# An Efficient Signal to Noise Approximation for Eccentric Inspiraling
Binaries

a Lisa<EMAIL_ADDRESS>b Alexandra
<EMAIL_ADDRESS>c Zhong-Zhi
<EMAIL_ADDRESS>
a  _Department of Physics, Harvard University, 17 Oxford St., Cambridge, MA
02138, USA_
b  _Institute of Physics, School of Basic Sciences, EPFL, 1015 Lausanne,
Switzerland_
c  _Department of Physics, Tsinghua University, Beijing 100084, China_

###### Abstract

Eccentricity has emerged as a potentially useful tool for helping to identify
the origin of black hole mergers. However, owing to the large number of
harmonics required to compute the amplitude of an eccentric signal, eccentric
templates can be computationally very expensive, making statistical analyses
to distinguish distributions from different formation channels very
challenging. In this paper, we outline a method for estimating the signal-to-
noise ratio for inspiraling binaries at lower frequencies such as those
proposed for LISA and DECIGO. Our approximation can be useful more generally
for any quasi-periodic sources. We argue that surprisingly, the signal-to-
noise ratio evaluated at or near the peak frequency (of the power) is well
approximated by using a constant noise curve, even if in reality the noise
strain has power law dependence. We furthermore improve this initial estimate
over our previous calculation to allow for frequency-dependence in the noise
to expand the range of eccentricity and frequency over which our approximation
applies. We show how to apply this method to get an answer accurate to within
a factor of two over almost the entire projected observable frequency range.
We emphasize this method is not a replacement for detailed signal processing.
The utility lies chiefly in identifying theoretically useful discriminators
among different populations and providing fairly accurate estimates for how
well they should work. This approximation can furthermore be useful for
narrowing down parameter ranges in a computationally economical way when
events are observed. We furthermore show a distinctive way to identify events
with extremely high eccentricity where the signal is enhanced relative to
naïve expectations on the high frequency end.

##  1 Introduction

The LIGO/Virgo detections of coalescing black hole binaries (BBHs) marked the
dawn of gravitational-wave (GW) astronomy [1, 2]. With increasing statistics
from the ongoing run of LIGO/Virgo, we expect to learn many properties of
stellar-mass BBHs. One of the important open questions is the formation
channel of these black hole pairs [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19].

Observing BBHs at lower frequencies in the millihertz range with LISA (and
sub-Hertz range with future detectors such as DECIGO) could provide far more
powerful information about the formation channels. Measurements of the orbital
eccentricity of BBHs [20, 21] in particular can help distinguish among
populations. Isolated BBHs typically possess little eccentricity while
dynamically formed BBHs could have observably large eccentricity [5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17]. Measuring eccentricity at different
frequencies in LIGO-Virgo-KAGRA (LVK), LISA, and DECIGO will be an even more
powerful method to identify formation channels.

However, the gravitational waves (GWs) radiated from inspiraling eccentric
binaries possess multiple harmonics and the width of the spectrum increases
with the eccentricity [22]. This complicates the calculation of the signal-to-
noise ratio (SNR) as one has to sum over those many harmonics. The spread of
frequencies makes the calculation of the average SNR more challenging in that
one can no longer just calculate the signal and divide it by a fixed noise.
This is especially problematic for large eccentricity since one needs to
evaluate many Bessel functions at large orders, which is computationally
expensive. Therefore it would be good to have a formula relating the SNR of an
eccentric binary to the SNR of a more tractable system.

In this paper we elaborate and extend the simplified calculation for the SNR
of eccentric binaries of [23], where two of us showed how to relate the SNR of
an eccentric binary to that of a circular binary with the same masses and peak
frequency $f_{p}$, the frequency of the peak harmonic in the GW power
spectrum. We proposed an approximating formula that assumed a constant noise
curve, using both an analytical argument at large $e$ and a numerical fit.
Treating the SNR $\varrho(f_{p},e)$ as a function of the peak frequency
$f_{p}$ and the eccentricity $e$, we showed

$\varrho(f_{p},e)/\varrho(f_{p},e=0)\propto(1-e^{2})^{3/4}$ (1)

for large $e$. We then provided a full formula working for all $e\in[0,1)$
through a numerical fit.

For noise curves that decrease with frequency (or ones that decrease over the
range of the signal), approximating the noise curve as a constant yields a
partial cancellation between the frequencies that are lower and higher than
the peak frequency. The corrections to our approximation would appear in the
second derivative, which means it depends on the spread of frequencies divided
by the peak frequency. So long as eccentricity is sufficiently far from unity,
this will be a small(ish) number. Near the turnaround of the noise curve,
there is less variation in noise from the get-go so the approximation there
also works very well. Here we show that for eccentricities less than 0.9, this
approximation works rather well (at least at the level of 50% accuracy and
often better). This accuracy should be adequate for distinguishing among
population distributions.

However, the partial cancellation does not necessarily work for the rising
part of the noise curve because of the $1/n$ factor in the amplitude of the
$n$’th harmonic, which manifests itself as a secondary peak in the spectrum at
low frequency where the noise can be relatively low, as can be seen in Fig. 2.
We deal with this by comparing the emission from an eccentric binary to the
known (and more simple to calculate) emission of a circular binary emitting at
the peak frequency determined from the eccentric system of interest but we
allow for a power law dependence on frequency for the noise.

Note that SNR involves the ratio of amplitude to noise, whereas the peak
frequency is evaluated for the emitted power. The difference is compensated by
powers of $n$ (where $n$ labels the harmonic) and we show how to incorporate
the $n$-dependence to get a simple formula. We show that our result is better
than anticipated. We derive a fairly precise formula for
$\varrho(f_{p},e)/\varrho(f_{p},e=0)$ for all $e\in[0,1)$ without assuming any
particular limit or using any numerical fit. We derive formulae for different
power laws of the noise curve. In particular, we consider three interesting
cases: $S_{N}(f)\propto(f^{0},f^{-4},f^{2})$. These correspond to realistic
noise curves such as LISA or DECIGO at the minimum noise, lower frequency, and
higher frequency. The noise curves as used at this point have essentially this
form (with some small deviations for LISA). Of course real noise curves might
have features that will have to be taken into account. But given the noise
curves in the literature [24, 25], our approximation works very well.

In this paper we clarify the original method, check in detail how well it
should work in LISA and DECIGO, and extend the analysis to power law noise
curves to show how accuracy can be improved even further, extending our
analysis to higher frequencies as well. We show that for periodic signals,
taking the sum over harmonics near peak frequency to get the signal we can to
a very good approximation treat the noise curve as flat.

The plan of this paper is as follows. In Sec. 2, we present our analytical
approximation based on SNR evaluated around the peak frequency, which then
evolves over time. We then generalize our formulae to power-law noise curves
in Sec. 3 and compare numerically the results of the approximation with those
of the true noise curve for LISA and DECIGO in Sec. 4. For the latter, we
allow for different redshifts (since DECIGO is sensitive to binaries at large
redshifts) which simply shifts the overall test to different observed
frequencies. We comment on the detection of highly eccentric binaries in Sec.
5 and conclude in Sec. 6.

##  2 LISA SNR of Eccentric BBHs

To set the stage, in this section we review the semi-quantitative arguments
presented in [23], which could help to gain some intuition about the SNR for
eccentric events.

We start with the time-domain formula for the SNR [22]. For a detector like
LISA with noise strain $S_{N}(f)$, the SNR $\varrho$ of eccentric BBHs can be
calculated as

$\varrho^{2}=2\int_{0}^{T_{O}}{\mathrm{d}}t\,\sum_{n=1}^{\infty}\displaystyle\frac{\,{h_{n}^{2}(t)}\,}{S_{N}(f_{n}(t))},$
(2)

where the summation is over all harmonic components of the GWs, $h_{n}$ is the
GW amplitude of $n$’th harmonic component with frequency $f_{n}=nf_{0}$, and
$f_{0}$ is the base frequency. $T_{O}$ is the total observation time. For
highly eccentric BBHs we need to sum over a large number of harmonics, which
can be numerically challenging. However, the noise strain $S_{N}(f)$ varies as
a power law over much of the range of observable frequencies (except at the
turnaround from one power law to another) and this observation can help to
simplify the harmonic summation. In fact, it turns out to be a good
approximation to pull $S_{N}$ out of the summation, essentially assuming
constant noise.

We then find that the signal would be proportional to $\sum h_{n}^{2}=\langle
h^{2}\rangle\equiv h_{c}^{2}$, which is simply the amplitude of GW radiation
averaged over one orbit. Having such a formula allows for a much better
understanding of eccentricity dependence of the signal to noise. To see how
this quantity depends on eccentricity, first consider the large $e$ limit
where we keep track of $\epsilon=1-e^{2}$ factors. Then the GW amplitude is
proportional to $\ddot{M}$ with $M\propto(a\epsilon)^{2}$, the mass quadrupole
of the binary, and where $a$ is the semi-major axis of the binary. (See the
appendix for more details.) To take the time derivative, we use the fact that
${\mathrm{d}}/{\mathrm{d}}t=\dot{\psi}({\mathrm{d}}/{\mathrm{d}}\psi)$ where
$\psi$ is the true anomaly of the binary orbit on which the mass quadrupole
has sinusoidal dependence – and $\dot{\psi}\propto(a\epsilon)^{-3/2}$.
Therefore,

$\langle
h^{2}\rangle\propto\langle\ddot{M}^{2}\rangle\propto\omega_{0}\int_{0}^{2\pi}{\mathrm{d}}\psi\,\dot{\psi}^{-1}\ddot{M}^{2}\propto
a^{-2}\epsilon^{-1/2}.$ (3)

Notice that the four time derivatives from $\ddot{M}^{2}$ introduce four
factors of $\dot{\psi}$ and there is one inverse $\dot{\psi}$ from converting
the time integral into an integral over the phase. So there are 3 factors of
$\dot{\psi}\propto(a\epsilon)^{-9/2}$ in total. In addition, the prefactor
$\omega_{0}\propto a^{-3/2}$. So we have the net result
$(a\epsilon)^{4}\cdot(a\epsilon)^{-9/2}\cdot a^{-3/2}\sim
a^{-2}\epsilon^{-1/2}$.

Now we rewrite $\varrho$ in terms of $f_{p}$ and $\epsilon$, so we use the
formula for $f_{p}$ derived in [26],

$f_{p}=\displaystyle\frac{\,{\sqrt{Gm}(1+e)^{\gamma}}\,}{\pi(a\epsilon)^{3/2}},~{}~{}~{}~{}\gamma=1.1954,$
(4)

where $m$ is the total mass of the binary. Below we will also use $n_{p}\equiv
f_{p}/f_{0}$. So, using $f_{p}\propto(a\epsilon)^{-3/2}$, we have $\langle
h^{2}\rangle\propto f_{p}^{4/3}\epsilon^{3/2}$. There we see if we treat the
SNR as a function of $f_{p}$ and $e$, that the SNR scales with $\epsilon$ like
$\epsilon^{3/4}$.

Note that this gives us only an approximate formula for the dependence on
$1-e$ as $e\rightarrow 1$. As in [23], to allow for all $0\leq e<1$, we treat
the dependence on $1+e$ as an unknown to be determined numerically. As was
shown in [23], a simple formula turns out to well approximate the sum over
harmonics for any value of eccentricity $0\leq e<1$:

$h_{c}^{2}(f_{p},e)=h_{c}^{2}(f_{p},e=0)\cdot(1-e)^{3/2}.$ (5)

Therefore, for binaries with little chirping during the whole observation time
meaning that $f_{p}$ and $e$ are relatively constant, we have,

$\varrho(f_{p},e)=\varrho(f_{p},e=0)\cdot(1-e)^{3/4}.$ (6)

For chirping binaries, we can extend to the following generalized expression,

$\displaystyle\varrho^{2}(f_{p},e)=4\int{\mathrm{d}}t\,\displaystyle\frac{\,{h_{c}^{2}(f_{p}(t),e=0)}\,}{S_{N}(f_{p}(t))}\big{[}1-e(t)\big{]}^{3/2}.$
(7)

Here $f_{p}(t)$ and $e(t)$ should be calculated using Peters’s equations (30)
and (31), as well as (4). In this way we can avoid the summation over GW
harmonics. We check the accuracy of our approximation at LISA and DECIGO (for
two different values of redshift) in Fig. 1, which shows a comparison of SNR
between our formula and an explicit numerical computation summing over
harmonics applicable to events at LISA and DECIGO for frequencies up to 0.1Hz
and 10Hz respectively. The inclusion of the redshift affects both the original
formula (2) and the approximation (7) in the same way (as expected). The only
difference between the two lower panels of Fig. 1 is the shift in the initial
frequency of the events as $f_{p0}/(1+z)$. We include this as a consistency
check of our result.

Figure 1: The SNR $\varrho_{\text{an}}$ calculated from (7) divided by the SNR
$\varrho_{\text{num}}$ calculated by harmonic summation, plotted as function
of initial peak frequency $f_{0}$ and eccentricity $e_{0}$. For this
calculation we assume a binary with $m_{1}=m_{2}=30M_{\odot}$ and four years
of observation time. UPPER LEFT: The noise curves of LISA [24] and DECIGO [25]
used for this computation. UPPER RIGHT:
$\varrho_{\text{an}}/\varrho_{\text{num}}$ for LISA at $z=0$. LOWER:
$\varrho_{\text{an}}/\varrho_{\text{num}}$ for DECIGO for events that occur at
$z=0$ (LOWER LEFT) and $z=5$ (LOWER RIGHT).

Eq. (6) shows that a nonzero eccentricity will decrease the SNR for a binary
with fixed peak frequency. One can again understand this suppression of SNR by
noting that the eccentricity lengthens the lifetime $\tau$ for fixed $f_{p}$
and thus suppresses the radiation power $\dot{E}\sim\tau^{-1}$.

A constant noise strain turns out to be a reasonably good assumption for
$e<0.9$ as we can see in Fig. 1. As we will now see, the correction comes from
deviations from a constant noise curve over frequency. It is nonetheless
remarkable that so long as we work in terms of the measurable signal
concentrated at and near the peak frequency, we can treat the noise curve as
constant, even with significant power law dependence. The result is accurate
to at least within a factor of two for eccentricities less than 0.9 over the
entire plotted parameter range. Beyond this, for large eccentricity $e\sim
0.9$, the analytical approximation and the precise formula agree less well,
but still within an order of magnitude (as can be seen in the upper right
panel of Fig. 5).

We see that our approximation of a constant noise curve works well in the
falling part of the noise curve and the turnaround point. In the next section,
we derive a formula that is even more accurate on the falling part and
improves the rising part of the noise curve as well.

##  3 Precise SNR Formulae for Eccentric Binaries with Power-Law Noise

The analysis of [23] reviewed above assumed a constant noise curve over
frequency. In reality, the noise curves for the two lower frequency GW
detectors we are considering are projected to have a falling power law
dependence ($f^{-4}$ until frequencies $f=5$mHz at LISA and 0.2Hz at DECIGO),
where they turn around to follow an $f^{2}$ dependence. In light of this,
assuming a constant noise curve might seem to be a terrible approximation. In
this section we will show that the additional dependence is well under control
at fixed peak frequency.

In Fig. 2 we show the (unnormalized) spectrum of eccentric binaries with
$f_{p}=0.01$Hz and $e=(0.1,0.5,0.9)$, along with the LISA noise curve. As $e$
gets larger, one can clearly observe the broadening of the spectrum due to an
increasing number of harmonics. We see that at larger eccentricity, the
spectrum develops a secondary peak at the lower frequency end, essentially
because of the $1/n$ factor enhancement associated with the amplitude (not the
power).

Figure 2: The spectrum of non-chirping eccentric binaries with $f_{p}=0.01$Hz.
The three sets of dots (in magenta, green, and blue) correspond to
$e=(0.1,0.5,0.9)$, respectively, with all other parameters are fixed. In the
same plot we show the LISA noise $S_{N}^{1/2}(f)$ in N2A5 configuration (solid
black curve). We also draw two curves representing power-law noises with
$f^{-4}$ (dashed line) and $f^{2}$ (dotted line).

The key insight of the analysis below is that the power-law dependences in the
noise is mathematically equivalent to taking time derivatives of the GW
amplitudes. We will show that including those additional power dependencies
does not heavily affect our previous result for decreasing noise $S_{N}(f)\sim
f^{-4}$ but does bring discrepancies for rising noise $S_{N}(f)\sim f^{2}$,
allowing us to correct the formula in this case as well.

We begin with the expression of the SNR in the time domain again, but this
time without the harmonic decomposition,

$\varrho^{2}=2\int_{0}^{T_{O}}{\mathrm{d}}t\,\displaystyle\frac{\,{1}\,}{S_{N}(f(t))}\langle
h_{ij}^{\text{TT}}(t)h_{ij}^{\text{TT}}(t)\rangle.$ (8)

The average $\langle\cdots\rangle$ in this expression consists of both the
angular average and the time average over one orbital motion. The angular
average is due to our ignorance of the direction of the orbital plane, and the
time average is to make sense of the GW amplitude $h_{c}$. By the standard
procedure, this average can be related to an average over the mass quadrupole
in the following way. (See the appendix for more details.)

$\displaystyle h_{c}^{2}(t)\equiv\langle
h_{ij}^{\text{TT}}(t)h_{ij}^{\text{TT}}(t)\rangle=$
$\displaystyle~{}\displaystyle\frac{\,{4G^{2}}\,}{d^{2}c^{8}}\displaystyle\frac{\,{\omega_{0}}\,}{2\pi}\int_{0}^{P}{\mathrm{d}}t\int\displaystyle\frac{\,{{\mathrm{d}}\Omega}\,}{4\pi}\,\Lambda_{ij,k\ell}\ddot{Q}_{ij}(t)\ddot{Q}_{k\ell}(t)$
$\displaystyle=$
$\displaystyle~{}\displaystyle\frac{\,{4G^{2}}\,}{d^{2}c^{8}}\displaystyle\frac{\,{\omega_{0}}\,}{2\pi}\int_{0}^{2\pi}{\mathrm{d}}\psi\,\dot{\psi}^{-1}\displaystyle\frac{\,{2}\,}{5}\Big{[}\ddot{M}_{11}^{2}+\ddot{M}_{22}^{2}+2\ddot{M}_{12}^{2}-\displaystyle\frac{\,{1}\,}{3}\big{(}\ddot{M}_{11}+\ddot{M}_{22}\big{)}^{2}\Big{]}.$
(9)

Here $d$ is the distance from the binary to the detector (neglecting the
redshift), $\omega_{0}=\sqrt{Gm/a^{3}}$ is the orbital frequency and
$P=2\pi/\omega_{0}$ is the orbital period. The tensor $\Lambda_{ij,k\ell}$ is
the projector onto transverse and traceless components, $M_{ij}=\mu
x_{i}x_{j}$ is the second moment of the mass, and
$Q_{ij}=M_{ij}-\delta_{ij}M_{kk}/3$ is the mass quadrupole.

To compute the SNR, we now decompose the above expression into harmonics, so
that the SNR becomes

$\varrho^{2}=2\int_{0}^{T_{O}}{\mathrm{d}}t\sum_{n=1}^{\infty}\displaystyle\frac{\,{1}\,}{S_{N}(f_{n}(t))}h_{c,n}^{2}(t),$
(10)

where $h_{c,n}^{2}(t)$ denotes the $n$’th harmonic component of
$h_{c}^{2}(t)$. In general, it is the harmonic sum in this expression that
makes the computation complicated. This is particularly true for eccentric
binaries, because their higher harmonic amplitudes are expressed in terms of
the Bessel function $J_{n}$ of order $n$, whose numerical evaluation can be
rather slow.

However, the computation can be greatly simplified if $S_{N}(f_{n}(t))$ is an
even integer power of the frequency, as we now show. For a space based
detector such as LISA or DECIGO, we recognize three distinct regions in the
noise spectrum $S_{N}(f)$. First, around the minimum, the noise curve
$S_{N}(f)$ can be approximated by a constant up to second order effects.
Second, at the low frequency part, the noise curve scales as $S_{N}(f)\sim
f^{-4}$. 444At even lower frequency, e.g., $f<10^{-4}$Hz in LISA, the noise
spectrum scales as $S_{N}(f)\sim f^{-5}$. Our method does not directly
generalize to odd powers so does not cover this region. But the noise here is
already high enough that this region is irrelevant to observing stellar-mass
binaries. Third, in the high frequency region, $S_{N}(f)\sim f^{2}$.
Therefore, we shall consider these three cases in turn.

### Constant noise, $S_{N}(f)\sim f^{0}$.

In this case, we can take the noise strain $S_{N}(f)$ out of the summation.
Then we only need to compute $\sum_{n}h_{c,n}^{2}(t)=h_{c}^{2}(t)$, which is
nothing but the quantity in (3). Using the explicit expression for $M_{ij}$
and $\dot{\psi}$, we find

$\displaystyle h_{c}^{2}(t)=$
$\displaystyle~{}\displaystyle\frac{\,{64(Gm_{c})^{10/3}(\pi
f_{p})^{4/3}}\,}{5c^{8}d^{2}}\mathcal{I}_{0}(e),$ (11)
$\displaystyle\mathcal{I}_{0}(e)\equiv$
$\displaystyle~{}(1-e^{2})^{3/2}(1+e)^{-4\gamma/3}\bigg{(}1+\displaystyle\frac{\,{1-\sqrt{1-e^{2}}}\,}{3}\bigg{)},$
(12)

where $\gamma\simeq 1.1954$ is the exponent that appears in the formula for
the peak frequency. So we find

$\displaystyle\frac{\,{\varrho(f_{p},e)}\,}{\varrho(f_{p},e=0)}\Big{|}_{S_{N}=\text{const}}=\mathcal{I}_{0}(e).$
(13)

### Decreasing noise, $S_{N}(f)\propto f^{-4}$.

In this case we can no longer take $S_{N}(f)$ out of the summation. But we
observe that $S_{N}(f_{n})$, where $f_{n}=nf_{p}/n_{p}$, is
$S_{N}(f_{n})=S_{N}(f_{p})(n/n_{p})^{-4}$, so $S_{N}(f_{p})$ can be taken
outside the summation. Here $n_{p}$ is the harmonic number of the peak
harmonic. So the SNR becomes

$\varrho^{2}=2\int_{0}^{T_{O}}{\mathrm{d}}t\,\displaystyle\frac{\,{1}\,}{S_{N}(f_{p})}\sum_{n=1}^{\infty}\displaystyle\frac{\,{n^{4}}\,}{n_{p}^{4}}h_{c,n}^{2}(t).$
(14)

To complete the summation here, we observe that
$h_{n,ij}^{\text{TT}}\propto\cos(n\omega_{0}t)$ or $\sin(n\omega_{0}t)$. So,

$\ddot{h}_{n,ij}^{\text{TT}}=-n^{2}\omega_{0}^{2}h_{n,ij}^{\text{TT}}=-(2\pi
f_{p})^{2}\displaystyle\frac{\,{n^{2}}\,}{n_{p}^{2}}h_{n,ij}^{\text{TT}}.$
(15)

Therefore,

$\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{\,{n^{4}}\,}{n_{p}^{4}}h_{c,n}^{2}(t)=$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{(2\pi
f_{p})^{4}}\langle\ddot{h}_{ij}^{\text{TT}}(t)\ddot{h}_{ij}^{\text{TT}}(t)\rangle$
$\displaystyle=$ $\displaystyle~{}\displaystyle\frac{\,{1}\,}{(2\pi
f_{p})^{4}}\displaystyle\frac{\,{4G^{2}}\,}{d^{2}c^{8}}\displaystyle\frac{\,{\omega_{0}}\,}{2\pi}\int_{0}^{2\pi}{\mathrm{d}}\psi\,\dot{\psi}^{-1}\displaystyle\frac{\,{2}\,}{5}\Big{[}\ddddot{M}_{11}^{2}+\ddddot{M}_{22}^{2}+2\ddddot{M}_{12}^{2}-\displaystyle\frac{\,{1}\,}{3}\big{(}\ddddot{M}_{11}+\ddddot{M}_{22}\big{)}^{2}\Big{]}.$
(16)

Finishing the integral, we get

$\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{\,{n^{4}}\,}{n_{p}^{4}}h_{c,n}^{2}=$
$\displaystyle~{}\displaystyle\frac{\,{64(Gm_{c})^{10/3}(\pi
f_{p})^{4/3}}\,}{5c^{8}d^{2}}\mathcal{I}_{-4}(e),$ (17)
$\displaystyle\mathcal{I}_{-4}(e)\equiv$
$\displaystyle~{}(1-e^{2})^{3/2}(1+e)^{-16\gamma/3}\Big{(}1+\displaystyle\frac{\,{85}\,}{6}e^{2}+\displaystyle\frac{\,{5171}\,}{192}e^{4}+\displaystyle\frac{\,{1751}\,}{192}e^{6}+\displaystyle\frac{\,{297}\,}{1024}e^{8}\Big{)}.$
(18)

So we get

$\displaystyle\frac{\,{\varrho(f_{p},e)}\,}{\varrho(f_{p},e=0)}\Big{|}_{S_{N}\propto
f^{-4}}=\mathcal{I}_{-4}(e).$ (19)

### Increasing noise, $S_{N}(f)\propto f^{2}$.

Using the reasoning similar to that above, we see that this time we need to
compute

$\displaystyle\sum_{n=1}^{\infty}\displaystyle\frac{\,{n_{p}^{2}}\,}{n^{2}}h_{c,n}^{2}=$
$\displaystyle~{}(2\pi
f_{p})^{2}\displaystyle\frac{\,{4G^{2}}\,}{d^{2}c^{8}}\displaystyle\frac{\,{\omega_{0}}\,}{2\pi}\int_{0}^{2\pi}{\mathrm{d}}\psi\,\dot{\psi}^{-1}\displaystyle\frac{\,{2}\,}{5}\Big{[}\dot{M}_{11}^{2}+\dot{M}_{22}^{2}+2\dot{M}_{12}^{2}-\displaystyle\frac{\,{1}\,}{3}\big{(}\dot{M}_{11}+\dot{M}_{22}\big{)}^{2}\Big{]}$
$\displaystyle=$ $\displaystyle~{}\displaystyle\frac{\,{64(Gm_{c})^{10/3}(\pi
f_{p})^{4/3}}\,}{5c^{8}d^{2}}\mathcal{I}_{2}(e),$ (20)
$\displaystyle\mathcal{I}_{2}(e)\equiv$
$\displaystyle~{}\displaystyle\frac{\,{1}\,}{1-e^{2}}(1+e)^{2\gamma/3}\Big{(}1-\displaystyle\frac{\,{1}\,}{3}e^{2}\Big{)}.$
(21)

Therefore, for this rising noise we have

$\displaystyle\frac{\,{\varrho(f_{p},e)}\,}{\varrho(f_{p},e=0)}\Big{|}_{S_{N}\propto
f^{2}}=\mathcal{I}_{2}(e).$ (22)

In Fig. 3 we show the three analytical formulae for SNR, (13), (19), and (22),
corresponding to three power laws $S_{N}\sim(f^{0},f^{-4},f^{2})$ of the noise
curve. We compare these results with explicit harmonic summation and find
perfect agreement as we should since the formula is exact in this case. This
figure also shows clearly why the constant-noise assumption works well even
when when the noise curve scales as $f^{-4}$: This is essentially because
$\mathcal{I}_{0}$ and $\mathcal{I}_{-4}$ are numerically very similar. On the
other hand, the SNR scales with $e$ in the opposite direction when $S\sim
f^{2}$, showing that the constant-noise assumption will not work well for the
rising part of the noise curve, and an improved formula (22) was called for.
We will confirm these observations in the next section by an explicit
numerical check.

Figure 3: The SNR of eccentric binaries with power-law noises, normalized by
the SNR of a circular binary with the same peak frequency. The three panels
from left to right correspond to
$S_{N}(f)\sim(f^{0},f^{-4},f^{2})$,respectively. The black dots are results of
harmonic summations, and the blue curves are computed using (13), (19), and
(22), respectively.

We note that the LISA noise strain $S_{N}(f)$ increases following a power law
at both sides of the LISA frequency band, namely $S_{N}(f)\propto f^{2}$ on
the high frequency side ($f>0.1$Hz) and $S_{N}(f)\propto f^{-4}$ at low
frequencies ($f<1$mHz). We can now substitute the above power law into (2) and
redo the summation over harmonics. The $n$th harmonic introduces additional
factors of powers of $n$ for the associated time derivative and we are not
left with the simple sum we had before. However, we saw that this
$n$-dependence can be replaced by a time derivative. For $a\epsilon$ constant
as it is for fixed peak frequency, any additional eccentricity dependence
introduced cancels (though the overall factor can change). This is a
remarkable result.

##  4 Comparing with Numerical Results

In this section we compare the SNR calculated using our approximating formulae
with the numerical results obtained by directly summing over harmonics. We do
this comparison for different choices of parameters and instruments with
increasing generality.

First, we consider a chirping binary of black hole with
$m_{1}=m_{2}=30M_{\odot}$ and compute its SNR collected in LISA. For this
computation we assume N2A5 configuration and four years of total observation
time, and assume the events arise at low redshift.555We generalize our
expression to higher redshift in a forthcoming publication [27]. We calculate
the ratio $\varrho_{\text{an}}/\varrho_{\text{num}}$ where
$\varrho_{\text{an}}$ is calculated using our analytical formulae, and
$\varrho_{\text{num}}$ is from the harmonic summation. During the four-year
observation, the binary will chirp ($\dot{f}_{p}>0$) and circularize
($\dot{e}<0$ if $e\neq 0$). The rates of chirping and circularization depend
on $f_{p}$ and $e$. To get a better intuition about this, we show in Fig. 4
the evolution of binaries in the $(f_{p},e)$ plane over four years. To remind
the reader that the actual spectra of these eccentric binaries have finite
width, we show in the right panel of Fig. 4 the “width” of the spectrum of an
eccentric binary. Here the “width” is defined to be the higher harmonic number
at which the radiation power is half of its peak value at $f_{p}$.

We note that on the decreasing part of the noise curve, the frequency changes
very little due to chirping. However, at higher frequencies and, in
particular, near the dip of the LISA noise curve, the chirping changes the
frequency significantly. This will be important below, where the approximation
works very well apart from events with high eccentricity at frequencies near
but higher than the frequency at the dip of the noise curve.

Figure 4: (Left) The evolution of binaries over fur years with various initial
peak frequencies $f_{p}$ and eccentricities $e$. (Right) The width of the GW
spectrum of a binary with eccentricity $e$.

We consider the three analytical formulae derived in the previous section. In
the two upper panels of Fig. 5 we show the result for the constant noise
formula (13). We split the result into two panels, corresponding to the low-
frequency/decreasing-noise and high-frequency/increasing-noise parts of the
LISA, respectively. We see that the constant noise formula gives a decent
estimate of the SNR for most parameter space. Except for the region with
$f_{p}\gtrsim 0.1$Hz and $e_{0}\gtrsim 0.5$, the constant-noise formula and
the numerical result always agrees within a factor of two. As expected, the
constant-noise formula works best around the minimum of the noise curve,
corresponding to $f_{p}\simeq 0.01$. The formula also works relatively well
for the decreasing part of the noise curve. This is explained by the fact that
the constant-noise formula and the decreasing-noise formula have the same
limiting behavior when $e\rightarrow 1$. The constant-noise formula does not
work so well for high frequencies ($f_{p}>0.1$Hz) because the noise curve is
rising here and the actual SNR has opposite asymptotic behavior as
$e\rightarrow 1$, as explicitly shown by the increasing-noise formula.
Finally, there some features in the region
$10^{-3}\text{Hz}<f_{p}<10^{-2}\text{Hz}$ and this is due to the galactic
noise in the noise curve, which cannot be modeled by a simple power.

We then compare the decreasing-noise formula (19) with the numerical result in
the lower-left panel of Fig. 5. We can see clear improvement compared to the
constant-noise formula in the low frequency part when $f_{p}<10^{-3}$Hz, where
the noise curve is well approximated by $f^{-4}$ so our analytical formula
becomes asymptotically exact. The lack of agreement compared to the constant-
noise curve for higher frequency $f_{p}\gtrsim 10^{-2.5}$Hz is due to the
galactic noise, which makes the power law approximation less accurate in this
region.

In the lower-right panel of Fig. 5 we show the comparison between the
increasing-noise formula (22) and the numerical result. This time we see clear
improvement at higher frequencies compared with the constant-noise formula.
And the quality of the approximation has the opposite asymptotic behavior in
the two right panels, as expected. For very large eccentricity, we see that
the increasing-noise formula always tends to overestimate the SNR, because
this formula assumes an increasing noise curve everywhere and thus would
underestimate the noise at sufficiently low frequencies, where the actual
noise curve deviates from the $S_{N}(f)\sim f^{2}$ behavior. This
overestimated SNR becomes more significant for higher eccentricity because a
more eccentric binaries will radiate a wider spectrum extending to lower
frequencies.

As can be seen from the two right panels of Fig. 5 there is a small region
($f_{p0}\sim 0.1$Hz, $e_{0}>0.8$) where the analytical formula does not work
as well, yielding more than a factor of two error. One needs to be careful
when applying our formula to compute SNR for binaries in this region. However,
here we benefit from the change of frequency due to chirping at high frequency
illustrated in Fig. 4. An event moves out of the regime where the
approximation works less well. Applying similar reasoning to DECIGO, the dip
in noise occurs at even higher frequency where the chirping is even more
significant so our approximation should apply at least as well in that
frequency regime.

Figure 5: The SNR calculated from analytical formulae in Sec. 3 divided by the
SNR calculated by harmonic summation, plotted as function of the peak
frequency $f_{0}$ and the eccentricity $e_{0}$ at the initial time of
observation. For this calculation we assume a binary with
$m_{1}=m_{2}=30M_{\odot}$ at $z=0$ and four years of observation time. The two
upper panels show the result of constant-noise formula (13) divided into two
panels corresponding to the approximation to the noise curve used in the lower
panels. The lower-left panel and lower-right panel show the results of
decreasing-noise formula (19) and increasing-noise formula (22), respectively.
Notice the coordinate range here extends further than in Fig. 1.

##  5 High Eccentricity Detection

We note a particularly interesting case when the eccentric binaries’ peak
frequency $f_{p}$ sits at the rising part of the noise curve where
$S_{N}(f)\propto f^{2}$. As shown above, the SNR diverges for large
eccentricity like $\rho^{2}\sim(1-e^{2})^{-1}$ with other parameters fixed.
This is due to the broadened GW spectrum at large $e$ so that the spectrum
extends to the lower frequency part where the noise is also lower. This
suggests an enhanced detectability of highly eccentric binaries at the rising
part of the noise curve over the naive expectation. This point was often
ignored in previous studies, but can be crucial for highly eccentric binaries
formed with frequencies where the detector noise curve is rising. This
corresponds to $f_{p}\gtrsim 0.01$ for LISA and $f_{p}\gtrsim 0.1$ for DECIGO.

It is interesting to check whether we expect it to be possible to detect any
eccentric binaries of this category, namely $f_{p}>0.01$ for LISA (or
$f_{p}>0.1$ for DECIGO) and $e>0.9$. A quick and dirty estimate for the number
of observable binaries of this sort can be done by looking at LIGO events with
(in principle) detectable nonzero eccentricity, namely, $e>0.01$ at
$f_{p}=10$Hz. It was derived in [23] that the eccentricity $e$ and the peak
frequency $f_{p}$ for a binary are related by

$\displaystyle\displaystyle\frac{\,{f_{p}}\,}{f_{p*}}=\displaystyle\frac{\,{\mathcal{H}(e)}\,}{\mathcal{H}(e_{*})},$
$\displaystyle\mathcal{H}(e)\equiv\displaystyle\frac{\,{(1+e)^{\gamma}}\,}{[(1-e^{2})\mathcal{G}(e)]^{3/2}},$
$\displaystyle\mathcal{G}(e)\equiv\displaystyle\frac{\,{e^{12/19}}\,}{1-e^{2}}\bigg{(}1+\displaystyle\frac{\,{121}\,}{304}e^{2}\bigg{)}^{870/2299}.$
(23)

From this we see that any binaries with $e>0.01$ at $f_{p}=10$Hz would have
$e>0.9$ at the rising part of the LISA noise.

Many dynamical channels predict the existence of such highly eccentric
binaries. For example, [28] predicts a local merger rate
$0.35\text{Gpc}^{-3}\text{yr}^{-1}$ for the dynamical binaries formed in dense
star clusters and $\sim 9\%$ of them would have $e>0.01$ at $10$Hz,
corresponding to large eccentricity in the sub-Hertz region. [28] argued that
around 80% percent of the KL binaries and 20% of non-KL binaries from this
channel can have large eccentricity at $f_{p}\gtrsim 0.01$Hz in LISA. There
are also other channels that can produce similar highly eccentric mergers. In
any case, when these mergers appear in the rising part of LISA or DECIGO
noise, they can carry significantly large eccentricity, so that our method
here could be quite useful.

##  6 Conclusions

A great deal of work has been devoted to detailed predictions of the wave
forms for GW signals and building templates to accommodate them. However, very
little attention has been paid to approximations that can be useful when
predicting distributions and also when narrowing the parameter space for
existing events. We have shown how to readily approximate SNR, even for highly
eccentric events, in future lower frequency detectors. We summarize our result
over the entire LISA frequency range in Fig. 6. Furthermore this type of
analysis can sometimes highlight interesting features of the proposed
detections, such as the anomalously big potential SNR for highly eccentric
events.

Figure 6: The LISA SNR calculated from analytical formulae in Sec. 3 divided
by the SNR calculated by harmonic summation, plotted as function of the peak
frequency $f_{0}$ and the eccentricity $e_{0}$ at the initial time of
observation. For this calculation we assume a binary with
$m_{1}=m_{2}=30M_{\odot}$ at $z=0$ and four years of observation time. We use
(19), (13), and (22) for $f_{p0}<10^{-2.6}$Hz,
$10^{-2.6}$Hz$<f_{p0}<10^{-1.3}$Hz, and $f_{p0}>10^{-1.3}$Hz, respectively.

In a future work we will follow up our analysis on relating distributions at
different frequencies when the events have initially high eccentricity. In
particular we can use the method outlined here to estimate the importance of
intermediate frequency bands when distinguishing populations.

In this paper our main focus has been on the eccentric stellar-mass binaries.
But we emphasize that our method can be directly generalized to any quasi-
period sources. Notable examples include intermediate-mass black hole
binaries, extreme mass ratio inspirals, and spinning compact objects. We look
forward to interesting predictions for those in the future.

### Acknowledgment

We thank Karan Jani and Nick DePorzio for useful discussions and Nick Deporzio
for helpful comments on the manuscript. LR was supported by an NSF grant
PHY-1620806, the Chan Foundation, a Kavli Foundation grant “Kavli Dream Team,”
the Simons Fellows Program, the Guggenheim Foundation, and a Moore Foundation
Fellowship. ZZX is supported by Tsinghua University Initiative Scientific
Research Program.

##  Appendix A More details

The GW amplitude is related to the mass quadrupole $Q_{ij}$ by

$h_{ij}^{\text{TT}}=\displaystyle\frac{\,{1}\,}{d}\displaystyle\frac{\,{2G}\,}{c^{4}}\Lambda_{ij,k\ell}\ddot{Q}_{k\ell}.$
(24)

Here $Q_{ij}=M_{ij}-\delta_{ij}M_{kk}/3$ and $M_{ij}=\mu x_{i}x_{j}$. The
projector $\Lambda_{ij,k\ell}$ projects a tensor to its symmetric, transverse,
and traceless part,

$\displaystyle\Lambda_{ij,k\ell}(\hat{\mathbf{n}})=P_{ik}P_{j\ell}-\displaystyle\frac{\,{1}\,}{2}P_{ij}P_{k\ell},$
$\displaystyle P_{ij}(\hat{\mathbf{n}})=\delta_{ij}-n_{i}n_{j}.$ (25)

To carry out the angular integral in (3), it is useful to note that

$\int\displaystyle\frac{\,{{\mathrm{d}}\Omega}\,}{4\pi}\Lambda_{ij,k\ell}=\displaystyle\frac{\,{1}\,}{30}(11\delta_{ik}\delta_{j\ell}-4\delta_{ij}\delta_{k\ell}+\delta_{i\ell}\delta_{jk}).$
(26)

So,

$\int\displaystyle\frac{\,{{\mathrm{d}}\Omega}\,}{4\pi}\Lambda_{ij,k\ell}Q_{ij}Q_{k\ell}=\displaystyle\frac{\,{2}\,}{5}Q_{ij}Q_{ij}.$
(27)

(Note that $Q_{ij}$ is traceless.) Rewriting $Q$ in terms of $M$ then gives
Eq. (3).

Then we need to know what is $M$ and how to compute its time derivative.

By rotation symmetry we can always bring the elliptical orbit into the $(x,y)$
plane. Then, using the polar coordinates $(r,\psi)$, the second moment of the
mass $M_{ij}$ is

$M_{ij}=\mu r^{2}\begin{pmatrix}\cos^{2}\psi&\cos\psi\sin\psi\\\
\cos\psi\sin\psi&\sin^{2}\psi\end{pmatrix}.$ (28)

For an elliptical orbit, $r$ can be rewritten as
$r=a(1-e^{2})/(1+e\cos\psi)^{2}$. So we have expressed $M_{ij}$, and thus
$h_{ij}$, in terms of $\psi$. It is this phase $\psi$ that has fast periodic
motion. So, to take time derivatives on $M_{ij}$, we only need to know how to
take time derivative on $\psi$, and this is nothing but Kepler’s second law:

$\dot{\psi}=\sqrt{\displaystyle\frac{\,{Gm}\,}{a^{3}(1-e^{2})^{3}}}(1+e\cos\psi)^{2}.$
(29)

The chirping and circularization of an eccentric binary by radiating GWs at
quadrupole order are described by Peters’s equations [29],

$\displaystyle\dot{a}=$
$\displaystyle-\displaystyle\frac{\,{64}\,}{5}\displaystyle\frac{\,{G^{3}\mu
m^{2}}\,}{c^{5}a^{3}}\displaystyle\frac{\,{1}\,}{(1-e^{2})^{7/2}}\bigg{(}1+\displaystyle\frac{\,{73}\,}{24}e^{2}+\displaystyle\frac{\,{37}\,}{96}e^{4}\bigg{)},$
(30) $\displaystyle\dot{e}=$
$\displaystyle-\displaystyle\frac{\,{304}\,}{15}\displaystyle\frac{\,{G^{3}\mu
m^{2}}\,}{c^{5}a^{4}}\displaystyle\frac{\,{e}\,}{(1-e^{2})^{5/2}}\bigg{(}1+\displaystyle\frac{\,{121}\,}{304}e^{2}\bigg{)}.$
(31)

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