The Origin of the Ultraluminous X-Ray Sources

We implemented a model based on nonlinear scaling of X-ray luminosity and mass accretion rate (e.g., Shakura & Sunyaev 1973),

$$\begin{eqnarray}&&{L}_{{\rm{X}},\mathrm{tot}}={L}_{\mathrm{Edd}}(1+\mathrm{ln}\,\dot{m}),\end{eqnarray} \tag{ 1 }$$

where $\,\dot{m}$ is the mass transfer (MT) rate in Eddington rate units. Mass accretion onto the compact object $(\,{\dot{M}}_{\mathrm{acc}})$ is limited to $\,{\dot{M}}_{\mathrm{Edd}}$ and the rest of the mass transferred to the system is lost in a wind from the inner disk region. Such outflows from ULXs were recently observed by Pinto et al. (2016).

The collimation of radiation from the innermost parts of the accretion disk, i.e., beaming, has a prodigious influence on the apparent isotropic luminosity. The beaming factor is defined as $b={\rm{\Omega }}/4\pi$, where Ω is the solid angle of emission (e.g., King et al. 2001). If we assume that there are two conical beams with opening angle θ, then ${\rm{\Omega }}=4\pi (1-\cos \theta /2)$.

The isotropic equivalent X-ray luminosity (${L}_{{\rm{X}}}$) can be expressed as

$$\begin{eqnarray}&&{L}_{{\rm{X}}}=\displaystyle \frac{{L}_{{\rm{X}},\mathrm{tot}}}{b}.\end{eqnarray} \tag{ 2 }$$

Under the assumption of the isotropic distribution of disk inclinations in space (see King 2009), the probability of observing a source along the beam is equal b.

The results of theoretical studies as well as the outcomes of detailed modeling of accretion disks favor the dependence of beaming on the MT rate (e.g., Lasota 2016). King (2009) proposed that b scales as

$$\begin{eqnarray}b & \sim & \displaystyle \frac{73}{\,{\dot{m}}^{2}}\quad \,\dot{m}\geqslant 8.5,\\ b & = & 1\qquad \,\dot{m}\lt 8.5.\end{eqnarray} \tag{ 3 }$$

For very high MT rates, this prescription may give an extremely small value of b, i.e., strong beaming. For example, a Galactic ULX candidate SS433 has $\,\dot{m}\approx 3000\mbox{--}{10}^{4}$ (e.g., Fabrika 2004) for which Equations (3) and (2) give $b\approx 7\times {10}^{-7}\mbox{--}8\times {10}^{-6}$ and ${L}_{{\rm{X}}}\approx {10}^{46}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. Therefore, we will assume saturation at $\,\dot{m}=150$. For higher mass accretion rates, we will consider that the beaming is constant and equal $b\approx 3.2\times {10}^{-3}\,(\theta \approx 9^\circ )$. Theoretical background for beaming saturation was presented by Lasota et al. (2016).

According to our results, NSULX are present in ULX populations of all ages and metallicities. The first NSULXs form as early as $\sim 6\,\mathrm{Myr}$ after the start of SF, but the ULX phase may occur also in a very old system (${t}_{\mathrm{age}}\sim 5\,\mathrm{Gyr}$). Dominant evolutionary routes leading to the occurrence of high MT rate and formation of an NSULX depend on the age of the system at the time of the ULX phase. Below, we describe the typical routes for early (${{ \mathcal R }}_{\mathrm{NS},\mathrm{MS}}$), mid-age (${{ \mathcal R }}_{\mathrm{NS},\mathrm{HG}}$), and old (${{ \mathcal R }}_{\mathrm{NS},\mathrm{RG}}$) stellar populations (see also Figure 1). Ranges represent 50% of values in present day populations.⁷

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6–59 $\,\mathrm{Myr}$; Route ${{ \mathcal R }}_{\mathrm{NS},\mathrm{MS}}$: Typical companion MS—On ZAMS, the masses of the primary and secondary are $7.2\mbox{--}12\,{M}_{\odot }$ and $1.3\mbox{--}2.0\,{M}_{\odot }$, respectively. The initial separation, $500\mbox{--}2600\,{R}_{\odot }$, shrinks during the CE phase commenced by the primary. After the CE phase, the primary forms an NS through a core collapse supernova explosion (SN). The natal kick leaves the system on an orbit that is tight enough for the secondary to fill its Roche lobe (RL) within about $2\,\mathrm{kyr}$ due to nuclear expansion and gravitational radiation (GR) before it leaves the main sequence (MS). During the RLOF, the companion is comparable in mass ($0.9\mbox{--}1.5\,{M}_{\odot }$) to the NS ($1.1\mbox{--}1.4\,{M}_{\odot }$), thus the MT is stable and may be sustained as long as $\sim 1.3\,\mathrm{Myr}$ (see Figure 1, top).

400–820 $\,\mathrm{Myr}$; Route ${{ \mathcal R }}_{\mathrm{NS},\mathrm{HG}}$: Typical companion HG—In approximately $54 \% \mbox{--}73 \%$ percent of the systems following the ${{ \mathcal R }}_{\mathrm{NS},\mathrm{MS}}$ route, the RLOF during the MS phase, if present, is not strong enough to power a ULX. These secondaries start to reach their terminal-age MS and expand rapidly several hundred $\,\mathrm{Myr}$ after the start of the SF. If the separation is short enough, they will fill their RL. Thus, at this time, a Hertzsprung-gap (HG) star becomes the most common NSULX companion.

1200–4300 $\,\mathrm{Myr}$; Route ${{ \mathcal R }}_{\mathrm{NS},\mathrm{RG}}$: Typical companion RG—The evolution of the late-time NSULXs is significantly different than that of the earlier ULXs. After a CE phase, the primary forms an oxygen–neon white dwarf (ONeWD) instead of an NS, and thus the mass of the primary is on average lower compared with routes ${{ \mathcal R }}_{\mathrm{NS},\mathrm{MS}}$ and ${{ \mathcal R }}_{\mathrm{NS},\mathrm{HG}}$. However, when the secondary becomes a red giant (RG) and fills its RL of $\sim 10\mbox{--}20\,{R}_{\odot }$, the primary accretes additional mass and forms an NS due to an accretion induced collapse (AIC). Afterward, the RG refills its RL and a short ULX phase occurs (Figure 1, middle).

3.1.1. Formation of BHULXs

By the number of ULXs (#ULX), we understand the predicted number of observed systems at the present time. We find that #ULX strongly depends on the SFR history (Figure 2).

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Constant SFR naturally leads to a constant growth in #ULX, which starts early after the start of SF ($\sim 5\,\mathrm{Myr}$) and after a few gigayears reaches a constant value of $\#\mathrm{ULX}=9.3$, 100, and 56 for ${Z}_{\odot }$, ${Z}_{\odot }$/10, and ${Z}_{\odot }$/100, respectively (see Section 4.2).

BHULXs appear first in both constant and burst SF scenarios. In the former case, the number of BHULXs increases to the age of a few hundred megayears, at which time it reaches a constant value of 3.8, 92, and 51 for ${Z}_{\odot }$, ${Z}_{\odot }/10$, and ${Z}_{\odot }/100$, respectively. In the case of a burst SF, the formation time of BHULXs is limited nearly exactly to the duration of the burst ($\lesssim 200\,\mathrm{Myr}$).

NSULXs resulting from different evolutionary routes described in Section 3.1 appear sequentially in their chronological order and form the characteristic steep sections (constant SF) or humps (burst SF) at $\sim 100\,\mathrm{Myr}$ (${{ \mathcal R }}_{\mathrm{NS},\mathrm{MS}}$), $\sim 200\,\mathrm{Myr}$ (${{ \mathcal R }}_{\mathrm{NS},\mathrm{HG}}$), and $\sim 1\,\mathrm{Gyr}$ (${{ \mathcal R }}_{\mathrm{NS},\mathrm{RG}}$).

ULX populations younger than $\sim 100\,\mathrm{Myr}$ are dominated by the BHULX. For constant SF, the NSULXs become more abundant than the BHULXs (58% of the ULX accretors) only if $Z={Z}_{\odot }$. For lower metallicities, the #BHULX grows rapidly during the first few hundred megayears, whereas the rate of the NSULXs formation is much lower, and they are not able to match the abundance of the BHULXs during the next $10\,\mathrm{Gyr}$. As a result, the NSULXs account for not more than $\sim 12 \%$ and $\sim 8 \%$ of all ULXs for ${Z}_{\odot }$/10 and ${Z}_{\odot }$/100, respectively. However, for burst SF and all investigated metallicities, the NSULXs exceed the BHULXs in numbers within $\lesssim 1\,\mathrm{Gyr}$, and after a few gigayears they become the only surviving ULXs.

Several surveys found a break in the X-ray luminosity function (XLF) at $\lesssim 2\times {10}^{10}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, which may point toward different populations below and above this luminosity (e.g., Grimm et al. 2003; Swartz et al. 2011; Mineo et al. 2012). Thus, it is interesting to point out that hyper-luminous X-ray sources (HLX) defined as ULXs with ${L}_{{\rm{X}}}\gtrsim {10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, thus clearly above the $2\times {10}^{10}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ threshold, have different donors than the ULXs with ${L}_{{\rm{X}}}\ll {10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. We find that $52 \% \mbox{--}84 \%$ of the ULXs (described in Section 3) have luminosities in the range of $(1\mbox{--}3)\times {10}^{39}$ erg s⁻¹.

Approximately 90% of HLXs are BHULXs with HG donors. NS accretors in HLXs may obtain nearly as high luminosities as BH accretors and are typically accompanied by evolved helium stars (evHeS). It is noteworthy that the evolutionary HLX routes match those reported for HLXs with ${L}_{{\rm{X}}}\geqslant {10}^{42}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ in Wiktorowicz et al. (2015). The current results, however, are based on a much wider range of models and a more physical treatment of NS/BH accretion, and we include here the HLX evolutionary paths for completeness (Table 2). Notice that the PULX NGC5907 ULX1 has a luminosity $\gtrsim {10}^{41}$ erg s⁻¹ (Fürst et al. 2016).

15–60 $\,\mathrm{Myr}$; Route ${{ \mathcal R }}_{\mathrm{BH},\mathrm{HG}}$: Typical companion HG—A typical system begins its evolution as a $25\mbox{--}50\,{M}_{\odot }$ primary with a $5\mbox{--}10\,{M}_{\odot }$ companion. After $\sim 5\,\mathrm{Myr}$ and the loss of $1\mbox{--}10\,{M}_{\odot }$ in stellar wind the primary fills the RL as a CHeB star and commences the CE phase. It loses a large fraction of its mass ($\sim 50 \%$), but the separation shrinks to $\sim 15\,{R}_{\odot }$. After additional $\sim 0.5\,\mathrm{Myr}$ a $\sim 7\mbox{--}18\,{M}_{\odot }$ BH forms in a direct collapse. The secondary becomes an HG star $\sim 10\mbox{--}55\,\mathrm{Myr}$ later and it expands rapidly. Shortly after, it fills its RL and starts the MT. For a short time ($\lt 0.1\,\mathrm{Myr}$), the MT is high enough to power a ULX with ${L}_{{\rm{X}}}\gt {10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (see Figure 3, top).

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17–75 $\,\mathrm{Myr}$; Route ${{ \mathcal R }}_{\mathrm{NS},\mathrm{evHeS}}$: Typical companion evHeS–The stars on ZAMS are less massive and have smaller mass ratios than in the case of ${{ \mathcal R }}_{\mathrm{BH},\mathrm{HG}}$. When the primary fills the RL as an RG, the MT is stable and proceeds for a few tens of $\,\mathrm{kyr}$. The donor losses most of its hydrogen envelope and after a few $\,\mathrm{Myr}$ at the age of $\sim 25\,\mathrm{Myr}$ it forms an NS through an SN explosion. The companion, being an RG, expands and fills its RL. This time the MT is unstable due to a high-mass ratio and the CE occurs. Afterward, the secondary becomes a low-mass (a few Solar masses) helium star (HeS). The secondary continues its evolution for the next few $\,\mathrm{Myr}$ s, then expands and fills its RL again. This time, although being very strong, the MT is stable and for $\sim 1\,\mathrm{kyr}$ it powers an HLX (see Figure 3, bottom).

Table 1.  Typical Parameters of ULXs

	Present					ZAMS
Route	t(Myr)	Δ t(Myr)	${M}_{{\rm{a}}}$(M⊙)	${M}_{{\rm{b}}}$(M⊙)	a(R⊙)	${M}_{\mathrm{ZAMS},{\rm{a}}}$(M⊙)	${M}_{\mathrm{ZAMS},{\rm{b}}}$(M⊙)	${a}_{\mathrm{ZAMS}}$(R⊙)
Z = ${Z}_{\odot }$
${{ \mathcal R }}_{\mathrm{BH},\mathrm{MS}}$	11–22	<0.2	7.7–8.6	5.9–7.2	18–22	40–50	6.6–12	3800–4600
ev. route: CE1(4-1; 7/8-1) SN1 MT2(14-1)
${{ \mathcal R }}_{\mathrm{NS},\mathrm{MS}}$	6–38	≲0.4	1.1–1.3	1.1–1.4	2.2–3.6	8.2–9.3	1.3–1.7	700–1200
ev. route: CE1(3/4/5-1; 7/8-1) SN1 MT2(13-1)
${{ \mathcal R }}_{\mathrm{NS},\mathrm{HG}}$	430–730	0.3–0.8	∼1.3	0.6–1.0	7.3–17	7.9–8.3	1.6–2.0	800–1200
ev. route: CE1(3/4/5-1; 7/8-1) SN1 MT2(13-1/2)
${{ \mathcal R }}_{\mathrm{NS},\mathrm{RG}}$	2300–4300	0.1–0.2	∼1.3	∼1.0	35–50	7.5–7.9	1.2–1.5	1100–1700
ev. route: CE1(6-1; 12-1) MT2(12-3) AICNS1 MT2(13-3)
Z = ${Z}_{\odot }$/10
${{ \mathcal R }}_{\mathrm{BH},\mathrm{MS}}$	4–37	<27	5.7–8.9	5.6–7.8	14–19	26–37	6.0–11	1800–3700
${{ \mathcal R }}_{\mathrm{NS},\mathrm{MS}}$	9–55	≲0.8	1.1–1.3	1.2–1.5	1.6–2.7	7.2–12	1.5–1.9	1000–2600
${{ \mathcal R }}_{\mathrm{NS},\mathrm{HG}}$	400–820	≲0.6	1.2–1.3	0.6–0.9	9.3–22	6–8	1.3–1.8	600–1500
${{ \mathcal R }}_{\mathrm{NS},\mathrm{RG}}$	1400–2600	≲0.1	∼1.3	∼1.0	35–45	6.3–6.8	1.3–1.6	1700–2700
Z = ${Z}_{\odot }$/100
${{ \mathcal R }}_{\mathrm{BH},\mathrm{MS}}$	8.7–19	<1.0	12–21	7–11	12–17	35–55	9.1–14	540–2000
${{ \mathcal R }}_{\mathrm{NS},\mathrm{MS}}$	15–59	0.6–1.3	1.2–1.4	0.9–1.0	2.8–3.5	7.2–8.2	1.7–2.0	500–900
${{ \mathcal R }}_{\mathrm{NS},\mathrm{HG}}$	400–690	≲0.4	∼1.3	0.6–0.8	17–35	6.0–6.3	1.7–2.1	700–900
${{ \mathcal R }}_{\mathrm{NS},\mathrm{RG}}$	1200–2100	≲0.1	∼1.3	∼1.0	30–45	6.0–6.2	1.3–1.7	900–1300

Note. Typical present and ZAMS parameters of most common ULX evolutionary routes. Ranges represent 50% of values in present day populations. The parameter columns represent, respectively: age of the system in ULX phase (t), duration of ULX phase (Δt), compact object mass (${M}_{{\rm{a}}}$), donor mass (${M}_{{\rm{b}}}$), separation (a), primary mass on ZAMS (${M}_{\mathrm{ZAMS},{\rm{a}}}$), secondary mass on ZAMS (${M}_{\mathrm{ZAMS},{\rm{b}}}$), separation on ZAMS (${a}_{\mathrm{ZAMS}}$). The schematic evolutionary routes provided for Z = ${Z}_{\odot }$ are the same for other metallicities. Symbols represent (Belczynski et al. 2008) CE1—common envelope (donor: primary); MT2—mass transfer (donor: secondary); SN1—supernova (primary); AICNS1—accretion induced collapse (primary); 1—MS; 2—HG; 3—RG; 4—CHeB; 5—early AGB; 6—thermal-pulsing AGB; 7—HeS; 8—evHeS; 12—ONeWD; 13—NS; 14—BH.

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Table 2.  Typical Parameters of HLXs

	Present					ZAMS
Route	t(Myr)	Δ t(Myr)	${M}_{{\rm{a}}}$(M⊙)	${M}_{{\rm{b}}}$(M⊙)	a(R⊙)	${M}_{\mathrm{ZAMS},{\rm{a}}}$(M⊙)	${M}_{\mathrm{ZAMS},{\rm{b}}}$(M⊙)	${a}_{\mathrm{ZAMS}}$(R⊙)
Z = ${Z}_{\odot }$
${{ \mathcal R }}_{\mathrm{BH},\mathrm{HG}}$	15–40	≲0.08	8.0–10	1.7–3.3	20–90	40–50	3.8–8.2	3700–4600
ev. route CE1(4/5-1; 7/8-1) SN1 MT2(14-1/2)
${{ \mathcal R }}_{\mathrm{NS},\mathrm{evHeS}}$	17–40	≲0.001	1.3–1.4	1.7–2.6	≲4.4	10–11	8.5–10	30–300
ev. route MT1(2/3/4/5/6-1/2/4) SN1 CE2(13-3/4/5; 13-7/8) MT2(13-8/9)
Z = ${Z}_{\odot }$/10
${{ \mathcal R }}_{\mathrm{BH},\mathrm{HG}}$	20–60	≲0.06	7.2–9.5	1.7–3.3	12–120	25–35	4.0–7.0	2000–5000
${{ \mathcal R }}_{\mathrm{NS},\mathrm{evHeS}}$	40–60	≲0.001	∼1.3	1.5–2.0	8.4–13	8.8–9.5	5.5–7.3	170–430
Z = ${Z}_{\odot }$/100
${{ \mathcal R }}_{\mathrm{BH},\mathrm{HG}}$	20–45	≲0.04	10–18	1.2–3.7	10–75	25–45	5.0–9.0	700–2600
${{ \mathcal R }}_{\mathrm{NS},\mathrm{evHeS}}$	50–75	≲0.001	∼1.3	1.9–2.5	2.1–6.9	6.4–6.8	5.5–6.2	170–280

Note. Typical present and ZAMS parameters of typical most luminous ULXs (HLX; routes ${{ \mathcal R }}_{\mathrm{BH},\mathrm{HG}}$ and ${{ \mathcal R }}_{\mathrm{NS},\mathrm{evHeS}}$). The table is organized in the same way as Table 1.

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Table 4.  #ULX for Burst and Constant SFR

		Time Since the Start of Star Formation
		Burst SFRa				Constant SFR
Metallicity		100 $\,\mathrm{Myr}$	1 $\,\mathrm{Gyr}$	5 $\,\mathrm{Gyr}$	10 $\,\mathrm{Gyr}$	10 $\,\mathrm{Gyr}$
	#ULX	4.0 × 102	1.8 × 101	9.5 × 10−1	1.1 × 10−1	9.3 × 100
${Z}_{\odot }$	#BHULX	3.7 × 102	2.8 × 10−7	⋯	⋯	3.8 × 100
	#NSULX	3.5 × 101	1.8 × 101	9.5 × 10−1	1.1 × 10−1	5.5 × 100
	#ULX	7.3 × 103	1.1 × 102	7.0 × 10−1	7.5 × 10−2	1.0 × 102
${Z}_{\odot }$/10	#BHULX	7.2 × 103	4.0 × 100	⋯	⋯	9.2 × 101
	#NSULX	8.1 × 101	1.1 × 102	7.0 × 10−1	7.5 × 10−2	1.3 × 101
	#ULX	5.0 × 103	2.9 × 101	1.0 × 10−1	5.0 × 10−7	5.6 × 101
${Z}_{\odot }$/100	#BHULX	4.9 × 103	⋯	⋯	⋯	5.1 × 101
	#NSULX	1.6 × 101	2.9 × 101	1.0 × 10−1	5.0 × 10−7	5.0 × 100

Note. Number of ULXs (#ULX) in the reference model with a division on BHULXs (#BHULX) and NSULXs (#NSULX) for different population ages. BHULX are also present late after the burst ($\sim 1\,\mathrm{Gyr}$), but #BHULX is negligible in that population's age in comparison to #NSULX or the #BHULX during the burst.

^aBurst duration is $100\,\mathrm{Myr}$.

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Table 5.  #ULX(${L}_{{\rm{X}},\min }$) $100\,\mathrm{Myr}$ after SF Start

		${L}_{{\rm{X}},\min }$
Metallicity		${10}^{39}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$	$3\times {10}^{39}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$	${10}^{40}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$	${10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$
	#ULX	4.0 × 102	6.5 × 101	1.4 × 101	5.0 × 10−1
${Z}_{\odot }$	#BHULX	3.7 × 102	4.0 × 101	4.7 × 100	3.8 × 10−1
	#NSULX	3.5 × 101	2.5 × 101	9.4 × 100	1.3 × 10−1
	#ULX	7.3 × 103	1.9 × 103	2.0 × 102	1.2 × 101
${Z}_{\odot }$/10	#BHULX	7.2 × 103	1.9 × 103	1.8 × 102	1.1 × 101
	#NSULX	8.1 × 101	4.0 × 101	1.2 × 101	4.3 × 10−1
	#ULX	5.0 × 103	2.4 × 103	3.9 × 102	1.5 × 101
${Z}_{\odot }$/100	#BHULX	4.9 × 103	2.4 × 103	3.8 × 102	1.5 × 101
	#NSULX	1.6 × 101	1.2 × 101	9.0 × 100	7.5 × 10−2

Note. Number of ULXs depending on the minimal luminosity (${L}_{{\rm{X}},\min }$) for different metallicities with a division on BHULXs and NSULXs for AD1 BK model.

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Table 6.  Number of ULXs and Maximal Luminosity (Constant SFR^a)

Modelb	Number per MWEGc			${L}_{{\rm{X}},\max }(\mathrm{erg}\,{{\rm{s}}}^{-1})$ d
	#ULX	#BHULX	#NSULX	BHULX	NSULX
	${Z}_{\odot }$
AD0 BN	3.1 × 101	3.1 × 100 (10%)	2.8 × 101 (89%)	8.4 × 1044	3.3 × 1044
AD0 B01	1.8 × 102	1.1 × 100 (0%)	1.8 × 102 (99%)	8.4 × 1045	3.3 × 1045
AD0 BK	2.2 × 101	2.7 × 100 (12%)	1.9 × 101 (87%)	2.6 × 1047	1.0 × 1047
AD0 BS	2.0 × 101	2.3 × 100 (11%)	1.8 × 101 (88%)	5.5 × 1045	2.2 × 1045
AD1 BN	1.5 × 101	4.4 × 100 (29%)	1.1 × 101 (70%)	3.0 × 1040	5.1 × 1039
AD1 B01	2.0 × 102	1.5 × 100 (0%)	2.0 × 102 (99%)	3.0 × 1041	5.1 × 1040
AD1 BK	9.3 × 100	3.8 × 100 (41%)	5.5 × 100 (58%)	9.2 × 1042	1.6 × 1042
AD1 BS	2.0 × 101	2.9 × 100 (14%)	1.7 × 101 (85%)	2.0 × 1041	3.4 × 1040
	${Z}_{\odot }/10$
AD0 BN	1.6 × 102	9.4 × 101 (58%)	6.7 × 101 (41%)	6.8 × 1045	2.3 × 1044
AD0 B01	3.7 × 102	4.3 × 101 (11%)	3.3 × 102 (88%)	6.8 × 1046	2.3 × 1045
AD0 BK	1.3 × 102	8.4 × 101 (63%)	5.0 × 101 (36%)	2.1 × 1048	7.1 × 1046
AD0 BS	1.1 × 102	6.4 × 101 (60%)	4.1 × 101 (39%)	4.5 × 1046	1.5 × 1045
AD1 BN	1.3 × 102	1.1 × 102 (83%)	2.1 × 101 (16%)	8.6 × 1040	4.5 × 1039
AD1 B01	4.2 × 102	5.0 × 101 (11%)	3.7 × 102 (88%)	8.6 × 1041	4.5 × 1040
AD1 BK	1.0 × 102	9.2 × 101 (87%)	1.3 × 101 (12%)	2.7 × 1043	1.4 × 1042
AD1 BS	1.1 × 102	6.8 × 101 (61%)	4.2 × 101 (38%)	5.7 × 1041	3.0 × 1040
	${Z}_{\odot }/100$
AD0 BN	8.5 × 101	6.1 × 101 (72%)	2.3 × 101 (27%)	8.9 × 1045	2.5 × 1044
AD0 B01	8.6 × 101	1.4 × 101 (16%)	7.2 × 101 (83%)	8.9 × 1046	2.5 × 1045
AD0 BK	7.2 × 101	5.6 × 101 (78%)	1.6 × 101 (21%)	2.7 × 1048	7.7 × 1046
AD0 BS	4.9 × 101	3.8 × 101 (78%)	1.1 × 101 (21%)	5.9 × 1046	1.6 × 1045
AD1 BN	6.6 × 101	5.9 × 101 (89%)	7.2 × 100 (10%)	1.3 × 1041	5.5 × 1039
AD1 B01	8.0 × 101	1.6 × 101 (19%)	6.4 × 101 (80%)	1.3 × 1042	5.5 × 1040
AD1 BK	5.6 × 101	5.1 × 101 (91%)	5.0 × 100 (8%)	4.0 × 1043	1.7 × 1042
AD1 BS	5.0 × 101	3.7 × 101 (73%)	1.3 × 101 (26%)	8.6 × 1041	3.6 × 1040

Notes.

^aAt the age of 10 $\,\mathrm{Gyr}$. ^bAD0/1—"upper limit"/logarythmic accretion model (Appendix A.1 and Section 2.1); BN/B01/BK/BS—beaming models (Appendix A.2 and Section 2.2). ^cMilky Way Equivalent Galaxy. ^dThe maximal obtained luminosity for the particular accretor. In all cases, the beaming that corresponds to these luminosities is saturated, so the beaming factor b = 1, 0.1, $\sim 3.2\times {10}^{-3}$, and ∼0.15 for BN, B01, BK, and BS, respectively. Corresponding opening angles are 180°, $\sim 52^\circ$, $\sim 9^\circ$, and $\sim 64^\circ$, respectively.

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Table 7.  General Parameters of the Simulational Results (Burst SFR $100\,\mathrm{Myr}$ Ago)

Modela	Number per MWEGa			${L}_{{\rm{X}},\max }(\mathrm{erg}\,{{\rm{s}}}^{-1})$b
	#ULX	#BHULX	#NSULX	BHULX	NSULX
	${Z}_{\odot }$
AD0 BN	3.4 × 102	2.8 × 102 (83%)	5.8 × 101 (16%)	8.4 × 1044	3.3 × 1044
AD0 B01	4.7 × 102	9.3 × 101 (19%)	3.8 × 102 (80%)	8.4 × 1045	3.3 × 1045
AD0 BK	2.8 × 102	2.6 × 102 (90%)	2.8 × 101 (9%)	2.6 × 1047	1.0 × 1047
AD0 BS	2.8 × 102	2.1 × 102 (75%)	6.8 × 101 (24%)	5.5 × 1045	2.2 × 1045
AD1 B01	5.7 × 102	1.3 × 102 (22%)	4.4 × 102 (77%)	3.0 × 1041	5.1 × 1040
AD1 BK	4.0 × 102	3.7 × 102 (91%)	3.5 × 101 (8%)	9.2 × 1042	1.6 × 1042
AD1 BS	3.8 × 102	2.8 × 102 (75%)	9.4 × 101 (24%)	2.0 × 1041	3.4 × 1040
	${Z}_{\odot }/10$
AD0 BN	7.5 × 103	7.2 × 103 (96%)	2.9 × 102 (3%)	6.8 × 1045	1.2 × 1044
AD0 B01	3.0 × 103	3.0 × 103 (98%)	4.9 × 101 (1%)	6.8 × 1046	1.2 × 1045
AD0 BK	6.6 × 103	6.4 × 103 (98%)	1.2 × 102 (1%)	2.1 × 1048	3.7 × 1046
AD0 BS	5.2 × 103	5.1 × 103 (98%)	6.7 × 101 (1%)	4.5 × 1046	7.9 × 1044
AD1 BN	8.6 × 103	8.5 × 103 (98%)	1.0 × 102 (1%)	8.6 × 1040	4.5 × 1039
AD1 B01	3.9 × 103	3.8 × 103 (96%)	1.4 × 102 (3%)	8.6 × 1041	4.5 × 1040
AD1 BK	7.3 × 103	7.2 × 103 (98%)	8.1 × 101 (1%)	2.7 × 1043	1.4 × 1042
AD1 BS	5.7 × 103	5.6 × 103 (97%)	1.5 × 102 (2%)	5.7 × 1041	3.0 × 1040
	${Z}_{\odot }/100$
AD0 BN	5.9 × 103	5.9 × 103 (99%)	2.8 × 101 (0%)	8.9 × 1045	2.1 × 1043
AD0 B01	1.3 × 103	1.3 × 103 (99%)	4.4 × 100 (0%)	8.9 × 1046	2.1 × 1044
AD0 BK	5.4 × 103	5.4 × 103 (99%)	1.1 × 101 (0%)	2.7 × 1048	6.5 × 1045
AD0 BS	3.7 × 103	3.7 × 103 (99%)	6.9 × 100 (0%)	5.9 × 1046	1.4 × 1044
AD1 BN	5.7 × 103	5.7 × 103 (99%)	1.4 × 101 (0%)	1.3 × 1041	5.5 × 1039
AD1 B01	1.5 × 103	1.5 × 103 (96%)	4.9 × 101 (3%)	1.3 × 1042	5.5 × 1040
AD1 BK	5.0 × 103	4.9 × 103 (99%)	1.6 × 101 (0%)	4.0 × 1043	1.7 × 1042
AD1 BS	3.7 × 103	3.6 × 103 (98%)	5.8 × 101 (1%)	8.6 × 1041	3.6 × 1040

Note. The same as in Table 6, but for burst SF, which started $100\,\mathrm{Myr}$ ago and lasted for $100\,\mathrm{Myr}$.

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Table 8.  General Parameters of the Simulational Results (Burst SFR $1\,\mathrm{Gyr}$ Ago)

Modela	Number per MWEGa			${L}_{{\rm{X}},\max }(\mathrm{erg}\,{{\rm{s}}}^{-1})$b
	#ULX	#BHULX	#NSULX	BHULX	NSULX
	${Z}_{\odot }$
AD0 BN	8.3 × 101	8.5 × 10−3 (0%)	8.3 × 101 (99%)	2.0 × 1042	2.2 × 1042
AD0 B01	3.9 × 102	3.6 × 10−1 (0%)	3.9 × 102 (99%)	2.0 × 1043	2.2 × 1043
AD0 BK	6.0 × 101	3.5 × 10−4 (0%)	6.0 × 101 (99%)	6.2 × 1044	6.8 × 1044
AD0 BS	5.5 × 101	1.5 × 10−3 (0%)	5.5 × 101 (99%)	1.3 × 1043	1.4 × 1043
AD1 BN	3.8 × 101	8.7 × 10−5 (0%)	3.8 × 101 (99%)	3.7 × 1039	3.9 × 1039
AD1 B01	5.2 × 102	1.2 × 10−1 (0%)	5.2 × 102 (99%)	3.7 × 1040	3.9 × 1040
AD1 BK	1.8 × 101	2.8 × 10−7 (0%)	1.8 × 101 (99%)	1.1 × 1042	1.2 × 1042
AD1 BS	8.0 × 101	1.3 × 10−5 (0%)	8.0 × 101 (99%)	2.4 × 1040	2.6 × 1040
	${Z}_{\odot }/10$
AD0 BN	6.8 × 102	4.1 × 100 (0%)	6.8 × 102 (99%)	4.8 × 1042	1.4 × 1043
AD0 B01	8.5 × 102	1.2 × 101 (1%)	8.4 × 102 (98%)	4.8 × 1043	1.4 × 1044
AD0 BK	5.7 × 102	3.9 × 100 (0%)	5.7 × 102 (99%)	1.5 × 1045	4.3 × 1045
AD0 BS	2.8 × 102	4.2 × 100 (1%)	2.8 × 102 (98%)	3.2 × 1043	9.2 × 1043
AD1 BN	1.2 × 102	4.0 × 100 (3%)	1.2 × 102 (96%)	9.0 × 1039	3.5 × 1039
AD1 B01	9.9 × 102	5.0 × 100 (0%)	9.8 × 102 (99%)	9.0 × 1040	3.5 × 1040
AD1 BK	1.1 × 102	4.0 × 100 (3%)	1.1 × 102 (96%)	2.8 × 1042	1.1 × 1042
AD1 BS	2.9 × 102	3.8 × 100 (1%)	2.9 × 102 (98%)	5.9 × 1040	2.3 × 1040
	${Z}_{\odot }/100$
AD0 BN	2.2 × 102	5.6 × 10−2 (0%)	2.2 × 102 (99%)	3.4 × 1042	7.3 × 1042
AD0 B01	1.7 × 102	1.6 × 100 (0%)	1.7 × 102 (99%)	3.4 × 1043	7.3 × 1043
AD0 BK	1.6 × 102	1.3 × 10−2 (0%)	1.6 × 102 (99%)	1.0 × 1045	2.2 × 1045
AD0 BS	7.2 × 101	1.1 × 10−2 (0%)	7.2 × 101 (99%)	2.2 × 1043	4.8 × 1043
AD1 BN	3.1 × 101	0.0 × 100 (0%)	3.1 × 101 (100%)	0.0 × 100	2.8 × 1039
AD1 B01	2.2 × 102	0.0 × 100 (0%)	2.2 × 102 (100%)	0.0 × 100	2.8 × 1040
AD1 BK	2.9 × 101	0.0 × 100 (0%)	2.9 × 101 (100%)	0.0 × 100	8.6 × 1041
AD1 BS	6.3 × 101	0.0 × 100 (0%)	6.3 × 101 (100%)	0.0 × 100	1.8 × 1040

Note. The same as in Table 6, but for burst SF, which started $1\,\mathrm{Gyr}$ ago and lasted for $100\,\mathrm{Myr}$.

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Table 9.  General Parameters of the Simulational Results (Burst SFR $5\,\mathrm{Gyr}$ Ago)

Modela	Number per MWEGa			${L}_{{\rm{X}},\max }(\mathrm{erg}\,{{\rm{s}}}^{-1})$b
	#ULX	#BHULX	#NSULX	BHULX	NSULX
	${Z}_{\odot }$
AD0 BN	7.3 × 100	0.0 × 100 (0%)	7.3 × 100 (100%)	0.0 × 100	7.2 × 1043
AD0 B01	6.6 × 101	0.0 × 100 (0%)	6.6 × 101 (100%)	0.0 × 100	7.2 × 1044
AD0 BK	4.9 × 100	0.0 × 100 (0%)	4.9 × 100 (100%)	0.0 × 100	2.2 × 1046
AD0 BS	3.5 × 100	0.0 × 100 (0%)	3.5 × 100 (100%)	0.0 × 100	4.7 × 1044
AD1 BN	7.2 × 10−1	0.0 × 100 (0%)	7.2 × 10−1 (100%)	0.0 × 100	2.9 × 1039
AD1 B01	5.0 × 101	0.0 × 100 (0%)	5.0 × 101 (100%)	0.0 × 100	2.9 × 1040
AD1 BK	9.5 × 10−1	0.0 × 100 (0%)	9.5 × 10−1 (100%)	0.0 × 100	8.9 × 1041
AD1 BS	2.0 × 100	0.0 × 100 (0%)	2.0 × 100 (100%)	0.0 × 100	1.9 × 1040
	${Z}_{\odot }/10$
AD0 BN	6.7 × 100	0.0 × 100 (0%)	6.7 × 100 (100%)	0.0 × 100	9.4 × 1042
AD0 B01	6.5 × 101	7.2 × 10−1 (1%)	6.4 × 101 (98%)	6.1 × 1039	9.4 × 1043
AD0 BK	4.9 × 100	0.0 × 100 (0%)	4.9 × 100 (100%)	0.0 × 100	2.9 × 1045
AD0 BS	4.4 × 100	0.0 × 100 (0%)	4.4 × 100 (100%)	0.0 × 100	6.2 × 1043
AD1 BN	1.4 × 100	0.0 × 100 (0%)	1.4 × 100 (100%)	0.0 × 100	3.0 × 1039
AD1 B01	5.1 × 101	0.0 × 100 (0%)	5.1 × 101 (100%)	0.0 × 100	3.0 × 1040
AD1 BK	7.0 × 10−1	0.0 × 100 (0%)	7.0 × 10−1 (99%)	0.0 × 100	9.2 × 1041
AD1 BS	3.2 × 100	0.0 × 100 (0%)	3.2 × 100 (100%)	0.0 × 100	2.0 × 1040
	${Z}_{\odot }/100$
AD0 BN	1.9 × 100	0.0 × 100 (0%)	1.9 × 100 (100%)	0.0 × 100	1.2 × 1043
AD0 B01	1.9 × 101	2.0 × 10−2 (0%)	1.9 × 101 (99%)	6.8 × 1039	1.2 × 1044
AD0 BK	1.4 × 100	0.0 × 100 (0%)	1.4 × 100 (100%)	0.0 × 100	3.7 × 1045
AD0 BS	2.0 × 100	0.0 × 100 (0%)	2.0 × 100 (100%)	0.0 × 100	7.9 × 1043
AD1 BN	3.0 × 10−4	0.0 × 100 (0%)	3.0 × 10−4 (100%)	0.0 × 100	3.0 × 1039
AD1 B01	4.9 × 100	1.2 × 10−1 (2%)	4.8 × 100 (97%)	6.5 × 1039	3.0 × 1040
AD1 BK	1.0 × 10−1	0.0 × 100 (0%)	1.0 × 10−1 (100%)	0.0 × 100	9.2 × 1041
AD1 BS	7.9 × 10−1	0.0 × 100 (0%)	7.9 × 10−1 (100%)	0.0 × 100	2.0 × 1040

Note. The same as in Table 6, but for burst SF which started $5\,\mathrm{Gyr}$ ago and lasted for $100\,\mathrm{Myr}$.

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Table 10.  General Parameters of the Simulational Results (Burst SFR $10\,\mathrm{Gyr}$ Ago)

Modela	Number per MWEGa			${L}_{{\rm{X}},\max }(\mathrm{erg}\,{{\rm{s}}}^{-1})$b
	#ULX	#BHULX	#NSULX	BHULX	NSULX
	${Z}_{\odot }$
AD0 BN	1.2 × 100	0.0 × 100 (0%)	1.2 × 100 (100%)	0.0 × 100	8.7 × 1042
AD0 B01	8.1 × 100	0.0 × 100 (0%)	8.1 × 100 (100%)	0.0 × 100	8.7 × 1043
AD0 BK	8.3 × 10−1	0.0 × 100 (0%)	8.3 × 10−1 (100%)	0.0 × 100	2.7 × 1045
AD0 BS	7.4 × 10−1	0.0 × 100 (0%)	7.4 × 10−1 (100%)	0.0 × 100	5.7 × 1043
AD1 BN	2.9 × 10−2	0.0 × 100 (0%)	2.9 × 10−2 (100%)	0.0 × 100	3.0 × 1039
AD1 B01	2.6 × 100	0.0 × 100 (0%)	2.6 × 100 (100%)	0.0 × 100	3.0 × 1040
AD1 BK	1.1 × 10−1	0.0 × 100 (0%)	1.1 × 10−1 (100%)	0.0 × 100	9.2 × 1041
AD1 BS	3.6 × 10−1	0.0 × 100 (0%)	3.6 × 10−1 (100%)	0.0 × 100	2.0 × 1040
	${Z}_{\odot }/10$
AD0 BN	1.5 × 100	0.0 × 100 (0%)	1.5 × 100 (100%)	0.0 × 100	5.4 × 1042
AD0 B01	1.2 × 101	0.0 × 100 (0%)	1.2 × 101 (100%)	0.0 × 100	5.4 × 1043
AD0 BK	1.1 × 100	0.0 × 100 (0%)	1.1 × 100 (100%)	0.0 × 100	1.7 × 1045
AD0 BS	9.4 × 10−1	0.0 × 100 (0%)	9.4 × 10−1 (100%)	0.0 × 100	3.6 × 1043
AD1 BN	2.1 × 10−1	0.0 × 100 (0%)	2.1 × 10−1 (100%)	0.0 × 100	2.8 × 1039
AD1 B01	7.3 × 100	0.0 × 100 (0%)	7.3 × 100 (100%)	0.0 × 100	2.8 × 1040
AD1 BK	7.5 × 10−2	0.0 × 100 (0%)	7.5 × 10−2 (100%)	0.0 × 100	8.6 × 1041
AD1 BS	2.9 × 10−1	0.0 × 100 (0%)	2.9 × 10−1 (100%)	0.0 × 100	1.8 × 1040
	${Z}_{\odot }/100$
AD0 BN	6.2 × 10−1	0.0 × 100 (0%)	6.2 × 10−1 (100%)	0.0 × 100	5.1 × 1042
AD0 B01	5.9 × 100	0.0 × 100 (0%)	5.9 × 100 (100%)	0.0 × 100	5.1 × 1043
AD0 BK	4.4 × 10−1	0.0 × 100 (0%)	4.4 × 10−1 (100%)	0.0 × 100	1.6 × 1045
AD0 BS	3.8 × 10−1	0.0 × 100 (0%)	3.8 × 10−1 (100%)	0.0 × 100	3.4 × 1043
AD1 BN	1.5 × 10−4	0.0 × 100 (0%)	1.5 × 10−4 (100%)	0.0 × 100	2.8 × 1039
AD1 B01	2.9 × 100	0.0 × 100 (0%)	2.9 × 100 (100%)	0.0 × 100	2.8 × 1040
AD1 BK	5.0 × 10−7	0.0 × 100 (0%)	5.0 × 10−7 (100%)	0.0 × 100	8.6 × 1041
AD1 BS	2.3 × 10−5	0.0 × 100 (0%)	2.3 × 10−5 (100%)	0.0 × 100	1.8 × 1040

Note. The same as in Table 6, but for burst SF, which started $10\,\mathrm{Gyr}$ ago and lasted for $100\,\mathrm{Myr}$. A total absence of BHULXs may be observed.

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Table 11.  Companions (Constant SFR)

Model	MS	HG	RG	CHeB	HeMS	HeWD	HybWD	MS	HG	RG	HeMS	HeHG	HeGB	HeWD	COWD	HybWD
									${Z}_{\odot }$
AD0 BN	0.82	0.18	0.01					0.03	0.05	0.05	0.01	0.08		0.32	0.09	0.37
AD0 B01	0.91	0.07	0.02					0.02	0.02	0.01	0.16			0.38	0.10	0.32
AD0 BK	0.91	0.09						0.01	0.04	0.04	0.01	0.08		0.34	0.10	0.39
AD0 BS	0.92	0.08						0.01	0.06	0.03	0.03	0.09		0.33	0.09	0.35
AD1 BN	0.85	0.15						0.04	0.10	0.02	0.02	0.09	0.01	0.22	0.08	0.42
AD1 B01	0.94	0.05						0.05	0.04	0.01	0.15	0.01		0.33	0.10	0.32
AD1 BK	0.94	0.06						0.13	0.10	0.08	0.04	0.04		0.17	0.07	0.37
AD1 BS	0.94	0.06						0.09	0.16	0.05	0.02	0.08		0.23	0.08	0.30
									${Z}_{\odot }/10$
AD0 BN	0.68	0.30		0.01				0.06	0.47	0.02		0.01		0.21	0.07	0.17
AD0 B01	0.87	0.10						0.02	0.03		0.03			0.47	0.15	0.29
AD0 BK	0.74	0.24	0.01	0.01				0.03	0.51	0.01				0.21	0.07	0.16
AD0 BS	0.79	0.20	0.01	0.01				0.03	0.49	0.01		0.01		0.22	0.07	0.17
AD1 BN	0.69	0.28		0.02				0.10	0.39	0.02		0.02		0.19	0.08	0.21
AD1 B01	0.91	0.08						0.06	0.04	0.01	0.03			0.42	0.15	0.30
AD1 BK	0.76	0.21	0.01	0.01				0.11	0.54	0.05				0.13	0.05	0.13
AD1 BS	0.81	0.17	0.01	0.01				0.08	0.52	0.03		0.01		0.16	0.06	0.15
									${Z}_{\odot }/100$
AD0 BN	0.77	0.09		0.13	0.01			0.02	0.48	0.04		0.06		0.25	0.01	0.13
AD0 B01	0.89	0.04		0.06				0.03	0.03	0.03	0.07			0.56	0.03	0.25
AD0 BK	0.83	0.05		0.12				0.01	0.48	0.04		0.04		0.27	0.01	0.14
AD0 BS	0.87	0.05		0.08				0.02	0.29	0.12		0.04		0.35	0.02	0.18
AD1 BN	0.76	0.10		0.13	0.01			0.04	0.45	0.05		0.13		0.14	0.01	0.18
AD1 B01	0.91	0.04		0.05				0.13	0.05	0.06	0.07			0.33	0.03	0.31
AD1 BK	0.84	0.06		0.10				0.12	0.47	0.20		0.03		0.07	0.01	0.10
AD1 BS	0.89	0.05		0.06				0.10	0.36	0.27		0.03		0.11	0.01	0.12

Note. Companions (donors) in ULXs formed in models with constant SFR. Fractions for different companion types are provided separately for BHULXs and NSULXs. The age of the population is $10\,\mathrm{Gyr}$.

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Table 12.  Companions (Burst SFR 100 $\,\mathrm{Myr}$ Ago)

Model	MS	HG	RG	CHeB	HeMS	HeWD	HybWD	MS	HG	RG	HeMS	HeHG	HeGB	HeWD	COWD	HybWD
									${Z}_{\odot }$
AD0 BN	0.89	0.11						0.42			0.02	0.49	0.01			0.06
AD0 B01	0.96	0.03						0.01			0.98	0.01
AD0 BK	0.97	0.03						0.29			0.04	0.57				0.10
AD0 BS	0.97	0.03						0.07			0.67	0.22				0.03
AD1 BN	0.90	0.10						0.28				0.59	0.12
AD1 B01	0.97	0.03						0.14			0.85	0.01
AD1 BK	0.98	0.02						0.89				0.10
AD1 BS	0.97	0.03						0.73			0.11	0.15	0.01
									${Z}_{\odot }/10$
AD0 BN	0.86	0.11		0.02	0.01			0.98			0.01	0.01
AD0 B01	0.97	0.03						0.97			0.02	0.01
AD0 BK	0.94	0.04		0.01				0.98			0.02	0.01
AD0 BS	0.95	0.03		0.01				0.96			0.03	0.01
AD1 BN	0.85	0.12		0.03	0.01			0.94			0.01	0.04	0.02
AD1 B01	0.97	0.03		0.01				0.99			0.01
AD1 BK	0.94	0.03		0.02				1.00
AD1 BS	0.95	0.04		0.01				0.99
									${Z}_{\odot }/100$
AD0 BN	0.81	0.05		0.13	0.01			0.97				0.02
AD0 B01	0.91	0.02		0.06				0.88			0.10	0.01
AD0 BK	0.86	0.01		0.12				0.98			0.01	0.01				0.01
AD0 BS	0.90	0.02		0.08				0.85			0.12	0.01				0.01
AD1 BN	0.79	0.06		0.14	0.01			0.60				0.27	0.13
AD1 B01	0.92	0.02		0.05				0.98			0.01	0.01
AD1 BK	0.87	0.02		0.11				1.00
AD1 BS	0.92	0.02		0.06				0.98				0.02

Note. Companions (donors) in ULXs formed in models with burst SF. Start of the burst was $100\,\mathrm{Myr}$ ago and its duration was $100\,\mathrm{Myr}$.

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Table 13.  Companions (Burst SFR 1 $\,\mathrm{Gyr}$ Ago)

Model	MS	HG	RG	CHeB	HeMS	HeWD	HybWD	MS	HG	RG	HeMS	HeHG	HeGB	HeWD	COWD	HybWD
									${Z}_{\odot }$
AD0 BN		1.00						0.02	0.07					0.49	0.09	0.33
AD0 B01	0.28	0.43	0.29					0.07	0.03					0.12	0.03	0.74
AD0 BK		1.00						0.01	0.07					0.49	0.09	0.33
AD0 BS		1.00						0.02	0.12					0.46	0.08	0.31
AD1 BN			1.00					0.01	0.31		0.01			0.35	0.07	0.25
AD1 B01	0.51	0.01	0.47		0.01			0.15	0.17	0.02				0.09	0.02	0.54
AD1 BK			1.00					0.03	0.41		0.01			0.29	0.06	0.20
AD1 BS			1.00					0.04	0.47					0.26	0.05	0.17
									${Z}_{\odot }/10$
AD0 BN	0.40	0.37	0.18					0.01	0.74					0.17	0.03	0.05
AD0 B01	0.32	0.44	0.23					0.03	0.08					0.32	0.05	0.51
AD0 BK	0.41	0.34	0.19						0.78					0.15	0.03	0.05
AD0 BS	0.40	0.37	0.18					0.01	0.58					0.28	0.05	0.08
AD1 BN	0.60	0.10	0.24					0.04	0.47					0.33	0.06	0.11
AD1 B01	0.64	0.22	0.13					0.08	0.11					0.28	0.05	0.47
AD1 BK	0.60	0.10	0.24					0.03	0.73	0.01				0.15	0.03	0.05
AD1 BS	0.58	0.12	0.25					0.02	0.63	0.01				0.23	0.04	0.07
									${Z}_{\odot }/100$
AD0 BN		0.87	0.03	0.10					0.79					0.17	0.01	0.03
AD0 B01	0.12	0.47	0.38				0.03	0.01	0.10	0.01				0.36	0.01	0.51
AD0 BK		1.00							0.79					0.17	0.01	0.03
AD0 BS		0.90	0.02	0.07					0.60					0.32	0.02	0.05
AD1 BN									0.46	0.02				0.43	0.03	0.07
AD1 B01								0.19	0.08	0.02				0.29	0.01	0.42
AD1 BK								0.12	0.57	0.10				0.17	0.01	0.03
AD1 BS								0.05	0.46	0.12				0.30	0.02	0.04

Note. Companions (donors) in ULXs formed in models with burst SF. Start of the burst was 1 $\,\mathrm{Gyr}$ ago and its duration was 100 $\,\mathrm{Myr}$.

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Table 14.  Companions (Burst SFR 5 $\,\mathrm{Gyr}$ Ago)

Model	MS	HG	RG	CHeB	HeMS	HeWD	HybWD	MS	HG	RG	HeMS	HeHG	HeGB	HeWD	COWD	HybWD
									${Z}_{\odot }$
AD0 BN										0.37				0.63
AD0 B01										0.01				0.75	0.24
AD0 BK										0.32				0.68
AD0 BS										0.19				0.81
AD1 BN										0.31				0.69
AD1 B01										0.02				0.63	0.35
AD1 BK										0.78				0.22
AD1 BS										0.61				0.39
									${Z}_{\odot }/10$
AD0 BN										0.16				0.56	0.28
AD0 B01								0.01		0.02				0.70	0.27
AD0 BK										0.17				0.55	0.28
AD0 BS										0.21				0.53	0.26
AD1 BN										0.02				0.43	0.54
AD1 B01								0.01		0.04				0.54	0.41
AD1 BK										0.25				0.34	0.41
AD1 BS										0.38				0.28	0.34
									${Z}_{\odot }/100$
AD0 BN								0.10		0.25				0.66
AD0 B01						1.00		0.02	0.03	0.17				0.77
AD0 BK								0.04		0.30				0.66
AD0 BS								0.02		0.60				0.38
AD1 BN										1.00
AD1 B01						0.17		0.09	0.10	0.68				0.08
AD1 BK								0.38		0.62
AD1 BS								0.03		0.97

Note. Companions (donors) in ULXs formed in models with burst SF. Start of the burst was 5 $\,\mathrm{Gyr}$ ago and its duration was 100 $\,\mathrm{Myr}$.

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Table 15.  Companions (Burst SFR 10 $\,\mathrm{Gyr}$ Ago)

Model	MS	HG	RG	CHeB	HeMS	HeWD	HybWD	MS	HG	RG	HeMS	HeHG	HeGB	HeWD	COWD	HybWD
									${Z}_{\odot }$
AD0 BN										0.40				0.60
AD0 B01										0.08				0.92
AD0 BK										0.38				0.62
AD0 BS										0.40				0.60
AD1 BN										0.19				0.81
AD1 B01										0.31				0.69
AD1 BK										0.92				0.08
AD1 BS										0.91				0.09
									${Z}_{\odot }/10$
AD0 BN														0.83	0.17
AD0 B01										0.07				0.73	0.20
AD0 BK														0.83	0.17
AD0 BS														0.83	0.17
AD1 BN														0.89	0.11
AD1 B01										0.11				0.56	0.34
AD1 BK														0.90	0.10
AD1 BS														0.89	0.11
									${Z}_{\odot }/100$
AD0 BN														1.00
AD0 B01								0.05	0.07	0.12				0.54
AD0 BK														1.00
AD0 BS														1.00
AD1 BN										1.00
AD1 B01								0.11	0.13	0.27				0.01
AD1 BK										1.00
AD1 BS										1.00

Note. Companions (donors) in ULXs formed in models with burst SF. Start of the burst was 10 $\,\mathrm{Gyr}$ ago and its duration was 100 $\,\mathrm{Myr}$. A total absence of BHULXs may be observed.

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Our results predict a large fraction of NS accretors among the ULXs. Current observational status is consistent with our results. The discovery of the first NSULX (Bachetti et al. 2014) proved that apparent luminosities orders of magnitude larger than the EL are possible in nature. Kluźniak & Lasota (2015) proposed that the M82 X-1 and the P13 in NGC 7793 may also be PULXs. The following discoveries (Fürst et al. 2016; Israel et al. 2017a) showed that NSs may indeed be widespread among the ULXs. Finally, a known X-ray pulsar SMC X-3 was reported to reach an outburst luminosity of $2.5\times {10}^{39}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$ (Tsygankov et al. 2017). Pintore et al. (2017) argued that the energy spectra of several ULXs can be explained with a "pulsator-like" continuum model used to describe Galactic XRBs containing NSs (e.g., White et al. 1995; Coburn et al. 2001).

Theoretical considerations further support our findings. King & Lasota (2016) showed that a ULX with a weakly magnetized NS and a ULX with a BH may be observationally hard to distinguish if the NS does not pulsate. King et al. (2017) argued that the conditions for pulsations in PULXs are relatively strict and a large population of NSULXs will mimic the BHULXs.

The significant number of NSULX that follows from most of our simulated models may be understood on the statistical ground. While it is easier to obtain large MT rates in the BHULXs, the NSULXs progenitors are more abundant in the ZAMS distributions. In general, NSs form from stars with ZAMS masses $\sim 8\mbox{--}22\,{M}_{\odot }$, whereas BHs form from more massive stars with $\sim 22\mbox{--}150\,{M}_{\odot }$. For ${\alpha }_{\mathrm{IMF}}=-2.7$, it gives the ratio of NS to BH progenitors of ∼4.8, and for ${\alpha }_{\mathrm{IMF}}=-2.3$ this ratio is 3.0. Moreover, in the NSULX route, which dominates in the old stellar populations (${{ \mathcal R }}_{\mathrm{NS},\mathrm{RG}}$), the primary on ZAMS may have an even lower mass of $\sim 6\,{M}_{\odot }$, which gives the ratio of NS to BH progenitors of 4.8 for ${\alpha }_{\mathrm{IMF}}=-2.3$. It first becomes a WD before accreting mass and collapsing into an NS. Additionally, donors in NSULXs are less massive than in BHULXs and, therefore, also more abundant on ZAMS. We note that low-mass donors may occur also in BHULXs, but it is far harder (and therefore less probable) for them to survive the CE. Even if they succeeded, the post-CE separation is so small that the RLOF occurs earlier than what is typical for NSULXs. All of these results are for a significantly smaller number of BHULXs with low-mass ($\lesssim 2\,{M}_{\odot }$) donors than in NSULXs.

During short SF episodes, the ULX population is dominated by sources with BH accretors. BHs may acquire stable MT from heavier companions than NSs. Heavier stars expand more and faster, therefore, BHULXs age during ULX phase is of the order of $\sim 10\,\mathrm{Myr}$, whereas for NSULX it is $\sim 100\,\mathrm{Myr}$ (route ${{ \mathcal R }}_{\mathrm{NS},\mathrm{MS}}$). As a result BHULXs appear in larger numbers in the first $\sim 100\,\mathrm{Myr}$. The ULX phase may appear in NSULX as early as $\sim 10\,\mathrm{Myr}$ after ZAMS. However, it demands a very specific pre-CE conditions, which occur in a relatively narrow ZAMS parameter space. For an average post-CE separation in early NSULXs (${{ \mathcal R }}_{\mathrm{NS},\mathrm{MS}}$) the star needs $\sim 100\,\mathrm{Myr}$ to fill its RL.

The picture is different for the constant SF or long bursts. As the SF continues, the #BHULX saturates for $\sim 200\,\mathrm{Myr}$ after the SF start, because at this time a comparable number of sources begin and end their ULX phase. However, a significant number of NSULXs have their ULX phases during the post-MS expansion of their light donors. Therefore, the #NSULX grows continuously for several gigayears and new NSULX formation routes appear successively (see Section 3.1 for the sequence). For $Z={Z}_{\odot }$, the #NSULX becomes higher than #BHULX after $\sim 1\,\mathrm{Gyr}$. However, for lower metallicities, #NSULX does not overcome #BHULX within $10\,\mathrm{Gyr}$.

When the SF is extinguished, ${{ \mathcal R }}_{\mathrm{BH},\mathrm{MS}}$ quickly disappear as their massive donors quickly end their lives ($\lesssim 500\,\mathrm{Myr}$). On the other hand, NSULXs with low-mass donors may commence the ULX phase after several gigayears of evolution when the secondary fills the RL during the post-MS expansion (${{ \mathcal R }}_{\mathrm{NS},\mathrm{HG}}$ and ${{ \mathcal R }}_{\mathrm{NS},\mathrm{RG}}$). Therefore, after the end of the SF, NSULXs quickly (in $\sim 100\,\mathrm{Myr}$) become the dominant ULXs and after $\sim 1\,\mathrm{Gyr}$ BHULXs disappear completely.

The population synthesis of NSULXs was recently performed with the use of the BSE code by Fragos et al. (2015). However, they omitted the pre-SN evolution and explored a far smaller parameter space than that presented in this work. Even though they utilized the MESA code to calculate the MT rates, their maximum values are similar to ours ($\,\dot{M}\lesssim {10}^{-2}\,\mathrm{Myr}$). We obtained less massive companions in NSULXs, but (Fragos et al. 2015) focused on M82 X-2 source with companion mass $\gtrsim 5.2\,{M}_{\odot }$ (Bachetti et al. 2014) and assumed that heavier stars are necessary to survive the CE phase.

Shao & Li (2015) also took the population synthesis approach to study NSULXs and found a significant population of these sources. They adopted a realistic envelope's binding energy from Xu & Li (2010), but considered only a solar metallicity and constant beaming of b = 0.1.

Observations show that ULXs are often associated with low metallicity environments (e.g., Pakull & Mirioni 2002; Soria et al. 2005; Luangtip et al. 2015; Mapelli et al. 2010). Many authors have demonstrated that in a low metallicity environment it is easier to form a massive BH (e.g., Mapelli et al. 2009; Zampieri & Roberts 2009; Belczynski et al. 2010), and it is more common to obtain an RLOF onto a compact object (Linden et al. 2010).

In this study, we find that the metallicity has a strong impact on #ULX, but only in SF regions (See Table 4). We show that the number of #BHULX increases significantly for ${Z}_{\odot }$/10 in comparison to ${Z}_{\odot }$. However, the #NSULX is virtually unaffected by metallicity. Interestingly, for models with the lowest investigated metallicity (${Z}_{\odot }/100$), we found fewer ULXs than for $Z={Z}_{\odot }/10$. Therefore, our results suggest that the relation $\#\mathrm{ULX}(Z)$ is not monotonic (see Prestwich et al. 2013). We plan to study this outcome in more detail in a forthcoming publication (G. Wiktorowicz et al. 2017, in preparation).

Several searches were performed to directly observe the ULX companions. However, only a handful have been detected among hundreds of known ULXs. Additionally, ULXs, being extra-galactic objects, are observationally biased toward more luminous and therefore more massive companions. In our results, NS counterparts in ULX are predominantly low-mass ($\lesssim 1.5\,{M}_{\odot }$) stars. If located in other galaxies, such objects would be difficult to detect. This may explain why it is hard to find counterparts to the majority of the ULXs.

Some ULXs were observed near OB stars (e.g., Soria et al. 2005). Such stars are too massive to provide a stable MT to an NS accretor, so they may only be accompanied by BH accretors, unless the MT occurs through a stellar wind (not considered in our study). OB stars evolve quickly ($\lt 100\,\mathrm{Myr}$), which suggests that they should be spatially associated with the SF regions, where the BHULXs are more numerous than the NSULXs according to our results. Typical BHULX companions in our simulations are MS stars of $5.6\mbox{--}11\,{M}_{\odot }$, which matches the B spectral type.

Roberts et al. (2008) found five potential optical ULX companions with the use of the Hubble Space Telescope. The optical search for companions in ULXs was performed also by Gladstone et al. (2013) who used the same instrument and found 13 ± 5 counterparts to ULXs located within a distance of $D\leqslant 5\,\mathrm{Mpc}$. Most of them were MS stars or RGs. A near-infrared search was performed by Heida et al. (2014, 2016) who observed 62 ULXs in 37 galaxies located within $10\,\mathrm{Mpc}$. They found 17 candidate counterparts, 13 of which are red supergiants. We note that red supergiants are present in our results in small numbers ($\lt 1 \%$). The relatively large number of these stars in observations may be an observational bias, as they are significantly brighter than the most typical companions predicted by our study ($\lt 2\,{M}_{\odot }$). Nevertheless, red supergiants may be an interesting topic for a separate study.

The companion to NSULX P13 in NGC 7793 was detected and estimated to be a BI9a star with mass $18\mbox{--}23\,{M}_{\odot }$ (Motch et al. 2014). The radius of such a star is $4\mbox{--}700\,{R}_{\odot }$, depending on its evolutionary stage. An orbital period estimate (${P}_{\mathrm{orb}}=64$ days) puts an RL radius of a donor at $\sim 120\,{R}_{\odot }$ (under the assumption of circular orbit and ${M}_{\mathrm{NS}}=1.4\,{M}_{\odot }$). The BI9a star may obtain such a radius during the CHeB phase. According to a standard prescription, in a high mass-ratio system, the RLOF will be unstable and the CE will occur. However, Pavlovskii et al. (2017) found that the range of stable mass ratios may be wider than previously thought. We will analyze this possibility in a separate study (Wiktorowicz et al. 2017, in preparation). BHULXs with high-mass donors may be predecessors of double BH mergers (Belczynski et al. 2016).

In addition to the accretion model described in Section 2.1, to which we will refer to as AD1, we investigated a possibility that all mass may be transferred from the donor and efficiently accreated by the accretor, i.e.,

$$\begin{eqnarray}&&\,{\dot{M}}_{\mathrm{acc}}=\,{\dot{M}}_{\mathrm{RLOF}}.\end{eqnarray} \tag{ 5 }$$

This corresponds to the highest accretion rate that could potentially occur in a given system, but is most likely not realistic for high $\,{\dot{M}}_{\mathrm{RLOF}}$. Therefore, this model should be considered as a rough upper limit for accretion.

The X-ray luminosity is calculated as

$$\begin{eqnarray}&&{L}_{{\rm{X}}}=\displaystyle \frac{\epsilon G\,{M}_{\mathrm{BH}/\mathrm{NS}}\ \,{\dot{M}}_{\mathrm{RLOF}}}{\,{R}_{\mathrm{acc}}}=\eta \,{\dot{M}}_{\mathrm{RLOF}}{c}^{2},\end{eqnarray} \tag{ 6 }$$

where is a conversion efficiency of gravitational energy into radiation equal 1.0 for an NS (surface accretion) and 0.5 for a BH (disk accretion), $\,{M}_{\mathrm{BH}/\mathrm{NS}}$ is the mass of the accretor, $\,{R}_{\mathrm{acc}}$ is the radius of an NS (assumed to be 10 km) or a BH (3 Schwarzschild radii, non-rotating BH), and η is the radiative efficiency of a standard thin disk equal ∼0.2 for an NS and 1/12 for a BH (e.g., Shakura & Sunyaev 1973).

In addition to the beaming model described in Section 2.2 to which we will refer to as BK, we investigated also the following models:

A.2.1. No Beaming (BN)

For this model, we assumed isotropic emission, therefore,

$$\begin{eqnarray}&&b=1\end{eqnarray} \tag{ 7 }$$

for all systems and all MT rates.

A.2.2. Constant Beaming, no Saturation (B01)

A.2.3. Sądowski's Model with Saturation (BS)

This beaming prescription is based mainly on the results of theoretical and numerical analyses presented in Lasota et al. (2016) and based on the results of KORAL (Sa̧dowski et al. 2014) GRRMHD simulations. They showed that super-Eddington disks never become geometrically thick, because the thickness of the slim disk does not depend on the MT rate. Even for very high $\,\dot{m}$ the ratio of photosphere height (H) to radius (R) is $H/R\lesssim 1.6$. We fitted a phenomenological model to approximate the relation between H/R and $\,\dot{m}$ found by Lasota et al. (2016; Figure 4), and we obtained

$$\begin{eqnarray}&&\displaystyle \frac{H}{R}=\displaystyle \frac{1.6}{1+\tfrac{4}{\,\dot{m}}}\end{eqnarray} \tag{ 8 }$$

for non-rotating BH accretion with non-saturated magnetic field. This relation corresponds to $R=30\,{R}_{{\rm{G}}}$, but H/R should not be significantly larger for other radii. The equation shows a moderate photosphere height also for sub-Eddington MT rates. We used the same prescription for NS accretors.

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The H/R is related to the opening angle θ as

$$\begin{eqnarray}&&{\left(\displaystyle \frac{H}{R}\right)}^{-1}=\tan \displaystyle \frac{\theta }{2},\end{eqnarray} \tag{ 9 }$$

which allows us to derive the beaming factor as

$$\begin{eqnarray}&&b=1-\cos \displaystyle \frac{\theta }{2},\end{eqnarray} \tag{ 10 }$$

and the apparent isotropic luminosity is calculated as in Equation (2).

The beaming becomes nearly constant for $\,\dot{m}\gtrsim 100$, which gives the minimum opening angle ${\theta }_{\min }\approx 64^\circ$ ($b\approx 0.15$). The prescription works for the entire range of $\,\dot{m}$ and shows a small collimation also for nearly Eddington MT ($\,\dot{m}\lt 1$). However, below $\,\dot{m}\approx 0.12$ the beaming factor is always greater than $b\gt 0.95(\theta \gt 170^\circ )$.

A.2.4. Additional Characteristics of the Simulated Populations of ULXs