Quasar (QSO) absorption lines provide a powerful probe of the variation of fundamental constants.
Absorption lines in intervening clouds along the line of sight of the QSO give access to the spectra
of the atoms present in the cloud, that it is to paleo-spectra. The method was first used by
Savedoff [447] who constrained the time variation of the fine-structure constraint from the doublet
separations seen in galaxy emission spectra. For general introduction to these observations, we refer
to [412, 474, 271].
Indeed, one cannot use a single transition compared to its laboratory value since the expansion of the universe induces a global redshifting of all spectra. In order to tackle down a variation of the fundamental constants, one should resort on various transitions and look for chromatic effects that can indeed not be reproduce by the expansion of the universe, which acts chromatically on all wavelengths.
To achieve such a test, one needs to understand the dependencies of different types of transitions, in a
similar way as for atomic clock experiments. [175, 169] suggested to use the convenient formulation
The shift between two lines is easier to measure when the difference between the -coefficients of the
two lines is large, which occurs, e.g., for two levels with large
of opposite sign. Many methods were
developed to take this into account. The alkali doublet method (AD) focuses on the fine-structure doublet of
alkali atoms. It was then generalized to the many-multiplet method (MM), which uses correlations
between various transitions in different atoms. As can be seen on Figure 3
, some transitions are
almost insensitive to a variation of
. This is the case of Mg ii, which can be used as an
anchor, i.e., a reference point. To obtain strong constraints one can either compare transitions of
light atoms with those of heavy atoms (because the
dependence of the ground state
scales as
) or compare
and
transitions in heavy elements (in that case, the
relativistic correction will be of opposite signs). This latter effect increases the sensitivity and
strengthens the method against systematic errors. However, the results of this method rely on two
assumptions: (i) ionization and chemical homogeneity and (ii) isotopic abundance of Mg ii close to the
terrestrial value. Even though these are reasonable assumptions, one cannot completely rule out
systematic biases that they could induce. The AD method completely avoids the assumption of
homogeneity because, by construction, the two lines of the doublet must have the same profile.
Indeed the AD method avoids the implicit assumption of the MM method that chemical and
ionization inhomogeneities are negligible. Another way to avoid the influence of small spectral
shift due to ionization inhomogeneities within the absorber and due to possible non-zero offset
between different exposures was to rely on different transitions of a single ion in individual
exposure. This method has been called the Single ion differential alpha measurement method
(SIDAM).
Most studies are based on optical techniques due to the profusion of strong UV transitions that are
redshifted into the optical band (this includes AD, MM, SIDAM and it implies that they can be applied
only above a given redshift, e.g., Si iv at , Fe ii
at
) or on radio techniques since
radio transitions arise from many different physical effects (hyperfine splitting and in particular H i 21 cm
hyperfine transition, molecular rotation, Lambda-doubling, etc). In the latter case, the line frequencies and
their comparisons yield constraints on different sets of fundamental constants including
,
and
. Thus, these techniques are complementary since systematic effects are different in optical and radio
regimes. Also the radio techniques offer some advantages: (1) to reach high spectral resolution
(
), alleviating in particular problems with line blending and the use of, e.g., masers allow
to reach a frequency calibration better than roughly 10 m/s; (2) in general, the sensitivity
of the line position to a variation of a constant is higher; (3) the isotopic lines are observed
separately, while in optical there is a blend with possible differential saturations (see, e.g., [109] for a
discussion).
Let us first emphasize that the shifts in the absorption lines to be detected are extremely small. For
instance a change of of order 10–5 corresponds a shift of at most 20 mÅ for a redshift of
,
which would corresponds to a shift of order
, or to about a third of a pixel at a spectral
resolution of
, as achieved with Keck/HIRES or VLT/UVES. As we shall discuss later, there
are several sources of uncertainty that hamper the measurement. In particular, the absorption lines have
complex profiles (because they result from the propagation of photons through a highly inhomogeneous
medium) that are fitted using a combination of Voigt profiles. Each of these components depends on
several parameters including the redshift, the column density and the width of the line (Doppler
parameter) to which one now needs to add the constants that are assumed to be varying. These
parameters are constrained assuming that the profiles are the same for all transitions, which
is indeed a non-trivial assumption for transitions from different species (this was one of the
driving motivations to use the transition from a single species and of the SIDAM method).
More important, the fit is usually not unique. This is not a problem when the lines are not
saturated but it can increase the error on
by a factor 2 in the case of strongly saturated
lines [91
].
The first method used to set constraint on the time variation of the fine-structure constant relies on fine-structure doublets splitting for which
Several authors have applied the AD method to doublets of several species such as, e.g., C iv, N v, O vi,
Mg ii, Al iii, Si ii, Si iv. We refer to Section III.3 of FVC [500] for a summary of their results (see
also [318]) and focus on the three most recent analysis, based on the Si iv doublet. In this particular case,
(resp. 362) cm–1 and
(resp. –8) cm–1 for Si iv
(resp.
) so that
. The method is based on a
minimization of multiple component Voigt profile
fits to the absorption features in the QSO spectra. In general such a profile depends on three
parameters, the column density
, the Doppler width (
) and the redshift. It is now extended to
include
. The fit is carried out by simultaneously varying these parameters for each
component.
One limitation may arise from the isotopic composition of silicium. Silicium has three naturally
occurring isotopes with terrestrial abundances 28Si:29Si:30Si = 92.23:4.68:3.09 so that each absorption line
is a composite of absorption lines from the three isotopes. However, it was shown that this effect of isotopic
shifts [377] is negligible in the case of Si iv.
A generalization of the AD method, known as the many-mulptiplet was proposed in [176]. It relies on the
combination of transitions from different species. In particular, as can be seen on Figure 3,
some transitions are fairly unsensitive to a change of the fine-structure constant (e.g., Mg ii or
Mg i, hence providing good anchors) while others such as Fe ii are more sensitive. The first
implementation [522
] of the method was based on a measurement of the shift of the Fe ii (the rest
wavelengths of which are very sensitive to
) spectrum with respect to the one of Mg ii.
This comparison increases the sensitivity compared with methods using only alkali doublets.
Two series of analyses were performed during the past ten years and lead to contradictory
conclusions. The accuracy of the measurements depends on how well the absorption line profiles are
modeled.
As we pointed out earlier, one assumption of the method concerns the isotopic abundances of Mg ii that
can affect the low- sample since any changes in the isotopic composition will alter the value of effective
rest-wavelengths. This isotopic composition is assumed to be close to terrestrial 24Mg:25Mg:26Mg =
79:10:11. No direct measurement of
in QSO absorber is currently feasible due
to the small separation of the isotopic absorption lines. However, it was shown [231], on the basis of
molecular absorption lines of MgH that
generally decreases with decreasing metallicity. In standard
models it should be near 0 at zero metallicity since type II supernovae are primarily producers of
24Mg. It was also argued that 13C is a tracer of 25Mg and was shown to be low in the case of
HE 0515-4414 [321]. However, contrary to this trend, it was found [552] that
can reach high
values for some giant stars in the globular cluster NGC 6752 with metallicity [Fe/H]
.
This led Ashenfelter et al. [18
] to propose a chemical evolution model with strongly enhanced
population of intermediate (
) stars, which in their asymptotic giant branch phase are
the dominant factories for heavy Mg at low metallicities typical of QSO absorption systems,
as a possible explanation of the low-
Keck/HIRES observations without any variation of
. It would require that
reaches 0.62, compared to 0.27 (but then the UVES/VLT
constraints would be converted to a detection). Care needs to be taken since the star formation
history can be different ine each region, even in each absorber, so that one cannot a priori use
the best-fit obtained from the Keck data to the UVES/VLT data. However, such modified
nucleosynthetic history will lead to an overproduction of elements such as P, Si, Al, P above current
constraints [192], but this later model is not the same as the one of Ref. [18] that was tuned to avoid these
problems.
In conclusion, no compelling evidence for a systematic effect has been raised at the moment.
Refs. [90, 470
] analyzed the observations of 23 absorption systems, fulfilling the above criteria, in
direction of 18 QSO with a S/N ranging between 50 and 80 per pixel and a resolution
. They
concluded that
This analysis was challenged by Murphy, Webb and Flambaum [372, 371
, 370
]. Using (quoting them)
the same reduced data, using the same fits to the absorption profiles, they claim to find different individual
measurements of
and a weighted mean,
On the basis of the articles [372, 371, 370] and the answer [471], it is indeed difficult (without having
played with the data) to engage one of the parties. This exchange has enlightened some differences in the
statistical analysis.
To finish, let us mention that [361] reanalyzed some systems of [90
, 470] by means of the SIDAM
method (see below) and disagree with some of them, claiming for a problem of calibration. They also claim
that the errors quoted in [367
] are underestimated by a factor 1.5.
On the one hand, it is appropriate that one team has reanalyzed the data of the other and challenged its analysis. This would indeed lead to an improvement in the robustness of these results. Indeed a similar reverse analysis would also be appropriate. On the other hand both teams have achieved an amazing work in order to understand and quantify all sources of systematics. Both developments, as well as the new techniques, which are appearing, should hopefully set this observational issue. Today, it is unfortunately premature to choose one data set compared to the other.
A recent data [523] set of 60 quasar spectra (yielding 153 absorption systems) for the VLT was used and
split at
to get
This method [320] is an adaptation of the MM method in order to avoid the influence of small spectral
shifts due to ionization inhomogeneities within the absorbers as well as to non-zero offsets between different
exposures. It was mainly used with Fe ii, which provides transitions with positive and negative
-coefficients (see Figure 3
). Since it relies on a single ion, it is less sensitive to isotopic abundances, and
in particular not sensitive to the one of Mg.
The first analysis relies on the QSO HE 0515-4414 that was used in [427] to get the constraint (77). An
independent analysis [361
] of the same system gave a weighted mean
The comparison of UV heavy element transitions with the hyperfine H i transition allows to extract [496]
Using 9 absorption systems, there was no evidence for any variation of [494
],
Following [147], we note that the systems lie in two widely-separated ranges and that the two samples
have completely different scatter. Therefore it can be split into two samples of respectively 5 and 4 systems
to get
In such an approach two main difficulties arise: (1) the radio and optical source must coincide (in the
optical QSO can be considered pointlike and it must be checked that this is also the case for the radio
source), (2) the clouds responsible for the 21 cm and UV absorptions must be localized in the same place.
Therefore, the systems must be selected with care and today the number of such systems is small and are
actively looked for [411].
The recent detection of 21 cm and molecular hydrogen absorption lines in the same damped
Lyman- system at
towards SDSS J1337+3152 constrains [472
] the variation
to
The H i 21 cm hyperfine transition frequency is proportional to (see Section 3.1.1). On the
other hand, the rotational transition frequencies of diatomic are inversely proportional to their reduced
mass
. As on the example of Equation (35
) where we compared an electronic transition to a
vibro-rotational transition, the comparison of the hyperfine and rotational frequencies is proportional
to
The constraint on the variation of is directly determined by comparing the redshift as determined
from H i and molecular absorption lines,
This method was first applied [513] to the CO molecular absorption lines [536] towards PKS 1413+135
to get
The radio domain has the advantage of heterodyne techniques, with a spectral resolution of 106 or more, and dealing with cold gas and narrow lines. The main systematics is the kinematical bias, i.e., that the different lines do not come exactly from the same material along the line of sight, with the same velocity. To improve this method one needs to find more sources, which may be possible with the radio telescope ALMA 3.
Using transitions originating from a single species, like with SIDAM, allows to reduce the systematic effects.
The 18 cm lines of the OH radical offers such a possibility [95, 272].
The ground state, , of OH is split into two levels by
-doubling and each of these
doubled level is further split into two hyperfine-structure states. Thus, it has two “main” lines (
)
and two “satellite” lines (
). Since these four lines arise from two different physical processes
(
-doubling and hyperfine splitting), they enjoy the same Rydberg dependence but different
and
dependencies. By comparing the four transitions to the H i hyperfine line, one can have access to
Using the four 18 cm OH lines from the gravitational lens at toward PMN J0134-0931 and
comparing the H i 21 cm and OH absorption redshifts of the different components allowed to set the
constraint [276
]
Another combination [300] of constants can be obtained from the comparison of far infrared fine-structure
spectra with rotational transitions, which respectively behaves as and
so that
they give access to
Using the C ii fine-structure and CO rotational emission lines from the quasars J1148+5251 and BR 1202-0725, it was concluded that
which represents the best constraints at high redshift. As usual, when comparing the frequencies of two different species, one must account for random Doppler shifts caused by non-identical spatial distributions of the two species. Several other candidates for microwave and FIR lines with good sensitivities are discussed in [299].
The satellite OH 18 cm lines are conjugate so that the two lines have the same shape, but with one line in
emission and the other in absorption. This arises due to an inversion of the level of populations within the
ground state of the OH molecule. This behavior has recently been discovered at cosmological distances and
it was shown [95] that a comparison between the sum and difference of satellite line redshifts probes
.
From the analysis of the two conjugate satellite OH systems at towards PKS 1413+135 and
at
towards PMN J0134-0931, it was concluded [95
] that
One strength of this method is that it guarantees that the satellite lines arise from the same gas, preventing from velocity offset between the lines. Also, the shape of the two lines must agree if they arise from the same gas.
As was pointed out in Section 3.1, molecular lines can provide a test of the
variation4 [488]
of since rotational and vibrational transitions are respectively inversely proportional to their reduce
mass and its square-root [see Equation (35
)].
H2 is the most abundant molecule in the universe and there were many attempts to use its absorption
spectra to put constraints on the time variation of despite the fact that H2 is very difficult to
detect [387
].
As proposed in [512], the sensitivity of a vibro-rotational wavelength to a variation of can be
parameterized as
We refer to Section V.C of FVC [500] for earlier studies and we focus on the latest results. The
recent constraints are mainly based on the molecular hydrogen of two damped Lyman-
absorption systems at
and
in the direction of two quasars (Q 1232+082 and
Q 0347-382) for which a first analysis of VLT/UVES data showed [262] a slight indication of a
variation,
It was further improved with an analysis of two absorption systems at and
in
the directions of Q 0405-443 and Q 0347-383 observed with the VLT/UVES spectrograph. The
data have a resolution
and a S/N ratio ranging between 30 and 70. The same
selection criteria where applied, letting respectively 39 (out of 40) and 37 (out of 42) lines for
each spectrum and only 7 transitions in common. The combined analysis of the two systems
led [261]
These two systems were reanalyzed [289], adding a new system in direction of Q 0528-250,
This method is subject to important systematic errors among which (1) the sensitivity to the laboratory
wavelengths (since the use of two different catalogs yield different results [431]), (2) the molecular
lines are located in the Lyman-
forest where they can be strongly blended with intervening
H i Lyman-
absorption lines, which requires a careful fitting of the lines [289
] since it is
hard to find lines that are not contaminated. From an observational point of view, very few
damped Lyman-
systems have a measurable amount of H2 so that only a dozen systems is
actually known even though more systems will be obtained soon [411]. To finish, the sensitivity
coefficients are usually low, typically of the order of 10–2. Some advantages of using H2 arise
from the fact there are several hundred available H2 lines so that many lines from the same
ground state can be used to eliminate different kinematics between regions of different excitation
temperatures. The overlap between Lyman and Werner bands also allow to reduce the errors of
calibration.
To conclude, the combination of all the existing observations indicate that is constant at the 10–5
level during the past 11 Gigayrs while an improvement of a factor 10 can be expected in the five coming
years.
It was recently proposed [201, 202] that the inversion spectrum of ammonia allows for a better sensitivity
to
. The inversion vibro-rotational mode is described by a double well with the first two levels below
the barrier. The tunneling implies that these two levels are split in inversion doublets. It was
concluded that the inversion transitions scale as
, compared with a rotational
transition, which scales as
. This implies that the redshifts determined by the two types of
transitions are modified according to
and
so
that
This method was also applied [323] in the Milky Way, in order to constrain the spatial variation of
in the galaxy (see Section 6.1.3). Using ammonia emission lines from interstellar molecular clouds
(Perseus molecular core, the Pipe nebula and the infrared dark clouds) it was concluded that
. This indicates a positive velocity offset between the ammonia inversion transition
and rotational transitions of other molecules. Two systems being located toward the galactic center while
one is in the direction of the anti-center, this may indicate a spatial variation of
on galactic
scales.
The detection of several deuterated molecular hydrogen HD transitions makes it possible to test the
variation of in the same way as with H2 but in a completely independent way, even though today it has
been detected only in 2 places in the universe. The sensitivity coefficients have been published in [263] and
HD was first detected by [387].
HD was recently detected [473] together with CO and H2 in a DLA cloud at a redshift of 2.418 toward
SDSS1439+11 with 5 lines of HD in 3 components together with several H2 lines in 7 components. It
allowed to set the 3 limit of
[412
].
Even though the small number of lines does not allow to reach the level of accuracy of H2 it is a very promising system in particular to obtain independent measurements.
Similar analysis to constrain the time variation of the fundamental constants were also performed with emission spectra. Very few such estimates have been performed, since it is less sensitive and harder to extend to sources with high redshift. In particular, emission lines are usually broad as compared to absorption lines and the larger individual errors need to be beaten by large statistics.
The O iii doublet analysis [24] from a sample of 165 quasars from SDSS gave the constraint
The method was then extended straightforwardly along the lines of the MM method and applied [238] to the fine-structure transitions in Ne iii, Ne v, O iii, O i and S ii multiplets from a sample of 14 Seyfert 1.5 galaxies to derive the constraint
This subsection illustrates the diversity of methods and the progresses that have been achieved to set robust
constraints on the variation of fundamental constants. Many systems are now used, giving access to
different combinations of the constants. It exploits a large part of the electromagnetic spectrum
from far infrared to ultra violet and radio bands and optical and radio techniques have played
complementary roles. The most recent and accurate constraints are summarized in Table 10 and
Figure 4.
Constant |
Method | System | Constraint (× 10–5) | Redshift | Ref. |
|
AD | 21 | (–0.5 ± 1.3) | 2.33 – 3.08 | [377![]() |
|
AD | 15 | (–0.15 ± 0.43) | 1.59 – 2.92 | [91] |
|
AD | 9 | (–3.09 ± 8.46) | 1.19 – 1.84 | [349] |
|
MM | 143 | (–0.57 ± 0.11) | 0.2 – 4.2 | [367] |
|
MM | 21 | (0.01 ± 0.15) | 0.4 – 2.3 | [90] |
|
SIDAM | 1 | (–0.012 ± 0.179) | 1.15 | [361![]() |
|
SIDAM | 1 | (0.566 ± 0.267) | 1.84 | [361] |
|
H i - mol | 1 | (–0.16 ± 0.54) | 0.6847 | [375![]() |
|
H i - mol | 1 | (–0.2 ± 0.44) | 0.247 | [375] |
|
CO, CHO+ | (–4 ± 6) | 0.247 | [536] | |
|
OH - H i | 1 | (–0.44 ± 0.36 ± 1.0syst) | 0.765 | [276] |
|
OH - H i | 1 | (0.51 ± 1.26) | 0.2467 | [138] |
|
H i - UV | 9 | (–0.63 ± 0.99) | 0.23 – 2.35 | [494] |
|
H i - UV | 2 | (–0.17 ± 0.17) | 3.174 | [472] |
|
C ii - CO | 1 | (1 ± 10) | 4.69 | [327![]() |
|
C ii - CO | 1 | (14 ± 15) | 6.42 | [327] |
|
OH | 1 | < 1.1 | 0.247, 0.765 | [95![]() |
|
OH | 1 | < 1.16 | 0.0018 | [95] |
|
OH | 1 | (–1.18 ± 0.46) | 0.247 | [273] |
|
H2 | 1 | (2.78 ± 0.88) | 2.59 | [431![]() |
|
H2 | 1 | (2.06 ± 0.79) | 3.02 | [431] |
|
H2 | 1 | (1.01 ± 0.62) | 2.59 | [289![]() |
|
H2 | 1 | (0.82 ± 0.74) | 2.8 | [289![]() |
|
H2 | 1 | (0.26 ± 0.30) | 3.02 | [289] |
|
H2 | 1 | (0.7 ± 0.8) | 3.02, 2.59 | [490] |
|
NH3 | 1 | < 0.18 | 0.685 | [366] |
|
NH3 | 1 | < 0.38 | 0.685 | [353] |
|
HC3N | 1 | < 0.14 | 0.89 | [250] |
|
HD | 1 | < 9 | 2.418 | [412] |
|
HD | 1 | (0.56 ± 0.55stat ± 0.27syst) | 2.059 | [342] |
At the moment, only one analysis claims to have detected a variation of the fine structure constant
(Keck/HIRES) while the VLT/UVES points toward no variation of the fine structure constant. It has led to
the proposition that may be space dependent and exhibit a dipole, the origin of which is not
explained. Needless to say that such a controversy and hypotheses are sane since it will help improve
the analysis of this data, but it is premature to conclude on the issue of this debate and the
jury is still out. Most of the systematics have been investigated in detail and now seem under
control.
Let us what we can learn on the physics from these measurement. As an example, consider the
constraints obtained on ,
and
in the redshift band 0.6 – 0.8 (see Table 10). They can be used to
extract independent constraints on
,
and
We mention in the course of this paragraph many possibilities to improve these constraints.
Since the AD method is free of the two main assumptions of the MM method, it seems important to increase the precision of this method as well as any method relying only on one species. This can be achieved by increasing the S/N ratio and spectral resolution of the data used or by increasing the sample size and including new transitions (e.g., cobalt [172, 187]).
The search for a better resolution is being investigated in many direction. With the current
resolution of , the observed line positions can be determined with an accuracy of
. This implies that the accuracy on
is of the order of 10–5 for lines
with typical
-coefficients. As we have seen this limit can be improved to 10–6 when more
transitions or systems are used together. Any improvement is related to the possibility to measure
line positions more accurately. This can be done by increasing
up to the point at which
the narrowest lines in the absorption systems are resolved. The Bohlin formula [62] gives the
estimates
The limitation may then lie in the statistics and the calibration and it would be useful to use more than
two QSO with overlapping spectra to cross-calibrate the line positions. This means that one needs to
discover more absorption systems suited for these analyses. Much progress is expected. For
instance, the FIR lines are expected to be observed by a new generation of telescopes such as
HERSCHEL6.
While the size of the radio sample is still small, surveys are being carried out so that the number of
known redshift OH, HI and HCO+ absorption systems will increase. For instance the future
Square Kilometer Array (SKA) will be able to detect relative changes of the order of 10–7 in
.
In conclusion, it is clear that these constraints and the understanding of the absorption systems will increase in the coming years.
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Living Rev. Relativity 14, (2011), 2
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