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arXiv:astro-ph/0602577v2 25 May 2006

CMB constraints on the simultaneous variation of the fine

structure constant and electron mass

Kazuhide Ichikawa, Toru Kanzaki, and Masahiro Kawasaki

Institute for Cosmic Ray Research,

University of Tokyo, Kashiwa 277-8582, Japan

(Dated: February 5, 2008)

Abstract

We study constraints on time variation of the fine structure constant α from cosmic microwave

background (CMB) taking into account simultaneous change in α and the electron mass mewhich

might be implied in unification theories. We obtain the constraints −0.097 < ∆α/α < 0.034 at

95% C.L. using WMAP data only, and −0.042 < ∆α/α < 0.026 combining with the constraint on

the Hubble parameter by the HST Hubble Key Project. These are improved by 15% compared

with constraints assuming only α varies. We discuss other relations between variations in α and

mebut we do not find evidence for varying α.

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I.INTRODUCTION

One of fundamental questions in physics is whether or not the physical constants are

literally constant. In fact, the physical constants may change in spacetime within the context

of some unification theories such as superstring theory and investigating their constancy

is an important probe of those theories. In addition to the theoretical possibility, some

observations suggest the time variation of the fine structure constant (the coupling constant

of electromagnetic interaction) α. Recent constraints including such non-null results are

briefly summarized as follows.

Terrestrial limits on α come from atomic clocks [1, 2, 3, 4, 5, 6, 7], the Oklo natural fission

reactor in Gabon [8] and meteorites. Ref. [7] derived the limit on the temporal derivative of α

at present as (−0.3±2.0)×10−15yr−1. Measurements of Sm isotopes in the Oklo provide two

bounds on the variation of α as ∆α/α = −(0.8±1.0)×10−8and ∆α/α = (0.88±0.07)×10−7

[8]. The former result is null, however, the latter is a strong detection that α was larger at

z ∼ 0.1. The meteorite bound obtained by measuring187Re decay rate is now controversial

although varying α is not suggested anyway. Ref. [9] obtained ∆α/α = (−8 ± 8) × 10−7,

whereas Refs. [10, 11] argued that the constraint should be much weaker due to uncertainties

in the decay rate modeling.

On the other hand, there are three kinds of celestial probes. One is to use big bang

nucleosynthesis (BBN) [12, 13, 14, 15] which provides constraints at very high redshifts

(z ∼ 109-1010), for example, −5.0×10−2< ∆α/α < 1.0×10−2(95% C.L.) [13] or |∆α/α| <

6 × 10−2[16]. The second is from the spectra of high-redshift quasars (z ∼ 1-3) [17, 18,

19, 20, 21, 22, 23], where there are conflicting results. Refs. [17, 18] suggested that α

was smaller at z ? 1, ∆α/α = (−0.543 ± 0.116) × 10−5[18]. This result, however, is

not supported by other observations [19, 20, 21, 22, 23]. For example, Ref. [19] obtained

the constraint as ∆α/α = (−0.6 ± 0.6) × 10−6and the others too found no evidence for

varying α. Finally, we can use Cosmic Microwave Background (CMB) to measure α at

z ∼ 1100 [24, 25, 26, 28]. Analyses using pre-WMAP data are found in Refs. [26, 27, 28].

Refs. [29, 30] derive a constraint using the WMAP first-year data, −0.05 < ∆α/α < 0.02

or −0.06 < ∆α/α < 0.01 (95% C.L.) respectively with or without marginalizing over the

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running of the spectral index [30].1

Recently, there are many studies on constraining the time variation in α which accompa-

nies the variation of the other coupling constants, as would occur rather naturally in unified

theories. Under such a framework, BBN has been studied in Refs. [13, 35, 36, 37, 38, 39,

40, 41, 42], quasar absorption systems in Ref. [43], meteorites in Ref. [9], the Oklo reactor

in Ref. [40], and atomic clock experiments in Refs. [43, 44]. For example, BBN constraint

improves by up to about 2 orders (−6.0×10−4< ∆α/α < 1.5×10−4[13]) although the fac-

tor may vary depending on how they are correlated to α. Hence, it is important to consider

the variation of other coupling constants along with α.

In this paper, we investigate constraint on the time variation of α from CMB using the

WMAP first-year data. In particular, we consider meto vary dependently to the variation

in α because, as mentioned above, such might be the case in some unified theory. Since such

theory is now under development, how their variations are related to each other can not be

predicted. Therefore, we work in a phenomenological way guided by a low energy effective

theory of a string theory and adopt to vary mein power law of α.2

In the next section, we briefly review the recombination process in the early universe and

make clear how it depends on α and me. In section III, we illustrate their effects on the

epoch of recombination and the shape of CMB power spectrum. In section IV, we describe

the relations between the variations in α and mewhich we adopt in our analysis. In section

V, we present our constraint and we conclude in section VI.

II.

α AND meDEPENDENCE OF THE RECOMBINATION PROCESS

Non-standard values of α and memodify the CMB angular power spectrum mainly by

changing the epoch of recombination. Thus, let us visit briefly the recombination process

in the universe and see where those constants appear. We follow the treatment of Ref. [47],

1Note that two results suggesting varying α, one of the Oklo results [8] and the quasar observation of

Ref. [18] have different signs of ∆α. These results, if correct, can not be explained by homogeneous and

monotonically time-varying α. They may indicate that α is not a monotonically varying function of time

or as investigated in Refs. [31, 32, 33], may suggest a spatial variation of α. In passing, we refer to Ref. [34]

for the effects of spatial varying α on CMB.

2Refs. [45, 46] studied CMB constraints on the variation of Higgs expectation value whose effect is assumed

to appear only in the change in meso there is some overlap between their analysis and ours. However,

they did not consider α variation at the same time.

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which is implemented in the RECFAST code. They have shown that the recombination

process is well approximated by the evolutions of three variables: the proton fraction xp,

the singly ionized helium fraction xHeII, and the matter temperature TM. Their equations

are given bellow. We denote the Boltzmann constant k, the Planck constant h and the

speed of light c. In addition to the variables above, xp= np/nHand xHeII= nHeII/nH, we

use the electron fraction xe= ne/nH= xp+ xHeIIas an auxiliary variable (note that nX

stands for the number density of species X but nHis defined as the total hydrogen number

density, including both protons and hydrogen atoms). z is used for the redshift and H for

the expansion rate.

Adopting the three level approximation, the time evolution of the proton fraction of xp

is described by

dxp

dz

=

CH

H(z)(1 + z)

?xexpnHRH− βH(1 − xp)e−hνH/kTM?,

(1)

where νH = c/(121.5682 nm) is the Lyα frequency and RH is the case B recombination

coefficient for H which is well fitted by

RH = 10−19F

atb

1 + ctdm3s−1

(2)

with t = TM/(104K), a = 4.309, b = −0.6166, c = 0.6703, d = 0.5300 [48], and the fudge

factor F = 1.14 introduced to reproduce the more precise multi-level calculation [47]. βHis

the photoionization coefficient

βH= RH

?2πmekTM

h2

?3

2

exp

?

−BH2s

kTM

?

, (3)

and CHis the so-called Peebles reduction factor

CH=

[1 + KHΛHnH(1 − xp)]

[1 + KH(ΛH+ βH)nH(1 − xp)], (4)

where the binding energy in the 2s energy level is BH2s= 3.4 eV, the two-photon decay rate

is ΛH= 8.22458 s−1and KH= c3/(8πν3

HH).

Here, comments on what Eq. (1) means may be in order. The first term in the square

brackets in Eq. (1) represents the recombinations to excited states of the atom, ignoring

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recombination direct to the ground state. The second term represents the rate of ionization

from excited states of the atom. The difference of those two terms is the net rate of produc-

tion of hydrogen atoms when one could ignore the Lyα resonance photons. These photons

reduce the rate by the factor CH, which can be written as the ratio of the net decay rate to

the sum of the decay and ionization rates from the n = 2 level,

CH=

ΛR+ ΛH

ΛR+ ΛH+ βH. (5)

We have rewritten Eq. (4) to derive this expression using the decay rate ΛR≡ (KHn1s)−1

allowed by redshifting of Lyα photons out of the line, and using n1s = nH− npwhich is

justified by the far greater occupation number of the hydrogen atom ground state than that

of the excited states altogether.

The evolution of the singly ionized helium fraction of xHeIIis similarly described by

dxHeII

dz

×?xHeIIxenHRHeI− βHeI(fHe− xHeII)e−hνHeI/kTM]?,

=

CHe

H(z)(1 + z)

(6)

where fHeis the total number fraction of helium to hydrogen (using primordial helium mass

fraction Yp, fHe= Yp/{4(1 − Yp)} where we take Ypto be 0.24), νHeI= c/(60.1404 nm) is

the frequency corresponding to energy between ground state and 21s state, and RHeIis the

case B HeI recombination coefficient for singlets [51]

RHeI =

q

??TM

T2

?

1 +TM

T2

?1−p?

1 +TM

T1

?1+p?−1

m3s−1,

(7)

with q = 10−16.744, p = 0.711, T1= 105.114K and T2fixed arbitrary 3K. βHeIis the photoion-

ization coefficient

βHeI= RHeI

?2πmekTM

h2

?3

2

exp

?

−BHeI2s

kTM

?

, (8)

CHeII=

[1 + KHeIΛHenH(fHe− xHeII)exp(∆E/kTM)]

[1 + KHeI(ΛHe+ βHeI)nH(fHe− xHeII)exp(∆E/kTM)],

(9)

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where the binding energy in the 2s energy level is BHeI2s= 3.97 eV and the two-photon decay

rate is ΛHe= 51.3 s−1, KHeI= c3/(8πν3

HeIH) and ∆E is the energy separation between 21s

and 21p, ∆E/h = c/(58.4334 nm)−νHeI. Note that, contrary to the case with H, the energy

separation ∆E between 21s and 21p is so large that we can not neglect it [50].

The matter temperature TMis evolved as

dTM

dz

=

8σTaRTR4

3H(z)(1 + z)me

xe

1 + fHe+ xe(TM− TR) +

2TM

(1 + z),

(10)

where TRis the radiation temperature, σT= 2α2h2/(3πm2

ec2) is the Thomson cross section,

and aR= k4/(120πc3h3) is the black-body constant. The Compton scattering makes TRand

TM identical at high redshifts. However, the adiabatic cooling becomes dominant at low

redshifts, which leads to the significant difference between TMand TR.

Now, we explain how quantities which appear in these equations depend on α and me.

Two-photon decay rates scale as α8me[52, 53]. Since binding energies scale as α2me, so do

νHand νHeI, and KHand KHeIscale as α−6m−3

e. The remaining task is to investigate how

the recombination coefficient R depends on α and me. To do this, we follow the treatment

of Ref. [25]. The recombination coefficient can be expressed as

R =

∗

?

×

n,l

[8π(2l + 1)]

?kTM

σnly2dy

exp(y) − 1,

2πme

?3

2

exp

?Bn

kTM

?

?∞

Bn/kTM

(11)

where Bnis the binding energy for n-th excited state and σnlis the ionization cross section

for (n,l) excited state. The asterisk in the upper bound of summation indicates that the

sum needs to be regulated, but since this regularization depends only weakly on α and me,

it can be neglected [54]. The cross section σnlscales as α−1m−2

e

[53]. Altogether,

∂R(TM)

∂α

∂R(TM)

∂me

=

2

α

?

R(TM) − TM∂R(TM)

?

∂TM

?

,(12)

= −1

me

2R(TM) + TM∂R(TM)

∂TM

?

.(13)

Combining with the fitting formulae (2) and (7), we obtain how RHand RHeIdepend on α

and me.

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FIG. 1: The ionization fraction xeas a function of redshift z for no change of α (solid curve), an

increase of α by 5% (dashed curve), a decrease of α by 5% (dotted curve).

FIG. 2: The ionization fraction xeas a function of redshift z for no change of me(solid curve), an

increase of meby 5% (dashed curve), a decrease of meby 5% (dotted curve).

III.EFFECTS ON THE EPOCH OF RECOMBINATION AND CMB ANGULAR

POWER SPECTRUM

We have investigated the equations which describe the process of recombination and

how they depend on the coupling constants in the previous section. We incorporate the

dependence on α and me into the RECFAST code [47] and solve the equations for the

ionization fraction xeas a function of redshift with several different values of α and me. The

results are shown in Figs. 1 and 2. We have assumed a flat universe and used cosmological

parameters (ωb,ωm,h) = (0.024,0.14,0.72), where ωb≡ Ωbh2is the baryon density, ωm≡

Ωmh2is the matter density, h is the Hubble parameter, and Ω denotes the energy density

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FIG. 3: The visibility function as a function of conformal time for no change of α (solid curve), an

increase of α by 5% (dashed curve), a decrease of α by 5% (dotted curve).

FIG. 4: The visibility function as a function of conformal time for no change of me(solid curve),

an increase of meby 5% (dashed curve), a decrease of meby 5% (dotted curve).

in unit of the critical density. The most important feature is the shift of the epoch of

recombination to higher z as α or meincreases. We can also see this by the rightward shift

of the peak of the visibility function shown in Figs. 3 and 4. This is easy to understand

because the binding energy Bnscales as α2meand photons should have higher energy to

ionize hydrogens.

In Figs. 5 and 6 we show the power spectrum of the CMB temperature anisotropy for

several different values of α and me as calculated by the CMBFAST code [55] with the

modified RECFAST. We consider a flat ΛCDM universe with a power-law adiabatic pri-

mordial fluctuation. The adopted cosmological parameter values are (ωb,ωm,h,ns,τ) =

(0.024,0.14,0.72,0.99,0.166) where τ is the reionization optical depth and nsis the scalar

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FIG. 5: The spectrum of CMB fluctuations for no change of α (solid curve), an increase of α by

5% (dashed curve), a decrease of α by 5% (dotted curve).

FIG. 6: The spectrum of CMB fluctuations for no change of me(solid curve), an increase of me

by 5% (dashed curve), a decrease of meby 5% (dotted curve).

spectral index. We fix the value of the amplitude of primordial power spectrum. We can see

two effects of varying α and mein Figs. 5 and 6. Increasing α or meshift the peak positions

to higher values of l and amplify the peak heights.

The peak position shift is understood as follows. Using lpto denote the position of a

peak, rθ(z) for the angular diameter distance and rs(z) for the sound horizon, one can write

lp∼rθ(zls)

rs(zls),(14)

where zls is the redshift of the last scattering surface. Increasing α or me increases the

redshift of the last scattering surface due to the larger binding energy, as in Figs. 3 and 4.

The higher zlsin turn corresponds to a smaller sound horizon and a larger angular diameter

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distance, which lead to a higher value of lp.

The changes in the peak heights are caused by modifications to the early ISW effect

and the diffusion damping. The larger zlsleads to the larger early ISW effect making the

first peak higher. To consider the effect beyond the first peak, we focus our attention on

the visibility function. The peak of the visibility function moves to a larger redshift since

the recombination occurs at higher redshift, when the expansion rate is faster. Hence, the

temperature and xedecreases more rapidly, making the peak width of the visibility function

narrower (see Figs. 3 and 4). Since the width of the visibility function corresponds to the

damping scale, an increase in α or medecreases the effect of damping. This is the reason

why the amplitude at larger l increases with increasing α and me. Moreover, as seen in

Figs. 3 and 4, α changes the visibility function width more than medoes (quantitatively, an

increase of α or meby 5% makes the full width at half maximum of the visibility function

narrower by 10% or 2% respectively) because the binding energy which scales as α2me. Thus

the damping scale is more sensitive to the change of α than me, as appears in Figs. 5 and 6.

Now we discuss the effects of varying α and me somewhat more quantitatively using

the following four quantities which characterize a shape of CMB power spectrum [56]: the

position of the first peak l1, the height of the first peak relative to the large angular-scale

amplitude evaluated at l = 10,

H1≡

?∆Tl1

∆T10

?2

(15)

the ratio of the second peak (l2) height to the first

H2≡

?∆Tl2

∆Tl1

?2

(16)

the ratio of the third peak (l3) height to the first

H3≡

?∆Tl3

∆Tl1

?2

(17)

where (∆Tl)2≡ l(l + 1)Cl/2π. Note that these four quantities do not depend on overall

amplitude. We calculate the response of these four quantities when we vary the parameters

ωb, ωm, h, τ, ns, α and me.When we vary one parameter, the other parameters are

fixed and flatness is always assumed (especially, increasing h means increasing ΩΛbecause

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ΩΛ= 1 − ωm/h2).

∆l1 = 16∆ωb

ωb

− 25∆ωm

+ 290∆α

ωm

− 47∆h

+ 150∆me

h

+36∆ns

ns

αme

,(18)

∆H1 = 3.0∆ωb

ωb

− 3.0∆ωm

+ 3.9∆α

ωm

− 2.2∆h

+ 1.4∆me

h

− 1.7∆τ

τ

+18∆ns

ns

αme

+ 0.41∆ns

,(19)

∆H2 = −0.30∆ωb

+0.91∆α

ωb

+ 0.015∆ωm

ωm

ns

α

+ 0.30∆me

me

, (20)

∆H3 = −0.19∆ωb

+0.57∆α

ωb

+ 0.21∆ωm

ωm

+ 0.56∆ns

ns

α

− 0.019∆me

me

,(21)

and values at the fiducial parameter values are l1 = 220, H1 = 6.65, H2 = 0.442 and

H3= 0.449. Derivatives of l1and H1with respect to α and meare positive as is expected

from the considerations above. Furthermore, ∆l1/∆α and ∆l1/∆meare much larger than

the other derivatives of l1while ∆H1/∆α and ∆H1/∆mehave relatively similar values to

the other derivatives of H1. Since such changes are most effectively mimicked by the change

in h, it is considered to be the most degenerate parameter with α and me. We have seen

above that when α or meincreases, the first peak is enhanced by larger ISW effect and the

second or higher peaks are enhanced by smaller diffusion damping. The derivatives of H2and

H3tell us which effect is important. Since ∆H2/∆α and ∆H3/∆α are positive and larger

than the derivatives with respect to me, we see that the effect on the diffusion damping

is more significant than that on the early ISW for varying α. They seem to somewhat

cancel each other for varying meespecially regarding H3. Such behavior is consistent with

the consideration at the end of the previous paragraph, that the diffusion damping is more

sensitive to the change in α than me.

IV.RELATION BETWEEN VARIATIONS OF α AND me

We expect a unified theory can predict the values of the coupling constants, how they are

related to each other and how much they vary in cosmological time scale. In string theory, a

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candidate for unified theory, there is a dilaton field whose expectation value determines the

values of coupling constants. However, since it is not fully formulated at present, we have

to assume how α and meare related to vary and constrain their variations. To be concrete,

following Ref. [35], let us start from considering the low energy action derived from heterotic

string theory in the Einstein frame. The action is written as

S =

?

1

2DµφDµφ − Ω−2V (φ)

− ¯ψγµDµψ − Ω−1mψ¯ψψ −

d4x√−g

?

1

2κ2R −1

2∂µΦ∂µΦ

−

α′

16κ2Ω2FµνFµν

?

,(22)

where Φ is the dilaton field, φ is an arbitrary scalar field, and ψ is an arbitrary fermion.

Dµis the gauge covariant derivative corresponding to gauge fields with field strength Fµν,

κ2= 8πG and Ω = e−κΦ/√2is the conformal factor which is used to move from string frame.

More concretely, φ is the Higgs field and V (φ) is its potential. The overall factor Ω before

the scalar potential means that the Higgs vacuum expectation value ?H? is independent of

the dilaton so it is taken to be constant. Fµνis the gauge field with gauge group including

SU(3)×SU(2)×U(1). We define its Lagrangian density for the gauge field as −(1/4g2)FµνFµν

where g is the unified coupling constant. Compared with equation Eq. (22),

1

g(Mp)2=α′e−√2κΦ

4κ2

,(23)

where Mpis the Planck scale. We can calculate the gauge coupling constants at low energy

using renormalization group equations. α almost does not run, and hence the α at low

energy

α ≃ α(Mp) =g(Mp)2

4π

=κ2e

√2κΦ

πα′

.(24)

As for the other gauge coupling constants, variation of the strong coupling constant may

affect CMB since its low energy value determines the QCD scale ΛQCDwhich in turn deter-

mines nucleon masses. However, how the variation of ΛQCDis related to that of α at low

energy can not uniquely be determined from eq. (22) and especially depends on the details

of unification scheme [57]. Therefore, for simplicity, we just assume ΛQCDdoes not vary.

The ψ’s are the ordinary standard model leptons and quarks. As we take ?H? = const, the

Yukawa couplings depend on the dilaton as eκΦ/√2. Therefore the relation between variations

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of α and meis given by

me+ ∆me

me

=

?α + ∆α

α

?1/2

.(25)

In this paper, we also consider other possibilities phenomenologically by adopting a power

law relation as

me+ ∆me

me

=

?α + ∆α

α

?p

,(26)

and compute constraints for several values of p. In addition to the case with changing only

α (p = 0) and the model described above (p = 1/2), we consider cases with p = 2 and 4.

V.CONSTRAINTS ON VARYING α AND me

We constrain the variation of α in the models described in the previous section using

the WMAP first-year data. CMB power spectrum is calculated by CMBFAST [55] with

RECFAST [47] modified as in Sec. II. The χ2is computed for TT and TE data set by the

likelihood code supplied by the WMAP team [58, 59, 60]. As defined in Sec. III, we consider

six cosmological parameters ωb, ωm, h, τ, nsand overall amplitude A in the ΛCDM model

assuming the flatness of the universe. We report A in terms of l(l+1)Cl/2π at l = 2 in unit

of µK2. In this paper we do not consider gravity waves, running of the spectral index and

isocurvature modes. We calculate χ2minimum as a function of α and derive constraints on

α. The minimization over six other parameters are performed by iterative applications of

the Brent method [61] of the successive parabolic interpolation. More detailed description

of this minimization method is found in Ref. [62]. We search for minimum in the region

τ < 0.3, which is a prior adopted in Refs. [29, 30]. We derive constraints with or without

the constraint on the hubble parameter h. When we combine it, we use the Hubble Space

Telescope (HST) Hubble Key Project value h = 0.72 ± 0.08 [63] whose error is regarded as

gaussian 1σ.

Fig. 7 shows our results of χ2minimization. It compares varying α only and varying α

and mewith the relation of eq. (25), respectively with or without the HST prior. Without

the HST prior, we find at 95% C.L. that −0.107 < ∆α/α < 0.043 with changing α only

and −0.097 < ∆α/α < 0.034 with the model described in the previous section. Although

the best fit α is 4% less than the present value, we find that ∆α = 0 is consistent with the

WMAP observation and evidence for varying α is not obtained. The effect of varying me

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FIG. 7: χ2function for variation in α only (solid curves) and α and mesimultaneously with the

relation of eq. (25) (dashed curves) . The HST prior is imposed for the thinner curves.

FIG. 8: χ2function for variation in α and mewith several values of p defined in Eq. (26). The

curves are for, from outside to inside, p = 0,1/2,2 and 4. The HST prior is imposed.

simultaneously is found to make the constraint more stringent by 13%. This rather small

effect is reasonable since, as is discussed in Sec. III, the effect of meon CMB power spectrum

is slightly smaller than α, and the relation of eq. (25) we adopt here does not change me

much relative to α.

We find that minimum χ2is given at ∆α/α = −0.04 with (ωb,ωm,h,ns,τ,A) =

(0.021,0.132,0.523,0.979,0.146,942) for the case of changing α only, and ∆α/α = −0.04

with (ωb,ωm,h,ns,τ,A) = (0.020,0.131,0.485,0.979,0.140,907) for the case of changing α

and metogether. Both cases have notably small values of h. Since h is considered to be the

most degenerate parameter with α or meas discussed in the end of Sec. III, it is instructive

to investigate how constraints tighten when h is limited to higher values such as the HST

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power law index constraint (95% C.L.)

p = 0−0.048 < ∆α/α < 0.032

−0.042 < ∆α/α < 0.026

−0.031 < ∆α/α < 0.017

−0.023 < ∆α/α < 0.011

p = 1/2

p = 2

p = 4

TABLE I: The constraints on α with mevaries according to various power law relation eq. (26) of

the index p. The HST prior is imposed.

measurement. From Fig. 7, we obtain, with the HST prior, that −0.048 < ∆α/α < 0.032

with changing α only, and −0.042 < ∆α/α < 0.026 with the model described in the previous

section. Compared with no HST prior constraints, they are stringent by about factor of 2

for both cases. Moreover, since low values of h which give good fit with ∆α/α ≈ −0.04 are

ruled out by the HST prior, the center of allowed region has shifted to larger ∆α.

Here, we comment on the constraint previously obtained by Refs. [29, 30] from the WMAP

data. As mentioned in Sec. I, they reported the constraint on α to be −0.06 < ∆α/α < 0.01

(95% C.L.). They fixed me when varying α and values quoted here is the case with no

running for the primordial power spectrum. This constraint seems to have been obtained

with marginalization on grid with 0 < ΩΛ< 0.95 [29] so it should be compared with our

constraint without the HST prior, −0.107 < ∆α/α < 0.043, which is much weaker than

theirs. The difference might be traced to the different analysis method but we could not

reproduce their results by our method.

Finally, we investigate the cases in which mevaries more than α. We consider the models

with p = 2 and 4 in eq. (26). We calculate constraints with the HST prior and results are

summarized in Fig. 8 and Table I along with p = 0 and 1/2 cases. Compared with p = 0

(only varying α) case, the constraints become smaller by 40% (p = 2) and 60% (p = 4).

Although those constraints are much smaller than the case with p = 1/2, they are still

consistent with ∆α = 0.

15