# Limits on a Gravitational Field Dependence of the Proton–Electron Mass Ratio from H in White Dwarf Stars

###### Abstract

Spectra of molecular hydrogen (H) are employed to search for a possible proton-to-electron mass ratio () dependence on gravity. The Lyman transitions of H, observed with the Hubble Space Telescope towards white dwarf stars that underwent a gravitational collapse, are compared to accurate laboratory spectra taking into account the high temperature conditions ( K) of their photospheres. We derive sensitivity coefficients which define how the individual H transitions shift due to -dependence. The spectrum of white dwarf star GD133 yields a constraint of for a local environment of a gravitational potential , while that of G2938 yields for a potential of .

###### pacs:

97.20.Rp, 06.20.Jr, 33.20.Lg, 14.20.DhTheories of high-energy physics with a non-unique vacuum state, that invoke extra dimensions, or contain new light scalar fields can permit or require space-time variations of the fundamental low-energy “constants” of nature Bekenstein1982 (); Uzan2011 (). Small time-variations of non-gravitational constants have negligible effects on the expansion dynamics of the universe but have potentially observable effects on astronomical spectra. Self-consistent scalar-tensor theories for the variation of these constants (analogous to Brans-Dicke theory BD1961 () for a varying gravitation “constant”, ) are needed to evaluate their full cosmological consequences. Theoretical studies have focused on a varying fine-structure constant , which is simplest to develop because of its gauge symmetry Sandvik2002 (); Barrow2012 (), and a varying proton-electron mass ratio , Barrow2005a (); Scoccola2008 (). Scaling arguments have been used to relate changes in , to changes in using the internal structure of the standard model, including supersymmetry Calmet2002 (). Typically (in the absence of unusual cancellations involving the rates of change of , and the supersymmetry-breaking and grand unification energy scales), they predict that changes in at low energies should be about an order of magnitude greater than those in . However, high-redshift cosmological bounds on variation are expected to be weaker than those from laboratory tests of the equivalence principle Barrow2005a (). Indications of possible variations of in time Webb1999 () and space Webb2011 () and time variations in Reinhold2006 () have been reported. Systematic investigations of the spectra of cold H towards quasar sources have now produced a constraint on -variation over cosmological time scales yielding at redshifts , corresponding to look-back times of 10-12 Gyr Malec2010 (); Bagdonaite2014 ().

Besides dependencies on cosmological scales, the couplings between light scalar fields and other fields can generate dependencies of coupling strengths on the local matter density Sandvik2002 (); Khoury2004 (), or on local gravitational fields Magueijo2002 (); Flambaum2008 (). Such couplings violate the Einstein equivalence principle that is fundamental to General Relativity BD1961 (); Will2014 (). The gravitational potential at distance from an object of mass is commonly expressed in dimensionless units of . A number of studies have been performed using ultrastable lasers and atomic clocks exploiting the eccentricity of the Earth’s orbit Ferrell2007 (); Fortier2007 (); Blatt2008 (); Shaw2008 () causing sinusoidal changes of . Recently, a spectroscopic study of Fe V and Ni V ions in the local environment of the photosphere of a white dwarf was employed to assess the dependence of in a strong gravitational field () Berengut2013 (). In the present study we use the spectrum of molecular hydrogen in the photosphere of two white dwarfs, GD133 (WD 1116026) and G2938 (WD 2326049), obtained with the Cosmic Origins Spectrograph on the Hubble Space Telescope Xu2013 (), to probe a possible dependence of on a gravitational potential that is times stronger than its value at the Earth’s surface (which is actually dominated by the contribution from the Sun’s potential).

In Fig. 1 an overview of the H absorption lines in the G2938 photosphere is shown for the wavelength range 1337–1347 Å. The total spectrum covers wavelengths from 1144 to 1444 Å. The data of both G2938 and GD133 were retrieved from the Hubble Space Telescope archive ^{1}^{1}1HST-archive, Cycle 18, program 12290, PI M. Jura.. The individual exposures (3 of G2938 and 5 of GD133) were rebinned to a common wavelength scale and combined using the same techniques as in Malec2010 (); Bagdonaite2014 (). For both stars, lines pertaining to the – Lyman band are solidly detected in the range 1298 – 1444 Å at a signal-to-noise ratio of 15. The – Werner band transitions fall in the range 1144 – 1290 Å at a lower signal-to-noise ratio of 5 and are only weakly detected and thus are not considered in the present analysis. Due to the high temperature in the photosphere, the observed H lines are from multiple vibrationally – and rotationally – excited levels of the ground electronic state. The most intense H Lyman transitions involve the () bands for and vibrational levels with the highest population in the level (at K).

The laboratory wavelengths are derived from combination differences using level energies in the states from Refs. Bailly2010 (); Abgrall1993 (). The ground state level energies used in the derivation are from ab initio calculations including relativistic and quantum electrodynamical effects Komasa2011 (), with estimated uncertainties better than 0.001 cm, which were tested in metrology laser experiments Salumbides2011 (); Dickenson2013 (). The most accurate transition wavelengths are those derived from Ref. Bailly2010 () (for ) with relative accuracies at the level, while those derived from Ref. Abgrall1993 (), for higher quantum numbers, exhibit relative accuracies of . When the level energies from Ref. Abgrall1993 () are used, an energy correction for each band is applied (typically 0.04-0.06 cm) based on the comparison of Refs. Bailly2010 (); Abgrall1993 () at low quantum numbers. The most intense transitions are listed in Table 1, and the complete list involves around 1500 lines Salumbides ().

Transition | Wavelength | |||||
---|---|---|---|---|---|---|

0 | 3 | R(9) | 1 313. | 376 43 (2) | -0.106 | 0.0494 |

0 | 3 | P(9) | 1 324. | 595 01 (2) | -0.115 | 0.0538 |

0 | 3 | P(11) | 1 345. | 177 88 (2) | -0.129 | 0.0508 |

0 | 4 | R(7) | 1 356. | 487 60 (2) | -0.114 | 0.0821 |

0 | 4 | R(9) | 1 371. | 422 41 (2) | -0.125 | 0.0816 |

0 | 4 | R(11) | 1 389. | 593 79 (2) | -0.138 | 0.0816 |

0 | 4 | R(13) | 1 410. | 648 (1) | -0.152 | 0.0821 |

0 | 4 | P(9) | 1 383. | 659 16 (2) | -0.134 | 0.0739 |

0 | 4 | P(11) | 1 403. | 982 60 (2) | -0.148 | 0.0765 |

0 | 4 | P(13) | 1 427. | 013 40 (2) | -0.163 | 0.0793 |

Sensitivity coefficients due to a variation in were calculated for each transition using a semi-empirical method based on the experimentally-determined level energies. The coefficient is separated into electronic (), vibrational () and rotational () contributions, which are calculated via

(1) |

with related to and to separate the and contributions. In the framework of the Born–Oppenheimer approximation the are set to zero. The method is related to the Dunham approach Ubachs2007 (), but turns out to be more robust due to the elimination of correlations in the fitting of the parameters of the Dunham matrix. This effect is more problematic for Dunham representations of levels with higher values of quantum numbers , where the dominant contribution of the higher-order terms in the expansion are susceptible to numerical errors. In contrast, the present method does not have this disadvantage and the accuracy of the values are just limited by the experimental data. The uncertainty of the -coefficients is estimated to be as good as for the observed Lyman transitions, especially because for there are no perturbations with the electronic state in the probed wavelength range Ubachs2007 (). The results of the present method were verified to agree with the Dunham approach Ubachs2007 () for low . -coefficients for the strongest Lyman bands observed in both white dwarfs, and , are plotted in Fig. 2. For comparison, the Lyman band observed in quasar absorption studies are also plotted, showing the higher sensitivity of the band despite the same .

To analyze the white dwarf spectra we do not follow the common procedure Malec2010 (); Bagdonaite2014 () of assigning and fitting individual transitions of H. Since they are relatively weak and self-blended we fit them simultaneously over most of the range between 1298 and 1444 Å. We only exclude regions where blends with atomic species occur: the geo-coronal O I transitions at 1298 – 1310 Å, the photospheric and interstellar atomic transitions listed in Xu2014 (), and part of the spectrum at 1411.7 Å where some previously unidentified atomic transitions have been found. The non-linear least squares Voigt profile fitting program VPFIT10.0 ^{2}^{2}2Developed by R. F. Carswell et al.; available at http://www.ast.cam.ac.uk/~rfc/vpfit.html is used to model the absorption spectrum of H. A Voigt profile represents an absorption lineshape involving Doppler broadening, due to thermal motion of the absorbing gas, and Lorentzian broadening arising from the finite lifetimes of the excited states, represented by the damping parameter ^{3}^{3}3For the present list of H transitions parameters correspond to the total radiative transition probabilities from H. Abgrall, E. Roueff, and I. Drira, Astron. Astrophys. Suppl. Ser. 141, 297 (2000)., convolved with an instrumental line spread function ^{4}^{4}4We use the COS/HST “lifetime position=2” instrumental profiles provided at: http://www.stsci.edu/hst/cos/performance/spectral_resolution/.

The intensities of the H absorption lines are described by the product of the oscillator strength and the normalized population of the ground ro-vibrational level, calculated from a partition function at a temperature :

(2) |

where is the nuclear statistical weight. Consequently, all lines probing odd- levels (ortho-H) benefit in relative strength from the 3:1 spin statistics ratio between ortho- and para-levels. Invoking this definition of line strengths leads to a model which essentially requires only 4 free parameters: total column density of the gas, , redshift of the absorbing cloud, , linewidth, , and temperature . Once the fit is optimized, we introduce an additional free parameter which allows for small relative line-shifts that are governed by the calculated sensitivity coefficients :

(3) |

where represents the transition wavelength observed in the white dwarf spectra and is a corresponding wavelength measured in the laboratory. Models of different temperatures were fitted to the data (Fig. 3), resulting in a best-fit temperature of K for GD133 and K for G2938. Displayed in Fig. 1 are fitting results of the G2938 spectrum, where the model is based on a total of 870 H transitions with the relative strengths defined for K. The derived H temperature is in good agreement with independent temperature determinations from Balmer-H lines for GD133 but differs for GD2938 Koester2009 (). For either star, measurements of are only slightly affected by the choice of temperature, as shown in Fig. 3. The best-fit model of GD2938 yields , and the one of GD133 results in . The adequacy of the fit is reflected by the reduced of , where the number of degrees of freedom is equal to 9633 for both spectra. A column density , a linewidth km s, and a redshift were measured for GD133. For G2938 the results are: , km s, and . The quoted widths are deconvolved from the instrument profile, km s.

The measured redshifts of are primarily determined by the gravitational redshift associated with the local potential in the white dwarf photospheres, with contribution from the proper motion of the objects and from the uncertainty of the absolute wavelength calibration of the COS instrument, amounting to of the measured redshift value. For a -analysis, the relative wavelength calibration accuracy is of the utmost importance. If not taken into account, velocity distortions – velocity shifts which change with wavelength – may have a significant effect on measurements Bagdonaite2014 (). We searched for such distortions by applying the ‘direct comparison method’ Evans2013 () to individual exposures against the combined spectra. The ‘direct comparison method’ is a model-independent technique of comparing pairs of spectra in order to detect and correct for velocity shifts. No evidence for relative distortions between exposures was found, with 1- limits of 25 m s nm per exposure. Applying artificial distortions of and to the combined 3 exposures of G2938 and the 5 of GD133 produces systematic shifts in of and , respectively. The same analysis also allows us to correct the combined spectra for small relative shifts (0.2 km s) between individual exposures. The corrected spectra, plus the above estimate of systematic errors, provide our fiducial measurements: for G2938 and for GD133.

The line broadenings of and km s are primarily determined by gas kinetics at the prevailing temperatures of K yielding km s, with , where is the Boltzmann constant, and the molecular mass. It can be estimated that an H absorption cloud of 10 km depth in the photosphere of GD133 would be subject to a “gravitational width” of or only km s with a similarly small estimate for G2938. From a photospheric model the maximum H molecular density ((H)/(H)=10) was found to coincide with a total material density of g cm Xu2014 () amounting to 3 mbar for an H-atmosphere. This would translate into a broadening of km s.

Stark and Zeeman broadening effects on the H lines are assumed to be small because the excited state for the Lyman bands is of valence character and only weakly susceptible to external fields; no laboratory measurements of Stark and Zeeman effects have been reported for the molecular Lyman bands. G2938 is a well-studied irregular pulsator, with time-evolving dominant periods of a few hundred seconds resulting in velocity shifts of atomic hydrogen transitions as large as 16.5 km s Thompson2003 (). In the case of GD133 pulsations occur at a dominant period of 120 s and are much weaker than in G2938 Silvotti2006 (). In either case, the pulsation period is smaller than the exposure times that exceed 2000 s and, thus, the H spectra in individual exposures and the combined spectra will be smeared out by this effect.

The broadening effects outlined here mainly affect the resolution of the spectra and the accuracy of the constraint on . They should not affect the symmetry of the H absorption lines, and even if they did, the effect would be the same for all transitions and thus it is unlikely to mimic -variation.

The above constraints on from the white dwarf spectra can be interpreted in terms of a dependence on a dimensionless gravitational potential Magueijo2002 (); Flambaum2008 (), , where a Taylor series expansion up to the second order can be used to specifically probe strong field gravitational phenomena:

(4) |

The linear term can be constrained most directly from a laboratory spectroscopic investigation of SF molecules aimed to detect a temporal variation of the proton-to-electron mass ratio Shelkovnikov2008 (). We estimate from the results presented in Ref. Shelkovnikov2008 () that the seasonal difference amounts to . Invoking a gravitational potential at the Earth’s surface of (due to the field produced by the Sun) and an Earth orbit eccentricity of leading to a 2.6 % effect on the difference in the potential between aphelion and perihelion in the current epoch, the laser spectroscopic experiment yields
^{5}^{5}5It is noted that most of the spectroscopic measurements of Shelkovnikov2008 () were fortuitously taken at the aphelion period (July 2005 and 2006), while some were recorded at perihelion (in 2004 and 2006), thus probing the maximum potential difference produced during the Earth’s orbit..
Other Earth-based spectroscopic investigations (and combinations thereof) yield even tighter constraints of Ferrell2007 (); Fortier2007 (), and Blatt2008 (), although with model-dependence. These results constrain the linear term, more than the present white dwarf study.

The analysis of H spectral lines in the white dwarfs yield . The physical properties of GD133 correspond to a gravitational potential in the photosphere at the white-dwarf surface, while that of G2938 is . This delivers a constraint of , which is several orders of magnitude more stringent than from the Earth-based experiments. This demonstrates that the high gravitational field conditions of white dwarfs (10,000 times that on the Earth’s surface) is a sensitive probe to constrain . Using the methods presented here, future studies of H in the photospheres of white dwarfs should provide further information on the possible variation of the proton-to-electron mass ratio under conditions of strong gravitational fields.

This work was supported by the FOM-Program “Broken Mirrors & Drifting Constant”, Science and Technology Facilities Council, Templeton Foundation and Australian Research Council (DP110100866). Tyler Evans (Swinburne University of Technology) is thanked for calculating velocity shifts between individual exposures.

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