Searching for cosmological variation of fundamental

constants using high-resolution quasar

spectroscopy

John K. WebbUniversity of New South WalesSydney, AustraliaKVI, Friday 31 August 2007

• 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

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.

Dimensionless ratios – the things we can actually check

To Earth

CIVSiIVCIISiII

Lyem

Ly forest

Lyman limit Ly

NVem

SiIVem

CIVem

Lyem

Ly SiII

quasar

Quasars: physics laboratories in the early universe

e

p

m

m

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

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:

Reinhold et al PRL April 2006

First quasar, Q0347-383

Single velocity component

H2 lines in the Lyman alpha forest of Q0347

H2 lines in the Lyman alpha forest of Q0347H2 lines in the Lyman alpha forest of Q0347

H2 lines in the Lyman alpha forest of Q0347H2 lines in the Lyman alpha forest of Q0347

H2 lines in the Lyman alpha forest of Q0347H2 lines in the Lyman alpha forest of Q0347

H2 lines in the Lyman alpha forest of Q0347

H2HI

H2 lines in the Lyman alpha forest of Q0347

A. “Raw” error bars

B. Constant added to error bars forcing (2 = 1

C. Raw error bars, iteratively 3 clipped forcing (2 = 1

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

Second quasar, Q0528-250

Multiple velocity components

J=0 J=1

J=3J=2

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

How many free parameters should be fitted?

How many free parameters should be fitted?

“Right” answer probably somewhere in there, but where?

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)

2 H2 components, “raw” error bars

3 H2 components, “raw” error bars

Results for 0528-250

3H2:VPFIT internal method (no clipping) :

Reduced redshift method (no clipping):

3 clipping:

2H2:

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

c

e

2

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

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,

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

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

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( ) ( )

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

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

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

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

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.

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)

Highly exaggerated illustration of how transitions shift in different directions by different amounts – unique pattern

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”

Wavelength precision and q values

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).

Low-z vs. High-z constraints:

/ = -5×10-5High-z Low-z

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)

Low redshift data: MgII and FeII (most susceptible to systematics)

Low-z MgII/FeII systems:

High-z damped Lyman- systems:

Webb, Flambaum, Churchill, Drinkwater, Barrow PRL, 82, 884, 1999

Webb, Murphy, Flambaum, Dzuba, Barrow, Churchill, Prochaska, Wolfe. PRL, 87, 091301-1, 2001

Murphy, Webb, Flambaum, MNRAS, 345, 609, 2003

Murphy, Webb, Flambaum, MNRAS, 345, 609, 2003

High and low redshift samples are more or less independent

/ = (-0.06 ± 0.06)×10-5

Chand, Srianand, Petitjean, Aracil (2004):

A re-analysis, using the SAME data and SAME models:

Results of re-fitting the SAME spectra:

Chand et al points

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