Fishing for Planets: A Comparative Analysis of EPRV Survey Performance in the Presence of Correlated Noise

With dedicated exoplanet surveys underway for multiple extreme-precision radial velocity (EPRV) instruments, the near-future prospects of RV exoplanet science are promising. These surveys’ generous time allocations are expected to facilitate the discovery of Earth analogs around bright, nearby Sun-like stars. But survey success will depend critically on the choice of observing strategy, which will determine the survey’s ability to mitigate known sources of noise and extract low-amplitude exoplanet signals. Here we present an analysis of the Fisher information content of simulated EPRV surveys, accounting for the most recent advances in our understanding of stellar variability on both short and long timescales (i.e., oscillations and granulation within individual nights, and activity-induced variations across multiple nights). In this analysis, we capture the correlated nature of stellar variability by parameterizing these signals with Gaussian process kernels. We describe the underlying simulation framework and the physical interpretation of the Fisher information content, and we evaluate the efficacy of EPRV survey strategies that have been presented in the literature. We explore and compare strategies for scheduling observations over various timescales, and we make recommendations to optimize survey performance for the detection of Earth-like exoplanets.


INTRODUCTION
For several decades, astronomers have been carrying out systematic radial velocity (RV) surveys to search for exoplanets around nearby stars.The first detection of an exoplanet orbiting a main sequence star by Mayor & Queloz (1995) was the result of one such search, and this discovery spurred decades of interest and accelerating progress in exoplanet science.As instrumentation and analysis techniques have improved, as evidenced by the growing class of spectrographs capable measuring Doppler signals with sub-m s −1 precision, so too have the prospects of exoplanet surveys.
Several of these extreme precision radial velocity (EPRV) spectrographs support exoplanet surveys with generous time allocations, totalling hundreds to thousands of hours each year for up to a decade.These include the EXPRES 100 Earths Survey (Jurgenson et al. 2016;Brewer et al. 2020), the NEID Earth Twin Survey (Schwab et al. 2016;Gupta et al. 2021), the upcoming Terra Hunting Experiment (Hall et al. 2018) with HARPS-3 (Thompson et al. 2016), the HARPS-N Rocky Planet Search (Cosentino et al. 2012;Motalebi et al. 2015), and a blind radial velocity survey with ESPRESSO (Pepe et al. 2021;Hojjatpanah et al. 2019).While the exact objectives of each survey vary, two common themes are the discovery and characterization of habitable-zone, Earth-mass exoplanets and the refinement of intrinsic exoplanet occurrence rate statistics, arXiv:2303.14571v2[astro-ph.EP] 24 Jun 2024 particularly for exoplanets in mass and period regimes that were inaccessible to previous instruments.
Even with substantial time allocations and precise instruments, detecting sub-m s −1 signals will be a challenge.In this regime, RV uncertainties are dominated by correlated noise from intrinsic stellar variability, which has been identified as the largest remaining hurdle for RV detection of Earth-like exoplanets (Crass et al. 2021).We expect our ability to mitigate these correlated noise contributions and extract exoplanet-induced signals will be greatly impacted by the enacted observing scheme (Dumusque et al. 2011;Hall et al. 2018;Chaplin et al. 2019;Luhn et al. 2023).Indeed, recent studies on the scheduling of RV observations to confirm and characterize transiting systems (Burt et al. 2018;Cloutier et al. 2018;Cabona et al. 2021) as well as for largescale planet searches (Luhn et al. 2023) have shown that the choice of observing strategy will impact survey efficacy in the presence of correlated noise.To identify an optimal strategy for Earth analog searches, a sizeable parameter space of survey strategies may need to be explored.
Various methods have been used to assess the efficiency of exoplanet detection with RV searches, including Bayesian inference for adaptive schedules (Ford 2008;Loredo et al. 2011) and injection and recovery of planetary and stellar signals (e.g., Hall et al. 2018).Here, we follow Baluev (2008) and Baluev (2009) and we use Fisher information analysis to assess and compare RV exoplanet search strategies.
In this work, we present a framework in which Fisher information is used to quantify the intrinsic information content of simulated RV observations and to predict the detection limits achieved by ongoing and future EPRV exoplanet surveys in a computationally efficient manner.We show that such a tool can be used to identify survey strategies that will optimize sensitivity to specific populations of exoplanets, such as Earth analogs.We describe the Fisher information calculation in Section 2 and the input Gaussian process (GP) kernels with which we parameterize contributions from stellar variability in Section 3. In Section 4, we compare the detection limits achieved by sets of simulated observations across various timescales to formulate an idealized survey design.We apply these findings to full survey simulations in Section 5, in which we also describe our observation simulation tool that accounts for practical observing constraints to ensure realistic survey realizations.And we discuss our results and their implications in Section 6 and caveats and prospects for future work in Section 7.

FISHER INFORMATION
The Fisher information content of a data set can be used to calculate the expected uncertainty on a set of parameters for a representative model (Fisher 1922).The Fisher information matrix, B, is given as the expectation value of the Hessian of the negative log likelihood function taken over data x B i,j = −E x ∂ 2 ln L(x; θ) ∂θ i ∂θ j (1 where θ is the parameter vector and the log likelihood is (2) for a model µ and time series x of length N , and a N ×N covariance matrix, C, that describes the expected noise properties of the measurements.Both µ and C may depend on θ in the general case.
Assuming that the data are drawn from a multivariate Gaussian distribution with mean µ and covariance C, the Fisher information can be written as For a derivation, see Kay (1993) or Malagò & Pistone (2015).If the covariance is independent of the model parameters, this reduces to The diagonal elements of the inverse of the Fisher information matrix represent the parameter uncertainties: It is important to note here that the time series x has dropped out of the equation because of the average taken over this value, and the Fisher information depends only on the model µ, the covariance matrix C, and the times t at which measurements are taken.That is, we do not need to know the measured values of x to determine the expected parameter uncertainties.The use of Fisher information analyses for astronomical data sets was first studied by Tegmark et al. (1997), who explored the utility of this metric in the context of cosmology, and later works applied Fisher information to the optimization of observing strategies for RV exoplanet science (e.g., Baluev 2008Baluev , 2009Baluev , 2013)).More recent applications include the work of Gomes et al. (2022), who assess sensitivity to gravitational effects of an as-yet undetected outer Solar System planet, and that of Cloutier et al. (2018), who used Fisher information to explore the impact of correlated noise signals on follow-up observations of transiting exoplanets.Here, we adopt the notation used in Cloutier et al. (2018) and present a framework for using Fisher information to calculate detection sensitivity limits for blind RV exoplanet surveys.The RV semi-amplitude induced by a single exoplanet with mass M p and period P orbiting a star with mass M ⋆ is For our model in this work, we assume circular, zeroeccentricity orbits and single-planet systems such that the exoplanet-induced RV signal can be represented with a simplified Keplerian where ϕ 0 is a phase offset and t is the set of observation times.The corresponding parameter vector is then and the derivative of the model with respect to this vector is The inclusion of a covariance matrix in Equation 4 allows us to capture the effect of correlated noise on the achieved RV sensitivity.In a pure white noise scenario, C will be a matrix with elements C n,m = σ 2 n δ nm , where σ n is the total measurement uncertainty for observation 1 ≤ n ≤ N and δ is the delta function That is, all the off-diagonal elements will be 0; independent measurements will have independent noise properties.But if correlated noise sources are present, some off-diagonal elements will be non-zero as well, representing covariances between the noise properties of pairs of observations.We discuss anticipated sources of noise and the construction of a covariance matrix for EPRV data sets in the following section.

CONSTRUCTING A COVARIANCE MATRIX FOR RADIAL VELOCITY OBSERVATIONS
The full covariance matrix to be used in Equation 4 will be made up of a sum of white noise and correlated noise components where I is the N × N identity matrix and the remaining terms are described below.The primary source of white noise is photon noise, or the limit on the precision of an individual RV measurement as determined by the Doppler information content of the observed stellar spectrum.The photon noise limit, σ photon , depends on the wavelength range and spectral features from which the RV measurement is derived and on the observed signal-to-noise ratio (S/N) across this spectral range, and it can be calculated analytically as shown by Bouchy et al. (2001).
Instrument systematics and RV variations due to intrinsic stellar variability constitute two important sources of correlated noise that will impact RV exoplanet discovery efforts (Crass et al. 2021).In this work, we will focus primarily on the correlated nature of stellar variability.Well-characterized instrument systematics are typically modeled and calibrated out during the data reduction process for EPRV spectrographs (e.g., Halverson et al. 2016;Petersburg et al. 2020).Though systematics persist at a level that obstructs the detection of 10 cm s −1 level signals, progress on calibration and reduction techniques (e.g., Zhao et al. 2021;Cretignier et al. 2023) may make them manageable in the near future.Here, we consider the post-calibration RV time series as our starting point, such that any systematics that can be removed have been removed, and we will reserve a more detailed discussion of instrument-related correlated noise for Section 7.2.
Intrinsic stellar RV variations are present in the final RV time series at the ∼ 1 m s −1 level for typical observations of quiet exoplanet search targets.To account for these stellar signals, it is common practice to model them using Gaussian process (GP) regression while simultaneously fitting for an exoplanet-induced signal (see review by Aigrain & Foreman-Mackey 2022).This approach provides for straightforward integration with the covariance matrix, as one can build individual sources of variability into this matrix with appropriate GP kernel functions.In the remainder of this section, we describe the dominant sources of stellar variability for typical Sun-like G-and K-dwarfs and we present the form of their respective GP kernels.
On short timescales, i.e., less than a single night, the most important sources of stellar RV variations are p-mode oscillations, which manifest as m s −1 -level variations on 5-10 minute timescales, and surface granulation, which has a similar amplitude on slightly longer timescales.For oscillations, we adopt the GP kernel described by Luhn et al. (2023) and based on work by Pereira et al. (2019) and Guo et al. (2022), which takes the form where ∆ is the N ×N matrix that represents the absolute value of the time delay between pairs of observations S osc is the power at the peak of the oscillation excess, ω osc is the characteristic oscillation frequency, Q is the quality factor, and We also adopt the following two-component granulation kernel from Luhn et al. (2023) and Guo et al. (2022): where S 1 and S 2 describe the power of each granulation component and ω 1 and ω 2 are the respective characteristic frequencies.We note that our granulation model does not account for supergranulation, which typically occurs on timescales of longer than a day.Supergranulation is not a negligible source of stellar RV variability (e.g., Meunier et al. 2015;Al Moulla et al. 2023), but we do not include it in this work because we do not have a suitable model.We show the form of the oscillation and granulation kernels in Figure 1, assuming perfectly Solar hyperparameters as given in Table 1.
Stellar RV variability resulting from rotational modulation of active regions and starspots is our primary source of correlated noise on longer timescales.Activityinduced variability is commonly modeled using a quasiperiodic GP kernel of the form where α is the RV amplitude, λ 1 is the coherence timescale, λ 2 represents the stellar rotation period, and Γ is the periodic complexity factor, or in this case, the complexity of the spot and active region coverage across the stellar surface (Aigrain et al. 2012;Haywood et al  2014).This kernel has been shown to be effective for disentangling exoplanet-and activity-induced RV signals, and each hyperparameter has a clear physical interpretation, making it a natural choice for modeling RV time series data in the presence of activity.However, recent work by Gilbertson et al. (2020b) based on simulated solar spectra with short-lived spots (Gilbertson et al. 2020a) has shown other kernels outperform the quasi-periodic kernel, particularly for detecting exoplanets with K ≤ 30 cm s −1 .We therefore also consider the following latent GP kernel which is a linear combination of the Matérn-5/2 kernel and its second derivative, with a single timescale hyperparameter, λ, and amplitude coefficients a 0 and a 1 .We refer the reader to Gilbertson et al. (2020b) 12; left) and for stellar surface granulation (Equation 14; right).We assume instantaneous exposures for both kernels.
We show that oscillations and granulation only introduce significant RV noise on short timescales of less than a single night.
et al. ( 2023) for the calculation of the second derivative of the Matérn-5/2 kernel and for a full derivation of Equation 16.We show both activity kernels in Figure 2.For the Matérn-5/2 kernel, we assume hyperparameters as derived by Gilbertson et al. (2020b) and for the quasiperiodic kernel, we assume hyperparameters from a fit to HARPS-N solar data by (Langellier et al. 2021).
Both sets of hyperparameters are given in Table 1.In subsequent sections, we test survey performance using each of the two activity kernels and comment on how our conclusions change based on the assumed kernel form.

FORMULATING AN IDEALIZED SURVEY STRATEGY
We begin our investigation of survey performance with a set of experiments intended to isolate each source of correlated noise and identify observing strategies that will best mitigate their respective impacts on exoplanet detection sensitivity.Separate sets of simulated observing schedules are generated to highlight the effects of stellar RV signals that vary on timescales of less than a single night for Sun-like stars, granulation and p-mode oscillations, and those that span multiple nights, namely active regions and spots.For each set of observations, we calculate the resulting Fisher information and assess the achieved sensitivity limits.In all cases, we adopt perfectly Solar hyperparameters for the GP kernels with which we build each covariance matrix.We assume that only a single planet is present in the system and we take the stellar noise model to be fixed.

Intra-night Observing Schedule Simulations
As we show in Figure 1, correlations in oscillation and granulation RV signals are not expected to remain coherent for longer than an hour for Sun-like stars, so observations collected on separate nights are effectively independent.To assess the impact of oscillations and granulation on detection sensitivity limits, we therefore explore different strategies for distributing observations within a single night.We consider the following intranight cadences: • six visits with one exposure per visit, • three visits with two exposures per visit, • two visits with three exposures per visit, • one visit with six exposures, • and one visit with a single continuous exposure.
For each of these cadences, we generate 100 nights of observation times across a 500-night baseline (average inter-night separation of 5 nights), and we repeat this for exposure times 15 s ≤ t exp ≤ 1200 s.For the three strategies with multiple visits, the start times of each visit are evenly distributed across a six hour window.For the three strategies with multiple exposures in a single visit, the exposures are separated by a readout time, t r .We then calculate the Fisher information of the simulated observations for covariance matrices consisting of correlated noise from (i) p-mode oscillations, (ii) granulation, (iii) p-mode oscillations and granulation, and (iv) p-mode oscillations, granulation, and photon noise.We do not include an instrumental jitter term for these tests.
For oscillations and granulation, we start with the GP kernels in Equations 12 and 14.However, as Luhn et al. (2023) note, exposures are not instantaneous, and the exposure times we consider here are comparable to the variability timescales for oscillations and granulation in the case of Sun-like stars.We therefore compute the double integral of these GP kernels over the exposure  15; left) and for the Matérn-5/2 kernel for rotationally-modulated stellar activity (Equation 16; right).While the covariance for both kernels persists across many nights, we note that the curves differ significantly in amplitude and shape.
times to calculate the true covariance between pairs of observations.A detailed discussion of the oscillation and granulation kernel integration is given in the Appendix of Luhn et al. (2023).The assumed hyperparameters are given in Table 1.We determine the photon noise using the NEID exposure time calculator, described in Gupta et al. (2021), for Solar analogs with V = 4 and 8 mag.
Finally, we use the covariance matrices and simulated observation times to evaluate the performance of each strategy, where the performance is defined as the expected uncertainty on the semi-amplitude, σ K , using Equations 4 and 5.We calculate this value for an exoplanet with orbital parameters K = 10 cm s −1 and P = 100 d.To marginalize over orbital phase, we perform the calculation for 10 values of ϕ 0 uniformly distributed on the interval 0 ≤ ϕ 0 < 2π and we use the mean value of the resulting Fisher information uncertainty estimates.The dependence of σ K on the total nightly time cost of each strategy is shown in Figure 3, where the time cost is calculated as the sum of the exposure times and associated overheads, i.e., readout and target acquisition time (t acq ): We assume here that the target need only be acquired once per visit and that the readout cost of the final exposure in a sequence can be folded into the cost of acquiring the subsequent target.We use overhead costs of 30 seconds and 180 seconds for readout and acquisition, respectively, to match typical overheads for observations of bright stars with the NEID spectrograph.

Intra-night Observing Strategy Results
In Figure 3, we show that in the presence of correlated noise from oscillations alone, the expected semiamplitude uncertainty, σ K , falls off much more rapidly for the single-visit strategies than for the multi-visit strategies across all nightly time allocations.These strategies take advantage of the significant negative covariance seen on timescales comparable to 1/ν max , rapidly averaging out the p-mode signal rather than allowing it to bin down like white noise when the covariance approaches zero (i.e., when exposures are widely separated relative to the oscillation periods for Sun-like stars).But it is interesting that neither σ K nor the relative performance of the different intra-night strategies changes monotonically with nightly time cost.Instead, each intra-night strategy produces several local minima and maxima in σ K , and these features emerge at different values of t night .For granulation, on the other hand, while σ K does decrease monotonically, the relative performance of each observing strategy is inverted.This can be attributed to the positive covariance of the granulation signal, which is not conveniently averaged out on short timescales.In contrast to oscillations, the correlated noise contribution from granulation is optimally minimized by taking exposures separated by ∆ ≳ 15 minutes, where the covariance asymptotically approaches zero.However, our results for the combined case with both oscillations and granulation do not consistently prefer the six-visit strategy.Instead, every strategy yields the lowest value of σ K for at least one value of t night when we include both covariance kernels.That is, the relative advantages of a single-visit strategy, which is optimal for averaging out the oscillation signal, and a multi-visit strategy, which is optimal for binning down the granulation signal, will win out on different time scales.It is also important to note that  18).We calculate σK for simulated observing schedules with 100 nights of observations distributed across 500 total nights, wherein the observations on each night are distributed according to each of the five strategies depicted in the upper left panel.These strategies are described in Section 4.1.In the remaining panels, we show the impact of contributions from various combinations of photon noise and correlated noise from oscillations and granulation.We also include inset plots to highlight the performance of each multi-exposure strategy relative to the single continuous exposure strategy for nightly time costs of less than an hour.
the optimal strategy for any given nightly time cost will depend on the oscillation and granulation kernel hyperparameter of the specific star being observed.Here, we have assumed the hyperparameters are fixed at the Solar values, but these will vary from star to star.
These results are largely consistent with those of previous studies that explore the RV noise contributions from oscillations and granulation.Chaplin et al. (2019) recently presented a model for predicting residual pmode oscillation amplitudes as a function of exposure time and stellar parameters, showing that local minima in the residual amplitude curve are present for exposure times close to integer multiples of ν max , the central frequency of the oscillation power excess.Their predictive model has since become widely used in the design of EPRV observations, both for exoplanet surveys (e.g., Brewer et al. 2020;Gupta et al. 2021) as well as for studies of stellar signals (e.g., Sulis et al. 2023;Gupta et al. 2022).The covariance behavior of the oscillation kernel we use here (Equation 12) produces these same local minima, albeit on different timescales due to our inclusion of overheads costs and noncontiguous observations.But while the Chaplin et al. (2019) model highlights the merits of a mitigation strategy that averages over the oscillations, our results demonstrate the relative advantage of sampling the oscillation signal instead.In the Fisher information framework, the strategy with six consecutive exposures outperforms the continuous exposure strategy on short timescales, because this sampling allows us to use the measured RVs and known GP kernel form to model out the oscillation signal.
We also compare our findings to those of Dumusque et al. (2011), who simulate measurements of stellar RV signals from fitted asteroseismic power spectra, including both oscillations and granulation, and calculate the predicted RV rms for several years of simulated data with different intra-night exposure times and cadences.For the Sun-like, G2V star α Cen A, Dumusque et al. (2011) find that they can achieve a lower RV rms by distributing observations across many hours instead of concentrating them in a single visit with the same total on-sky exposure time.We see the same trend for the granulation-only case in Figure 3, in which the six-visit strategy outperforms the other strategies by a significant margin for nightly time costs up to one hour.However, our results differ from theirs for the case with both oscillations and granulation.We attribute this to two factors.First, Dumusque et al. (2011) do not account for overheads when comparing the performances of different observing cadences.Cadences with more visits are not penalized for the additional costs that will be incurred, so their performance relative to cadences with fewer visits appears enhanced.On the other hand, their study includes the effects of supergranulation and a constant instrument noise floor, both of which introduce correlated noise on timescales longer than oscillations and granulation alone.Because we neglect these sources of correlated noise, we diminish the relative benefits of increasing the spacing between observations.We comment on correlated noise from instrument systematics in more detail in Section 7.2.We also note that other differences between the Dumusque et al. ( 2011) methodologies and our own, such as the precise oscillation and granulation models used, may have an effect on the differences between the results.
The final noise source we consider here is photon noise.In Figure 3, we see that the multi-visit strategies are penalized more heavily due to the additional overhead costs.That is, for a given time allocation, increasing the number of visits will decrease the total exposure time and thus degrade the RV precision and achieved uncertainty on K.The single-visit strategies are again preferred across most nightly time costs.This preference is stronger for fainter stars, for which photons noise contributes a larger share of the total uncertainty.

Inter-night Observing Schedule Simulations
Unlike oscillations and granulation, rotationally modulated, activity-induced RV signals exhibit correlations on timescales longer than a single night.This is true for both activity GP kernels shown in Figure 2. To explore how these signals impact long term RV precision, we simulate sets of observations with one visit per night and various inter-night distributions across a typical observing season, which we assume here to be eight months, or 240 days.The following strategies are considered: • Uniform Sampling: Observations are uniformly distributed across each observing season • Centered: All observations occur on consecutive nights at the center of each observing season • Single Burst: A mixture of the uniform and centered strategies, in which 60% of the observations take place on consecutive nights at the center of each season and the remainder are uniformly distributed • Double Burst: Similar to single burst, but with two sets of consecutive nights bracketing the center of the season, each containing 30% of the observations • Monthly Runs: Observations occur in equal blocks of consecutive nights at the start of each month • On / Off: Observations occur in five equally spaced blocks of consecutive nights These strategies are depicted in Figure 4.For each strategy, we build a 10 year observing schedule with annual allocations ranging from 60 observations per year to 240 observations per year, and we use Equation 4 to calculate the expected mass precision for an exoplanet with orbital parameters K = 10 cm s −1 and P = 300 d, again marginalizing over orbital phase as in the previous section.We calculate σ K separately for the quasiperiodic and Matérn-5/2 activity GP kernels, and we do not include any other sources of noise in the covariance matrix.

Inter-night Observing Strategy Results
The results from the two activity kernels show a stark contrast in the preferred strategy, the overall magnitude of σ K , and the degree to which σ K changes with increasing number of observations (Figure 5).When we build the covariance matrix using the quasiperiodic kernel given in Equation 15, the total length of the observing baseline is the overriding factor in determining σ K .The Centered strategy, which uses just a fraction of the full 240-day seasonal baseline each year, performs significantly worse than the other strategies.For the other strategies, performance appears to be dictated by the uniformity of the observations across the seasonal baseline; the Uniform strategy produces the smallest uncertainty, followed by the Double Burst, Single Burst, On / Off, and Monthly schedules.But for the Matérn-5/2 kernel, Equation 16, dense coverage is much more important than broad coverage.Strategies with long runs of high-cadence observations (Centered, Single Burst, Two Bursts) consistently produce the smallest σ K for a given number of observations.And while the performance of all six strategies improves monotonically with increasing number of observations per year, these improvements are much more significant for Matérn-5/2 activity kernel than for the quasiperiodic activity kernel.The predicted value of σ K decreases by less than 7% for the best performing strategy for the quasiperiodic kernel when the number of observations increases from 60 to 240 per year, while the equivalent change in σ K for the best performing strategy for the Matérn-5/2 kernel is greater then 50%.In addition, we show that although these two kernels nominally represent the covariance behavior of the same physical process, they result in uncertainties that differ by nearly an order of magnitude regardless of observing strategy.
The differences in the predicted uncertainty for each activity kernel can be explained by their respective covariance behaviors.The Matérn-5/2 kernel's negative covariance on short time scales (≲ 10 days) favors highcadence observations, which will efficiently average out activity-induced RV variations.The covariance of the quasiperiodic kernel remains positive on all time scales, so the preferred strategies will be those with separations no smaller than the correlation time.The covariance is close to zero for these strategies, so the measurement uncertainties are in practice equivalent to white noise.The local maximum near the stellar rotation period penalizes the Monthly strategy, for which consecutive sets of observations are separated by this same amount, and the relatively poor performance of the high-cadence Centered strategy naturally follows, as clustering all observations in the subset of a season leads to high covariance. .Expected semi-amplitude uncertainty, σK , for a K = 10 cm s −1 signal and an orbital period of P = 300 days as a function of number of observations per year.We calculate σK for simulated observing schedules with 10 years of observations and a 240-day annual observing season, wherein the observations for each season are distributed according to each of the strategies depicted in Figure 4 and described in Section 4.3.We include correlated noise contributions from rotationally-modulated stellar activity represented by a quasiperiodic kernel (dot-dashed lines) and by a Matérn-5/2 kernel (solid lines).
The time scale for the covariance of the quasi-periodic kernel to fall to zero can also explain the significantly lower σ K values we expect to retrieve for the Matérn-5/2 kernel.The covariance for the Matérn-5/2 kernel and its derivative not only reaches a white noise approximation at much smaller separations, but the negative covariance allows the activity signal to be averaged out at a rate much faster than white noise binning.

Comparison to Posterior Sampling
To test the accuracy of our methods, we perform a set of injection-recovery tests and compare the resulting parameter uncertainties to the uncertainties we predict from the expected Fisher information content.We simulate a 10-year RV time series consisting of a planet signal with θ in = {K in , P in , ϕ 0,in } = {1 m s −1 , 300 days, 0.6} and a stellar activity signal drawn from a GP seeded by the quasiperiodic kernel given in Equation 15.We then generate a set of 1000 measurements randomly sampled across the 10-year baseline, and we introduce white noise by drawing each RV from a Gaussian centered on the true value and with a width of 1 cm s −1 .These steps are repeated with different random seeds to produce 10 independent simulated data sets.We also repeat the process for input semi-amplitudes of K in = 10 cm s −1 and K in = 10 m s −1 as well as for the Matérn-5/2 activity kernel given in Equation 16.
For each simulated data set, we use the numpyro library (Phan et al. 2019) to fit the measured RVs and sample our model parameters, θ, via Markov Chain Monte Carlo (MCMC) analysis.We set Gaussian priors on K and P , centered on their input values and with widths of K in /2 and 50 days, respectively, and we sample ϕ 0 uniformly on [−π, π] with cyclic boundary conditions.The sampler is conditioned on the input GP kernel using tinygp (Foreman-Mackey et al. 2024) with the hyperparameters fixed to their input values, and we initialize each fit at θ = {K, P, ϕ 0 } = {K in , 280 days, 1.0}.The sampler consistently failed to converge for the trials with the quasiperiodic activity kernel and K in = 10 cm s −1 , but we successfully recover the input parameters to better than 2σ in all other cases.We compute σ K and σ P for each trial as the standard deviations of the posterior parameter distributions for K and P .
In Table 2, we list the recovered parameter uncertainties for each test alongside the expected parameter uncertainties as calculated via Fisher information analysis.The results of the two analyses are fully consistent with each other in all cases for which the sampler successfully recovered the input planet signal, confirming that Fisher information analysis is indeed a valid method for predicting parameter uncertainties.We note that the case for which injection-recovery tests failed to converge (K in = 10 cm s −1 , quasiperiodic kernel), is also the only one for which the expected σ K from the Fisher information calculation is greater than the injected K.That is, the expected detection significance is < 1σ, which likely explains the failure to recover the injected signal.We test values of K in between 10 cm s −1 and 1 m s−1 with the quasiperiodic kernel to determine whether this is indeed the case, and we find that our fit begins to converge on the input signal for some simulations with K in > 0.6 m s−1 (K/σ K ≈ 4) and it reliably converges by K in = 0.9 m s−1 (K/σ K ≈ 5).When the fit converges, the estimated uncertainties are consistent with those predicted by Fisher information analysis.

APPLICATION TO A REALISTIC SURVEY
Our analysis in the previous section sheds some light on the relative performance of different observing cadences both within a night and across many nights.However, the implications of the absolute precision we report, quantified by the predicted uncertainty on K, are not as easy to interpret given that we have isolated individual noise sources.That is, these results alone do not tell us how each strategy would fare in the presence of all sources of correlated noise.In addition, we ignored practical constraints on our ability to reproduce each intended survey strategy.Real observations are subject to constraints from telescope and observatory schedules, weather losses, nightly and seasonal observing windows, and intra-survey competition.In this section, we explore how our choice of survey strategy affects sensitivity to Earth analogs in the presence of all sources of correlated stellar signals described in Section 3. Accounting for realistic observing constraints, we apply several of the strategies assessed in Section 4 to simulations of a full EPRV exoplanet survey and calculate the resulting Fisher information content.

Survey Parameters and Schedule Simulation
As a template for our survey simulations, we adopt the target list of the NEID Earth Twin Survey (NETS; Gupta et al. 2021) and the parameters and constraints of the Terra Hunting Experiment (Hall et al. 2018).The Terra Hunting Experiment will nominally have access to 50% of the total observing time on the HARPS-3 spectrograph (Thompson et al. 2016) for 10 years.We compute the total time expected to be available for observations and allocate this evenly across all targets.We consider both a 40-star target list as well as a narrower, 20-star subset of this list, so that we may evaluate the effects of sample size on survey performance.Targets are set to be observed 100 times per year for the former sample and 200 times per year for the latter sample.The targets for both lists are given in Table 3.Following our findings in Section 4.2, just one visit is allocated on each night that a star is observed.See Appendix for a detailed description of the target selection criteria and time allocation breakdown.
Because of the wide variation in our results in Section 4.4, we consider four different inter-night observing strategies for our survey simulations.These include the best-and worst-case strategies for the quasiperiodic and Matérn-5/2 activity kernels (Uniform, Centered, and Monthly) as well as the Double Burst strategy as an intermediate reference point.We simulate a 10year survey for each of these strategies using a custom scheduling framework.In brief, our scheduler accounts for the following practical observing constraints: target observability, seeing, weather losses, telescope and facility maintenance, and intra-survey competition (i.e., we can not observe two or more targets simultaneously).Quantitative descriptions of the scheduling algorithms are given in Appendix .Values are given as the median ± standard deviation across the 10 trials.We do not list the standard deviation of the Fisher information results for the quasiperiodic activity kernel, as the spread in both σK and σP is negligible (< 1 part per thousand) for all values of Kin.

RV Noise Contributions
As in Section 4, we do not need to generate RV measurements to accompany our simulated survey schedules.The Fisher information content does not depend on the measurements themselves.However, we do need to account for the associated noise properties.Here, we describe the computation of the covariance matrix for each target star, accounting for all four sources of noise in Equation 11.

Photon Noise
Photon noise contributes a diagonal term to each stars covariance matrix.The photon noise precision of an exposure, σ photon , depends on the RV information content of the observed spectrum as described by Bouchy et al. (2001).For observations with a given spectrograph, the precision will vary as a function of the spectral type of the star and the achieved SNR, which in turn depends on the exposure time, stellar brightness, and observing conditions.To determine σ photon for our simulated observations, we first calculate the expected SNR as a function of T eff , V -band magnitude, and exposure time using the NEID exposure time calculator.We then apply two multiplicative corrections to account for the simulated seeing conditions and for the effective system throughput of HARPS-3 relative to NEID The seeing correction is calculated by taking the cross section of the fiber (on-sky diameter d f = 1.4 ′′ ; Thompson et al. 2016) and the seeing disk, assuming that the star is perfectly centered and comparing this to the same cross section in median seeing conditions (FWHM median = 1.3 ′′ ).To scale the throughput, we calculate the photon noise precision precision we would expect to achieve for some test values of R system , and find we can match the HARPS-3 results calculated by Thompson et al. (2016) if we apply a correction factor of R system = 1.375.However, Thompson et al. ( 2016) assume a fairly conservative system throughput of 5%.Here, we assume a more optimistic 10% average efficiency, so we set R system = 2.75.The final photon noise is then estimated from SNR scaled using the NEID exposure time calculator as described in Gupta et al. (2021).We note that for NEID, the detector saturates at SNR≈ 625.In median seeing conditions, we find that SNR scaled will exceed this threshold in a 320 second exposure for stars brighter than V = 4.68, hence our decision to use shorter exposures for the brightest stars in our sample.We do not account for the dependence of throughput on target airmass.

Stellar Variability
We calculate the correlated noise contributions from stellar variability using Equations 12 and 14 for oscillations and granulation, respectively, again integrating over the exposure time as in Section 4.1, and using Equations 15 and 16 for spot-induced activity.In Section 4, we assumed perfectly known solar hyperparameters as given in Table 1.We continue to assume Solar values for the activity kernels here, but to enable a more even comparison of their absolute performance, we set the amplitude of the quasiperiodic kernel to be α = 0.6 m s −1 such that it matches that of the Matérn-5/2 kernel at ∆ = 0.For the asteroseismic signals, we take advantage of known stellar parameter scaling relations (given in Luhn et al. 2023) to estimate more accurate oscillation and granulation kernel hyperparameters for each of our target stars.As inputs to these scaling relations, we take spectroscopically derived effective temperatures, T eff , and surface gravities, log g, from Brewer et al. (2016).These stellar parameters are listed in Table 3.For each simulated survey schedule, we treat the two activity kernels independently just as we did in Section 4, computing separate covariance matrices and analyzing separate sets of results.

Detection Sensitivity and Survey Success Metrics
With the simulated schedules and covariance matrices in hand, we can use Equations 4 and 5 to calculate σ K and assess the achieved sensitivity of each survey strategy.We first compute K as in Equation 6for a broad range of M p sin i − P parameter combinations, assuming circular orbits and taking stellar masses listed in Table 3 (taken from Brewer et al. 2016), and then we calculate σ K for the corresponding parameter vector θ, marginalizing over orbital phase at each point.We use these results to determine the expected detection significance, K/σ K , which we show for two representative cases in Figure 6.
To compare performance across different survey strategies, we define two quantitative success metrics.The first metric is simply the detection significance for an Earth analog, [K/σ K ] ⊕ .Here, we define an Earth analog to be a M p sin i = M ⊕ exoplanet with an orbital period of 300 days (green 'x' in Figure 6).We choose P = 300 days rather than P = 365 days so as not to bias our results against strategies that are more susceptible to annual aliasing.For each simulated survey, we calculate and tabulate (Table 4) the number of stars for which the detection significance exceeds a threshold H, i.e., N [K/σ K ]⊕>H , setting H = [3, 5, 10] for the Matérn-5/2 activity kernel results and H = 1 for the quasiperiodic kernel results.We also show the distribution of [K/σ K ] ⊕ values for each simulated survey in Figure 7.
We adopt a slightly looser definition of an Earth analog for our second success metric, D ⊕ , which we will refer to as the fractional Earth analog discovery space:  The given values represent the number of stars, N , and the average fractional parameter space, D⊕, above each specified detection significance threshold.
We set P min = 200 days and P max = 500 days and we calculate the minimum detectable mass, M p,min , as where [M p sin i] H is the mass at which K/σ K = H for a given orbital period.The value of D ⊕ thus represents the fraction of parameter space to which each survey is sensitive in the region bounded by200 days < P < 500 days and 0.5M ⊕ < M p sin i < 2M ⊕ .We again set thresholds of H = 3, 5, and 10 for the Matérn-5/2 activ-ity kernel and a H = 1 threshold for the quasiperiodic activity kernel.By expanding our definition of an Earth analog to include all exoplanets in this region, indicated by the green box in Figure 6, we reduce our sensitivity to small K/σ K fluctuations in M p sin i − P space.For each simulated survey, we take the mean value of D ⊕ for all stars and show the results in Table 4.We divide this metric in half for the 20-star sample results to account for the smaller survey sample.

Quasiperiodic activity model
In the case that our stellar activity model is best represented by the quasiperiodic kernel given in Equation 15, all of the survey strategies tested here are expected to yield marginal detections of Earth analogs, at best ([K/σ K ] ⊕ < 1.5; Figure 7).Still, we can assess the relative performance of the different strategies.As we show in Table 4 and Figure 7, the N [K/σ K ]⊕>1 success metric is fairly sensitive to our choice of survey strategy for the full 40-star sample.While nearly every star reaches the [K/σ K ] ⊕ > 1 threshold for the Uniform and Double Burst strategies, far fewer stars reach this threshold for the Monthly strategy and only a handful do for the Centered strategy.Yet in the case of the 20-star subsample, this performance gap is nearly erased, and nearly every star reaches the [K/σ K ] ⊕ > 1 threshold for all strategies.This is entirely consistent with our interpretation of the quasiperiodic kernel analysis in Section 4.4.For this activity kernel, the achieved detection sensitivity is strongly correlated with the length of the observing baseline within each observing season.The increased number of observations only significantly benefits the Centered strategy, for which this increase also translates to a greatly extended annual baseline (Figure 7).There is a nearly negligible change in the distribution of [K/σ K ] ⊕ values between the 40-star and 20-star samples for the Uniform and Double Burst strategies, despite the latter sample having twice the number of observations per star.
The D ⊕ success metrics paint a more granular picture of relative survey performance than do the [K/σ K ] ⊕ values, but the results of these two metrics in Table 4 are largely consistent with each other.The only stark discrepancy is in the results of the Centered strategy with the 40-star sample, for which only the D ⊕ metric is comparable to the other strategies and sample sizes but far fewer stars reach the [K/σ K ] ⊕ > 1 threshold.This suggests that many stars fall just short of the threshold in this case, which is confirmed in Figure 7.

Matérn-5/2 activity model
For the Matérn-5/2 activity kernel, the absolute performance of each survey strategy is much greater than for the quasiperiodic kernel.Nearly every star reaches the H = 3 threshold for all simulated survey strategies and even the H = 5 threshold for each of the 20-star simulations.And while only half of the observed stars reach [K/σ K ] ⊕ > 5 for the 40-star simulations with the Centered, Uniform, and Double Burst strategies, each of these strategies will still yield close to 20 stars that are sensitive to Earth analogs at this level.The relatively poor performance of the Monthly strategy here is consistent with our results in Section 4.4, as is the lack of variation from strategy to strategy for the 20-star sample.But contrary to what one might expect based on the earlier unit tests, the Centered strategy fails to outpace the Uniform and Double Burst strategies for the 40-star sample.The difference between the results presented here and those presented in Section 4.4, which uses an idealized observing schedule, likely stems from the practical constraints imposed on the full survey simulations.By restricting the available observing nights to account for these constraints, we have effectively extended the annual observing baseline of the Centered strategy such that it is no longer as distinct.
The larger sample size of 40 stars will be preferred for a survey with a target detection threshold of H = 3.But as H increases to H = 5 and again to H = 10, N [K/σ K ]⊕>H drops off much more steeply for the 40-star sample than for the 20-star sample.This same effect is illustrated in Figure 7, which shows that the average detection significance is significantly higher for the 20-star sample, thus demonstrating that preferred sample size is inversely proportional to the stringency of our detection significance requirements.This is in stark contrast to our interpretation of the N [K/σ K ]⊕>H success metric for the quasiperiodic kernel, for which we see only a marginal increase in detection significance when we reduce the sample size and double the number of observations per target.
As with the quasiperiodic activity model, the D ⊕ success metric is more granular than -but largely consistent with -the [K/σ K ] ⊕ metric.There is again one instance in which the performance of a single strategy appears to perform poorly based on the number of stars above a [K/σ K ] ⊕ threshold (in this case, the Monthly strategy with the 40-star sample), but the D ⊕ values are more comparable to those of the other strategies.

Cadence vs. Baseline
The factors driving the relative performance of the survey strategies explored here can largely be distilled into to three key parameters.These are the frequency with which each star is observed, the total number of stars in the sample, and the observing baseline for each star both within each season and across the entire survey.Within the constraints of a survey with a fixed total time allocation, these parameters cannot be independently tuned.But we can still examine how our results change when two or more parameters are adjusted.Indeed, we have already shown that, concerning the trade-off between sample size and cadence, the results for the two activity kernels are at odds with each other.For the quasiperiodic kernel, increasing the cadence adds little value and is not worth the decrease in sample size, while for the Matérn-5/2 kernel, a higher cadence and smaller sample size is increasingly preferred as we raise our target detection significance threshold.
To shed light on the trade-off between observing cadence and total survey baseline, we re-compute the Fisher information and resulting [K/σ K ] ⊕ values for each of the 20-star survey simulations using just the first 5 years of observations for each star.By fixing the total number of observations to about 1000 per star, we can directly compare these results the results for these same stars over the full 10-year duration of the 40-star survey simulations.As Figure 8 shows, our strategy recommendation will again depend on the assumed activity kernel.For the quasiperiodic model, the longer 10-year survey outperforms the shorter 5-year survey for all strategies, and for the Matérn-5/2 kernel the shorter, higher-cadence survey wins out.In both cases, the Centered observing strategy is shown to be least sensitive to this trade-off between baseline and cadence.This is consistent with expectations given that changing the num-ber of observations for this strategy serves only to adjust the length of the baseline within each season rather than the frequency of observations.Based on our results, the preferred survey strategies differ significantly for each of the two activity kernels we have applied, as does the absolute detection significance we can expect to achieve.But when interpreting these results, one should be careful not to attribute the differences purely to the kernel forms.As we show in Figure 2 and Table 1, we assume significantly different correlation length hyperparameters for the quasiperiodic and Matérn-5/2 kernels.While the form of each kernel undoubtedly plays a role in the resulting detection sensitivities, the stark contrast we predict could likely be reduced by adjusting these hyperparameters.We comment on this further in the following section.⊕ for a 10-year survey with 100 observations per year (vertical axis) and a 5-year survey with 200 observations per year (horizontal axis).We compute the latter values using the first 5 years of observations from the 20-star survey simulations.On the left, we show that for the quasiperiodic kernel, the longer 10-year survey delivers a higher detection significance than the higher cadence 5-year survey.On the right, we show that the opposite is true for the Matérn-5/2 kernel, which performs better with a higher observing cadence.
One shortcoming of the work presented herein is that our analysis framework relies on perfect knowledge of the stellar variability kernels and hyperparameters with which we construct the covariance matrices for our Fisher information calculations.The target lists for ongoing EPRV exoplanet searches are selectively composed of intrinsically "quiet" stars (Brewer et al. 2020;Gupta et al. 2021), as is the target list used for the survey simulations in this work.This is advantageous in that it naturally reduces the impact of variability on exoplanet detection, but it has also limited our ability to directly measure and model these stars' intrinsic variability signals, as the amplitudes of these signals are at a level below the precision floor achieved by surveys with older spectrographs.As such, the stellar variability models we use here are informed primarily by Solar observations and models and, in the case of oscillations and granulation, extended to other stars via scaling relations.While physically motivated, these scaling relations are not perfectly accurate, and oscillation and granulation timescales and amplitudes will often depend on additional stellar physics that these relations do not capture (Cattaneo et al. 2003;Jiménez-Reyes et al. 2003;Bonanno et al. 2014;García et al. 2014;Mathur et al. 2019;Gupta et al. 2022).The results of our stellar activity kernel analysis should be treated with caution as well; the achieved survey sensitivity and our interpretation of the preferred strategy will depend on the kernel form and hyperparameters and differs greatly between the two kernels we test.Empirical investigations of the activity signals of quiet stars, such as the collaborative EXPRES Stellar Signals Project (Zhao et al. 2020(Zhao et al. , 2022) ) will enable informed observing schemes that are tailored to individual stars.
We also note that we have assumed that each of the stellar variability kernel hyperparameters are not only perfectly accurate, but also perfectly precise.Even if one were to apply this analysis to stars with well constrained asteroseismic and activity signals, the resulting orbital parameter uncertainties would not include propagated uncertainties on the measured correlated noise hyperparameters.Extending this framework to accommodate hyperparameter uncertainties once empirical models are in hand will improve the accuracy of our results and enable us to quantify how tightly the kernel hyperparameters need to be constrained to adequately model out stellar variability signals and enable Earth analog exoplanet detection.

Accounting for Instrumental Noise
We have chosen to ignore noise contributions from spectrograph and pipeline systematics in this study so that we could isolate the effects of other sources of noise and assess relevant mitigation strategies.As we state in Section 3, we assumed to first order that well-understood instrumental variations are calibrated out prior to com-puting the RV time series from observed stellar spectra, and we ignore residuals and other poorly understood systematic signals that remain.A thorough discussion of the cumulative covariance contributions from numerous sub-m s −1 instrument systematics is well beyond the scope of this work, but it is worth considering their impact on our conclusions.
We repeat the Fisher information analysis for our intra-night observing schedule simulations in Section 4 with the addition of a single instrumental covariance contribution that is constant for all observations taken within a single night and perfectly uncorrelated across multiple nights.Noise such as this might stem from variations in a wavelength solution that is anchored to a nightly calibration sequence.This covariance contribution can be represented by a simple step function kernel: We perform this analysis for internal single epoch precisions of σ instrument = 30 cm s −1 and σ instrument = 10 cm s −1 , where the former is based on recent best estimates for current generation EPRV spectrographs (Halverson et al. 2016;Blackman et al. 2020) and the latter is an optimistic projection for future instruments.
As expected, the addition of k instrument to our covariance matrix neither changes the preferred intra-night observing strategy nor accentuates any differences between strategies.Because the covariance timescale is set such that each simulated schedule receives the same number of independent epochs, this instrumental noise contribution serves only to raise the achieved precision floor for all strategies.We find that the increase in σ K can be approximated as This will have a negligible effect on sensitivity to Earth analogs when σ K is on the order of 10 cm s −1 or greater (e.g., for our full simulation results using the quasiperiodic activity kernel).But even with thousands of observations and an instrumental precision of ≲ 30 cm s −1 , this can significantly degrade the achieved Earth analog detection significance when [K/σ K ] ⊕ ≳ 5 (e.g., for our results using the Matérn-5/2 activity kernel).Future studies that account for realistic instrumental noise contributions in simulations such as those discussed in Section 5 are strongly encouraged.

Stellar Mass Dependence
Survey performance will vary from star to star as a function of stellar mass, and we expect to achieve better sensitivity to exoplanets orbiting low mass stars given that K ∝ M −2/3 ⋆ .As we show in Figure 9, this dependence is reflected in our results for the quasiperiodic activity kernel.The achieved detection significance, [K/σ K ] ⊕ , shows little scatter about a simple M −2/3 ⋆ relation.However, this relation does not adequately explain the Matérn-5/2 activity kernel detection significance results, which exhibit a much steeper mass dependence.This suggests that σ K depends on mass as well for this model.Luhn et al. (2023) see a similar trend in their survey analysis results and arrive at the conclusion that this is a consequence of the stellar parameter dependence of the oscillation and granulation correlated noise signals.Low-mass stars thus carry even more of an advantage for low-mass exoplanet detection than the RV semi-amplitude scaling would suggest.We note that this effect is only observed in the Matérn-5/2 activity kernel results because the correlated noise contributions from the quasiperiodic activity kernel are so large that they mask any strong dependence on the asteroseismic correlated noise.

Sensitivity to assumed definition of an Earth analog
Our definition of an Earth analog as applied to the D ⊕ survey success metric is somewhat arbitrary and does not fully capture the commonly cited objective of ambitious EPRV surveys, which is to detect and measure the masses and orbits of Earth-mass planets in the Habitable Zones of their host stars (Crass et al. 2021).Because we fix the period range to 200 days < P < 500 days for all stars, we do not account for the change in the location of the Habitable Zone as a function of stellar luminosity.Similar arguments can be made to change our mass limits of 0.5M ⊕ < M p sin i < 2M ⊕ .However, we show in Figure 10 that while certain features in the detection significance contours are more pronounced for some strategies than for others, the local slope of these contours over our specified period and mass ranges is quite insensitive to survey strategy.Instead, the differences between the success metrics calculated for each strategy result primarily from linear displacements of the detection significance levels.Changing the boundaries or even the shape of the region we use to calculate D ⊕ will therefore have little to no effect on the relative performance of different survey strategies.

RV Model Complexity
In this work, we employ a simple 3-parameter model for the exoplanet-induced RV signal, assuming only one planet with a circular orbit.future investigations in this vein could benefit from additional model complexity (e.g., non-zero eccentricity or multiple planet signals).Our approach also implicitly assumes perfect knowledge of γ, the absolute RV of the stellar rest frame, and of the RV signals imparted by any other exoplanets in the same system as the nominal Earth analog we are interested in detecting.In the case that γ is the same for all observations, it can be accounted for by adding a constant offset term to Equation 7, which will have no effect on the resulting Fisher information.But in practice, it is often necessary to combine data from multiple instruments to enable the detection of RV signals from low-mass, longperiod exoplanets.The slight differences in the absolute RV scale for each instrument will necessitate the inclusion of instrument-specific γ terms.A time-dependent γ term can also be used to account for RV zero-point offsets in the data stream from a single instrument, such as those introduced by the SOPHIE octagonal fiber upgrade (Bouchy et al. 2013) or the HIRES detector upgrade.Post-survey detection sensitivity analyses should take care to explicitly account for a varying γ term.
The assumption that we will have detected and characterized all other significant exoplanet-induced signals in each system is also likely too optimistic.First, we note that several studies have found evidence for elevated co-occurrence rates for inner terrestrial planets and cool gas giants (Bryan et al. 2019;Rosenthal et al. 2022) and multi-planet systems in general have been shown to be fairly common (Fang & Margot 2012;Zhu 2022).While EPRV surveys may easily detect and characterize a Jupiter analog with K ∼ 10 m s −1 , uncertainties on the mass and orbit will still complicate the detection of smaller RV signals.In addition, signals from multiple small planets with orbital periods close to 1 year, e.g., an Earth analog and a Venus analog, may be particularly difficult to disentangle.Though it is beyond the scope of the present work, expanding our RV model to account for signals from multiple planets within the Fisher information framework will enable us to determine the magnitude of this effect on Earth analog detection.

CONCLUSIONS
We outline a detailed framework for analyzing the detection sensitivity of EPRV exoplanet surveys and we use this to assess the efficacy of various survey strategies in the correlated noise dominated regime.By using Fisher information analysis to quantify the impact of intrinsic stellar variability on exoplanet detection we can capture these correlated noise contributions with GPs and we show that this method accurately reproduces the results of injection and recovery tests of synthesized RV time series measurements.
We describe the design and execution of several tests with which we isolate and assess the impact of sources of noise on different timescales.From our simulations of survey schedules with different intra-night visit distributions, we show that: • distributed multi-visit strategies outperform single-visit strategies across all nightly time costs in the presence of granulation alone, but this advantage is reversed for oscillations • in the oscillations-only case, one can achieve higher precision for visits shorter than ≲ 20 minutes by sampling the oscillation signal with a sequence of consecutive exposures instead of averaging over the signal with a single continuous exposure • in the presence of correlated noise from oscillations and granulation together, single-visit strategies are generally preferred • when we for photon noise, multi-visit strategies are more heavily penalized by the precision loss incurred by overhead costs Analogous survey schedule simulations are used to test the performance of different distributions of observations across an observing season, and we find that: • dense, high-cadence observing strategies are preferred when rotationally modulated stellar activity signals are best represented by a Matérn-5/2 kernel with a relatively short correlation length • sparser observing strategies that maximize the seasonal observing baseline are preferred when these same signals are represented by a quasiperiodic kernel with a correlation length on par with the Solar rotation period These findings are applied to simulations of 10-year EPRV exoplanet surveys with realistic target lists and time constraints as well as practical observing constraints such as weather losses, intra-survey competition, and seasonal observability.We calculate and compare the expected parameter uncertainties for each survey and show that the results of these more comprehensive simulations are largely consistent with those of the isolated tests.We also quantify the absolute detection sensitivity limits we will expect to achieve and discuss our findings.For the Matérn-5/2 activity kernel, we expect to be sensitive Earth analog exoplanets around as many as ∼ 20 quiet stars at the [K/σ K ] ⊕ > 5 level

Figure 2 .
Figure 2. Covariance between pairs of observations as a function of separation in time for the quasiperiodic kernel for rotationally-modulated stellar activity (Equation15; left) and for the Matérn-5/2 kernel for rotationally-modulated stellar activity (Equation16; right).While the covariance for both kernels persists across many nights, we note that the curves differ significantly in amplitude and shape.

Figure 3 .
Figure3.Expected semi-amplitude uncertainty, σK , for a K = 10 cm s −1 signal and an orbital period of P = 100 days as a function of total nightly time cost (as calculated via Equation18).We calculate σK for simulated observing schedules with 100 nights of observations distributed across 500 total nights, wherein the observations on each night are distributed according to each of the five strategies depicted in the upper left panel.These strategies are described in Section 4.1.In the remaining panels, we show the impact of contributions from various combinations of photon noise and correlated noise from oscillations and granulation.We also include inset plots to highlight the performance of each multi-exposure strategy relative to the single continuous exposure strategy for nightly time costs of less than an hour.

Figure 4 .
Figure 4. Simulated inter-night observing schedules over a single observing season.We show the idealized distribution of observations across a 240-night baseline for each of the strategies described in Section 4.3.Isolated points represent individual observing nights and lines represent blocks of consecutive observing nights.For each strategy, we show the distribution of observing nights for different numbers of annual observations, increasing in the vertical direction from 60 to 240 in increments of 20 observations per year.
Figure5.Expected semi-amplitude uncertainty, σK , for a K = 10 cm s −1 signal and an orbital period of P = 300 days as a function of number of observations per year.We calculate σK for simulated observing schedules with 10 years of observations and a 240-day annual observing season, wherein the observations for each season are distributed according to each of the strategies depicted in Figure4and described in Section 4.3.We include correlated noise contributions from rotationally-modulated stellar activity represented by a quasiperiodic kernel (dot-dashed lines) and by a Matérn-5/2 kernel (solid lines).

Figure 6 .
Figure 6.Expected exoplanet detection sensitivity limits as calculated using our Fisher information framework for a typical target star with approximately 100 observations per year for 10 years and the Centered inter-night observing strategy.We include contributions from photon noise, oscillations, granulation, and stellar activity.We plot K/σK as a function of minimum mass, Mp sin i, and orbital period, P .For reference, we also plot the 3-, 5-, and 10-σ detection significance contours and we indicate the locations of the Solar System planets Venus, Earth, and Jupiter with the ⊙ symbol.The green 'x' and green dashed box indicate the mass and period values and ranges we use to define an Earth analog for the [K/σK ]⊕ and D⊕ survey success metrics, respectively.Results for the quasiperiodic activity kernel are shown on the left and results for the Matérn-5/2 activity kernel are shown on the right.

Figure 7 .
Figure7.Expected Earth analog detection significance, [K/σK ]⊕, as a function of number of visits for each of our survey simulations.The darker lines that terminate in filled circles correspond to the results for the 20-star simulations and the lighter lines that terminate in open circles correspond to the results for the 40-star simulations.The size of each circle scales with stellar mass.We note that each of these lines is exhibits a relatively smooth dependence on number of observations within an observing season, but that these smooth changes are interrupted by sharper kinks between observing seasons.Dashed vertical lines mark the expected number of observations per star for an idealized observing schedule, i.e., 1000 total observations for each of the 40-star simulations and 2000 total observations for each of the 20-star simulations.The final distributions of [K/σK ]⊕ values for each survey are shown as histograms in the narrow panels to the right side.

Figure 8 .
Figure8.Comparison of [K/σK ]⊕ for a 10-year survey with 100 observations per year (vertical axis) and a 5-year survey with 200 observations per year (horizontal axis).We compute the latter values using the first 5 years of observations from the 20-star survey simulations.On the left, we show that for the quasiperiodic kernel, the longer 10-year survey delivers a higher detection significance than the higher cadence 5-year survey.On the right, we show that the opposite is true for the Matérn-5/2 kernel, which performs better with a higher observing cadence.

Figure 9 .
Figure 9. Dependence of expected Earth analog detection significance on stellar mass for each of our simulated surveys.On the left, we show that for the quasiperiodic kernel, [K/σK ]⊕ follows a K ∝ M

Figure 11 .
Figure11.First two years of simulated observations for the Double Burst inter-night observing strategy as applied to the 40-star target sample with practical constraints on target observability.Stars are shown in order of increasing RA from top to bottom.Orange markers represent nights during which each target was observed, grey markers show observations that were scheduled and then dropped due to simulated weather losses and telescope closures, and grey lines indicate the nights on which these losses occurred.For most stars, we are able to achieve a reasonable approximation of the idealized version of this strategy shown in Figure4in spite of these practical constraints.

Table 1 .
Solar Hyperparameter Values for Stellar Variability Gaussian Process Kernels

Table 2 .
Parameter uncertainties from posterior sampling and Fisher information analysis

Table 4 .
Survey Simulation Success Metrics While our model allows us to focus on and compare specific survey strategy choices,