ABSTRACT
In an effort to resolve the discrepancy between two measurements of the fundamental constant μ, the proton to electron mass ratio, at early times in the universe we reanalyze the same data used in the earlier studies. Our analysis of the molecular hydrogen absorption lines in archival Very Large Telescope/Ultraviolet and Visible Echelle Spectrometer (UVES) spectra of the damped Lyman alpha systems in the quasi-stellar objects Q0347-383 and Q0405-443 yields a combined measurement of a Δμ/μ value of (−7 ± 8) × 10−6, consistent with no change in the value of μ over a time span of 11.5 Gyr. Here, we define Δμ as (μz − μ0) where μz is the value of μ at a redshift of z and μ0 is the present-day value. Our null result is consistent with the recent measurements of King et al., Δμ/μ = (2.6 ± 3.0) × 10−6, and inconsistent with the positive detection of a change in μ by Reinhold et al. Both of the previous studies and this study are based on the same data but with differing analysis methods. Improvements in the wavelength calibration over the UVES pipeline calibration is a key element in both of the null results. This leads to the conclusion that the fundamental constant μ is unchanged to an accuracy of 10−5 over the last 80% of the age of the universe, well into the matter dominated epoch. This limit provides constraints on models of dark energy that invoke rolling scalar fields and also limits the parameter space of supersymmetric or string theory models of physics. New instruments, both planned and under construction, will provide opportunities to greatly improve the accuracy of these measurements.
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1. INTRODUCTION
The values of the fundamental constants determine the nature of the physical universe, from the size of mountains on Earth to the eventual fate of the universe as a whole. Historically, we have assumed that these constants are invariant in space and time. Speculation on the possibility of a time variation of the constants was first discussed by Dirac (1937), Teller (1948), and Gamow (1967). In very rare cases, such as the Oklo mine (Damour & Dyson 1996), there exists a terrestrial laboratory to test for time varying constants. It has been known for over 30 yr (Thompson 1975) that damped Lyman alpha systems (DLAs; Wolfe et al. 2005) offer the opportunity to measure the value of the fundamental constant μ, the proton to electron mass ratio,7 at early times in the universe. The opportunity stems from the direct dependence of the rotational energy of molecules on μ and the square root dependence on μ of the vibrational energy relative to the electronic energy.8 Each absorption line has a unique shift for a change in μ that depends on the vibrational and rotational quantum numbers of the upper and lower energy states. At the time of Thompson (1975), however, the observational capabilities of astronomical spectroscopy and the accuracy of molecular hydrogen laboratory spectroscopy allowed only very crude determinations of μ at relatively modest look back times. The high line density of atomic hydrogen lines in DLAs and the rarity of DLAs with measurable amounts of molecular hydrogen further complicated progress.
Foltz et al. (1988) and Cowie & Songaila (1995) made early measurements of μ at significant look back times and found no change to accuracies of Δμ/μ ⩽ 2 × 10−4 and 7 × 10−4 in the spectrum of PKS 0528-250 at a redshift of 2.811. At the same time calculations of the expected shifts were made by Varshalovich & Levshakov (1993) who developed a method of sensitivity constants for each line that will be discussed later in this work. An additional constraint of 2 × 10−4 was obtained on the same object by Potekhin et al. (1998). An excellent review of studies relevant to a determination of the time history of μ and other fundamental constants is given in Uzan (2003).
Three advances now provide the opportunity to measure μ at large look back times and at accuracies that are starting to impact other areas of physics such as dark energy and string theory. The first advance is the construction of large telescopes such as the Keck telescopes, the Very Large Telescopes (VLTs) and now the Large Binocular Telescope (LBT). A second advance is the installation of stable, high-resolution and sensitive spectrometers such as High Resolution Echelle Spectrometer (HIRES; Vogt et al. 1994) at Keck and Ultraviolet and Visible Echelle Spectrometer (UVES; Dekker et al. 2000) at the VLT. The third key advance is the measurement of the wavelengths of the H2 Lyman and Werner electronic transitions to accuracies of a few parts in 108 (Ubachs et al. 2007). In addition Ubachs et al. (2007) have recalculated the sensitivity constants, taking into account both adiabatic and nonadiabatic perturbations, to provide an invaluable set of wavelengths at the present-day value of μ and wavelength sensitivities to μ for the evaluation of the astronomical observations.
The most recent efforts to measure μ at high redshifts have centered on the UVES on the VLT. The spectra of two quasars observed in 2002 January (Q0347-383) and 2003 January (Q0405-443) contain H2 absorption lines at redshifts of 3.0249 and 2.5947. The first observations of Q0347-383 were commissioning observations carried out in 1999 and described by D'Odorico et al. (2001). Ivanchik et al. (2002) used these data along with a UVES spectrum of Q 1232+082 to investigate possible changes in μ. They found two results, Δμ/μ = (5.7 ± 3.8) × 10−5 and Δμ/μ = (12.5 ± 4.5) × 10−5 at the 3σ level for two different sets of thorium argon wavelength lists. A subsequent analysis by Levshakov et al. (2002) using just the Q0347-383 spectra found a result of −1.5 × 10−5 ⩽ Δμ/μ ⩽ 5.7 × 10−5. A later reanalysis of the Q0347-382 data by Ivanchik et al. (2003) produced a limit at a confidence level of 95% of |Δμ/μ| < 8 × 10−5. Ubachs & Reinhold (2004) combined the line lists of Ivanchik et al. (2002) and Levshakov et al. (2002) and found that Δμ/μ = (−0.5 ± 3.6) × 10−5 at the 2σ level.
The 2002 and 2003 UVES VLT observations of Q0347-383 and Q0405-443 (Ivanchik et al. 2005) had higher signal to noise than the 1999 observations. Using new laser determined H2 wavelengths from Philip et al. (2004) and the UVES pipeline reduction of the spectra they found Δμ/μ = (1.64 ± 0.74) × 10−5. Reinhold et al. (2006) subsequently utilized a new set of laser determined H2 wavelengths and the pipeline data to find a change in μ of Δμ/μ = (2.4 ± 0.6) × 10−5. Ubachs et al. (2007) detail the determination of the H2 parameters and gives a more complete list of laser determined wavelengths that slightly alters the result to Δμ/μ = (2.45 ± 0.59) × 10−5. The Ubachs et al. (2007) H2 parameters essentially remove the properties of H2 from the error budget leaving the data reduction and signal to noise of the observed spectrum as the primary error contributors. The Reinhold et al. (2006) result for Q0347-383 was examined by Wendt & Reimers (2008) who concluded that the data were consistent with −0.7 × 10−5 ⩽ Δμ/μ ⩽ 4.9 × 10−5 at the 95% confidence level. King et al. (2009) have taken the same data set as Reinhold et al. (2006) with the addition of spectra of Q0528-250 and found a value of Δμ/μ = (2.6 ± 3.0) × 10−6 for the combined data set. A key element in their analysis is an improved wavelength calibration as described in Murphy et al. (2008b). At this time there are two analyzes of the same data that lead to two different conclusions. Our independent analysis of the same data concludes that there is no evidence for a change in μ, consistent with the results of King et al. (2009).
For completeness we consider radio frequency measurements of μ that are more precise but at significantly lower redshift. Although the wavelength determinations are more precise, transitions in different molecules must be compared to provide information on any change in μ. Recently, Flambaum & Kozlov (2007) have looked for variations in μ using the radio emission lines of ammonia and carbon monoxide. They take advantage of the high sensitivity of the inversion spectrum of ammonia to changes in μ with where zinv is the redshift of the inversion lines of ammonia, zrot is the redshift of the rotational lines of CO, and z0 is the cosmological redshift of the galaxy. For the galaxy B0218+357 at a redshift of 0.68470 they find Δμ/μ = (0.6 ± 1.9) × 10−6. Murphy et al. (2008a) have improved this result to Δμ/μ ⩽ 0.18 × 10−6. This result is at relatively low redshift and it depends on ammonia and carbon monoxide having identical kinetic velocities in the molecular clouds. This is probably unlikely since, unlike the ubiquitous CO molecule, NH3 is concentrated in the colder denser cores of molecular clouds. The fact that it is a null result, however, adds credence to the result since an offset in kinetic velocity would have to accurately match any change in μ to produce a null result. The result is also for a relatively low redshift, placing it within the current dark energy dominated epoch of the universe. Some dark energy theories predict that the fundamental constants only roll during the matter dominated epoch and freeze out at their present values once dark energy becomes dominant around a redshift of 1 (e.g., Barrow et al. 2002). Table 1 provides a summary of the astronomical determinations of μ.
Table 1. Recent Astronomical μ Measurements
Object | Reference | Redshift | Δμ/μ |
---|---|---|---|
PKS 0528-250 | Foltz et al. (1988) | 2.811 | || ⩽ 2 × 10−4 |
PKS 0528-250 | Cowie & Songaila (1995) | 2.811 | || ⩽ 7 × 10−4 |
PKS 0528-250 | Potekhin et al. (1998) | 2.811 | || ⩽ 2 × 10−4 |
Q0347-383 + Q1232+082 | Ivanchik et al. (2002) | 3.0249 | (5.7 ± 3.8) × 10−5 |
Q0347-383 | Levshakov et al. (2002) | 3.0249 | −1.5 × 10−5 ⩽ 5.7 × 10−5 |
Q0347-383 | Ivanchik et al. (2003) | 3.0249 | || ⩽ 8 × 10−5 |
Q0347-383 | Wendt & Reimers (2008) | 3.0249 | −0.7 × 10−5 ⩽ 4.9 × 10−5 |
Q0347-383 + Q0405-443 | Ubachs & Reinhold (2004) | 3.0249, 2.5974 | (−0.5 ± 3.8) × 10−5 |
Q0347-383 + Q0405-443 | Ivanchik et al. (2005) | 3.0249, 2.5974 | (1.64 ± 0.74) × 10−5 |
Q0347-383 + Q0405-443 | Reinhold et al. (2006) | 3.0249, 2.5974 | (2.4 ± 0.6) × 10−5 |
Q0347-383 + Q0405-443 | Reinhold et al. (2006) | 3.0249, 2.5974 | (2.45 ± 0.59) × 10−5 |
Q0347-383 + Q0405-443 | This work | 3.0249, 2.5974 | (−7 ± 8) × 10−6 |
Q0347-383 + Q0405-443 + PKS 0528-250 | King et al. (2009) | 3.0249, 2.5974, 2.811 | (2.6 ± 3.0) × 10−6 |
B0218+357 | Flambaum & Kozlov (2007) | 0.6847 | (0.6 ± 1.9) × 10−6 |
B0218+357 | Murphy et al. (2008a) | 0.6847 | || ⩽ 0.18 × 10−6 |
Milky Way | Levshakov et al. (2008) | 0.0 | (4 − 14) × 10−8 |
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In our own Galaxy, Levshakov et al. (2008) have reported variations in μ based on the same ammonia transition along different lines of sight. In this case, the variation is relative to the CCS molecule and is manifested by a general positive velocity offset between the ammonia and CCS emission lines. Their result gives Δμ/μ = (4–14) × 10−8. Slight errors in the line frequencies could mimic such a result.
Laboratory experiments have set limits on the present-day rate of change of μ. Even though their time base is brief by cosmological standards, their wavelength accuracy is far better than can be achieved in astronomical observations. The current best laboratory limits appear to be the results of Blatt et al. (2008) which give a result of . To put this in perspective if the rate of change is constant at then the change at the 11 Gyr look back time of Q0347-383 would be 1.1 × 10−5, similar to the astronomical results given in this work. There is no real expectation that the rate of change would be constant so both the astronomical and laboratory results work in concert to constrain possible physical models that predict changes in time of the values of the fundamental constants. The results of Blatt et al. (2008) depend on the Schmidt model for the nuclear magnetic moment and therefore may be deemed as model dependent. A laboratory result that is independent of the Schmidt model is given by Shelkovnikov et al. (2008) who give . Other limits on the present rate of variation in μ based on the weak equivalence principle and various theories of particle physics are discussed by Dent et al. (2008).
The remainder of the paper addresses the measurement of μ in the spectra of Q0347-383 and Q0405-443. The wavelength calibration and data reduction to produce the spectra used in this work will only be summarized since it is discussed in detail in Thompson et al. (2009). That separate publication is intended to give a full description of the data analysis in order to allow the reader to concentrate on the measurement of μ described here without a lengthy data reduction description at the beginning. In this paper, we bring those analysis methods to bear in an effort to discriminate between the positive and null results for a variation in μ.
2. OBSERVATIONS
The observations of Q0347-383 and Q0405-443 with UVES on VLT occurred during the nights of 2002 January 7–9 for Q0347-383 and 2003 January 4–6 for Q0405-443.9 The emission line redshifts for these quasi-stellar objects (QSOs) are 3.22 and 3.02, respectively (Ivanchik et al. 2005). The data were retrieved from the VLT archive along with the MIDAS based UVES pipeline reduction procedures. On each of the nights three separate spectra of the QSO were taken with accompanying long slit calibration lamp integrations at the same grating setting. The slit width and length for both object and calibration line observations are 0.8 and 6.6 arcsec. The grating angle for the Q0347-383 observations had a central wavelength of 4300 Å and for Q0405-443, 3900 Å. The images are 2 × 2 pixel binned on chip with a size of 1024 by 1500 binned pixels. A single pixel is 15 μm in size and 0.22 arcsec on the sky. In the following, the word pixel refers to the 2 × 2 binned pixels (0.44 arcsec) in the images obtained from the archive. At 4000 Å a pixel is approximately 0.0416 Å which is about 3 km s−1. Both the calibration and object images are binned identically. Exposure times and other observational parameters are given in Tables 2 and 3 and are described in Ivanchik et al. (2005).
Table 2. Observational Parameters for Q0347-383
Archive File | Date | Exposure (s) |
---|---|---|
UVES_2002_01_08T00:46:05_351_b.fits | 2002 Jan 8 | 4500 |
UVES_2002_01_08T02:03:41_018_b.fits | 2002 Jan 8 | 4500 |
UVES_2002_01_08T03:21:18_348_b.fits | 2002 Jan 8 | 4500 |
UVES_2002_01_09T00:43:43_109_b.fits | 2002 Jan 9 | 4500 |
UVES_2002_01_09T02:02:11_833_b.fits | 2002 Jan 9 | 4500 |
UVES_2002_01_09T03:19:58_841_b.fits | 2002 Jan 9 | 4500 |
UVES_2002_01_10T00:48:56_171_b.fits | 2002 Jan 10 | 4500 |
UVES_2002_01_10T02:06:28_725_b.fits | 2002 Jan 10 | 4500 |
UVES_2002_01_10T03:24:33_981_b.fits | 2002 Jan 10 | 4500 |
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Table 3. Observational Parameters for Q0405-443
Archive File | Date | Exposure Time |
---|---|---|
UVES_2003_01_04T00:43:06_274_b.fits | 2003 Jan 4 | 4500 |
UVES_2003_01_04T02:09:06_464_b.fits | 2003 Jan 4 | 4500 |
UVES_2003_01_04T03:34:08_623_b.fits | 2003 Jan 4 | 4500 |
UVES_2003_01_05T00:48:35_827_b.fits | 2003 Jan 5 | 4500 |
UVES_2003_01_05T02:16:14_922_b.fits | 2003 Jan 5 | 4500 |
UVES_2003_01_05T03:46:36_522_b.fits | 2003 Jan 5 | 4500 |
UVES_2003_01_06T00:45:18_207_b.fits | 2003 Jan 6 | 4500 |
UVES_2003_01_06T02:15:26_790_b.fits | 2003 Jan 6 | 4500 |
UVES_2003_01_06T03:46:49_242_b.fits | 2003 Jan 6 | 4500 |
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There are differences in the way the long slit calibration spectra were taken between the two objects. In the case of Q0405-443 there was a long slit calibration spectrum taken immediately after the object spectrum in all but one case. The time tags of the grating position encoder readouts for the object and calibration spectra are identical as are the values of the grating position encoder readings. This indicates that there was no adjustment of the grating position between the paired object and calibration spectra. The exception is the night of 2003 January 5, where there is no long slit calibration spectrum for the second object observation.
For Q0347-383 two long slit calibration spectra were taken in between the three object spectra for each night. The encoder readings indicate that there were no grating resets performed between the object spectrum and the calibration spectrum for the first two pairs of object and calibration observations for each night. The time tags, however, on the third night of 2002 January 10, pairs the long slit calibration spectrum with the third object spectrum. The proper pairings of observations are important in calculating the shifts needed to accurately combine the observations as is discussed in Section 3.1.
3. DATA REDUCTION
The spectra described by Ivanchik et al. (2005) and used by Reinhold et al. (2006) were produced by the standard UVES pipeline. The pipeline produces excellent spectra for most observations, however, Murphy et al. (2008b) points out that the thorium argon line list used in the wavelength calibration may not be accurate enough for the precise determination of fundamental constants. We reached similar conclusions as discussed in Section 3.1. The final output of the UVES pipeline is an interpolated spectrum with equal wavelength intervals as opposed to an intensity and wavelength on a pixel-by-pixel basis. In what follows we only use images and spectra on a pixel-by-pixel basis. The wavelength calibration and the production of the spectra used in this study are described in detail in Thompson et al. (2009). The descriptions given here are short summaries of the methods.
3.1. Wavelength Calibration
Independently of Murphy et al. (2008b), we became aware that the standard Th/Ar line list used in the UVES pipeline analysis provides good wavelengths for most studies but is the primary limiting factor in obtaining the accuracy required for a determination of μ at the 10−5 level. In fact only about 1/4 of the lines are free of blending and other problems. We then recalibrated the wavelength solutions using the long slit calibration line spectra taken during the observations of the two QSOs. This is described in detail in Thompson et al. (2009) which is intended to serve as the record of the wavelength calibration used in this study and therefore will not be repeated here. The new wavelength calibration is the primary reason for a null result in this study. It should be noted that this recalibration differs from the recalibration used in King et al. (2009) in two ways. First, this calibration is based on the calibration spectra taken during the observation of the analyzed spectra. Second, the calibration is done order by order. This results in some lines being declared good in one order but unusable in another order where they fall in low signal-to-noise areas.
The wavelength calibration described in Thompson et al. (2009) tracks the shifts in the wavelength positions between observations and between different observing nights. The final wavelength calibration is relative to a master solution which is set at a single position determined by a master long slit calibration lamp image that is the median of all of the calibration lamp images shifted to the position of the first calibration lamp image. The shifts are small, a few hundredths of a pixel width, but important in this study. The shifts are carried out by cubic interpolation and are rigid. The wavelength solution for each order of the master long slit calibration image is a six term Legendre polynomial whose coefficients are different for each order. The wavelength solution for the object spectra will differ from the master solution first due to small shifts in the actual grating position from the position appropriate to the master solution and due to the motion of the observatory about the barycenter of the Earth–Sun system. These are corrected for in the production of the individual spectra.
3.2. Spectrum Production
The order by order final spectrum for each object is a three-dimensional array of dimensions [np, nord, 6] where np is the number of pixels in the dispersion direction (1500), nord is the number of orders, and the six last dimensions are flux, wavelength, variance, continuum, fit, and the fit convolved with the instrumental profile for each pixel in the spectrum. In this case, the fit is the continuum minus the best fit to the H2 lines at their natural line width. This is what the spectrum would look like if the instrument profile was infinitely narrow. The first two are derived from the object and calibration spectra and calculated for each spectrum. The last four are calculated after the spectra are combined into a single spectrum but could be, in principle, calculated for the individual spectra. The observational parameters for Q0347-383 and Q0405-443 are listed in Tables 2 and 3, respectively.
3.2.1. Flux
At this stage there are nine spectra for each of the two objects. In each order of the spectra, the flux is distributed over several pixels in the cross-dispersion direction. We tested several optimal extraction methods for combining the flux in the cross-dispersion direction into a single value. These tests indicated that the UVES MIDAS Version 2.2.0 pipeline extraction did as good or better job of combining the flux than any of the methods we tested. We therefore used this intermediate product of the pipeline to produce the nine flux versus pixel spectra for the two objects. These are not the interpolated to constant delta wavelength values spectra that are the final product of the pipeline. The next step is to assign a proper wavelength designation to each of the pixel positions in each order. Our goal is a proper vacuum wavelength at rest with respect to the barycenter of the Earth–Sun system. Again note that our reference to pixels is to the 2 × 2 pixel binned output available in the archive.
3.2.2. Wavelength
As mentioned in Section 3.1, the true wavelengths of the pixels are slightly different for each spectrum due to small shifts in the spectrometer configuration and different barycentric velocities. The wavelengths are calculated by first shifting the master wavelength solution by the amount calculated from the associated long slit calibration spectrum and then correcting the wavelengths for barycentric velocity. The associated long slit calibration line observations are given in Thompson et al. (2009).
The wavelength shift due to small differences in the grating position is calculated from the shifts found during the wavelength calibration. As described in Section 2, care was taken to not reset the grating position between a long slit calibration observation and its associated object exposures. For these observations, the shift of the object spectrum wavelength position is identical to the shift calculated for the associated calibration line spectrum given in Thompson et al. (2009). There are, however, two object observations where it appears that the grating position was reset without an associated long slit calibration line exposure. They are the observations in Table 2 for Q0347-383 that end in 109_b and 981_b. For these observations, our only recourse was to take the shift as the average between the shifts immediately preceding and immediately after the observations. The correlation between the encoder readouts and the shifts calculated during the wavelength calibration did not appear to be accurate enough to be used as a direct indicator of the amount of shift. Once the shift is determined the master wavelength solution is interpolated to account for the shift in pixel position between the associated calibration line image and the master image. The wavelengths associated with the flux in the pixels now have the correct observed vacuum wavelength as observed but must still be corrected for the barycentric velocity. Note that the handling of the shift values is different from the description given in Thompson et al. (2009). At that time it was not known that there was a procedure of resetting the grating position between observations. It does not affect the master wavelength calibration since the shifts were calculated directly from the calibration observations, but it does matter for the object spectra.
The component along the direction to the object of the barycentric velocity of the observatory was calculated using the date and time of the midpoint of the integration. This velocity is due to the Earth's orbit and rotation relative to the barycenter of the Earth–Sun system. The wavelength scale was then corrected for this motion so that the final wavelengths are vacuum wavelengths as observed in a reference frame at rest relative to the barycenter. This is slightly different than the heliocentric wavelengths used in Ivanchik et al. (2002).
3.3. Co-addition of the Spectra
At this point the individual spectra have the flux, wavelength, and noise values populated in their respective arrays. The wavelengths, however, are slightly different for each spectrum due to the grating shifts and different barycentric velocities. Accurate co-addition of the spectra requires that they all be on the same wavelength scale. We choose to shift all of the spectra to the wavelength scale of the master wavelength solution for the grating angle setting as described in Thompson et al. (2009).
3.3.1. Shift to Common Wavelength Scale
The spectrum shift was accomplished by interpolation of the flux from the wavelength scale of the spectrum to the master wavelength scale of the master solution using the IDL10 code procedure INTERPOL in double precision mode. In the following, we will write IDL provided procedure names in capital italic letters and procedures written by the authors using IDL code in lower case italics. INTERPOL uses linear interpolation which is appropriate for this case since the shifts are only a few hundredths of a pixel. For cases with significant fractions of a pixel other interpolation methods may be more appropriate. At the end of this procedure all spectra are on a common wavelength scale, the master wavelength solution for each order.
3.3.2. Addition of the Spectra
All of the observed spectra for each of the two objects were combined to produce the final two spectra for analysis. The excellent and uniform observing conditions at the VLT produced a suite of individual spectra with remarkably similar signal-to-noise characteristics. In other words, the weight of each of the spectra were essentially indistinguishable from each other. For this reason we simply produced two spectra for each object, one that is the mean of all the flux values at a given wavelength and the second which is the median of the flux values. Again the differences between these two spectra were minimal. The median spectrum was judged to have slightly better signal to noise in both objects and is the spectrum that is used in the analysis.
3.3.3. The Variance
The UVES pipeline calculates the variance for each of the spectrum fluxes but the documentation is not clear on the exact method of calculation. The variance is an important quantity in calculating the χ2 values for the wavelength and density fits so uncertainty in how it was calculated is worrisome. The variance is therefore calculated explicitly form the nine spectra normalized to a common total flux value. The normalizations varied between 0.8 and 1.2 for the nine spectra that were combined to make the final spectrum for each object. The normalization, therefore, does not have a large effect on the calculated variance.
3.4. H2 Line Parameters
A primary component of this study is the use of accurate molecular data provided by several recent studies of the H2 molecule (Ubachs et al. 2007; Ivanov et al. 2008, and the references therein). The data from these references include the vacuum wavelengths and calculated sensitivity factors Ki where the index i indicates the line. The sensitivity factor to a variation in μ is different for each line and is defined as
The precise vacuum wavelengths from these references have average errors on the order of 5 × 10−6 Å which produce a negligible contribution to the errors in determining the redshift of each line. The oscillator strengths for each transition were calculated using the Einstein A coefficients from Abgrall et al. (1993a) for the Lyman transitions and Abgrall et al. (1993b) for the Werner transitions.
3.5. Output Products
The calculation maintains two primary output products. The first is a six component spectrum of each order. The six components are the double precision wavelength, the flux, the standard deviation of the flux, the continuum fit, the line fit, and the line fit convolved with the instrument profile. The second output is an IDL structure array which contains the line information. An IDL structure is a multiformat data set that can contain text, integer, floating point, and arrays of any of these formats. There is a structure for each line which contains the molecular data such as oscillator strength and vacuum rest wavelength as well as the calculated data from the fit such as redshift and density. Information on whether the fit for the line converged is also in the structure.
4. ANALYSIS OF THE H2 LINES
The purpose of this analysis is to determine as accurately as possible the true vacuum wavelength of the observed H2 absorption lines in the spectra of Q0347-383 and Q0405-443 produced by the procedures discussed in Section 3 and Thompson et al. (2009). These wavelengths are then used in Section 5 to measure the value of μ at the epoch represented by the redshift of the DLA absorption line system. The procedures described in the following are procedures written in IDL.
4.1. Establishing the Continuum
Our definition of the continuum in this section is not the true continuum of the quasar spectrum but rather the true spectrum without the H2 absorption lines. This is the canvas that the H2 spectrum is painted on. In an analogy to preparing a canvas by sizing it we refer to this as sizing the continuum. The procedure starts with a first guess of the column densities of the first five rotational levels of the H2 electronic and vibrational ground state and calculates the expected line width, defined as all regions less than 90% of the continuum, using a Voigt function, the oscillator strength of the transition, a temperature of 350 K and the redshift of the H2 absorption line system. The kinetic temperature of 350 K is equivalent to Doppler parameter of 1.7 km s−1. Initially, the redshift was taken from previous studies but subsequent iterations used the best redshift from the previous iteration. There is only one velocity component for Q0347-383 and only the strongest component of the two velocity components in Q0405-443 was used. The natural line shape is convolved with the UVES instrumental profile to produce the observed line profile. This procedure simply defines the spectral region that must the replaced with the continuum estimate. The spectral region inside the line width is then replaced with the expected continuum calculated by interpolating the spectrum on either side of the line which has undergone a 5 pixel smoothing. It is important to note that as mentioned at the beginning of the section this is a local continuum fit which represents what the spectrum would be if there were no molecular hydrogen lines, not the true continuum flux. Since the fit is local there is an independent continuum fit for each line. The typical line spans two to three pixels. The automated continuum calculation gives a rough, first-order fit to the sized continuum. At this point a hard copy plot of the spectrum and continuum is produced and examined.
The continuum fit of the selected lines is then interactively refined in an IDL based procedure that displays the fit for each line and allows the user to adjust the fit interactively. The complex spectrum of the Lyman alpha forest insures that most of the continuum fits must be adjusted. Since the goal of this study is the most precise possible wavelength fit, rather than a column density fit, the continuum is usually adjusted preferentially to lower values that emphasize the line center as opposed to the lower signal-to-noise wings of the line. Tests with various continuum fits indicated that adjustment of the continuum to higher levels produced significant errors in wavelength for some lines but that adjustment to a reasonable range lower levels did not produce wavelength changes larger than the wavelength errors determined in Section 5.2.3. Apparently, the larger number of pixels in the high continuum cases allowed noise in the wings of the line to have a greater influence on the fit than with the low continuum case. The line shape convolved with the instrumental profile is about twice as wide as the natural line shape for unsaturated lines. A mosaic of the spectral regions of all of the lines used in the analysis along with the continuum and line fits are shown in Figures 1 and 2. These spectra are displayed to allow the reader to judge the quality of the continuum and line fits. The spectra at this point are ready for the Δμ/μ determination.
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Standard image High-resolution imageTable 4. Q0347-383 Line List
Trans.a | Order | K factor | Obs. Wavelengthb | Pos. Error | Neg. Error | χ2 | Rest Wavelengthb | Redshiftc |
---|---|---|---|---|---|---|---|---|
L15P1 | 123 | 0.05147000 | 3782.21819 | 0.0112 | −0.0094 | 3.56 | 939.70672 | 3.02489213 |
L14R1 | 123 | 0.04625000 | 3811.49618 | 0.0069 | −0.0089 | 7.84 | 946.98040 | 3.02489448 |
W3Q1 | 123 | 0.02149000 | 3813.28002 | 0.0110 | −0.0062 | 2.73 | 947.42188 | 3.02490179 |
W3Q1 | 122 | 0.02149000 | 3813.28420 | 0.0079 | −0.0055 | 3.59 | 947.42188 | 3.02490620 |
W3P3 | 122 | 0.02097000 | 3830.37745 | 0.0056 | −0.0049 | 6.23 | 951.67186 | 3.02489304 |
L13R1 | 121 | 0.04821000 | 3844.04623 | 0.0046 | −0.0043 | 11.31 | 955.06582 | 3.02490189 |
L13P1 | 121 | 0.04772000 | 3846.62792 | 0.0085 | −0.0076 | 6.33 | 955.70827 | 3.02489760 |
W2Q1 | 120 | 0.01396000 | 3888.44675 | 0.0052 | −0.0058 | 30.09 | 966.09608 | 3.02490687 |
W2Q2 | 120 | 0.01272000 | 3893.21423 | 0.0097 | −0.0078 | 8.29 | 967.28110 | 3.02490468 |
L12R3 | 120 | 0.03682000 | 3894.80256 | 0.0085 | −0.0088 | 1.16 | 967.67695 | 3.02489959 |
W2Q3 | 120 | 0.01088000 | 3900.33218 | 0.0042 | −0.0040 | 15.21 | 969.04922 | 3.02490617 |
W1Q1 | 118 | 0.00487000 | 3971.76790 | 0.0037 | −0.0054 | 3.82 | 986.79800 | 3.02490469 |
W1Q1 | 117 | 0.00487000 | 3971.76771 | 0.0075 | −0.0130 | 2.04 | 986.79800 | 3.02490450 |
L9R1 | 117 | 0.03753000 | 3992.75767 | 0.0034 | −0.0038 | 8.86 | 992.01637 | 3.02489091 |
L8R0 | 116 | 0.03475000 | 4032.24698 | 0.0066 | −0.0075 | 6.32 | 1001.82387 | 3.02490607 |
L8R1 | 116 | 0.03408000 | 4034.76532 | 0.0051 | −0.0041 | 16.47 | 1002.45210 | 3.02489587 |
L8R1 | 115 | 0.03408000 | 4034.75032 | 0.0104 | −0.0097 | 10.49 | 1002.45210 | 3.02488091 |
W0R2 | 115 | −0.00525000 | 4061.22312 | 0.0083 | −0.0104 | 11.53 | 1009.02492 | 3.02489873 |
W0Q2 | 115 | −0.00710000 | 4068.92599 | 0.0067 | −0.0070 | 27.08 | 1010.93845 | 3.02489982 |
W0Q2 | 114 | −0.00710000 | 4068.91220 | 0.0138 | −0.0157 | 4.71 | 1010.93845 | 3.02488618 |
L7R1 | 114 | 0.03027000 | 4078.98076 | 0.0063 | −0.0093 | 7.84 | 1013.43701 | 3.02489816 |
L7P3 | 114 | 0.02460000 | 4103.38732 | 0.0046 | −0.0053 | 2.64 | 1019.50224 | 3.02489289 |
L6P3 | 112 | 0.02033000 | 4150.43809 | 0.0083 | −0.0182 | 2.55 | 1031.19260 | 3.02489126 |
L5P1 | 112 | 0.02064000 | 4178.48451 | 0.0086 | −0.0089 | 12.21 | 1038.15713 | 3.02490566 |
L5R2 | 112 | 0.01997000 | 4180.62771 | 0.0074 | −0.0064 | 4.82 | 1038.69027 | 3.02490313 |
L4P2 | 110 | 0.01346000 | 4239.36224 | 0.0061 | −0.0046 | 2.54 | 1053.28426 | 3.02489850 |
L4P3 | 110 | 0.01051000 | 4252.19544 | 0.0070 | −0.0065 | 4.61 | 1056.47144 | 3.02490335 |
L3R1 | 109 | 0.01099000 | 4280.32103 | 0.0042 | −0.0042 | 45.60 | 1063.46014 | 3.02490030 |
L3P1 | 109 | 0.01001000 | 4284.92877 | 0.0059 | −0.0058 | 8.87 | 1064.60539 | 3.02489862 |
L3R2 | 109 | 0.00953000 | 4286.49249 | 0.0073 | −0.0073 | 12.69 | 1064.99481 | 3.02489519 |
L3R3 | 109 | 0.00719000 | 4296.48519 | 0.0041 | −0.0042 | 9.06 | 1067.47855 | 3.02489136 |
L2P2 | 107 | 0.00184000 | 4351.98530 | 0.0113 | −0.0135 | 8.78 | 1081.26603 | 3.02489783 |
L2P3 | 107 | −0.00115000 | 4365.24899 | 0.0138 | −0.0117 | 18.47 | 1084.56034 | 3.02490192 |
L1R1 | 106 | −0.00143000 | 4398.14064 | 0.0054 | −0.0052 | 22.80 | 1092.73243 | 3.02490172 |
L1P1 | 106 | −0.00259000 | 4403.45255 | 0.0038 | −0.0036 | 22.80 | 1094.05198 | 3.02490250 |
Notes. aTransitions are labeled with L or W for Lyman or Werner, then the vibrational quantum number of the upper state, next R, Q, or P transitions and finally the rotational quantum number of the lower state. bVacuum wavelength. cBarycentric redshift.
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5. DETERMINATION OF THE Δμ/μ VALUE
5.1. Selection of Suitable H2 Lines for the Measurement of μ
Most of the H2 lines are unusable due to the Lyman alpha forest. Appropriate lines are picked at this time based on freedom from interference by other lines and signal to noise. A basic selection rule is lines that have greater than 50% asymmetry between the height of their short wavelength and long wavelength shoulders are rejected. This limits the number of lines used that lie on the shoulders of other lines. Lines that have asymmetric profiles, indicating a blend of two lines, are also rejected. Lines that have profiles broader by 50% than expected from a single line are similarly rejected. Finally, lines that do not converge to a stable redshift value in the following analysis are not used to determine the μ value. During the course of the analysis slightly different selection rules were applied. More lenient rules led to larger errors as did more stringent rules that reduced the number of lines in the analysis. In no case, however, did the results exceed a 2σ excursion from a null result. The lists of lines used in this analysis are given in Tables 4 and 5. These line lists do not directly correspond to the lines used by Reinhold et al. (2006) who did not publish their selection criteria but were guided by lines selected by Ivanchik et al. (2005). King et al. (2009) do not list the lines that they used but the text indicates that they were not the same as those use by Reinhold et al. (2006). They did, however, conduct an analysis using the same lines as Reinhold et al. (2006) and obtained a result of Δμ/μ = (12.0 ± 14.0) × 10−6. As in the analysis of Reinhold et al. (2006) only the stronger of the double system of H2 lines in Q0405-443 were used. The two systems are separated by 13 km s−1 which is a separation of roughly four of the double binned pixels.
Table 5. Q0405-443 Line List
Trans.a | Order | K Factor | Obs. Wavelengthb | Pos. Error | Neg. Error | χ2 | Rest Wavelengthb | Redshiftc |
---|---|---|---|---|---|---|---|---|
L16P1 | 139 | 0.05297000 | 3351.25678 | 0.0040 | −0.0066 | 5.30 | 932.26621 | 2.59474230 |
W4P2 | 139 | 0.02569000 | 3352.45726 | 0.0052 | −0.0066 | 2.66 | 932.60468 | 2.59472489 |
W4P3 | 139 | 0.02350000 | 3360.32455 | 0.0060 | −0.0072 | 1.27 | 934.79006 | 2.59473715 |
L15P3 | 138 | 0.04676000 | 3394.61670 | 0.0083 | −0.0058 | 10.34 | 944.33046 | 2.59473389 |
W3R2 | 137 | 0.02287000 | 3404.62607 | 0.0037 | −0.0041 | 13.42 | 947.11169 | 2.59474612 |
L14R2 | 137 | 0.04715000 | 3409.50987 | 0.0049 | −0.0043 | 13.11 | 948.47125 | 2.59474246 |
W3Q3 | 137 | 0.01828000 | 3416.42562 | 0.0029 | −0.0028 | 5.69 | 950.39773 | 2.59473251 |
L13P2 | 136 | 0.04577000 | 3442.49784 | 0.0049 | −0.0049 | 14.30 | 957.65223 | 2.59472649 |
L12P2 | 134 | 0.04341000 | 3473.51556 | 0.0142 | −0.0071 | 6.53 | 966.27550 | 2.59474659 |
W2P3 | 134 | 0.00992000 | 3488.92549 | 0.0077 | −0.0080 | 28.26 | 970.56332 | 2.59474279 |
L11P2 | 133 | 0.04092000 | 3506.10669 | 0.0088 | −0.0053 | 3.54 | 975.34576 | 2.59473208 |
L9R2 | 131 | 0.03594000 | 3571.54986 | 0.0023 | −0.0020 | 7.52 | 993.55061 | 2.59473370 |
L9P2 | 131 | 0.03489000 | 3576.31685 | 0.0031 | −0.0045 | 9.02 | 994.87408 | 2.59474322 |
L9P2 | 130 | 0.03489000 | 3576.30117 | 0.0026 | −0.0028 | 18.84 | 994.87408 | 2.59472746 |
L9P3 | 130 | 0.03202000 | 3586.92017 | 0.0078 | −0.0057 | 8.06 | 997.82718 | 2.59473087 |
L8R2 | 130 | 0.03251000 | 3609.06404 | 0.0076 | −0.0083 | 6.95 | 1003.98545 | 2.59473739 |
L8R2 | 129 | 0.03251000 | 3609.06183 | 0.0073 | −0.0052 | 5.13 | 1003.98545 | 2.59473519 |
L8P2 | 129 | 0.03137000 | 3614.13175 | 0.0037 | −0.0038 | 12.62 | 1005.39320 | 2.59474457 |
W0R3 | 128 | −0.00631000 | 3631.14901 | 0.0049 | −0.0046 | 7.27 | 1010.13025 | 2.59473346 |
W0Q2 | 128 | −0.00710000 | 3634.05522 | 0.0035 | −0.0041 | 23.82 | 1010.93845 | 2.59473440 |
L7P2 | 128 | 0.02750000 | 3653.90105 | 0.0045 | −0.0044 | 15.12 | 1016.46125 | 2.59472734 |
L6P2 | 127 | 0.02324000 | 3695.76334 | 0.0040 | −0.0125 | 21.67 | 1028.10609 | 2.59472954 |
L6P2 | 126 | 0.02324000 | 3695.77191 | 0.0034 | −0.0079 | 6.73 | 1028.10609 | 2.59473788 |
L5P2 | 125 | 0.01857000 | 3739.84184 | 0.0039 | −0.0037 | 11.57 | 1040.36733 | 2.59473210 |
L5R3 | 125 | 0.01759000 | 3742.68792 | 0.0040 | −0.0038 | 17.41 | 1041.15892 | 2.59473260 |
L5P3 | 125 | 0.01564000 | 3751.12621 | 0.0030 | −0.0043 | 26.05 | 1043.50319 | 2.59474340 |
L5P3 | 124 | 0.01564000 | 3751.12050 | 0.0031 | −0.0031 | 10.63 | 1043.50319 | 2.59473793 |
L4R2 | 124 | 0.01497000 | 3779.85697 | 0.0048 | −0.0040 | 15.03 | 1051.49857 | 2.59473334 |
L4R3 | 123 | 0.01261000 | 3788.77180 | 0.0054 | −0.0038 | 25.20 | 1053.97610 | 2.59474166 |
L3P2 | 122 | 0.00790000 | 3835.21789 | 0.0037 | −0.0040 | 17.12 | 1066.90068 | 2.59472813 |
L3R3 | 122 | 0.00719000 | 3837.30732 | 0.0033 | −0.0035 | 10.22 | 1067.47855 | 2.59473951 |
L3P3 | 122 | 0.00493000 | 3846.87296 | 0.0057 | −0.0044 | 5.16 | 1070.14088 | 2.59473508 |
L3P3 | 121 | 0.00493000 | 3846.87940 | 0.0040 | −0.0054 | 16.28 | 1070.14088 | 2.59474110 |
L2R2 | 121 | 0.00360000 | 3879.52758 | 0.0031 | −0.0030 | 8.81 | 1079.22542 | 2.59473332 |
L2R2 | 120 | 0.00360000 | 3879.53298 | 0.0021 | −0.0022 | 7.05 | 1079.22542 | 2.59473832 |
L1P2 | 119 | −0.00475000 | 3941.40916 | 0.0044 | −0.0034 | 20.26 | 1096.43894 | 2.59473657 |
L1R3 | 119 | −0.00509000 | 3942.44064 | 0.0034 | −0.0042 | 8.44 | 1096.72534 | 2.59473835 |
L1P2 | 118 | −0.00475000 | 3941.41164 | 0.0031 | −0.0043 | 8.44 | 1096.43894 | 2.59473884 |
L1P3 | 118 | −0.00775000 | 3953.45055 | 0.0061 | −0.0050 | 8.44 | 1099.78718 | 2.59474144 |
L0R0 | 117 | −0.00800000 | 3983.42760 | 0.0040 | −0.0090 | 8.44 | 1108.12733 | 2.59473816 |
L0P2 | 117 | −0.01191000 | 3999.12069 | 0.0028 | −0.0028 | 8.44 | 1112.49600 | 2.59472815 |
L0R3 | 117 | −0.01202000 | 3999.42503 | 0.0084 | −0.0105 | 8.44 | 1112.58000 | 2.59473029 |
Notes. aTransitions are labeled with L or W for Lyman or Werner then the vibrational quantum number of the upper state, R, Q, or P transitions and finally the rotational quantum number of the lower state. bVacuum wavelength. cBarycentric redshift.
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5.2. Fitting the Lines
The line-fitting procedures are IDL based double precision procedures written and developed by authors. Each line is fit individually rather than calculating a complete synthetic spectrum for several reasons. The first is that most of the large number of molecular hydrogen lines are unusable due to blends with other lines or due to complete obliteration by the Lyman α forest. They would simply contribute noise to the fit. Second, we are looking for shifts away from the expected wavelengths that a global fit would wash out. Finally, we allow the column density to be an independent parameter for each line and use anomalous densities to find lines that are blended with other lines.
The individual selected H2 lines are fit iteratively with alternate adjustments of the wavelength and the column density. The fit function is a Voigt function calculated with the IDL function VOIGT that is convolved with the instrument profile. The IDL VOIGT function is calculated with double precision parameters and returns a double precision result. The instrument profile is represented by a Gaussian of half-width 0.037014 Å at 3900 Å digitized in units of 0.001 Å (R. Carswell 2005, private communication). The half-width is adjusted at other wavelengths to be directly proportional to the wavelength. Changes in the width of the Gaussian by plus or minus 10% changed the derived column density but had no effect on the derived wavelength within the wavelength errors attributable to signal to noise. The kinetic and excitation temperature of the gas is not varied but held at 350 K for both objects for the initial fitting. During the iteration of the fits, we did not require all lines with the same lower state to have the same column density. This is similar to letting the excitation temperature vary from line to line. The kinetic temperature is held fixed at 350 K. Changes in the kinetic temperature by ±100 K did not alter the derived wavelengths within the 1σ bounds.
The fit is started with an initial guess at the column density for each ground-state rotational level and an initial guess at the redshift. After a few runs these initial guesses were refined to produce a better starting solution. The fit procedure starts with the wavelength adjustment followed by a column density adjustment. This procedure is iterated six times. Lines that have not converged after six iterations are then rerun with another six tries at convergence. Any lines that have not converged in both column density and wavelength after the two iterations are not used in the analysis. Convergence is declared when two tries in sequence return the same column density and wavelength values.
5.2.1. Wavelength Iteration
The first of the six tries in each iteration starts with a sweep of the wavelength in 200 × 10−5 Å steps on either side of the starting wavelength. The starting wavelength is either the wavelength from the initial guess at the redshift for the first iteration or the best wavelength from the previous try in the second and subsequent iterations. At each of the 200 wavelengths on either side of the initial wavelength the line fit is calculated as a Voigt function superimposed upon the continuum spectrum as discussed in Section 4.1 and Section 5.2. The calculated spectrum is then convolved with the instrument profile as described in Section 5.2. A χ2 value for the difference between the fit and the observed flux is then calculated for all spectral points in the line that are deeper than 95% of the continuum. After the sweep over the 400 wavelength positions the best wavelength is taken to be the wavelength with the minimum χ2 value. If the best wavelength is not at either extremum of the wavelength sweep the number of test wavelengths is reduced by a percentage that is proportional to the distance of the best wavelength from the extremum. The minimum number of test wavelengths is 10. Subsequent tries at the wavelength fit are all centered at the best wavelength from the previous try.
5.2.2. Column Density Iteration
After each try at the wavelength fit there is an adjustment of the column density. The wavelength is fixed at the best wavelength from the last wavelength iteration. The column density is varied over 200 values ranging from 1% of the initial density to twice the initial density in 1% increments. The initial column density is either the initial guess column density or the density found in previous column density iteration. A χ2 value is then calculated for all column density points in the same way as the wavelength iteration. The best column density is taken as the density with the minimum χ2 value. Although the column density range is asymmetric between the high and low ends it usually converges in the first 2–3 iterations.
5.2.3. The χ2 Values
Mosaic plots of the χ2 values calculated for each line are shown in Figures 3 and 4. The values are shown for 100 wavelengths on either side of the best wavelength. The wavelength values are spaced by 10−4 Å rather than the 10−5 Å spacing in the actual analysis. Note that the χ2 values are smoothly varying in a semiparabolic shape with a definite minimum. Murphy et al. (2008c) point out that this is a necessary criterion for valid χ2 values. Although the values are generally symmetric about the minimum there is significant asymmetry in some of the plots. This is expected from line profiles that are also not symmetric. Only the wavelength, which is the primary parameter of interest, is varied in the plots, as opposed to both the wavelength and the column density in the fitting procedure, so that the minimum χ2 value is for 1 degree of freedom. The χ2 values range from values almost as low as 1.1 to values as high as 45.6, with Q0347-383 having in general lower χ2 values than Q0405-443 which has a lower signal-to-noise spectrum. There does not appear to be any correlation between the individual χ2 values and the deviation of the reduced redshift from the average reduced redshift for each object. The lines with the highest deviation from the average reduced redshift are not the ones with the highest χ2 values.
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Standard image High-resolution imageThe error bars for each line in Figures 5 and 6 are calculated by running the line fit calculations on either side of the minimum χ2 wavelength until the χ2 value increases by unity over the minimum χ2 value. This produces in some cases a significant difference between the positive and negative error bars. The positive and negative errors along with the χ2 values for the individual line fits are listed in Tables 4 and 5.
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Standard image High-resolution image5.3. Results
Tables 4 and 5 give the results of the line fitting for Q0347-383 and Q0405-443, respectively. Note that some transitions are repeated since they appear in two different orders. In our analysis we treat these as independent measures. Figures 5 and 6 are the plots of reduced redshift ζ versus sensitivity factor Ki for Q0347-383 and Q0405-443 in manner similar to that first used by Varshalovich & Levshakov (1993). Reinhold et al. (2006) and Ubachs et al. (2007) both display their results with similar plots. The reduced redshift ζ is defined by
where zQ is the true redshift of the system taken as the median redshift of all of the lines, zi is the redshift of individual lines, and Ki is defined in Equation (1). The median redshifts of the H2 absorptions in Q0347-383 and Q0405-443 are 3.0248996 and 2.5947366, respectively, relative to the Earth–Sun barycenter. The slope in this plot is the value of Δμ/μ.
The thick black dash dot and dash triple dot lines in the figures are the weighted and unweighted linear least-squares fits to the combined data for all rotational levels for each object where the weights are determined by the standard deviations of the individual data points. The colored light solid and dotted lines are the fits to only lines with the same rotational level ground states given by the color codes in the caption. The weighted and unweighted fits to Q0347-383 are Δμ/μ = (−28 ± 16) × 10−6 and Δμ/μ = (−19 ± 15) × 10−6. The weighted and unweighted fits for Q0405-443 are Δμ/μ = (0.55 ± 10) × 10−6 and Δμ/μ = (3.7 ± 14) × 10−6. For the combined data set shown in Figure 7 the weighted fit yields Δμ/μ = (−7.0 ± 8) × 10−6 and the unweighted fit gives Δμ/μ = (−6 ± 10) × 10−6. Both of these results are consistent with no variation in μ at the 68% confidence level. Our result is consistent with the findings of Wendt & Reimers (2008) giving −7.0 × 10−6 ⩽ Δμ/μ ⩽ 49 × 10−6 and King et al. (2009) which give Δμ/μ = (2.6 ± 3.0) × 10−6. They are inconsistent at a roughly 3σ level with that of Reinhold et al. (2006) and Ubachs et al. (2007) who found Δμ/μ = (24.5 ± 5.9) × 10−6.
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Standard image High-resolution imageThe stated errors between the three measurements vary widely. It should be noted that the error quoted by Wendt & Reimers (2008) is a 2σ error so the value ±25 × 10−6 should be used in comparison to the other two measurements. King et al. (2009) perform an analysis where they use the same lines as Ubachs et al. (2007) and their error grows to about ±9 × 10−6 for the individual objects. This error is similar but slightly smaller than our errors. King et al. (2009) also include a third object Q0528-250 which has a significantly smaller quoted error than the two objects common to all of the studies. In the weighted mean of errors Q0528-250 has a dominant effect on the quoted error. In the following, we discuss our error analysis.
5.4. Error Analysis
The least-squares linear fit to the weighted combined data of 77 lines gives a χ2 value of 104.9. Assuming N-2 degrees of freedom this is a χ2 value of 1.4 per degree of freedom. In the combined data, there are 10 pairs of lines of the same transition but observed in a different orders. Q0347-383 has four pairs and Q0405-443 has six pairs. We have treated these as independent measurements since they have independent photon noise and read noise statistics. Systematic effects such as unknown blended lines and continuum shape may introduce systematics into the measurements. Individual inspection of the line pairs indicates that the dispersion in reduced redshift between the two measurements is consistent with the dispersion between independent lines and should not bias the results.
5.4.1. Bootstrap Analysis
As an alternative check on the statistical significance of the null result we performed a bootstrap analysis on the combined data set. We produced 10,000 new data sets by drawing the same number of lines but randomly selected allowing duplication from the original data set. Linear least-square fits were performed on the data sets and the result plotted as a histogram shown in Figure 8. The smooth curve in the figure is a Gaussian fit to the histogram. The peak of the Gaussian is at a Δμ/μ value of −6.4 × 10−6 and the half-width at half-maximum is 12 × 10−6, both of which are consistent with the results of the χ2 analysis. The histogram values conform to the Gaussian fit quite well indicating the appropriateness of the assumption of Gaussian distributed errors.
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Standard image High-resolution image5.4.2. Ground-state Rotational Levels
Since transitions from four different rotational levels (J = 0, 1, 2, 3) are used it is possible that the transitions could arise from physically offset regions of the molecular cloud that could also have velocity offsets. Our null result is less prone to this type of error, however, velocity offsets could be in a direction to reduce a Δμ/μ signal. To check for this the solutions for lines with the same rotational ground level and the same object are plotted in Figures 5 and 6. In Q0347-383 there is only one line with a ground rotational level of 0 so no solution is plotted for it. The J = 1 and 3 solutions have roughly the same slope while the J = 2 solution has a different slope with the opposite sign. This would appear difficult to achieve with physical velocity offsets which would be presumed to have a smooth gradient of velocity with temperature. In Q0405-443, there are no J = 0 lines and only one J = 1 line so no solutions exist for those systems. The J = 2 and J = 3 slopes are of opposite sign but within the error bars of each other. From this analysis, we conclude that it is unlikely that velocity gradients with excitation level are masking a change in μ.
5.4.3. Combination of the Data Sets
To improve the statistics of the sample we have combined the lines from the two systems. The higher redshift system associated with Q0347-383 shows a shift in μ at the 1σ level in the unweighted fit and at a 1.75σ level in the weighted fit, both indicating a decrease in the value of μ. The system associated with Q0405-443 at a lower redshift shows an increase in μ in both the unweighted and weighted fit but at levels significantly less that 1σ. The Q0405-443 system has seven more lines than the Q0347-383 system. It could be argued that the combination of the two data sets dilutes the signal of a real shift in the Q0347-383 system at the higher redshift and earlier time in the universe. Although we would not claim a real shift in μ with a 1.75σ result we cannot rule out that the combined data set is diluting the evidence for a change in μ. The higher than expected value of the χ2 per degree of freedom could be due to the difference in fitted slopes of the two systems taken separately.
5.4.4. Systematics
The method of sensitivity coefficient fitting is subject to systematic errors in the wavelength scale. In general, the sensitivity coefficients increase with increasing vibrational energies in the upper level of the transitions. This means that for a given electronic transition system the sensitivity coefficient increases with decreasing transition wavelength. That means that any systematic error that produces an erroneous gradient in the wavelength calibration will mimic a change in μ. This is mitigated to some degree by the mixture of Lyman and Werner bands. The higher electronic energy of the upper level of the Werner system places low values of the sensitivity coefficient at the same wavelengths as high sensitivity coefficient Lyman transitions. At wavelengths longer than the longest wavelengths of the Werner system, however, there is no mitigating effect. It may be this effect coupled with the systematic errors in the older UVES pipeline reductions found by Murphy et al. (2008b) that produced the positive detection of a change in μ by Reinhold et al. (2006). The analysis in Thompson et al. (2009), Figures 7 and 8, indicates that the wavelength calibration used in this analysis is not subject to systematic errors of the magnitude cited in Murphy et al. (2008b). In addition our wavelength calibration is on an order by order basis which resets the solution for each order, making it more difficult to have systematic effects over the whole wavelength solution.
5.4.5. Comparison of Lyman and Werner Lines
The Werner lines with a higher upper electronic level provide lines with low upper state vibrational levels at wavelengths that are close to Lyman lines with high upper state vibrational levels. Since the sensitivity factors are roughly proportional to upper state vibrational level this mixes lines with low sensitivity factors with those with high factors. Under the assumption that any possible systematic wavelength errors are minimized for lines that lie close together we have looked at the redshift differences between all of the Werner lines and the Lyman lines that are adjacent to them in the same order. There are a limited number of line pairs that satisfy this criterion, eight for Q0347-383 and seven for Q0405-443. Histograms of the distribution are given in Figure 9. The distribution of delta redshift values (z(Lyman)−z(Werner)) for Q0405-443 are roughly evenly distributed around zero but the delta redshift values for Q0347-383 are all negative. This is consistent with the negative slope of the fit in Figure 5.
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Standard image High-resolution image5.5. The Marginal Possibility of a Shift in Q0347-383
The analysis results for Q0347-383 show a negative shift in the value of μ at the 1.75σ level which has a statistical probability of being a true shift at the 91% level if the errors are Gaussian distributed. This is certainly not a level which justifies declaring a change in a fundamental constant but raises the marginal possibility that there might be a change. In addition the comparison between the Werner and Lyman lines in Figure 9 shows a negative delta between the Werner and Lyman lines for a seven cases which has a probability of 2−7 = 0.008 chance of happening randomly. If a monotonically rolling scalar field is invoked for the change, the higher redshift of Q0347-383 could be why a change is seen in Q0347-383 and not Q0405-443. We consider this evidence as suggestive but in no way conclusive. It does point out the need for observations of systems at higher redshift.
6. CONCLUSIONS AND IMPLICATIONS
Our basic conclusion, based on the combination of data from Q0347-383 and Q0405-443, is that there has been no change in the value of μ to 1 part in 105 over a time span of 11.5 Gyr. This is approximately 80% of the age of the universe. The accuracy of the limit on Δμ/μ is set by both the spectral resolution and the signal-to-noise ratio of the flux. This conclusion is consistent with the results of King et al. (2009) but inconsistent with the results of Reinhold et al. (2006). Starting with the same raw data, the primary difference in this analysis is the use of improved wavelength calibration techniques that eliminated the systematic variations in the calibration used in the UVES pipeline at the time of the Reinhold et al. (2006) analysis. The line selection is also most likely different from Reinhold et al. (2006) but without a list of those lines it is difficult to assess the influence of the lines chosen. There is a marginal possibility of the detection of a change in μ based on the Q0347-383 data alone. We, however, feel that this result while suggestive is certainly not conclusive.
What implications does a limit on Δμ/μ of 10−5 have on theories of dark energy that invoke a rolling scalar field potential as the source of the dark energy? Chongchitnan & Efstathiou (2007) have despaired about distinguishing between a universe with a cosmological constant relative to a universe with a quintessence rolling scalar field, however, the former predicts no change in μ while the latter predicts a change even though the magnitude or even the sign of the change is not presently calculable. Detection of a change in μ or its companion the fine structure constant α would be strong evidence for quintessence as opposed to a cosmological constant.
Quintessence is usually expressed in terms of a potential V(ϕ) that is a function of the rolling scalar ϕ. The change in μ is then expressed as
where κ is , mPl is the Planck mass and ζμ is a parameter of unknown value (Avelino et al. 2006, and references therein). Determination of the value of μ at high redshift is therefore a direct way to distinguish between quintessence and a cosmological constant. In grand unified theories (GUTs) the rolling of μ is typically given by
where ΛQCD is the quantum chromodynamics (QCD) scale, ν is the Higgs vacuum expectation value (VEV), R is a model-dependent value (Avelino et al. 2006, and references therein), and α is the fine structure constant. In many GUT models the value of R is large and negative ∼−50 (Avelino et al. 2006).
Our current results limit the value of ζμκ(ϕ − ϕ0) in Equation (3) to be on the order of 10−5 or less, but does not tell us the individual values of (ϕ − ϕ0) or ζμ. The results do, however, rule out Model A of Avelino et al. (2006) at about the 4σ level where the potential is given by
which predicts a value of at a redshift of 3. This means that even at the current level of accuracy significant bounds on the quintessence models are being established. In all fairness to the model, it must be pointed out that it was designed to achieve that result to match the findings of Reinhold et al. (2006).
If the claim of a detected change in the fine structure constant α (; Murphy et al. 2003) is accepted then this implies a value of R of ⩽2 which is significantly different in sign and magnitude that the typical GUT value quoted above. This would mean that either the roll of both the QCD scale and the Higgs VEV is small or that they are equal to each other by less than a factor of 2. Of course if the claim for a change in α is not accepted the current limitation on places no limit on R.
R.I.T. acknowledges interesting and useful conversations with Wim Ubachs, Dimitrios Psaltis, Feryal Ozel, and Michael Murphy on theory and technique. C.M. acknowledges very useful discussions with Paolo Molaro. The work of C.M. is funded by a Ciencia2007 research grant.
Footnotes
- *
Based on observations made with ESO Telescopes at the La Silla or Paranal Observatories under program IDs 68.A-0106 and 70.A-0017.
- 7
Although the literature is approximately equally divided in usage we designate μ as the proton to electron mass ratio rather than the inverse to be consistent with the other recent astronomical determinations of μ discussed here.
- 8
See Shu (1991), Chapter 28, for an alternative derivation of the dependence.
- 9
Based on observations made with ESO telescopes at the Paranal Observatory under program IDs 68.A-0106 and 70.A-0017.
- 10
IDL stands for Interactive Data Language registered by ITT Visual Information Solutions.