Detailed abstract

Were the laws of nature the same ten billion light years from us?
Theories unifying gravity and other interactions suggest the
possibility of spatial and temporal variation of physical ``constants''
 in the Universe. Current interest is high because in superstring
theories which have additional  dimensions, compactified
on tiny scales, any variation of the  size of the extra dimensions
results in changes of the 3-dimensional observed coupling constants.
At present no mechanism for keeping the internal spatial scale static
has been found (for example,
our three ``large'' spatial dimensions increase in size).  
Therefore, unified theories applied to cosmology suffer generically
from a problem of predicting time-dependent coupling constants.
Moreover, there is a mechanism which makes all coupling constants
and masses of elementary particles to be both space and time
dependent, and influenced by local circumstances [1].

The strongest terrestial constraint on the time
evolution of the fine structure constant, $\alpha$,
comes from a natural uranium nuclear fission
reactor in Gabon, West Africa, which was active 1.8 billion years
ago. The relative change of $\alpha$ during this time interval does not
exceed  1.2$\times 10^{-7}$ [2]. However, this limit is based on certain
assumptions and  covers a relatively small fraction of the age of 
the Universe. Also, it does not exclude oscillatory dependence  of $\alpha$.  

Astrophysical measurements enable us to push the epoch probed back
to much earlier times.
Any change in $\alpha$ could be detected
via shifts in the rest wavelengths of resonance transitions
in quasar absorption systems. For example, the ratio
of fine structure splitting of an alkali-type doublet to the
mean transition frequency is proportional to  $\alpha^2$.
A comparison of these ratios in cosmic spectra with laboratory values
provides powerful constraints on variability.
This method was proposed by J. Bachall and M. Schmidt in 1967.

Recently a new approach has been developed which improves the 
sensitivity to a variation of $\alpha$ by more than an order of
magnitude [3,4].
The relative value of any relativistic corrections to atomic
transition frequencies is proportional to $\alpha^2$. These corrections
can exceed the fine structure interval between the excited levels
by an order of magnitude (for example, an s-wave electron does not
have the spin-orbit splitting  but it has the maximal
relativistic correction to energy). The relativistic corrections
vary very strongly from atom to atom and can
have opposite signs in different transitions (for example,
in s-p and d-p transitions). Thus, any variation of $\alpha$
could be revealed by comparing different transitions
in different atoms in cosmic and laboratory spectra.

This method provides an order of magnitude precision gain compared to 
measurements of the fine structure interval.
Relativistic many-body calculations are used to reveal the dependence of
atomic frequencies on the fine structure constant, for a range 
of atomic species observed in
quasar absorption spectra [3].
It is convenient to present results for the transition frequencies
as functions of  $\alpha^2$ in the form
\begin{equation}
        \omega = \omega_0 + q_1 x + q_2 y,
\label{omega}
\end{equation}
where
$x = (\frac{\alpha}{\alpha_l})^2 - 1, y = (\frac{\alpha}{\alpha_l})^4 - 1$
and $\omega_0 $ is a laboratory frequency of a particular transition.
New and accurate laboratory measurements of $\omega_0 $ have been carried out
specifically for this work
by Imperial College (London), Lund University and NIST groups.
We stress that the second and third terms contribute only if   $\alpha$
deviates from the laboratory value $\alpha_l$.
The initial observational
results  [4] for two Mg II lines and five Fe II lines
suggest that $\alpha$ may have been smaller in the past.

This work has been continued in Ref. [5]. A large set of data consists
of 49 quasar absorption systems located between 4 and 11
billion light years from us (starting from
10\% of the age of the Universe after Big Bang).   Many
lines of  Mg I, Mg II, Fe II, Zn II, Cr II, Ni II, Si II, Al II and Al III have been included and a study of both temporal and spatial dependence
of  $\alpha$ has been performed. For the whole sample,
$ \Delta \alpha/\alpha =(- 6.2 \pm 1.5) \times 10^{-6}$.
We should stress that only statistical error is presented here.
 This error is now small and the main efforts
are directed towards the study of various systematic effects.

 This cosmic spectroscopy method has been extended to study variation
of other fundamental parameters. The ratio of the hydrogen atom hyperfine
transition frequency to a  molecular (CO, CN, CS, HCO$^+$, HCN)  rotational
frequency is proportional to $y=\alpha^2 g_p$ where $g_p$ is the proton
 magnetic $g$-factor. A preliminary result here [5]
 is $y=(- 2.4 \pm 1.8) \times 10^{-6}$ about 4 billion light years from us.  
 The ratio of rotational and optical frequencies
is sensitive to the ratio of the electron and proton masses,
 hyperfine/optical comparison constrains $\alpha^2 g_p m_e/m_p$.
Note that the proton $g$-factor and mass are functions of the
strong interaction constant $\alpha_s$ and vacuum condensates of
the quark and gluon fields.

Another method to search for  the time variation of $\alpha$
is to study variation of the ratio of frequencies in the laboratory.
The strongest laboratory limit on the $\alpha$ variation was
obtained by comparing H-maser vs Hg~II microwave atomic clocks over
140 days [6]. This yielded an upper limit
$\dot \alpha/\alpha \le 3.7 \times 10^{-14}/\mbox{yr}$.

Another possibility is to use optical atomic frequency standards.
Any evolution of $\alpha$ in time would lead to the frequency shift.
To establish the connection between $\dot \alpha$ and $\dot \omega$
relativistic calculations of the $\alpha$ dependence of the 
relevant frequencies for
Ca I, Sr II, Ba II, Yb II, Hg II, In II, Tl II and Ra II have been
 performed [3]. The  $\alpha$ dependence of the microwave frequency standards
(Cs, $Hg^+$) has also been accurately calculated.
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\begin{center}
{\bf \underline{References}}
\end{center}

1. T. Damour, A.M. Polyakov. Nucl. Phys. B {\bf 423}, 532 (1994).

2. A.I. Shlyakhter, Nature {\bf 264}, 340 (1976); T. Damour and
 F. Dyson, Nucl. Phys. B {\bf 480} , 37 (1996).

3. V.A. Dzuba, V.V. Flambaum, J.K. Webb. Phys. Rev. Lett.
 {\bf 82}, 888 (1999); Phys. Rev. A{\bf 59}, 230 (1999); Phys. Rev A,
 in press.

4. J.K. Webb, V.V. Flambaum, C.W. Churchill, M.J. Drinkwater,
and J.D. Barrow, Phys. Rev. Lett. {\bf 82}, 884 (1999). 

5. J.K. Webb, M.T. Murphy, V.V. Flambaum, C.W. Churchill, J.X.
Prochaska, A.M. Wolfe and V.A. Dzuba, to be submitted to Phys. Rev. Lett.

6. J. D. Prestage, R. L. Tjoeker, and L. Maleki,
Phys. Rev. Lett. {\bf 74}, 3511 (1995).