Statistical tests exploiting simulations of the colored noise: theoretical anal- ysis and numerical study 

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The radial-velocity technique

The radial velocity technique is based on the Doppler-Fizeau e ect introduced by Doppler in 1842 [Doppler, 1842]. This e ect links the stellar motion (due to the in uence of an orbiting planet) on the light-of-sight of the observer to a wavelength shift in the stellar spectra. When the star is going toward the observer, the stellar lines are blueshifted and redshifted when the star is moving away (cf. Fig. 1.3). By ignoring relativistic e ects (as VR c), the relationship between the RV and the wavelength shift is:
with c the speed of light, 0 the wavelength in the reference frame, the observed wavelength and VR the projected velocity (i.e. the radial velocity).
The shape of a RV planet signature depends on the planet orbital parameters. The motion is described by the Keplerian parameters of an elliptical planet orbit. These signatures are described in the following section, which is largely inspired by [Perryman, 2011].

Planetary signatures

The RV semi-amplitude (K) due to a planet orbiting a star is related to the stellar and planetary characteristics by:
with G the gravitational constant, T the planet period, M the planet mass, M? the stellar mass, i the inclination of the planet orbit with respect to the reference frame (cf. Fig. 1.6) and e the orbit eccentricity12 (for the detailed calculations, see p.12 of [Perryman, 2011]). According to (1.2), for a given stellar mass, it is easier to detect massive planets, short period planets, and planets with small eccentric orbit. These remarks are indeed observed in Fig. 1.2, where the detected planets are mostly massive and with short period orbit (generally combined with a small eccentric orbit due to the tidal e ects of the host star).
The rst rows of Table. 1.1 shows the RV semi-amplitude of the Solar System planets, evaluated from (1.2) in the perfect observational conditions (i = 90 ). For example, an Earth-like planet orbiting around a Sun at 1 AU (row 3) has a signal amplitude of 8.9 cm.s 1 whereas a Jupiter-like planet at 5:2 AU (row 5) produces a RV signal of 12.4 m.s 1 amplitude. To give an order of magnitude, a radial velocity variation of 12 m.s 1 corresponds to the detection of a wavelength shift in the spectral lines of= 2 10 4 stability of the spectrograph. Rows 9 and 10 illustrate K for an Earth-like planet in close orbit and a Super-Earth planet (at 1 AU) to be compared to Earth (row 3). When the planet is 10 times closer (or 5 times more massive), K is 3 (or respectively 5) times larger. The last three rows correspond to some known exoplanet signatures. First, 51 Pegasi b (row 11) evoked in Sec. 1.1 for which the mass of the star is M? = 1:04 M and the inclination of the orbit i = 80 leading to a RV semi-amplitude of 55:9 m.s 1. As an illustration, the observed RV curve of 51 Peg b is shown in the left panel of Fig. 1.4 as function of the phase13. Follows Centauri B b (row 12), which is a debated planet detection around a solar mass star (M? = 0:907 M ) of spectral type K1V [Dumusque et al., 2012, Hatzes, 2013, Rajpaul et al., 2016], and Proxima Centauri b (row 13) orbiting a M dwarf star (M? = 0:12 M ) of spectral type M5.5Ve [Anglada-Escude et al., 2016].
Figure 1.4 { Left: Observed radial velocity curve of the planet 51 Pegasi b (extracted from
[Mayor and Queloz, 1995]). Right: RV signatures of the Earth (blue), Jupiter (red) and the entire Solar System planets (assuming no interaction between them, black) evaluated for i = 90 . The RV of the entire Solar System is completely dominated by the Jupiter’s in uence, which with Saturn, contains more than 90% of the whole Solar System planets mass.
13The phase is t t0 with t the time, t0 the time of the planet passage at periastron, and T the planet period.
Considering now the orbit characterization, the signatures for Np planets with Keplerian orbits can be expressed by:
with Kp the semi-amplitude given by (1.2), ep the eccentricity, !p the argument of periastron14 (0 !p < 2 ), V0 the velocity of the system barycentre, and ap(t) the true anomaly. In practice, only the Keplerian parameters Tp; !p; ep; Mp, and Kp can be derived from the observations. The shape of the RV planet signature in (1.3), with respect to the di erent Keplerian parameters, will be studied in details in Chap. 2 (Sec. 2.3).
with Ep(t) the eccentric anomaly. The eccentric anomaly can be calculated using the mean anomaly M(t) which is, at a given time, the angular distance of the planet from periastron:
with t0 the time of the planet passage at periastron. The mean anomaly can be seen as a ctive mean motion around the planetary orbit and can be inferred directly from the data (see Fig. 1.5). The second equality in (1.5) is called the Kepler equation and has to be solved by iterative numerical methods such as the Newton-Raphson algorithm. At a given time, the planet position (indicated by the blue circle in Fig. 1.5) can be described both in terms of the true anomaly ap (with respect to the ellipse centre, i.e. the system center of mass, here indicated by the star symbol) and the eccentric anomaly Ep (with respect to the auxiliary circle, i.e. the circle which circumscribes the planetary ellipse).
As seen in Sec. 1.1, the detectability of the RV planet signature depends on the planet’s or-bit inclination on the light-of-sight. The inclination parameter is involved in (1.3) through Kp / Mp sin(i) (cf. (1.2)) and cannot be determined from RV observations (but by transit for example). This leads only to a lower limit of the planet mass when the planet is only detected by the RV technique.
Fig. 1.6 shows the e ect of the inclination on the RV curves for three di erent con gurations:
a) The \pole-on » geometry (i = 0 ), where the Doppler signature is minimum (VR = 0). In this con guration the planet signature is not detectable and we have to use astrometric measurements to detect the planet.
b) An intermediate case (i = 45 ), where the Doppler signature increases.
c) The \edge-on » geometry (i = 90 ), where the RV semi amplitude is maximum (max VR). This is the best con guration.

Method of measurement and instrumental errors

s discussed in the beginning of this section, RV measurements are based on the systematic mea-sure of Doppler shifts in the stellar absorption lines. Ignoring the relativistic e ects (VR c), the velocities can be determined by the classical form of the Doppler e ect given in (1.1). Ac-cording to Hatzes & Cochran, the RV precision (or dispersion, instr) is related to the resolution R of the spectrograph, the signal-to-noise ratio (SNR) S=N, and the spectral range B of the spectrograph as [Hatzes and Cochran, 1992]:
The resolution and the spectral range of the spectrograph xed, the instrumental precision depends on the SNR. The SNR depends on a lot of parameters, which modify the saturation level (e.g. the signal magnitude, the detector characteristics, the stellar magnitude and spectral type, the exposure time of the observations, the considered wavelength range or the quality of the atmosphere). Achieving a RV precision of some m.s 1 is possible for two reasons.
First, a long-term instrumental stability, both in temperature, pressure and humidity, over months or years. This is possible with echelle spectrographs15 operating at high resolution.
The principle of any spectrograph is to disperse the received starlight into its wavelength components. Echelle spectrographs use a re ecting grating (i.e. a dispersive element), composed of equally spaced grooves (facets).
Today (2017), the more precise spectrograph is HARPS, which is installed on the ESO 3.6-m telescope, La Silla, since 2003. HARPS operates in the optical domain (378 691 nm) with a resolution of R = 120 000. According to the extreme stability and the characteristics of HARPS (e.g. captors and telescopes sizes, spectrograph resolution, calibration by Thorium-Argon lamps), and considering a photon noise limit evaluated in nominal condition (e.g. for stars with MV = 7), this spectrograph is able to provide long term (1 month) wavelength ref-erence allowing to achieve a precision around 1 m.s 1 until the saturation limit of the CCD is reached (300 000 electron peaks intensity) [Bouchy et al., 2001, Lovis et al., 2006]. This limit-ing value of 1 m.s 1 has to be seen as the intrinsic scatter on the observed RV. Signal with amplitudes below such limit are undetectable (whatever the number of observed data).
The detection also takes bene t from the instantaneous combination of many absorption lines in the stellar spectrum through the cross-correlation technique, which improves the measured SNR of the target Doppler shift (e.g. [Allende Prieto, 2007]). To do that, stars with spectral type latter than F 6 are the privileged targets as, at a given metallicity, they present a lot of absorption lines (see Fig. 1.7). For earlier-type stars, complications appear as they are often fast rotators making their (few) spectral lines wider (and consequently the wavelength shifts is less detectable). One notes however that adapted crossed correlation techniques using speci c tem-plates (e.g. [Galland et al., 2005]) or other dedicated techniques (e.g. [Astudillo-Defru, 2015]) exist for these stars.
Figure 1.7 { RV accuracy depending on the spectral type. Early type stars, from A0 to F5, and late type stars, from K5 to M0, lead to poor RV accuracy. Main sequence stars, from F5 to K5, are more favoured for RV technique. Figure adapted from [Bozza et al., 2016].
leaned to a speci c angle (the blaze angle) with respect to the grating normal. This angle of di raction is chosen to make constructive interference and to di ract most of the light into high order (as a zero order corresponds to the white light which bring no information). One can report to [Vogt, 1987] for details on this instrument.
Following the Hatzes’ chapter in [Bozza et al., 2016], the more common methodology of RV extraction is based on the cross-correlation technique. The main steps are:
1. Pixel measurements of the absorption lines position in the stellar spectrum by an echelle spectrograph (e.g. HIRES, HARPS). In the spectrograph, many di raction orders disperse the light and are recorded in a CCD where the absorption lines can be measured. The accuracy of the measurements depends on the instrumental resolution and on the target star. For example, Fig. 1.7 (extracted from [Bozza et al., 2016]) shows an estimate of RV errors as a function of the stellar spectral type (evaluated by taking into account the stellar mean rotational velocity and approximate stellar line densities). Cool stars (spectral type later than F6) have a good accuracy while more massive stars with high rotational rates and high e ective temperature induce large RV errors.
2. Conversion in wavelength shift with a calibration method. The rst technique consists in calibrating the observed spectrum with a simultaneous calibration done with a Thorium lamp. This is the method used with spectrographs like ELODIE and HARPS [Bouchy, 2015]. Another techniques consist in using an absorption cell [Gri n and Gri n, 1973] as for the spectrograph of the McDonald Observatory (US), or in using the Earth atmosphere as a reference (through telluric spectral lines). All these calibration methods need high stability to minimize the instrumental drifts. Some errors can happen during the data calibration step and one notes that a future promising method, the \Laser frequency Combs », is in development for the next generation of spectrograph like ESPRESSO [Pepe et al., 2010].
3. Cross-correlation of the entire spectrum. Thousands of lines are measured in the spectra of FGK stars and the wavelength shift is deduced by cross-correlation of the entire spectrum with templates [Baranne et al., 1996]. Again, this is only possible by combining thousands of lines. Indeed, since HARPS’ camera pixel size is about 15 m (corresponding to 800 m.s 1), it would be impossible to measure a 1 m.s 1 displacement with a unique spectral line.
4. Translation in radial velocities. Using (1.1), the wavelength shift is measured with respect to (w.r.t.) the reference frame of the observer. As the referential of the measure-ment is linked to the spectrograph, we have to correct the shifts due to the Earth motion depending on the instrument location and epochs of measurements. These barycentric corrections include (among other things) the Earth’s orbital motion (up to 30 km.s 1), the Earth’s rotation ( 4:6 km.s 1), the Solar System motion, and the proper motion of the star w.r.t. the Solar System (some milli arcseconds per year for the majority of the stars16 [Rouan, 2011]). These motions are evaluated with the ephemerides, the observer position, the distance to the Earth centre and the position of the observed star.
During the process of RV extraction, instrumental noises can hide the searched planet signatures. Fig. 1.8 shows the RV semi-amplitude Kp evaluated by (1.2) as function of the planet periods Tp (for ep = 0 and M? = 1 M ) for di erent planet masses (1 MJ ; 5 M ; 1 M ). The two RV amplitudes indicated at right represent the limit range of the most accurate spectrograph in activity, HARPS (blue shade region), and the future spectrograph ESPRESSO (yellow shade region). One can observe that Earth-mass planets are not detectable with HARPS (at least for period > 1 hour), which is only able to detect Super-Earth with close orbit (period 1 year). On the contrary, the future high-precision spectrograph ESPRESSO will decrease the instrumental noise level by a factor of 10 allowing the detection of Earth-like planets (e.g. the blue curve corresponding to 1 M with 1 year period is at the expected detection limit of ESPRESSO). These performance will be achieved thanks to the reduction of the instrumental errors (by the spectrograph stability) and the improvement in the data calibration (through the laser frequency comb technique).
For these modern stable spectrographs, the instrumental noise will not prevent the detection of telluric RV planet signatures. Indeed, another source of noise appears when looking for small planets: the noise created by the star itself, which becomes the main limiting factor at such low instrumental levels.
Figure 1.8 { RV planet semi-amplitude as function of periods for ep = 0 and a solar mass star. The blue and yellow shade regions represent approximately the HARPS and ESPRESSO instrumental limits.

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Stellar noises

Astrophysical noise sources arise from the stellar photospheric activity of the host star. Indeed, the light observed to detect the RV planet signatures is formed at the stellar surface and therefore is very a ected by its dynamical environment. These plasma motions alter the shape of the stellar absorption lines and a ect the RV under the form of correlated noises.
Ignoring these noises, or misunderstanding them, had lead to several false planet detections in the past (examples of debated detections are [Queloz et al., 2001, Figueira et al., 2010, Dumusque et al., 2012]). Among the stellar activity, one can distinguish three main sources of noise:

Table of contents :

I Introduction 
1 Exoplanet detection 
1.1 Methods and discoveries
1.2 The radial-velocity technique
1.2.1 Planetary signatures
1.2.2 Method of measurement and instrumental errors
1.3 Stellar noises
1.3.1 Magnetic activity
1.3.2 Oscillations
1.3.3 Convection
1.3.4 Summary
1.4 Simulations of the convective stellar activity
1.5 Conclusions on the astrophysical context of the thesis
2 Detection of periodic signatures in noise: tools, models and tests 
2.1 Hypothesis testing
2.1.1 Design of hypotheses
2.1.2 Statistical tests and decision rule
2.1.3 Frequentist approaches: tests based on likelihood functions
2.1.4 A brief comment on Bayesian approaches
2.2 Periodogram(s)
2.2.1 The classical periodogram
2.2.2 Periodogram of a single sinusoid
2.2.3 Parseval’s identity
2.2.4 Periodogram resolution
2.2.5 Bias and variance of the periodogram
2.2.6 Asymptotic distribution in the case of a pure random process
2.2.7 Asymptotic distribution of P in the case of a periodic signal in noise
2.2.8 A note on some modied periodograms
2.3 Planetary signatures in the periodogram
2.3.1 Signatures in the time domain
2.3.2 Signatures in the Fourier domain
2.3.3 Discussion and signal model
2.4 Statistical tests applicable to the periodogram
2.4.1 Preliminary notations
2.4.2 Classical tests
2.4.3 Tests designed for multiple sinusoids
2.5 Adaptive tests that can be applied to the periodogram
2.5.1 Introduction of the problem
2.5.2 Example: Kolmogorov-Smirnov test
2.5.3 Higher Criticism
2.5.4 Berk-Jones test
2.6 Sinusoid detection in unknown noise statistics: related works
2.6.1 Non parametric approches
2.6.2 Parametric methods
2.7 Conclusions
2.7.1 Main conclusions
2.7.2 This thesis
II Proposed detection method 
3 Statistical tests exploiting simulations of the colored noise: theoretical anal- ysis and numerical study 
3.1 Statement, assumptions and objectives of the detection problem
3.1.1 Model under both hypotheses
3.1.2 Assumptions and objectives
3.2 Statistical distribution of the proposed standardized periodogram
3.2.1 Standardized periodogram eP
3.2.2 Statistics of eP
jPL under H0
3.2.3 Statistics of eP
jPL under H1
3.2.4 Mean and variance of eP
jPL under H0
3.3 Statistical tests applied to the standardized periodogram
3.3.1 Test of the maximum periodogram value (TM)
3.3.2 Fisher’s test and its variations (TF , TCh, TF;rob)
3.3.3 Testing the Nth C largest periodogram value (TC)
3.3.4 Adaptive tests (HC, BJ)
3.4 Standardization using parametric estimates of SE
3.4.1 AR parameters estimation
3.4.2 Distribution of ePjbS E;AR
3.5 Numerical studies on synthetic noise processes
3.5.1 Classical tests based on eP jPL
3.5.2 Adaptive tests based on ePjPL
3.5.3 Eect of ignoring noise correlations
3.5.4 Comparison of the proposed standardization with a parametric approach
3.5.5 Application to the design of observational strategies
3.6 Summary and conclusions
4 Extension to the case of uneven sampling 
4.1 Eect of uneven sampling on the periodogram
4.1.1 Considered sampling patterns
4.1.2 Classical periodogram and uneven sampling
4.2 Variants of the periodogram
4.2.1 The Lomb-Scargle periodogram
4.2.2 Least-square tting and other related periodograms
4.3 Generalized Extreme Value distribution
4.3.1 Generalities
4.3.2 Parameters estimation
4.3.3 Diagnostic plots
4.4 False alarm rates evaluation
4.4.1 Method of \independent » frequencies
4.4.2 Bootstrap methods
4.4.3 FA evaluation using theory of the extremes
4.4.4 Conclusions
4.5 Proposed bootstrap procedure
4.5.1 Direct bootstrap approach
4.5.2 \Accelerated » bootstrap approach
4.6 Numerical studies on a synthetic noise process
4.6.1 Validation on known processes
4.6.2 \GEV accelerated » algorithm
4.7 Conclusions
Conclusions and perspectives


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