Searching for chameleon-like scalar fields with the ammonia method.††thanks: Based on observations obtained with the 100-m telescope at Effelsberg/Germany which is operated by the Max-Planck Institut für Radioastronomie on behalf of the Max-Planck-Gesellschaft (MPG).
Key Words.:Line: profiles – ISM: molecules – Radio lines: ISM – Techniques: radial velocities – elementary particles
Context:In our previous work we found a statistically significant offset m s between the radial velocities of the HCN and NH transitions observed in molecular cores from the Milky Way. This may indicate that the electron-to-proton mass ratio, , increases by when measured under interstellar conditions with matter densities of more than 10 orders of magnitude lower as compared with laboratory (terrestrial) environments.
Aims: We map four molecular cores L1498, L1512, L1517, and L1400K selected from our previous sample in order to estimate systematic effects in due to possible velocity gradients or other sources across the cloud, and to check the reproducibility of the velocity offsets on the year-to-year time base.
Methods: We use the ammonia method, which involves observations of inversion lines of NH complemented by rotational lines of other molecular species and allows us to test changes in due to a higher sensitivity of the inversion frequencies to the -variation as compared with the rotational frequencies.
Results: We find that in two cores L1498 and L1512 the NH (1,1) and HCN (2–1) transitions closely trace the same material and show an offset of (HCN) – (NH) = m s throughout the entire clouds. The offsets measured in L1517B and L1400K are m s, and m s, respectively, and are, probably, subject to Doppler shifts due to spatial segregation of HCN versus NH. We also determine frequency shifts caused by external electric and magnetic fields, and by the cosmic black body radiation-induced Stark effect and find that they are less than 1 m s.
Conclusions: The measured velocity offset in L1498 and L1512, being expressed in terms of , gives = . Although this estimate is based on a limited number of sources and molecular pairs used in the ammonia method, it demonstrates a high sensitivity of radio observations in testing fundamental physics. The non-zero signal in should be further examined as larger and more accurate data sets become available.
The present paper continues our study (Levshakov et al. 2008; Molaro et al. 2009; Levshakov et al. 2010a, hereafter Paper I) on differential measurements of the electron-to-proton mass ratio, , by means of high resolution spectral observations (FWHM m s) of narrow emission lines (FWHM m s) of N-bearing molecules arising in cold and dense molecular clouds ( K, cm). This study is aimed, in general, at testing the Einstein equivalence principle of local position invariance (LPI) which states that outcomes of physical non-gravitational experiments should be independent of their position in space-time (e.g., Dent 2008). However, the violation of LPI is anticipated in non-standard physical theories associated with varying fundamental constants and, in particular, in those dealing with dark energy. A concept of dark energy with negative pressure appeared in physics long before the discovery of the accelerating universe through observations of nearby and distant supernovae type Ia (Perlmutter et al. 1998; Riess et al. 1998). Early examples of dark energy in a form of a scalar field with a self-interaction potential can be found in a review by Peebles & Ratra (2003). To explain the nature of dark energy, many sophisticated models have been suggested (see, e.g., Caldwell et al. 1998), and among them the scalar fields which are ultra-light in cosmic vacuum but possess an effectively large mass locally when they are coupled to ordinary matter by the so-called chameleon mechanism (Khoury & Weltman 2004a,b; Brax et al. 2004; Avelino 2008; Burrage et al. 2009; Davis et al. 2009; Brax 2009; Upadhye et al. 2010; Brax & Zioutas 2010). A subclass of such models considered by Olive & Pospelov (2008) predicts that fundamental physical quantities such as elementary particle masses and low-energy coupling constants may also depend on the local matter density.
In the Standard Model of particle physics there are two fundamental parameters with the dimension of mass – the Higgs vacuum expectation value (VEV GeV), which determines the electroweak unification scale, and the quantum chromodynamics (QSD) scale MeV, which characterizes the strength of the strong interaction. The masses of the elementary particles such as the electron and the quarks are proportional to the Higgs VEV, and the mass of the composite particle proton is proportional to . Therefore, by probing the dimensionless mass ratio we can experimentally test of the ratio of the electroweak scale to the strong scale at different physical conditions (Chin et al. 2009).
In this regard, the masses of the elementary particles mediated by the chameleon-like scalar fields should depend on the local density of baryons and electrons. Among the observationally accessible media the most dramatic difference of local densities occurs between terrestrial laboratories and cosmic space. For instance, a typical interstellar molecular cloud with the kinetic temperature K and the gas number density cm has the local density of g cm. Laboratory vacuum spectrographs usually operate in the pressure range from 1 to 10 mtorr and at the temperature regime of K (e.g., Willey et al. 2002; Ronningen & De Lucia 2005; Ross & Willey 2005). This gives g cm if K, mtorr, and the vacuum chamber is filled by H. In this case, the ratio exceeds . Besides, the size of the vacuum chamber is cm, and its walls are naturally made of extremely heavy materials, i.e. the chameleon fields in the chamber are subject to high density boundary conditions.
|Core||R.A.||Dec.||Offset||rms (mK)||rms (mK)||Date||UT|
|(J2000.0)||(km s)||(arcsec)||23 GHz||18 GHz||(2010)||(h:m)||#|
It follows from the above considerations that the ratio measured in low density environments of the interstellar and/or intergalactic medium may differ from its terrestrial (laboratory) value. To estimate the fractional changes in , , the so-called ammonia method was elaborated by Flambaum & Kozlov (2007, hereafter FK) which involves observations of inversion lines of NH complemented by rotational lines of other molecular species. The inversion transitions of ammonia and its isotopologues strongly depend on (van Veldhoven et al. 2004; FK; Kozlov et al. 2010). For NH, the fractional change in frequency is , whereas for rotational lines, . Therefore, the comparison of radial velocities of NH inversion lines, , with radial velocities of rotational transitions, , provides a sensitive limit on the variation of (FK):
where is the speed of light. Thus, measuring the line shifts in cosmic objects where the local matter density is significantly lower than in laboratory experiments one can probe whether the mass-ratio is position-invariant or not. Such tests will complement current cosmological probes of dark energy and its couplings to matter.
NH was detected in numerous molecular clouds within the Milky Way (e.g., Jijina et al. 1999) and even at significant redshifts (Henkel et al. 2005, 2008). In particular, observations of two molecular absorption-line systems at redshift and towards, respectively, the gravitationally lensed quasars B0218+357 and PKS 1830–211 yielded constraints on down to 1 ppm (FK; Murphy et al. 2008; Menten et al. 2008; Henkel et al. 2009)111Hereafter, 1 ppm = , and 1 ppb = .. However, can be probed at orders of magnitude more sensitive levels if nearby Galactic clouds are studied. Then a spectral resolution of FWHM m s allows us to measure the line position with uncertainties of a few m s. In 2008-2009, we observed 41 molecular cores in the disk of the Milky Way in three molecular transitions NH , HCN , and NH at the radio telescopes in Medicina (32-m), Nobeyama (45-m), and Effelsberg (100-m) and obtained a statistically significant positive velocity offset between the rotational HCN (2–1), NH and inversion NH (1,1) lines. Since these transitions show a good correlation in their spatial distributions, such an offset might indicate that at low interstellar densities increases compared with its terrestrial value (Paper I). The most accurate estimate was obtained from the Effelsberg dataset consisting of twelve pairs of NH and HCN lines: = m s which translates into ppb. We note that because of a rounded value for the fast Fourier transform spectrometer (FFTS) channel separation ( kHz for NH and kHz for HCN instead of 1.220703125 kHz) not taken into account in Paper I, the corrected is to be ppb.
These observations were carried out in a single position mode, i.e. we observed only the central parts of molecular cores showing strongest NH emission. The aim was to find suitable targets for most precise measurements of the velocity offsets between rotational and inversion transitions. In the present paper, we report on new observations where we measure at different positions across individual clouds in order to test the reproducibility of the velocity offsets in the presence of the large-scale velocity gradients. For this purpose, from the list of molecular cores observed in our previous study with the Effelsberg telescope, we selected the objects which comply with the following criteria: () profiles of the NH (1,1) and HCN (2–1) hyperfine structure (hfs) transitions are symmetric and well-described by a single-component Gaussian model, and () the line widths are thermally dominated, i.e., the parameter (NH)(HCN) , where is the velocity dispersion. The chosen targets are molecular cores L1498, L1512, L1517B and L1400K which were already extensively studied in many molecular lines (e.g. Benson & Myers 1989; Lee et al. 2001; Lee et al. 2003; Tafalla et al. 2004, 2006; Crapsi et al. 2005). In the present paper we describe the high-precision measurements of the radial velocities of NH (1,1) and HCN (2–1) hyperfine transitions in these clouds from 31 lines of sight in total.
We carried out our observations targeting a sample of well-studied nearby molecular cores that are essentially devoid of associated IR sources (so-called starless cores with narrow emission lines). All of our objects lie in the Taurus-Auriga-Perseus molecular complex, i.e., at the distance of about 140 pc (L1498, L1512, L1517BC) and 170 pc (L1400K) from the solar system (Jijina et al. 1999). Observations were performed with the 100-m Effelsberg radio telescope in January 2010. The positions observed are listed in Table 1. All sources, except L1517B, have the same central coordinates as used in Paper I. For L1517B, the central position is shifted at with respect to the previous coordinates, and the source is called L1517BC. The () = (1,1) inversion line of ammonia (NH) at 23.694 GHz and the rotation line of cyanoacetylene (HCN) at 18.196 GHz were measured with a K-band HEMT (high electron mobility transistor) dual channel receiver222At 18 GHz we used only one channel Rx2 since Rx1 was not operating during our observations., yielding spectra with angular resolution of HPBW in two orthogonally oriented linear polarizations. Averaging the emission from both channels gives typical system temperatures of 100–150 K for NH and 80–100 K for HCN on a main beam brightness temperature scale.
The measurements were carried out in frequency switching mode using a frequency throw of 5 MHz. The backend was an FFTS operated with its minimum bandwidth of 20 MHz providing simultaneously 16 384 channels for each polarization. The resulting channel separations are 15.4 m s and 20.1 m s for NH and HCN, respectively. We note, however, that the true velocity resolution is about twice as lower, FWHM m s and 40 m s, respectively (Kein et al. 2006).
The sky frequencies were reset at the onset of each scan and the Doppler tracking was used continuously to track Doppler shifts during the observations. In the Effelsberg driving program, the position of the center of gravity of Earth is taken from the JPL ephemerides including the influence of the Moon, and the rotation of Earth at the telescope position is added to this center of gravity. The velocity of the source with respect to Local Standard of Rest (LSR), , is corrected for these motions and the corrected value of determines the current sky frequency. The stability of the LO-frequency during 5 minutes exposures was about .
Observations started by measuring the continuum emission of calibration sources (NGC 7027, W3(OH), 3C 286) and continued by performing pointing measurements toward a source close to the spectroscopic target. Spectral line measurements were interspersed with pointing measurements at least once per hour. The calibration is estimated to be accurate to 15% and the pointing accuracy to be superior to 10 arcsec. Each molecular transition was observed in a set of four sequential 5 min exposures, which were repeated twice at 18 GHz to compensate for the non-operating channel Rx1. Thus, the total exposure time at one coordinate point was one hour. Some points were observed several times to control the accuracy of the measurements. All spectra were obtained with a comparable noise level which is indicated by the rms values in Table 1. The position offsets within each individual cloud are shown in Fig. 1.
|Offset||#||satellite||all||all||m s||m s||K||m s|
|Note. — Numbering in column 2 is the same as in the last column of Table 1.|
|Offset||#||satellite||all||all||m s||m s||K||m s|
We used the CLASS reduction package333http://www.iram.fr/IRAMFR/GILDAS for standard data reduction. After corrections for the rounded frequencies (see Sec. 1), the individual exposures were co-added to increase the signal-to-noise ratio, S/N. The spectra were folded to remove the effects of the frequency switch, and base lines were determined for each spectrum. The resolved hfs components show no kinematic sub-structure and consist of an apparently symmetric peak profile without broadened line wings or self-absorption features, as shown in Fig. 2 where the observed profiles of the NH (1,1) and HCN (2–1) lines toward L1498 are plotted. The line parameters such as the total optical depth in the transition, (i.e., the peak optical depth if all hyperfine components were placed at the same velocity), the radial velocity, , the line broadening Doppler parameter, , and the amplitude, , were obtained through the fitting of the one-component Gaussian model to the observed spectra as described in Paper I:
which transforms for optically thin transitions into
The rest-frame frequencies of the NH hfs components are taken from Kukolich (1967) and listed in Table 2 (Paper I). We checked these frequencies using the last JPL model444http://spec.jpl.nasa.gov/ftp/pub/catalog/catdir.html which takes into account the fit to all published ammonia data. The difference between various estimates does not exceed 0.05 kHz. In addition, Lapinov et al. (2010) carried out new laboratory measurements of the frequencies of the NH (1,1) hfs components and their results coincide with Kukolich’s frequencies within the 0.2 kHz range.
The rest-frame frequencies of the HCN (2–1) hfs components listed in Table 3 (Paper I) are based on Lapinov (2008, private comm.) calculations which combine all available laboratory measurements. These frequencies are in agreement with the data from the Cologne Database for Molecular Spectroscopy (Müller et al. 2005) within uncertainty interval, but Lapinov’s values have a slightly higher precision: kHz versus 0.2 to 0.7 kHz by Müller et al.. The recent laboratory measurements of the HCN (2–1) frequencies by Lapinov et al. (2010) did not show any systematic shifts between the previously obtained and new measured values: all frequencies agree within 0.2 kHz. Note that at the moment the accuracy with which the frequencies of NH (1,1) and HCN (2–1) are known is the highest among all observable molecular transitions, with the uncertainties being (NH) m s ( Hz), and (HCN) m s ( Hz).
We also estimated the kinetic temperature, , and the nonthermal (turbulent) velocity dispersion, , based on the line broadening Doppler parameters, , of the NH (1,1) and HCN (2–1) lines. Here is the line of sight velocity dispersion of the molecular gas within a given cloud. If the two molecular transitions trace the same material and have the same non-thermal velocity component, then is the quadrature sum of the thermal and turbulent velocity dispersions. In this case a lighter molecule with a mass should have a wider line width as compared with a heavier molecule with .
For thermally dominated line widths (, see Sect. 1) and co-spatially distributed species one has the following relations (e.g., Fuller & Myers 1993):
where is Boltzmann’s constant, and the thermal velocity dispersion, , is given by
However, HCN is usually distributed in a larger volume of the molecular core as compared with NH: in general, N-bearing molecules trace the inner core, whereas C-bearing molecules occupy the outer part (e.g., Di Francesco et al. 2007). Such a chemical differentiation and velocity gradients within the core may cause a larger nonthermal component in the velocity distribution of HCN. If both molecules are well shielded from the external incident radiation and the gas temperature is mainly due to the heating by cosmic rays, then a formal application of Eqs.(5) and (6) to the apparent line widths provides a lower limit on and an upper limit on .
We now consider the relative radial velocities of HCN and NH, (HCN) – (NH). As discussed in Paper I, the velocity offset can be represented by the sum of two components
where is the shift due to a putative -variation, and is the Doppler noise — a random component caused by possible spatial segregation of HCN versus NH and their different irregular Doppler shifts. The Doppler noise can mimic or obliterate a real signal and, hence, should be minimized. This can be achieved either by detailed mapping of the velocity field in molecular lines which closely trace each other as performed in the present study, or by averaging over a large data sample as realized in Paper I.
|Offset||#||satellite||all||all||m s||m s||K||m s|
Equations (2)-(4) define our model of the hfs line profiles. The model parameters were determined by using a nonlinear least-squares method (LSM). The best fit values of the radial velocity and the Doppler width are given in Tables 2-5, columns 3, 4, and 5. An example of the fit is shown in Fig. 2. The residuals depicted below the profiles are normally distributed with zero mean and dispersion equal to the noise dispersion in the observed spectrum. While dealing with NH, we calculated two sets of the fitting parameters: () based on the analysis of only optically thin satellite lines with (col. 3), and () obtained from the fit to the entirety of the NH (1,1) spectrum including the main transitions with which have optical depths (col.4), as can be inferred from the relative intensities of the hfs components. The corresponding radial velocity offsets are marked by and . All hfs transitions of the HCN (2–1) line from our sample are optically thin () and their and values are listed in col. 5. The calculated limiting values of the kinetic temperature (lower limit) and the nonthermal rms velocity dispersion (upper limit) of the material presumably traced by the NH and HCN emission are given in columns 8 and 9. Column 10 presents the signal-to-noise ratio per spectral channel at the maximum intensity peak of NH/HCN.
The errors of the fitting parameters were calculated from the diagonal elements of the covariance matrix at the minimum of . In addition, the error of the radial velocity was calculated independently by the method. However, because of the spectral leakage (Klein et al. 2006), the fluxes in neighboring spectral channels are correlated (correlation coefficient ), and these errors must be further corrected. The correlation length of spectral channels can be deduced from the analysis of the autocorrelation function (ACF) of the intensity fluctuations in the spectral regions free from emission lines (computational details are given in Levshakov et al. 1997, 2002). The data become less correlated (, ) after averaging over two neighboring spectral channels. The LSM is known to be an unbiased and minimum variance estimator (Gauss-Markov theorem) irrespective of the distribution of the measurements including correlated data as well (Aitken 1934). This means that the best fit model parameters deduced from both the strongly and less correlated spectra should coincide. Numerical tests confirm this assumption: from both spectra we obtained exactly the same and values, but in case of strongly correlated data the errors representing the ellipsoid were a factor of smaller as compared with the errors based on less correlated data. Thus, errors given in parentheses in Tables 2-5, columns 3-5, represent calculated values multiplied by this factor. These errors were further transformed into uncertainties of the quantities listed in columns 6-9. We also checked both folded and unfolded spectra and found that the measured velocity offsets are not affected by the folding procedure.
Since HCN was observed with one K-band channel (Rx2) and NH with both (Rx1 and Rx2), we compared the Rx2 data of HCN with NH from the same channel and found that changes in the radial velocities were insignificant, within uncertainty range.
Besides the formal statistical (model fitting) uncertainties there can be additional errors caused by instrumental imperfections. To control this type of errors, we carried out repeated observations at 10 offsets having the same coordinates: 3, 8 (L1498), 1, 10, 12, 13 (L1512), 1, 6 (L1517BC), and 1, 2 (L1400K) in Table 1. The dispersions resulting from these repeated measurements (Tables 2-5) are m s and m s. These dispersions are systematically lower than the reported errors of and , which ensures that we are not missing any significant instrumental errors at the level of a few m s.
The comparison of the velocity dispersions determined from the NH (1,1) and HCN (2–1) lines (Tables 2-5) does not show any significant variations with position within each molecular core. All data are consistent with thermally dominated line broadening, i.e., as expected, for both sets ‘’ and ‘’ of the data points. In particular, the latter set gives the following weighted mean values: , , , and . The maximum contribution of the nonthermal motions to the velocity dispersion is observed in the L1517BC core. The weighted mean values of the velocity dispersions for NH range between m s and m s, and for HCN between m s and m s. This can be compared with the speed of sound inside a thermally dominated region of a cold molecular core which is defined as (e.g., Shu 1977)
where is the mean molecular mass. With amu for molecular clouds, one has m s, which shows that at the typical kinetic temperature of 10 K, the nonthermal velocities are in general subsonic, and that the selected targets do represent the quiescent material at different distances from the core centers.
The nonthermal velocity dispersions derived from the apparent line widths depend on the adopted gas temperature. The gas temperature in a molecular cloud is determined by the balance between heating and cooling. If the only source of heating is the cosmic rays and the cooling is due to the line radiation mainly from CO, then a lower bound on the kinetic temperature is about 8 K (Goldsmith & Langer 1978). In cloud cores where the gas is well-shielded from the background ionizing radiation the gas temperature is about 10 K (Goldsmith 2001). In particular, just this value is measured in the L1498, L1517B and L1512 cores considered here (Benson & Myers 1989; Lee et al. 2003; Tafalla et al. 2004).
|Offset||#||satellite||all||all||m s||m s||K||m s|
In the L1498 cloud, we obtain a lower limit on the kinetic temperature K (average over 8 points) which is slightly lower than K measured by a different method from the relative population of the and (1,1) levels of NH described by the rotational temperature (Tafalla et al. 2004). For L1512, the temperature averaged over 11 points (Table 3) is K which is consistent with the value of 10 K. The kinetic temperatures in the L1517BC core is well below the Tafalla et al.’s value of 9.5 K in all scanned points which means that the non-thermal velocity dispersions of NH and HCN differ significantly and that both species do not trace the same material. The three scanned points in the L1400K core show K – close to the expected value (measurements of the gas temperature in this core were not performed in previous studies). We note, however, that when the linewidths of NH and HCN are comparable, the estimate by means of Eq. (5) becomes unstable and leads to unphysical results (e.g., points , 8 in Table 3). In general, the spatial fluctuations of do not exceed a few kelvin implying the uniform heating and the absence of the localized heat sources. The kinetic temperature tends to rise with the distance from the core center (points in Table 2, in Table 3, and in Table 4) which is in line with the results of Tafalla et al. (2004).
The radial velocity profiles along the different diagonal cuts toward the selected targets are depicted in Figs. 3 and 4. The diagonal cut in L1498 between and (Fig. 3a) exhibits coherently changing velocities of (HCN) and (NH) (except the point at the core edge) with a turn at . The velocity gradient is small, km s pc. This picture coincides with the previously obtained results based on observations of CO, CS, NH, and NH in this core and was interpreted as an inward flow (Lee et al. 2001; Tafalla et al. 2004). Taking together, all available observation classify the L1498 cloud as one of the most quiet molecular core. Thus, we can expect that in this core the Doppler noise, i.e., the irregular random shifts in the radial velocities between different transitions, is minimal.
In the L1512 core, the (NH) and (HCN) distributions are almost parallel (Fig. 3b, d). The same kinematic picture was obtained for this core by Lee et al. (2001) from observations of CS and NH lines and interpreted as a simple rotation around the center. The velocity gradients derived from both the NH and HCN lines are similar, km s pc, and consistent with the gradient based on NH measurements by Caselli et al. (2002). This means that NH, HCN, and NH trace the same gas and the Doppler shifts between them should be insignificant.
In the core L1517BC which is known to be very compact (Lee et al. 2001) we observed only a small central part where the velocities of NH and HCN along two perpendicular cuts (Fig. 4a, b) do not change much: (NH) km s pc, (HCN) km s pc (panel a), and (NH) (HCN) km s pc (panel b). However, a wider area () observation of this core revealed an outward gas motion at the core periphery with a higher velocity gradient, (NH) km s pc (Tafalla et al. 2004) which is consistent with earlier results on NH observations by Goodman et al. (1993). This can lead to an additional shift in the radial velocity of HCN line since, in general, the C-bearing molecules occupy a lower density gas ( cm) in the envelope of the molecular core. A higher non-thermal velocity dispersion of the HCN line as compared to NH was already mentioned above in regard with the temperature measurements in this core.
In the L1400K core we observed only three positions. Both molecules trace the same gradient of km s pc which is in line with (NH) km s pc derived by Caselli et al. (2002). The mapping in different molecular lines by Tafalla et al. (2002) revealed that L1400K deviates significantly from spherical symmetry and exhibits quite complex kinematic structure. In particular, distributions of NH and NH do not coincide: NH has an additional component to the west from the center. This explains why Craspi et al. (2005) report for this core a blue-ward skewness, , of the the NH (1–0) hfs profiles, whereas in our observations the NH (1,1) hfs transitions are fully symmetric: at the central position, the skewness of the resolved and single hfs component of NH is .
The mean values of the velocity offsets between HCN and NH measured across each of 4 cores are presented in Table 6. For each individual molecular core we calculated the unweighted and weighted mean (weights inverse proportional to the variances) by averaging over available offsets for two datasets: based on only optically thin hfs transitions of NH (marked by index ), and on the all transitions including the strong main components of NH (marked by index ). We also used a robust redescending -estimate (Maximum-likelihood) for the mean and the normalized Median Absolute Deviation (MAD) for the scale of the distribution as described in the Appendix of Paper I. These statistics work well for inhomogeneous data sets with outliers and deviations from normality. The median values, which are robust estimates independent on the assumed distribution of data, are also presented to demonstrate the consistency of different calculations of the mean values.
We obtain very similar velocity shifts m s and m s (-estimates) for, respectively, the cores L1498 and L1512 where the minimal level of the Doppler noise is expected. A larger shift m s is observed in the L1517BC core – again in accord with the revealed kinematic structure of this core which allows us to expect a higher radial velocity for the HCN line. On the other hand, a lower value m s in L1400K may be due to irregular kinematic structure of the core center which could increase the radial velocity of the NH line. Thus, as the reference velocity offset we choose the most robust -estimate of the mean value from the L1498 and L1512 cores: m s. Taking into account that the uncertainty of the HCN (2–1) rest frequency is about 3 m s, whereas that of NH (1,1) is less than 1 m s, we finally have m s. Being interpreted in terms of the electron-to-proton mass ratio variation, as defined in Eq.(1), this velocity offset provides = ppb.
In Fig. 5, we compare the velocity offsets obtained in our observations with the 100-m Effelsberg telescope in Feb 2009 and Jan 2010. For L1498, L1512 and L1517BC, the reproducibility is very good, whereas for L1400K the results differ significantly. The reason for this discrepancy is unclear.
5.1 Kinematic velocity shifts
In two molecular cores, L1498 and L1512, with lowest Doppler noise, we register very close values of the velocity offset m s between the rotational transition HCN (2–1) and inverse transition NH (1,1), and these values coincide with the most accurate estimate obtained from the Effelsberg dataset on 12 molecular clouds in the Milky Way (Paper I). Two other cores, L1517B and L1400K, exhibit velocity shifts which are either higher ( m s in L1517B) or lower ( m s in L1400K) than the mean value, but the positive (L1517B) and negative (L1400K) deflections from the mean can be explained from the observed kinematics in these cores.
The rotational transition HCN (2–1) was chosen for several reasons: its laboratory frequency is known with an accuracy of m s, it closely traces the NH (1,1) emission, and its frequency of 18.2 GHz is close to 23.7 GHz of NH (1,1) yielding the similar angular resolution for both species. However, it is well known that the C-bearing molecules freeze out in the central core parts whereas the N-bearing molecules survive. Thus, observing the NH and HCN molecules we cannot completely reject the possibility that the velocity shifts between them are of kinematic nature.
It is clear that the simultaneous observation of the NH (1,1) inverse transition and rotational transitions of some N-bearing molecules such as, e.g., NH (1-0) and ND (1-0) would give a more stringent test. The main obstacle on this way is that the laboratory frequencies of NH and ND are known with accuracy not better than 14 m s (Paper I, Sect.4.2). Using the NH rest frequency from the Cologne Database for Molecular Spectroscopy (CDMS; Müller et al. 2005) and observing with the Nobeyama 45-m telescope we obtained a velocity shift between NH and NH of m s in L1498, m s in L1512, and m s in L1517B (Paper I). We note that similar shifts are indicated for the central parts of L1498 and L1517B in Fig. 11 from Tafalla et al. (2004) who observed these targets with the 30-m IRAM telescope and used the NH rest frequency very close to the CDMS value.
In the prestellar core L183, Pagani et al. (2009) observed NH (1–0) and ND (1–0) and found that along two perpendicular cuts the two lines are systematically shifted with respect to NH (1,1). The velocity offset is about 28 m s, when adopting the CDMS catalogue rest frequencies (Paper I; Molaro et al. 2009), but m s if the frequency of NH is taken from Caselli et al. (1995).
Thus, almost the same velocity offset between the NH inversion transition and rotational lines from different molecular species (HCN, NH, ND), different targets, and different telescopes is observed. Tentatively, this supports our initial hypothesis that the equivalence principle of local position invariance may break in low-density environment. Obviously, for more definite conclusions new laboratory measurements of the rest frequencies and new observations involving other targets and other rotational transitions are required.
In this context we note that the matter density in diffuse interstellar clouds, where the UV lines of the molecular hydrogen H are observed, is about 5 orders of magnitude lower than in molecular cores. This implies that the expected value of based on the H line position measurements should not be less than . This makes future experiments with CODEX and ESPRESSO spectrograph for extremely stable Doppler measurements (Pasquini et al. 2009) of a great importance since current limits on obtained with the UVES/VLT and HERES/Keck are at the level of a few ppm (e.g., Malec et al. 2010).
5.2 Frequency shifts caused by external fields
In this section we consider possible systematic effects caused by external fields: we start with a static electric field, then consider a thermal background radiation, and, finally, discuss a static magnetic field. We deduce all estimates for the inversion transition in ammonia. Rotational transitions can be treated in a similar way.
5.2.1 Stark effect induced by static electric fields
Static electromagnetic fields are always present and can affect the observed frequencies of microwave transitions in the laboratory and in astrophysical environments (see, e.g., Bethlem et al. 2008).
The static Stark displacement of the inversion frequency in NH is given by (Townes & Schawlow 1955)
Here is the reduced Planck constant, is the electric dipole amplitude for the NH (1,1) inversion transition, D is the molecular dipole moment, is the electric field strength, , , and are, respectively, the molecular angular momentum and its projections on the external electric field and on the molecular axis. Averaging over the projection of , one finds555The projection of ranges between and . Assuming equal population of these levels (all of them have the same energy), we have :
where is the line-of-sight velocity offset caused by the Stark effect. The largest offset occurs for the transitions. In this case the shift of m s requires a field of V cm. For laboratory conditions, this is a sufficiently large field which is easily controlled. For dense interstellar molecular clouds, such a field would accelerate electrons to ultra-relativistic energies ( eV) within their mean free path ( cm). Thus, we conclude that static electric fields cannot cause any systematic effects on this scale.
We note that the collision shift when molecules get close to each other and the NH (1,1) inversion line is shifted due to the electric dipole-induced interactions between collision partners is also less than 1 m s for the physical conditions realized in the most accurate laboratory studies with molecular fountains and in cold interstellar molecular cores (Bethlem et al. 2008).
5.2.2 Stark effect induced by black body radiation
Now we estimate the frequency shift due to the black body radiation (BBR) at a given radiation temperature . According to Farley & Wing (1981), the BBR-induced Stark shift is given by:
where is the Boltzmann constant, and is the speed of light. In Eq.(12), we take into account that the levels of the inversion doublet are shifted from each other and sum over Cartesian components of the BBR field. The universal function has the form
and is restricted to the interval . At the radiation temperature K, we are interested in the limit , where
Using this expression and averaging again over the projection of , we obtain:
We see, that the BBR shift grows with temperature and it is larger for a warm laboratory environment as compared with the interstellar 3 K background field. However, even for K the BBR shift for the inversion transition is negligible, . The BBR shift for the rotational transition of the HCN molecule is of the same order of magnitude.
5.2.3 Zeeman effect induced by static magnetic fields
Finally, we estimate the effect of a static magnetic field. Dense molecular cores have weak magnetic fields with typical strengths of G or less (Crutcher et al. 2010). For example, CCS Zeeman observations of L1498 set an upper limit on G for the line of sight component of the magnetic field (Levin et al. 2001). The line shapes can be sensitive to random motions at the Alfvén velocity associated with the magnetic field , (here is the strength in G of the line-of-sight component of the magnetic vector, is the ion density in g cm, and is in cm s). The presence of the magnetic field can be felt by the neutral material through collisions with charged particles. If we assume that in our observations the nonthermal line widths are due to Alfvénic turbulent motions whose kinetic energy is equal to the static magnetic field energy, so that , then an order of magnitude estimate of the magnetic field strength at cm, and m s (see Sect. 3) provides an upper limit on G. Since in this estimate we substitute the ion density by the total gas density, the strength of the static magnetic field in the interiors of L1498, L1512, L1517B, and L1400K should be considerably lower than this upper limit. This is more than five orders of magnitude smaller than the unscreened Earth’s magnetic field which is about half a Gauss at the surface of the Earth. Typical laboratory magnetic shields reduce the Earth’s field by 2–3 orders of magnitude to a mG level.
The order of magnitude estimate of the Zeeman splitting for NH, where magnetic interaction is dominated by 3 protons since the nitrogen -factor is small (), can be obtained as follows. For a single proton, we have
where is the nuclear magneton, is the proton g-factor, is the proton spin and is the strength of the magnetic field. Roughly, the maximum shift is realized when all three proton spins are aligned (ortho-NH), so we can use the above equation and put . With erg G and G, we obtain Hz. Since the line belongs to the para-species of the ammonia molecule, where the proton spins are not aligned, the expected frequency shift is even lower for the transition observed by us.
One should also keep in mind that the Zeeman shifts turn to zero after averaging over magnetic quantum number . Therefore, in the first approximation, magnetic fields induce a line broadening, but not a frequency shift. Thus, the systematic shift of the ammonia inversion transition of about kHz detected in the starless molecular cores cannot be explained by a static magnetic field effect.
We conclude that static fields and thermal background radiation produce systematic velocity shifts much smaller than 1 m s. Thus, field effects do not contribute significantly to the systematic error of our observations.
The present paper continues our studies where we use the ammonia inversion transition in conjunction with low-lying rotational transitions of other molecules to probe the dependence of the electron-to-proton mass ratio on the ambient matter density. As experimental tests, we suggest to observe dense prestellar molecular cores located in the disk of the Milky Way. Here we report on mapping of four molecular cores (L1498, L1512, L1517B, and L1400K) in two molecular transitions of NH and HCN with high-spectral resolution at the 100-m Effelsberg telescope. The main results are as follows.
The completely resolved hfs components of the NH (1,1) and HCN (2–1) transitions allow us to determine the line centers with the precision of m s, which is comparable with laboratory uncertainties of the rest frequencies of these transitions.
In two cores, L1498 and L1512, with lowest Doppler noise, we obtain a statistically significant positive velocity offset between the rotational HCN and inversion NH transitions of m s. In two other cores, L1517B and L1400K, the velocity offsets are, respectively, higher and lower, but these deflections can be explained by the observed kinematic structure in L1517B and L1400K. If we assume that the measured velocity offset is caused by the electron-to-proton mass ratio variation, then = ppb. The non-zero implies that at deep interstellar vacuum the electron-to-proton mass ratio increases by as compared with its terrestrial value.
The reproducibility of the velocity offset at the same facility (Effelsberg telescope) on the year-to-year base is very good, except for the L1400K target where the central point is probably an outlier.
We show that the effects of static electric and magnetic fields and the BBR-induced Stark shifts are not larger than 1 m s, and can be neglected in the total error budget.
The results obtained tentatively support the hypothesis that in low-density environment the equivalence principle may break. This may be a consequence of the chameleon-like scalar field. New laboratory measurements of the rest frequencies and new observations involving other rotational transitions and other targets (e.g., Levshakov et al. 2010b) are required to reach more definite conclusions.
Acknowledgements.We are grateful to the staff of the Effelsberg radio observatory for assistance in our observations and to Drs. Peter Müller, Jürgen Neidhöfer, and Johann Schramml for technical details on the 100-m telescope instrumentation. The project has been supported in part by DFG Sonderforschungsbereich SFB 676 Teilprojekt C4, the RFBR grants No. 09-02-12223, 09-02-00352, and 08-02-92001, the Federal Agency for Science and Innovations grant NSh-3769.2010.2, the Program IV.12/2.5 of the Physical Department of the RAS, and by the Chinese Academy of Sciences visiting professorship for senior international scientists grant No. 2009J2-6.
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