,
proportional to the relative entropy $S_{\rm SM,DM}$. Since this is assumed to
be initially vanishing, $\delta Q_{\rm intr,DM}$ generates no isocurvature
perturbation.

###  3.2 Intrinsic non-adiabatic pressure

The second kind of source term, the non-adiabatic part of the pressure
perturbation intrinsic of each fluid $\alpha$, is given by [10]

$\delta\wp_{{\rm
intr},\alpha}=\delta\wp_{\alpha}-c_{\alpha}^{2}\delta\rho_{\alpha}\qquad\hbox{where}\qquad
c_{\alpha}^{2}=\dot{\wp}_{\alpha}/\dot{\rho}_{\alpha}$ (11)

is its adiabatic speed of sound. This term vanishes when the pressure and
energy inhomogeneities respect the equation of state of the fluid,
$\wp_{\alpha}(\rho_{\alpha})$.

Freeze-in particle-physics processes contribute as $\delta\wp_{{\rm
intr,SM}}\neq 0$ because they convert SM particles into DM particles, thereby
inducing an energy and momentum loss of the SM fluid, as dictated by the
specific freeze-in interaction, that generically does not follow the equation
of state of the SM fluid.

As a simple example, freeze-in via the decay into DM particles of some SM
particle (or, in its extensions, of some speculative new-physics particle
tightly coupled to the SM) transfers more energy than pressure
($\dot{\rho}_{\rm SM}/\dot{\wp}_{\rm SM}>\rho_{\rm SM}/\wp_{\rm SM}$) because
it must be massive and because particles decay slower when they have higher
relativistic energy. Coming to freeze-in scatterings, they are described by a
cross-section $\sigma($SM SM $\rightarrow$ DM DM$)$ that, at leading order,
depends on invariant energy $\sqrt{s}$. The energy dependence of $\sigma$ can
either result in a larger energy transfer when the colliding SM particles have
higher energy $E\,\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}\,T$ (this can happen in UV-dominated freeze-in, via
non-renormalizable interactions, for example gravitational [11]) or when the
colliding SM particles have lower energy $E\,\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}\,T$ (this can
happen in IR-dominated freeze-in, via renormalizable interactions). As another
possibility, $\delta\wp_{{\rm intr,SM}}$ is nearly-vanishing in freeze-in
models that only lead to the disappearance of ultra-relativistic SM particles,
as they (on angular average) satisfy the same equation of state $\wp=\rho/3$
as the radiation-dominated SM fluid.

However, the fact that freeze-in scatterings can contribute as
$\delta\wp_{{\rm intr,SM}}\neq 0$ is inconsequential, as we must also take
into account the self-interactions of the SM fluid. A multitude of SM particle
processes allow the SM fluid to locally re-thermalize to its equation of state
with rates $\Gamma$ much faster than the Hubble rate and than the freeze-in
rate. Typically $\Gamma\sim g^{2}T$ where $g\sim 1$ is a typical SM coupling.
The re-thermalizion processes conserve the SM energy $\rho_{\rm SM}$: $\delta
Q_{\rm SM}$ remains given by freeze-in processes only, so that $\delta Q_{{\rm
intr,SM}}=0$ remains as in section 3.1. On the other hand, the SM pressure
$\wp_{\rm SM}$ changes such that the sum of the two processes (freeze-in and
re-thermalization) leads to $\delta\wp^{\rm total}_{{\rm intr,SM}}=0$.

This leaves $\delta\wp_{{\rm intr,DM}}$ as a possible source of iso-
curvatures. A self-thermalization argument parallel to what discussed for the
SM plasma implies $\delta\wp_{{\rm intr,DM}}=0$ if DM has significant self-
interactions just after being produced during freeze-in. This happens, for
example, if DM is a multiplet under a dark gauge group [12] that confines at a
scale $\Lambda$ and if freeze-in happens at $T\gg\Lambda$. If instead DM self-
interactions are negligible, a formalism extended to higher moments may be
needed, but the physics is simple: DM particles free stream on sub-horizon
scales, but not on large scales $k\rightarrow 0$. The non-thermal DM
distribution $f(\vec{x},t,q)=f_{0}(q)+\delta f$ produced by freeze-in
redshifts with scale factor $a$ as [13]

$\rho_{\rm DM}=\frac{1}{a^{4}}\int\frac{d^{3}q}{(2\pi)^{3}}E\,f,\qquad\wp_{\rm
DM}=\frac{1}{a^{4}}\int\frac{d^{3}q}{(2\pi)^{3}}\frac{q^{2}}{3E}\,f$ (12)

where $q$ and $E=\sqrt{q^{2}+a^{2}M^{2}}$ are the comoving momentum and
energy. Two limits are of special interest. If freeze-in is IR-dominated, DM
is only mildly relativistic, so that DM motion is soon red-shifted down to
negligible pressure, $\wp_{\rm DM}\ll\rho_{\rm DM}$. UV-dominated freeze-in
can produce ultra-relativistic DM with $\wp_{\rm DM}/\rho_{\rm
DM}\simeq\dot{\wp}_{\rm DM}/\dot{\rho}_{\rm DM}\simeq 1/3$, that becomes non-
relativistic only later when the SM cools down to temperatures comparable to
the DM mass $M$, while the horizon reaches larger scales.

##  4 Conclusions

We considered generic models of freeze-in (from decays, from scatterings, IR-
dominated, UV-dominated…) finding that the generated Dark Matter inherits the
Standard Model adiabatic inhomogeneities on the cosmological scales probed by
current observations, that were super-horizon during freeze-in. In section 2
we presented an intuitive argument based on the well-known ‘separate universe’
picture. This was substantiated in section 3 by checking the explicit sources
of iso-curvature perturbations on super-horizon scales.

Iso-curvature perturbations can only be generated on small scales that were
sub-horizon during freeze-in: this effect can perhaps be relevant in models
where freeze-in happens at the lowest possible temperature $T\sim M\sim{\rm
keV}$, possibly in the presence of dark long-range forces.

In conclusion, freeze-in appears a viable mechanism for generation of the
cosmological DM abundance. Similar arguments hold for other particle-physics
mechanisms such as ‘cannibalism’ [14] or ‘freeze-out and decay’.

#### Acknowledgements

We thank Guido d’Amico, Alessio Notari, Paolo Panci, Michele Redi, Andrea Tesi
for discussions. This work was supported by the MIUR grant PRIN 2017L5W2PT.

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