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![Page 1: Searching for cosmological variation of fundamental constants using high- resolution quasar spectroscopy John K. Webb University of New South Wales Sydney,](https://reader035.fdocuments.in/reader035/viewer/2022062421/56649f555503460f94c79006/html5/thumbnails/1.jpg)
Searching for cosmological variation of fundamental
constants using high-resolution quasar
spectroscopy
John K. WebbUniversity of New South WalesSydney, AustraliaKVI, Friday 31 August 2007
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• Why our particular set of (numerical values for) the fundamental constants (given the system of units)? Anthropic explanation? Should we be able to explain the numerical values from first principles?
• Fine tuning: Eg, electromagnetic coupling constant, = 1/137. Slight decrease: molecular bonds break at lower T, number of stable elements in periodic table increases. A 4% change would shut down stellar heavy element production – no life. Many other examples.
• Were our present-day constants laid down a the beginning or have they evolved into a more finely-tuned set?
• What is a fundamental constant? We formulate the “physical laws” in terms of observables, which = measurements wrt to some standard, but is that fundamental or merely “human”?
• Dimensional vs dimensionless. Does it make any sense to generate theories in which, eg, c varies? What changes can we actually look for?
• Should (can?) all physical laws be satisfactorily re-expressed in terms of dimensionless constants?
“Fundamental” constants
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History: Dirac, Milne, 1937. The first to ask “Do the constants of Nature vary?”
Theoretical motivation: unification models – more dimensions (“compactified”). Cosmological evolution? String theories generically allow for variations.
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Dimensionless ratios – the things we can actually check
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To Earth
CIVSiIVCIISiII
Lyem
Ly forest
Lyman limit Ly
NVem
SiIVem
CIVem
Lyem
Ly SiII
quasar
Quasars: physics laboratories in the early universe
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e
p
m
m
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DLAs with molecular hydrogen
• ~17,000 quasars with z>2.3 now known. Only ~15 known to have H2
• Claim that ~1/5 of all DLAs have H2
QSO 2348-011Q1443+272
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Extracting from high redshift quasar spectra (mp/me)
The Born-Oppenheimer approximation relates vibrational-rotational and electronic frequencies to , can be used to express the wavelength dependence of an observed H2 transition on a fractional change in , = (z – 0)/0
ii
zi Kz
1)1(
0 Calculated
Measured in laboratory experiment
Observed (rest-frame) H2 wavelength from quasar spectrum
This can be conveniently expressed as a simple linear relationship:
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Reinhold et al PRL April 2006
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First quasar, Q0347-383
Single velocity component
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H2 lines in the Lyman alpha forest of Q0347
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H2 lines in the Lyman alpha forest of Q0347H2 lines in the Lyman alpha forest of Q0347
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H2 lines in the Lyman alpha forest of Q0347H2 lines in the Lyman alpha forest of Q0347
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H2 lines in the Lyman alpha forest of Q0347H2 lines in the Lyman alpha forest of Q0347
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H2 lines in the Lyman alpha forest of Q0347
H2HI
H2 lines in the Lyman alpha forest of Q0347
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A. “Raw” error bars
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B. Constant added to error bars forcing (2 = 1
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C. Raw error bars, iteratively 3 clipped forcing (2 = 1
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A. Reduced redshift method, raw error bars2)/ = 1.61 B. Reduced redshift method, grown error bars2)/ = 1.0 (forced) C. 3 clip, raw errors 2)/ = 1.0 (forced) D. Solving directly for internal to VPFIT:
Results for 0347-383
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Second quasar, Q0528-250
Multiple velocity components
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J=0 J=1
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J=3J=2
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J=4
Note:a) Decrease in component
line width of the lines as J increases (not seen before?)
b) Increased strength in right hand component as J increases
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How many free parameters should be fitted?
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How many free parameters should be fitted?
“Right” answer probably somewhere in there, but where?
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How many free parameters should be fitted?
Aikaike Information Criterion: 0528-250: There are clearly at least 2 velocity components, very probably 3 (all previous analyses had fitting one single component)
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2 H2 components, “raw” error bars
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3 H2 components, “raw” error bars
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Results for 0528-250
3H2:VPFIT internal method (no clipping) :
Reduced redshift method (no clipping):
3 clipping:
2H2:
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1) Re-computation of Voigt function, H(a,u), finer interpolation.
2) Calculating Voigt profile, sub-divide spectral pixels, integrate over each pixel.
3) Calculating d/dp prone to error, can result in poor chisq-parameter curves.
4) Problem requires “physical constraints”- H2 lines can be unresolved, keeping b free for every line can severely ill-condition the matrices, rendering solutions meaningless. Solution: “tie” parameters, or sometimes fix uninteresting parameters, but must explore the consequences external to the non-linear least-squares procedure.
5) Best method probably to solve for explicitly within the non-linear least-squares procedure, but:
- Ensure matrix equations are well-conditioned; - All algorithmic parameters (eg stopping criterion, finite-difference derivatives)
should account for machine precision limits.- Absorption line parameters, where necessary, are re-scaled to minimise dynamic
range in Hessian and gradient matrices;- Check everything with synthetic spectra;- Improve systematic errors estimating, and chisq-parameter space exploration,
probably via Markov-Chain Monte-Carlo methods (work in progress)
Some technical points for the numerical-methods enthusiast
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c
e
2
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Metal absorption
Over 60 000 data points!
Quasar Q1759+75
H absorption
H emission
C IV doublet
C IV 1550ÅC IV 1548Å
QSO absorption lines:
• A Keck/HIRES doublet
Separation 2
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Parameters describing ONE absorption line
b (km/s)
1+z)rest
N (atoms/cm2)
3 Cloud parameters: b, N, z
“Known” physics parameters: rest, f,
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Cloud parameters describing TWO (or more) absorption lines from the same species (eg. MgII 2796 + MgII 2803 A)
z
b
bN
Still 3 cloud parameters (with no assumptions), but now there are more physics parameters
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Cloud parameters describing TWO absorption lines from different species (eg. MgII 2796 + FeII 2383 A)
b(FeII)b(MgII)
z(FeII)
z(MgII)
N(FeII)N(MgII)
i.e. a maximum of 6 cloud parameters, without any assumptions
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However…
bobserved2 b b
kT
mcons tthermal bulk
2 2 2tan
T is the cloud temperature, m is the atomic mass
So we understand the relation between (eg.) b(MgII) and b(FeII). The extremes are:
A: totally thermal broadening, bulk motions negligible,
B: thermal broadening negligible compared to bulk motions,
b MgIIm Fe
m Mgb FeII Kb FeII( )
( )
( )( ) ( )
b MgII b FeII( ) ( )
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We can therefore reduce the number of cloud parameters describing TWO absorption lines from different species:
bKb
z
N(FeII)N(MgII)
i.e. 4 cloud parameters, with assumptions: no spatial or velocity segregation for different species
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How reasonable is the previous assumption?
FeII
MgII
Line of sight to Earth
Cloud rotation or outflow or inflow clearly results in a systematic bias for a given cloud. However, this is a random effect over and ensemble of clouds.
The reduction in the number of free parameters introduces no bias in the results
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The “alkali doublet method”
Resonance absorption lines such as CIV, SiIV, MgII are commonly
seen at high redshift in intervening gas clouds. Bethe & Salpeter 1977
showed that the of alkali-like doublets, i.e transitions of the
sort
are related to by
which leads to
:
:
2
1
221
2
)(
Note, measured relative to same ground state
2/12
2/12
2/32
2/12
PS
PS
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In addition to alkali-like doublets, many other more complex species are seen in quasar spectra. Note we now measure relative to different ground states
Ec
Ei
Represents differentFeII multiplets
The “Many-Multiplet method” (Webb et al. PRL, 82, 884, 1999; Dzuba et al. PRL, 82, 888, 1999) - use different multiplets simultaneously - order of magnitude improvement
Low mass nucleusElectron feels small potential and moves slowly: small relativistic correction
High mass nucleusElectron feels large potential and moves quickly: large relativistic correction
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Advantages of the Many Multiplet method
1. Includes the total relativistic shift of frequencies (e.g. for s-electron) i.e. it
includes relativistic shift in the ground state
(Spin-orbit method: splitting in excited state - relativistic correction is smaller, since excited electron is far from the nucleus)
2. Can include many lines in many multiplets
Ji
Jf
(Spin-orbit method: comparison of 2-3 lines of 1 multiplet due to selection rule for E1 transitions - cannot explore the full multiplet splitting)
1 fi JJ
3. Very large statistics - all ions and atoms, different frequencies, different
redshifts (epochs/distances)
4. Opposite signs of relativistic shifts helps to cancel some systematics.
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1. Zero Approximation – calculate transition frequencies using complete set of Hartree-Fock energies and wave functions;
2. Calculate all 2nd order corrections in the residual electron-electron interactions using many-body perturbation theory to calculate effective Hamiltonian for valence electrons including self-energy operator and screening; perturbation V = H-HHF.
This procedure reproduces the MgII energy levels to 0.2% accuracy (Dzuba, Flambaum, Webb, Phys. Rev. Lett., 82, 888, 1999)
Dependence of atomic transition frequencies on
Important points: (1) size of corrections are proportional to Z2, so effect is small in light atoms, greatest in heavy atoms;(2) greatest precision will be achieved when considering all relativistic effects (ie. including ground state)
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Highly exaggerated illustration of how transitions shift in different directions by different amounts – unique pattern
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Relativistic shift of the central line in the multiplet
Procedure1. Compare heavy (Z~30) and light (Z<10) atoms, OR
2. Compare s p and d p transitions in heavy atoms.
Shifts can be of opposite sign.
1qEE2
0
z0zz
Ez=0 is the laboratory frequency. 2nd term is non-zero only if has changed. q is derived from relativistic many-body calculations.
)S.L(KQq K is the spin-orbit splitting parameter. Q ~ 10K
Numerical examples:
Z=26 (s p) FeII 2383A: = 38458.987(2) + 1449x
Z=12 (s p) MgII 2796A: = 35669.298(2) + 120x
Z=24 (d p) CrII 2066A: = 48398.666(2) - 1267xwhere x = z02 - 1 MgII “anchor”
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Wavelength precision and q values
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Biggest shifts are around 300 m/s. Doppler searches for extra-solar planets reach ~3 m/s at similar spectral resolution (but far higher s/n).
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Low-z vs. High-z constraints:
/ = -5×10-5High-z Low-z
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Numerical procedure: Use minimum no. of free parameters to fit the data
Unconstrained optimisation (Gauss-Newton) non-linear least-squares method (modified version of VPFIT, explicitly included as a free parameter);
Uses 1st and 2nd derivates of with respect to each free parameter ( natural weighting for estimating ;
All parameter errors (including those for derived from diagonal terms of covariance matrix (assumes uncorrelated variables but Monte Carlo verifies this works well)
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Low redshift data: MgII and FeII (most susceptible to systematics)
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Low-z MgII/FeII systems:
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High-z damped Lyman- systems:
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Webb, Flambaum, Churchill, Drinkwater, Barrow PRL, 82, 884, 1999
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Webb, Murphy, Flambaum, Dzuba, Barrow, Churchill, Prochaska, Wolfe. PRL, 87, 091301-1, 2001
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Murphy, Webb, Flambaum, MNRAS, 345, 609, 2003
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Murphy, Webb, Flambaum, MNRAS, 345, 609, 2003
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High and low redshift samples are more or less independent
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/ = (-0.06 ± 0.06)×10-5
Chand, Srianand, Petitjean, Aracil (2004):
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A re-analysis, using the SAME data and SAME models:
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Results of re-fitting the SAME spectra:
Chand et al points
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Concluding remark
The bottom line:
: current best data suggests a deviation from present day value, but large statistical result in preparation
: most probably a null result, but obviously better data is vital