1. Cosmological variation of the proton-to-electron mass ratio and the spectrum of H 2 Ruth Buning Bachelor project 2004

2. Has changed on a cosmological timescale? Most recent result:

3. Overview Introduction to the topic: measuring  Introduction to the topic: measuring  This project: This project: –Modelling of H 2 spectra –Mass dependence of spectral lines –Recalculation of  /  Conclusion Conclusion

4. Empirical search for a change in  Compare spectra of different epochs: Compare spectra of different epochs: Use the spectrum of H 2 Use the spectrum of H 2 Each spectral line i of H 2 depends in a different way on  Each spectral line i of H 2 depends in a different way on  Quantified by sensitivity coefficients K i Quantified by sensitivity coefficients K i Quasars 12 Gyr ago Lab today

5. Goals Calculate new K i values from new lab data Calculate new K i values from new lab data Determine new constraint on  /  from new K i Determine new constraint on  /  from new K i

6. Energy levels in a diatomic molecule Electronic Electronic Vibrationalv Vibrationalv RotationalJ RotationalJ

7. Representation of the energy levels Rotation: Rotation: Vibration: Vibration: (Rigid rotor) (Nonrigid rotor) (Harmonic oscillator) (Anharmonic oscillator)

8. Dunham representation Energies derived from Schrödinger equation and general potential function Energies derived from Schrödinger equation and general potential function These contain the parameters  e and B e, which are mass-dependent These contain the parameters  e and B e, which are mass-dependent

9. The sensitivity coefficients Dependence of transition wavelengths (or energies) on  quantified by sensitivity coefficients: Dependence of transition wavelengths (or energies) on  quantified by sensitivity coefficients: E and dE/d  derived from Dunham expansion: E and dE/d  derived from Dunham expansion:

10. The H 2 data Available lab data: Available lab data: –Lyman (B-X): v’=0-18, v’’=0 –Werner (C-X): v’=0-4, v’’=0 –J between 0 and 5 Fitted Y kl coefficients: Fitted Y kl coefficients: –Lyman: Y 00 -Y 130, Y 01 -Y 41, Y 02 -Y 32, Y 03 –Werner: Y 00 -Y 40, Y 01 -Y 31, Y 02 -Y 12 Ground state Y kl from literature Ground state Y kl from literature

11. Perturbations B- and C- states perturb each other B- and C- states perturb each other Decreased weight in Dunham fit Decreased weight in Dunham fit Unperturbed lines:  =0.001-0.1 cm -1 Unperturbed lines:  =0.001-0.1 cm -1 Perturbed lines:  =1-15 cm -1 Perturbed lines:  =1-15 cm -1

12. Corrections I Adiabatic correction Adiabatic correction –Mass-dependence of the electronic energy Lyman:Werner:

13. Corrections II Approximation on mass dependence of Y kl coefficients Approximation on mass dependence of Y kl coefficients +/- 1% of ad cor. Pos or neg

14. Results: K i values

16. Determine  /  Data from 3 quasars with different redshifts Data from 3 quasars with different redshifts Individual redshift of lines within a quasar: Individual redshift of lines within a quasar: Reduced redshift: Reduced redshift:

17. Results: Variation of  / 

18. Conclusions  /  can be derived from comparison quasar and lab spectra H 2  /  can be derived from comparison quasar and lab spectra H 2 Different and more accurate K i follow from Dunham fit of lab data Different and more accurate K i follow from Dunham fit of lab data Corrections are larger than previously claimed uncertainties Corrections are larger than previously claimed uncertainties Improvement on K i is significant if a nonzero  /  will be found Improvement on K i is significant if a nonzero  /  will be found

19. Questions?