Gaia, the PPN-Γ Parameter and Scalar-Tensor Cosmologies a Quick Overview

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Gaia, the Global Sphere Reconstruction and GR The PPN-γ and scalar-tensor cosmologies Gaia, the PPN-γ and limits on scalar-tensor cosmologies

Gaia, the PPN-γ parameter and scalar-tensor cosmologies A quick overview

A. Vecchiato1

1INAF - Astrophysical Observatory of Torino

PONT 2017 Avignon, April 24, 2017

A. Vecchiato Gaia and scalar-tensor cosmologies Gaia, the Global Sphere Reconstruction and GR The PPN-γ and scalar-tensor cosmologies Gaia, the PPN-γ and limits on scalar-tensor cosmologies Outline

1 Gaia, the Global Sphere Reconstruction and General Relativity Astrometry and gravity Gaia and the global sphere reconstruction

2 The PPN-γ and scalar-tensor cosmologies Scalar-tensor theories and cosmologies The PPN Framework PPN-γ and scalar-tensor theories

3 Gaia, the PPN-γ and limits on scalar-tensor cosmologies Constraining scalar-tensor cosmologies with Gaia Problems and pitfalls

A. Vecchiato Gaia and scalar-tensor cosmologies Gaia, the Global Sphere Reconstruction and GR Astrometry and gravity The PPN- and scalar-tensor cosmologies γ Gaia and the global sphere reconstruction Gaia, the PPN-γ and limits on scalar-tensor cosmologies Outline

1 Gaia, the Global Sphere Reconstruction and General Relativity Astrometry and gravity Gaia and the global sphere reconstruction

2 The PPN-γ and scalar-tensor cosmologies Scalar-tensor theories and cosmologies The PPN Framework PPN-γ and scalar-tensor theories

3 Gaia, the PPN-γ and limits on scalar-tensor cosmologies Constraining scalar-tensor cosmologies with Gaia Problems and pitfalls

First experimental evidences of General Relativity came from Astrometry.

One can thus hope that the Thus the results of the discussion of the observations expeditions to Sobral and of Mercury simply confirm Principe can leave little doubt previous research. Now this is that a deflection of light takes not negligible: we see here place in the neighbourhood of that the approximately 3-fold the Sun and that it is of the secular movement of the amount demanded by eccentricity, added to the Einstein’s generalised theory of secular movement of the relativity, as attributable to the perihelion, gives a sum in sun’s gravitational field. which the observations are (Dyson, Eddington and greater by 3900 than those Davidson, 1920) which result from calculation. (Le Verrier, 1859)

Body δαM (µas) δαQ (µas)

Sun 1.75 106 1 101 × ∼ Mercury 83 Sun Jupiter 100 Saturn Uranus Venus 493 Neptune 10 µas 10 1 1 µas Earth 574 0.6 − (mas) 2 Moon 26 δα 10−

Mars 116 0.2 3 10− Jupiter 16270 240 10 4 Saturn 5780 95 −

Uranus 2080 8 0 10 20 30 40 50 α (deg) Neptune 2533 10

1 Gaia, the Global Sphere Reconstruction and General Relativity Astrometry and gravity Gaia and the global sphere reconstruction

2 The PPN-γ and scalar-tensor cosmologies Scalar-tensor theories and cosmologies The PPN Framework PPN-γ and scalar-tensor theories

3 Gaia, the PPN-γ and limits on scalar-tensor cosmologies Constraining scalar-tensor cosmologies with Gaia Problems and pitfalls

What: ESA satellite, orbiting around the Sun-Earth L2 point (1.5 106 km from Earth) When: launched Dec 19th, 2013, 5yrs operations ·

Why: 6D map (positions and velocities) of 109 Milky Way objects @∼ 10 100 µas accuracy −

What: ESA satellite, orbiting around the Sun-Earth L2 point (1.5 106 km from Earth) When: launched Dec 19th, 2013, 5yrs operations ·

Why: 6D map (positions and velocities) of 109 Milky Way objects @∼ 10 100 µas accuracy − How: scanning mode measurements (as opposed to “step-and-stare” mode)

What: ESA satellite, orbiting around the Sun-Earth L2 point (1.5 106 km from Earth) When: launched Dec 19th, 2013, 5yrs operations ·

Why: 6D map (positions and velocities) of 109 Milky Way objects @∼ 10 100 µas accuracy −

How: scanning mode measurements (as opposed to “step-and-stare” mode) combination of the three independent motions of the scanning law allows for a complete coverage of the sky every 6 months

Create a “geodetic” network of measurements

N = 5 ∗

Nunk = 10

Narcs = 10

Network closed! Solve an Equation System

Observational errors ⇒ 1 solution in the least-squares sense; 2 overdetermined system of equations. S

Observational errors ⇒ 1 solution in the least-squares sense;

2 overdetermined system of S¯ equations. σS S

Observational errors ⇒ 1 solution in the least-squares sense;

2 overdetermined system of S¯ equations. σS S

8 Nunk N 10 ∼ ∗ ' 2 10 Nobs 10 Nunk 10 ∼ ∼

The Gaia basic observable is the abscissa φ between the x axis and one viewing direction ea r cosψ = ˆ · (1) (ˆa,r) r | | cosψ(ˆx,r) cosφ = q (2) 1 cos2 ψ − (ˆz,r)

Depends on the coordinates of one star(S) and on the satellite attitude(A) at the time of the observation The aberration enters in the same way as for the arcs

Equivalent of Eq. (1): the relativistic viewing direction and kν (null four-vectors) have to be Γ(s) projected in the spatial hypersurface relative to uµ the observer uµ

Tµν = gµν + uµ uν

µ ν µν Tµν E k T cosψ = aˆ (ˆa,k) p µ ν Tµν k k (3) It is possible to include one or more PPN parameters like γ (Global parameters)

In principle, each observation is a function of Astrometric (S), Attitude (A), Instrument (C), and Global (G) parameters.

ν ν k = k (α ,δ ,π , µα , µδ ,γ) ∗ ∗ ∗ ∗ ∗ ν ν (a) (a) (a) Eaˆ = Eaˆ σ1 ,σ2 ,σ3 ,γ

S A C G cosψ(ˆa,k) = Faˆ x ,x ,x ,x

Estimation of PPN-γ The dependence on γ gives the estimation of this parameter as a by-product of the sphere reconstruction

In principle, each observation is a function of Astrometric (S), Attitude (A), Instrument (C), and Global (G) parameters.

ν ν k = k (α ,δ ,π , µα , µδ ,γ) ∗ ∗ ∗ ∗ ∗ ν ν (a) (a) (a) Eaˆ = Eaˆ σ1 ,σ2 ,σ3 ,γ

S A C G cosψ(ˆa,k) = Faˆ x ,x ,x ,x

Estimation of PPN-γ The dependence on γ gives the estimation of this parameter as a by-product of the sphere reconstruction

A. Vecchiato Gaia and scalar-tensor cosmologies Gaia, the Global Sphere Reconstruction and GR Scalar-tensor theories and cosmologies The PPN-γ and scalar-tensor cosmologies The PPN Framework Gaia, the PPN-γ and limits on scalar-tensor cosmologies PPN-γ and scalar-tensor theories Outline

1 Gaia, the Global Sphere Reconstruction and General Relativity Astrometry and gravity Gaia and the global sphere reconstruction

2 The PPN-γ and scalar-tensor cosmologies Scalar-tensor theories and cosmologies The PPN Framework PPN-γ and scalar-tensor theories

3 Gaia, the PPN-γ and limits on scalar-tensor cosmologies Constraining scalar-tensor cosmologies with Gaia Problems and pitfalls

Scalar fields minimally or non-minimally coupled with gravity (metric tensor) arise naturally when attempting to Incorporate Mach’s Principle in gravity theories (e.g. Brans-Dicke theory) Include inflationary stages or Dark Energy-like effects in Cosmological models (Wang et a. 2016, PhLA 380, 3761 and references therein) Attempts to unify gravity with other interactions and/or to formulate quantum theories of gravity BD theory can be derived from Kaluza-Klein theories by compactification of extra dimensions Low-energy limit of some string theories is a BD-like theory where the scalar field is represented by the string dilaton

1 Mach’s Principle roughly means that in empty space mI = 0 2 Historically, Einstein included the Cosmological constant to incorporate Mach’s Principle in General Relativity, thus eliminating any trace of an absolute spacetime (believing that Λ > 0 in empty space g = 0) ⇒ αβ 3 It failed because GR requires G = mG/mI 4 Substituting the constant G with a variable coupling parameter G (xα ), namely a scalar ﬁeld φ, allows for a constant mG and a variable mI 1 LGR = √ g R 2Λ + LM Ψ,g , − 2κ − αβ 4 c ω (φ) µ LST = √ g φR ∂µ φ∂ φ 2Λ(φ) +LM Ψ,g . 16π − − φ − αβ

1 Gaia, the Global Sphere Reconstruction and General Relativity Astrometry and gravity Gaia and the global sphere reconstruction

2 The PPN-γ and scalar-tensor cosmologies Scalar-tensor theories and cosmologies The PPN Framework PPN-γ and scalar-tensor theories

3 Gaia, the PPN-γ and limits on scalar-tensor cosmologies Constraining scalar-tensor cosmologies with Gaia Problems and pitfalls

The Parametrized Post-Newtonian formalism is a framework which has been designed to describe the Post-Newtonian limit of all the metric theories of gravity by means of a set of 10 parameters. These parameters are associated to speciﬁc physical properties.

Parameter Meaning Parameter Meaning

How much space-curvature γ α3 produced by unit rest mass?

How much “nonlinearity” in the Violation of conservation of total β ζ1 superposition law for gravity? momentum?

ξ Prefered-location eﬀects? ζ2

α1 ζ3

α2 Prefered-frame eﬀects? ζ4

α3

Parameter Meaning Parameter Meaning

How much space-curvature γ α3 produced by unit rest mass?

How much “nonlinearity” in the Violation of conservation of total β ζ1 superposition law for gravity? momentum?

ξ Prefered-location eﬀects? ζ2

α1 ζ3

α2 Prefered-frame eﬀects? ζ4

α3

The Post-Newtonian limit of a single theory can be reconstructed by assigning speciﬁc values to each of the parameters, i.e. each theory is identiﬁed/characterized by the values assumed by these parameters.

PPN Parameters Theory γ β ξ α1 α2 α3 ζ1 ζ2 ζ3 ζ4

General Relativity 1 1 0 0 0 0 0 0 0 0 1+ω Brans-Dicke 2+ω 100000000 Bergmann-Wagoner- 1+ω 1 + Λ 00000000 Nordtvedt 2+ω

General Vector-Tensor γ0 β 0 0 α10 α20 0 0 0 0 0 2 1 2 Will-Nordtvedt 1 1 0 0 K / 1 + 2 K 0 0 0 0 0

Rosen 1 1 0 0 (c0/c1) 1 0 0 0 0 0 −

Lee-Lightmann-Ni ac0/c1 β 0 ξ 0 α10 α20 0 0 0 0 0

f (R) γ0 β 0 0 0 0 0 0 0 0 0

1 Gaia, the Global Sphere Reconstruction and General Relativity Astrometry and gravity Gaia and the global sphere reconstruction

2 The PPN-γ and scalar-tensor cosmologies Scalar-tensor theories and cosmologies The PPN Framework PPN-γ and scalar-tensor theories

3 Gaia, the PPN-γ and limits on scalar-tensor cosmologies Constraining scalar-tensor cosmologies with Gaia Problems and pitfalls

The estimation of the PPN γ parameter thus has precise theoretical and observational implications.

Fulﬁlling theoretical needs

It is the phenomenological “trace” of a scalar ﬁeld coupled with gravity which is related to:

theories fully compatible with the Mach principle; cosmological scenarios with inﬂationary stage and/or DE-like expansion; theories aiming to provide a formulation of a quantum theory of gravity.

Explaining observational evidences

It can provide constraints on theories which claim of being able to account for several astrophysical and cosmological problems without any need for DM or DE like, e.g.:

uniformity of CMB (inﬂuation) and/or acceleration of cosmological expansion (attributed to DE); galactic rotation curves, galaxy cluster masses and dynamics (attributed to DM); observational data from gravitational lensing; Tully-Fisher relation.

The estimation of the PPN γ parameter thus has precise theoretical and observational implications.

Fulﬁlling theoretical needs

It is the phenomenological “trace” of a scalar ﬁeld coupled with gravity which is related to:

theories fully compatible with the Mach principle; cosmological scenarios with inﬂationary stage and/or DE-like expansion; theories aiming to provide a formulation of a quantum theory of gravity.

Explaining observational evidences

It can provide constraints on theories which claim of being able to account for several astrophysical and cosmological problems without any need for DM or DE like, e.g.:

The estimation of the PPN γ parameter thus has precise theoretical and observational implications.

Fulﬁlling theoretical needs

It is the phenomenological “trace” of a scalar ﬁeld coupled with gravity which is related to:

theories fully compatible with the Mach principle; cosmological scenarios with inﬂationary stage and/or DE-like expansion; theories aiming to provide a formulation of a quantum theory of gravity.

Explaining observational evidences

It can provide constraints on theories which claim of being able to account for several astrophysical and cosmological problems without any need for DM or DE like, e.g.:

A. Vecchiato Gaia and scalar-tensor cosmologies Gaia, the Global Sphere Reconstruction and GR Constraining scalar-tensor cosmologies with Gaia The PPN- and scalar-tensor cosmologies γ Problems and pitfalls Gaia, the PPN-γ and limits on scalar-tensor cosmologies Outline

1 Gaia, the Global Sphere Reconstruction and General Relativity Astrometry and gravity Gaia and the global sphere reconstruction

2 The PPN-γ and scalar-tensor cosmologies Scalar-tensor theories and cosmologies The PPN Framework PPN-γ and scalar-tensor theories

3 Gaia, the PPN-γ and limits on scalar-tensor cosmologies Constraining scalar-tensor cosmologies with Gaia Problems and pitfalls

Present best results for γ

γ 1 lower Experiment Effect Technique | −bound| Light Global HIPPARCOS 3 10 3 deflection Astrometry · − Light Radio VLBI 4.5 10 4 deflection Interferometry · − Round-trip Shapiro Cassini travel time of 2.3 10 5 time delay − radar signals ·

References: Froeschlé et al. (1997), Shapiro et al. (2004), Bertotti et al. (2003).

The Gaia accuracy on γ can be roughly estimated with some simpliﬁed order-of-magnitude considerations: ﬁrst of all, for each observation, σ σ γ ∆α , γ ∼ ∆α

where ∆α is the light deflection effect and σ∆α the angular measurement error; on average, it is reasonable to assume that for Gaia, σ∆α 100 µas, ∼ 2 ∆α 5 mas for at least 1 million stars, which gives σγ 2 10− for each' observation; ∼ · each star will be observed about 400 times during the 5 years 6 mission, which yields a final σγ 10 . ∼ − Estimated accuracy This result is confirmed by current simulations, which suggest a 6 final accuracy of σγ 10 . ∼ − A. Vecchiato Gaia and scalar-tensor cosmologies Gaia, the Global Sphere Reconstruction and GR Constraining scalar-tensor cosmologies with Gaia The PPN- and scalar-tensor cosmologies γ Problems and pitfalls Gaia, the PPN-γ and limits on scalar-tensor cosmologies Estimated Accuracy on γ for Gaia

The Gaia accuracy on γ can be roughly estimated with some simpliﬁed order-of-magnitude considerations: ﬁrst of all, for each observation, σ σ γ ∆α , γ ∼ ∆α

The foreseen Gaia accuracy can help to put more stringent constraints on cosmological scenarios derived from scalar-tensor theories: The so-called runaway-dilaton scenario (Damour, Piazza, Veneziano 5 7 2002, PhRevD 66, 046007) predicts 10− . γ 1 . 10− at present time, depending on the specific choice of the| inflation− | potential. In the framework of the so-called Screened Modified Gravity (SMG), a dilaton scenario with a quadratic coupling potential (Zhang et al., 2016, PhRevD 93, 124003) gives the relation γ 1 = 4Φ(β 1), which implies γ 1 β 1 . This means that− the current− − constraints on|β− 1| from| Solar− | System ephemerides already give γ 1 6.6 10 −10, a level not affected by the Gaia capabilities. | − | . · − The chameleon mechanism (original or exponential) of the SMG can 12 be constrained to √8πGξφ∞ . 10− , namely one order of magnitude better than the present value (Zhang et al., 2016, cit.).

A class of quintessence models with a scalar ﬁeld non-minimally coupled with gravity is potentially able to explain both the inﬂationary and the late time accelerated expansion (Demianski et al. 2008, A&A 481, 279). This is characterized by a quadratic coupling function F (φ) = ξ (s)φ 2, where s is a real number and

(2s + 3)2 ξ (s) = . 48(s + 1)(s + 2)

The constraints on the PPN-γ parameter then translates into a constraint on s via the relation (γ 1) = 4ξ (s)/[1 + 8ξ (s)], which evaluates to 0 for s = 1.5.− − −

Current Cassini experiment results constrain s in the range [ 1.504, 1.496]. − − Gaia limit to 6 γ 1 . 10− would bring this| − range| to 1.50086 < s < 1.49914.− −

1 Gaia, the Global Sphere Reconstruction and General Relativity Astrometry and gravity Gaia and the global sphere reconstruction

2 The PPN-γ and scalar-tensor cosmologies Scalar-tensor theories and cosmologies The PPN Framework PPN-γ and scalar-tensor theories

3 Gaia, the PPN-γ and limits on scalar-tensor cosmologies Constraining scalar-tensor cosmologies with Gaia Problems and pitfalls

Correlations among: parallaxes Basic Angle variations (Γ) Parametrized Post-Newtonian parameter γ

Gaia can hope to provide an estimation of the PPN-γ parameter one order of magnitude better than the current best estimation as a by-product of the global astrometric sphere reconstruction, namely the procedure used to establish the global astrometric celestial reference frame. This parameter is related to relativistic scalar-tensor theories of gravity and cosmologies in several ways. The accuracy foreseen for Gaia is signiﬁcant from the cosmological point of view, since it can put to test scenarios like the dilaton-runaway or scalar-tensor cosmologies with both inﬂationary and accelerated expansion phases. Although appealing, this perspective have to be pursued with extreme care because of the potential pitfalls of the method used.

A. Vecchiato Gaia and scalar-tensor cosmologies