Search for the primordial gravitational waves with Very Long Baseline Interferometry
11 Search for the primordial gravitational waves with VeryLong Baseline Interferometry
Oleg TitovGeoscience Australia, PO Box 378, Canberra, ACT, AustraliaS´ebastien LambertSYRTE, Observatoire de Paris, PSL Research University, CNRS,Sorbonne Universit´es, UPMC Univ. Paris 06, LNEFebruary 15, 2016
Abstract
Some models of the expanding Universe predict that the astrometric proper motion of distantradio sources embedded in space-time are non-zero as the radial distance from observer to thesource grows. Systematic proper motion effects would produce a predictable quadrupole patternon the sky that could be detected using Very Long Baseline Interferometry (VLBI) technique. Thisquadrupole pattern can be interpreted either as an anisotropic Hubble expansion, or as a signatureof the primordial gravitational waves in the early Universe. We present our analysis of a large setof geodetic VLBI data spanning 1979–2015 to estimate the dipole and quadrupole harmonics in theexpansion of the vector field of the proper motions of quasars in the sky. The analysis is repeatedfor different redshift zones.
Very long baseline interferometry (VLBI) measures the differential arrival times of signals from extra-galactic radio sources. It is currently the most powerful technique for measuring absolute positionsof thousands of radio sources, the orientation and rotation speed of the Earth and ground-based sta-tion coordinates, with an accuracy of about 1 cm (or 0.1 mas). VLBI has been extensively used forastrometry and geodesy for about 30 years and, since 1998, is operated by the International VLBIService (IVS, Schuh & Behrend 2012). VLBI allows an astrometric precision of ∼ µ as (Fey et al.2015).A dipole systematic caused by the galactocentric acceleration of the Solar System Barycentre wasdetected for the first time in VLBI data by Titov et al. (2011, 2013). The measured amplitude ofthe aberration drift is in good agreement with the value predicted by Galactic models (e.g., Reid etal. 2009). The quadrupole component presents some systematics that have to be clarified. A majorinterest of this component is that it can constrain Hubble Constant anisotropy or the amplitude ofprimordial gravitational waves (Kristian & Sachs 1966; Gwinn et al 1997). The Galactic aberration, or secular aberration drift, is a small proper motion of a few microarc secondsaffecting distant bodies induced by the rotation of the Solar system about the Galactic center, whichtakes about 250 Myr (Kovalevsky 2003). This systematic effect appears as a dipolar deformation ofthe proper motion field towards the Galactic center ( α = 266 ◦ , δ = − ◦ ) with a magnitude about6 µ as/yr, corresponding to a Solar system acceleration of 3 × − km/ s in accordance with theequation as follows ∆ µ α cos δ = − d sin α + d cos α, (1)28th Texas Symposium on Relativistic AstrophysicsGeneva, Switzerland – December 13-18, 2015c (cid:13) Commonwealth of Australia (Geoscience Australia) 2016 a r X i v : . [ a s t r o - ph . I M ] M a r ∆ µ δ = − d cos α sin δ − d sin α sin δ + d cos δ, (2)where the d i are the components of the acceleration vector in units of the proper motion, and whichcorresponds to the degree 1 spheroidal (or electric) development of (e.g., Mignard & Morando 1990) (cid:126)µ = (cid:88) l,m (cid:16) a El,m (cid:126)Y
El,m + a Ml,m (cid:126)Y
Ml,m (cid:17) , (3)where d = a E , , d = a E , − , and d = a E , , and Y El,m and Y Ml,m the vector spherical harmonics of electricand magnetic types of degree l and order m . In addition to the aberration distortion, there may also be a small global rotation which can bedescribed by the toroidal (or magnetic) harmonics of degree 1:∆ µ α cos δ = r cos α sin δ + r sin α sin δ − r cos δ, (4)∆ µ δ = − r sin α + r cos α, (5)where the r i can be expressed in terms of vector spherical harmonics coefficients as r = a M , , r = a M , − ,and r = a M , .Along with the aberration and rotation, more advanced cosmological effects may be detected usingthe proper motion of distant quasars. In particular, the anisotropic expansion of the Universe wouldresult in the degree 2 vector spherical harmonics of electric type, and the primordial gravitation waveswould be an origin of the degree 2 harmonics of electric and magnetic types. To investigate a possiblequadrupolar anisotropy of the velocity field, let us give the development of the degree 2 vector sphericalharmonics (i.e., l = 2 in Eq. (3)):∆ µ α cos δ = − ( a E, Re2 , sin 2 α − a E, Im2 , cos 2 α ) cos δ + ( a E, Re2 , sin α − a E, Im2 , cos α ) sin δ +( a M, Re2 , sin 2 α − a M, Im2 , cos 2 α ) sin δ cos δ + ( a M, Re2 , sin α − a M, Im2 , cos α ) cos 2 δ − a M , sin δ cos δ, (6)∆ µ δ = − ( a E, Re2 , cos 2 α + a E, Im2 , sin 2 α ) sin δ cos δ − ( a E, Re2 , cos α + a E, Im2 , sin α ) cos 2 δ +( a M, Re2 , cos 2 α + a M, Im2 , sin 2 α ) cos δ − ( a M, Re2 , cos α + a M, Im2 , sin α ) sin δ + a E , sin δ cos δ. (7) About 10 million VLBI observations since 1979 were analyzed with the geodetic VLBI analysis softwareCalc/Solve to generate astrometric coordinate time series of about 3800 radio sources. Amplitudesand direction of dipole, rotation and second order harmonics are displayed in Table 1 (below) forvarious subsets of radio sources. The figures show the proper motion in right ascension and dipolesystematics for the most observed sources, as well as the electric part of the quadrupole systematicsfor closest sources.Table 1 displays the magnitude of the dipole, rotation and second order harmonics for differentsets of reference radio sources. As the magnitude estimates of the second order harmonics exceed the3- σ formal errors in some cases, they vary from one solution to another. More VLBI data needs to becollected to obtain more reliable results.28th Texas Symposium on Relativistic AstrophysicsGeneva, Switzerland – December 13-18, 2015c (cid:13)(cid:13)
Ml,m (cid:17) , (3)where d = a E , , d = a E , − , and d = a E , , and Y El,m and Y Ml,m the vector spherical harmonics of electricand magnetic types of degree l and order m . In addition to the aberration distortion, there may also be a small global rotation which can bedescribed by the toroidal (or magnetic) harmonics of degree 1:∆ µ α cos δ = r cos α sin δ + r sin α sin δ − r cos δ, (4)∆ µ δ = − r sin α + r cos α, (5)where the r i can be expressed in terms of vector spherical harmonics coefficients as r = a M , , r = a M , − ,and r = a M , .Along with the aberration and rotation, more advanced cosmological effects may be detected usingthe proper motion of distant quasars. In particular, the anisotropic expansion of the Universe wouldresult in the degree 2 vector spherical harmonics of electric type, and the primordial gravitation waveswould be an origin of the degree 2 harmonics of electric and magnetic types. To investigate a possiblequadrupolar anisotropy of the velocity field, let us give the development of the degree 2 vector sphericalharmonics (i.e., l = 2 in Eq. (3)):∆ µ α cos δ = − ( a E, Re2 , sin 2 α − a E, Im2 , cos 2 α ) cos δ + ( a E, Re2 , sin α − a E, Im2 , cos α ) sin δ +( a M, Re2 , sin 2 α − a M, Im2 , cos 2 α ) sin δ cos δ + ( a M, Re2 , sin α − a M, Im2 , cos α ) cos 2 δ − a M , sin δ cos δ, (6)∆ µ δ = − ( a E, Re2 , cos 2 α + a E, Im2 , sin 2 α ) sin δ cos δ − ( a E, Re2 , cos α + a E, Im2 , sin α ) cos 2 δ +( a M, Re2 , cos 2 α + a M, Im2 , sin 2 α ) cos δ − ( a M, Re2 , cos α + a M, Im2 , sin α ) sin δ + a E , sin δ cos δ. (7) About 10 million VLBI observations since 1979 were analyzed with the geodetic VLBI analysis softwareCalc/Solve to generate astrometric coordinate time series of about 3800 radio sources. Amplitudesand direction of dipole, rotation and second order harmonics are displayed in Table 1 (below) forvarious subsets of radio sources. The figures show the proper motion in right ascension and dipolesystematics for the most observed sources, as well as the electric part of the quadrupole systematicsfor closest sources.Table 1 displays the magnitude of the dipole, rotation and second order harmonics for differentsets of reference radio sources. As the magnitude estimates of the second order harmonics exceed the3- σ formal errors in some cases, they vary from one solution to another. More VLBI data needs to becollected to obtain more reliable results.28th Texas Symposium on Relativistic AstrophysicsGeneva, Switzerland – December 13-18, 2015c (cid:13)(cid:13) Commonwealth of Australia (Geoscience Australia) 2016Figure 1: The proper motion in right ascension (left) and dipole systematics (right) for the radiosources observed in more than 1000 sessions. The green dot indicates the position of the Galacticcenter. Table 1: Parameters estimated from different subsets of radio sources.Dipole only Dipole + rotation 16 parameters55 radio sources observed in more than 1000 sessions.Amplitude ( µ as/yr) 5 . ± . . ± . . ± . α ( ◦ ) 257 ±
19 277 ±
20 270 ± δ ( ◦ ) − ± − ± − ± µ as/yr) 4 . ± . . ± . µ as/yr) 5 . ± . µ as/yr) 5 . ± . . ± . . ± . α ( ◦ ) 273 ±
13 278 ±
14 289 ± δ ( ◦ ) − ± − ± − ± µ as/yr) 2 . ± . . ± . µ as/yr) 4 . ± . z > . µ as/yr) 7 . ± . . ± . . ± . α ( ◦ ) 267 ±
15 277 ±
14 297 ± δ ( ◦ ) − ± − ± − ± µ as/yr) 3 . ± . . ± . µ as/yr) 4 . ± . z < . µ as/yr) 3 . ± . . ± . . ± . α ( ◦ ) 285 ±
23 284 ±
31 283 ± δ ( ◦ ) − ± − ± − ± µ as/yr) 3 . ± . . ± . µ as/yr) 7 . ± . (cid:13)(cid:13)
31 283 ± δ ( ◦ ) − ± − ± − ± µ as/yr) 3 . ± . . ± . µ as/yr) 7 . ± . (cid:13)(cid:13) Commonwealth of Australia (Geoscience Australia) 2016Figure 2: The electric part of the quadrupole systematics for closest sources ( z < . The paper is published with the permission of the CEO, Geoscience Australia.
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