Peeps: Gravitational Bursts and Peer Effects

• Peeps are a polysemous term describing brief, recurring gravitational-wave bursts in extreme mass-ratio inspirals and specific peer effects in economic choice models.
• In astrophysics, peeps are modeled using Numerical Kludge methods to capture periapsis features and assess their impact on LISA detection under different background scenarios.
• In discrete-choice contexts, peeps quantify how social interactions alter both consideration sets and preference rankings, aiding in the separation of intertwined peer effects.

“Peeps” is a polysemous technical term in recent research usage. In gravitational-wave astrophysics, a peep is the recurring, short-duration periapsis burst emitted during the early, very highly eccentric phase of an extreme mass-ratio inspiral (EMRI), when a stellar-mass compact object repeatedly sweeps through periapsis around a massive black hole yet remains silent for most of each long orbit (Oliver et al., 25 Jul 2025). In microeconomic theory, “peeps” is shorthand for peer effects that operate through either consideration sets, preference rankings, or both in dynamic discrete-choice environments, with identification based on how social interactions distort observed conditional choice probabilities (Kashaev et al., 2023). The two usages are unrelated substantively, but both concern mechanisms whose individual realizations can be weak or intermittent while their aggregate consequences are analytically decisive (Oliver et al., 2023).

1. Definitions and terminological scope

In the gravitational-wave literature, a peep is defined by recurrence. The underlying system is a stellar-mass compact object of mass $\mu\sim 10\!-\!30\,M_\odot$ orbiting a massive black hole of mass $M\sim 10^{5.2}-10^8\,M_\odot$, with mass ratio $q\equiv \mu/M \sim 10^{-7}-10^{-4}$. After capture into a near-parabolic bound orbit, the object spends most of its orbital period far from the massive black hole and radiates little in the LISA band, then emits a brief millihertz burst at periapsis; because the orbit is bound with $e\lesssim 1$ rather than exactly parabolic, the burst repeats once per orbit (Oliver et al., 25 Jul 2025). This distinguishes a peep both from an extreme-mass-ratio burst treated as a one-off parabolic flyby and from the later, quasi-continuous EMRI waveform near merger.

In the discrete-choice literature, “peeps” denotes peer effects in a model where social interactions can change either what an agent considers or how she ranks what she considers. The core objects are two directed edge sets, $\Gamma=(\Gamma_C,\Gamma_R)$, where $\Gamma_C$ affects consideration and $\Gamma_R$ affects preferences. A peer can therefore affect only consideration, only preferences, or both (Kashaev et al., 2023). This terminology is narrower than generic “social interactions,” because the paper ties it to a specific identification program based on behavioral signatures in observed choices.

A common misconception in both literatures is to collapse the term into a nearby but distinct category. In the astrophysical case, peeps are not merely parabolic bursts with slightly different notation; the bound recurrence is the central physical feature. In the economic case, peer effects are not restricted to preference spillovers; limited consideration is a distinct margin with different empirical implications (Oliver et al., 25 Jul 2025).

2. Gravitational-wave peeps as an EMRI regime

The gravitational-wave peep arises in the standard EMRI capture channel through stellar dynamical interactions in galactic nuclei. Two-body or multi-body scattering occasionally places a compact object on an extremely eccentric bound orbit around a central massive black hole, after which gravitational radiation reaction removes energy and angular momentum and converts the capture into an inspiral (Oliver et al., 25 Jul 2025). The key observational fact is a strong separation of timescales: the orbital period is set mainly by the very large apoapsis of the nearly parabolic orbit, whereas the LISA-band emission is concentrated into a short periapsis passage.

This regime can persist for most of the inspiral lifetime. One explicit capture-stage example has $T \approx 185000\ \text{years}$ for an orbit with $a=0.9M$, $p=120M$, and $M\sim 10^{5.2}-10^8\,M_\odot$0, while a full inspiral example lasts about $M\sim 10^{5.2}-10^8\,M_\odot$1; early in that evolution, a 7-day zoom-in contains one peep, whereas near plunge the same interval contains about 115 peaks and the signal becomes nearly continuous (Oliver et al., 2023). The early peep phase is therefore not a different source class from EMRIs, but an early dynamical regime of the same system.

The morphology is governed primarily by periapsis conditions. The emitted waveform can exhibit zoom-whirl or homoclinic behavior, with a long-period structure plus a short, rapid periapsis feature, and the amplitude evolves only weakly over much of the inspiral because the periapsis remains roughly fixed while the apoapsis shrinks (Oliver et al., 25 Jul 2025). This is why peeps can remain astrophysically long-lived yet individually faint.

The distinction from parabolic-burst modeling is consequential. For fixed parameters, a bound source with $M\sim 10^{5.2}-10^8\,M_\odot$2 can complete 3 orbits in one year and emit 3 peeps, while an almost parabolic source emits only one burst; in the example reported, the peep spectrum has a peak amplitude about 4 times larger and extends to higher frequencies, and at $M\sim 10^{5.2}-10^8\,M_\odot$3 the peep source gives about 50 complete orbits in one year (Oliver et al., 2023). Approximating peeps as single-pass parabolic bursts therefore undercounts the finite-time frequency-domain content relevant to space-based detection.

3. Modeling peeps for LISA and confusion-noise estimation

The waveform model used for peeps is the Numerical Kludge (NK) model of Babak et al. (2007). The parameter space is reduced to the 12 quantities

$M\sim 10^{5.2}-10^8\,M_\odot$4

with redshift $M\sim 10^{5.2}-10^8\,M_\odot$5 as input and conversion to luminosity distance $M\sim 10^{5.2}-10^8\,M_\odot$6. The workflow is to define the orbit through the constants of motion $M\sim 10^{5.2}-10^8\,M_\odot$7, integrate the Kerr geodesic equations to obtain the trajectory in Boyer–Lindquist coordinates $M\sim 10^{5.2}-10^8\,M_\odot$8, map that trajectory into flat-space Cartesian coordinates, and then apply the quadrupole formula to compute the metric perturbation (Oliver et al., 25 Jul 2025). The model is taken to retain sufficient strong-field structure for the peep regime, especially for periapses $M\sim 10^{5.2}-10^8\,M_\odot$9, corresponding roughly to $q\equiv \mu/M \sim 10^{-7}-10^{-4}$0 at peep eccentricities.

The source population is built from the Illustris-1 cosmological hydrodynamic simulation out to $q\equiv \mu/M \sim 10^{-7}-10^{-4}$1, with the EMRI-relevant massive-black-hole mass range restricted to

$q\equiv \mu/M \sim 10^{-7}-10^{-4}$2

mass functions tabulated in $q\equiv \mu/M \sim 10^{-7}-10^{-4}$3-dex bins, and shell-by-shell integration in $q\equiv \mu/M \sim 10^{-7}-10^{-4}$4 increments beyond the local $q\equiv \mu/M \sim 10^{-7}-10^{-4}$5 volume (Oliver et al., 25 Jul 2025). Baseline EMRI rates are taken from Babak et al. (2017), using the median rate from their Eqs. 23–31 and plunge-to-EMRI ratio $q\equiv \mu/M \sim 10^{-7}-10^{-4}$6. For each massive black hole in the synthesized population, a compact-object mass is drawn, a 4-year EMRI occurrence probability is assigned, orbital and angular parameters are sampled, an NK waveform is generated, the signal is cosmologically redshifted via

$q\equiv \mu/M \sim 10^{-7}-10^{-4}$7

and the result is passed through the LISA response using fastlisaresponse with second-generation time-delay interferometry and ESA trailing orbits, retaining only the $q\equiv \mu/M \sim 10^{-7}-10^{-4}$8 and $q\equiv \mu/M \sim 10^{-7}-10^{-4}$9 channels because $e\lesssim 1$0 is approximately null (Oliver et al., 25 Jul 2025).

The study does not construct the background via the standard $e\lesssim 1$1 or $e\lesssim 1$2-based stochastic-background formalism. Instead it performs direct time-domain Monte Carlo superposition of many individually unresolvable detector responses. Characteristic strain is taken as

$e\lesssim 1$3

and single-channel SNR is evaluated with the standard matched-filter-like expression

$e\lesssim 1$4

with the $e\lesssim 1$5 and $e\lesssim 1$6 channel SNRs then combined in quadrature (Oliver et al., 25 Jul 2025).

Three background scenarios summarize the astrophysical uncertainty. Background 1 assumes $e\lesssim 1$7, $e\lesssim 1$8, and at most one highly eccentric EMRI per massive black hole during the mission. Background 2 keeps the one-or-fewer-per-massive-black-hole assumption but broadens the capture distribution to $e\lesssim 1$9 and $\Gamma=(\Gamma_C,\Gamma_R)$0. Background 3 adopts Background 2 and then multiplies it by $\Gamma=(\Gamma_C,\Gamma_R)$1 to approximate an incoherent sum of 1000 signals per host, motivated by the much more abundant early-EMRI picture discussed by Amaro-Seoane et al. (2024) (Oliver et al., 25 Jul 2025).

The reported Monte Carlo realization generates 1470 unique peep waveforms. In Background 1, the per-channel SNRs are of order $\Gamma=(\Gamma_C,\Gamma_R)$2 and the combined SNR is $\Gamma=(\Gamma_C,\Gamma_R)$3. In Background 2, the per-channel SNRs are of order $\Gamma=(\Gamma_C,\Gamma_R)$4 and the combined SNR is $\Gamma=(\Gamma_C,\Gamma_R)$5. In Background 3, the $\Gamma=(\Gamma_C,\Gamma_R)$6 and $\Gamma=(\Gamma_C,\Gamma_R)$7 channels each have SNR $\Gamma=(\Gamma_C,\Gamma_R)$8, with combined SNR $\Gamma=(\Gamma_C,\Gamma_R)$9 (Oliver et al., 25 Jul 2025). Under the first two, peeps produce only a slight rise in the LISA noise floor; under the third, they would constitute a directly detectable foreground capable of obscuring other sources.

4. “Peeps” in dynamic discrete choice

In the economic model, there is a finite set of agents

$\Gamma_C$0

a finite menu

$\Gamma_C$1

and a choice configuration $\Gamma_C$2. Time is continuous, and each agent $\Gamma_C$3 receives revision opportunities according to an independent Poisson clock with rate $\Gamma_C$4 (Kashaev et al., 2023). When a clock rings, the decision problem is two-stage: the agent first forms a consideration set $\Gamma_C$5, then chooses an option from $\Gamma_C$6.

The model’s distinctive feature is the separation of social influence into two directed networks. For agent $\Gamma_C$7, $\Gamma_C$8 denotes peers affecting consideration and $\Gamma_C$9 denotes peers affecting preferences. Consideration for alternative $\Gamma_R$0 is summarized by $\Gamma_R$1, with the baseline restriction that inclusion is independent across alternatives: $\Gamma_R$2 and $\Gamma_R$3. For $\Gamma_R$4, the probability of considering $\Gamma_R$5 depends on the number, not the identity, of consideration peers choosing $\Gamma_R$6: $\Gamma_R$7 This makes consideration effects alternative-specific and non-crossing.

Conditional on $\Gamma_R$8, the agent chooses according to

$\Gamma_R$9

again with an anonymity restriction: the rule depends on counts of peers choosing each alternative in $T \approx 185000\ \text{years}$0, not their identities. Unlike consideration, preference effects are intrinsically cross-alternative because peer shifts among alternatives change relative desirability. The paper gives a logit example,

$T \approx 185000\ \text{years}$1

and explicitly allows $T \approx 185000\ \text{years}$2 itself to affect utilities, so IIA can fail (Kashaev et al., 2023).

The observed conditional choice probability is

$T \approx 185000\ \text{years}$3

which factorizes into the probability that $T \approx 185000\ \text{years}$4 is considered times the probability that $T \approx 185000\ \text{years}$5 is chosen conditional on being considered (Kashaev et al., 2023). This decomposition is the basis for identification.

5. Identification, recovery, and empirical implementation

Because only one agent updates at a time almost surely, the model induces a continuous-time Markov chain over configurations. Under positive consideration and positive choice probabilities, the paper establishes existence of a unique full-support invariant distribution $T \approx 185000\ \text{years}$6 (Kashaev et al., 2023). More important than equilibrium existence, however, is the identification logic: consideration effects are separable and alternative-specific, whereas preference effects generate cross-order effects across alternatives.

The first step is to detect whether another agent $T \approx 185000\ \text{years}$7 is any kind of peer by examining

$T \approx 185000\ \text{years}$8

where $T \approx 185000\ \text{years}$9. If this is nonzero, $a=0.9M$0 belongs to the relevant peer set. The second step uses the cross-alternative double difference

$a=0.9M$1

Because the consideration term for $a=0.9M$2 does not depend on peers choosing $a=0.9M$3, this object vanishes for consideration-only links and is nonzero when the link affects preferences. A further same-alternative double-difference argument, together with the exclusion condition

$a=0.9M$4

separates preference-only links from links affecting both margins (Kashaev et al., 2023).

Once the network structure is known, the paper shows how to recover ratios of consideration probabilities, synthetic reduced-menu conditional choice probabilities, and—under sufficient support—both the full consideration mechanism $a=0.9M$5 and the choice rule $a=0.9M$6. The key constructive device is that variation in consideration-only peers mimics menu variation. If a consideration-only peer switches to $a=0.9M$7, only the chance that $a=0.9M$8 enters the consideration set changes; preferences do not. This makes it possible to identify counterfactual reduced-menu probabilities without actual menu variation (Kashaev et al., 2023).

The paper also distinguishes two data environments. With continuous-time observations, the transition-rate matrix is directly observable, so $a=0.9M$9 is identified and summing over $p=120M$0 recovers $p=120M$1. With discrete-time panel observations at fixed interval $p=120M$2, the transition matrix satisfies $p=120M$3, and under a generic distinct-eigenvalue condition from Blevins (2018), $p=120M$4, $p=120M$5, and $p=120M$6 are generically identified (Kashaev et al., 2023).

The empirical application studies Starbucks and Luckin expansion across 152 Chinese markets, treating each firm-market pair $p=120M$7 as an agent in a binary-choice environment. Preference peers are assumed known from industrial-organization structure and are same-market only, whereas consideration peers can come from neighboring markets and are estimated. The recovered consideration network has 960 directed links, average in-degree and out-degree about 6.3, and about 30% of markets have no incoming consideration effects from other markets, though almost all affect some other market’s consideration (Kashaev et al., 2023). The main findings are that Starbucks is close to full consideration, Luckin exhibits substantial limited consideration, ignoring limited consideration misreads profitability, and forcing full consideration causes duopolies to emerge about 17 months sooner.

6. Caveats, misconceptions, and broader analytical significance

In gravitational-wave analysis, the central caveat is that the peep “background” is not a standard isotropic Gaussian stochastic gravitational-wave background derived from $p=120M$8-type integrals. It is a direct waveform-summation estimate of a confusion foreground generated by many individually unresolvable deterministic sources (Oliver et al., 25 Jul 2025). Over the affected band, the frequency range is $p=120M$9, the 4-year bin width is $M\sim 10^{5.2}-10^8\,M_\odot$00, and the number of bins is therefore $M\sim 10^{5.2}-10^8\,M_\odot$01. In the first two scenarios there are only $M\sim 10^{5.2}-10^8\,M_\odot$02 peep signals, far fewer than the number of bins, so the foreground is better described as sparse, burst-train, or “popcorn-like” than as naturally Gaussianized noise (Oliver et al., 25 Jul 2025). Additional uncertainties arise from EMRI rate prescriptions, assumed $M\sim 10^{5.2}-10^8\,M_\odot$03 and $M\sim 10^{5.2}-10^8\,M_\odot$04 distributions, Illustris massive-black-hole demographics, the neglect of orbital evolution over 4 years, and the approximation of the most abundant case by multiplying Background 2 by $M\sim 10^{5.2}-10^8\,M_\odot$05.

In the discrete-choice setting, the main limitation is that the clean cross-alternative logic requires $M\sim 10^{5.2}-10^8\,M_\odot$06. In the binary case $M\sim 10^{5.2}-10^8\,M_\odot$07, one can identify the total peer set but cannot separate consideration and preference effects without additional structure, such as one known network component, stronger network restrictions, or a non-exponential curvature condition for preference ratios (Kashaev et al., 2023). The baseline recovery of consideration probabilities also relies on the independent-inclusion consideration structure, although the paper provides a separate large-support identification result for a fully general consideration mechanism using excluded covariates that affect only preferences.

A plausible commonality across the two literatures is methodological rather than substantive. In both, the relevant micro-events are individually easy to misclassify as negligible: a single peep waveform is usually unresolvable, and a single peer perturbation can be confounded with standard utility variation. Yet in both cases the aggregate object—confusion noise in one literature, nonparametrically identified network structure in the other—depends precisely on taking those weak or intermittent effects seriously (Oliver et al., 25 Jul 2025).