Mem. S.A.It. Vol. 83, 1033c© SAIt 2012 Memorie della

Astrometric cosmology

M. G. Lattanzi

Istituto Nazionale di Astrofisica – Osservatorio Astronomico di Torino, Via Osservatorio20, I-10025 Pino Torinese, Italy e-mail: lattanzi@oato.inaf.it

Abstract. The accurate measurement of the motions of stars in our Galaxy can provide ac-cess to the cosmological signatures in the disk and halo, while astrometric experiments fromwithin our Solar System can uniquely probe possible deviations from General Relativity.This article will introduce to the fact that astrometry has the potential, thanks also to im-pressive technological advancements, to become a key player in the field of local cosmol-ogy. For example, accurate absolute kinematics at the scale of the Milky Way can, for thefirst time in situ, account for the predictions made by the cold dark matter model for theGalactic halo, and eventually map out the distribution of dark matter, or other formationmechanisms, required to explain the signatures recently identified in the old componentof the thick disk. Final notes dwell on to what extent Gaia can fulfill the expectations ofastrometric cosmology and on what must instead be left to future, specifically designed,astrometric experiments.

1. Introduction

With the Gaia mission approaching launch, theidea of organizing a workshop dedicated tothe implications of micro-arcsecond (µas) as-trometry in QSO astronomy and FundamentalPhysics finally turned into reality. I was hav-ing the irresistible opportunity to put forth myconcept of “Astrometric Cosmology”. This isthe ancient observational science of measuringangles among sources on the celestial spherebrought by technology extraordinary advance-ments to new accuracy levels. These are able tocontribute, in some cases uniquely, to modernCosmology by confronting its detailed zero-redshift predictions to the observed complex-ities in the stellar phase space of our Galaxy.

Send offprint requests to: M. G. Lattanzi

2. The new astrometry

There have been several definitions of “new”(or “modern”) astrometry over the years in an-ticipation of, or right after, the release of thedata from the Hipparcos mission in 1997 (ESA1997), the event that brought this ancient ob-servational branch of Fundamental Astronomyto a new life.

Kovalevsky in his seminal essay Prospectsfor space stellar astrometry (Kovalevsky 1984)used the size of the accessible field-of-view(FOV) to classify astrometry 1 thus emphasiz-ing the critical role of technology for provid-ing an ancient science with a prime spot intonext century astrophysics: if the prospects ofnarrow field astrometry could still be grantedfrom the ground (with space as the obvious, al-

1 From the few arc-seconds of narrow-field as-trometry to the 4π of global, all-direction, astrom-etry.

1034 Lattanzi: Astrometric cosmology

though expensive, alternative) by utilizing in-terferometry and adaptive optics against atmo-spheric limitations, going into space appearedas the only option for the future of wide fieldand global astrometry even at visible wave-lengths.Here, I will adopt a purely operational defi-nition for “new astrometry”, so that its differ-ent forms discussed in the following sectionswill appear as logical transitions. Specifically,the following five “key words” are all it isneeded for my definition of new astrometry:all-sky, faint magnitude, completeness, wave-length coverage, and accuracy. These keywords synthesize what is necessary today tomake progress in wide-field or, better, globalastrometry: the science of the materializationof the Reference Frame (RF), as I like tocall it. Indeed, the definition and realizationof the RF, through the implementation of theInternational Coordinate Reference System2

(ICRS), is probably one of the most far reach-ing tasks of fundamental astronomy in the 21stcentury.

Progress in catalog astronomy3 has twodistinct but equally important sides: one isoperational, the other scientific. Common toboth is the realization of the RF across theelectromagnetic spectrum (wavelength cov-erage) and its extension, or densification, tofaint magnitudes through a complete census ofall the sources to a specified magnitude limit.Also, in order of increasing accuracy, i.e. thephysics of gravitation utilized, the RF wouldbe called inertial, absolute, non-rotating, or,simply, local (relative to the observer).

Deep space navigation and full (optimal)exploitation of the largest ground-based andspace-borne observatories are operational ex-amples that cannot do without faint and com-plete astronomical catalogs. This is the caseof the 8-m class telescopes at facilities likeESO VLT and the Gemini Observatory, whoseoptimal operations require the acquisition of

2 See www.iers.org/IERS/EN/Science/ICRS/ICRS.html

3 This is the name often utilized as a synonym forglobal astronomy.

suitable reference sources (both in magnitudeand color) for top performances of their activeand adaptive optic systems; or like the ChineseLAMOST (Xiang-Qun Cui et al 2012), whichneeds to feed and real time operate its 4000-fiber-fed robotic facility.As for orbiting observatories, operations of theHubble Space Telescope (HST), including ob-serving proposals preparation, require reliableomnidirectional blind pointing capabilities tominimize orbital maneuvers (especially real-time ones) thus maximizing efficiency (i.e., thefraction of orbital time spent on science ex-posures) and cost effectiveness. Moreover, themost sensible instrumentation aboard requiresprotection from stars as faint as V = 18−20 in aprocedure that in the Hubble operations jargonis called “bright objects alert”4. This in turn re-quires 4π completeness of the supporting cata-log.And supporting HST operations was the maindriver for the realization of the Second GuideStar Catalog (GSC2): released to the commu-nity in 2006 and published in the Summer of2008 (Lasker, Lattanzi, McLean et al. 2008),the GSC2 lists approximately 1 billion objects,i.e. unique entries, and is complete to the redmagnitude R = 20. GSC2 is one of the bestproducts ground based catalog astrometry hasto offer (see also Monet et al. 2003) possess-ing all of the traits of modernity introducedabove except for accuracy that is limited by theground-based material from which the catalogwas made. Its scientific relevance resides in be-ing both the most detailed multi-color view ofthe Milky Way at optical wavelengths to date,and the faintest and densest materialization atthese wavelengths of the ICRF, whose primaryrealization has been dominated by radio astron-omy.

Global astrometry is also the science ofmeasuring absolute stellar distances. Its poten-tial for astrophysics comes from the ability tocontribute to the direct, i.e. model independent,calibration of radiant and gravitational energy,the two forms of energy that dominate in the

4 Stellar objects in that magnitude range are po-tentially dangerous sources, as way too “bright” inlong exposures imaging of extremely deep fields.

Lattanzi: Astrometric cosmology 1035

Universe. However, stars are far away and theirangular motions are small as they decreasewith the inverse of distance itself. Earth’s at-mosphere has then always posed a serious lim-itation to increasingly accurate (absolute) dis-tances; and if astrometry was to move awayfrom the immediate solar neighborhood, it wasessential to go into space, irrespective of wave-length, therefore requiring tremendous techno-logical challenges.The success of the ESA mission Hipparcosdemonstrated that technological advancementsin the last two decades of the 20th century ma-tured to a level sufficient for a first giant stepforward. Space-borne global astronomy had fi-nally come of age through the realization ofa self-calibrating dual-FOV scanning satellite(ESA 1997).The numbers delivered by the Hipparcos satel-lite remain unmatched even today, three lustresafter the publication of the catalog: 118,218positions, annual proper motions, and paral-laxes down to V = 12.45, with a median pre-cision of better than 1 mas. This is 100 timesin number and not less than a 10–fold in-crease in precision to what produced from theground since the beginning of modern astrom-etry in 1838, when the German astronomerFredrich Bessel measured the first trigonomet-ric parallax establishing the distance to the star61 Cygni. With Hipparcos, astrometry had fi-nally acquired the capability of producing largenumbers of accurate parallaxes throughout thesky. And it was thanks to such an accuracyof genuinely absolute distances that space be-came undoubtedly the new frontier for globalastrometry, which, in turn, established itself asindispensable for solving the open problems instellar and galactic astrophysics.

2.1. Astrophysical astrometry

What is modern astrometry to do? Its job isto measure, for individual stars and indepen-dently from models, the following fundamentalquantities with increasing accuracy and for thelargest samples possible:

1. distance (absolute parallax p);5 All–sky completeness was limited to ∼ 9 mag.

2. angular position (coordinates);3. velocity (proper motion µ)6;4. mass;5. photospheric (angular) size φ.

This is the realm of astrophysical astrometry.Except for photospheric sizes, which are

the objective of sub–mas (kilometric) opticalinterferometry and can be achieved also fromthe ground, thanks to atmospheric coherence(isoplanatic patches) over the small angles in-volved, the other quantities must all rely onspace astrometry for best results.The importance of distance is obvious as itprovides absolute calibration of radiant energycritical for understanding stellar interiors andatmospheric models; also, together with inter-ferometric (photospheric) diameters, distancecan constrain the physics governing the unsta-ble phases of intrinsically variable stars, like,e.g., pulsating stars. The challenge is to reachsufficiently large volumes of the Galaxy en-compassing samples representative of the com-plexity of the HR diagram. If distances accu-rate to 10% are deemed adequate, reaching thekpc scale implies parallaxes to better than 100µas.Binary systems can provide access to massesacross stellar types and evolutionary stages;however, for the mass to be known to betterthan 3%, i.e. what is required to discriminateamong the different stellar models, their dis-tance must be known with a relative accuracythree times better. Therefore, reaching the kpcscale requires, this time, parallaxes ten timesbetter to 10 µas, way beyond what achievedwith Hipparcos.

Accurate coordinates, or angular position,fix directions on the celestial sphere thus phys-ically materializing the RF. But why is RF ma-terialization important in astrophysics? The an-swer is that it allows accurate alignment ofpoint-like or resolved emissions at differentwavelengths through RF registration, as op-posed to “blind” (non-physical) overlap of en-

6 Proper motion provides, with distance, the tan-gential component of stellar motion; the radial com-ponent needed for a full reconstruction of spatialvelocity is assumed to be complemented by spec-troscopy.

1036 Lattanzi: Astrometric cosmology

ergy peaks. This is the way for model inde-pendent characterization of high energy galac-tic and extra-galactic phenomena, which are“bright” emitters at, say, radio, X or gammafrequencies but often quite faint at opticalwavelengths.This is the case of the nearby (250 pc away)radio quiet 237–millisec pulsar Geminga. ThisINS7 pulsar was optically identified on deepHST images at V=26 mag; this allowed the ac-curate registration, to 5 mas, of a historical se-ries of X and gamma ray pulses after a 10-mag,5-step transfer of the Hipparcos (ICRF) RFonto the same HST images (Caraveo, Lattanziet al. 1998). Theory predicts that pulsar rota-tion slows down as dΩ/dt = −k Ωn,where Ω =2π/P is the rotational frequency (P the corre-sponding period), k is a positive constant, andn is the breaking index that describes how thepulsar spins down. Simple modelling of pulsarradiation predicts n = 3; therefore, the pos-sibility of determining n independently fromtheory provides direct insight into the physicsof pulsars. The registration of the pulse tim-ing through accurate astrometry allows the de-termination of the first and second derivativesof the spin frequency (or period) and there-fore of the breaking index via the relation n =(Ω Ω)/Ω2, and of the pulsar characteristic ageas τ = 1

1−nΩ

Ω, or its analogue in terms of the

spin period τ = 1n−1

PP .

In the radio domain, VLBI and VLA havesince a few decades reached accuracies andresolutions exceeding the milli–second of arc.And RF registration between radio maps andoptical images of the same objects, or, bet-ter, of the same regions of the sky, has beeninstrumental in testing structural and energymodels of active extragalactic objets. In 1997,Lattanzi, Capetti & Macchetto extended thepoint–like source technique utilized for theaccurate absolute positioning of Geminga toSeyfert 2 galaxies. The bright Hipparcos-basedICRS frame was first transferred to faint HSTimages of the class prototype NGC 1068 takenin visible light (continuum and [OIII] nar-row line emissions) through specifically pro-cured ground–based material for extending,

7 Isolated Neutron Star

and therefore densify, the primary RF to faintermagnitudes. Then, the registered space–borneimages were overlaid onto MERLIN 6-cmmaps of the same radio loud galaxy to ex-plore the spatial relationship of the two typesof emission. The results were quite telling: sig-nificant anti–correlations existed in the angu-lar energy distributions at the different frequen-cies, bringing solid evidence to the validity ofthe Unified Model theory for Seyfert galaxies(Capetti et al. 1997).

Distances to the stars are not only the keyto absolute radiant energy but also to the gravi-tation energy content of the Milky Way (MW).For, distances fix, along with space velocities,the initial (boundary) conditions for MW dy-namics. This time, reaching the kilo–parsecscale requires that not only parallax, but alsoproper motion must be known to better than7% for spatial velocities, Vt, accurate to 10%,as per the error propagation formula (σVt/Vt) =√

(σµ/µ)2 + (σp/p)2 (the σ’s indicating stan-dard errors). Measuring individual stars, i.e.mapping the MW phase space in situ, providesaccurate probing of its most prominent con-stituents (bulge, disk, halo) required to con-front with the most sophisticated, cosmologydriven, high resolution models of MW dynam-ics. I will briefly go back to this point in thefollowing sections.

2.2. Relativistic astrometry: the Gaia era

As mentioned earlier, thanks to Hipparcos,space astrometry demonstrated its technologi-cal maturity. With the new century approach-ing and technology advancing fast, precisionastrometry was ready for a second giant step.However, one gap needed to be closed: extend-ing accuracy to faint magnitudes. This meantrealizing in space a large scale catalog (simi-lar to, e.g., the GSC2) following the preceptsof the Hipparcos mission: the Gaia missionconcept, visually shown in Fig. 1, was finallyborn (see Perryman et al. 2001, and referencestherein)!

The Gaia portal at www.rssd.esa.int/index.php?project=GAIA\&page=indexprovides a detailed and constantly updated

Lattanzi: Astrometric cosmology 1037

Gaia

Hipparcos

Fig. 1. The Gaia paradigm: build an all-sky largeand deep catalog, pretty much like the latest gen-eration of ground-based surveys (e.g., the GSC2),by going into space (for maximum accuracy) fol-lowing a mission profile similar to that proved withHipparcos.

account of the status of the mission, whichwill be launched by ESA not earlier thanOctober 2013. Of relevance here is the ex-pected end-of-life astrometric performanceshown in Tab. 1 and released by ESA late lastyear following the positive conclusion of thepayload critical design review. Comparing thelisted figures to the requirements discussedin sec. 2.1, these appear largely satisfieddepending on target magnitude and color.Astronomers wishing to assess the impact ofthe Gaia mission on their own research fieldsshould start from Table 1. This is the resultof averaging over the five-year mission andacross the latitude dependent number–of–observations distribution. Therefore, scienceprograms investing variable phenomena, i.e.requiring epoch observations, or concentratedalong particular lines of sight should refer tothe detailed information available at the Gaiawww portal (section on Gaia’s error budget)before establishing feasibility. There, theycan also find more information on the actualerror statistics, with indication on the expectedpresence, and level, of possible correlationsand of residual systematic errors.

Because of the choice to operate Gaiain the visual domain, interstellar extinction

will cause difficulties in penetrating the kilo–parsec scales at all longitudes in the MW disk,which requires going to redder wavelengths. Inthis context, the effort by the Japanese SpaceAgency (JAXA), through the JASMINE pro-gram (Gouda 2011), is of particular value:after a demonstrator, nano-JASMINE, to belaunched in November 2013, two more satel-lites will follow, with the medium-class mis-sion JASMINE, anticipated to be launched in2020, operating in the Kw band (1.5 - 2.5 µm)and targeting 10 µas at Kw = 11.And, hopefully, NASA will rethink its strategyby going back to pursue missions like JMAPand the sub-µas capabilities of the SIM-lightproject (Unwin et al. 2008).

At those accuracies light does not propa-gate in straight lines and time does not beat thesame everywhere: photons follow geodesicsand physical time is only that of the observer:welcome to the land where Einstein’s GeneralRelativity rules!Astrometry becomes fully relativistic even inweak gravitational fields, where the corre-sponding weakly relativistic metric tensor, gαβ,can be given in terms of the perturbation, hαβ(|hαβ| 1), to the flat Minkowskian metric,ηαβ, as gαβ = ηαβ + hαβ + O(h2), i.e., the back-ground geometry is sufficiently small that non–linear terms can be neglected. In a gravitation-ally bound system, the virial theorem assuresthat all forms of energy density within the sys-tem cannot exceed the maximum amount ofits gravitational potential U; therefore, in thepresence of a weak field it must be |hαβ| .U/c2 ∼ v2/c2 1, where U = GM/D, v is thecharacteristic velocity within the bound sys-tem, and D represents the system linear size.That this situation applies to our Solar System(SS) can be easily verified by recalling that v ∼30 km/sec is its typical internal velocity, andD ∼ 70 au its characteristic extension.

This conference dedicated a full day tothe subject of relativistic reference framesand the description of space–time fabric (seethe contributions by Klioner, Kopeikin, andCrosta in this volume). In the context ofthe Gaia mission, two different formulationsof relativistic light propagation have beendeveloped to model astrometric observations

1038 Lattanzi: Astrometric cosmology

Table 1. Gaia End-of-life averaged astrometric performance; Req.= Required and Perf. = per-formance, as estimated after the payload acceptance review (courtesy of Jos de Bruijne).

B1 V G2 V M6 Vµas Req. Perf. Req. Perf. Req. Perf.

V < 10 mag < 7 8.4 < 7 8.6 < 7 10.6V = 15 mag < 25 26.3 < 24 24.4 < 12 9.4V = 20 mag < 300 328.7 < 300 292.8 < 100 97.7

of distant sources by a SS observer: the GREMformulation, the well known coordinate basedapproach described by Klioner (2003 andreferences therein), and the RAMOD model,the approach fully compliant with the preceptsof local measurement in a relativistic settingdiscussed by Crosta (2011, and referencestherein). Their theoretical equivalence, tothe 1-µas accuracy level suitable for Gaia,has been recently demonstrated (Crosta andVecchiato 2010, Crosta 2011)8 and will beexploited, in a process called, in the Gaiajargon, Astrometric Verification (or AVU;see Lattanzi et al. 2006), by comparing theresults of two fully independent astrometricreconstructions of the celestial sphere toassess all-sky scientific reliability on positions,including parallax, and proper motions.However, what I wish to emphasize hereis that these recent investigations in lightpropagation and direction measurement havefinally provided what I like to call the basicequation of relativistic astrometry:

l(i) = n(i)(1 − h00

2) + O(

v4

c4 ) , (1)

where h00 = 2U/c2, which is Eq. (25) in Crostaand Vecchiato (2010). This equation providesthe components of the spatial light directionl(i) in terms of its Euclidean counterparts n(i)

(used in classical astrometry) at the observer(satellite) location in the gravitational field of

8 This is one of the most important results ofthe Gaia Data Processing and Analysis Consortium(DPAC) collaboration in the field of relativistic as-trometry

the solar system (h00). l is the direction for thelocal barycentric observer, i.e., for the observerwhose frame is at rest relative to, and its axeshave the same orientation as, that at the SSbarycenter. With reference to Fig. 2, the actual(observed) direction in GR, i.e. the analogueof vector S , the aberrated direction seen at thetime of observation by a classically modelledobserver on the moving satellite (SRS)9, is ob-tained by a simple, Minkowskian, boost trans-formation (see Crosta and Vecchiato, 2010).The generalization to the full problem, i.e., tothe solution of the photon trajectory is given inCrosta (2011).

Fig. 2. The vectors representing the light directionin the pM/pN approaches inside the near-zone of thesolar system.

9 At the instant of observation the origin of thelocal barycentric frame coincides with that of themoving observer (satellite)

Lattanzi: Astrometric cosmology 1039

3. Astrometry is “local” anyway!

The µas accuracy achievable with space–borneastrometry allows astronomers to reach thekilo–parsec (kpc) scale of the Milky Way, yetthis is not enough to probe directly the Mega–parsecs of extragalactic distances (see Tab. 2);this would require angular accuracies in thenano–arcsec regime, beyond today’s and, prob-ably, near future technology. Therefore, de-spite its tremendous improvements over thelast very few decades, space astrometry contin-ues to share one key trait with its ground-basedtraditions: individual distances or, better, ”insitu” investigations can only access the localuniverse.

On the other hand, improvements in cos-mological models have recently producedquantitative predictions on the present–dayconsequences of the evolution of the Universe,something that is often referred to as LocalCosmology (LC).

These “cosmological consequences” couldmanifest themselves as characteristic signa-tures in the main constituents of the MW (halo,disk) or small perturbations in the gravity inaction within our SS. As we will see, some ofthese perturbations are well within the reach ofGaia’s astrometry. Therefore, astrometry can,once again, contribute direct and model inde-pendent tests not just of astrophysics, but, thistime, of cosmology!

4. Cosmology at the local scale

Is it really possible to investigate through“local measurements” on the nature of theUniverse? Does LC really exists?

4.1. Fossils in the Milky Way

To find out about signatures at z = 0, one has toturn to predictions from the most sophisticatedcosmological simulations, which are built fol-lowing the adoption of a universe model.

Among the available universe models thestandard (cosmological) model (SCM), or con-cordance model, is the one that has convimc-ingly explained the tiny (in amplitude) cosmicmicrowave background (CMB) fluctuations as

observed by the WMAP satellite and otherCMB experiments (Seife 2003).

Therefore we looked at the predictions ofavailable simulations based on such a model,i.e., a flat universe whose total relative den-sity (Ω =1) is dominated, ΩΛ ' 0.70, bythe dark energy, or Λ, component, while thepart contributed by matter, Ωm, is mostly madeof cold dark (CD) particles (25%), leaving ameager ' 5% to ordinary (baryonic) matter(Sawangwit and Shanks 2010).

The paradigm of these Lambda Cold DarkMatter (ΛCDM) models, in keeping with the“merger tree” of Lacey and Cole (1993), isthat smaller DM haloes merge at higher red-shifts (z ≥ 3) to form the large structures ob-served today (z = 0). This could be the way thehalo of a massive spiral like the MW formed.The key fact here is that the imprint of thesemerger events remain as conspicuous “fossil”signatures in the phase space of the MW halohas shown in Fig. 3. The Gaia satellite hasbeen conceived as the ultimate phase-spacemachine: by measuring distance and velocityof individual stars to sufficient accuracy andwithin e few kpc from the Sun, the satelliteis required to reveal and characterize in-situthe patterns emerging form the halo accretionhistory shown in Fig. 3, i.e. map the stars be-longing to the different “strings”, thus accu-rately testing present-time predictions of theΛCDM Universe. In fact, the Gaia end-of-lifeperformance table presented in sec. 2.2 tellsthat a 10% error on distance is still achievedat 10 kpc from the Sun (that is the galacto-centric distance from 10 to 20 kpc in Fig. 3);this confirms that Gaia will examine with un-precedented clarity, through its individual mea-surements of the proper tracers, the fine six-dimensional structure of the inner halo, i.e.,within 3 ÷ 4 kpc of the Sun location.

The halo is not the only MW structuralcomponent with remnants of, or clues to, theformation and early evolution of our Galaxy.Within a distance envelope similar to that ofthe inner halo, i.e. for distances from the galac-tic plane in the range 1 ÷ 3 kpc, Spagna,Lattanzi et al. (2010) found evidence of a cor-relation between (galactocentric) circular ve-locity (Vφ) and metallicity ([Fe/H]) with an

1040 Lattanzi: Astrometric cosmology

Table 2. The local Universe. Distances to some of the closest and best known sources outside theMW. The parallax error indicated, set to 10% of the corresponding parallax (p), is the minimumrequired for astrometric, in situ, access to the scale of the Universe just around our Galaxy. Thenano-arcsecond (nas ≡ nano-arcsec, i.e., 10−9 arcsec, or ∼ 4 × 10−14rad) is required to reachastrometrically the closest quasars

Object Sky direction redshift distance σp (10% p) Comment(z) (Mpc) (µas)

LMC Dorado/Mensa 0 ∼ 0.1 ∼ 1M31 Andromeda 0 ∼ 0.8 ∼ 0.1M33 Triangulum 0 ∼ 0.9 0.1M81 UMa ∼ 0 ∼ 4 ∼ 0.03M77 (NGC1068) Cetus ∼ 0.01 ∼ 19 ∼ 0.01 Seyfert 2 prototype3C405 Cygnus 0.04 ∼ 150 ≤ 1 nas Closest quasar.3C273 Virgo 0.2 ∼ 749 Optically brightest quasars

in the sky (at V' 12.9).

estimated gradient of ∂Vφ/∂[Fe/H] ' 40 ÷50 km/sec/dex derived from a sample of about27,000 stars with metallicity -1 < [Fe/H] < -0.5 dex that was adopted as an unbiased tracerof the thick disk. In a recent paper Curir,Lattanzi et al. (2012) explained, through ex-tensive numerical simulations, that this strongrotation-metallicity relation can be the resultof: (i) the natural dynamical evolution (associ-ated to heating and radial migration) of a long-lived disk population of main sequence dwarfstars in the gravitational field of a massiveMW-like DM halo consistent with a ΛCDMmodel; and (ii) the presence of a cosmologi-cally plausible radial metallicity gradient withlower metallicity in the inner regions (i.e.,an “inverse”, positive, gradient) in the earlyGalaxy. Indeed, a positive rotation-metallicitycorrelation , in all similar to that observed bySpagna, Lattanzi et al. (2010), develops ratherquickly without the need of any merging eventsand remains stable over several Giga years toan age compatible with that of the observeddisk population (∼ 10 Gyr).That an inverse (positive) chemical gradient,whose fossil manifestation shows up in today’sthick disk, could exist in the early MW isproved by the findings of Cresci et al. (2010),who studied a sample of distant galaxies atredshift z ≥ 3 and found evidence for an in-

verse metallicity gradient, possibly producedby the accretion of primordial gas. If we uti-lize the formula for the Einstein-de Sitter (flat)universe

1 + z = (t0te

)2/3 , (2)

(Shu 1982, p.378), then the “emission” time teof the photons at z = 3 is ∼ 2 Gyr given thatthe epoch of observation (present time)10 t0 is≈ 10 Gyr; this is consistent with the epoch atwhich Curir, Lattanzi et al. (2012) injected thepositive chemical gradient in their purely N-body simulation.Although the Spagna, Lattanzi et al. (2010)discovery has been confirmed by several inde-pendent studies (see Curir, Lattanzi et al. 2012for a review), the significant errors that afflictthe extant, limited, data on which these stud-ies are based are still the main limitation to adetailed theoretical understanding of the struc-tural and chemical history of the Galaxy’s thickdisk, although indications are already there thatthe ΛCDM model seems, at best, to be able toaccount for only part of the story that made theMW what it is today. This is something Gaia isin a unique position to settle once and for all,

10 In the same universe t0 = (2/3)×H0−1, i.e., ∼ 10

Gyr for a Hubble constant of H0 = 70 km/sec/Mpc

Lattanzi: Astrometric cosmology 1041

as anticipated in Re Fiorentin et al. (2012, andreferences therein).

Fig. 3. Milky Way reduced phase space (galacto-centric line-of-sight velocity component,RVgc, vsdistance from the galactic center, Rgc) showingthe highly structured fossile features (“streams” or“strings”) resulting from a ΛCDM simulation of 100dwarf galaxies merging with the MW DM halo. Thediffrent strings-like structures clearly differentiatethe individual merging events.

4.2. Quasar astrometry

Why are quasars mentioned here? It was justconcluded that the closest quasar is so far awaythat even nano-arcsec astrometry could not ’lo-cate’ it to the required (better than 10%) accu-racy. The answer is in the fact that cosmologyis usually associated not with large distances,but rather with expansion, i.e., a cosmologicalvelocity. Now, let us follow the suggestion, the“temptation”, that the cosmological velocity isnot just pure recession (spherical expansion)but is more general in nature admitting a trans-verse component, i.e., “perpendicular” to thetraditional line-of-sight component, measuredthrough spectra and used in the Hubble lawVrec = H0 × d, with the proper distance d ex-pressed in Mpc when H0 is in km/sec/Mpc. Ifsuch a rotational (tangential) component to thecosmic expansion exists, then it would have todo with the appearance of proper motions ofthe distant quasars through the conversion rela-tion Vrot ∝ µ×d, with Vrot in km/sec, µ the cor-

responding cosmic proper motion, in arcsec/yr,and d usually in pc.This is the kind of astrometric cosmology men-tioned by Eubanks (1991); he reconsidered ra-dio interferometric quasar astrometry for use incosmology thanks to the fact that, back then,VLBI was already able to pin point quasarsources to z∼ 3 with sub-mas precision in a sin-gle observing session.Therefore, if we admit that recession and ro-tational components contribute equally to thecosmic expansion, then we can set the scalarrelation Vrec ' Vrot, that does not dependon distance and provides the means to eval-uate the level of cosmological proper motiongiven the current estimates for H0. SettingH0 = 75 km/sec/Mpc one has µ ' H0/(5 ×106) ' 15 µas/yr, with the effect increasing forlarger values of the Hubble constant.According to Table 1, this number is certainlywithin the capability of the Gaia mission, oncewe factor in the large number of quasars thatthe satellite will observe up to redshift z < 3.And the suggestion is there that the previousrelation can be inverted for an independent es-timate of H0 from quasar’s all-sky astrometry,the only serious limitation being the actual ac-curacy (as opposed to precision) of the mea-sured proper motions.

Finally, it is worth emphasizing that onlythe most astrometrically stable quasars couldbe used in this kind of cosmology work andtheir existence, i.e. the precise characterizationof their astrometric history (extensively dis-cussed at this meeting), might enable anotherlong sought for discovery, that of fossil gravita-tional waves, echoes of catastrophic (energetic)events reaching us from the deep Universe.

4.3. What if gravity deviates from GR?

If GR is the correct description of gravity, thanordinary matter is just a small portion of what“there is” in the Universe. The concordancemodel rules, GR is valid everywhere, and acosmological constant is associated to the ex-otic components of dark matter and dark en-ergy, as mentioned in sec. 4.1.

The small (∼ 10−5) amplitude fluctuationsmeasured by the WMAP satellite in the CMB

1042 Lattanzi: Astrometric cosmology

temperature 11 are thought to be the seeds thatled to the formation of larger and larger struc-tures and eventually to galaxies through themerger tree paradigm discussed in sec. 4.1 inthe context of the ΛCDM model. However,the physical explanation for the origin of suchCMB variations seems to elude GR. For, thoseripples could be the result of fluctuations in ascalar field (see below) that drove inflation, i.e.,the phase of accelerated expansion in the evo-lution of the Universe just before the afterglowlight pattern that we observe as CMB today.Therefore, there might have been an epoch inthe early Universe when gravity was not fol-lowing GR. This of course does not directlychallenge Einstein’s theory supporting SCM,today unquestionably the best in describing theUniverse that emerged afterward.Yet, the existence of large galaxies (possiblythe largest) and clusters of galaxies at high red-shift (Sawangwit & Shanks 2010), and the ap-parent lack of, or less than anticipated, merg-ing activity in the formation of the MW haloand thick disk (sec. 4.1), are mounting astro-physical evidence against the ΛCDM model.Also, the same accelerated expansion observedat an epoch (0.2 . z . 0.6) much more re-cent than CMB, through Type Ia Supernovae(Riess et al. 1998), has required the introduc-tion of exotic matter terms, including dark en-ergy responsible for repulsive gravity, in orderfor GR to hold. Might it be, then, that cos-mological (CMB) and extragalactic observa-tions are probing the breakdown of GR at largescales?

The SCM new components of the universe,i.e., the new sources of gravity, can be dealtwith through the use of the modified field equa-tions Einstein introduced himself, i.e.,

Gµν + Λ gµν =8πGc4 Tµν , (3)

11 These fluctuations in the microwave sky aresmall in amplitude but quite conspicuous in angu-lar size reaching the 1, or twice the size of the fullMoon (Sawangwit & Shanks 2010, and referencestherein)

where Gµν is the Einstein tensor12, gµν isthe metric tensor introduced before, Λ the cos-mologial constant, and Tµν the stress-energy(or energy-momentum) tensor representing thegravity sources. Tµν can ’absorb’ the dark mat-ter component of the SCM universe and, inthis same context, the term in Λ is the same(to within a proportionality factor) as an intrin-sic energy density of the vacuum; as such, it iscommonly moved onto the right-hand side ofEq. 3 becoming part of the field sources. For, ifthis kind of energy density is positive, the as-sociated negative pressure can drive an accel-erated expansion of the universe, as observed.

Now, while waiting to unravel the physicalnature of those new constituents dominating(over regular - baryonic - matter) the SCMuniverse13, it could well be, using similararguments, that it is the geometric terms on theleft-hand side of Eq. 3 that need modification.Different possibilities for modifications ofgravity have been put forth that can contributeboth at galactic (e.g., galaxy rotational curves)and beyond galactic (i.e., cosmological) scales,and they all challenge GR and seek validationthrough the same observational data that haveso far corroborated the success of the GRsupported SCM.

The subject of modified gravity theorieshas received quite a lot of attention at this con-ference(see Bertolami and Capozziello in thisvolume) in view of the special role that µasastrometric observations can have in testinggravity at the local scale, adding and comple-menting to the data available, or soon to be-come available, at the galactic, extragalactic,and cosmological scales.

The f(R) gravity theories have the poten-tial to realize realistic cosmology (cosmologi-cal acceleration), galactic dynamics, etc. with-out non-baryonic dark matter and dark en-ergy. They emerge from the consideration thatthere is no fundamental reason for the Ricci

12 Gµν ≡ Rµν − 12 R gµν, Rµν is the Ricci tensor and

R the Ricci scalar or scalar curvature.13 The burden is indeed on the Standard Model of

particle physics to break the degeneracy and find theexotic particles (e.g., the WIMPS) in the lab!

Lattanzi: Astrometric cosmology 1043

scalar to appear in its simplest (linear) form,i.e. f ′(R) = 0, in the field equations of GR(Capozziello & Faraoni, 2011). This is perhapswhat makes f(R) gravity appealing, it goes be-yond GR without giving up its fundamentalstructure. This generalization of the scalar met-ric R can then be constrained by, e.g., astromet-ric measurements from within the SS throughexpressions like

γPPNR − 1 =

−[f ′′(R)]2

f ′(R) + 2[f ′′(R)]2 , (4)

relating the first and second derivatives off(R) to the deviation of the PPN parameter γfrom its GR (=1) value (Capozziello & Troisi,2005). Such a deviation can be measured asan extra astrometric component to the rela-tivistic direction of distant objects as seen bya SS observer like the Gaia satellite. Similardeviations are predicted by scalar-tensor the-ories (e.g., Fuji & Maeda, 2003; Ferreira &Starkman 2009, and references therein), an-other class of alternatives to GR capable of aunified picture (i.e. from the cosmological tothe local scale).Therefore the measurement of the PPN mainparameters can be used to test at the very localscale, i.e. within the ’cosmological bubble’ ofthe MW at z = 0, these modified gravities, pro-viding local verifications to cosmological con-straints. The possibility to execute these testswith Gaia is discussed in the next section.

4.4. Cosmology from within the SolarSystem

As mentioned earlier, the ripples present in theCMB observed distribution could be the resultof fluctuations in a scalar field that drove infla-tion, and this is the view of the scalar-tensortheories (see, e.g., Damour and Nordtvedt,1993). There is the possibility that this infla-tion field, which couples with gravity, fadeswith time. Today (z=0), its residue would man-ifest itself through very small deviations fromEinstein’s GR. Astrometric observations canbe the means to trace back the presence of thisscalar field, through accurate measurements ofdeflection of the light coming from bright and

angularly intrinsically stable sources. The de-viation (γPPN − 1) predicted in Damour, Piazza& Veneziano (2002) is particularly tiny, inthe range 10−8 ÷ 10−5 (see also Vecchiato etal. 2003, and references therein). The largervalue has already been excluded by the resultsbased on radio links to the Cassini spacecraft(Bertotti et al. 2003). Vecchiato, Lattanzi et al.(2003) showed that the Gaia nominal missioncould measure γ, i.e. its deviation from unity,to ∼ 10−7 (1 σ) after 5 years of continuous ob-servations, and using a subset of approximately106 stars chosen as the most astrometricallystable among the millions available in the mag-nitude range V ≤ 12 of the GAIA survey.Those results, scaled to the predicted per-formance of the as-built astrometric payload(Table 1), brings the error on γ closer to 10−6.This number implies that GAIA will certainlyextend by more than an order of magnitudethe chance of probing possible local deviationsfrom GR, yet significantly away from the lowerbound of the interval above. Unquestionablythen, closing the gap with the local effects pre-dicted by a scalar-tensor gravity cosmology re-quires the 1 µas accuracy level, therefore estab-lishing the need to go beyond Gaia, possiblytoward an astrometric mission fully dedicatedto the physics of gravitation.

What if not even the smallest predicteddeviations from GR are detected at the localscale? It could be evidence not just of prob-lems with scalar-tensor theories over GR, asGR appears to be inadequate at epochs closerto the Big Bang, but that ’reality’ is more com-plicated and that is why we will not know untilgravitation and quantum mechanics are recon-ciled (brought together). Or, until a completelydifferent view of the physical world (perhapsGR and QM cannot be reconciled!) will comeinto play.

5. Micro-arcsec precsion and beyond

The previous sections have definitively andquantitatively established the realm of LC;they have also made clear that LC wouldgreatly benefit from actually reaching the 1-µasaccuracy level, already exceeding Gaia capa-

1044 Lattanzi: Astrometric cosmology

bilities in certain circumstances, like the exper-iments from within the SS.

Cosmology or fundamental physics are notthe only reasons for reaching the µas and sub-µas astrometric precision. The detailed charac-terization (i.e., accurate ephemeris) of rockyand down to Earth-mass planets for atmo-spheric analysis and, in a non-too-distant fu-ture, imaging of those systems more likely tobe similar to our own requires 1-µas singleepoch measurements and possibly better: giventhat the reflex motion induced by Jupiter onour Sun, when seen by an observer at 10 pc,amounts to 500 µas, the pull due to Earth fromthe same distance scales as the product of theplanet mass ratio (∼1/300) times the variationof the orbital sizes (∼1/5.2) to 0.3 µas!

It is therefore time we ask ourselves thequestion: what is a “micro-arc-second”?

Unlike resolution in imaging astronomy orinterferometry, astrometric accuracy, or astro-metric resolution, can be understood as theability to reconstruct, or maintain, the line-of-sight of a given imaging system to within thedesired angular accuracy, so that any changesin direction of the light coming from a dis-tant source is intrinsic to the source and not tothe photon collecting (and recording) device,thus available for further astrophysical investi-gations.

Fig. 4 is an attempt at “materializing” theconcept of one-micro-arcsecond astrometricaccuracy. It shows the sketch of a 2-m flat mir-ror, roughly the size of the two Gaia’s pri-maries, with the vector normal to the surfacerepresenting its line-of-sight. A change of ∼1µas, i.e. ∼ 5 pico-radians, in the line-of-sightdirection translates into a mirror displacement,actually the quantity that makes sense metro-logically, of ∼ 5 pico-meters, 100 times smallerthan the atomic dimensions!

Is the pico-meter level of stabilization ac-tually accessible?There are two possibilities for lowering in-strumental line-of-sight jitter of large payloads(say, a ’cube’ of ≈ 4 m on a side, like in thecase of Gaia): passive and active stabilization.

Active stabilization to the pico-meter levelwas the first to be proven in the laboratoryboth in Europe, as part of a technology en-

Fig. 4. Visual representation of the µarcsec as-trometric accuracy, i.e., of the line-of-sight stabi-lization/reconstruction, in terms of linear quantitiesroughly representing the typical size of optical pay-loads of satellites like Gaia (main mirror size ∼ 2m).

abling program funded by ESA in the contextof the development phase of the Gaia program(Bertinetto & Canuto 2001), and in the US, bythe outstanding work of Shao’s JPL team de-signing and experimenting technology for thedifferent variants of a (sub-) µas space interfer-ometry mission (Shao 2006; see also Unwin,Shao et al. 2008, and references therein).However, active stabilization requires spatial-ization of sophisticated technology needingreal time control logics (like, e.g., servo loops)to be operational for the relatively long timespans usually employed by astrometry (to re-solve time dependent quantities, i.e., parallaxand space velocities). This considerably risesthe risk level, let alone costs, of already chal-lenging programs.That is why passive means have been preferredso far, like in the case of the Gaia satellite (pay-load and service module). Nevertheless, pas-sive technology has required its own devel-opments and validations, relying on the prop-erties of new materials (SiC was selected forGaia) produced and assembled in structurallyhomogeneous payloads, and enclosed in envi-ronments tens of cubic meters in volume tobe kept thermally quite everywhere, again pas-sively, to better than 10−5 K, a daunting chal-lenge in itself.

Lattanzi: Astrometric cosmology 1045

Finally, the two space instrumentationbuilding methods considered for reducing in-strumental jitter also differ in their notion ofobserving strategies and systematic errors con-trol.

Active stabilization schemes emphasize theuse of specific on-board technology for con-tinuous instrumental calibration and monitor-ing, thus endowing astrometry with the capa-bility to reach 1 µas in a single epoch measure-ment (time resolved µas astrometry) and, con-sequently, that to average multiple epoch mea-surements to well below 1 µas 14.

In a pure passive stabilization strategy, be-sides the passive thermal control, it is the astro-metric payload itself that “does it all” duringscience operations, being the only hardwaredesigned for high precision measurements;also, instrumental design minimizes systemat-ics. The required error on the program objectsis reached only at the end of the on-ground dataprocessing, when all of the elementary expo-sures are assembled together.Also, the same high precision measurementson the detected sources (stars or star-like ob-jected) are utilized for assessing and, possi-bly, modelling the residual systematic biasesthrough what is called, in the case of a spinningsatellite akin Gaia, the closure condition: as thesatellite re-observes the same sky regions af-ter completing one revolution some hours later,any changes in direction over this time inter-val for the most astrophysically stable sourcesin the field cannot be due to parallax effectsor intrinsic motions, they must be attributed totiny changes within the instrument. Thus, thefine instrumental behavior, and therefore sys-tematic errors, can be accounted for by addingto the astrometric equations15 a part describing

14 The presence of extra hardware does not nec-essarily mean an active payload, i.e., extra movingparts and closed control loops. Calibration and mon-itoring measurements, for reducing systematic er-rors, can be taken at the same time as science ob-servations and later reduced on the ground.

15 These are the relativistic equations relating thesatellite epoch observations to the evolution of stel-lar positions with time due to source finite distancesand space motions, i.e., the unknown astrophysi-cal quantities to be determined via inversion (in the

instrumental evolution based, e.g., on the bestpossible physical characterization of the instru-mentation itself. Note that a similar schemecan be used to model residual, unknown, satel-lite attitude deviations from nominal, whichwould otherwise degrade the final astrometricerror budget (Vecchiato 2012, and referencestherein).It should now be clear why systems like theseare named self calibrating instruments and, inHipparcos, they helped overcoming technolog-ical limitations of the time. However, whenunnecessary, self calibrating instruments carrythe potential drawback, besides the need ofsubstantially increasing their number, of in-troducing pernicious correlations among theastronomical and instrumental parameters uti-lized in the model equations describing the ob-servations.

Active or passive designs apart, somethingsimilar to what happened with the Hipparcosmission (when technology took optical astrom-etry into space delivering milli-arcsec preci-sion) must repeat itself: it is once again tech-nology that can push space astrometry to accu-racy levels unthinkable just a few years back.

5.1. Beyond Gaia

The pure passive approach adopted for theGaia payload and its service module is al-lowing space astrometry to push toward the10 µas accuracy level, 100 times better thanHipparcos. As we have seen from the consider-ations elaborated in the previous sections, thisis certainly sufficient to open the field of as-trometric cosmology, however it represents aserious limitation to, e.g., astrometric (i.e., di-rect) cosmology and fundamental physics ex-periments at optical wavelengths from withinour SS (sec. 4.4). The use of stars or, morein general, astrophysical sources for instru-ment self calibration cannot support require-ments on astrometric resolutions approachingthe 1-µas level (Makarov et al. 2010 and ref-erences therein). Luckily enough, technologyexists (Bertinetto & Canuto 2001; Zhai et al.

least-squares sense) of the system of observationequations.

1046 Lattanzi: Astrometric cosmology

2011) that can, once put on board, disentanglefine instrumental behavior from intrinsic, andtherefore astrophysically rewarding, astromet-ric changes of the program sources, thus allow-ing for the extra push forward!

In a far reaching conference like thisGREAT workshop, with the Gaia launch fastapproaching, inevitably the very future of as-trometry after Gaia is put into question.There is indeed a huge science gap left open byGaia’s 10-µas astrometry, e.g.:

1. (near)Earth-like extrasolar planets andtheir ephemeris call for truly µas (or better)single-exposure accuracy;

2. Local Cosmology will undergo a revolu-tion if objects could be “pin pointed” to10%, or better, of their positions and ve-locities up to a scale of 100 Kpc (∼ 10times further than the ∼ 5 kpc covered withGaia);

3. Fundamental physics would see gravita-tional theories astrometrically tested and/orupper limits placed on them to unprece-dented levels (e.g., PPN γ down to 10−8).

Scientists and engineers will have to dis-cuss again the tenets of space astrometry: isall-sky coverage always needed16 (e.g. accu-rate reference frame)? Is a survey required (e.g.completeness)?Can differential astrometry, andtherefore ultimate systematic error control, dothe job or is absolute astrometry (i.e., abso-lute parallaxes) a necessity? Is long operations(several years) required?In defining scientific goals, designers mightfind out that multiple, smaller missions mightbe better, i.e. more scientific rewarding, andcheaper than multipurpose larger facilities.

Finally, are people actively thinking of anafter Gaia? Two designs for astrometric mis-sions where submitted to ESA in response totheir latest call for medium class missions!One, the GAME mission, was discussed at thismeeting and is primarily dedicated to funda-mental physics (see Gai, this volume; see alsoGai et al. 2012); the other, NEAT, concen-trates on the sub-µas astrometric characteriza-

16 To be distinguished from all/sky accessibilitythat should always be provided.

tion of Earth-like extrasolar planets (Malbet etal. 2012).

Hopefully, ESA will not wait for too longbefore initiating dedicated, both scientific andtechnological, assessment studies; and NASAwill reconsider sooner than not their positionon US astrometric missions, and to strengthensynergies and collaborations. That the scien-tific community is ready is crystal clear!

6. Conclusions

The role of astrometry has been revampedthanks to technology that has provided accessto space and the possibility to implement theGaia concept. For this, over the next decadeor two we will know more of the real storyof dark matter (and dark energy) and the va-lidity of GR, i.e., we will be practicing a newbranch of fundamental astronomy: AstrometricCosmology!

The hope is that the actual geometry of theUniverse, which astrometry might help unveil-ing through tests from within the SS, will re-gain ordinary matter, the baryons of which weare made, some of its role that the story toldtoday by the concordance model (Seife 2003)assigns almost entirely to the mystery of darkmatter and dark energy.

From the technological standpoint, spaceappears once again as the key to the next break-through in space astrometry, the place for: (a)producing the first “extrasolar ephemerides” ofa distant planetary system SS alike, or (b) themost direct and extreme tests of matter’s lightbending properties to probe the validity of GR.A repetition of the 1919 experiment by Dyson,Eddington, and Davidson (1920), but this timeto challenge, with 21st century technology, thelast of classical theories, Einstein’s GeneralRelativity itself.

Acknowledgements. I am indebted to B.Bucciarelli, M. Crosta, F. de Felice, M. Gai,P. Re Fiorentin, A. Spagna, and A. Vecchiato fortheir precious help through the preparation of thiscontribution and for giving me access to material inadvance of publication. My gratitude to M. Crostafor clarifying concepts that would otherwise haveremained obscure to me.Finally, I dedicate this article to Prof. P.L. Bernacca

Lattanzi: Astrometric cosmology 1047

for his forward vision and dedication that providednew impetus to modern astrometry in Italy.Support for this work comes from the Italian SpaceAgency (ASI) under contract I/058/10/0.

References

Bertinetto, F. & Canuto, E. 2001, Optic.Engineer., 40(1), 76

Bertotti, B., Iess, L. & Tortora, P. 2003, Nature,425, 374

Capetti, A., Macchetto, F. & Lattanzi, M. G.1997, ApJ, 476, L67

Capozziello, S. & Faraoni, V. 2011, BeyondEinstein Gravity, Springer

Capozziello, S., Troisi A. 2005 Phys. Rev. D72, 044022

Caraveo, P., et al. 1998, A&A, 329, L1Cresci, et al. 2010, Nature, 467, 811Crosta, M. 2011, Class. Quantum Grav., 28,

235013Crosta, M. & Vecchiato, A. 2010, A&A509,

A37Crosta, M. & Mignard, F. 2006, Class.

Quantum Grav., 23, 4853Curir, A., et al. 2012, A&A, 545, A133Damour, T., & Nordtvedt, K. 1993, Phys. Rev.

Lett., 70, 2217Damour, T., Piazza, F., & Veneziano, G. 2002,

Phys. Rev. D, 66, 046007Dyson, F. W., Eddington, A. S., & Davidson,

C. 1920, Philo. Trans. of the Royal Societyof London, 220A, 291

ESA 1997, The Hipparcos and TychoCatalogues, ESA SP 1200

Eubanks, T. M. 1991, BAAS, 23, 1465Ferreira, P. G. & Starkman, G. D. 2009,

Science, 326, 812Gai, M., et al. 2012, Exp Astron 34,165180Gouda, N. 2011, JASMINE, Scholarpedia,

6(10), 12021Klioner, S. 2003, AJ, 125, 1580Kovalevsky, J. 1984, Space Science Reviews,

39, 1–63

Lacey, C. & Cole, S. 1993, MNRAS, 262, 627Lasker, B. et al. 2008, AJ, 136, 735Lattanzi, M. G., et al. 2006, Astrometric

Verification Unit, DPAC Tech. rep., GAIA-C3-TN-INAF-ML-001-2

Lattanzi, M. G. Capetti, A. & Macchetto, F.1997, A&A, 318, 997

Makarov, V. V., Parker, D. & Ulrich, R. K.2010, ApJ717, 1202

Fabien, M., et al 2012, Exp Astron 34,385413Monet, D. G., et al. 2003, AJ, 125, 984Perryman M. A. C. et al. 2001, A&A, 369, 339Re Firentin, P., et al. 2012, Societe Francais

d’Astronomie et d’Astrophysique (SF2A)2012, S. Boissier, P. de Laverny, N.Nardetto, R. Samadi, D. Valls-Gabaud andH. Wozniak (eds)

Riess, A. et al. 1998, AJ, 116, 1009Sawangwit, U. & Shanks, T. 2010, A&G 51

5.14Seife, C. 2003, Science, 302, 2038Shao, M. 2006, in Advances in Stellar

Interferometry, Proc. SPIE, 6268, 62681ZShu, F. H. 1982, The Physical Universe,

University Science Books, Mill Valley,California (USA)

Spagna, A., Lattanzi, M. G., Re Fiorentin, P. &Smart, R. L. 2010, A&A, 510, L4

Unwin, S.C., et al. 2008, PASP, 120, 38Vecchiato A. 2012, Putting Einstein to test:

astrometric experiments from space, fun-damental physics, and local cosmology. InAstrophysics, Prof. Ibrahim Kucuk (Ed.),InTech, ISBN:978-953-51-0473-5

Vecchiato, A., et al. & Gai, M. 2003, A&A399,337

Xiang-Qun Cui, et al. 2012, Res. Astron.Astrophys., 12, 1197

Zhai, C., Shao, M., Goullioud, R., Nemati,B. 2011, Micro-pixel accuracy centroiddisplacement estimation and detectorcalibration, Proc. R. Soc. A (2011).doi:10.1098/rspa.2011.0255