2. The Planck frequency comb

2.1. Powers of α

The most salient problem with using the Planck time as a frequency standard is that it is so short. A photon with the Planck energy $E_{\rm P} = \sqrt {\hbar \! c^5/G}$ has of order 10²³ eV, which is far above the energy of the most energetic photons known. Indeed, if such photons can exist at all, they can interact with cosmic microwave background photons and produce particle–antiparticle pairs, meaning that space is actually opaque to them.

There is, however another physical constant fundamental to electromagnetic (and weak force) radiation: the charge of the electron e. Note that this constant is not particular to the electron: all charges in nature are integer multiples of e, with the exception of the quarks, and even those come in multiples of e/3. Combined with the constants used in Planck's units, one can construct the dimensionless fine structure constant α ≈ 1/137, expressed in electrostatic units as $\alpha = e^2/\lpar c\hbar \rpar$.

One can therefore construct an array of logarithmically spaced frequencies – a ‘frequency comb’Footnote ⁵ – defined by the Planck energy multiplied by integer powers of the fine structure constant.

This array will include photons at a large number of frequencies, some of them at convenient places across the electromagnetic spectrum. That is, one can write that there exists an array of natural photon energies

(1)$$E = E_{\rm P} \alpha^n$$

corresponding to frequencies

(2)$$\nu = {\alpha^n\over 2\pi t_{\rm P}}$$

for many integer values of n. Equivalently, one can think of α as the ‘base’ of a multiplicative counting system in Planck units. We can then define a dimensionless quantity n describing a photon with energy E or frequency ν or wavelength λ as

(3)$$n = \log_\alpha{\lpar E/E_{\rm P}\rpar } = \log_\alpha{\lpar 2\pi\nu t_{\rm P}\rpar } = \log_\alpha{\lpar 2\pi l_{\rm P}/\lambda\rpar }$$

where $l_{\rm P} = \sqrt {\hbar \!G / c^3}$ is the Planck length. We can then identify the relevant Schelling points as frequencies or energies for which n is an integer. I list those frequencies with integer values of n in Table 1.

Table 1. The Planck frequency comb

$n = \log _\alpha {\lpar E/E_{\rm P}\rpar }$.

These values are uncertain at the level of 7 ppm, limited by our knowledge of G. This is comparable to the precision with which we know the frame of the CMB and LSR.

For completeness, I include a wide range of frequencies here. I note in this and subsequent tables the frequency ranges of some space observatories capable of detecting these photons, and I mark the approximate energy of the highest energy cosmic rays (but these are presumably atomic nuclei, not photons).

Except in the midinfrared where airborne or space missions are necessary, from the optical through the radio there exist a wide array of ground-based telescopes that could perform spectroscopy to search for these signals. I note in the tables some of the optical/near-infrared bands, the frequency range of ALMA, the various microwave bands and the frequencies typical of telescopes designed to detect the Epoch of Reionization (Furlanetto et al., Reference Furlanetto, Oh and Briggs2006).

Below a few MHz radio waves cannot penetrate the ionosphere, however space missions such as the Netherlands-China Low-Frequency Explorer and SunRISE are planned to observe this part of the EM spectrum. Past this point I list in the notes the approximate frequencies corresponding to the gravitational wave detection experiments at LIGO, the planned space gravitational wave antenna LISA, and the pulsar timing arrays (PTAs).

Interestingly, there is a comb tooth in the optical, n = 13 at 6103 Å. Then next tooth, at n = 14, has λ = 83.632μ, and has a high background contamination from the Earth's atmosphere but is observable by the FIFI-LS instrument on SOFIA and was in the bandpass of Herschel. In the microwave, the n = 15 tooth is at 26.158 GHz, in K band and accessible to many radio telescopes. The n = 16 tooth at 190.89  MHz is subject to significant terrestrial radio frequency interference, but is accessible by, for instance, the Murchison Widefield Array and the Five hundred meter Aperture Spherical Telescope. There are no comb teeth in the frequency range spanned by ALMA.

3. Anthropecentrism of ‘Natural’ units, and alternative formulations

Planck felt that his units were ‘natural’, meaning that they transcended not just the physics tradition he was trained in, but humanity itself because they used only fundamental constants. This makes sense: we refer to these constants of nature as ‘fundamental’ because they describe properties of space, time and the fundamental forces, as opposed to the properties of the content of the universe (atomic constants, for instance) which can, in principle at least, be derived from them. One can easily imagine that all physics traditions, including extraterrestrial ones, would come to similar conclusions.

However, one can imagine many other ways to formulate the construction of natural units and the fundamental constants, and this complicates their use as Schelling points. To give a few examples:

• • Planck formulated his constants in terms of $\hbar$, so that the inverse of the Planck time corresponds to an angular frequency, which is a common convention in physics. But astronomers, for instance, typically express the energetic quality of light in cycles per second or wavelength, and so prefer to write E = hν or E = hc/λ instead of $E = \hbar \!\omega$. This feels awkward to some physicists because of the way we formulate physics in terms of algebra and our desire to be parsimonious in the number of symbols we use (avoiding lots of ‘2π's flying around) but the traditions of physics and engineering of other species may teach differently. One should therefore consider a set of natural units and α formulated with h instead of $\hbar$, and this would yield a different set of frequencies.

• • When determining the ‘base’ by which to multiply the Planck units, instead of the fine structure constant (which uses e, a constant of nature not found in the Planck formulation) one could use the base of the natural logarithm (also e!). One could, of course, use any dimensionless number.

• • In formulating the fine structure constant, one could also choose the smallest unit of charge in nature, that of the quark, e/3, instead of the charge of the electron.

• • The dark energy appears to provide an additional fundamental scale to the universe. Its energy density of ~7 × 10⁻³⁰g cm⁻³ can be used to construct a frequency known to any species with an understanding of cosmology (roughly 10⁻¹⁹ Hz, comparable to the present-day value of the Hubble constant). Our current best measurements of this quantity are probably not sufficiently precise for its use as a Schelling point in frequency, however.

• • One can multiply any apparently natural constant by an arbitrary dimensionless constant and have an equally valid constant. It is possible and perhaps likely that another species would have additional factors of 2, 3 or even π in their formulations of ‘fundamental’ physical constants.

• • For instance, even our use of π as a fundamental constant of mathematics is somewhat arbitrary: much or most of mathematics and physics can be expressed in a smaller number of algebraic symbols with the substitution 2π → τFootnote ⁷.

• • Other species may find it more obvious to transmit and receive at non-integer values of n, for instance half-integers or powers of 2π.

• • The use of the electron charge e in constructing the Planck Frequency Comb is most self-consistent when used to measure the frequency of photons (or neutrinos), but in principle one could use these same frequencies as a Schelling point for a beacon in gravitational waves. Alternatively, one might in this case use instead the gravitational fine structure constant, α_G = (m _e/m _P) ² ≈ 1.752 × 10⁻⁴⁵ where m _e is the electron mass and m _P is the Planck mass. In additional to somewhat arbitrarily using the electron's mass in a Schelling point for gravitational waves, it produces comb spacings so wide that there is barely even a single useful frequency: n = 1 corresponds to ν = 5.1715 mHz.

• • There are other constants of nature one could use. The detection of electromagnetic radiation always involves interactions with electrons, so it might be obvious to use m _e as a basis for a frequency comb. Indeed, the comb defined by $E = \lpar {1\over 2}m_{\rm e}c^2\rpar \alpha ^n$ contains another well-known unit of energy at n = 2: the Rydberg (R _∞ =13.6 eV, almost exactly the ionization potential of ground-state hydrogen).

Making different choices than Planck and I have made amount to changing the constant factor (2π/t _P) and scaling factor (α) of the Planck Frequency Comb. Because of this, the use of natural units as a Schelling point is perhaps best applied as a search for combs of signals of arbitrary constant and spacing, removing the need for one to guess at another species’ preference for the above choices. This complicates efforts to search a particular frequency, but implies that beacons may transmit at more than one frequency, corresponding to multiple teeth of their comb (so that the constant and spacing can be identified as obviously artificial and tied to the physical constants).

I provide in Tables 2–5 of the Appendix lists of Planck frequencies for different choices of $\hbar$ versus h (in both E _P and α, for consistency), for the natural logarithm versus α, and also the ‘Rydberg Frequency Comb’. To distinguish these alternative schemes from the one in Table 1, one can index n by which version of Planck's constant and which base it uses, yielding:

(4)$$n \equiv n_{{\hbar}\comma \alpha} = {\log{\lpar E/E_{\rm P}\rpar }\over \log{\alpha}}$$

(5)$$n_{\hbar\comma e} = \ln{\lpar E/E_{\rm P}\rpar }$$

(6)$$n_{h\comma \alpha} = {\log{\lpar E/\sqrt{hc^5/G}\rpar }\over \log{\lpar e^2/\lpar ch\rpar \rpar }}$$

(7)$$n_{h\comma e} = \ln{\lpar E/\sqrt{hc^5/G}\rpar }$$

(8)$$n_{R} = {\log{\lpar E/\lpar m_{\rm e}c^2/2\rpar \rpar }\over \log{\alpha}} = {\log{\lpar E/R_\infty\rpar }\over \log{\alpha}}-2$$

Significant wavelengths and frequencies appearing in these tables include 6332 Å and 6867 Å in the optical, 1.721 and 1.867μ in H band, and 1.07, 1.278, 2.528, 2.683, 2.909 and 7.908 GHz in the microwave.

Doubtless, other formulations of the Planck Frequency Comb can be made. Indeed, one might equally strongly argue that the empirically observed frequencies of strong atomic and molecular lines are more obvious as Schelling points than those constructed using abstract fundamental constants, and that a beacon would multiply these by ‘important’ mathematical constants to tellingly distinguish them from astrophysical sources (‘π times hydrogen’, for instance. See Blair and Zadnik, Reference Blair and Zadnik1993, for a list of examples.) Put another way, to an observational astronomer the frequencies nature provides observationally are the most obvious bases for units because they imagine their alien counterparts using radio telescopes observe the same thing, while for a theoretical physicist it is the fundamental constants of nature that provide that commonality because to build such a transmitter requires an understanding of physics.

Despite these difficulties, the Planck Frequency Comb I present in Table 1 is, I think, closely aligned with the spirit of Planck's original suggestion and among the most parsimonious in terms of number of algebraic symbols and physical constants.

But the number of choices one needs to make to construct the comb and choose a reference frame – and the fact that other physics traditions might use different units or formulations of physics or choices of frame – highlights the difficulty in applying Schelling's insight to SETI. One wishes to find certain points of commonality that are not specific to humans, but it can be challenging to identify which aspects of our physics traditions are truly universal, which are particular to how humans think, and which are simply accidents of how our physics has developed.

It is possible that there is some way to measure the parsimony of expression of quantities that would be present in all physics and mathematical traditions, including extraterrestrial ones, and so help guide work identifying such Schelling points. Such a possibility could be worthwhile exploring as an interdisciplinary effort among mathematics, physics, complexity theory and anthropology.

4. Applications

The most straightforward application of the Planck Frequency Comb as a Schelling point is to search for beacons at the frequencies of its teeth.

The importance of identifying specific search frequencies is not as important as it once was, because modern astronomical spectroscopy can search a large number of frequencies simultaneously. For instance, the Breakthrough Listen backend (MacMahon et al., Reference MacMahon, Price, Lebofsky, Siemion, Croft, DeBoer, Enriquez, Gajjar, Hellbourg, Isaacson, Werthimer, Abdurashidova, Bloss, Brandt, Creager, Ford, Lynch, Maddalena, McCullough, Ray, Whitehead and Woody2018) has a bandwidth of over 6 GHz, allowing it to perform high resolution spectroscopy across the entirety of any of the Green Bank Telescope's lower frequency receivers in a single integration.

Setting aside the issues of anthropogenic radio interference and background from the Earth's atmosphere, the sensitivity of such work to a narrowband signal is in principle set by the noise of the instrument and the length of the integration (Siemion et al., Reference Siemion, Demorest, Korpela, Maddalena, Werthimer, Cobb, Howard, Langston, Lebofsky, Marcy and Tarter2013). The threshold one chooses to identify candidate detections can then be set by the number of candidates one wishes to screen or, almost equivalently, the probability that signal of a given strength would have been observed by chance. In either case, the threshold is a function of the number of independent frequencies one observes.

For instance, the Breakthrough Listen radio search (e.g. Enriquez et al., Reference Enriquez, Siemion, Foster, Gajjar, Hellbourg, Hickish, Isaacson, Price, Croft, DeBoer, Lebofsky, MacMahon and Werthimer2017; Price et al., Reference Price, Enriquez, Brzycki, Croft, Czech, DeBoer, DeMarines, Foster, Gajjar, Gizani, Hellbourg, Isaacson, Lacki, Lebofsky, MacMahon, Pater, Siemion, Werthimer, Green, Kaczmarek, Maddalena, Mader, Drew and Worden2020; Sheikh et al., Reference Sheikh, Siemion, Enriquez, Price, Isaacson, Lebofsky, Gajjar and Kalas2020) searches billions of 2.7 Hz channels simultaneously, and typically sets a 25-σ threshold for detection above the instrumental noise. The high-resolution optical continuous-wave laser search of Tellis and Marcy (Reference Tellis and Marcy2015) and Tellis and Marcy (Reference Tellis and Marcy2017) had a bandwidth of a factor of 2 (from 3640–7890 Å) and a resolution of 60 000, meaning they searched ~45 000 independent frequency channels simultaneously. They chose a threshold equivalent to ~7-σ to generate a manageable number of candidates.

Application of a ‘magic frequency’ search using the Planck Frequency Comb or other list of frequencies allows for a much more sensitive search, because the number of trial frequencies is considerably lower. For instance, there are only two or three frequencies in the tables in this work in a given radio or optical band, meaning that they could each be examined individually for signals at the noise limit without any preliminary candidate thresholding, improving the sensitivity of the above searches by factors of several.

Another application would be to search for signals transmitted at multiple comb teeth, especially simultaneously. For the frequencies suggested in this work, this would usually involve observations with different kinds of telescopes or receivers, and then combining the significance of any signals found at multiple tooth frequencies. One might do such observations simultaneously, or one could prioritize observations by following up candidate signals at one comb tooth with observations at another. Indeed, there is little reason to suppose that a species would choose only a single mode of communication. Especially if success requires the receiver to guess the transmitter's frequency, transmitting all along a comb of frequencies makes strategic sense as a way to maximize the chances that they will guess correctly.

One could also acknowledge the uncertainty in correctly guessing the constant and spacing of the comb, and instead perform a search similar to that above but for all possible combs, including tightly-spaced combs with many teeth in the spectral grasp of single broadband instrument. This would be similar to the approach of Borra (Reference Borra2012) who advocated searching for light from laser frequency combs (which would evenly spaced in frequency, not log frequency as in the Planck Frequency Comb).