Interpreting gravitational-wave burst detections: constraining source properties without astrophysical models

We can set an upper limit on the characteristic size (D) of a GW source ⁸ using the central frequency of the observed GW signal, ${f}_{0}^{\text{GW}}$ (see appendix A). Based on the quadrupole nature of the dominant term in GW emission (see e.g. chapter 3.3.5 in [15] for details), we can assume that:

$$\begin{equation}{f}_{0}^{\text{GW}}\simeq 2{f}^{\text{s}},\end{equation} \tag{ 1 }$$

where f^s is the frequency of the motion generating the GW. (1) becomes an exact equivalence if the motion is a harmonic oscillation at non-relativistic speeds. Causality requires that the typical velocity in the source, v, must at all time be less than the speed of light, c, i.e. c > v ≡ πf^s D, where D is defined as the characteristic size (and D/2 as the characteristic radius) of the GW source. Combining this criterion with (1) gives the following upper limit on D:

$$\begin{equation}{D}_{\mathrm{max}}=\frac{2c}{\pi }\frac{1}{{f}_{0}^{\text{GW}}}\simeq 1909{\left(\frac{{f}_{0}^{\text{GW}}}{100\;\mathrm{H}\mathrm{z}}\right)}^{-1}\;\mathrm{k}\mathrm{m}.\end{equation} \tag{ 2 }$$

It has been shown that ${f}_{0}^{\text{GW}}$ can be estimated robustly for a wide range of signal morphologies with a median statistical error of ∼10% of the signal bandwidth (see [13]). Thus determining D_max, which only depends on ${f}_{0}^{\text{GW}}$, is straightforward for any detected transient GW signal.

In most cases the cosmological redshift (z) of the source is unknown. It has been neglected in (2), because taking it into account would introduce a (1 + z)⁻¹ factor in the expression of D_max, which would reduce the D_max value we get in (2). Thus, the way we define D_max in (2) remains a valid upper limit for all sources, regardless of their z ⁹ . Alternatively, we can say that with (2), we can set an upper limit on the characteristic size measured in the detector frame, which is (1 + z) times larger than in the source frame.

It should be noted, that there is an inescapable limitation of this constraint: it can only give information about the GW source, but not necessarily about the entire astrophysical object, e.g. for an eccentric BBH system in the phase of repeated bursts (see e.g. [21]), we would only observe GWs at pericenter passages. This means that the upper limit on the characteristic size obtained with (2) would only be an upper limit on the pericenter distance, but not on the semi-major axis or on the apocenter distance of the system.

Knowing the previously described upper limit on D, we can also set an upper limit on the GW source's ⁸ characteristic mass, M. This is based on the fact that there is an absolute upper bound on the compactness of the GW source:

$$\begin{equation}\frac{M}{R}{\leqslant}\frac{{c}^{2}}{G},\end{equation} \tag{ 3 }$$

where R = D/2 is the characteristic radius of the source, G is the gravitational constant, and the equality corresponds to a maximally spinning Kerr black hole. Combining this with (2), we get the following upper limit on M:

$$\begin{equation}{M}_{\mathrm{max}}=\frac{{c}^{3}}{\pi G}\frac{1}{{f}_{0}^{\text{GW}}}\simeq 646{\left(\frac{{f}_{0}^{\text{GW}}}{100\;\mathrm{H}\mathrm{z}}\right)}^{-1}\;{\mathrm{M}}_{\odot },\end{equation} \tag{ 4 }$$

where M_⊙ is the mass of the Sun.

Similarly to the case of D_max, it is straightforward to calculate M_max for any observed transient GW signal, and it gives a valid upper limit on M regardless of the z of the source. Also similarly to D_max, M_max is only a valid upper limit on the GW source, but not necessarily on the entire astrophysical object. E.g. in case of a CCSN, M_max constrains the mass of the GW-emitting core, but not the entire exploding star. For more details see the discussion at the end of section 2.1.

General relativity implies an upper limit on the luminosity of any emission process regardless of the mass of the source, which comes from the fact that a higher luminosity would require such a high energy density in the source, that it would collapse into a black hole (see e.g. chapter 9.4 of [19] for details):

$$\begin{equation}{L}_{\mathrm{max}}=\frac{{c}^{5}}{G}\simeq 3.6{\times}1{0}^{52}\;\mathrm{W}.\end{equation} \tag{ 5 }$$

We can set an upper limit on the luminosity distance of a GW source ⁸ , d_L , by comparing the observed GW luminosity, L_GW, to L_max. For a GW signal, assuming isotropic emission, we cancalculate the time-dependent luminosity of the source using the following formula (see e.g. equation (17) in [22]):

$$\begin{equation}{L}_{\text{GW}}\left(\overline{t}\right)=\frac{{c}^{3}}{4G}{d}_{L}^{2}{\left(\dot {h}\left(t\right)\right)}^{2},\end{equation} \tag{ 6 }$$

where (t) is the time derivative of the detected waveform, h(t), and $\overline{t}$ and t are measured in the source-frame and the detector-frame, respectively.

Combining (5) with (6), and neglecting the cosmological Doppler-effect (i.e. assuming that $\overline{t}-t$ is constant), we get an upper limit on the luminosity distance of the source:

$$\begin{equation}{d}_{L,\mathrm{max}}=2c{\left({\dot {h}}^{{\ast}}\right)}^{-1}\simeq 19.4{\left(\frac{{\dot {h}}^{{\ast}}}{1{0}^{-18}\;{\mathrm{s}}^{-\mathrm{1}}}\right)}^{-1}\;\mathrm{G}\mathrm{p}\mathrm{c},\end{equation} \tag{ 7 }$$

where * is the maximum of (t) throughout the observed signal (see appendix A). Note that a similar analysis can give a distance estimate specifically for BBHs [23].

The cosmological Doppler-effect can be taken into account by introducing a (1 + z)⁻¹ factor in (7), which gives the following equation:

$$\begin{equation}{d}_{L,\mathrm{max}}=2c\frac{1}{{\dot {h}}^{{\ast}}\left(1+{z}_{\mathrm{max}}\right)},\end{equation} \tag{ 8 }$$

where z_max is the redshift that corresponds to a luminosity distance of d_L,max. As z_max depends on d_L,max, (8) cannot be solved analytically. The dependence of z_max on d_L,max is set by the cosmological model, for which we assume a flat ΛCDM cosmology with parameters reported in the 'Planck + WP + highL + BAO' column of table 5 in [24]. The self-consistent solution of (8) can be found by iteration (see section 3.2), which we have found to be converging in all cases, and which significantly lowers the upper limit given by (7).

In this section we examine what constraints we can set on a GW source ⁸ for which we know its comoving distance, d_C, e.g. from identification of its host galaxy (as it happened for GW170817, see [25]).

The + and × polarizations of the emitted GW are given by the quadrupole formula ¹⁰ :

$$\begin{equation}{h}_{+}\left(t\right)=\frac{G}{{c}^{4}{d}_{\mathrm{C}}}\left({\ddot {M}}_{11}-{\ddot {M}}_{22}\right),\end{equation} \tag{ 9 }$$

$$\begin{equation}{h}_{{\times}}\left(t\right)=\frac{2G}{{c}^{4}{d}_{\mathrm{C}}}{\ddot {M}}_{12},\end{equation} \tag{ 10 }$$

where M_ij are the second momenta of mass in the source frame where the observer is at the positive z direction, and the two dots denote the second time derivative. The second time derivative of all M_ij components must in all cases satisfy:

$$\begin{equation}{\ddot {M}}_{ij}{\leqslant}M{c}^{2},\end{equation} \tag{ 11 }$$

where M is the characteristic mass of the source. From (11), we can set an upper limit on the GW amplitudes:

$$\begin{equation}{h}_{+,{\times}}\left(t\right)< sim \frac{2GM}{{d}_{\mathrm{C}}{c}^{2}},\end{equation} \tag{ 12 }$$

which is valid for all t.

Combining h₊ and h_× in the output of a GW detector implies the following upper limit on the maximum amplitude value of the observed h(t) GW signal (see appendix A):

$$\begin{equation}{h}^{{\ast}}< sim \frac{4GM}{{d}_{\mathrm{C}}{c}^{2}}.\end{equation} \tag{ 13 }$$

Thus by knowing the comoving distance, we can set a lower limit on the GW source's ⁸ characteristic mass:

$$\begin{equation}{M}_{\mathrm{min}}=\frac{{c}^{2}}{4G}{h}^{{\ast}}{d}_{\mathrm{C}}\simeq 0.52\left(\frac{{h}^{{\ast}}}{1{0}^{-21}}\right)\left(\frac{{d}_{\mathrm{C}}}{100\;\mathrm{M}\mathrm{p}\mathrm{c}}\right)\;{\mathrm{M}}_{\odot },\end{equation} \tag{ 14 }$$

Using (3), we can follow a similar argument to the one we used in section 2.2, and we can convert M_min into the following lower limit on the characteristic size of the GW source:

$$\begin{equation}{D}_{\mathrm{min}}=\frac{1}{2}{h}^{{\ast}}{d}_{\mathrm{C}}\simeq 1.54\left(\frac{{h}^{{\ast}}}{1{0}^{-21}}\right)\left(\frac{{d}_{\mathrm{C}}}{100\;\mathrm{M}\mathrm{p}\mathrm{c}}\right)\;\mathrm{k}\mathrm{m},\end{equation} \tag{ 15 }$$

The cosmological Doppler-effect does not affect h*, so there is no need for a redshift correction in (14) and (15).

Note that when the source distance is known, we can also refine our upper limits on the characteristic size and mass by including a (1 + z)⁻¹ factor in the expressions of D_max and M_max (see (2) and (4)).

We used three kinds of GW signals to test our constraints: BBH signals observed by the LIGO and Virgo detectors during O1 and O2; simulated BBH signals; and simulated CCSN signals ¹¹ . We analyzed all these signals with the BayesWave algorithm [11, 20], which robustly reconstructs the waveforms and central moments of signals for a wide range of signal morphologies [13]. We reconstructed all simulated signals from samples of simulated Gaussian noise corresponding to the detectors' design sensitivities.

We tested our constraints on five BBH signals detected during O1 and O2: GW150914 [28], GW151226 [2], GW170104 [26], GW170608 [27], and GW170814 [6]. We compared our constraints to the reference values of the parameters, which we equate with the point estimates obtained from the model-dependent analysis. We defined the reference value of the characteristic mass as the observed source-frame total mass: ${M}_{\text{ref}}^{\left(\mathrm{B}\mathrm{B}\mathrm{H}\right)}\equiv {m}_{1}+{m}_{2}$, where m₁ and m₂ are the observed source-frame component masses. We chose the reference value of the characteristic size to be ${D}_{\text{ref}}^{\left(\mathrm{B}\mathrm{B}\mathrm{H}\right)}\equiv 2\left({r}_{1}+{r}_{2}\right)$, where r₁ and r₂ are the Schwarzschild radii associated with the observed source-frame component masses. Finally, we defined the reference value of the luminosity distance, ${d}_{L,\mathrm{r}\mathrm{e}\mathrm{f}}^{\left(\mathrm{B}\mathrm{B}\mathrm{H}\right)}$, as the luminosity distance estimation from the model-dependent analysis. We listed these reference values of parameters, alongside with the constraints, in table 1.

Table 1.  Constraints derived for BBH signals GW150914 [2], GW151226 [2], GW170104 [26], GW170608 [27], GW170814 [6] observed by the LIGO–Virgo detector network during the O1 and O2 observing runs. For a detailed description of the constraints, see section 2. We also show the reference values (medians with 90% credible intervals) of the parameters. To calculate the lower limits, we used the point estimate of the distance from the model-dependent analysis. Note that our constraints are consistent with the model-based parameter estimates, and they would provide useful information were there no source models available.

	GW150914	GW151226	GW170104	GW170608	GW170814
${D}_{\text{ref}}^{\left(\mathrm{B}\mathrm{B}\mathrm{H}\right)}\left(\mathrm{k}\mathrm{m}\right)$	$38{6}_{-20}^{+24}$	$12{9}_{-10}^{+35}$	$30{0}_{-30}^{+35}$	$11{2}_{-6}^{+30}$	$33{0}_{-16}^{+20}$
Dmax(km)	1563	1112	1418	1457	1930
Dmin(km)	25	21	33	17	36
${M}_{\text{ref}}^{\left(\mathrm{B}\mathrm{B}\mathrm{H}\right)}\left({\mathrm{M}}_{\odot }\right)$	$65.{3}_{-3.4}^{+4.1}$	$21.{8}_{-1.7}^{+5.9}$	$50.{7}_{-5.0}^{+5.9}$	$19.{0}_{-1.0}^{+5.0}$	$55.{9}_{-2.7}^{+3.4}$
Mmax (M⊙)	529.0	376.6	480.3	493.2	653.3
Mmin (M⊙)	8.6	7.0	11.2	5.8	12.0
${d}_{L,\mathrm{r}\mathrm{e}\mathrm{f}}^{\left(\mathrm{B}\mathrm{B}\mathrm{H}\right)}\left(\mathrm{M}\mathrm{p}\mathrm{c}\right)$	${420}_{-180}^{+150}$	${440}_{-190}^{+180}$	${880}_{-390}^{+450}$	${340}_{-140}^{+140}$	${540}_{-210}^{+130}$
dL,max (Mpc)	8450	21 690	12 860	20 260	13 960

We also tested our constraints on simulated BBH signals, because for these we can directly compare our constraints to the preset parameters of the simulations without relying on results of a model-dependent analysis. We define the reference values of parameters in the same way as for observed BBHs, but using the preset parameter values instead of the observed ones. We used 300 simulated BBH signals for these tests produced with LIGO algorithm library's injection infrastructure [29]. We used the SEOBNRv2threePointFivePN waveform approximant, which applies the aligned-spin effective-one-body model described in [30]. We set a uniform distribution between 20 M_⊙ and 50 M_⊙ for the BBH component masses, and for simplicity, we set the spins of all black holes to zero.

We have also tested our constraints on CCSNe, which are less energetic systems than BBHs, but are among the most promising sources of GW bursts [31–33]. Also, because the GW waveform emitted during a CCSN is non-deterministic [34], our constraints may help understand the physics of the explosion if we detect such a signal. We used GW waveforms B15-WH07 and B20-WH07 from [34], which represent simulated GWs emitted from non-rotating 15 M_⊙ and 20 M_⊙ CCSN progenitors, respectively. Note that these are two-dimensional simulations, where only one polarization is available, and the GW amplitude could differ from the one found in three-dimensional simulations. We compared our characteristic size constraints to the diameter of the iron core right after collapse, which is about 100 km for both waveform simulations (see e.g. figure 5 of [34]), i.e. we set ${D}_{\text{ref}}^{\left(\mathrm{C}\mathrm{C}\mathrm{S}\mathrm{N}\right)}\equiv 100$ km. We compared our constraints on the characteristic mass to the mass of the iron core, which is about 1.4 M_⊙ for all massive stars, i.e. ${M}_{\text{ref}}^{\left(\mathrm{C}\mathrm{C}\mathrm{S}\mathrm{N}\right)}\equiv 1.4\;{\mathrm{M}}_{\odot }$. We scaled the simulated GW signals to make them consistent with a CCSN exploding at a distance of 1 kpc, and we compared our distance constraints to this distance, so ${d}_{L,\mathrm{r}\mathrm{e}\mathrm{f}}^{\left(\mathrm{C}\mathrm{C}\mathrm{S}\mathrm{N}\right)}\equiv 1$ kpc.

Figure 1 show our upper limits along with simulated and observed signals we tested them on. Panel (a) shows D_max as a function of ${f}_{0}^{\text{GW}}$, panel (b) shows M_max as a function of ${f}_{0}^{\text{GW}}$, and panel (c) shows d_L,max as a function of *. The shaded regions represent the parameter spaces consistent with the upper limits. Simulated signals are also plotted with their D_ref and ${f}_{0}^{\text{GW}}$ values for panel (a), M_ref and ${f}_{0}^{\text{GW}}$ values for panel (b), and d_L,ref and * values for panel (c). ${f}_{0}^{\text{GW}}$ and * values were reconstructed by BayesWave for all the signals (see section 3.1 for details). Panel (c) also shows d_L,max without redshift correction (dashed blue line). We can see that taking the cosmological Doppler-effect into account makes the upper limit significantly more restrictive (especially at low * values). Luminosity distance values corresponding to z = 1 and z = 10 are marked with red solid (horizontal) lines on panel (c).

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All three panels of figure 1 show that all plotted data points fall into the region allowed by our upper limit. This means that our upper limits are valid for all GW signals we used in our tests. This is consistent with our expectations based on the theoretical arguments we presented in section 2.

To characterize the efficiency of our upper limits, we compare them to the corresponding reference values. The closer the limit and reference value are to each other, the more we learn about the source from that constraint. Thus we focus on: the ${\mathcal{R}}_{D}\equiv {D}_{\text{ref}}/{D}_{\mathrm{max}}$ ratio for the characteristic size limit, the ${\mathcal{R}}_{M}\equiv {M}_{\text{ref}}/{M}_{\mathrm{max}}$ ratio for the characteristic mass limit, and the ${\mathcal{R}}_{d}\equiv {d}_{L,\mathrm{r}\mathrm{e}\mathrm{f}}/{d}_{L,\mathrm{max}}$ ratio for the luminosity distance limit. These ratios should be below 1 in all cases, and the closer they are to 1, the better our constraints are in characterizing the source. Figure 2 show the distributions of these ratios: ${\mathcal{R}}_{D}$ in panel (a), ${\mathcal{R}}_{M}$ in panel (b), and ${\mathcal{R}}_{d}$ in panel (c). For simulated signals, we show the histogram of the $\mathcal{R}$ values, while for observed BBHs, we plot their corresponding ${\mathcal{R}}_{D}$ values as dashed vertical lines.

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From figure 2(a), it is immediately visible that ${\mathcal{R}}_{D}$ is above 5% for all tested signals, and reaching ∼40% for some of the BBH signals. Also note that the distributions for CCSNe and BBHs are overlapping with each other, which means that our assumptions for the size constraint's limiting case are similarly met for both type of signals. Figure 2(b) shows that ${\mathcal{R}}_{M}$ is above 0.8% and 2.5% for all simulated CCSN and BBH signals, respectively, and reaching ∼20% for some of the simulated BBH signals. Note that ${\mathcal{R}}_{M}$ is significantly lower for CCSN than for BBH signals. This can be understood if we consider that motions in CCSNe are similarly relativistic as in BBHs, but CCSNe are less compact than BBHs (see (3)). Figure 2(c) shows that while the median ${\mathcal{R}}_{d}$ is 4.0% for simulated BBHs and 3.9% for observed BBHs, it is only (6.7 × 10⁻⁵)% for CCSN signals. So we can see that CCSN signals have ${\mathcal{R}}_{d}$ ratios orders of magnitudes smaller than BBHs. This difference can be understood as CCSNe are less effective GW emitters than BBHs. While the six orders of magnitude difference between the upper limit and the reference value of distance we see for CCSNe may suggest that the luminosity distance upper limit is not informative, one should keep in mind that for signals coming from an unexpected new source type, these constraints are the only quantitative information we can derive about the source.

Figure 3 shows our lower limits with solid blue lines as a function of h*d_C: M_min in panel (a), and D_min in panel (b). Shaded regions represent the parameter spaces consistent with the lower limits. The simulated and observed signals are also plotted with their M_ref or D_ref values, and h*d_C values, where h* was reconstructed by BayesWave. All plotted data points fall into the regions allowed by our lower limits, which means that our lower limits are valid for all signals we used in our tests. This is consistent with our expectations based on the theoretical arguments we presented in section 2.

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Similarly to upper limits, to characterize our constraint's efficiency, we should examine the ratio of the lower limit and the reference value. Figure 4(a) shows this ratio for the characteristic mass (${\overline{\mathcal{R}}}_{M}\equiv {M}_{\mathrm{min}}/{M}_{\text{ref}}$), and figure 4(b) shows it for the characteristic size (${\overline{\mathcal{R}}}_{D}\equiv {D}_{\mathrm{min}}/{D}_{\text{ref}}$). For simulated signals, we show the histogram of $\overline{\mathcal{R}}$ values, while for observed BBHs, we plot their corresponding $\overline{\mathcal{R}}$ values as dashed vertical lines.

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The median value of ${\overline{\mathcal{R}}}_{M}$ is 5.8% for simulated BBHs, 5.4% for observed BBHs, and (3.1 × 10⁻³)% for CCSNe. The median value of ${\overline{\mathcal{R}}}_{D}$ is 2.9% for simulated BBHs, 2.7% for observed BBHs, and (1.3 × 10⁻⁴)% for CCSNe. So we can see that the distributions of ${\overline{\mathcal{R}}}_{M}$ and ${\overline{\mathcal{R}}}_{D}$ are clearly different for BBHs and CCSNe, with CCSNe being farther away from $\overline{\mathcal{R}}=1$, which can be understood as CCSNe are less energetic GW sources then BBHs.

Reliable measurements of signal parameters (which include uncertainties) are essential for giving meaningful constraints on source parameters. These can be obtained from a reconstructed waveform which has corresponding confidence intervals (see appendix A), i.e. it covers all the possible waveforms that are consistent with the data at some confidence level. These confidence intervals on the waveform can always be translated to confidence intervals on signal parameters, and thus to confidence intervals on our constraints. In the analysis of an observed signal, one would want to use e.g. the 90th percentile for upper limits and 10th percentile for lower limits to get constraints at the 90% confidence level. For simplicity, in this paper we have chosen to use the median values instead. We do not expect this to have a significant effect on our results, because the expected systematic errors are at the 10% level (see [13]), which is much smaller than the difference between our constraints and the true value of the parameters (see section 3.2).

Model-independent waveform reconstructions have systematic errors too. Both algorithms and the detectors themselves have varying sensitivities throughout the time-frequency region we analyze, which can lead to systematic inaccuracies in the reconstructions of the waveforms. For example, algorithms providing unmodeled waveform reconstructions (including BayesWave) tend to miss parts of the inspiral phase of low signal-to-noise ratio GWs emitted by low-mass binary black hole systems. Increasing the signal-to-noise ratio would allow the detection of earlier parts of the inspiral, but at any sensitivity, we will always miss the part of the signal where the frequency is outside the sensitive band of the detector. Modeled searches can extract the signal even at frequencies where the detectors are less sensitive by utilizing some waveform model, which we do not have in the scenario examined in this work.

Note that the data can also contain transient non-Gaussian noise features (see e.g. [35]), which we have not considered in our analysis. Such a noise transient could bias the estimated parameters of the GW signal if they coincide both in time and frequency [36]. However, there are ongoing efforts to use the BayesWave algorithm to safely remove such noise transients without compromising the GW signal [37–39]. Also note that, since glitches typically occur on the order of once per minute [38], a coincidence is relatively unlikely for short GW bursts with a duration of less than a few seconds.

The sensitivity of our detectors and algorithms is a limitation in reconstructing the GW waveform model-independently, and should be kept in mind when interpreting a detection. This caveat, however, is not specific to our methods described in this paper. It will be present in any analysis that uses the model-independently reconstructed waveform to interpret a signal of unknown origin.

Since we do not assume any source model during their derivation, our constraints can be applied to any GW burst detection, with the caveats that we make a few simplifying assumptions (e.g. isotropic emission, see details in section 2). If a particular source family is not naturally described by our constrained parameters (D, M, and d_L ), those can be mapped onto the parameters relevant for the given source. Note that we already show an example for such a mapping in section 3.1, where we transform the 'characteristic size' to the Schwarzschild radii of components of a BBH system. Another example is cosmic strings, which are actively being searched for by the LVC [40], but conventionally are not parameterized with masses. The nominal parametrization uses the string tension (Gμ) and invariant length (l), or invariant radius (R) if a circular geometry is assumed. Nevertheless, R and Gμ can be used to associate a mass (called 'loop mass') to a string as M_loop = 2πRμ = lμ (see e.g. [41]), which then can be constrained by our characteristic mass parameter we described in section 2.2. This demonstrates that even if a source is not naturally characterized by the same parameters we aim to constrain, our constraints can always be mapped onto the parameter space of the source model.

Note that the constraints presented in this paper can safely be used for GW memory signals too, where the parent signal is not detected (see e.g. [42]). These memory signals will always be detected at lower frequencies compared to their undetected parent burst signals, which means that our mass and size upper limits will give less stringent constraints [42]. Also showed that the amplitude of the memory signal is always smaller than the parent signal, so our mass and size lower limits will give a smaller (more conservative) constraint. The lower frequency and amplitude also implies that our measured * will be smaller compared to the parent signal, so our luminosity distance upper limit will also give a higher (more conservative) constraint. This analysis shows that if the parent signal does not violate our constraints, the memory signal will not violate it either.