A Search for Technosignatures Around 11,680 Stars with the Green Bank Telescope at 1.15–1.73 GHz

The VEGAS instrument splits the 800 MHz recorded bandwidth into 256 coarse channels of 3.125 MHz each. In the process of doing so, it applies a bandpass filter to each coarse channel. This filter reduces the amplitude of the baseline near both edges of the spectra. We restored an approximately flat baseline by dividing each spectrum by a model of the bandpass filter response. This model was obtained by fitting a sixteen-degree Chebyshev polynomial to the median bandpass response of 28 scans in the 1664.0625–1667.1875 MHz frequency range, which is generally devoid of interference because it falls in the middle of the radio astronomy protected band (1660.6–1670.0 MHz) for the hydroxyl radical. We enforced an even response by setting the odd coefficients of the polynomial to zero.

Over the 150 s duration of our scans, narrowband signals from fixed-frequency transmitters are well approximated at the receiver by linear frequency modulated (FM) "chirp" signals, where the rate of change in frequency is dictated by the orbital and rotational motions of both the emitter and the receiver. The linear FM waveform is characterized by a signal of the form

$$\begin{eqnarray}&&s(t)=A\cos (2\pi ({f}_{0}t+{{Kt}}^{2}/2)),\hspace{2cm}0\leqslant t\leqslant \tau ,\end{eqnarray} \tag{ 1 }$$

where A is the signal amplitude, f₀ is the frequency at t = 0, K is the rate of change of the frequency, and τ is the duration of the scan. In complex exponential notation

$$\begin{eqnarray}&&s(t)=A\exp (j2\pi ({f}_{0}t+{{Kt}}^{2}/2))=A\exp (j\theta (t)),0\leqslant t\leqslant \tau .\end{eqnarray} \tag{ 2 }$$

The instantaneous frequency is the time derivative of the phase, i.e.,

$$\begin{eqnarray}&&f(t)=\displaystyle \frac{1}{2\pi }\displaystyle \frac{d\theta (t)}{{dt}}={f}_{0}+{Kt}.\end{eqnarray} \tag{ 3 }$$

The frequency increases linearly as f(t) = f₀ + Kt, with a total frequency excursion equal to K τ. An audible signal with this time–frequency behavior would sound like a chirp, hence the name commonly attributed to the waveform.

Doppler dechirping consists of compensating for a signal's drift in time–frequency space to facilitate integration of the signal power over the scan duration. In SETI searches, the drift rate is not known a priori. We used a tree algorithm of complexity O($N\mathrm{log}N$) (Taylor 1974; Siemion et al. 2013) to integrate the signal power at 1023 trial drift rates over the range ±8.86 Hz s⁻¹ with a drift rate resolution of ${\rm{\Delta }}\dot{f}=0.0173$ Hz s⁻¹. This approximate technique, known as incoherent dechirping, does not recover 100% of the signal power. Margot et al. (2021) quantified this signal loss with a dechirping efficiency factor η_D (0 ≤ η_D ≤ 1) for a variety of settings, including searches that utilize incoherent summing of power spectra prior to signal detection. In this and previous UCLA SETI work, we do not use incoherent averaging (i.e., N_{INC_SUMS} = 1) and the dechirping efficiency ranges between 60% and 100% with an average η_D ≃ 72% over the ±8.86 Hz s⁻¹ drift rate range. For searches with N_{INC_SUMS} = 51 over a ±4 Hz s⁻¹ drift rate range (e.g., Price et al. 2020; Gajjar et al. 2021) the dechirping efficiency ranges between 4% and 100% with an average η_D ≃ 16%. Importantly, the nominal performance of the tree algorithm for such searches is maintained in only a fairly narrow range of drift rates up to ±0.15 Hz s⁻¹, and the efficiency drops precipitously at larger drift rates due to Doppler smearing of the signal (Margot et al. 2021, Figure 7).

The received frequency f_r of a monochromatic transmission at frequency f_t experiences a time rate of change that depends on the line-of-sight acceleration $\dot{v}$ between transmitter and receiver. To first order

$$\begin{eqnarray}&&\displaystyle \frac{{\dot{f}}_{r}}{{f}_{t}}=\displaystyle \frac{\dot{v}}{c},\end{eqnarray} \tag{ 4 }$$

where the overdot denotes a time derivative and c is the speed of light. Our selection of a range of trial drift rates with maximum value ${\dot{f}}_{r,\max }=\pm 8.86$ Hz s⁻¹ corresponds to a fractional drift rate of ±6.24 nHz at f_t = 1.42 GHz (maximum accelerations ${\dot{v}}_{\max }$ of 1.87 m s⁻²). It is an appropriate choice because it accommodates line-of-sight accelerations due to the spins and orbits of most exoplanets. It can handle accelerations due the orbits of ∼73% of confirmed exoplanets with known semimajor axes and orbital periods, ∼93% of confirmed exoplanets with semimajor axes greater than 0.05 au, and 100% of confirmed exoplanets with semimajor axes greater than 0.1 au (NASA Exoplanet Archive 2019). It can also handle accelerations due to the spins of Earth-size planets at arbitrary periods (above the rotational breakup period) and Jupiter-size planets with spin periods greater than 11.5 hr. Transmitters located on exotic platforms that somehow exceed these limits could escape detection by our pipeline if the transmitted waveforms were not compensated to account for the platform's acceleration (Figure 1).

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At each frequency bin, our algorithm selects the trial drift rate that yields the greatest integrated signal power and determines whether the prominence of the signal (Margot et al. 2021) exceeds 10 times the standard deviation of the noise. The properties of signals that exceed the threshold are stored in a structured query language (SQL) database.

In practice, the algorithm proceeds in order of decreasing prominence in each coarse channel.

Signals with ${\dot{f}}_{r}=0$ are marked as anthropogenic RFI because they imply zero line-of-sight acceleration between the transmitter and receiver. Signals that are detected in more than one direction on the sky are also marked as RFI because a signal emitted beyond the solar system can appear in only one telescope beam. Finally, sources that are detected in only one of the two scans are also marked as intermittent RFI.

The direction-of-origin filter, also known as a directional filter or sky localization filter, can be run efficiently by retrieving the signal properties from our SQL database. A more stringent filter can be obtained by running the machine-learning (ML) algorithm of Pinchuk & Margot (2022).

Signals that remain after the line-of-sight distance and acceleration elimination process are marked as candidate technosignatures. All such candidates that fall outside of permanent RFI bands (Pinchuk et al. 2019) are visually inspected.

The flux from a transmitter with an equivalent isotropic radiated power (EIRP) at distance r is S = EIRP/(4π r²).

The signal-to-noise ratio (S/N) of a narrowband radio link has been computed by, e.g., Friis (1946), Kraus (1986), Enriquez et al. (2017), and Margot et al. (2018). It reads

$$\begin{eqnarray}&&{\rm{S}}/{\rm{N}}=\displaystyle \frac{S}{\mathrm{SEFD}}\sqrt{\displaystyle \frac{{n}_{\mathrm{pol}}\tau }{{\rm{\Delta }}f}},\end{eqnarray} \tag{ 5 }$$

where S is the observed flux, SEFD is the system equivalent flux density, a common measure of telescope and receiver performance, n_pol is the number of polarizations summed incoherently, τ is the integration time, and Δf is the channel receiver bandwidth (i.e., frequency resolution).

In a more rigorous formulation, the S/N includes the quantization efficiency η_Q due to imperfect digitization of the voltage signals and the dechirping efficiency η_D due to imperfect integration of the signal power (Margot et al. 2021)

$$\begin{eqnarray}&&{\rm{S}}/{\rm{N}}={\eta }_{Q}{\eta }_{D}\displaystyle \frac{S}{\mathrm{SEFD}}\sqrt{\displaystyle \frac{{n}_{\mathrm{pol}}\tau }{{\rm{\Delta }}f}}.\end{eqnarray} \tag{ 6 }$$

Quantization efficiency approaches unity with 8 bit sampling and is η_Q = 88.25% for an optimal 2 bit sampler (Kogan 1998). Dechirping efficiency with an O($N\mathrm{log}N$) algorithm depends on the frequency drift rate and ranges between 60% and 100% for the data acquisition and processing choices in this and previous UCLA SETI work, and between 4% and 100% for a BL-like process with N_{INC_SUMS} = 51 and ${\dot{f}}_{r}$ within ±4 Hz s⁻¹. It is possible to improve the dechirping efficiency if one is willing to use a costly O(N²) incoherent dechirping algorithm. For signals of interest with known frequency drift rates, UCLA SETI has the capability to apply a coherent dechirping algorithm to the raw voltage data, in which case η_D ≃ 1.

For the UCLA SETI program at the GBT, we have η_Q = 0.8825, SEFD = 10 Jy, n_pol = 2, τ = 150 s, and Δf ≃ 3 Hz. Our usual detection threshold is set at S/N = 10, such that signals with flux ${S}_{\det }=11.3\times {10}^{-26}$ W m⁻² are detectable. With these parameters, an Arecibo Planetary Radar (EIRP = 2.2 × 10¹³ W) is detectable at 415 ly and a thousand Arecibos can be detected at 13,123 ly. Conversely, transmitters located 326 ly (100 pc) away are detectable with 0.62 Arecibos (EIRP = 1.35 × 10¹³ W), transmitters located 20,000 ly (6132 pc) away are detectable with 2323 Arecibos (EIRP = 5.08 × 10¹⁶ W), and transmitters located at the Galactic center are detectable with 4130 Arecibos.

To quantify the end-to-end efficiency of the UCLA SETI pipeline, we injected 10,000 artificial chirp signals in raw voltage data from our 2021 search, processed the data as we normally do, and quantified the number of injected signals that were recovered by the pipeline.

We used Equation (2) to inject the artificial signals in complex voltage data sampled with 8 bit quantization, and we adjusted the signal amplitudes to achieve an S/N upon recovery of approximately 20. The starting frequencies of the signals were drawn randomly from a uniform distribution in the range 1.15–1.73 GHz, with the exclusion of the 1.20–1.34 GHz range that is blocked by a notch filter at the GBT. The drift rates were drawn randomly from a uniform distribution in the range ±8.86 Hz s⁻¹ (Figure 2).

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We used the exact same data files (raw voltage data files injected with 10,000 artificial signals) to estimate the end-to-end efficiency of a process that imitates the BL pipeline. Specifically, we computed power spectra with the FFTW library (Frigo & Johnson 2005) and a transform length of 2²⁰, yielding a frequency resolution Δf = 2.98 Hz that approximates the finest frequency resolutions (2.79 Hz, 2.84 Hz, and 2.93 Hz) of BL spectra (Lebofsky et al. 2019, Table 4). We applied a bandpass correction appropriate for the BL data acquisition backend. We then summed 51 consecutive power spectra incoherently, yielding a time resolution of 17.11 s, to approximate the 51-fold incoherent summing and time resolutions (17.40, 17.98, and 18.25 s) of the High Spectral Resolution (HSR) BL spectra (Lebofsky et al. 2019, Table 4). Finally, we ran BL's version of the Doppler dechirping tree algorithm, as implemented in turboSETI (Enriquez et al. 2017), with a maximum drift rate of ±8.88 Hz s⁻¹ and minimum S/N of 10, to identify signals and quantify the number of injected signals that were recovered (turboSETI -M 8.881784197 -s 10).

A signal was deemed to be recovered if two conditions were jointly met: (1) the recovered frequency was within ±6 Hz of the injected frequency, and (2) the recovered drift rate was within ±0.05 Hz s⁻¹ of the injected drift rate. These tolerances were designed to accommodate slight mismatches of up to two bins in the frequency dimension and the frequency drift rate dimension. The probability of an accidental match is less than one in a billion. Because of differences in algorithm implementation, the drift rate bins were 0.0173 Hz s⁻¹ for UCLA SETI and 0.0249 Hz s⁻¹ for turboSETI.

Ma et al. (2023) used a β-convolutional variational autoencoder and random forest analysis to identify eight promising signals not previously identified by the BL pipeline. They named these candidates MLc1 to MLc8, in reference to the ML process used in their analysis. We were interested in finding out whether these signals could be natively detected by the UCLA SETI pipeline, without any ML assistance. We downloaded the BL HSR power spectra of the MLc candidates and corresponding OFF scans, applied a bandpass correction (Section 3.1) appropriate for the BL data acquisition backend, and ran the resulting spectra through the UCLA SETI pipeline. We used only the first four of six scans of each pair of sources to mimic the A-B-A-B sequence used in UCLA SETI observations.

We detected 41.2 million narrowband signals with the data from our 2020–2023 observations. Almost all (99.43%) of these signals were rejected automatically by the UCLA SETI pipeline as RFI, which left approximately 230,000 signals warranting further consideration. Tens of thousands of these signals are being inspected by thousands of volunteers on the website http://arewealone.earth as part of a citizen science collaboration (Li & Margot 2023). Almost all (99.78%) of the remaining signals were detected in regions of dense RFI. We visually inspected all ∼500 candidate technosignatures that were detected outside of dense RFI regions and determined that they were all anthropogenic.

It is remarkable that, in over 82 million narrowband signal detections obtained during an 8 yr period (Table 1), not a single signal has merited follow-up observations. There have been plenty of instances where promising signals were only marginally detected in the corresponding OFF scans. We eliminate such signals from consideration.

The UCLA SETI pipeline recovered 9400 signals out of 10,000 injections, yielding an end-to-end pipeline efficiency for narrowband chirp signals of 94%. When regions of dense RFI were excluded, the UCLA SETI pipeline recovered 6716 signals out of 6807 injections, for an improved recovery rate of 98.7% (Figure 3). The distributions of recovered S/Ns and drift rates match those of the injected population (Figure 4). Signals that were not recovered are usually found near the bandpass edges, where the bandpass response and correction may be less than ideal, or intersect other signals in time–frequency space.

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A process designed to imitate the BL pipeline recovered a much smaller fraction of the injections. Specifically, only 570 signals out of 10,000 injections were recovered, with S/N and drift rate distributions that do not match the injected population and illuminate the reasons for the poor performance (Figure 5). Almost all (99.1%) signals recovered by the BL-like process have drift rates within ±1 Hz s⁻¹. Injections with larger drift rates are rarely recovered. This result is entirely consistent with the theoretical expectation of low dechirping efficiency for high drift rate signals observed in incoherently summed power spectra. In this situation, the signal power gets smeared across multiple frequency resolution cells because of Doppler drift during the longer integration times. For BL incoherent sums of 51 spectra at ∼3 Hz resolution, which extend the integration times from ∼0.3 s to ∼17 s, drift rates that exceed ±0.15 Hz s⁻¹ experience Doppler smearing. The dechirping efficiency falls rapidly, reaching 16% for drift rates of 1 Hz s⁻¹ (Margot et al. 2021), making recovery of signals at larger drift rates challenging. The diminishing performance as a function of drift rate is evident when plotting the S/Ns of signals recovered by the BL-like process as a function of drift rate (Figure 6).

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Detailed signal counts are listed in Table 2. These counts provide reasonable estimates of the end-to-end pipeline efficiency of radio SETI pipelines. The efficiency of a BL-like process is ∼5.7% for drift rates within ±8.88 Hz s⁻¹. Because injected signals have uniformly distributed drift rates and because all recovered signals have drift rates within the ±4 Hz s⁻¹ range used by Price et al. (2020) and Gajjar et al. (2021), we can estimate the end-to-end pipeline efficiency of their searches at 5.7% × 8.88/4 = 12.7%. Likewise, we find 5.7% × 8.88/2 = 25.3% for the work of Enriquez et al. (2017), who sampled drift rates within ±2 Hz s⁻¹.

The UCLA SETI pipeline successfully detected MLc3, MLc4, MLc5, MLc7, and MLc8 without invoking our own ML algorithms (Pinchuk & Margot 2022). We did not attempt to detect MLc1, MLc2, and MLc6 because the drift rates reported by Ma et al. (2023) for these signals (1.11 Hz s⁻¹, 0.44 Hz s⁻¹, and 0.18 Hz s⁻¹, respectively) exceed the nominal range of the O($N\mathrm{log}N$) tree algorithm given the incoherent summing of 51 consecutive spectra in BL HSR data products. Based on the characteristics and appearance of the signals, we predict that if the raw voltage data had been preserved, we could have recovered MLc1, MLc2, and MLc6 by processing the data without incoherent summing.

The characteristics of the signals detected by the UCLA SETI pipeline generally match those of the ML detections well (Table 3). The magnitudes of the drift rates match, but the signs differ, which we attribute to an error in Ma et al. (2023)'s report because our values are consistent with the signal slopes in their supplemental figures.

MLc3 was detected by the UCLA SETI pipeline but correctly identified automatically as RFI because the same signal is detected in the OFF scan at S/N ∼ 12 (Figure 7, left).

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MLc4 was detected by the UCLA SETI pipeline and identified as a candidate warranting visual inspection. Visual inspection clearly reveals the presence of the signal in the OFF scan (Figure 7, center), indicating that this candidate is RFI.

MLc5 was detected by the UCLA SETI pipeline and identified as a candidate warranting visual inspection. Detection of the signal in the OFF scan is less compelling (Figure 7, right), but the similarity in signal morphology with MLc4 indicates that this candidate is RFI. The frequency spacing between MLc4 and MLc5 is almost exactly 4 MHz, which suggests a common interferer.

MLc7 was detected by the UCLA SETI pipeline but correctly identified automatically as RFI because the same signal is detected in the OFF scan.

MLc8 was detected by the UCLA SETI pipeline and identified as a candidate warranting visual inspection. The signal is detected in the OFF scan and therefore labeled as RFI.

In summary, none of the MLc signals detected in our work warrant further examination.

The DFM (Drake 1984) provides an estimate of the search volume of a SETI program that captures almost all essential elements: frequency coverage, sky coverage, and sensitivity. It is defined as

$$\begin{eqnarray}&&\mathrm{DFM}=\displaystyle \frac{B\,{\rm{\Omega }}}{{S}_{\det }^{3/2}},\end{eqnarray} \tag{ 7 }$$

where B is the total bandwidth examined, Ω is the fractional area of the sky covered, and ${S}_{\det }$ is the minimum flux required for a detection. Typical units are GHz m³ W^−3/2 (e.g., Horowitz & Sagan 1993).

For transmitters of a given EIRP, the ${S}_{\det }^{-3/2}$ factor is proportional to the total volume that can be examined by a search with minimum detectable flux ${S}_{\det }(\propto \mathrm{EIRP}/4\pi {r}_{\max }^{2})$. The fraction of this volume that is actually sampled by the search is proportional to the fraction Ω of a 4π solid angle. Multiple observations of the same patch of sky with similar observing parameters can easily be accounted for by rewriting Ω = ∑_i Ω_i , where the index i represents individual observations. As such, Ω should be viewed as an effective solid angle and not a physical solid angle. In this context, one observation is defined as a complete set of scans, e.g., two scans of source A for the UCLA SETI cadence. The fraction of the entire radio spectrum that is captured by the search is proportional to the bandwidth B.

In its original form, the DFM misses two essential elements. First, it assumes that pipelines are perfect with end-to-end pipeline efficiencies of 100%, whereas the efficiency of different programs can vary by more than an order of magnitude (Table 2). Second, it ignores the frequency drift rate coverage, i.e., range of line-of-sight accelerations sampled in a search, whereas this range is an obvious indicator of the thoroughness of the search. We propose the modified DFM (MDFM) to address these limitations

$$\begin{eqnarray}&&\mathrm{MDFM}={\eta }_{P}\,\displaystyle \frac{{\dot{v}}_{\max }}{c}\displaystyle \frac{B\,{\rm{\Omega }}}{{S}_{\det }^{3/2}},\end{eqnarray} \tag{ 8 }$$

where η_P is the end-to-end pipeline efficiency for the detection of signals of interest and ${\dot{v}}_{\max }/c$ is the maximum fractional frequency drift rate (with respect to the center of the band) considered in the search (Equation (4)). We express the latter in units of nHz and show MDFM values in units of Hz² m³ W^−3/2. We chose a metric that is linear in the range of frequency drift rates examined because we cannot predict the locations, sizes, or spins of preferred transmitter platforms. In the absence of reliable information, a uniform prior distribution for the frequency drift rate seems reasonable. One could design the distribution to accommodate the majority of exoplanet settings, with an upper limit of 26 nHz that accommodates 95% of confirmed exoplanets with known semimajor axes and orbital periods.

Values of the DFM and MDFM metrics for the UCLA SETI search are compared to those of select surveys in Table 4 and Figure 8. We have assumed η_P = 100% for the surveys of Horowitz & Sagan (1993) and Harp et al. (2016) and the estimates of Section 4.2 for the surveys of Enriquez et al. (2017), Price et al. (2020), and UCLA SETI. The drift rate coverage of Horowitz & Sagan (1993) is unlike those of modern surveys. It is large but samples only three distinct values (local standard of rest, galactic barycenter, and cosmic microwave background rest frame). We have assumed a fractional drift rate of 1 nHz as a compromise, which is the same value that Harp et al. (2016) used. We found that the MDFM of the UCLA SETI search falls in between the survey of 692 primary stars of Enriquez et al. (2017) and the survey of primary 1327 stars of Price et al. (2020).

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Table 4. Search Volume Characteristics of Select Surveys

	Horowitz & Sagan (1993)	Harp et al. (2016)	Enriquez et al. (2017)	Price et al. (2020)	UCLA SETI
Freq. coverage B (GHz)	4 ×10–4	variable a	0.660	variable b	0.439
Sky fraction Ω	0.7	1.17 × 10–3 c	2.88 × 10–4	5.03 × 10–4	4.91 × 10–5
Sensitivity ${S}_{\det }$ (W m−2)	1700 × 10–26 d	260 × 10–26 e	17.7 × 10–26 f	variable g	11.3 × 10–26 h
Pipeline efficiency ηP	100%	100%	25.3%	12.7%	94.0%
Drift rate coverage (nHz)	1	1	1.33	2.66	6.24
DFM (GHz m3 W−3/2)	1.23 × 1031	1.70 × 1032	2.56 × 1033	2.59 × 1034	5.74 × 1032
MDFM (Hz2 m3 W−3/2)	1.23 × 1031	1.70 × 1032	8.63 × 1032	1.21 × 1034	3.37 × 1033

Notes. The DFM does not account for pipeline efficiency nor frequency drift rate coverage, but the MDFM does.

^a We used 8 GHz for 65 stars, 2.04 GHz for 1959 stars, 0.337 GHz for 2822 stars, and 0.268 GHz for 7459 stars (Harp et al. 2016). ^b We used 0.66 GHz for GBT L band, 0.94 GHz for GBT S band, and 0.85 GHz for Parkes 10 cm (Price et al. 2020). ^c Based on 3''×6'' synthesized beam. The solid angle appears to have been overestimated by a factor of 4 in Enriquez et al. (2017). ^d For S/N = 30 in 20 s (Horowitz & Sagan 1993). ^e For S/N = 6.5 in 93 s (Harp et al. 2016). We used a system temperature of 108 K, which is the average across all four bands. ^f For S/N = 25 in 300 s (Enriquez et al. 2017). ^g For S/N = 10 in 300 s (Price et al. 2020). We used 7.1 × 10^–26 for the GBT and 24.0 × 10^–26 for Parkes. ^h For S/N = 10 in 150 s. This values takes the quantization efficiency η_Q = 0.8825 into account.

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Another possible disadvantage of the DFM is that it assumes a uniform distribution of transmitters on the sky, whereas transmitters may be preferentially located near stars, which are not uniformly distributed. At ∼1% of the Galactic scale, the assumption of spatial uniformity holds reasonably well. For instance, the Gaia catalog of nearby (100 pc) stars is expected to be volume complete for all stars of spectral type earlier than M8 and shows a roughly uniform spatial distribution of the 331,312 objects (Gaia Collaboration et al. 2021). At larger distances, the assumption breaks down, especially for directions perpendicular to the plane of the Galactic disk. Drake (1984) had anticipated this problem by considering distances <1 kpc.

As our Galactic models and star catalogs improve, we can refine the MDFM by replacing the physical volume covered by a search with the actual number of stars sampled by each observation, assuming again that transmitters may be preferentially located near stars. Let us consider the number of stars dn_* in an elemental volume of sky

$$\begin{eqnarray}&&{{dn}}_{* }(r,\theta ,\phi )={\rho }_{* }(r,\theta ,\phi ){r}^{2}\sin \theta d\theta d\phi {dr},\end{eqnarray} \tag{ 9 }$$

where ρ_* is the stellar density (number of stars per unit volume) and (r, θ, ϕ) describe spherical coordinates in a frame centered at the solar system barycenter. The figure of merit for transmitters of a fiducial EIRP can then be written

$$\begin{eqnarray}&&\mathrm{MDFM}{| }_{\mathrm{EIRP}}={\eta }_{P}\,\displaystyle \frac{{\dot{v}}_{\max }}{c}\,B\,\displaystyle \sum _{i}{n}_{* ,i}\end{eqnarray} \tag{ 10 }$$

where the number of stars in each observation i is extracted from a catalog query that includes distance and angular bounds or computed from a Galactic or extragalactic model

$$\begin{eqnarray}&&{n}_{* ,i}={\int }_{o}^{{r}_{\max }}{dr}{\iint }_{{{\rm{\Omega }}}_{i}}{\rho }_{* }(r,\theta ,\phi ){r}^{2}\sin \theta d\theta d\phi ,\end{eqnarray} \tag{ 11 }$$

with ${r}_{\max }=\sqrt{\mathrm{EIRP}/(4\pi {S}_{\det })}$ and Ω_i is the full width at half maximum (FWHM) solid angle of the telescope beam. Note that the quantization and dechirping efficiencies are properly taken into account via ${S}_{\det }$ and, therefore, ${r}_{\max }$. Multiple observations of the same stars are allowed in these expressions to account for the fact that repeated observations are valuable.

We describe a formalism to calculate upper bounds on the prevalence of civilizations operating transmitters detectable in SETI surveys. Our calculation presupposes that the observed stars form a representative sample of the population of stars in the relevant search volume.

We write the total number of observed stars for transmitters of a fiducial EIRP

$$\begin{eqnarray}&&{n}_{\mathrm{obs}}{| }_{\mathrm{EIRP}}=\displaystyle \sum _{i}{n}_{* ,i},\end{eqnarray} \tag{ 12 }$$

where the number of stars is calculated as in Section 5.1. When including all stars within the solid angle defined by the antenna beam FWHM, the EIRP ought to be augmented from its nominal value to account for emissions detected in off-axis directions. However, we ignore this small correction, which is at most a factor of 2 at the antenna beam's half maximum.

We consider the fraction f_TX of stars in the observed sample that host a detectable transmitter, such that the number of detectable transmitters in the observed sample is n_TX = f_TX × n_obs∣_EIRP. If the observed sample is representative of the entire search volume, f_TX can be used as an estimate that applies to the entire search volume. We wish to place an upper limit on f_TX on the basis of our observations and the fact that we did not detect a technosignature.

We acknowledge the fact that SETI pipelines are not 100% efficient. For each observation of a detectable transmitter, the probability of success for detection of the transmitter is <100%. For data analysis pipelines that have been characterized with an injection-and-recovery analysis, we can set this probability to η_P , the end-to-end pipeline efficiency.

We also acknowledge that a transmitter may not be detectable at all times by considering the duty cycle D of the transmitter, i.e., the fraction of time that the transmitter is beaming in Earth's direction.

We write the probability of detecting a transmitter in each observation of a star as

$$\begin{eqnarray}&&p={f}_{\mathrm{TX}}{\eta }_{P}D.\end{eqnarray} \tag{ 13 }$$

We consider the result of our observations as the result of n = n_obs∣_EIRP independent trials, each with the same probability of success p. The number of successes in such an experiment is given by the binomial distribution B(n, p).

We determine the largest possible value of f_TX that is consistent with obtaining zero successes in n attempts at a confidence level (CL). This value is obtained by solving

$$\begin{eqnarray}&&{\left(1-p\right)}^{n}=1-\mathrm{CL},\end{eqnarray} \tag{ 14 }$$

i.e.,

$$\begin{eqnarray}&&{f}_{\mathrm{TX}}^{u}=\displaystyle \frac{1-{\left(1-\mathrm{CL}\right)}^{1/n}}{{\eta }_{P}D},\end{eqnarray} \tag{ 15 }$$

where we have labeled the superscript u to denote the upper limit. At the 95% CL and for η_P ≃ 1, D ≃ 1, and n > 20, this result is well approximated by the "rule of three" (Jovanovic & Levy 1997): ${f}_{\mathrm{TX}}^{u}\simeq 3/n$. This rule was derived but not named as such in the Cyclops Report (Oliver & Billingham 1971, p. 53).

Our signal injection-and-recovery analysis indicates that the UCLA SETI pipeline would have at most a 94.0%–98.7% probability of detecting a narrowband technosignature in any given observation of a star hosting a detectable transmitter. If the transmission frequencies are uniformly distributed in the range 1.15–1.73 GHz, the probability is closer to 94.0%. If the transmission frequencies happen to fall among radio astronomy protected bands or regions where RFI is less severe, the probability is closer to 98.7%. We evaluate upper limits in the conservative case with η_p = 94%.

For this survey with 62 observations and the fiducial EIRP of 0.62 Arecibos (1.35 × 10¹³ W) corresponding to detectability up to 100 pc, we find N_obs∣_EIRP = 47 in the Gaia catalog of nearby stars (Gaia Collaboration et al. 2021). In this observed sample, the maximum transmitter fraction that is compatible with our nondetection at a CL = 95% is 6.6%, assuming a transmitter duty cycle of 100%. If this result is generalizable to the entire search volume, there are at most 6.6% of the 331,312 stars within 100 pc that host a transmitter detectable in our survey. If we consider a fiducial EIRP = 5.08 × 10¹⁶ W that enables detection of transmitters around any of the 10,230 observed stars located within 20,000 ly, we find that the fraction of stars with detectable transmitters is at most 3 × 10⁻⁴. This limit is more stringent than those published by Wlodarczyk-Sroka et al. (2020), considering the necessary revisions to their estimates (Section 5.3).

A detectable transmitter has the following sufficient characteristics: (1) it emits in the frequency range 1.15–1.73 GHz (excluding the range 1.20–1.34 GHz), (2) it has a line-of-sight acceleration with respect to the GBT that results in a frequency drift rate within ±8.86 Hz s⁻¹, and (3) it emits a fixed-frequency or chirp waveform with bandwidth <3 Hz, 100% duty cycle, and minimum EIRP as stated above. Characteristics (1) and (2) are necessary for detection, but characteristics (3) are not. For instance, we could detect more complex or broader waveforms (e.g., pulsed waveforms, nonchirp waveforms, or waveforms with >3 Hz bandwidth) provided that the integrated power exceeded our detection threshold. We could also detect an intermittent transmitter provided that we observed at a favorable time. For transmitter duty cycles below 100%, the upper limits on transmitter prevalence are degraded (Figure 9).

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Many previous works ignored the dechirping efficiency and provided estimates of SETI search volumes or upper limits on the fraction of stars hosting transmitters under the assumption that the end-to-end efficiency of their pipeline was 100% or 50%. Our results show that the efficiency of a BL-like process is closer to 5.7% for drift rates within ±8.88 Hz s⁻¹, 12.7% for drift rates within ±4 Hz s⁻¹, and 25.3% for drift rates within ±2 Hz s⁻¹, suggesting that the published estimates of search volumes and transmitter limits in previous works need revisions. We propose revised estimates below.

We suggest that the statement of Enriquez et al. (2017) that "fewer than ∼0.1% of the stellar systems within 50 pc possess these types of transmitters" ought to be rephrased as "fewer than ∼1.7% of the stellar systems within 50 pc possess the types of transmitters detectable in this search." Focusing on their 692 primary targets only, which are stars within 50 pc, we can apply the formalism of Section 5.2 with η_p = 25.3%, which is appropriate for frequency drift rates up to ±2 Hz s⁻¹ and BL data products with 51-fold incoherent averaging. We find that the upper limit on f_TX is 1.7% at the 95% CL for their 692 primary targets, which is a more realistic upper limit on the transmitter fraction.

Price et al. (2020) described the results of a search of 882, 1005, and 189 primary targets at L band, S band, and 10 cm, respectively, with drift rates up to ±4 Hz s⁻¹ in data products with 51-fold incoherent averaging. They used a detection probability p = 50% in their prevalence calculations, which yields 95% CL prevalence estimates of 0.68% at L band, 0.59% at S band, and 3.1% at 10 cm. These values are approximately 50% larger than those published by Price et al. (2020), which are 0.45%, 0.37%, and 2.0%, respectively, i.e., ∼2/n. With the more realistic pipeline efficiency of η_p = 12.7%, we find an upper limit of f_TX = 2.7% at L band, suggesting that their upper limits must be revised upwards by a factor of ∼6.

Wlodarczyk-Sroka et al. (2020) improved the prevalence estimates by accounting for stars captured in the telescope beam in addition to the primary targets of Enriquez et al. (2017) and Price et al. (2020). However, they combined two data sets that are heterogeneous in their probabilities of detection success, which makes revision of their estimates difficult. Nevertheless, we can safely infer that their upper limit estimates must be revised upwards by approximately an order of magnitude, i.e., between the correction factors of 6 and 17 that we identified for these two surveys. For instance, their estimate of the incidence of star systems with detectable transmitters within 100 pc ought to be revised from 0.061% to ∼0.6%. Likewise, their estimate for main-sequence stars within 1 kpc of 0.005% ought to be revised to 0.05%, slightly larger than our 0.03% upper limit for stars within 20,000 ly. We found 1732 stars within 1 kpc in our observed sample (EIRP = 1.35 × 10¹⁵ W), which leads to an upper limit of transmitter prevalence of 0.18% for these stars with our observations. For this 1 kpc search volume, the prevalence estimate of Wlodarczyk-Sroka et al. (2020), with the correction presented here, is approximately three times more stringent than ours but required approximately 100 times more telescope time (506.5 hr versus 5.2 hr of GBT L-band on-source integration time). Garrett & Siemion (2023)'s estimates for extragalactic transmitters require similar revisions.

Traas et al. (2021) observed 28 TESS targets with 28 cadences at L, S, C , and X band each, totaling ∼56 hr of on-source GBT time. They used a detection probability p = 50% in their transmitter prevalence calculations. With their assumed pipeline efficiency, we find an upper limit to the fraction of observed stars with transmitters of 20.3% at the 95% CL, which is not identical to their estimate of 12.72%. Because they searched drift rates of up to ±4 Hz s⁻¹ in data products with 51-fold incoherent averaging, a pipeline efficiency of η_p = 12.7% is more appropriate, and their upper limits must be revised upwards by a factor of ∼6, leading to a weak limit of ∼80%.

Franz et al. (2022) searched 5, 17, 21, and 23 cadences of observations at L, S, C , and X band, capturing a total of 61 TESS TOIs in transit with ∼30 hr of GBT on-source time. If the pipeline efficiency were η_p = 50%, one might expect an upper limit on the fraction of stars with transmitters of 90.0%, 32.3%, 26.6%, and 24.4%, respectively, which differ from their values by more than ∼60%. However, the pipeline efficiency is closer to η_p = 12.7%, which leads to an inability to place an upper limit at the 95% CL in the L, S , and C bands, and a weak limit of ∼96% at X band.

Shortly after Cocconi & Morrison (1959) proposed to search for interstellar communications in the radio part of the spectrum, Schwartz & Townes (1961) argued that similar searches could be conducted in the optical part of the spectrum. Townes (1983) suggested that searches in both the microwave and infrared parts of the spectrum were warranted. In this section, we recognize the value of searching at multiple wavelengths and briefly describe the results of a few optical SETI initiatives, being mindful that a review of this field is well beyond the scope of this work. Radio and optical SETI searches are most likely to be successful if presumptive civilizations operate beacons to emit distinctive signals, continuous or pulsed. Other information carriers such as charged particles, massive particles, gravitational waves, and neutrinos have been considered in the SETI context but are deemed inferior to photons for communication purposes (Hippke 2018).

Howard et al. (2004) and Stone et al. (2005) described the results of multiyear targeted searches of several thousand stars for nanosecond pulses emitted by laser beacons. Tellis & Marcy (2017) adopted a different approach and searched for laser emission lines in high-resolution spectra of 5600 nearby stars. Maire et al. (2019) described the results of a search for near-infrared pulses from 1280 celestial objects.

The prevalence calculations from these optical SETI surveys outperform the radio SETI limits calculated for distances to 100 pc but not those calculated for larger distances. Tellis & Marcy (2017) ruled out "models of the Milky Way in which over 0.1% of warm, Earth-size planets harbor technological civilizations that, intentionally or not, are beaming optical lasers toward us." Howard et al. (2004) published transmitter limits as a function of the transmitter repetition time. The fraction of stars with transmitting civilizations is also at most 0.1% in their work if one assumes repetition periods of ∼3 × 10⁴ hr, which may be reasonable considering that it would take us approximately that long to beam a laser at all 331,312 stars within 100 pc with a 5 minute dwell time (equivalent duty cycle D_opt ≃ 3 × 10⁻⁶). The upper limit on transmitter prevalence that we obtained is 3 × 10⁻⁴, but only for transmitters with D_radio = 100%.

From the point of view of the transmitting civilization, the requirements that an optical beacon be pointed at each one of tens of billions of stars is considerably more onerous than the equivalent requirement at radio wavelengths, where the entire sky can be covered much faster. For instance, the ratio of broadcast solid angles for beamwidths of 84 (GBT-class telescope at L band) and 1'' (optical telescope; Townes 1983) is 2.5 × 10⁵. This ratio also dictates the ratio of duty cycles D_radio/D_opt, which may justify the comparison with different values of the duty cycles in the paragraph above. To a civilization intent on broadcasting its existence, this intrinsic advantage of radio beacons may eclipse other considerations. However, we are unable to anticipate the choices of presumptive civilizations, and it behooves the SETI community to pursue a variety of search modalities.