Spatial and temporal variations of fundamental constants

Spatial and temporal variations of fundamental constants

S. A.  Levshakov Ioffe Physical-Technical Institute, St. Petersburg, Russia
lev@astro.ioffe.rssi.ru
   I. I.  Agafonova Ioffe Physical-Technical Institute, St. Petersburg, Russia
lev@astro.ioffe.rssi.ru
   P.  Molaro INAF – Osservatorio Astronomico di Trieste, Trieste, Italy    D.  Reimers Hamburger Sternwarte, Hamburg, Germany
Abstract

Spatial and temporal variations in the electron-to-proton mass ratio, , and in the fine-structure constant, , are predicted in non-Standard models aimed to explain the nature of dark energy. Among them the so-called chameleon-like scalar field models predict strong dependence of masses and coupling constants on the local matter density. To explore such models we estimated the parameters and in two essentially different environments, – terrestrial (high density) and interstellar (low density), – from radio astronomical observations of cold prestellar molecular cores in the disk of the Milky Way. We found that , and . If only a conservative upper limit is considered, then . We also reviewed and re-analyzed the available data on the cosmological variation of obtained from Fe i and Fe ii systems in optical spectra of quasars. We show that statistically significant evidence for the changing at the level of has not been provided so far. The most stringent constraint on was found from the Fe ii system at towards the bright quasar HE 0515–4414. The limit of corresponds to the utmost accuracy which can be reached with available to date optical facilities.

Line: profiles – ISM: molecules – Radio lines: ISM – Techniques: radial velocities – quasars: absorption lines – Cosmology: observations
\idline

00001

\subtitle\offprints

S. A. Levshakov

1 Introduction

Spatial and temporal variations in the electron-to-proton mass ratio, , and in the fine-structure constant, , are not present in the Standard Model of particle physics but they arise quite naturally in grant unification theories, multidimensional theories and in general when a coupling of light scalar fields to baryonic matter is considered (Uzan, 2003; Martins, 2008; Chin et al., 2009). The 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). However, scalar fields cause a problem since they could violate the equivalence principle which has never been detected in local tests (Turyshev et al., 2004). A plausible solution of this dissension was suggested with a so-called ‘chameleon’ model which assumes that a light scalar field acquires an effective potential and an effective mass due to its coupling to matter that strongly depends on the ambient matter density (Khoury & Weltman, 2004; Mota & Shaw, 2007; Olive & Pospelov, 2008). In such a way the scalar field may evade local tests of the equivalence principle since the range of the scalar-mediated force is too short ( mm) to be revealed at the terrestrial matter densities. This is not the case, however, for the 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 very large ( pc).

Calculations of atomic and molecular spectra show that different transitions have different sensitivities to changes in fundamental constants (Varshalovich & Levshakov, 1993; Dzuba et al., 1999; Kozlov et al., 2008). Thus, measuring relative radial velocities, , between such transitions one can probe the hypothetical variation of physical constants. For instance, a spatial dependence of and on the ambient matter density can be tested locally using terrestrial measurements of molecular transitions and comparing them with radio astronomical observations of molecular clouds in the disk of the Milky Way (here we have a typical difference between the environmental gas densities of 19 orders of magnitude). Complementary to these measurements, a temporal dependence of and on cosmic time can be tested through observations of high-redshift intervening absorbers seen in quasar spectra.

In this presentation we summarize our recent results on spatial and temporal variations of and obtained at high redshifts (QSO absorption systems) and at (the Milky Way disk).

2 Cosmological tests ()

The cosmological variability of can be probed by different methods. One of them, — the many-multiplet method, — was suggested by Dzuba et al. (1999) and extensively used in the analysis of quasar metal absorbers (Webb et al., 1999; Murphy et al., 2003; Chand et al., 2004). This method is based on the comparison of wavelengths of different transitions in different ions having different sensitivity coefficients to the variation of and rising from the same QSO absorption-line system. However, the approach to estimate  on base of different ions seems not to be very favorable: it requires many absorption systems in order to suppress kinematic effects caused by irregular velocity shifts between different ions, and its averaging procedure over many sight lines and over large range of redshifts may smear out a putative weak signal in . It is clear that in order to obtain  estimates at every particular redshift the kinematic effects must be suppressed. Such a procedure can be realized if lines of only one ion and arising from the same atomic levels are utilized (Levshakov, 2004). The corresponding method, — the single ion differential measurement (SIDAM), — and its applications to Fe i and Fe ii systems are described below.

Fe i and Fe ii are represented in optical spectra of quasars by several resonance transitions having different sensitivity coefficients. Under these conditions the accuracy of the  estimates is limited by only two factors: (1) uncertainties in the calibration of the wavelength scale (see, e.g., Centurión et al., these proceedings), and (2) unknown abundances of iron isotopes in a particular absorption system. For quasar spectra taken with the VLT, it is possible to reach the accuracy in the wavelength calibration of 30-50  (Molaro et al., 2008a) which translates into the error in  of 2. The boundaries on isotopic composition of iron do not exceed (Kozlov et al., 2004; Porsev et al., 2009). Thus, a single absorption system with iron ions can provide the accuracy of the  estimate at the level of .

2.1 Fe i at  = 0.45

Many resonance transitions of neutral iron Fe i associated with the absorption-line system at  = 0.45 were identified in the spectrum of HE 0001–2340 by D’Odorico (2007). Among them there are Fe i lines with quite different sensitivity coefficients which makes such systems suitable for individual  estimates. The sensitivity coefficients for Fe i transitions were calculated by Dzuba & Flambaum (2008) on our request and revealed to be highly sensitive to change.

Figure 1: Combined absorption-line spectra of Fe i associated with the = 0.45 absorption-line system towards HE 0001–2340 (normalized intensities are shown by dots with error bars). The smooth curves show the model profiles. The normalized . The zero radial velocity is fixed at .

The profiles of the six selected Fe i lines are shown in Fig. 1. In this case a simple one-component model can describe adequately the observed intensities. The most probable value of  = was calculated from the -curve presented in Fig. 2. Here we combined the Fe i 2967, 3021 Å lines having and the Fe i 2484, 3441, 3720, 3861 Å lines for which . In Fig. 2, the parameter is the velocity offset between these two groups of Fe i lines. In this case the estimate of  is given by

(1)

where is the difference between the radial velocities of these two groups of lines, and .

We expect that with higher spectral resolution and higher S/N the accuracy of the  estimate at  = 0.45 can be increased by several times since the Fe i lines in this system are extremely narrow (the Doppler width ) and have very simple profiles (D’Odorico, 2007).

Figure 2: as a function of the velocity difference between the Fe i 2967, 3021 Å lines and 2484, 3441, 3720, 3861 Å lines for the one-component model shown in Fig. 1. The confidence level is determined by which gives .
Figure 3: Profiles of Fe ii lines associated with the = 1.84 damped Ly system towards Q 1101–264 (normalized intensities are shown by dots with error bars). The ratios per pixel (from the top panel to the bottom) are 100, 138, 120, 120. The smooth curves show the model profiles. Parts of the Fe ii 2374 Å line are contaminated in the ranges  and  with telluric absorptions and not included in the analysis. The positions of the subcomponents of the equidispersion 17-component model are marked by the dotted vertical lines. The normalized . The zero radial velocity is fixed at .

2.2 Fe ii at  = 1.84

The only QSO spectrum which was obtained especially with the objective to measure  is the VLT/UVES spectrum of a bright quasar Q 1101–264 (Levshakov et al., 2007). This spectrum was observed with a high resolution (FWHM = 3.8 ) and S/N . Analyzing the Fe ii 2600, 2382, and 1608 Å lines from the absorption system with  = 1.84, we found  between the Fe ii 1608 Å line which has negative sensitivity coefficient and the Fe ii 2600, 2382 Å transitions having almost identical positive sensitivity coefficients. In terms of  this shift corresponds to  = .

When new sensitivity coefficients for the Fe ii lines became available (Porsev et al., 2007), we repeated the analysis of this system including the unblended parts of the Fe ii 2374 Å line. The profile of this line is blended in the velocity range  with telluric absorption and because of that it was not considered in Levshakov et al. (2007). This line has positive sensitivity coefficient like the Fe ii 2600, 2382 Å lines, but its oscillator strength is close to that of the weak line Fe ii 1608 Å. This allows us to control possible saturation effects in the strong lines Fe ii 2600, 2382 Å caused by unresolved components which may lead to some radial velocity shifts between the apparent positions of the strong and weak Fe ii lines. The profiles of all four Fe ii lines (Fig. 3) were fitted by a series of mathematical models differing both in the number of components (from 8 to 17) and in the types of components: either all components have individual -parameters or all components have identical -parameters — the so-called equidispersion deconvolution (Levshakov et al., 1999).

Figure 4: as a function of the velocity difference between the Fe ii Å and Å lines (shown in Fig. 3) for the 8- and 17-component equidispersion models. The confidence level is determined by which gives  for both represented models.

The 1 confidence limit for the velocity shift between the Fe ii 1608 Å and Fe ii 2600, 2383, 2374 Å lines was calculated via the -curve (Fig. 4) and by means of Monte-Carlo simulations (Fig. 5). In turn, modeling of the data points in the Monte-Carlo simulations occurred also in two ways: (1) the method of bootstrapping residuals, and (2) the residuals were modeled by a Markov chain. The bootstrap procedure breaks correlations between the data points presented in the original QSO spectra (see, e.g., Appendix A in Levshakov et al. 2002), whereas the Markov chain approximation to the residuals allows us to save the original correlation in the simulated data. In all approaches the negative velocity shift between the Fe ii 1608 Å line and the Fe ii 2600, 2382, 2374 Å lines was stable reproduced with the most probable value of  (as a 1 limit the most conservative value from all trials is taken). In terms of  this corresponds to  = . We consider the result robust in this respect in disagreement with what claimed by Murphy et al. (2008) who have not provided a quantitative analysis of the system to support their claim.

Figure 5: Histograms of the simulated samples generated by the bootstrapping residuals (panel a) and by the Markov chain approximation (panel b) for the 17-component equidispersion model. The sample size in both cases. The smooth curves are Gaussians defined by the first two moments of the corresponding samples: (panel a), and (panel b).

2.3 Fe ii at  = 1.15

The most accurate to date estimate of  was obtained for the Fe ii absorption system at = 1.15 detected in the spectrum of one of the brightest high-redshift quasar HE 0515–4414 (Reimers et al., 1998). Quast et al. (2004) used co-added exposures taken with the VLT/UVES and reported  = . Chand et al. (2006) re-observed this system with the high resolution and temperature stabilized spectrograph HARPS mounted on the ESO 3.6 m telescope at the La Silla observatory and obtained  = . We note that in both publications a wavelength calibration error was not taken into account, and once considered this their errors would be a factor two larger.

Due to the brightness of HE 0515–4414, the ratio in individual exposures of the VLT spectra is sufficiently high which allows us to process them separately, i.e. without co-adding. Levshakov et al. (2006) analyzed the best quality individual exposures and then averaged  values produced from the pairs of Fe ii lines (combinations of the blue line Fe ii 1608 Å and different red Fe ii lines). The resulting mean value was  = . However, it was overlooked that in this procedure the individual  values from different pairs become correlated. Accounting for these correlations, as described in Molaro et al. (2008b), and using the updated sensitivity coefficients from Porsev et al. (2007) we correct the latter  value to  = . Thus, all  estimates at  = 1.15, which are the most accurate so far, exclude any cosmological variations of at the level of .

3 Local tests ()

3.1 Estimate of

Among numerous molecules observed in the interstellar medium, ammonia NH is of a particular interest for the  tests due to high sensitivity of the inversion frequency at 23.7 GHz to a change in . Here  . The sensitivity coefficient () of the inversion transition was calculated by (Flambaum & Kozlov, 2007):

(2)

where and are the frequencies corresponding to the laboratory value of and to an altered in a low-density environment, respectively. For the rotational frequencies we have and

(3)

Comparing the apparent radial velocities for the NH inversion transition, , with rotational transitions, , of other molecules arising co-spatially with NH one finds

(4)

where is the speed of light.

The parameter in (4) could be considered as the sum of two components, , with being the shift due to -variation, and is the so-called Doppler noise. The input of the Doppler noise to a putative  signal can be reduced to some extent if the velocity shifts caused by inhomogeneous distribution of molecules and instrumental imperfections are minimized (for details, see Levshakov et al. 2009a).

To probe  under different local environments, we analyzed at first high resolution spectra (FWHM = 25 ) of the NH and CCS transitions obtained with the 100-m Green Bank Telescope (GBT) by Rosolowsky et al. (2008) and Rathborne et al. (2008), and moderate resolution spectra (FWHM = ) of the NH , HCN , and NH transitions observed with the 45-m Nobeyama radio telescope by Sakai et al. (2008). These spectra showed a systematic velocity shift between rotational and inversion transitions with the most accurate value of  based on the NH and CCS lines (Levshakov et al., 2008a).

Then we carried out our own observations of ammonia and other molecules with the 32-m Medicina, 100-m Effelsberg, and 45-m Nobeyama radio telescopes (Levshakov et al., 2009a).

Observations at the Medicina telescope were performed with both available digital spectrometers ARCOS (ARcetri COrrelation Spectrometer) and MSpec0 (high resolution digital spectrometer) which have channel separations of 4.88 kHz and 2 kHz, respectively. For ARCOS, this resolution corresponds to 62  at the position of the ammonia inversion transition (23 GHz) and 80  for the rotational HCN (2–1) line (18 GHz). For MSpec0, it is 25  and 32  at the corresponding frequencies.

The Effelsberg data were obtained with the HEMT (High Electron Mobility Transistor) dual channel receiver. The measurements were obtained in a frequency switching mode, with a frequency throw of 5 MHz. The backend was an FFTS (Fast Fourier Transform Spectrometer), operated with its minimum bandwidth of 20 MHz and simultaneously providing 16384 channels for each polarization. The resulting channel widths are 15.4 and 20.1  for NH and HCN, respectively.

At Nobeyama we used a low-noise HEMT receiver, H22, for the NH observations and the two sideband-separating SIS (Superconductor-Insulator-Superconductor) receiver, T100 (Nakajima et al., 2008), for the NH observations. Both of them are dual polarization receivers. We observed two polarization simultaneously. Autocorrelators were employed as a backend with bandwidth and channel separation of 4 MHz and 4.375 kHz, respectively. This corresponds to the channel widths of 57  at 23 GHz, and 14  at 93 GHz.

The most accurate estimate in this case was based on the comparison of the radial velocities of the NH inversion line with the rotational transition of HCN observed with the 100-m Effelsberg telescope: . The distribution of radial velocity offsets between the rotational and inversion transitions is shown in Fig. 6. The points represent molecular clouds with symmetric line profiles which can be described by a single-component Gaussian model. This requirement provides a better accuracy of the line center measurements. In the selected clouds the widths of the emission lines do not exceed greatly the Doppler width due to thermal motion of material (typical kinetic temperatures K). This ensures that turbulent motion does not dominate in the line broadening and thus the selected molecular lines sample the same kinetic temperature and arise most likely co-spatially.

Figure 6: Radial velocity offsets, , between the HCN and NH transitions for the data obtained with the 32-m and 100-m telescopes, and between the NH and NH transitions for the 45-m telescope (1 statistical errors are indicated). is the ratio of the Doppler -parameters: (NH)(HCN), or (NH)(NH). Given in parentheses are errors. The filled circles mark sources with thermally dominated motions.

The difference between the GBT and Effelsberg values is probably connected to uncertainties in the laboratory frequencies. The most accurate frequencies for the ammonia inversion transitions are measured with the uncertainty  (Kukolich, 1967; Hougen, 1972). The HCN laboratory frequencies are known with the error  (Müller et al., 2005). The worst known is the frequency of the CCS transition. The GBT radial velocities of CCS were measured with the frequency MHz estimated by Yamamoto et al. (1990) from radio astronomical observations (1 kHz at 22.3 GHz corresponds to ). The only laboratory measurement available gives MHz (Lovas et al., 1992). With this frequency we have . The CDMS catalogue value MHz (Müller et al., 2005) yields  which is in line with the Effelsberg measurements based on the NH and HCN lines.

These results show the importance of high precision laboratory measurements which are expected to provide radio frequencies with uncertainties 1  to be consistent with contemporary radio astronomical observations.

If we interpret the most accurate to date Effelsberg measurements of in terms of the electron-to-proton mass ratio variation, then  .

3.2 Constraint on

Complementary to the  estimate, constraints on -variation can be obtained from the comparison of the atomic fine-structure (FS) and molecular rotational transitions (Kozlov et al., 2008; Levshakov et al., 2008b). In this case the velocity difference between the rotational and FS lines is sensitive to the combination of and , :

(5)

where .

For practical applications we may compare the FS transitions of carbon [C i] with rotational transitions of CO. The spatial distribution of CO is closely traced by the [C i] FS lines (Spaans & van Dishoeck, 1997; Ikeda et al., 2002; Papadopoulos et al., 2004). The rest hyper-fine frequencies of low- rotational transitions of CO are known with very high accuracy,  (Cazzoli et al., 2004). The most precise rest frequencies to date for the [C i] transition 492160.651(55) MHz (Yamamoto & Saito, 1991) and transition 809341.97(5) MHz (Klein et al., 1998) restrict the uncertainties of the line centers to  and 18.5 , respectively. Thus, if we take  as the most conservative error of the line centering in a pair of CO–[\ionCi] lines, then the limiting accuracy for is about ppm.

The analysis of the published results on [C i] and CO low resolution observations (FWHM = ) of cold molecular cores gives  (Levshakov et al., 2009b), which leads to a limit . With the obtained estimate of , one finds from (5) that .

This result is already an order of magnitude more sensitive than the bound on the cosmological -variation found from absorption systems of quasars. The present estimate of  at can be considerably improved if higher resolution spectra of molecular cores will be obtained and the rest frequencies of the [C i] FS transitions will be measured with better accuracy.

4 Conclusions

Our current results of astrophysical tests on spatial and temporal variations of fundamental constants can be summarized as follows.

  1. The relative radial velocities of the rotational transitions in HCN and the inversion transition in NH measured with the 100-m Effelsberg radio telescope reveal a velocity offset of . This offset is regularly reproduced in observations of different cold molecular cores with different facilities at the 32-m Medicina and 45-m Nobeyama radio telescopes.

  2. Being interpreted in terms of the electron-to-proton mass ratio variation, the found velocity offset corresponds to  = . To cope with negative results obtained in laboratory experiments, a chameleon-like scalar field is required to explain the non-zero value.

  3. A new approach to probe -variations using the fine-structure transitions in atomic carbon [C i] and low- rotational transitions in CO is suggested, and a strong constraint on the dependence of on the ambient matter density is deduced at : .

  4. Another method to measure the cosmological variability of from resonance transitions of neutral iron Fe i is discussed and applied to the Fe i system at  = 0.45. The resulting constraint is  = .

  5. The Fe ii system at  = 1.84 is re-analyzed and a corrected value of  is obtained:  = .

  6. For the Fe ii system at  = 1.15 the overlooked correlations between individual  values are now taken into account that gives  = .

  7. It is shown that spectral radio observations in the Milky Way are at present the most sensitive tool to probe the values of fundamental constants at different ambient physical conditions. The internal (statistical) error of our Effelsberg data is which is about 1000 times lower as compared with the error in optical observations of quasars.

  8. Since extragalactic molecular clouds have gas densities similar to those in the interstellar medium, the value of  in high- molecular systems is expected to be at the same level as in the Milky Way, i.e. 10 — providing no temporal dependence of the electron-to-proton mass ratio is present.

To conclude, we note that in order to be completely confident that the revealed velocity shift between rotational and inversion lines is not due to kinematic effects in the molecular clouds but reflects a density-modulated variation of , new high precision radio-astronomical observations are needed for a wider range of objects. Besides, such observations should include essentially different molecules with tunneling transitions sensitive to the changes in and molecules with -doublet lines which also exhibit enhanced sensitivity to variations of and (Kozlov 2009). It is also very important to measure the rest frequencies of molecular transitions with an accuracy of about 1  in laboratory experiments — in some cases the present uncertainties in the rest frequencies are larger than the errors of radio-astronomical measurements. In addition, search for variations of the fine-structure constant in the Milky Way disk using mid- and far-infrared fine-structure transitions in atoms and ions, or search for variations of the combination of using the [C ii] and/or [C i] fine-structure transitions and low-lying rotational lines of CO would be of great importance for cross-checking the results.

Acknowledgements.
The authors are grateful to the staffs of the Medicina, Effelsberg, and Nobeyama radio observatories for technical assistance in observations. The project has been supported by DFG Sonderforschungsbereich SFB 676 Teilprojekt C4, the RFBR grants 09-02-12223 and 09-02-00352, and by the Federal Agency for Science and Innovations grant NSh 2600.2008.2.

References

  • Caldwell et al. (1998) Caldwell, R. R., Dave, R., & Steinhardt, P. J. 1998, Phys. Rev. Lett., 80, 1582
  • Cazzoli et al. (2004) Cazzoli, G., Puzzarini, C., & Lapinov, A. V. 2004, ApJ, 611, 615
  • Chand et al. (2006) Chand, H., Srianand, R., Petitjean, P., et al. 2006, A&A, 451, 45
  • Chand et al. (2004) Chand, H., Srianand, R., Petitjean, P., & Aracil, B. 2004, A&A 417, 853
  • Chin et al.  (2009) Chin, C., Flambaum, V. V., & Kozlov, M. G. 2009, NJPh, 11, 055048
  • D’Odorico (2007) D’Odorico, V. 2007, A&A, 470, 523
  • Dzuba & Flambaum (2008) Dzuba, V. A., & Flambaum, V. V. 2008, Phys. Rev. A, 77, 012514
  • Dzuba et al. (1999) Dzuba, V. A., Flambaum, V. V., & Webb, J. K. 1999, Phys. Rev. A, 59, 230
  • Flambaum & Kozlov (2007) Flambaum, V. & Kozlov, M. G. 2007, Phys. Rev. Lett., 98, 240801
  • Hougen (1972) Hougen, J. T. 1972, J. Chem. Phys., 57, 4207
  • Ikeda et al. (2002) Ikeda, M., Oka, T., Tatematsu, K., et al. 2002, ApJS, 139, 467
  • Khoury & Weltman (2004) Khoury, J., & Weltman, A. 2004, Phys. Rev. Lett., 93, 171104
  • Klein et al. (1998) H. Klein, H., Lewen, F., Schieder, R., et al. 1998, ApJ, 494, L125
  • Kozlov (2009) Kozlov, M. G. 2009, Phys. Rev. A, 80, 022118
  • Kozlov et al. (2008) Kozlov, M. G., Porsev, S. G., Levshakov, S. A., et al. 2008, Phys. Rev. A, 77, 032119
  • Kozlov et al. (2004) Kozlov, M. G., Korol, V. A., Berengut, J. C., et al. 2004, Phys. Rev. A, 70, 062108
  • Kukolich (1967) Kukolich, S. G. 1972, Phys. Rev., 156, 83
  • Levshakov et al. (2009a) Levshakov, S. A., Molaro, P., Lapinov, A. V., et al. 2009a, A&A, in press (astro-ph/0911.3732)
  • Levshakov et al. (2009b) Levshakov, S. A., Molaro, P., Reimers, D. 2009b, A&A, submitted
  • Levshakov et al. (2008a) Levshakov, S. A., Molaro, P., & Kozlov, M. G. 2008, arXiv: astro-ph/0808.0583
  • Levshakov et al. (2008b) Levshakov, S. A., Reimers, D., Kozlov, M. G., et al. 2008, A&A, 479, 719
  • Levshakov et al. (2007) Levshakov, S. A., Molaro, P., Lopez, S., et al. 2007, A&A, 466, 1077
  • Levshakov et al. (2006) Levshakov, S. A., Centurión, M., Molaro, P., et al. 2006, A&A, 449, 879
  • Levshakov (2004) Levshakov, S. A. 2004, in Astrophysics, Clocks and Fundamental Constants eds. S. G. Karshenboim and E. Peik (Springer, Berlin), p. 151
  • Levshakov et al. (2002) Levshakov, S. A., Dessauges-Zavadsky, M., D’Odorico, S., & Molaro, P. 2002, ApJ, 565, 696
  • Levshakov et al. (1999) Levshakov, S. A., Takahara, F., & Agafonova, I. I. 1999, ApJ, 517, 609
  • Lovas et al. (1992) Lovas, F. J., Suenram, R. D., Ogata, T., & Yamamoto, S. 1992, ApJ, 399, 325
  • Martins (2008) Martins, C. J. A. P. 2008, in Precision Spectroscopy in Astrophysics, eds. N. C. Santos, L. Pasquini, A. C. M. Correia, and M. Romaniello (Springer, Berlin), p. 89
  • Molaro et al. (2008a) Molaro, P., Levshakov, S. A., Monai, S., et al. 2008a, A&A, 481, 559
  • Molaro et al. (2008b) Molaro, P., Reimers, D., Agafonova, I. I., & Levshakov, S. A. 2008b, EPJST, 163, 173
  • Mota & Shaw (2007) Mota, D. F., & Shaw, D. J. 2007, Phys. Rev. D, 75, 063501
  • Murphy et al. (2008) Murphy, M. T., Webb, J. K., & Flambaum, V. V. 2008, MNRAS 384, 1053
  • Murphy et al. (2003) Murphy, M. T., Webb, J. K., & Flambaum, V. V. 2003, MNRAS 345, 609
  • Müller et al. (2005) Müller, H. S. P., Schlöder, F., Stutzki, J., & Winnewisser, G. 2005, J. Mol. Struct., 742, 215
  • Nakajima et al. (2008) Nakajima, T., Sakai, T., Asayama, S., et al. 2008, PASJ, 60, 435
  • Olive & Pospelov (2008) Olive, K. A., & Pospelov, M. 2008, Phys. Rev. D, 77, 043524
  • Papadopoulos et al. (2004) Papadopoulos, P. P., Thi, W.-F., & Viti, S. 2004, MNRAS, 351, 147
  • Peebles & Rata (2003) Peebles, P. J. E., & Rata, B. 2003, Rev. Mod. Phys., 75, 559
  • Porsev et al. (2009) Porsev, S. G., Kozlov, M. G., & Reimers, D.  2009, Phys. Rev. A, 79, 032519
  • Porsev et al. (2007) Porsev, S. G., Koshelev, K. V., Tupitsyn, I. I., et al. 2007, Phys. Rev. A, 76, 052507
  • Quast et al. (2004) Quast, R., Reimers, D., & Levshakov, S. A.  2004, A&A, 415, L7
  • Rathborne et al. (2008) Rathborne, J. M., Lada, C. J., Muench, A. A., et al. 2008, ApJS, 174, 396
  • Reimers et al. (1998) Reimers, D. Hagen, H.-J., Rodriguez-Pascual, P., & Wisotzki, L.  1998, A&A, 334, 96
  • Rosolowsky et al. (2008) Rosolowsky, E. W., Pineda, J. E., Foster, J. B. et al. 2008, ApJS, 175, 509
  • Sakai et al. (2008) Sakai, T., Sakai, N., Kamegai, K., et al. 2008, ApJ, 678, 1049
  • Spaans & van Dishoeck (1997) Spaans, M., & van Dishoeck, E. F. 1997, A&A, 323, 953
  • Turyshev et al.  (2004) Turyshev, S. G., Williams, J. G., Nordtvedt, K., Jr., et al. 2004, Lect. Notes Phys., 648, 311
  • Uzan (2003) Uzan, J.-P. 2003, Rev. Mod. Phys., 75, 403
  • Webb et al. (1999) Webb, J. K., Flambaum, V. V., Churchill, C. W., et al. 1999, Phys. Rev. Lett. 82, 884
  • Varshalovich & Levshakov (1993) Varshalovich, D. A., & Levshakov, S. A. 1993, JETP Lett., 58, 231
  • Yamamoto & Saito (1991) S. Yamamoto, S., & Saito, S. 1991, ApJ, 370, L103
  • Yamamoto et al. (1990) Yamamoto, S., Saito, S., Kawaguchi, K., & Ohishi, M. 1990, ApJ, 361, 318
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
354205
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description