Upper limits on the temperature of inspiraling astrophysical black holes

We present a method to constrain the temperature of astrophysical black holes through detecting the inspiral phase of binary black hole coalescences. At sufficient separation, inspiraling black holes can be regarded as isolated objects, hence their temperature can still be defined. Due to their intrinsic radiation, inspiraling black holes lose part of their masses during the inspiral phase. As a result, coalescence speeds up, introducing a correction to the orbital phase. We show that this dephasing may allow us to constrain the temperature of inspiraling black holes through gravitational-wave detection. Using the binary black-hole coalescences of the first two observing runs of the Advanced LIGO and Virgo detectors, we constrain the temperature of parental black holes to be less than about $ 10^9 $ K. Such a constraint corresponds to luminosity of about $ 10^{-16} M_{\odot} \rm s^{-1} $ for a black hole of $ 20 M_{\odot} $, which is about 20 orders of magnitude below the peak luminosity of the corresponding gravitational-wave event, indicating no evidence for strong quantum-gravity effects through the detection of the inspiral phase.

However, multipole moments may not be the only contribution to the luminosity of binary black hole systems. During the early inspiral phase, when black holes are sufficiently separated, they can be regarded as isolated objects, and hence can radiate [22] [23]. Thus, intrinsic black hole radiation may also contribute to the total gravitationalwave luminosity and affect the behaviour of binary black hole (BBH) coalescence. Note that particle creation of the surrounding spacetime, is negligible, hence not considered.
In what follows, we study the effects of intrinsic black hole radiation on a BBH coalescence by means of post-Newtonian (PN) calculations. Although there are alternative gravitational theories which also predict possible reduction of individual masses during the inspiral phase, such as the scalar-tensor-vector theory [24] [25], a maximalreach-analysis can still be done to estimate the upper bound of the temperature of the parental black holes by assuming that the intrinsic black-hole radiation is the only effect reducing black-hole masses during the inspiral phase of the detected events. Moreover, to thoroughly constraint the temperature, we regard the temperature of inspiraling black holes as a free parameter. We find that the intrinsic black-hole radiation results to a correction to the orbital phase, opening up a possibility to probe black hole radiation through inspiral phase detection. We then constrain the temperature of the parental black holes of the detected BBH coalescences during the first and second observing runs of the Advanced LIGO and Virgo detectors. We round up with a short discussion on the implications of our findings. Throughout this analysis, unless explicitly stated otherwise, we use c = G = = 1.

Orbital phase of BBH coalescence
The inspiraling phase of a BBH coalescence can be observed by detecting gravitational waves at spatial infinity. Gravitational waves from the inspiral phase of a compact binary coalescence (CBC) can be expressed, in the frequency domainh(f ), as where A(f ) is the amplitude function and Ψ (f ) the orbital phase of the CBC; they are both functions of frequency f .
Using the PN expansion of the Einstein field equation, Ψ (f ) obeys the differential equation [1]: where ν = (πM ω) 1/3 is the characteristic velocity of the binary with M the total mass of the system and ω the instantaneous orbital frequency, E(ν) is the binding energy (per unit mass) with a prime denoting its derivative with respect to the argument, and F(ν) is the energy flux. During the inspiraling phase of a BBH coalescence, ν 1, hence we can keep only the leading order term of the 3PN expression for E(ν) and the leading order of the 3.5PN expression for F(ν). More precisely [1] [2] where η = M 1 M 2 /(M 1 + M 2 ) 2 denotes the symmetric mass ratio. During the inspiral phase, when the parental black holes are at sufficient separation, they can be considered as isolated objects emitting radiation [22] [23]. Hence, they lose some of their mass and their horizon flux changes [26] [27]. We assume that the intrinsic black hole radiation is the only mechanism to reduce the black hole mass during the inspiral phase. Let us denote the luminosity of intrinsic radiation of each inspiraling black hole by F BHR,1 and F BHR,2 , and the total luminosity by F BHR = F BHR,1 +F BHR,2 , which depends on the mass and temperature of the black holes (see Eq. 5). The radiation power of black holes in the source frame is given by the Stefan-Boltzmann law [44], where σ B ≈ 5.67×10 −8 Wm −2 K −4 is the Stefan-Boltzmann constant, A and T are respectively the event-horizon area and temperature of the black hole. Hence, in the detector frame where M 1,2 are the red-shifted masses, χ 1,2 are dimensionless spins of the parental black holes and T 1,2 the corresponding red-shifted temperatures.
Considering the intrinsic radiation to be small (F BHR F) and to remain small throughout the inspiral phase (F BHR t M ), the total mass as a function of time, to leading order of F BHR t, reads where M is the initial total mass when the coordinate time t = 0. Similarly, ν and η are also functions of time. Since F BHR F and F BHR t M , we can write ν(t) and η(t) up to the leading order in F BHR , where ν and η are respectively the initial characteristic velocity of the binary and symmetric mass ratio at t = 0. The evolution of the total mass, Eq. (6), would effect both the amplitude and the phase of the emitted GWs. As the Advanced LIGO and Virgo detectors are expected to be more sensitive to phases than amplitudes [20] [21], we shall focus on the effects on the former. To calculate the change of phase, we first express the phase Ψ (ν) as where Ψ (0) (ν) is the orbital phase without considering the intrinsic black hole radiation and Ψ BHR (ν) is the leading order correction due to black hole radiation, almost of the same order as F BHR . Since F BHR F and F BHR t M , we can expand ν 3 E (ν)/F(ν), the RHS of Eq. (2), up to leading order F BHR t. Post-Newtonian calculations show that t can also be expressed in terms of ν. During the early inspiral phase when ν 1, one has [2] where t ref stands for the reference time. Thus, where There are a few points to note about Ψ BHR . Firstly, Ψ BHR has opposite sign to the ν-dependent terms of Ψ (0) (negative by convention), implying that coalescence will take place earlier than in the absence of black hole radiation. In other words, the intrinsic black hole radiation speeds up the coalescence process. This can be explained by the fact that the BBH system loses energy more rapidly due to the intrinsic black hole radiation. Secondly, Ψ BHR is larger for smaller ν, suggesting that the intrinsic black hole radiation affects the inspiral more significantly during the early inspiral phase when gravitational-wave luminosity is of the similar order with the intrinsic black hole radiation. Thirdly, Eq. (10) is invariant if we exchange the labels (1 ⇐⇒ 2), as expected, and its the right-hand side is dimensionless. Equations (5), (10) imply that the inspiral phase of binary black hole coalescences depends on the temperature, opening up the possibility to constrain the black hole temperature through the signal of the inspiral phase.

Parameter Estimation
To constrain the temperature in a model-independent way, we consider T 1 and T 2 as free parameters independent of the black hole mass and spin. We also assume the Stefan-Boltzmann law [42] [43] to be valid for astrophysical black holes even if their temperature is not the Hawking temperature.

Methodology
We estimate T 1 , T 2 and the parameters of the source BBH θ 1 (e.g. masses, spins, luminosity distance and sky location etc) for the detected BBH using Bilby [28]. We implement the correction of orbital phase into the IMRPhenomPv2 waveform model [29] [30], [31]. Specifically, we estimate the posterior of the base-10 log of T 1,2 (denoted as log 10 T 1,2 ) given measured strain datad using Bayes' theorem, p(log 10 T 1 , log 10 T 2 , θ|d, H 1 , I) ∝ p(d| log 10 T 1 , log 10 T 2 , θ, H 1 , I) × p(log 10 T 1 , log 10 T 2 , θ|H 1 , I); H 1 stands for our hypothesis that the parental black holes are emitting intrinsic radiation at temperatures T 1 and T 2 , p(d| log 10 T 1 , log 10 T 2 , θ, H 1 , I) is the likelihood function, p(log 10 T 1 log 10 T 2 , θ|H 1 , I) is the prior probability of log 10 T 1,2 and θ. Note that T 1,2 are measured in Kelvin. Specifically, we prescribe a uniform prior for log 10 T 1,2 ∈ [0, 12]. The lower limit of the prior corresponds to the order of magnitude of the cosmological microwave background, which should be the lower limit of the temperature measurement of all celestial objects in the Universe. We extend the upper limit to a value corresponding to the radiation power of about 10 2 M s −1 for a black hole of mass 10M , which is the typical order of magnitude of the peak luminosity of gravitational-wave events. Note that the Hawking temperature ∼ 10 −9 K is not within in our prior.
1 From now on we denote the more massive black hole by the subscript "1" and the less massive one by "2".
The likelihood of p(d| log 10 T 1 , log 10 T 2 , θ, H 1 , I) is [33] p(d| log 10 T 1 , log 10 T 2 , θ, H 1 , I) tilde is the Fourier transform,h D =h D (log 10 T 1 , log 10 T 2 , θ) is the waveform generated by the modified waveform given log 10 T 1,2 and θ, and by a|b we denote the noise-weighted inner product [34], with S h (f ) the power-spectral density of a given detector D (H, L,V for Hanford, Livingston and Virgo, respectively).

GW150914
We perform the analysis described in Sec. 3.1 to obtain the posterior p(log 10 T 1,2 |data) of log 10 T 1 and log 10 T 2 of the parental black holes of GW150914, the first detected gravitational wave event. The top panel of Fig. 1 shows the p(log 10 T 1 |data) (solid blue line) and p(log 10 T 2 |data) (dashed red line), marginalized over other parameters, including the chirp mass, spins, calibration error of detectors and the temperature of the companion. The posteriors are sampled with 1024 live points. The posteriors of log 10 T 1,2 have a in step-function shape around ∼ 10 7 K. This result indicates that the intrinsic radiation power of GW150914's parental black holes is smaller than 10 −17 M s −1 , which is ∼ 10 −19 of the peak luminosity. Hence, there is no evidence of intrinsic radiation from the progenitor black holes of GW150914. This constraint can be understood by a simple orderof-magnitude estimation. The parental masses of GW150914 are M 1 ∼ M 2 ∼ 30M . Thus, η ∼ 0.25. For its inspiral phase, ν = (πM ω) 1/3 ∼ 10 −1 . Since the parental masses are of similar order of magnitudes, approximate T 1 = T 2 = T est . Thus, Eq. (10) implies If we detect no signature of intrinsic radiation, Ψ BHR 1, implying T est 10 10 K, consistent with our result. The bottom panel of Fig. 1 shows the 90% confidence contour of the two-dimensional posterior of log 10 T 1 (solid blue line) and log 10 T 2 (dashed red line) vs. M c , where M c = (M 1 M 2 ) 3/5 /(M 1 + M 2 ) 1/5 is the chirp mass of GW150914 in the detector frame. The contours show no strong correlation between log 10 T 1 (or log 10 T 2 ) and M c . Therefore, log 10 T 1 and log 10 T 2 are not degenerate with M c . The shaded region shows the 90% percentile of M c of GW150914 estimated by the Advanced LIGO and Virgo detectors [11]. The 90% confidence contour overlaps with  the shaded region significantly, suggesting that the estimated M c is consistent with those of known studies. From Fig. 1, we conclude that our test puts reasonable constraints on parental black holes' temperature while estimates M c accurately.

O1 and O2 Events
We extend the analysis to all O1 and O2 binary black-hole events. In Fig. 2, we plot the posterior of log 10 T 1 (red violin plots in left hand side) and log 10 T 2 (violet violin plots in right hand side) for each detection. For the O2 events which were detected after the Virgo detector had been online, its data were also included for the analysis. All posteriors show no support for of T 1 and T 2 > 10 9 K. Most of the posteriors are of step-function shape, which are consistent with the results of GW150914 (see Fig. 1). Similar to the case of GW150914, all posteriors show no particularly strong support at any specific values of T 1 and T 2 . From Fig. 2, we conclude that the temperature of the parental black holes of all detected gravitationalwave events are 10 9 K and we have found no evidence of intrinsic radiations due to the progenitor black holes of O1 and O2 events.
Our constraints of temperature correspond to extremely small black-hole radiation luminosity. Table 1 summarises the 90% confidence interval of the posteriors of T 1 , T 2 and the corresponding power of intrinsic black hole radiation, as computed using Eq. (5). The 90% percentiles correspond to black hole radiation powers which are much smaller than those of the peak luminosity of gravitational waves (about 10 2 M s −1 for GW150914 [11]), consistent with our assumption of F BHR F and keeping only the first terms in the expansion of ν 3 E (ν)/F(ν). The constraints we have derived are consistent with our assumption F BHR t M . As shown in Table 1, for the analysed events F BHR < 10 −16 M s −1 and the inspiral phase lasted for at most ∼ 1s [14]. Hence, the mass loss of the parental black holes is less than 10 −16 M , which is much smaller than the smallest parental mass of the analysed events (∼ 10M ) [14]. Thus, Eqs. (6) and (10) are valid throughout the whole analysis. Since the constraint on the power of the intrinsic black hole radiation corresponds to only a tiny portion of the corresponding peak gravitational-wave luminosity, our results imply that we did not detect any evidence for abnormally strong black-hole radiations.

Concluding remarks
We have presented a method to constrain the temperature of astrophysical black holes through the inspiral phase detection, without assuming a specific dependence of the temperature on either mass or spin. The masses of inspiraling black holes are reducing during the inspiral phase due to black hole radiation, speeding up the coalescence process and introducing a correction to the orbital phase.
By parameterising the dephasing in terms of temperature, one can constrain the temperature of the parental black holes of the gravitational-wave events detected by the Advanced LIGO and Virgo detectors during the first and second observing runs, to be less than about 10 9 K. Constraints of this order of magnitude correspond to black hole radiation of about 10 −16 M s −1 by a black hole of about 10M , which is a tiny fraction of the peak luminosity of the corresponding gravitational-wave event. Note that throughout the paper, we have assumed that the intrinsic black hole radiation is the only effect that can reduce the masses of inspiraling black holes during BBH coalescence. Our constraints can be further improved if more detail investigation of the tidal effects due to alternative gravitational effects [27] [26] [36] [37], are included.
We find no evidence of intrinsic black hole radiation. This is expected because the predicted temperature of the progenitor black holes detected by the Advanced LIGO and Virgo detectors are ∼ 10 −9 K, much lower than the cosmic microwave background temperature. Unless there exist BBHs which consist of companion black holes of < 10 −9 M (such that their temperature > 3K ), it is unlikely the intrinsic radiation will produce visible effects on the inspiral phase for our test to measure.
Despite these difficulties, intrinsic black hole radiation by astrophysical black holes is worth to be investigated. Although we focus on black hole thermodynamics in General Relativity, our analysis can also be adapted for black holes in alternative theories. In particular, the Hawking radiation increases significantly for black holes in higherdimensional theories [51] [52] [53], while it may not follow the usual Stefan-Boltzmann law [55] [56] [57]. Our analysis may lead to meaningful constraints higher-dimensional theories and provide a more thorough test of black-hole thermodynamics.
In the work we have presented here, we considered a simple phenomenological approach, similar to [27] [26], to derive the leading order term of the dephasing. In a future study we plan to investigate how the emission of Hawking radiation can effect the dynamics of the BBH system in more detail. Table 1. The 90% of confidence interval (C.I.) of T eff , in the detect frame, for the O1 and O2 binary black hole coalescence detected by Advanced LIGO and Virgo. The black hole radiation luminosity at the 90% C.I. of T eff is also computed. The corresponding black hole radiation power is considerably smaller that the order of magnitude of the peak GW luminosity, suggesting no evidence of strong quantum gravity effects.