Searching for spatial variations of \alpha^{2}/\mu in the Milky Way

Searching for spatial variations of in the Milky Way

S. A. Levshakov INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34131 Trieste, Italy
lev@astro.ioffe.rssi.ru Key Lab. for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, CAS, 80 Nandan Road, Shanghai 200030, P.R. China Ioffe Physical-Technical Institute, Polytekhnicheskaya Str. 26, 194021 St. Petersburg, Russia
   P. Molaro INAF-Osservatorio Astronomico di Trieste, Via G. B. Tiepolo 11, 34131 Trieste, Italy
lev@astro.ioffe.rssi.ru
   D. Reimers Hamburger Sternwarte, Universität Hamburg, Gojenbergsweg 112, D-21029 Hamburg, Germany
Received 00 ; Accepted 00
Key Words.:
Line: profiles – ISM: molecules – Radio lines: ISM – Techniques: radial velocities
Abstract

Context:

Aims: We probe the dependence of on the ambient matter density by means of submm- and mm-wave bands spectral observations in the Milky Way.

Methods:A procedure is suggested to explore the value of , where is the electron-to-proton mass ratio, and is the fine-structure constant. The fundamental physical constants, which are measured in different physical environments of high (terrestrial) and low (interstellar) densities of baryonic matter are supposed to vary in chameleon-like scalar field models, which predict that both masses and coupling constant may depend on the local matter density. The parameter can be estimated from the radial velocity offset, , between the low-laying rotational transitions in carbon monoxide CO and the fine-structure transitions in atomic carbon [C i]. A model-dependent constraint on  can be obtained from  using  independently measured from the ammonia method.

Results: Currently available radio astronomical datasets provide an upper limit on m s (). When interpreted in terms of the spatial variation of , this gives . An order of magnitude improvement of this limit will allow us to test independently a non-zero value of = recently found with the ammonia method. Taking into account that the ammonia method restricts the spatial variation of at the level of and assuming that  is the same in the entire interstellar medium, one obtains that the spatial variation of does not exceed the value . Since extragalactic gas clouds have densities similar to those in the interstellar medium, the bound on  is also expected to be less than at high redshift if no significant temporal dependence of is present.

Conclusions:

1 Introduction

The dimensionless physical constants like the electron-to-proton mass ratio, , or the fine-structure constant, , are expected to be dynamical quantities in modern extensions of the standard model of particle physics (Uzan 2003; Garcia-Berro et al. 2007; Martins 2008; Kanekar 2008; Chin et al. 2009). Exploring these predictions is a subject of many high precision measurements in contemporary laboratory and astrophysical experiments. The most accurate laboratory constraints on temporal - and -variations of yr, and yr were obtained by Rosenband et al. (2008), and Blatt et al. (2008), respectively.

In case of monotonic dependence of and on cosmic time, at redshift (corresponding look-back time is yr) the changes of and would be restricted at the level of and . Here  (or ) is a fractional change in between a reference value and a given measurement obtained at different epochs or at different spatial coordinates: .

These constraints are in line with geological measurements of relative isotopic abundances in the Oklo natural fission reactor which allows us to probe at yr (). Assuming possible changes only in the electromagnetic coupling constant, Gould et al. (2006) obtained a model dependent constraint on . However, when the strength of the strong interaction, – the parameter , – is also suggested to be variable, the Oklo data does not provide any bound on the variation of (Flambaum & Shuryak 2002; Chin et al. 2009).

Current astrophysical measurements at higher redshifts are as follows. There was a claim for a variability of at the 5 confidence level: ppm (Murphy et al. 2004)111Hereafter, 1 ppm = ., but this was not confirmed in other measurements which led to the upper bound ppm (Quast et al. 2004; Levshakov et al. 2005; Srianand et al. 2008; Molaro et al. 2008a).

Measurements of the cosmological -variation exhibit a similar tendency. Non-zero values of ppm, ppm (Ivanchik et al. 2005), and ppm (Reinhold et al. 2006) found at (Q 0405–443) and (Q 0347–383) from the Werner and Lyman bands of H were later refuted by Wendt & Reimers (2008), King et al. (2008) and Thompson et al. (2009) who used the same optical absorption-line spectra of quasars and restricted changes in at the level of ppm. The third H system at towards the quasar J2123–0050 also does not show any evidence for cosmological variation in : ppm (Malec et al. 2010). More stringent constraints were obtained at lower redshifts from radio observations of the absorption lines of NH and other molecules: ppm at (Murphy et al. 2008), and ppm at (Henkel et al. 2009). Two cool gas absorbers at (Q 2337–011) and (Q 0458–020) were recently studied in the H i 21cm and C i absorption lines providing a constraint on the variation of the product (here is the proton gyromagnetic ratio): ppm (Kanekar et al. 2010). Thus, the most accurate astronomical estimates restrict cosmological variations of the fundamental physical constants at the level of 1-2 ppm.

The estimate of fractional changes in  and  by spectral methods is always a measurement of the relative Doppler shifts between the line centers of different atoms/molecules and their comparison with corresponding laboratory values (Savedoff 1956; Bahcall et al.1967; Wolfe et al. 1976; Dzuba 1999, 2002; Levshakov 2004; Kanekar & Chengalur 2004). To distinguish the line shifts due to radial motion of the object from those caused by the variability of constants, lines with different sensitivity coefficients, , to the variations of and/or are to be used222 is a dimensionless coefficient showing a relative change of the atomic transition frequency in response to a change of the physical constant : .. It is clear that the larger the difference between two transitions, the higher the accuracy of such estimates.

Optical and UV transitions in atoms, ions and molecular hydrogen H have similar sensitivity coefficients with not exceeding 0.05 (Varshalovich & Levshakov 1993; Dzuba 1999, 2002; Porsev et al. 2007). For atomic spectra, the estimate of  is given in linear approximation () by (e.g., Levshakov et al. 2006):

(1)

where are the radial velocities of two atomic lines, and is the speed of light. It was shown in Molaro et al. (2008b) that the limiting accuracy of the wavelength scale calibration for the VLT/UVES quasar spectra at any point within the whole optical domain is about 30 m s, which corresponds to the limiting relative accuracy between two lines measured in different parts of the same spectrum of about 50 m s. Taking into account that , it follows from Eq.(1) that the limiting accuracy of is 2 ppm, which is the utmost value that can be achieved in observations of extragalactic objects with present optical facilities.

A considerably higher sensitivity to the variation of physical constants is observed in radio range. For example, van Veldhoven et al. (2004) first showed that the inversion frequency of the level of the ammonia isotopologue ND has the sensitivity coefficient . Compared to optical and UV transitions, the ammonia method proposed by Flambaum & Kozlov (2007) provides 35 times more sensitive estimate of  from measurements of the radial velocity offset between the NH inversion transition at 23.7 GHz and low-lying rotational transitions of other molecules co-spatially distributed with NH:

(2)

The ammonia method was recently used to explore possible spatial variations333Hereafter, the term ‘spatial variation’ means a possible change in between its terrestrial and interstellar values. of physical constants from observations of prestellar molecular cores in the Taurus giant molecular cloud (Levshakov et al. 2010, hereafter L10), the Perseus cloud, the Pipe Nebula, and Infrared dark clouds (Levshakov et al. 2008b; Molaro et al. 2009). In contrast to the mentioned above laboratory constraints on temporal variations, this method reveals a tentative spatial variation of  at the level of = (L10). The corresponding conservative upper limit in this case is equal to .

In the present paper we consider fractional changes of a combination of two constants and , , which are estimated from the comparison of transition frequencies measured in different physical environments of high (terrestrial) and low (interstellar) densities of baryonic matter. The idea behind this experiment is that some class of scalar field models — so-called chameleon-like fields — predict the dependence of both masses and coupling constant on the local matter density (Olive & Pospelov 2008; Upadhye et al. 2010). Chameleon-like scalar fields were introduced by Khoury & Weltman (2004a,b) and by Brax et al. (2004) to explain negative results on laboratory searches for the fifth force which should arise inevitably from couplings between scalar fields and standard model particles. The chameleon models assume that a light scalar field acquires both an effective potential and effective mass because of its coupling to matter that depends on the ambient matter density. In this way, the chameleon scalar field may evade local tests of the equivalence principle and fifth force experiments since the range of the scalar-mediated fifth force for the terrestrial matter densities is too small to be detected. Similarly, laboratory tests with atomic clocks for -variations are performed under conditions of constant local density and, hence, they are not sensitive to the presence of the chameleon scalar field (Upadhye et al. 2010). This is not the case for space-based tests, where the matter density is considerably lower, an effective mass of the scalar field is negligible, and an effective range for the scalar-mediated force is large. Light scalar fields are usually attributed to a negative pressure substance permeating the entire visible Universe and known as dark energy (Caldwell et al. 1998). This substance is thought to be responsible for a cosmic acceleration at low redshifts, (Peebles & Rata 2003; Brax 2009).

2 [C i] and CO lines as probes of

The variations of the physical constants can be probed through atomic fine-structure (FS) and molecular rotational transitions (Levshakov et al. 2008a; Kozlov et al. 2008). The corresponding lines are observed in submm- and mm-wavelength ranges. Along with a gain in sensitivity, the use of such transitions allows us to estimate constants at very high redshifts () which are inaccessible to optical observations.

Let us consider radial velocity offsets between molecular rotational and atomic FS lines, . The offset is related to the parameter  as follows (Levshakov et al. 2008a):

(3)

The velocity offset in Eq.(3) can be represented by the sum of two components

(4)

where is the shift due to -variation, and is the Doppler noise — a random component caused by possible local offsets since transitions from different species may arise from slightly different parts of a gas cloud, at different radial velocities.

The Doppler noise yields offsets which can either mimic or obliterate a real signal. Nevertheless, if these offsets are of random nature, the signal can be estimated statistically by averaging over a large data sample:

(5)

Here we assume that the noise component has zero mean and a finite variance.

The Doppler noise component can be minimized if the chosen species are closely trace each other. An appropriate pair in our case is the atomic carbon FS transitions and rotational transitions of carbon monoxide CO. The spatial distributions of CO and [C i] are known to be well correlated (Keene et al. 1985; Meixner & Tielens 1995; Spaans & van Dishoeck 1997; Ikeda et al. 2002; Papadopoulos et al. 2004). The carbon-bearing species C, C, and CO are observed in photodissociation regions (PDRs) – neutral regions where chemistry and heating are regulated by the far-UV photons (Hollenbach & Tielens 1999). The PDR is either the interface between the H ii region and the molecular cloud or a neutral component of the diffuse interstellar medium (ISM). Far-UV photons ( eV eV) are produced by OB stars. Photons with energy greater than 11.1 eV dissociate CO into atomic carbon and oxygen. Since the C ionization potential of 11.3 eV is quite close to the CO dissociation energy, neutral carbon can be quickly ionized. This suggests the chemical stratification of the PDR in the line C/C/CO with increasing depth from the surface of the PDR. Then, one can assume that in the outer envelopes of molecular clouds neutral carbon lies within a thin layer determined by the equilibrium between photoionization/recombination processes on the C/C side, and photodissociation/molecule formation processes on the C/CO side. However, observations (Keene et al. 1985; Zhang et al. 2001) do not support such a steady-state model which predicts that C should arise only near the edges of molecular clouds. To explain the observed correlation between the spatial distributions of C and CO, inhomogeneous PDRs with clumping molecular gas were suggested. The revealed ubiquity of the [C i] transition in molecular clouds is in agreement with clumpy PDR models (Meixner & Tielens 1995; Spaans et al. 1997; Papadopoulos et al. 2004).

The ground state of the C atom consists of the triplet levels. The energies of the fine-structure excited levels relative to the ground state are K, and K, and the transition probabilities are s, and s (Silva & Viegas 2002). The excitation rates of the [C i] and levels for collisions with H at K are cms (Schröder et al. 1991). This implies that for the and levels the critical densities are 1000 cm and 3000 cm, respectively. The low- rotational transitions of CO trace similar moderately dense ( cm) and cold ( K) gas. It is not completely excluded, however, that some heterogeneity of spatial distributions of [C i] and CO may occur resulting in the radial velocity offsets.

In the chameleon-like scalar field models for density-dependent and the fractional changes in these constants arise from the shift in the expectation value of the scalar field between high and low density environments. Since the matter density in the interstellar clouds is times lower than in terrestrial environments, whereas gas densities between the molecular clouds themselves are much smaller ( cm), all interstellar clouds can be considered as having similar physical conditions irrespective of their location in space. This means that the noise component in Eq.(5) can be reduced by averaging over individual  values obtained from an ensemble of clouds for which the measurements of both [C i] and CO lines are available.

Equations (2) and (3) show that in order to estimate  and  with a comparable relative error the uncertainty of the velocity offset in (3) must be 3.5 times smaller than that in the ammonia method (5 m s, see L10). At the moment such data do not exist. Both laboratory and astronomical measurements of the [C i] frequencies have much larger uncertainties. For example, the rest frequencies of the [C i] transition 492160.651(55) MHz (Yamamoto & Saito 1991) and transition 809341.97(5) MHz (Klein et al. 1998) are measured with the uncertainties of m s and 18.5 m s, respectively. For CO the rest hyper-fine frequencies of low- rotational transitions are known with good accuracy: GHz, and GHz, i.e., m s (Cazzoli et al. 2004). Suggesting that the laboratory error m s dominates over the errors from measurements, one obtains that the limiting accuracy is ppm. To put in another words, if both species arise from the same volume elements and their radial velocities are known with a typical error of m s (e.g., Ikeda et al. 2002), then the mean can be estimated with a statistical error of m s from an ensemble of independent measurements.

Unfortunately, available observational data do not allow us to probe  at the 0.1 ppm level. First at all, only a handful of sources are known where both [C i] and CO radial velocities were measured (Schilke et al. 1995; Stark et al. 1996; Ikeda et al. 2002; Mookerjea et al. 2006a,b). The line profiles from these observations were usually fitted with single Gaussians in spite of apparent asymmetries seen in some cases (e.g., Fig. 7 in Ikeda et al. 2002). Besides, the measured radial velocities were not corrected for different beamsizes. As a result, the scatter in becomes large, and the accuracy of the  estimate deteriorates.

3 The estimate

In this section we consider constraints on the spatial variations of which can be obtained from observations of emission lines of atomic carbon and carbon monoxide in submm- and mm-wave regions. The FS [C i] lines and low- rotational lines of CO are observed towards many galactic and extragalactic objects (Bayet et al. 2006; Omont 2007). For our purpose we selected a few molecular clouds located at different galactocentric distances where the radial velocities of these species were measured with a sufficiently high precision ( m s).

Table 1 lists molecular clouds with both [C i] and CO line measurements which are available in literature. The data were obtained under the following conditions.

TMC-1 — the Taurus Molecular Cloud ( pc). This dark molecular cloud was studied with the Caltech 10.4m submillimeter telescope on Mauna Kea, Hawaii (Schilke et al. 1995). The beamsize at the [C i] (1-0) frequency was , while at the CO (2-1) frequency it was about . Schilke et al. observed similar shapes of the [C i] (1-0) and CO (2-1) profiles at five positions perpendicular to the molecular ridge close to the cyanopolyyne peak. The line parameters listed in Table 1 were derived by Gaussian fits, although the line shapes were not exactly Gaussians. Therefore the errors of the line parameters are the formal errors of the fitting procedure.

L183 — is an isolated quiescent dark cloud at a distance of about 100 pc (Mattila 1979; Franco 1989). The observations of the [C i] and CO lines at six positions along an east-west strip through the center of the cloud were obtained with the 15m James Clerk Maxwell Telescope (JCMT) on Mauna Kea, Hawaii (Stark et al. 1996). The beamsize at 492 GHz was and (A-band) and (B-band) at 220 GHz. The [C i] and CO data were smoothed to a resolution of 0.4 km s and 0.2 km s, respectively. These emission lines show similar asymmetric profiles which can be attributed to two kinematically different components closely spaced in velocity with central velocities around 1 km s and 2 km s. These components are marginally resolved in the [C i] spectra at two positions (# 7 and 8 in Table 1). But since CO lines were not resolved at these positions, we include in Table 1 the results of one component Gaussian fits of both CO (2-1) and [C i] (1-0) spectra from Stark et al. (1996).

Ceph B — is a giant Cepheus molecular cloud at a distance of pc located to the south of the Cepheus OB3 association of early-type stars (Blaauw 1964). Cepheus B, the hottest CO component of this complex (Sargent 1977, 1979), is surrounded by an ionization front driven by the UV radiation from the brightest members of the OB3 association (Felli et al. 1978). The observations of the [C i] (1-0) line were obtained using the KOSMA 3m submillimeter telescope on Gornergrat, Switzerlaand (Mookerjea et al. 2006a). This dataset was complemented with CO observed with the IRAM 30m telescope (Ungerechts et al. 2000). All data were smoothed to the spatial resolution of and the velocity resolution of 0.8 km s. Table 1 includes [C i] and CO (2-1) lines arising around of km s at the position of the hotspot in Cepheus B. The values of the [C i] (1-0) and CO (2-1) positions derived from Gaussian fitting were reported in Table 2 of Mookerjea et al. (2006a) without their errors. However, since the lines look symmetric (Fig. 3, Mookerjea et al. 2006a), we assign them an error of 0.1 km s. This is slightly larger than the uncertainty of 1/10 of the resolution element, – a typical error of the line position for symmetric profiles, – but does not affect significantly the sample mean value of .

Orion A,B — are giant molecular clouds located at pc (Genzel & Stutzki 1989). The observations of the [C i] (1-0) line towards 9 deg area of the Orion A cloud and 6 deg area of the Orion B cloud with a grid spacing of were carried out with the 1.2m Mount Fuji submillimeter telescope (Ikeda et al. 2002). These observations were complemented with the CO (1-0) dataset presented in Table 3 in Ikeda et al.. At the frequency 492 GHz the spatial and velocity resolutions were, respectively, and 1.0 km s, whereas at the frequency 110 GHz they were and 0.3 km s. The profiles of the [C i] (1-0) and CO (1-0) lines were found to be very similar. All spectra were well fitted with one or two Gaussian functions, and the velocity centers of the [C i] and CO lines are almost the same: km s. The results of the Gaussian fitting are given in Table 1.

Cas A — is a supernova remnant at a distance of kpc (Braun et al. 1987). It was mapped in the [C i] (1-0) line on the KOSMA 3m submillimeter telescope with the beamwidth of and the velocity resolution of 0.6 km s (Mookerjea et al. 2006b). These observations have been compared with the CO (1-0) observations (beamsize , spectral resolution km s) taken from Liszt & Lucas (1999). Both the [C i] (1-0) and CO (1-0) emission spectra were averaged over the disk of Cassiopeia A. The results of Gaussian fitting of subcomponents resolved in the [C i] (1-0) and CO (1-0) spectra are included in Table 1. Two strong emission feature observed in both [C i] and CO (1-0) lines were identified with the Perseus arm at km s ( kpc distant) and with the local Orion arm at km s( pc distant).

Source No. , , , , Ref.
km s km s km s km s km s
TMC-1 1 6. 1(1) 2. 0(1) 6. 0(1) 1. 5(2) 0. 1(1) Schilke et al. 1995
2 6. 1(1) 1. 6(1) 6. 1(1) 1. 5(1) 0. 0(1)
3 6. 1(1) 1. 7(1) 5. 9(1) 1. 6(1) 0. 2(1)
4 6. 1(1) 1. 6(1) 5. 8(1) 1. 2(2) 0. 3(1)
5 6. 2(1) 1. 5(1) 6. 4(1) 2. 0(2) –0. 2(1)
L183 6 2. 83(3) 2. 00(7) 2. 2(2) 1. 7(3) 0. 63(20) Stark et al. 1996
7 2. 41(4) 1. 81(9) 1. 6(2) 2. 2(4) 0. 81(20)
8 2. 16(3) 1. 84(7) 2. 2(1) 1. 6(3) –0. 04(10)
9 2. 05(4) 2. 6(1) 1. 6(1) 1. 8(3) 0. 45(11)
10 2. 20(4) 2. 0(1) 2. 4(1) 1. 4(3) –0. 20(11)
Ceph B 11 –13. 4(1) 2. 1 –14. 1(1) 1. 9 0. 7(1) Mookerjea et al. 2006a
Ori A,B 12 8. 6(1) 4. 0(1) 9. 3(1) 4. 7(1) –0. 7(1) Ikeda et al. 2002
13 7. 1(1) 2. 4(1) 7. 6(1) 3. 4(1) –0. 5(1)
14 4. 8(1) 2. 5(1) 5. 2(2) 3. 3(4) –0. 4(2)
15 10. 4(1) 2. 9(1) 10. 9(3) 3. 3(1) –0. 5(3)
16 11. 2(1) 2. 6(1) 11. 0(1) 3. 3(1) 0. 2(1)
17 9. 6(1) 2. 7(1) 9. 9(1) 3. 2(1) –0. 3(1)
18 9. 6(1) 4. 4(1) 9. 5(1) 5. 0(1) 0. 1(1)
19 8. 1(1) 3. 3(1) 8. 3(1) 3. 7(1) –0. 2(1)
20 5. 3(1) 2. 3(1) 5. 5(1) 3. 0(1) –0. 2(1)
21 10. 3(2) 3. 4(4) 10. 6(1) 4. 2(1) –0. 3(2)
22 10. 6(1) 1. 8(1) 10. 4(2) 2. 3(4) 0. 2(2)
23 9. 6(1) 2. 7(1) 9. 8(1) 3. 4(1) –0. 2(1)
Cas A 24 –46. 80(5) 2. 1(1) –47. 4(1) 2. 9(2) 0. 60(11) Mookerjea et al. 2006b
25 –1. 10(6) 1. 8(1) –1. 7(1) 3. 1(4) 0. 60(12)
.
Table 1: Parameters derived from Gaussian fits to the CO and [C i] emission line profiles observed towards Galactic molecular clouds. is the line center, is the line width (FWHM). The numbers in parentheses are the standard deviations in units of the last significant digit (see text for more details). Col. 7 lists velocity offsets , and their estimated errors.

The velocity offsets between the CO and [C i] lines are given in Col. 7 of Table 1, the corresponding linewidths, , are shown in Cols. 4 and 6. When both transitions trace the same material, the lighter element C should always have larger linewidth. If the line broadening is caused by thermal and turbulent motions, i.e., , then for two species with masses we have

(6)

In practice, this inequality is fulfilled only approximately. Except for the pure thermal and turbulent broadening there are many other mechanisms which can give rise to the broadening of atomic and molecular lines. These are saturation broadening (lines have different optical depths), the presence of unresolved velocity gradients (nonthermal distribution is not normal), the increasing velocity dispersion of the nonthermal component with increasing map size (the higher angular resolution is realized for the higher frequency transitions), etc. Thus, the consistency of the apparent linewidths defined by Eq.(6) is a necessary condition for two species with different masses to be co-spatially distributed, but is not a sufficient one.

From Table 1 it is seen that the inequality (6) is fulfilled for all selected pairs CO/[C i] within the estimated uncertainties of the linewidths. Thus, the whole sample of values can be used to estimate .

The averaging of the velocity offsets over the dataset gives the unweighted mean (C i) = km s. With weights inversionally proportional to the variances, one derives km s. The median of the sample is km s, and the robust -estimate (L10) is km s. The statistical error for the mean velocity offset measurement is larger than that expected from the published values of the statistical errors from the one component Gaussian fits: the mean error of the individual is 0.13 km s, and the expected error of the mean is 0.026 km s. A possible reason for such a high Doppler noise has been discussed in Sect. 2. The systematic error in this case is dominated by the uncertainty of the rest frequency of the [C i] (1-0) transition, m s. Thus, taking the -estimate as the best measure of the velocity offset, we have km s, and the upper limit on km s.

This estimate restricts the spatial variability of at the level of ppm. Recently we obtained a constraint on the spatial change of the electron-to-proton mass ratio ppm based on measurements in cold molecular cores in the Milky Way (L10). Combining these two upper limits, the fine-structure constant can be bound as ppm.

4 Conclusion

The level of 0.2 ppm represents a model-dependent upper limit on the spatial variations of . Under model-dependence we assume here that both  and  do not change significantly from cloud to cloud, since astrophysical measurements of these parameters are made in low density regions of the interstellar medium with .

For comparison, the upper limit on the temporal -variation obtained from high-redshift quasar absorbers is ppm (Sect. 1). If dependence of constants on the ambient matter density dominates over temporal (cosmological), as suggested in chameleon-like scalar field models, then one may expect that ppm at high redshifts as well, since quasar absorbers have gas densities similar to those in the interstellar clouds. Taking into account that the predicted changes in and are not independent and that -variations may exceed variations in (e.g., Calmet & Fritzsch 2002; Langacker et al. 2002; Dine et al. 2003; Flambaum et al. 2004), even a lower bound of ppm is conceivable within the framework of the chameleon models.

We note that if a theoretical prediction is valid, then , and, hence, the -estimate with a further order of magnitude improvement in sensitivity will provide an independent test of the tentative change of .

The factors limiting accuracy of the current estimate of  at are a relatively low spectral resolution of the available observations in submm- and mm-wave bands, a rather large uncertainty of the rest frequencies of the [C i] FS lines, and a small number of objects observed in both [C i] and CO transitions.

Modern telescopes like the recently launched Herschel Space Observatory can provide for Galactic objects the spectral resolution as high as 30 m s (e.g., the Heterodyne Instrument for the Far Infrared, HIFI, has resolving power ). This means that the positions of the [C i] FS lines can be measured with the uncertainty of 3 m s. In the near future, high precision measurements will be also available with the Atacama Large Millimeter/submillimeter Array (ALMA), the Stratospheric Observatory For Infrared Astronomy (SOFIA), the Cornell Caltech Atacama Telescope (CCAT) and others. Thus, any further advances in exploring depend crucially on new laboratory measurements of the [C i] FS frequencies. If these frequencies will be known with uncertainties of a few m s, then the parameter can be probed at the level of which would be comparable with the non-zero signal in the spatial variation of the electron-to-proton mass ratio .

Acknowledgements.
We thank our anonymous referee for valuable comments on the manuscript. The project has been supported in part by DFG Sonderforschungsbereich SFB 676 Teilprojekt C4, the RFBR grants 09-02-12223 and 09-02-00352, by the Federal Agency for Science and Innovations grant NSh-3769.2010.2, and by the Chinese Academy of Sciences visiting professorship for senior international scientists under grant No. 2009J2-6.

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