© The Authors 2023

1. Introduction

Black holes (BHs) are objects of great interest in astrophysics as final stages of stellar evolution (e.g., Chan et al. 2018) and also due to the fact that active Galactic nuclei are visible already in the earliest times of the Universe (e.g., Wang et al. 2021). BHs are often described within the Schwarzschild or Kerr solutions of general relativity. However, those solutions approach a flat spacetime at infinite spatial distance from the BH. This is inconsistent with the standard cosmological assumption that, on large scales, spacetime approaches a Friedman-Robertson-Walker (FRW) metric. As the field equations of general relativity are nonlinear, this cannot be resolved with a simple superposition of a “local” Kerr solution and a “global” FRW metric.

Finding BH solutions that approach an FRW metric at infinte spatial distance is the subject of ongoing research. In such a solution, the relativistic material (i.e., the BH) can become coupled to the cosmological expansion of the Universe (e.g., Faraoni & Jacques 2007; Croker & Weiner 2019). This would affect BHs that contain vacuum energy, which may apply to stellar BHs as well as supermassive BHs (SMBHs).

Recently, Farrah et al. (2023a) report that they find SMBHs in quiescent elliptical galaxies to grow in mass by factors of 7–20 between cosmological redshifts 0.7 ≲ z ≲ 2.5 and z ∼ 0, while at the same time no growth in stellar mass is observed. Farrah et al. (2023b) seek to explain this observation with cosmological coupling of the SMBH masses of the form

(1)

where M denotes the mass of the BH, a and a_i denote the scale factors at the current time and the initial time when the BH formed, and k is the cosmological coupling strength. Using a = 1/(1 + z) and assuming current time z = 0 for observation, we can translate Eq. (1) into cosmological redshift such that

(2)

with z_i denoting the redshift when the BH formed. Within the physically realistic range of −3 ≤ k ≤ +3 (e.g., Croker et al. 2021), different values of k are plausible: k = 0 corresponds to the scenario where there is no coupling at all. An analysis of the LIGO/Virgo BH mergers by Croker et al. (2021) estimates k ∼ 0.5, which is also compatible with results from Rodriguez (2023) assuming that there is no mass gap between neutron stars and BHs. Then there is the comoving scenario of k = 1, where the Schwarzschild radius expands together with the scale factor (Faraoni & Jacques 2007). Finally, k = 3 is the scenario producing a constant BH energy density. Given the results in Farrah et al. (2023a), Farrah et al. (2023b) estimate , and the authors emphasise that they exclude the no-coupling scenario of k = 0 at 99.98% confidence (∼3.9σ), including random errors as well as their best estimate of systematic effects. In particular, Farrah et al. (2023b) suggest that stellar-mass BHs are also subject to such cosmological coupling and may be related (potentially even causally) to the accelerated expansion of the Universe. These authors also propose that cosmologically growing stellar BHs formed at early times could be the progenitors of SMBHs observed later.

Taking the results reported by Farrah et al. (2023a) at face value, we want to investigate whether or not a cosmological coupling of BH masses of the form of Eq. (2) is consistent with observed stellar-mass BHs in binary systems. It is generally assumed that, in such binaries, the BH progenitor and the star that survives today formed at (nearly) the same point in time from the same gas reservoir. The more massive primary star quickly evolves into a BH within at most a few tens of millions of years. The less massive secondary, on the other hand, evolves much more slowly and is still observable today. We are particularly interested in binary systems that we can date, either through the visible secondary star or through the binary system being part of a larger stellar system of known age. This allows us to infer the masses of the BHs at the time of their formation, under the assumption of k = 3 cosmological coupling. Stellar-remnant BHs are not expected to form with masses below the Tolman–Oppenheimer–Volkoff (TOV) limit of ≈2.2 M_⊙ (e.g., Rezzolla et al. 2018) because the would-be progenitors of such BHs become neutron stars or white dwarfs instead. However, given the unknowns in the equation of state for neutron stars, the exact value of the TOV limit remains somewhat uncertain: Rocha et al. (2021) suggest that it could be as high as 2.6 M_⊙ and even more massive neutron stars supported by fast rotation are conceivable (Cho 2018; Rezzolla et al. 2018). Inferred initial BH masses below the TOV limit would therefore disfavor the cosmological coupling hypothesis. Additionally, there may be a mass gap between the most massive NS and the least massive BH (e.g., Özel et al. 2010; Farr et al. 2011), that is, between the TOV limit and a possible minimal BH mass of 5.4 M_⊙ (Ye & Fishbach 2022). However, the existence of such a mass gap is disputed.

In this Letter, we present an investigation of the binary system Gaia BH2, for which astrometry and spectroscopy allowed El-Badry et al. (2023a) to directly infer the mass of the BH (without any ambiguity due to unknown inclination angle). In Sect. 2, we estimate the age of Gaia BH2, which allows us to infer the original BH mass under the cosmological-coupling hypothesis. We apply the same analysis to Gaia BH1 in Sect. 3. In Sect. 4, we compare our findings to other results and briefly discuss further possible tests of the cosmological coupling hypothesis.

2. Gaia BH2

Using Gaia DR3 data (Gaia Collaboration 2023) and follow-up spectroscopy, El-Badry et al. (2023a) report their finding of a binary system comprised of a red giant star and a BH with mass M_BH = (8.9 ± 0.3) M_⊙. The red giant star has an apparent magnitude of G ∼ 12.3 mag and resides slightly above the Milky Way disk (b ∼ 2.8°) and about 50° away from the Galactic center. Despite its low Galactic latitude, El-Badry et al. (2023a) estimate the reddening of the red giant star as only E(B − V)∼0.2 mag, which is most likely due to it being only 1.16 kpc away.

El-Badry et al. (2023a) consider it unlikely that the red giant has undergone significant mass transfer to the BH, given the binary’s wide and noncircular orbit, the lack of evidence for ongoing mass transfer (no counterpart was found in X-ray or radio), and the prediction that any mass transfer from the BH progenitor to the secondary would have been unstable and short-lived. This also makes it very unlikely that the BH started out as a neutron star and formed only later through mass transfer.

We refine the original analysis of the secondary star in El-Badry et al. (2023a) by refitting its broadband spectral energy distribution (SED) using a Markov chain Monte Carlo (MCMC) algorithm. We place a Gaussian prior on the metallicity motivated by their spectroscopic observations, with [M/H] ∼ 𝒩(−0.02, 0.05), and a prior on the extinction of E(B − V)∼𝒩(0.2, 0.03). We also leave the distance as a free parameter, constrained by the Gaia parallax. Our approach is quite similar to that adopted by El-Badry et al. (2023a), but with the advantage that we track correlations between parameters; most significantly, the temperature and radius of the star. This produces the estimates T_eff = 4627 K, R = 7.73 R_⊙, and [M/H] = −0.021 and the following covariance matrix (in the same order of parameters):

(3)

where the numbers on the diagonal correspond to the squared uncertainties of each parameter. Our constrains are consistent within 1σ with those reported by El-Badry et al. (2023a).

Using isochrones from PARSEC 1.2S Colibri S37 models (Tang et al. 2014; Chen et al. 2015; Pastorelli et al. 2020, and references therein) and the isochrone forward model described in Andrae et al. (2023), we draw Monte Carlo samples from the likelihood specified by the temperature, radius and [M/H] estimates and their covariance above, where the age is one of the free model parameters. An example isochrone is shown in Fig. 1a and Fig. 1b shows the resulting age distribution for the red giant in Gaia BH2. Its median age estimate is 7.9 Gyr (5.2–11.1 Gyr central 68.3% confidence interval, 3.5–12.8 Gyr central 95.5% confidence, 2.5–13.5 Gyr central 99.7% confidence). This is consistent with the α-high chemical composition found spectroscopically by El-Badry et al. (2023a). Neglecting the time it took the massive primary star to evolve into a BH, we can approximate the BH formation with the age of the red giant star.

Fig. 1. Age determination of Gaia BH2. Panel (a): temperature–radius diagram with Gaia BH2 marked by red error ellipses of 1, 2, and 3σ, with PARSEC isochrones for [M/H] = −0.0244 in black. Panel (b): age distribution estimated from Monte Carlo sampling. Panel (c): distribution of original mass of Gaia BH2, which is the current mass divided by (1 + z)3 according to Eq. (2). The vertical dashed line indicates 2.2 M⊙, below which a BH is unlikely to form.

Using the results from Planck Collaboration VI (2020), we can convert the age distribution from Fig. 1b into a distribution of cosmological redshift z_form at which the formation of the BH would have occurred and from that we obtain the distribution of (1 + z_form)³. Quantitatively, we find a median mass inflation factor of (1 + z_form)³ ∼ 8.0 from our Monte Carlo samples, with a central 68.3% confidence interval ranging from 3.4 to 39.8 (2.2–321 central 95.5% confidence, 1.7–765 central 99.7% confidence). We finally obtain the distribution of the original mass of Gaia BH2 shown in Fig. 1c by dividing the current mass by (1 + z_form)³. Here, we also propagate the uncertainty of the current mass of Gaia BH2, (8.9 ± 0.3) M_⊙, by drawing Monte Carlo samples. As is evident from Fig. 1c, most of these original masses (76.92%) are lower than 2.2 M_⊙. Specifically, the median formation mass estimate is 1.11 M_⊙ with a central 68.3% confidence interval ranging from 0.223 to 2.64 M_⊙ (0.028–4.07 M_⊙ central 95.5% confidence, 0.012–5.17 M_⊙ central 99.7% confidence). This suggests that, if cosmological coupling with k = 3 applies, the original mass at the time of BH formation would have been lower than the mass of any known BH, and most likely too low to actually form a BH through known astrophysical channels. Table 1 provides the confidence levels rejecting various other values of k.

3. Gaia BH1

We repeat the exact same procedure for Gaia BH1 (El-Badry et al. 2022), which is a binary system of a BH of mass (9.62 ± 0.18) M_⊙ and a main sequence G-dwarf that is slightly hotter than the Sun and has a lower metallicity of [M/H] = −0.2. Its apparent magnitude is G ∼ 13.8 mag and it is 480 pc away. With Galactic coordinates (ℓ,b) = (22.63° ,18.05° ) it is also well above the Galactic plane. El-Badry et al. (2022) estimate its reddening as E(B − V)∼0.29.

Reanalysing Gaia BH1 using the same approach as for Gaia BH2, we estimate T_eff = 5885 K, R = 0.99 R_⊙ and [M/H] = −0.1963 with the following covariance matrix:

(4)

Again, using the isochrone forward model from Andrae et al. (2023), we draw Monte Carlo samples from the likelihood in (T_eff, R, [M/H]) space and thus infer the age of Gaia BH1, which we then convert to redshift using Planck Collaboration VI (2020). The results are shown in Fig. 2. Despite the visible secondary being a G-dwarf on the main sequence, we find the age to be reasonably well constrained with a median age of 7.1 Gyr (5.4–9.1 Gyr central 68.3% confidence interval, 3.8–11.4 Gyr central 95.5% confidence, 2.3–12.8 Gyr central 99.7% confidence). Thus, our age estimate of Gaia BH1 is younger and more precise than Gaia BH2. The reason for this surprisingly precise age estimate is that the secondary G-dwarf is not exactly on the main sequence but rather appears to be entering the turn-off phase, as is evident from Fig. 2a. Here, we emphasise that the uncertainty on the age estimate is mainly propagated from the atmospheric parameters, that is, the uncertainty on the secondary genuinely being in the turn-off phase versus it still being on the main sequence is included. Concerning the original BH mass, as shown in Fig. 2c, it was below 2.2 M_⊙ with 69.96% confidence. As with Gaia BH2, this suggests that, if cosmological coupling with k = 3 applies, the original mass at the time when Gaia BH1 formed would have been lower than the mass of any known BH, and most likely, too low to actually form a BH through known astrophysical channels. Again, Table 1 provides the confidence levels rejecting various other values of k.

Fig. 2. Same as Fig. 1 but for Gaia BH1. The PARSEC isochrone in panel (a) is for [M/H] = −0.1963.

4. Discussion

If the cosmological coupling strength of k = 3 applies, this implies an original BH mass below the TOV limit of 2.2 M_⊙ at 76.92% confidence for Gaia BH2 and 69.96% confidence for Gaia BH1. As both estimates for Gaia BH1 and BH2 are statistically independent, the combined probability that both BHs could have been above the TOV limit is only 6.9%. Stellar cores below ∼1.4 M_⊙ will rather form a white dwarf and cores below ∼2.2 M_⊙ will rather form a neutron star, but not a BH. Indeed, Rocha et al. (2021) suggest that neutron stars could remain stable up to 2.5 − 2.6 M_⊙. Consequently, Gaia BH1 and BH2 disfavor the cosmological coupling with k = 3 put forward by Farrah et al. (2023b). Table 1 provides the probabilities for additional values of coupling constants k and Fig. 3 shows a graphical representation of how Gaia BH1 and BH2 constrain k. While Fig. 3 cannot distinguish between values of k ≲ 1, it clearly disfavors k > 3, which is outside of the physically realistic range (e.g., Croker et al. 2021). If we allow a higher TOV limit of 2.6 M_⊙ or a mass gap between NS and BHs reaching 5.4 M_⊙, the likelihood of k > 1 decreases even further. Our finding is in agreement with similar results reported by Rodriguez (2023), who used two dynamically confirmed BHs in the globular cluster NGC 3201 to show that both of these BHs would need to have near face-on inclinations in order to be consistent with the k = 3 cosmological coupling, which is a configuration that Rodriguez (2023) assessed to have a probability of at most 10⁻⁴. While the results from Rodriguez (2023) are considerably more restrictive, nearby astrometrically characterized BHs such as Gaia BH1 and BH2 offer constraints on BH coupling that are complimentary to systems in globular clusters: their masses are better-constrained, but their ages are more uncertain. Furthermore, globular clusters often contain blue stragglers formed by stellar encounters. While we are not suggesting that the two BHs in NGC 3201 necessarily formed this way, it is in principle possible that BHs in globular clusters form later through collisions or capture of a neutron star followed by accretion. This is extremely unlikely for BHs detected astrometrically in the field outside of globular clusters.

Fig. 3. Likelihood of original BH mass at formation being below the TOV limit of 2.2 M⊙ as a function of coupling constant k. The blue line is for Gaia BH1, red for Gaia BH2, and black for the combination of both. As the measurements of Gaia BH1 and BH2 are statistically independent, the combination is simply the product of the two individual likelihoods. Additionally, the gray-dashed line shows the combined results for a higher TOV limit of 2.6 M⊙ (Rocha et al. 2021) while the gray dotted line shows the combined results for a mass gap with a minimal BH mass of 5.4 M⊙ (Ye & Fishbach 2022).

At the fundamental level, Rodriguez (2023) and our work rely heavily on the assumption that cosmological coupling and the TOV limit both apply at the same time to stellar-remnant BHs. This may not be the case, as the TOV limit theoretically requires the existence of an event horizon, which may not be the case for singularity-free BHs containing vacuum energy. However, as discussed by Rodriguez (2023, Sect. 4.1 therein), the formation of horizonless BHs of masses below the TOV limit is difficult to reconcile with observations of BHs in high-mass X-ray binaries.

Firmer conclusions for Gaia BH1 and Gaia BH2 would require a more accurate age estimate of the red giant branch companion in Gaia BH2. Alternatively, the discovery of more BHs in binary systems using Gaia astrometry is likely to provide additional constraints. El-Badry et al. (2023b) estimate that Gaia DR4 will allow the astrometrical discovery of a few dozen BHs in binary systems.

We finally note that we know of young BHs with masses M > 10 M_⊙ in high-mass X-ray binaries like Cygnus X-1 (e.g., Miller-Jones et al. 2021). It is reasonable to assume that such massive stellar BHs should also have formed in the early Universe. If a k = 3 coupling indeed applies, we should eventually find BHs in the mass range 70 − 160 M_⊙, where our current understanding of pair-instability supernovae predicts that there actually should be no BHs formed by stellar evolution (Woosley & Heger 2021). Similarly, if k = 3 cosmological coupling holds and Gaia BH2 formed with a mass of for example 3 M_⊙, then Gaia should soon discover many BHs with similar masses that formed more recently. It is therefore likely that Gaia will be able to test the cosmological coupling of BHs and their potential link to dark energy conclusively within the next few years.

Acknowledgments

We thank the anonymous referee for very useful feedback that greatly improved this manuscript. RA also thanks Knud Jahnke for detailed discussions on selection biases and their possible impact on SMBH masses in ellipticals at different redshifts.

References

All Tables

All Figures

Fig. 1. Age determination of Gaia BH2. Panel (a): temperature–radius diagram with Gaia BH2 marked by red error ellipses of 1, 2, and 3σ, with PARSEC isochrones for [M/H] = −0.0244 in black. Panel (b): age distribution estimated from Monte Carlo sampling. Panel (c): distribution of original mass of Gaia BH2, which is the current mass divided by (1 + z)3 according to Eq. (2). The vertical dashed line indicates 2.2 M⊙, below which a BH is unlikely to form.

In the text

	Fig. 2. Same as Fig. 1 but for Gaia BH1. The PARSEC isochrone in panel (a) is for [M/H] = −0.1963.
In the text

Fig. 3. Likelihood of original BH mass at formation being below the TOV limit of 2.2 M⊙ as a function of coupling constant k. The blue line is for Gaia BH1, red for Gaia BH2, and black for the combination of both. As the measurements of Gaia BH1 and BH2 are statistically independent, the combination is simply the product of the two individual likelihoods. Additionally, the gray-dashed line shows the combined results for a higher TOV limit of 2.6 M⊙ (Rocha et al. 2021) while the gray dotted line shows the combined results for a mass gap with a minimal BH mass of 5.4 M⊙ (Ye & Fishbach 2022).

In the text