New Consistency Relations between Averages and Variances of
Weakly Lensed Signals of Gravitational Waves

The main results presented above, i.e., Eqs. (21) and (22), are based on the computation of the average $\braket{\cdots}$ without paying particular attention to its meaning. However, it is important to revisit the meaning of the average to ensure a precise understanding of its implications, particularly in relation to the energy conservation law. In addition, it is also essential for determining how the average should be practically taken in future experimental settings.

The average considered to this point in this paper is referred to as the ensemble average (Ellis.Maartens.ea2012, ; Breton.Fleury2021nov, ), which hypothetically assumes the existence of multiple universes, each with different matter density configurations. In this scenario, we can compute the lensing signal $X(\chi_{s},\bm{\theta}_{s})$ (e.g., $S,K$ in wave optics and $\kappa,\gamma$ in geometric optics) by considering the GW(or light) signals from the same source at a fixed distance $\chi_{s}$ in each realization. Note that since $X$ describes the lensing effect, it does not depend on the physical property of the source. The ensemble average is then obtained by taking the average value of $X$ over the ensemble of universes. This is the original meaning of the ensemble average that we implicitly assumed in the previous discussion. In cosmology, it is presumed that the universe is statistically homogeneous and isotropic, meaning that the average of all realizations of universes is homogeneous and isotropic, even if each individual realization is not necessarily so. This implies that the spatial derivative of $\braket{X}$ with respect to the true location of the source always vanishes; thus, we obtain the following:

However, in reality, we only have access to a single realization of the universe, thereby making the true ensemble average unattainable. Therefore, it becomes necessary to replace the ensemble average with a statistically computable averaging process. In a statistically homogeneous and isotropic universe, one can find that the ensemble average is approximated by the average over the observers which represents the mean value of $X$ measured by a number of observers uniformly populated on the surface of a sphere with radius $\chi_{s}$ surrounding a single source. This allows us to rewrite $\braket{X}$ as follows:

Note that $\bm{\theta}$ is the location of the observers on the surface of a sphere with radius $\chi_{s}$ surrounding the source.

However, we can only observe the source from the Earth; thus, it remains unfeasible to directly compute the average over the observers. In practice, $\braket{X}$ is taken as the average over the sources, which represents the mean value of $X$ computed from various sources located at the same fixed distance $\chi_{s}$. It is obtained by simply summing all lensing signals $X$ from the sources at $\chi_{s}$ and dividing the sum by the number of the sources. As long as each individual source is fully resolved, the average over the sources can be identified as the ensemble average. In our context, we focus on a GW signal from binary systems where each individual source can be identified; thus, the ensemble averages of the lensing signals derived in the previous section ($\braket{S^{2}},\braket{K^{2}}$, etc.) should be taken as the average over the sources.

It is important to emphasize that $\braket{X}$, which, as discussed above, should not be confused with the average over the apparent directions of the sources within the framework of geometric optics. The average over the directions is another approach commonly used in cosmology to compute the average of $X$ (Kibble.Lieu2005oct, ; Fleury.Clarkson.ea2017mar, ) and is computed in a practical manner by dividing the celestial sphere into small patches with equal area and averaging $X$ over these patches.

The difference between the average over the sources and the average over the directions may seem subtle and indeed can be disregarded within the Born approximation (i.e., the first-order approximation of $X$). However, when the higher-order terms are taken into account, making the distinction between these two becomes crucial, and failure to do so results in erroneous outcomes (Bonvin.Clarkson.ea2015jun, ; Kaiser.Peacock2016feb, ).

Before delving into the main discussion, it is important to consider the meaning of the energy of GWs. Although defining the energy of GWs is not as simple as the case of electromagnetic waves, it is still possible to assign energy to GWs as a conserved quantity when there is a clear separation of scales (Isaacson1968feba, ). In the context of gravitational lensing, there are two types of metric perturbations: the gravitational potential $\Phi$ due to the presence of matter inhomogeneity and the metric perturbation caused by the GWs themselves. Here we assume that the wavelength of GWs is much shorter than the typical curvature radius of the gravitational potential; thus, the metric perturbation associated with GWs can be separated from the background metric. As a result, we can treat GWs as a classical field just like any other fields living in an inhomogeneous universe described in Eq. (1). This approach enables us to identify a conserved quantity corresponding to the energy of GWs (Maggiore2018apr, ).

With this in mind, we can observe that Eq. (2) is essentially a wave equation with the lensing effect included as an interaction between GWs and the gravitational potential $\Phi$. Thus, it can be rewritten as follows:

Now, let us consider the volume integral over the region $V$ whose surface is denoted as $S$. Here, since the energy of GWs in a certain region is given by Eq. (B) in Appendix B, we can connect Eq. (27) with the energy conservation law by taking the time average of Eq. (27)³³3The time average is defined as $\braket{A}_{t}=(1/T)\int_{t}^{t+T}dt^{\prime}A(t^{\prime})$ for an arbitrary time-dependent quantity $A(t)$. Note that $\braket{A}_{t}$ is still a function of time. in addition to the spatial integral. Then, Eq. (27) can be rewritten as follows:

where $\bm{n}$ is a unit normal vector at each point on $S$ and $T$ is a range of time average, which is taken sufficiently longer than the period of GWs. From this expression, it is clear that the left-hand side represents the average rate at which the total energy in the region $V$ varies, and the right-hand side represents the average energy flow going into $V$. Thus, when the sign of the right-hand side is flipped, it is interpreted as the the energy going out from $V$.

Suppose the GW source is at the origin of the coordinate and $\phi$ is the superposition of the different frequency modes:

where $h(\omega)$ is the Fourier transform of the original waveform. Next, we consider a sphere with the radius $\chi$. By taking the volume integral over this region and the time average, we obtain the following:

where $\Omega$ is a solid angle. Therefore, we obtain the following:

When the GW source is completely confined in the region $V$ and there are no objects in $V$ that absorb or produce GWs, then the right-hand side, especially $\int d\Omega F(\omega)F^{*}(\omega)$, becomes independent of the radius of a sphere $\chi$ surrounding the source. In addition, the left-hand side is independent of the matter distribution in the region $V$ assuming that the gravitational potential does not significantly change over time; thus, the right-hand side is also not subject to this dependence. Given that $F=1$ when there are no lensing effects, $\int d\Omega FF^{*}$ needs to be normalized as follows:

The right-hand side is the average of $FF^{*}$ over the observers, and it is identical to both the ensemble average and the average over the sources; thus, we obtain the following relation for the average of the absolute square of $F$:

In our notation, the magnification $K$ and the phase modulation $S$ are defined as $F=e^{K+iS+i\omega\Delta t_{s}}$, which allows us to rewrite the energy conservation condition as $\Braket{e^{2K}}=1$. In a weak lensing regime, $K$ is sufficiently smaller than unity and the Taylor expansion of $e^{2K}$ up to second order in $K$ provides $e^{2K}=1+2K+2K^{2}+\mathcal{O}(K^{3})$. From this expression, it is clear that, up to second order in $\Phi$, $\Braket{K^{2}}+\Braket{K}$=0 needs to hold. One noteworthy aspect of the relation $\braket{FF^{*}}=1$ is its generality. It is the full-order result and does not assume any specific distribution of matter.

In the following, we explain a more general way to derive the consistency relations (22) and (23). The physical interpretation of these relations may not be as clear as the consistency relation associated with energy conservation; however, they can still be derived from a more general, full-order condition, similar to how $\Braket{K}=-\braket{K^{2}}$ is directly derived from $\braket{FF^{*}}=1$.

By observing Eq. (3), it is clear that the expression takes the same form as the Schrödinger equation with time-varying mass. Therefore it is possible to obtain the formal solution to this equation using the path integral method, as presented by Nakamura.Deguchi1999jan :

where the normalization factor is absorbed in $\mathcal{D}[\bm{\theta}(\chi)]$ and is determined to satisfy $F=1$ when $\Phi=0$. Now, we consider taking the ensemble average of this expression. When the ensemble average is taken, the only random variable that appears in this expression is $\Phi$. Thus, $\braket{F}$ is given as follows:

Here, the computation of the $n$ point correlation function $\Braket{\Phi(\chi_{1},\bm{\theta}(\chi_{1}))\cdots\Phi(\chi_{n},\bm{\theta}(\chi_{n}))}$ is required to obtain $\Braket{\exp{\left[-2i\omega\int_{0}^{\chi_{s}}\Phi(\chi,\bm{\theta}(\chi))d\chi\right]}}$. By considering the spatial homogeneity and the assumption that the potential $\Phi$ evaluated at different $\chi$ is uncorrelated (the Limber approximation), we obtain $\Braket{\exp{\left[-2i\omega\int_{0}^{\chi_{s}}\Phi(\chi,\bm{\theta}(\chi))d\chi\right]}}=\Braket{\exp{\left[-2i\omega\int_{0}^{\chi_{s}}\Phi(\chi,\bm{\theta}_{s})d\chi\right]}}$⁴⁴4Because $\Braket{\Phi(\chi_{1},\bm{\theta}(\chi_{1}))\cdots\Phi(\chi_{n},\bm{\theta}(\chi_{n}))}=\delta^{D}(\chi_{1}-\chi_{2})\cdots\delta^{D}(\chi_{n-1}-\chi_{n})\Braket{\Phi(\chi_{1},\bm{\theta}(\chi_{1}))\cdots\Phi(\chi_{1},\bm{\theta}(\chi_{1}))}_{\perp}=\Braket{\Phi(\chi_{1},\bm{\theta}_{s})\cdots\Phi(\chi_{1},\bm{\theta}_{s})}$, where $\braket{\cdots}_{\perp}$ indicates the ensemble average on the plane perpendicular to the line of sight.. Then, $\braket{F}$ is further simplified as follows:

This is a surprisingly simple relation that is accurate to full-order. Here, $F$ is written as $F=e^{K(\omega)}e^{iS(\omega)+i\omega\Delta t_{s}}$; thus, this expression can be formally expanded in $\Phi$ as follows:

From this relation and Eqs. (4)–(10), we obtain the following expression up to second order in $\Phi$:

This is nothing more than Eqs. (22) and Eq. (23). In addition, the expressions of the consistency relations (22) and (23) are based partly on the Limber approximation, which was not assumed in the derivation of the consistency relation associated with energy conservation.

A notable difference between the consistency relations (22) and (23) and the one related to energy conservation (21) is that Eqs. (22) and (23) establish a nontrivial connection between the real and imaginary parts of the amplification factor (i.e., magnification $K$ and the phase modulation $S$ in weak lensing). Here, we propose that this non-trivial relation arises from the Shapiro time delay. As observed in Eq. (III.3), the amplification factor $F$ is obtained by the superposition of all waves traveling along various possible paths. Since the presence of the gravitational potential in a particular region only induces a phase shift to the GWs passing through that area, the resulting $F$ undergoes changes in both the magnification and the phase modulation. However, these changes are only due to constructive and destructive interference. Thus, it is expected that there is a nontrivial connection between the magnification and the phase modulation, and it appears that this connection becomes apparent in the form of the consistency relations when the average is taken ⁵⁵5The nontrivial relation between the real and the imaginary parts of the amplification factor has been reported by Tanaka.Suyama2023aug where the relation arises from the causality of GWs..

To obtain a more intuitive understanding of this nontrivial connection between the real and imaginary parts of $F$, we provide a simple toy model that demonstrates this effect. Suppose two GWs with the same amplitude travel along different paths of equal length and arrive at the location of an observer. Without any lensing objects, the amplification factor is $F=1$. However, if one of the GWs passes through a region with nonzero gravitational potential $\Phi$ that extends over a length $\Delta\chi$, the resulting amplification factor can be written as follows:

From this amplification factor, we obtain the expressions for the magnification $K_{\omega}$ and phase modulation $S_{\omega}$:

By expanding these expressions up to second order in $\Phi$, we can verify the following:

This relation is identical to Eq. (38) with the only difference being the absence of the averaging process. Therefore, it is reasonable to conclude that the Shapiro time delay is responsible for the origin of the consistency relation Eq. (38).

The consistency relations we derived above are satisfied irrespective of the shape of the matter power spectrum; thus, it is constructive to investigate the circumstances under which the consistency relations might be compromised, especially due to the violation of the fundamental physics principle we assumed rather than as a result of observational errors and biases. Given that Eq. (21) arises as a result of the energy conservation and Eqs. (22) and (23) (equivalently Eq. (38)) are attributed to the Shapiro time delay under the assumption of GWs propagating at the speed of light, it is expected that Eq. (21) remains to be satisfied whereas Eq. (38) may be violated if the speed of GW propagation is changed. In order to see if this expectation is indeed correct, let us consider the case of massive gravitons since this is the simplest modification to GR to account for the change in the propagation speed of GWs. When the mass of a graviton $m$ is considered, the wave equation for the amplification factor $F$ is rewritten as follows:

This expression indicates that a newly defined function $F^{\prime}=Fe^{\frac{im^{2}\chi_{s}}{2\omega}}=e^{K+i(S+\frac{m^{2}\chi_{s}}{2\omega})+i\omega\Delta t_{s}}$ satisfies the equation for a massless graviton (3). As we have shown above, the magnification and phase modulation for a massless graviton satisfy Eq. (38), and in this case, the corresponding magnification and phase modulation are $K_{\omega}$ and $S_{\omega}+\frac{m^{2}\chi_{s}}{2\omega}$; thus, the modified version of the consistency relation when the mass of a graviton is included is obtained by simply replacing $S_{\omega}$ with $S_{\omega}+\frac{m^{2}\chi_{s}}{2\omega}$ as follows:

This modified version of the consistency relation implies that the deviation from Eq. (22) is of the order $\frac{m^{2}\chi_{s}}{\omega}$ while the deviation from Eq. (23) is of the order $(\frac{m^{2}\chi_{s}}{\omega})^{2}$ when the mass of a graviton is considered. Note that the consistency relation originating from the energy conservation (i.e., $\braket{K}=-\braket{K^{2}}$ or $\braket{FF^{*}}=\braket{e^{2K}}=1$) is unchanged even when the mass of a graviton is considered since $FF^{*}=\braket{e^{2K}}=1$ is unaffected by replacing $F$ with $Fe^{\frac{im^{2}\chi_{s}}{2\omega}}$. Physically, this is a consequence of the fact that energy conservation is still satisfied despite the presence of massive gravitons; thus, the associated consistency relation (21) also remains unchanged.

The weak lensing signals $S$ and $K$ can be used to probe the small scale matter power spectrum (Takahashi2006jun, ; Oguri.Takahashi2020sep, ). In order to achieve this, it is of critical importance to accurately extract correct $S$ and $K$ from the observational data. As suggested in (Inamori.Suyama2021sep, ; Tanaka.Suyama2023aug, ), consistency relations have the potential to serve as a means to verify the reliability of the lensing signal obtained from observational data. By confirming the satisfaction of the consistency relations, we can independently confirm the correctness of the observed lensing signals without assuming the shape of matter power spectrum, enabling us to use the lensing signals as probes for small-scale matter density fluctuations. In addition, satisfaction of the consistency relations will confirm the validity of the general relativistic formulation of the lensing signals. Conversely, any deviation from the consistency relations serves as a warning sign that the estimation of $S$ and $K$ may not have been performed correctly, which prevents incorrect results from being inferred from unreliable data. While the primary objective of this paper is to present the new consistency relations and discuss their physical implications, it is worth providing a rough estimate of how well the presented consistency relations are satisfied under more realistic scenarios.

Therefore, we consider the feasibility of confirming the consistency relations following a similar method presented in (Inamori.Suyama2021sep, ; Mizuno.Suyama2023aug, ). In practical situations, the average $\braket{\cdots}$ is taken as the average over the sources, which requires a number of GWs from various sources, e.g., binary black holes located at a fixed redshift. However, in principle, it is impossible to collect a sufficient number of lensing signals from the sources with exactly the same redshift $z_{s}$; thus, it is necessary to redefine the average by allowing the inclusion of signals whose redshift falls within a range $z_{s}-\Delta z<z<z_{s}+\Delta z$. The redshift dependence of the lensing signal $X(=S,K)$ suggests that the observed variance at $z_{s}+\Delta z$ is roughly given by $\braket{X(z_{s}+\Delta z)^{2}}=\braket{X(z_{s})^{2}}(1+\mathcal{O}(\Delta z))$ (Takahashi2006jun, ; Oguri.Takahashi2020sep, ; Mizuno.Suyama2023aug, ). With this in mind, we define the estimators $\mathcal{E}_{A}$ and $\mathcal{E}_{B}$ as

where $K_{i}$ and $S_{i}$ are assumed to contain independent Gaussian noise $n_{i}$ with zero mean and variance $1/\mathrm{SNR}^{2}$, where $\mathrm{SNR}$ is the signal-to-noise ratio of the detectors for a particular frequency of GWs. In addition, the products of the signals, e.g., $K_{i}^{2}(\omega,z_{i})$ and $S_{i}(\omega,z_{i})K_{i}(\omega,z_{i})$, are assumed to be computed using the two values obtained from different detectors with independent noise. Under this assumption, we can immediately obtain $\braket{\mathcal{E}_{A}}=\braket{\mathcal{E}_{B}}=0$. Furthermore, under the assumptions of weak lensing, small $\Delta z$ ($\braket{X(z_{s}+\Delta z)^{2}}\sim\braket{X(z_{s})^{2}}$), and $|K|,|S|<1/\mathrm{SNR}$, we obtain, $\braket{\mathcal{E}_{A}^{2}}^{1/2}\sim\braket{\mathcal{E}_{B}^{2}}^{1/2}\sim\frac{1}{\mathrm{SNR}}\frac{1}{\sqrt{N}}$, which provides the estimated fluctuations in $\mathcal{E}_{A}$ and $\mathcal{E}_{B}$.

The number of GW events expected to be observed per year within a redshift range $2.9<z_{s}<3$ can be estimated as $N\sim 10^{3}$ under the assumption that the merger rate at $z_{s}=3$ is $R=20~{}\mathrm{Gpc^{-3}yr^{-1}}$ (Abbott.Abbott.ea2023mar, ). In the $\rm SNR=50$ case, $\frac{1}{\mathrm{SNR}\sqrt{N}}\sim 6\times 10^{-4}$. Since $\sqrt{\braket{K^{2}}}\sim\mathcal{O}(10^{-2})$ and $\sqrt{\braket{S^{2}}}\sim\mathcal{O}(10^{-3})$ at $z_{s}\sim 3$ and $f\sim 1$ Hz, in this scenario, the consistency relation (21) can be confirmed with an accuracy of approximately $\mathcal{O}(1)$% of $\braket{K},\braket{K^{2}}$, and the consistency relation (22) can be confirmed with an accuracy of up to $\mathcal{O}(10)$% of $\braket{S},\braket{SK}$. Note that the value of the merger rate $R$ used here is an estimated value at a fiducial redshift $z=0.2$ (rather than $z=3$). Since $R$ is expected to take a larger value at higher redshift, the number of GW events we estimated might be moderately underestimated. Thus, in reality, the consistency relation can be even more tightly confirmed.