EDP Sciences
Free Access
Issue
A&A
Volume 541, May 2012
Article Number A69
Number of page(s) 8
Section Cosmology (including clusters of galaxies)
DOI https://doi.org/10.1051/0004-6361/201218862
Published online 30 April 2012

© ESO, 2012

1. Introduction

The standard model of particle physics contains several fundamental constants whose values cannot be predicted by theory and need to be measured through experiments (Fritzsch 2009). They are the mass of the elementary particles and the dimensionless coupling constants which are assumed time-invariant although in theoretical models which seek to unify the four forces of nature they vary naturally on cosmological scales. The fine-structure constant α ≡ e2/(4πϵ0ħc) and the proton-to-electron mass ratio, μ = mp/me are two constants that can be probed in the laboratory as well as in the Universe by means of observations of absorption lines due to intervening systems in the spectra of distant QSOs and have been the subject of numerous studies. The former is related to the electromagnetic force while the latter is sensitive primarily to the quantum chromodynamic scale (see Flambaum 2004).

A probe of the variation of μ could be obtained by comparing relative frequencies of the electro-vibro-rotational lines of H2as first applied by Varshalovich & Levshakov (1993) after Thompson (1975) proposed the general approach to utilize molecule transitions for μ-determination. The original paper by Thompson (1975) did not take into account the different sensitivities within the molecular bands, which is the key of the modern approach.

The method is based on the fact that the wavelengths of vibro-rotational lines of molecules depend on the reduced mass, M, of the molecule. For molecular hydrogen M = mp/2 so that the comparison of an observed vibro-rotational spectrum with its present analog will give information on the variation of mN and me. Comparing electro-vibro-rotational lines with different sensitivity coefficients gives a measurement of Δμ/μ.

The observed wavelength λobs,i of any given line in an absorption system at the redshift z differs from the local rest-frame wavelength λ0,i of the same line in the laboratory according to the relation (1)where Ki is the sensitivity coefficient of the ith component computed theoretically for the Lyman and Werner bands of the H2 molecule (Varshalovich & Levshakov 1993; Varshalovich & Potekhin 1995; Potekhin et al. 1998; Meshkov et al. 2007; Ubachs et al. 2007).

It is useful to measure variations in velocities with comparison to the redshift of a given system defined by the redshift position of the lines with Ki ≈ 0, then introducing the reduced redshift ζi: (2)The velocity shifts of the lines are linearly proportional to Δμ/μ which can be measured through a regression analysis in the ΔVi − Ki plane.

This method was used to obtain a bound on the secular variation of the proton-to-electron mass ratio at Δμ/μ = (2.1 ± 3.6) × 10-5 from observations of the newly discovered H2 absorption systems at zabs = 3.0 towards QSO 0347-383 (Levshakov et al. 2002). Subsequent measures of the absorption systems of QSO 0347-383 and QSO 1232+082 provided a strong indication of a variation (2.4 ± 0.6) × 10-5, at 3.5σ (Reinhold et al. 2006; Ubachs et al. 2007). Earlier works (Ivanchik et al. 2005) also find hints for variation, but are still dominated by inaccuracies of the laboratory wavelengths. However, more recently King et al. (2008), Wendt & Reimers (2008), Thompson et al. (2009a), Wendt & Molaro (2011), King et al. (2011) with Δμ/μ = (0.3 ± 3.7) × 10-6 at zabs = 2.811 towards PKS 0528-250 and Bagdonaite et al. (2012) with Δμ/μ = (−6.8 ± 27.8) × 10-6 at zabs = 2.426 towards QSO 2348-011 reported a result in agreement with no variation.

The more stringent limits on Δμ/μ have been found from the combination of three H2 systems at Δμ/μ = (2.6 ± 3.0stat) × 10-6 (King et al. 2008) and taking into account additional transitions from deuterated molecular hydrogen (HD) in King et al. (2011). A fourth system has provided Δμ/μ = (+5.6 ± 5.5stat ± 2.9sys) × 10-6 (Malec et al. 2010).

An independent method relies on the inversion spectrum of ammonia as shown by Flambaum & Kozlov (2007). Ammonia NH3 inversion transitions are very sensitive to changes in μ due to a tunneling effect. The sensitivity coefficient of the inversion transition can be almost two orders of magnitude more sensitive to μ-variation than H2 molecular rotational frequencies. Thus by comparing the inversion frequency of NH3(1,1) with a rotational frequency of another co-spatial molecule it is possible to bind a variation of μ.

Flambaum & Kozlov (2007) combine three detected NH3 absorption spectra from B0218+357 with rotational spectra of CO, HCO+, and HCN to place a limit of (0.6 ± 1.9)  ×  10-6 for a look-back time of 6 Gyr (redshift z = 0.68). More recently Kanekar (2011) obtained Δμ/μ < 3.6 × 10-7 (3-σ level) for the same object. Murphy et al. (2008) with newly obtained high signal-to-noise rotational spectra of HCO+ and HCN obtained  < 1.8  ×  10-6 at a 95% CL.

Henkel et al. (2009) obtained a firm upper limit of 10-6 for a look-back time of 7 Gyr (z = 0.89) towards PKS 1830-211. This method is limited to low redshifts due to the small number of NH3 sources in general and to the large line widths and chemical segregation of different molecules at higher redshifts. For sources in the local Milky Way, however, a very strict upper limit of |Δμ/μ| < 3 × 10-8 was found utilizing both the ammonia method (Levshakov et al. 2010a,b), and the methanol method (Levshakov et al. 2011). Using ammonia and methanol, Ellingsen et al. (2012) found Δμ/μ < 4 × 10-7 at z ~ 0.886 towards B1830-210 at a 95% CL. Laboratory experiments by comparing the rates between clocks based on hyperfine transitions in atoms with a different dependence on μ restrict the time-dependence of μ at the level of yr-1 (Blatt et al. 2008).

In the following we will concentrate on the H2system observed towards QSO 0347-383 to trace the proton-to-electron mass ration μ at high redshift (zabs = 3.025). The motivation for re-analysis of QSO 0347-383 is given by the dramatically enhanced quality of the recent data of this quasar for the purpose of setting constraints on Δμ/μ. The single velocity component in H2absorption renders QSO 347-383 a prime target to further investigate the impact of wavelength calibration issues. Trying to reach a sensitivity of few parts per million everything becomes important and the special requirements of the observations as described in the following section are absolutely mandatory.

2. Data

2.1. Observations

The recent observations of QSO 0347-383 were performed with UVES on VLT on the nights of September 20–24, 2009. The journal of these observations is given in Table 1. The DIC 2 setting was used with blue setting and the 437 nm grating. The CCDs were not binned with pixel size of 0.013−0.015 Å, or 1.12 km s-1 at 400 nm along the dispersion direction. The observations are comprised of 11 exposures on four successive nights, of which 10 exposures were of 5400 s and one of 3812 s. Eight of the spectra with setting 437+760 and three the 437+860 setting, providing a coverage in the blue spectral ranges between 373 − 500 nm. QSO 0347-383 has no flux below 370 nm due to the Lyman discontinuity of the zabs = 3.023 absorption system.

Table 1

Journal of the observations (2009 data). Before and after each spectrum, a 30 s calibration frame was recorded.

The slit width was set to 0.7′′  for all observations providing a mean resolving power of λλ  ≈  66   000. Within each order the resolving power varies by about 15–20% being higher at the starting wavelength of each order. The average seeing along the exposures as recorded by the DIMM at Paranal is given in the last column of Table 1. We note, however, that the actual seeing at the UT2 of VLT was significantly better than that recorded by the DIMM.

The 11 different spectra and the corresponding co-added data (bottom) are shown in a region around L4R1 in Fig. 1.

thumbnail Fig. 1

The 11 single spectra and the corresponding co-added data (bottom) are plotted around the region of L4R1 and L4P1 (vertical lines). The stronger H2features can be merely distinguished in single spectra.

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2.2. Reduction

The last version of the UVES pipeline1 has been used for data reduction. The pipeline first uses a set of five bias to make a mean bias free of cosmic ray hits which is then subtracted to all the two dimensional format images. A pinhole lamp image is used for identifying the precise location of the Echelle orders which are curved and tilted upwards.

The pixel-wavelength conversion of the wavelength calibration is done by using the corresponding calibration spectrum. Murphy et al. (2008) and Thompson et al. (2009) independently showed that the standard Th/Ar line list used in the old UVES pipeline was a primary limiting factor. The laboratory wavelengths of the calibration spectrum were only given to three decimal digits (in units of Å) and, in many cases, the wavelengths were truncated rather than rounded from four decimal places (see Murphy et al. 2007).

Thompson et al. (2009) re-calibrated the wavelength solutions using the calibration line spectra taken during the observations of the QSOs and argued that the new wavelength calibration was a key element in their null result.

The new data UVES pipeline has solved these problems.

The UVES blue frame comprise 32 orders, from absolute number 96 to 124, covering the wavelength range 374−497 nm while the molecular lines are spread over 18 UVES echelle orders (from 106–122) covering the wavelength range 380−440 nm.

More than 55% of the  ≈400 ThAr lines in the region were used to calibrate the lamp exposure by means of a polynomial of the 5th order. Typical residuals of the wavelength calibrations were of  ~0.34 mÅ or  ~24 m s-1  at 400 nm and were found symmetrically distributed around the final wavelength solution at all wavelengths. By comparison in Malec et al. (2010) the wavelength calibration residuals have been rms  ~80 m s-1.

In our set of observations, calibration spectra were taken before and after the object spectra for each night. Calibration lamps taken immediately before and after object observation provides accurate monitoring of physical variations. Moreover, the calibration frames were taken in special mode to avoid automatic spectrograph resetting at the start of every exposure. Since Dec. 2001 UVES has implemented an automatic resetting of the Cross Disperser encoder positions at the start of each exposure. This implementation has been done to have the possibility to use daytime ThAr calibration frames for saving night time. If this is excellent for standard observations, it is a problem for the measurement of fundamental constants which requires the best possible wavelength calibration. Only calibration spectra that are attached as template to the OB allow to take the calibration exposure in exactly the same instrument setting as the science exposure. These calibrations can be taken upon user’s request in addition to the ones from the calibration plan. Thermal-pressure changes move in the cross dispersers in different ways, thus introducing relative shifts between the different spectral ranges in different exposures.

It should be emphasized that this effect has not been taken into account in the analysis performed so far on UVES data for μ variability.

There are no measurable temperature changes for the short exposures of the calibration lamps but during the much longer science exposures the temperature drifts generally by 0.1 K, and in two cases the drift is of 0.2 K while in other two there is no measurable change. Pressure values are surveyed at the beginning and end of the exposures and changes range from 0.2 to 0.8 mbar. The estimates for UVES are of 50 m s-1 for ΔT = 0.3 K or a ΔP = 1 mbar (Kaufer et al. 2004), thus assuring a radial velocity stability within  ~50 m s-1.

Individual spectra are corrected for the motion of the observatory about the barycenter of the Earth-Sun system and then reduced to vacuum. The velocity 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 to minimize the influence of changes. The changes in radial velocity during exposure induce a symmetric modification of the line profile. The absorption profile is not strictly Gaussian (or Voigt) anymore but rather slightly squared-shaped (since the FWHM of the line is  ≈5 km s-1 and the smearing of the line by Earth motions of  ±40 m s-1 the effect is negligible. The line shapes remain symmetric in any case and possible changes of radial velocities during exposure effects only the quality of the fit, it does not influence the measured centroid of an absorption line. 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.

The air wavelengths have been transformed into vacuum by means of the dispersion formula by Edlen (1966). Drifts in the refractive index of air inside the spectrograph between the ThAr and quasar exposures will therefore cause miscalibrations. According to the Edlen formula for the refractive index of air, temperature and atmospheric pressure changes of 1 K and 1 mbar would cause differential velocity shifts between 370 nm and 440 nm of  ~10 m s-1.

3. Preprocessing

3.1. Spectral radial velocity shifts

Wendt & Molaro (2011) showed the presence of overall shifts between spectra obtained with slit spectrographs such as UVES. For checking such a possibility in the new data set we obtained a median velocity per each exposure by fitting as many lines as possible in each of the eleven spectra. The 50 H2 lines were initially selected for that, but due to the relatively low SN spectra in single exposures, not all could be fitted, and for instance most of the weak J = 0 transitions have been missed.

Table 2

Median velocities of single spectra.

The median radial velocity in respect to the chosen absorber redshift derived from all detected lines within an individual exposure ranges from  −200 m s-1 to +354 m s-1 for the 11 spectra. Due to the low quality of the fits to single lines of individual spectra with low signal-to-noise, the median was chosen. Despite the low quality of the individual fits the median velocities listed in Table 2 are well defined as shown in Fig. 2. Bootstrapping is a useful practice of estimating properties of an estimator (i.e. its error). For example 48 line positions were determined in spectrum 8 (see Table 2). Their median velocity offset corresponds to 310 m s-1. The bootstrap histograms were implemented by constructing a number of resamples of the observed line positions (and of equal size to the observed data set), each of which is obtained by random sampling with replacement from the original data set.

For the data at hand the obtained velocity offsets bear no statistical significance because of the large scatter of the individual position measurements per single spectrum. The resulting offsets are, however, comparably well defined which made us confident to apply this procedure. The described method had no significant impact on the final result at the current level but it might provide a helpful tool in the future to check for potential inter-spectra shifts. The resulting velocities reflect the offsets of the individual spectra to a reference redshift of the H2absorber. Of the 11 shifts a mean velocity offset (relative to the assumed zabs) of 46 m s-1 was computed2. This low residual offset verifies the assumed redshift for the absorption system.

thumbnail Fig. 2

Exemplary bootstrap histogram of the median position of lines in spectrum 8 with respect to z = 3.02489817. The Gaussian fit corresponds to 302 ± 4 m s-1 (compare Table 2).

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4. Selection of lines and line fitting

For the analysis a total set of 50 H2lines are fitted. This preselection is based on earlier line identifications (see Wendt & Molaro 2011) including curve of growth analysis to avoid blends with the Lyman-α forest or other H2lines. The separate spectra are not coadded in this step since the fitting algorithm works on the different data sets simultaneously. This allows us to omit further rebinning of the data. The fitting code is based on an evolutionary algorithm, which tracks the global minimum via an interactive process of covering the parameter space (see Quast et al. 2005). Each set of fitting parameters is evaluated via χ2 in every single spectra. Thus, there is no need to redistribute the flux of each exposure to pixels of equal central wavelengths. Constant velocity offsets, however, potentially influence the shapes of the absorption features as would be the case of coadded spectra. Evolutionary fitting algorithms are less prone to converge in a local minimum rather than find the global minimum. An advantage over Monte Carlo chain methods as applied for the purpose of line fitting in simple cases for example in King et al. (2008) is the drastically reduced need for computer power. Additionally, the principle of evaluating multiple groups of parameters independently allows for consequent parallel computing.

For each set of lines sharing the same rotational level a common column density and a common broadening parameter is fitted with simplified pseudo-Voigt-function profiles. For weak lines a mere Gaussian profile would suffice since natural line broadening has no noticeable impact on the line shapes. In case of QSO 0347-383, only a single component is observed in H2. The only free parameter per each individual line is the radial velocity with respect to the absorber redshift. Out of the 50 H2lines analyzed, eight were excluded since they showed a comparably large positioning error of more than 300 m s-1. This is mostly due to a continuum highly contaminated by the presence of hydrogen absorption in the environment of the affected lines.

We note that our procedure is different from the one followed by King et al. (2008). They fitted numerous additional components in a region of H2absorption to narrow down the χ2 of the fit to the data.

Figure 3 compares a portion of spectrum used in the analyses by Ivanchik et al. (2005), King et al. (2008), Thompson et al. (2009), and Wendt & Molaro (2011) with the same portion of the new data we are analyzing here. The solid red vertical lines mark the H2component L4R1 and L4P1 and the dotted red lines indicate the 12 additional lines in that region3. The upper plot corresponds to the data of 2009 and reveals that some of the extra components clearly recreate the flux observed in 2002 but evidently do not correspond to factual properties of the absorber.

In King et al. (2008) the evolution of χ2 with an increasing number of additional free lines is the main criterion to fix the total number of components. While that approach clearly reduces the residuals of the fit, it may not reflect the physical properties of the absorber. Though it is likely that the absorber structure is too complex to be represented by a single component, we prefer to integrate the uncertainty of the true nature of the velocity components into the fitting uncertainty rather than “generating” components to fill up the flux in a poorly known continuum. Higher resolution spectra may verify or falsify some of the decisions on additional components and help to distinguish between apparent precision (lower χ2) and reached accuracy (better description of the physical conditions of the absorber) or rather the limit on information on the absorber.

thumbnail Fig. 3

Comparison between the 2009 data (top graph) and, with an offset, the original single observation run data of 2002 (9 frames, bottom) as used by Ivanchik et al. (2005), King et al. (2008), Thompson et al. (2009), and Wendt & Molaro (2011). The vertical lines indicate the positions of the H2component (solid) and the 12 additional lines fitted in King et al. (2008). In green the single line fits (L4R1 and L4P1) with their corresponding local continuum polynomial fit from this work. Note, that the fits for the analysis were not carried out on the rebinned and coadded data as shown in this figure.

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5. Results

In Fig. 4 the measured radial velocities of the 42 H2lines are plotted against the sensitivity towards k coefficients of the corresponding transition. Any correlation therein would indicate a variation of μ at z = 3.025 with respect to laboratory values. Table 3 lists the broadening parameter and the column density for all the lines from one particular J level fit together and consistently with one N and one b per rotation level.

Table 3

Line parameters for the observed three rotational levels of H2.

The data give no hint towards variation of the proton-to-electron mass ratio in the course of cosmic time. The uncertainties in the line positions of the H2features due to the photon noise are estimated by the fitting algorithm. These are shown in the errorbars in Fig. 4 and reported in Table 4. The mean error in the line positioning is of 152 m s-1. Even at first glance the given errorbars in Fig. 4 appear to be too small to explain the observed scatter.

Figure 5 shows the same line data as is Fig. 4 but lines with similar k values are binned. For a better overview, errorbars are omitted. The red data points (crosses) reflect the radial velocity within a small sensitivity range (given as x-errorbar). The y-errorbars correspond to the standard deviation of the mean value of the scatter within such a bin. The scatter within the bins can not be attributed to possible variations of μ since it is present for basically the same sensitivity parameter. The scatter is of the order of 220 m s-1 and thus larger than the positioning error of the individual lines.

thumbnail Fig. 4

Measured radial velocity vs. sensitivity for all 42 lines. Any correlation would indicate a change in μ. The different symbols and colors used correspond to the observed three rotational levels. The errorbars reflect the 1σ errors.

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thumbnail Fig. 5

The observed 42 H2lines with their corresponding radial velocities. In blue, bins of similar sensitivity are plotted. Blue y-errorbars reflect the standard deviation of radial velocities within an interval (x-errorbar).

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That is also reflected by a reduced χ2 of 2.7 for a weighted linear fit to the data (corresponding to Δμ/μ = (1.8 ± 8.2) × 10-6 at zabs = 3.025). The true scatter of the data is of the order of 220 m s-1 and constitutes an absolute limit of precision. The above mentioned errors of the fitting procedure require an additional systematic component to explain the observed scatter: (3)with σobs = 220 m s-1, σpos = 152 m s-1, and σsys = 151 m s-1.

Table 4

List of 42 H2lines fitted in QSO 0347-383. Radial velocities given relative to a redshift of z = 3.02489817.

A direct linear fit to the unweighted data yields:

(4)Bootstrap analysis is a robust approach to obtain a linear fit to the data in Fig. 4 and estimate an error based on the true scatter of the data. The corresponding bootstrap histogram including Gaussian fit is illustrated in Fig. 6. The Gaussian fit gives: (5)

thumbnail Fig. 6

Bootstrap histogram of 50.000 samples based on all 42 lines. The dashed line shows the bootstrap fit of the 2002 data of QSO 0347-383.

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6. Systematics

Any process which influences the measured redshift in dependence with the excitation energy would mimic a variation in μ since the transitions from higher excited states naturally show a stronger sensitivity towards changes in μ (see Varshalovish & Levshakov 1993). Figure 7 marks the wavelength ranges covered by the different orders of the CCD spectrum. Decreasing wavelengths are plotted rightwards since K sensitivity factors are increasing almost linearly with decreasing wavelengths and therefore Fig. 7 is comparable with Fig. 4. This new figure shows no trend but again a rather high scatter within the individual orders can be perceived.

thumbnail Fig. 7

All 42 lines with their radial velocity against the observed wavelength. Different orders are distinguished by colors and symbols.

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In the following, we consider a possible correlation of the line position uncertainty and its relative position within an order. Spectral distortions within the spectral orders have been investigated at the Keck/HIRES spectrograph by comparing the ThAr wavelength scale with a second one established from I2-cell observations of a bright quasar by Griest et al. (2010). In the wavelength range  ~ 5000 − 6200 Å  covered by the iodine cell absorption they found both absolute offsets which can be as large as 500 − 1000 m s-1  and an additional saw-tooth distortion pattern with an amplitude of about 300 m s-1. The distortions are such that transitions at the order edges appear at different velocities with respect to transitions at the order centers when calibrated with a ThAr exposure. This would introduce relative velocity shifts between different absorption features up to a magnitude the analysis with regard to Δμ/μ is sensitive to.

Whitmore et al. (2010) recently repeated the same test for UVES with similar finding though the saw-tooth distortions show slightly reduced peak-to-peak velocity variations of  ~ 200 m s-1. The physical explanation for those distortions is not yet known, so it is still to be examined whether the deviations are the same at other wavelengths or depend on the specific exposure.

The available solar atlas can be used to check UVES interorder distorsions as suggested in Molaro & Centurion (2011). For this purpose UVES observations of the solar spectrum reflected by the asteroid Iris were taken on Sep. 2009 with a resolving power R ≈    = 85   000.

The differences of the positions in the UVES spectrum and the absolute positions of the solar atlas for 238 solar photospheric lines in the region between 500 − 530 nm (Orders 121 to 116) are shown in Fig. 8. The schematic saw-tooth pattern detected by Whitmore et al. (2010) is also sketched in the figure. The stochastic distribution of the data points does not allow for any conclusion to be drawn about an underlying pattern. The observed dispersion is of 82 m s-1. Since the typical error in the measurement of lines in the UVES Iris spectrum is of  ≈ 30 m s-1 and wavelength calibration residuals are of 25 m s-1, there is an excess in the observed dispersion which suggests the presence of local deviations in the UVES spectrum. The saw-tooth pattern detected by Whitmore et al. (2010) is not revealed by our test.

thumbnail Fig. 8

Radial velocity shifts of individual lines Molaro et al. (2011).

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Figure 9 sorts the observed lines according to their relative position within their order. The origin of the abscissa reflects the central position within an order. All observed orders are stretched to an identical scale and over-plotted for this purpose. Lines near  − 1/2 on the X-axis are positioned near the left rim of the order, and so on. The distribution of obtained radial velocities seems to show a certain periodic pattern. The blue curve shows a fitted cosine with an amplitude of 151 m s-1. Considering the errorbars of the individual lines, this is no more than a slight indication which supports the presence of local distortions resulting from a non perfect calibration.

thumbnail Fig. 9

All 42 lines with their radial velocity against their relative position within their order. A cosine fit with an amplitude of 151 m s-1 is shown in blue.

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Systematic errors at the level of few hundred m s-1  have been revealed also in the UVES data by comparison of relative shifts of lines with comparable response to changes of fundamental constants (Centurion et al. 2009) and Molaro et al. (in prep.). Molaro et al. (in prep.) suggest that these distortions may originate from the block stiching of the CCDs. A CCD device is built-up by means of several sub-unit blocks with typical sizes of 512 pixels. The stitching of the blocks process produces misalignments of the order of few 0.01 of the pixel size in the block conjunctions. The ThAr has not enough lines to follow these imperfections which are therefore flattened in the pixel-to-wavelength conversion by a low order polynomial resulting into the observed spectral distortions.

7. Discussion

The result of Δμ/μ = (4.3 ± 7.2) × 10-6 we obtained is consistent with no variation of μ between zabs = 3.025 and z = 0. The null-result is in agreement with recent publications on Δμ/μ by King et al. (2011) and Weerdenburg et al. (2011) at zabs = 2.811 and zabs = 2.059, respectively. However, the present work utilizes the line-by-line fitting method (as in Ivanchik et al. 2005) in contrast to the other works which applied a comprehensive fitting method (CFM). The H2system in the spectrum of QSO 0347-383 has the particular advantage of comprising a single velocity component, which renders observed transitions independent of each other. For absorption systems with two or more closely and not properly resolved velocity components many systematic errors may influence distinct wavelength areas. The CFM fits all H2components along with additional HI lines and handles an artificially applied Δμ/μ as free parameter in the fit. The best matching Δμ/μ is then derived via the resulting χ2 curve. The CFM aims to achieve the lowest possible via additional velocity components. In this approach the information of individual transitions is lost though since merely the overall quality of the comprehensive model is judged.

In Weerdenburg et al. (2011) the number of velocity components is increased as long as the composite residuals of several selected absorption lines differ from flat noise. The residuals therein do not take into account the known inaccuracy of the estimated flux error (see Wendt & Molaro 2011; King et al. 2011).

As pointed out by King et al. (2011), for multi-component structures with overlapping velocity components the errors in the line centroids are heavily correlated and a simple χ2 regression is no longer valid. The same principle applies for co-added spectra with relative velocity shifts. The required rebinning of the contributing data sets implements further autocorrelation of the individual “pixels”.

The uncertainties of the oscillator strengths fi that are stated to be up to 50% (Weerdenburg et al. 2011) might further affect the criteria for additional velocity components. The method of CFM was applied for QSO 0347-383 by King et al. (2008). Section 4 and in particular Fig. 3 reveal some of the mentioned difficulties. The approach to fit individual lines with common physical properties which was applied here allows us to carry out an error analysis which reproduces the impact of differential shifts within the spectral orders. This yields a higher transparency of the error-budget for individual lines but at the possible cost of larger scatter resulting in a slightly larger error-estimate. For the small number of H2lines observed in the spectrum of QSO 0347-383 we prefer the method applied in this work. The immediate advantage is that we are not forced to estimate the different systematic errors based on assumptions. Instead the true limiting error can be gathered directly from the data distribution and we are further able to attribute it to different sources.

The new set of UVES observations of QSO 0347-383 this analysis is based has been taken with special care aimed to improve the measurement of Δμ/μ in a robust manner.In particular the observations have been taken with higher resolution, a 1 × 1 binning and calibration lamp spectra in direct combination with the main exposures. These boost the precision of the analysis roughly by a factor two with comparison with Wendt et al. (2008). We have shown that at the current level, calibration issues become the dominant source of error. In addition to positioning errors, which are related to the signal-to-noise-ratio of the data, we observe for the first time inter order distortions which seem to be of the same order of magnitude of the uncertainties in the line positions. The conclusion is that we do not detect change in the value of μ to 1 part in 105 over a time span of 11.5 Gyr, which is approximately 80% of the age of the universe.

High resolution data and attached calibration spectra are the key to understanding and handling the systematics which limit the precision of Δμ/μ measurements. It is important to fully control the analysis of individual absorption systems and to minimize the errors involved wherever possible before extending the Δμ/μ analysis to multiple systems. Different characteristics of individual absorbers tend to get lost while not all errors of the measurements are likely to average out.

Until new ways of wavelength calibration such as optical laser frequency combs (see Steinmetz et al. 2008) are installed for large optics, the data at hand is of the best qualityavailable. High resolution VLT-data with special care regarding the

calibration frames allows for the best precision that can be reached nowadays. In the context of Δμ/μ measurements, optical spectra of H2at high redshifts are still of high importance to complement high precision determinations of Δμ/μ in the local universe via observations in the radio regime.


1

Version 4.9.5.

2

A non-zero mean radial velocity directly reflects a deviation from the assumed absorber redshift.

3

Their individual positions are extracted from the plot in King et al. (2008).

Acknowledgments

We are thankful for helpful discussions on this topic with S. A. Levshakov and D. Reimers.

References

All Tables

Table 1

Journal of the observations (2009 data). Before and after each spectrum, a 30 s calibration frame was recorded.

Table 2

Median velocities of single spectra.

Table 3

Line parameters for the observed three rotational levels of H2.

Table 4

List of 42 H2lines fitted in QSO 0347-383. Radial velocities given relative to a redshift of z = 3.02489817.

All Figures

thumbnail Fig. 1

The 11 single spectra and the corresponding co-added data (bottom) are plotted around the region of L4R1 and L4P1 (vertical lines). The stronger H2features can be merely distinguished in single spectra.

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In the text
thumbnail Fig. 2

Exemplary bootstrap histogram of the median position of lines in spectrum 8 with respect to z = 3.02489817. The Gaussian fit corresponds to 302 ± 4 m s-1 (compare Table 2).

Open with DEXTER
In the text
thumbnail Fig. 3

Comparison between the 2009 data (top graph) and, with an offset, the original single observation run data of 2002 (9 frames, bottom) as used by Ivanchik et al. (2005), King et al. (2008), Thompson et al. (2009), and Wendt & Molaro (2011). The vertical lines indicate the positions of the H2component (solid) and the 12 additional lines fitted in King et al. (2008). In green the single line fits (L4R1 and L4P1) with their corresponding local continuum polynomial fit from this work. Note, that the fits for the analysis were not carried out on the rebinned and coadded data as shown in this figure.

Open with DEXTER
In the text
thumbnail Fig. 4

Measured radial velocity vs. sensitivity for all 42 lines. Any correlation would indicate a change in μ. The different symbols and colors used correspond to the observed three rotational levels. The errorbars reflect the 1σ errors.

Open with DEXTER
In the text
thumbnail Fig. 5

The observed 42 H2lines with their corresponding radial velocities. In blue, bins of similar sensitivity are plotted. Blue y-errorbars reflect the standard deviation of radial velocities within an interval (x-errorbar).

Open with DEXTER
In the text
thumbnail Fig. 6

Bootstrap histogram of 50.000 samples based on all 42 lines. The dashed line shows the bootstrap fit of the 2002 data of QSO 0347-383.

Open with DEXTER
In the text
thumbnail Fig. 7

All 42 lines with their radial velocity against the observed wavelength. Different orders are distinguished by colors and symbols.

Open with DEXTER
In the text
thumbnail Fig. 8

Radial velocity shifts of individual lines Molaro et al. (2011).

Open with DEXTER
In the text
thumbnail Fig. 9

All 42 lines with their radial velocity against their relative position within their order. A cosine fit with an amplitude of 151 m s-1 is shown in blue.

Open with DEXTER
In the text

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