Axi-Higgs cosmology: Cosmic Microwave Background and cosmological tensions

Hoang Nhan Luu1, ∗

1Department of Physics and Jockey Club Institute for Advanced Study,The Hong Kong University of Science and Technology, Hong Kong S.A.R., China

(Dated: November 3, 2021)

Non-canonical cosmology with an uplifted Higgs vacuum expectation value (Higgs-VEV) in theearly universe is believed to provide the solution for existing tensions within the ΛCDM regime. Werecently proposed a theoretical model called axi-Higgs to explore this framework. The axi-Higgsmodel features an ultralight axion with mass ma ∼ 10−29 eV, which couples to the Higgs fieldsuch that the Higgs-VEV is driven by the axion background evolution. In this paper, we performMarkov Chain Monte Carlo (MCMC) analyses with our modified Boltzmann solver to investigatethe parameter space of axi-Higgs beside other models including ΛCDM, ΛCDM+me, ΛCDM+ωa.Combining cosmological data from Cosmic Microwave Background (CMB), Baryon Acoustic Oscil-lations (BAO), weak-lensing (WL) cosmic shear survey, we found H0 = 69.3+1.2

−1.4 km/s/Mpc andS8 = 0.797 ± 0.012, which reduces the Hubble tension to approximately 2.5σ and slightly alleviatesthe S8 tension. The presence and behavior of this Higgs-VEV-driving axion may be tested by atomicclock measurements in laboratory and/or quasar spectral measurements in the near future.

I. INTRODUCTION

The standard model of cosmology under the nameΛ-cold-dark-matter (ΛCDM) has long become one ofthe cornerstones in modern cosmology. The ΛCDMmodel with six basic parameters successfully explainsvastly different cosmological/astrophysical observations,spanning from the primordial state of Big Bang Nucle-osynthesis (BBN) to the afterglow of Cosmic MicrowaveBackground (CMB) and the large-scale structure (LSS)of galaxies at the present time. Despite most of itsremarkable accomplishments, multiple tensions betweentheory and observations as well as between differentobservations have emerged over the years as the obser-vational sensitivity gradually improves. Among them,the Hubble tension between the late-time and early-timemeasurements, probably has caught most of attentionso far, most notably H0 = 73.2 ± 1.3 km/s/Mpc fromSH0ES 2020 [1] versus H0 = 67.36 ± 0.54 km/s/Mpcfrom Planck 2018 [2]. This 4-6σ discrepancy [3] arisesfrom whether H0 is directly measured from the localHubble flow or indirectly inferred from CMB acousticstandard rulers assuming ΛCDM is correct.

Many alternative models have attempted to resolvethe Hubble tension by introducing exotic dark mat-ter (DM), dark energy (DE), dark radition (DR), whicheither modify pre-recombination or post-recombinationphysics. The representatives of the first category areEarly Dark Energy [4, 5], extra relativistic degrees offreedom [6, 7]. Meanwhile, the second category includesmodels with DM-DE interactions [8], decaying DM [9],emergent DE with parametrized equation of state [10],etc. The list goes on with models of modified grav-ity [11], inflationary [12], modified recombination [13, 14]

∗ hnluu@connect.ust.hk

and many more, see [15, 16] and references therein foran updated overview of the landscape of Hubble tensionsolutions. Although most of these solutions could helpimprove H0, they are not sustainable for other cosmo-logical tensions, such as the S8 tension [17] between theCMB-inferred value of S8 [2] and the S8 measured bylow-redshift weak-lensing experiments [18]. This 2-3σdiscrepancy is not statistically significant in ΛCDMbut could be intensified in some of the aforementionedmodels.

Recently, the so-called axi-Higgs model [19] is pro-posed to alleviate the H0 and S8 tensions as well asthe tension of Li7 abundance in BBN and the isotropiccosmic birefringence. The model is a simple extension ofthe standard model of electroweak interactions (EW) toinclude an ultra-light axion field with ma ' 10−29 eV,in addition to the axion (m ' 10−22 eV) responsiblefor Fuzzy Dark Matter (FDM). The presence of theultra-light axion lifts the Higgs-VEV at the percentagelevel during the cosmological recombination time, andrelaxes it to today’s value at late times. In comparingthe model with data, we have proposed a new approachto study the observational constraints on the parame-ters of the axi-Higgs model, namely the leading-orderperturbative approach (LPA) [20]. This quantitativetransparent approach is semi-analytic, providing moreunderstanding of how small variations of cosmologicalparameters, specifically the Higgs-VEV, impacts datafitting.

Motivated by the lack of a formal analysis besides theLPA results, the standard Boltzmann analysis (SBA) isconducted for the axi-Higgs model in this paper. Thistraditional approach allows a complete implementationof the latest available data to determine the preciseconstraints of cosmological parameters. SBA providesa valuable check on the validity of LPA and the twoapproaches complement each other.

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The rest of the paper is structured as follows. We beginwith a review of the axion and Higgs-VEV cosmologicalevolution from Sec. II A to Sec. II C, then discuss CMBphenomenology of the axi-Higgs model in Sec. II D. Aftersetting up the data sets and methodology in Sec. III Aand Sec. III B, we present the main findings of the paperin Sec. III C where the axi-Higgs model is compared withother three models and Sec. III D where the axi-Higgsmodel is compared with different masses and cmb data.Sec. III E is devoted to an essential check of the previ-ous LPA results. Sec. IV gives conclusion and outlook onsome questions beyond the scope of this work. App. Acompares the axi-Higgs model with its phenomenologi-cal model ΛCDM+me+ωa. App. B justifies the effectivefluid approximation applied for axion equations. App. Cprovides some supplementary figures and tables for themain text.

II. AXI-HIGGS PHYSICS

Since our focus in this paper is on physics of CMB,we shall concentrate on the cosmic epoch around the re-combination era when the FDM axion has already rolleddown, contributing to the effective CDM density ωc. As aresult, the simplified axi-Higgs model is given by the ax-ion φ coupled to the electroweak Higgs doublet Φ, whichreads

V (φ) = m2φ2/2 +∣∣m2

sF (φ)− κΦ†Φ∣∣2 , (1)

F (φ) = (1 + δv)2 = 1 + Cφ2

M2Pl

, (2)

where the parameters ms and κ are fixed by the Higgsvacuum expectation value (Higgs-VEV) today v0 = 246GeV and the Higgs boson mass mΦ = 125 GeV; δv =∆v/v0 = (v − v0)/v0 is the fractional shift of v from itspresent-day value; MPl ' 2.4 × 1018 GeV is the reducedPlanck mass.

A. Axion Background Dynamics

The perfect square of the Higgs potential in Eq. (1)assures that its contribution to the vacuum energydensity stays at exactly zero, a necessary (but notsufficient) requirement for a naturally exponentiallysmall cosmological constant Λ. As shown in [19], thisperfect square form is crucial in the axi-Higgs model fortwo reasons: (i) Since the axion φ and the Higgs fieldΦ are coupled, their cosmological evolution is closelytied together. (ii) The impact of the Higgs field on theaxion’s evolution is negligible, so we can treat the axionevolution as if it is decoupled from the Higgs.

The standard axion regime is described by the La-grangian of a scalar field with a simple harmonic po-

tential spontaneously generated by non-perturbative dy-namics

L = −1

2(∂φ)2 −m2

af2a

[1− cos

(φ

fa

)](3)

' −1

2(∂φ)2 − 1

2m2aφ

2. (4)

In a spatially flat and homogeneous Friedmann-Lemaitre-Robertson-Walker (FLRW) universe, the axion back-ground equation of motion reads

φ+ 2Haφ+m2aa

2φ = 0, (5)

with its energy density and pressure given by

ρa =1

2a−2φ2 +

1

2m2aφ

2, (6)

pa =1

2a−2φ2 − 1

2m2aφ

2, (7)

where dots denote derivatives with respect to con-formal time d/dη. The conformal Hubble defined asH ≡ a/a = aH.

Initially, due to large Hubble friction, the axion slowlyrolls down the potential in (4) and contributes to thedark energy density until some transition scale factoraosc, which roughly satisfies

ma ' ξH(aosc). (8)

It then starts oscillating rapidly over Hubble time scaleswith a damping amplitude and contributes to the darkmatter density. In this context, we switch to the effectivetreatment as in [21, 22] to avoid resolving these oscilla-tions on time scales much smaller than Hubble time, i.e.

1

2a−2φ2 ' 1

2m2aφ

2, ρa 'ρa(aosc)a3

osc

a3, pa ' 0. (9)

The transition coefficient ξ is typically of order one,which is chosen to be ξ = 3 as the standard value inthis work. We discuss the implications of some alterna-tive choices in App. B.

B. Higgs-VEV on electron mass

The Higgs is coupled to the axion field so that its VEVis driven by axion background evolution [19, 20]. Ex-panding Eq. (2) up to the lowest order yields

δv =Cφ2

2M2pl

. (10)

When H(z) ma, the initial field φini stays unchanged,so it raises v > v0 during recombination. The value of φstarts dropping when H(z) . ξma, which drives v evolu-tion towards v0 as

δv ' δvosc(aosc/a)3, (11)

3

since δv ∝ φ2 ∝ ρa ∝ a−3 after the transition. Thetime dependence of the Higgs-VEV leads to a universalevolution of particle masses, electroweak coupling con-stants, QCD confinement scales, etc [23–28]. Early timesphysics, especially BBN and CMB, requires extensivemodifications to account for this unconventional Higgs-VEV. This work, however, is devoted to investigating itsimplications on CMB alone. We also omit the variationof the proton mass, which presumably makes our resultsunaffected 1. Eventually, varying the Higgs-VEV afterBBN era only induces an identical shift of the electronmass

δv = δme, (12)

which in turn alters various atomic constants involvedin recombination. Specifically, in an effective three-levelatomic model [29–31] those constants are: the hydrogenand helium energy levels Ei, the Thompson scatteringcross section σT , the two-photon decay rate A2γ , the ef-fective Lyman-α transition rate PSA1γ , the baryon tem-perature Teff, the effective recombination and photoion-ization rates α and β respectively; their correspondingdependencies on the electron mass read [32–34]

Ei ∝ me, σT ∝ m−2e , A2γ ∝ me, PSA1γ ∝ m3

e,

α ∝ m−2e , β ∝ me, Teff ∝ m−1

e .

We note that varying the fine-structure constant yieldssimilar effects [35], though not being motivated by theaxi-Higgs model.

When the axion mass ma is fixed, the axi-Higgs sec-tor can be characterized by two theoretical parameters:φini and C, namely the initial axion field value and the“coupling constant” of the Higgs with the axion. We canequivalently convert them to two phenomenological pa-rameters more relevant to cosmology, which are the axionrelic density ωa and the initial Higgs-VEV ratio (v/v0)ini

so that

ωa = F (φini), (v/v0)ini = 1 + Cφ2ini/(2M

2pl). (13)

Here F (φini) ∝ φ2ini if the axion abundance is negligi-

ble compared to the total cosmic budget at any time sothat the axion backreaction on the Hubble flow can beapproximately neglected in Eq. (5).

C. Axion Perturbations

The presence of an axion field sources scalar perturba-tion components of the metric, which is expressed in thesynchronous gauge as

ds2 = a2(η)[−dη2 + (δij + hij)dx

idxj]. (14)

1 δmp/δv ∼ O(10−3)

The axion perturbation equations can then be de-rived [21, 22, 36]

δa = −kua − (1 + wa)h/2

− 3H(1− wa)δa − 9H2(1− c2ad)ua/k, (15)

ua = 2Hua + kδa + 3H(wa − c2ad)ua, (16)

where h is the trace of hij . The density contrast and heatflux are defined, respectively, as

δa ≡ δρa/ρa, ua ≡ (1 + wa)va. (17)

Here, the equation of state and adiabatic sound speedare background-dependent quantities

wa ≡paρa, c2ad ≡

paρa

= 1 +2m2

aa2φ

3Hφ. (18)

After axion oscillations commence, Eqs. (15) and (16) areeffectively transformed to

δa = −kua − h/2− 3Hc2aδa − 9H2c2aua/k, (19)

ua = −Hua + kc2aδa + 3c2aHua, (20)

where the effective axion sound speed is

c2a =k2/(4m2

aa2)

1 + k2/(4m2aa

2). (21)

The above two effective perturbation equations are de-rived under the ansatz: (i) the background axion field andits perturbation can be expanded in terms of harmonicfunctions with a fixed frequency of order ma [37, 38]

φ = a−3/2 [A(η) cos(maη) +B(η) sin(maη)] , (22)

δφ = Ck(η) cos(maη) +Dk(η) sin(maη), (23)

satisfying the oscillation averaged condition ADk = BCk;(ii) the metric perturbations evolve on time scales muchlonger than the intrinsic axion Compton period, i.e.η m−1

a .

For simplicity, we focus only on adiabatic initial con-ditions throughout this work, i.e.

δa(ηini) = 0, ua(ηini) = 0 . (24)

The isocurvature mode of the primordial axion scalar per-turbations produced during inflation is assumed to benegligible.

D. Axi-Higgs cosmological implications

The presence of the Higgs-VEV-driving axion affectsvarious cosmological observables from early to late times.Firstly, the electron mass variation crucially alters thestandard recombination history. More massive electronsspeed up hydrogen and helium recombination, which

4

end up broadening the width of the photon visibilityfunction and shifting its peak to a higher redshift.Recombination will last longer and occur earlier as aconsequence. Secondly, the ultra-light axion influencesthe cosmic budget at both background and perturbationlevel, see Fig. 4. Due to DE-like nature, the axion lighterthan 10−28 eV has minor impacts on the backgroundexpansion before recombination era when the universewas dominated by radiation and matter. On the otherhand, axion perturbations start evolving even at veryearly times. The modes with horizon-sized wavelengthsalways grow while the modes whose wavelengths areshorter than the Jeans scale are suppressed initially.More axion leads to faster growth, which disturbs thegravitational potential in a non-trivial way. After theonset of oscillations, the axion background densitydilutes as CDM, therefore it mimics the CDM behavioron the late-time Hubble flow. The axion mode evolutionalso follows the same redshift scaling as CDM at thesetimes.

Fig. 1 shows the auto-correlation functions of temper-ature, polarization and lensing deflection with respect tonumerous values of (v/v0)ini and ωa. Input parameters,except for the varying parameter, are fixed to the fiducialvalues, which are chosen as follows

ωb = 0.02238, ωc = 0.1201, H0 = 67.32, (25)

109As = 2.1, ns = 0.9660, τreio = 0.0543, (26)

(v/v0)ini = 1.01, ωa = 0.001, ma = 10−29 eV, (27)

with the dimension [km/s/Mpc] of H0 will be implicitlyassumed from now on. The phenomenology of theaxi-Higgs model could leave distinctive imprints on theCMB spectra and matter power spectrum.

As the initial Higgs-VEV (v/v0)ini is uplifted, theacoustic peaks of CMB spectra shift to the high-l values.Meanwhile, the surge of ωa drags them back to thereverse direction, which implies that the original peakpositions may retain once the two parameters are turnedon simultaneously. However, the peak heights will bedragged up as well, which requires variations of otherparameters to keep the whole spectrum unchanged, e.g.increasing ωc pulls down the peaks again, see Fig. 5. Wealso notice that the spectrum variations with respect toωa are analogous to the variations with respect to H0,see Fig. 5. Apart from the peaks raise of the former andthe low-l plateau upswing of the latter, increasing ωaor H0 pushes the acoustic peaks toward larger angularscales. These two parameters are oppositely degenerate,hence anti-correlated if the other parameters are fixed.Compared with the TT case, the EE spectra are moreprone to the variations of (v/v0)ini and ωa since thesame order-of-magnitude deviations can be seen over thesmaller ranges of the corresponding parameters.

The lensing spectrum is stable with non-standard

(v/v0)ini values but extremely sensitive with any amountof ωa. In that case, the suppression of the lensing poweris caused by the smaller matter perturbation amplitude,which can be understood via the following scenarios: (i)The dark energy density ωΛ decreases when adding ωagiven a fixed value of H0. The universe with a loweramount of DE appears to have a shortened age, whichis not old enough for matter clustering to form large-scale structure. As a result, the matter power spectrumis overall suppressed on every scales, which is illustratedby the blue curves in Fig. 2. (ii) If we instead replace afraction of CDM by axion to keep DE fixed, the suppres-

sion of Cφφl is still similar to the last panel of Fig. 1 butthe matter power spectrum is only suppressed on scales

smaller than the Jeans scale kj ∝ ρ1/4a m

1/2a [39, 40], as

highlighted by the red curves in Fig. 2. Eventually, wewill observe the CMB photons propagating from the lastscattering surface less likely to be deflected by the in-termediate structure before reaching the Earth. There-fore, a substantial amount of axion relic density withma ∼ 10−29 eV is strictly prohibited by the current cos-mological observations, especially CMB lensing data.

III. STANDARD BOLTZMANN ANALYSIS

We modify the standard Boltzmann solver CAMB [41,42] to feature 2:

• One additional species of axion at background ex-pansion and linear perturbations with adiabaticinitial conditions. We treat the axion with effec-tive fluid approximation after its transition red-shift with detailed implementation shown in Sec. II,which matches the previous code axionCAMB [21].Note that we limit the axion mass to 10−26 eV <ma < 10−32 eV, so axion perturbations do notcontribute to the matter power spectrum but stillsource the metric perturbations, or equivalently thegravitational potential. We still count the axionbackground field as matter in terms of relic den-sity, i.e. ωm = ωb + ωc + ων + ωa.

• The free electron fraction xe ≡ ne/nH,0 calculatedfrom Recfast++ [43–47], which is modified to in-corporate the axion-driving time dependence of theelectron mass and the axion evolution inside theHubble function. In addition, we scale the Thomp-son scattering cross-section, the Compton coolingterm appearing in the baryon-photon coupled equa-tions and the reionization optical depth with re-spect to (v/v0) accordingly.

In our numerical implementation, we use the shootingmethod to obtain the φini given ωa as inputs, with a

2 Our modified version, namely aHCAMB, is publicly available athttps://github.com/lhnhan/aHCAMB.git

5

101 102 103

l

1000

2000

3000

4000

5000D

TT l[

K2 ]

0.7

0.8

0.9

1.0

1.1

1.2

1.3

(v/v

0)in

i

101 102 103

l

1000

2000

3000

4000

5000

6000

DTT l

[K

2 ]

0.000

0.025

0.050

0.075

0.100

0.125

0.150

0.175

0.200

a

103

l

10

20

30

40

DEE l

[K

2 ]

0.900

0.925

0.950

0.975

1.000

1.025

1.050

1.075

1.100

(v/v

0)in

i

103

l

5

10

15

20

25

30

35

40

DEE l

[K

2 ]

0.00

0.01

0.02

0.03

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0.05

0.06

0.07

a

101 102 103

l

1

2

3

4

5

6

7

8

107 l

(l+

1)D

l

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

(v/v

0)in

i

101 102 103

l

1

2

3

4

5

6

7

8

107 l

(l+

1)D

l

0.0000

0.0025

0.0050

0.0075

0.0100

0.0125

0.0150

0.0175

0.0200

a

FIG. 1. From top to bottom panels: the TT, EE and φφ power spectra in the axi-Higgs model generated by aHCAMB. There-scaled spectrum is related to the original one by Dl ≡ l(l + 1)Cl/2π. The color bars illustrate the range of (v/v0)ini (left)and ωa (right) over which the spectra variations are plotted in each corresponding panel.

tolerance of ∆ωa ' 10−5 for numerical efficiency, so theaHCAMB code will treat ωa < 10−5 as ωa = 0.

A. Data sets

We take into account the following data sets:

• CMBTT: Planck 2018 low-l (Commander) and high-l temperature power spectra (Plik), supplementedby low-l EE polarization spectra (SimAll) of pho-ton anisotropies emitted from Cosmic MicrowaveBackground

• CMBbase: same as CMBTT with an addition of

6

10 3 10 2 10 1 100

k [h Mpc 1]

102

103

104

P(k)

[h3

Mpc

3 ]

(v/v0)ini = 0.5(v/v0)ini = 1(v/v0)ini = 1.5

10 4 10 3 10 2 10 1 100

k [h Mpc 1]

100

101

102

103

104

P(k)

[h3

Mpc

3 ]

a = 0a = 0.1 c, fid

a = 0.2 c, fid

a = 0.2 , fid

a = 0.4 , fid

FIG. 2. The matter power spectrum observed at z = 0 in the axi-Higgs model. (Left) The curves are plotted with three differentvalues of (v/v0)ini and almost indistinguishable from each other. (Right) The curves are plotted with ωa = ωa,fid + ∆ωa. Thered ones and the blue ones have ωc = ωc,fid − ∆ωa and ωΛ = ωΛ,fid − ∆ωa, respectively. Other parameters are fixed to theirfiducial values.

Planck 2018 high-l EE polarization spectrum andhigh-l cross-correlation TE spectrum

• CMBfull: same as CMBbase with Planck 2018 con-servative lensing spectrum added

• BAO: baryon acoustic oscillation measurements oflarge-scale structure from 6DF galaxy survey [48]at z = 0.106, main galaxy sample from SDSSDR7 survey [49] at z = 0.15. We also con-sider the consensus data from BOSS DR12 SDSSsurvey [50] at z = 0.38, 0.51, 0.61, including theredshift-space distortion (RSD) constraints on thestructure growth fσ8.

• WL: cosmic shear weak lensing data from a jointanalyis of Kilo-Degree Survey (KV450) and DarkEnergy Survey (DES-Y1) [51]. We represent thisdata set, for simplicity, by a split-normal priors on

S8 = 0.755+0.019−0.021, (28)

where S8 ≡ σ8(Ωm/0.3)0.5, which has beenproven by [52] to give an equivalent result to thecomputationally-expensive analysis of galaxy two-point correlations.

• SN: the Pantheon [53] data set of 1048 supernovaeType Ia in the redshift range of 0.01 < z < 2.3,which is given in terms of luminosity distances.

• R19: a Gaussian prior of the local Hubble parame-ter [54] from the recent SH0ES measurement usingdistance ladder with Cepheids as calibrators

H0 = 74.03± 1.42 km/s/Mpc. (29)

B. Methodology

We run several Monte-Carlo Markov chains (MCMC)using CosmoMC [55, 56] to obtain constrains in the follow-ing models:

• ΛCDM: the standard cosmological model with sixvarying parameters

ωb, ωc, θMC, τreio, ln(1010As); (30)

• ΛCDM+me: an extension of ΛCDM with the elec-tron mass me varying constantly throughout thecosmic history

ωb, ωc, θMC, τreio, ln(1010As),me/me,0; (31)

• ΛCDM+ωa: an extension of ΛCDM with an ultra-light axion. We implicitly choose ma = 10−29 eVas the reference value, otherwise specifically stated.

Ωbh2,Ωch2, θMC, τreio, ln(1010As), ωa

with ma = 10−29 eV; (32)

• Axi-Higgs: an extension of ΛCDM with an ultra-light axion driving the Higgs-VEV (hence me)

ωb, ωc, θMC, τreio, ln(1010As), ns, (v/v0)ini, ωawith ma = 10−29 eV. (33)

Notice that axi-Higgs with the axion massma . 10−29 eV is approximately equivalent toΛCDM+me+ωa model as discussed in our previ-ous works [19]. The only distinction with the cur-rent study is that we precisely set the electron mass

7

nearly constant initially, i.e. (v/v0)ini = me/me,0,then traces the axion background evolution after-ward. We examine the difference between these twoframeworks in App. A.

Uniform priors are assumed on the varying cosmologi-cal parameters in all models, particularly for two newparameters as follows

0.9 < (v/v0)ini < 1.1, 0 < ωa < 0.1. (34)

The chains converge under Gelman-Rubin criterionwhere R − 1 < 0.03 with the first 30% of total stepsare discarded as burn-in. Marginalized posterior distri-butions and their plots are generated by GetDist pack-age [57]. The CMB lensing spectrum and the matterpower spectrum are computed with non-linear correc-tions although the default halofit [58] module used byCAMB is tuned for the halo model in pure CDM cosmol-ogy 3. Tab. VIII in App. C tests the reliability of aHCAMBin reproducing the previous constraints of parameters inthe ΛCDM+me and ΛCDM+ωa model.

C. Axi-Higgs with other models

We start with the comparison of the four mod-els introduced in Sec. III B with the base data setCMBbase+BAO+WL. The marginalized constraints ofcosmological parameters are shown in Tab. I and Fig. 6with several highlights:

• ΛCDM prefers a lower value of ωc and a higherH0 than the ones predicted by the analysis withonly CMBbase+BAO, see Tab. VII. This shift isobviously due to the strong preference for a smallerS8 value of WL data. Since S8 ∝ σ8

√ωm/h2 by

definition, reducing ωc (hence reducing ωm as wellas σ8) and increasing H0 suppresses S8.

• ΛCDM+ωa gives similar constraints to the analysisfrom [21, 22, 36] with older Planck data releasesand Tab. VII. Axion density is constrained with astringent bound even at 2σ level, which shows thatthere is no preference for axion abundance from thecurrent data resolution. Compared to Tab. VII, themean value of H0 slightly moves to the lower side,which is the result of the geometry degeneracy withωa as discussed in Sec. II D.

• ΛCDM+me has been believed to relieve H0 ten-sion when being fitted with the combination ofCMB and BAO data, see [14] or Tab. VII. Theinclusion of WL data slightly hinders this possi-bility although H0 is still mildly higher than the

3 Ref. [36] proved that the halofit effect is small for axions ofmass up to 10−26 eV. However, non-linear corrections of CDM-like axions require a dedicated code, e.g. WARMANDFUZZY [59]

one in ΛCDM. Moreover, the electron mass fractionme/me,0 in this case no longer exhibits convincingdeviation from unity.

• Axi-Higgs notably yields an enhanced value of H0

and a suppressed value of S8, which makes it a po-tential candidate in resolving both tensions existingin ΛCDM. A significant increase of (v/v0)ini is pre-ferred at 1σ level, apparently implying a 1% upliftof the electron mass at recombination as found pre-viously in [19]. Besides, a more relaxing bound ofωa compared with ΛCDM+ωa shows that a finiteamount of axion is viable, even though not likelyat 1σ level.

Notice that the angular sound horizon at recombi-nation θ∗ is in agreement across all four models, withuncertainties of 0.3%, further proves that it is the mostaccurately measured CMB observables [2].

Data CMBbase+BAO+WLModel ΛCDM ΛCDM+ωa ΛCDM+me axi-Higgs

ωb 0.02248 ± 0.00013 0.02250 ± 0.00013 0.02253 ± 0.00016 0.02269+0.00018−0.00022

ωc 0.11806 ± 0.00088 0.11779 ± 0.00091 0.1186 ± 0.0018 0.1206+0.0023−0.0027

100θMC 1.04108 ± 0.00028 1.04111 ± 0.00030 1.0427 ± 0.0046 1.0501+0.0061−0.0083

τreio 0.0510 ± 0.0075 0.0529 ± 0.0076 0.0509 ± 0.0076 0.0526 ± 0.0077ln(1010As) 3.031 ± 0.015 3.035 ± 0.015 3.032 ± 0.015 3.038 ± 0.016

ns 0.9688 ± 0.0036 0.9692 ± 0.0037 0.9682 ± 0.0042 0.9673 ± 0.0041

(v/v0)ini 1 1 1.0023 ± 0.0066 1.0131+0.0088−0.012

ωa 0 < 0.00133 0 < 0.00328

H0 68.19 ± 0.40 67.80+0.53−0.47 68.6 ± 1.1 69.3+1.2

−1.4

S8 0.805 ± 0.010 0.800 ± 0.011 0.805 ± 0.010 0.797 ± 0.012σ8 0.8001 ± 0.0062 0.790+0.011

−0.0078 0.803 ± 0.011 0.793 ± 0.013φini 0 1.68+0.67

−0.79 0 2.6 ± 1.1100θ∗ 1.04129 ± 0.00028 1.04132 ± 0.00029 1.04126 ± 0.00031 1.04120 ± 0.00030

TABLE I. Marginalized distributions of cosmological pa-rameters in four models fitted with the base data setCMBbase+BAO+WL. The uncertainties and the upperbounds of parameters are shown at 68% and 95% confidencelevels, respectively. The parameter (v/v0)ini = me/me,0 inthe ΛCDM+me model. The initial misaligned axion field φini

is quoted in unit of 1017 GeV.

We proceed by presenting the constraints of thesame set of models extended with the late-time localmeasurements of SN and H0 in Tab. II and Fig. 7. Asthe main results of this work, we compute the detailedstatistics of each individual model and plot their best-fitspectra in Fig. 3 for comparison.

Generally, ΛCDM shows no significant differencewith the extra data sets. ΛCDM+ωa has ωa severelydisfavored at less than 0.5% of total matter densitywhile H0 moderately improves due to the H0 prior fromR19, which proves again the anti-correlation betweenthe Hubble constant and the DE-like axion density. Ourspeculation is that having multiple axion species mayexaggerate Hubble tension unless their densities aregreatly suppressed in this model.

On the other hand, ΛCDM+me and axi-Higgs see anotable rise of background parameters ωb, ωc, H0 and

8

(v/v0)ini. Meanwhile, S8 and σ8 are brought up as a re-sult of having more matter in the cosmic budget. There-fore, the Hubble tension is greatly alleviated in bothmodels but the S8 tension is almost untouched, but notworsen. Compared to the above axi-Higgs constraintswith the base data set, the upper bound of axion den-sity is tighten but it notably peaks at a positive valueof ωa ∼ 0.001. Thus, the existence of the Higgs-VEV-driving axion is present with the whole set of observa-tional data. We anticipate that future WL data withlow S8 and better systematic errors will place improvedconstraints on ωa.

D. Axi-Higgs with different data sets

Firstly, Tab. III and Fig. 8 compare parametersconstraints in three different CMB data sets in order tounderstand the role of CMB spectra on the axi-Higgsmodel. Starting from CMB temperature-only data,we observe an expected trend where the parameteruncertainties shrink as the extra information of CMBpolarization and lensing is incorporated. Interestingly,the hint of small axion density found in CMBTT andCMBbase is flatten in CMBfull. This may be due tothe preference of CMB lensing data over high lensingdeflection amplitudes [61]. The fact that the lensingspectrum significantly declines with just a tiny amount ofaxion put a tight constraint on ωa, which makes S8 lesssuppressed, as explained in Sec. II D. Yet, the evidenceof a finite axion amount is recovered if the late-timeSN+R19 data is taken into account, see Sec. III C.

We further observe that positive Higgs-VEV deviation(v/v0)ini becomes most pronounced with CMBTT but issignificantly dragged down with CMBfull. The Hubbleconstant also follows a similar trend. These propertiesreveal important relations between cosmological param-eters, stated as follows: an increase of H0 and a declineof S8 is induced by a positive ωa and a rise of (v/v0)ini.The axi-Higgs model with these correlations is capableof resolving H0 and S8 tensions, as discussed in [20].

Secondly, we explore parameter constraints with twomore axion masses other than the reference value 10−29

eV in Tab. IV and Fig. 9. The axion mass in the axi-Higgs model must lie within 10−30 ≤ ma ≤ 10−29 [19]to maintain a significant uplift of Higgs-VEV beforerecombination, so that the sound horizon is suppressedto resolve Hubble tension. Considering δvini of orderO(1%), an axion heavier than the upper bound wouldyield negligible deviation of me by recombinationbecause its amplitude started damping at earlier times,i.e. aosc < arec. On the contrary, the axion lighterthan the lower bound would have been detected inquasar observations [62] or atomic clock experiments [63]because its amplitude was not sufficiently attenuated bythe present day.

500 1000 1500 2000 2500

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axi-Higgs

2 10 30300

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0255075100

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100.10

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1)2

C l/(2

)

FIG. 3. Residuals of TT, TE, EE, φφ spectra with respectto CMB data in ΛCDM, ΛCDM+me, ΛCDM+ωa, axi-Higgs.We define ∆CX

l ≡ CXl,Y −CX

l,fid where X’s are spectral indicesand Y ’s are model indices. Cl,Y ’s are computed with the best-fit parameters from Tab. II while the reference spectra Cl,fid’sare obtained with the best-fit of ΛCDM with CMBfull onlyas provided by Planck 2018 [2]. The dashed vertical lines atl = 30 separate the low-l and high-l region. The left x-axis isgiven in log scale with the y-axis on the left for l ≤ 30; theright x-axis is given in linear scale with the y-axis on the rightfor l ≥ 30. One exception is the last panel with one commony-axis and the x-axis scales switching at l = 10. The low-lTE data points are shown for illustration but not included inour data fitting.

As expected, we find that the 10−28-eV axion con-straints are almost identical the ones from ΛCDM+ωain Sec. III C except for slightly more suppression of σ8

due to mass difference. The reason is because this axiontransits at redshift zosc ' 1090 barely before recombina-tion (zosc ' 1100), which implies roughly one order ofmagnitude reduction of the initial deviation (v/v0)ini atrecombination, i.e. δv/δvini ∼ 0.1. This result strength-ens our statement that the initial uplift of Higgs-VEV

9

Data CMBfull+BAO+WL+SN+R19Model ΛCDM ΛCDM+ωa ΛCDM+me axi-Higgs

ωb 0.02259 (0.02261) ± 0.00013 0.02260 (0.02261) ± 0.00013 0.02276 (0.02275) ± 0.00015 0.02292 (0.02294) ± 0.00018ωc 0.11737 (0.11730) ± 0.00082 0.11723 (0.11735) ± 0.00080 0.1211 (0.1211) ± 0.0016 0.1235 (0.1235) ± 0.0023

100θMC 1.04120 (1.04120) ± 0.00029 1.04123 (1.04122) ± 0.00028 1.0511 (1.0511) ± 0.0037 1.0585 (1.0585)+0.0055−0.0064

τreio 0.0569 (0.0570)+0.0066−0.0073 0.0579 (0.0579)+0.0065

−0.0076 0.0509 (0.0510) ± 0.0073 0.0550 (0.0552) ± 0.0078ln(1010As) 3.044 (3.044) ± 0.014 3.046 (3.046) ± 0.014 3.036 (3.036) ± 0.014 3.049 (3.049) ± 0.016

ns 0.9708 (0.9720) ± 0.0036 0.9712 (0.9719) ± 0.0036 0.9660 (0.9666) ± 0.0041 0.9649 (0.9649) ± 0.0042

(v/v0)ini 1 1 1.0143 (1.0144) ± 0.0053 1.0252 (1.0251)+0.0080−0.0094

100ωa 0 < 0.0623 (∼ 0) 0 0.137 (0.142)+0.047−0.12

H0 68.57 (68.61) ± 0.37 68.42 (68.59) ± 0.40 70.75 (70.73) ± 0.90 71.33 (71.24) ± 0.97S8 0.8019 (0.8013) ± 0.0087 0.7998 (0.8026) ± 0.0087 0.8049 (0.8050) ± 0.0087 0.8021 (0.8017) ± 0.0096σ8 0.8031 (0.8032) ± 0.0055 0.7990 (0.8041)+0.0066

−0.0058 0.8205 (0.8205) ± 0.0083 0.8133 (0.8119) ± 0.0098φini 0 1.10 (∼ 0)+0.36

−0.63 0 2.70 (2.92)+1.1−0.84

100θ∗ 1.04140 (1.04141) ± 0.00029 1.04143 (1.04141) ± 0.00028 1.04115 (1.04106) ± 0.00030 1.04108 (1.04106) ± 0.00031

χ2tot 3843.8 3844.0 3836.6 3837.4

∆χ2tot 0 0.2 −7.2 −6.4

χ2TTTEEE high-l 2350.4 2350.2 2346.7 2350.2χ2

TT low-l 22.2 22.2 23.2 23.7χ2

EE low-l 396.3 396.6 395.7 396.1χ2

lensing 9.8 9.6 9.42 10.1χ2

6dF ∼ 0 ∼ 0 0.2 0.1χ2

MGS 2.2 2.2 3.2 2.8χ2

BAO DR12 6.1 6.1 9.2 7.7χ2

WL 5.9 6.3 6.9 6.1χ2

SN 1034.7 1034.7 1035.0 1034.8χ2

R19 14.6 14.7 5.4 3.9

TABLE II. Same as Tab. I with the full data set CMBfull+BAO+WL+SN+R19. The best-fit values of cosmological parametersare quoted in the parentheses next to the mean values, which is found by BOBYQA minimization routine [60] embedded in CosmoMC.The χ2 statistics are calculated for the best-fit parameters of each model, where ∆χ2

tot indicates χ2tot difference between the

corresponding model and ΛCDM.

Model axi-HiggsData CMBTT+BAO+WL CMBbase+BAO+WL CMBfull+BAO+WL

ωb 0.02252+0.00026−0.00029 0.02269+0.00018

−0.00022 0.02267 ± 0.00018ωc 0.1216+0.0036

−0.0045 0.1206+0.0023−0.0027 0.1206+0.0020

−0.0023

100θMC 1.0553+0.0097−0.013 1.0501+0.0061

−0.0083 1.0488+0.0056−0.0066

τreio 0.0510 ± 0.0079 0.0526 ± 0.0077 0.0548 ± 0.0077ln(1010As) 3.035 ± 0.017 3.038 ± 0.016 3.045 ± 0.015

ns 0.9660 ± 0.054 0.9673 ± 0.0041 0.9667 ± 0.0039

(v/v0)ini 1.021+0.014−0.019 1.0131+0.0088

−0.012 1.0112+0.0080−0.0096

ωa 0.00233+0.00058−0.0023 < 0.00328 < 0.00220

H0 69.6+1.5−1.8 69.3+1.2

−1.4 69.2 ± 1.2S8 0.787 ± 0.014 0.797 ± 0.012 0.8064 ± 0.0094σ8 0.782 ± 0.015 0.793 ± 0.013 0.803 ± 0.010φini 3.5+1.4

−1.2 2.6 ± 1.1 2.13 ± 0.90100θ∗ 1.04108 ± 0.00046 1.04120 ± 0.00030 1.04118 ± 0.00030

TABLE III. Marginalized distributions of cosmological pa-rameters in axi-Higgs fitted with combinations of differentCMB data set and BAO+WL.

needs to remain constant in a narrow window of red-shift around recombination to distinguish axi-Higgs andthe standard axion cosmology. Axi-Higgs essentially con-verges to ΛCDM+ωa model when the axion mass is heav-ier. This limit may not apply to ΛCDM+me+ωa wherethe electron mass is not coupled to axion dynamics, seeApp. A for more details. On the other hand, the 10−30-eV axion constraints resemble their 10−29-eV counter-parts. Howeber, the peak of ωa distribution is shiftedtowards zero and its upper bound is slighted relaxed.

We anticipate this trend continuing for even lighter ax-ions because they tend to become more interchangeablewith DE. Axi-Higgs and ΛCDM+me are basically indis-tinguishable for the lightest axion of ma . 10−33 eV,which suggests that axi-Higgs only manifests itself whenma falls within the range of 10−30 − 10−29 eV regard-less of other constraints from observations. Our choiceof ma = 10−29 eV as the reference mass is justified torepresent this whole mass range.

Model axi-HiggsData CMBbase+BAO+WLma 10−28 eV 10−29 eV 10−30 eV

ωb 0.02245 ± 0.00018 0.02269+0.00018−0.00022 0.02271+0.00019

−0.00023

ωc 0.11785 ± 0.00090 0.1206+0.0023−0.0027 0.1205+0.0021

−0.0025

100θMC 1.04103 ± 0.00092 1.0501+0.0061−0.0083 1.0503+0.0060

−0.0081

τreio 0.0527 ± 0.0077 0.0526 ± 0.0077 0.0521 ± 0.0077ln(1010As) 3.034 ± 0.016 3.038 ± 0.016 3.037 ± 0.016

ns 0.9673 ± 0.0056 0.9673 ± 0.0041 0.9670 ± 0.0043

(v/v0)ini 0.9995 ± 00040 1.0131+0.0088−0.012 1.0131+0.0086

−0.012

ωa < 0.00164 < 0.00328 < 0.00362

H0 67.69 ± 0.54 69.3+1.2−1.4 69.4 ± 1.3

S8 0.797 ± 0.011 0.797 ± 0.012 0.799 ± 0.011σ8 0.785+0.013

−0.0099 0.793 ± 0.013 0.796 ± 0.012φini 1.69 ± 0.65 2.6 ± 1.1 2.9 ± 1.2

100θ∗ 1.04129 ± 0.00030 1.04120 ± 0.00030 1.04120 ± 0.00030

TABLE IV. Marginalized distributions of cosmological pa-rameters in axi-Higgs fitted with the base data setCMBbase+BAO+WL for different axion masses.

10

E. Comparison with Linear Perturbative Approach

Linear Perturbative Approach (LPA) was first intro-duced in Ref. [19] to roughly estimate the change ofcosmological parameters providing there is an upshiftof Higgs-VEV at early times. It was formulated sys-tematically as a semi-analytical method to derive theapproximate constraints of cosmological parameters inthe axi-Higgs model later on [20].

Let us shortly recap the method: we start withthe reference point where ωb, ωc and H0 are alreadydetermined in the ΛCDM model with Planck 2018 dataand allow them to vary together with the additionalparameters ωa and (v/v0)ini. By calculating the deriva-tives of the observables representing observational data,we can infer the directions along which these variablesmove in the multi-dimensional parameter space as weextend the ΛCDM to axi-Higgs model. This is carriedout where the other parameters such as As, ns, τreio

etc. remain unchanged for simplicity. In Ref. [20],we have found that the marginalized distributions ofωb, ωc, H0 could be perfectly reproduced in the ΛCDMand ΛCDM+me models. Thus, LPA has been appliedto constrain axi-Higgs in a similar manner given the factthat Boltzmann code for axi-Higgs was not available atthe time.

In this section, we check the previous results ob-tained by LPA by using the standard Boltzmann anal-ysis outlined in the previous sections. To provide anobjective test, we consider two data combinations withtwo different axion masses: (i) ma = 10−29 eV withCMBfull+BAOnoRSD+WL where the BAO data is de-fined as in Sec. III A but without RSD measurements;(ii) ma = 2× 10−30 eV with the equivalent data set usedin [20], i.e. CMBfull+BAOnew+WLnew in which

• CMBfull: as defined in Sec. III A for SBA. ForLPA, it is compressed and represented by four phe-nomenological observables: la, leq, lD, SL, see [20].

• BAOnew: including the same 6dF and MGS dataat low redshifts but replacing BOSS DR12 by therecent eBOSS DR16 survey which probes Lumi-nous Red Galaxies (LRG) sample at z = 0.698 [64],Emission Line Galaxy (ELG) sample at z =0.845 [65], quasar sample at z = 1.48 [66], Lyman-αforests at z = 2.334 [67].

• WLnew: given in terms of S8 priors from DES-Y1 3x2 pt [68] and the joint analysis of KiDs-1000+BOSS+2dFLenS [18], which reads

S8,DES = 0.773+0.0026−0.0020,

S8,KiDs = 0.766+0.0020−0.0014.

Marginalized constraints of parameters are comparedbetween the two approaches SBA and LPA in Tab. V

and Fig. 10.

At the first glance, the correlations of cosmologicalparameters obtained by SBA are in good agreementto the ones in LPA, which proves that the derivativesX|Y computed in [20] have correct signs. However, weobserve noticeable disparity in ωa and H0 constraints,especially in scenario (ii) where LPA yields a tighterbound of ωa and overestimates the mean value of H0.This inconsistency is presumably originated from: (1)the backreaction of the axion evolution on the Hubbleexpansion was neglected in LPA; (2) LPA only samplesωb, ωc, H0, (v/v0)ini, ωa while leaving As, ns, τreio fixedto their reference values, which leads to less ωa isallowed; (3) the non-linearity of the derivatives X|Ywhen being away from the reference point, for instances∆σ8|a ∼ O(1) is found along the ωa direction; (4) theCMB spectral information may not entirely encodedin their compressed observables. Remember that thevalidity of LPA is the basic assumption that any effectsfrom these sources of errors are next to leading ordercorrections and can be ignored at this level of accuracy.Thus, the approach is only reliable when the variationsof the parameters in the axi-Higgs model against thosein ΛCDM are of the order of a few percent.

Nonetheless, we further investigate the (4) possibilityby replacing SL by S8 or σ8 and make a comparisonin Fig. 11. Interestingly, the resulting posterior distri-butions changes dramatically even though these quan-tities are just slightly different in terms of the power ofΩm. This test suggests that the CMB observable SL mayget non-trivial corrections in the axi-Higgs model, whichare not yet understood in the current LPA implemen-tation. Provided the reasonable agreement between twoapproaches, LPA is still helpful to sketch a qualitativepicture while SBA is necessary to provide quantitativeresults.

Model axi-HiggsData CMBfull+BAOnoRSD+WL CMBfull+BAOnew+WLnew

ma 10−29 eV 2 × 10−30 eVMethod SBA LPA SBA LPA

ωb 0.02271 ± 0.00019 0.02269 ± 0.00017 0.02269 ± 0.00022 0.02273 ± 0.00021ωc 0.1208+0.0021

−0.0023 0.1204 ± 0.0018 0.1204 ± 0.0022 0.1206 ± 0.0019(v/v0)ini 1.0127+0.0082

−0.0099 1.0108 ± 0.0072 1.0108+0.0093−0.011 1.0127 ± 0.0089

ωa < 0.00209 < 0.00128 < 0.00210 < 0.00128τreio 0.0551 ± 0.0074 0.0543 0.0550 ± 0.0076 0.0543

ln(1010As) 3.046+0.014−0.015 3.045 3.044+0.013

−0.015 3.045ns 0.9669 ± 0.0042 0.9660 0.9664 ± 0.0040 0.9660H0 69.6 ± 1.3 69.6 ± 1.2 69.4 ± 1.7 70.0 ± 1.6S8 0.8052 ± 0.0095 0.8065 ± 0.0091 0.8069 ± 0.0097 0.8042 ± 0.0098

100θ∗ 1.04118 ± 0.00030 1.04116 ± 0.00030 1.04118 ± 0.00031 1.04114 ± 0.00030

TABLE V. SBA and LPA marginalized posteriors of cos-mological parameters in the axi-Higgs model fitted withCMBfull+BAOnoRSD+WL and CMBfull+BAOnew+WLnew.Note that CMBfull for LPA is represented by the compressedobservables while it denotes the complete spectra in SBA.

11

IV. CONCLUSION AND DISCUSSION

We have comprehensively studied the cosmologi-cal implications of the axi-Higgs model, where theHiggs-VEV is driven by the ultralight axion with massma ∼ 10−29 eV. Under presently available data fromcosmic microwave background, large-scale structure andlocal astrophysical measurements, we place constraintson the cosmological parameters of axi-Higgs and makesystematic comparisons with other models, includingΛCDM; ΛCDM+me, ΛCDM+ωa. The results suggestthat axi-Higgs may potentially resolve the Hubbletension to restore the cosmic concordance, either with orwithout H0 priors. In the landscape of Hubble tension,the axi-Higgs model falls into the category of mixedpre-recombination and post-recombination solutions.

The same conclusion has been reached in Ref. [20]using the newly-developed linear perturbative approach.This work, therefore, has complemented and formallytested LPA. Nevertheless, we believe that LPA is stillin its infancy stage as mentioned in Sec. III E. Oneof our goals is to extend the minimal set of the CMBcompressed obvervables with others which determinesAs, ns, τreio when they are allowed to vary. Another goalis to adapt the partial derivatives computation withmultiple reference points to improve the accuracy whendealing with non-linear variations. Eventually, LPA isan efficient tool to estimate cosmological parameters,which requires significantly shorter computational timecompared to the traditional analysis 4.

There also remain many questions that we havenot elaborated on. Firstly, on the subject of BBN,Ref. [19] has predicted that the Higgs-VEV must stayroughly 1% above its present-day value to alleviate the7Li puzzle [69], which turns out to be consistent withthe (v/v0)ini constraints obtained independently withthe base data set in Tab. I. Secondly, the interactionterm of axion and photon gF F has been ignored in theLagrangian (1). This term is responsible for rotatingthe photon polarization plane [70], which may inducethe non-vanishing isotropic cosmic birefringence (ICB)signal recently detected from the cross-correlation CEBlspectrum of CMB data [71]. The constraint of theICB angle β = 0.35 ± 1.4 deg roughly translates tofa ∼ 1017 − 1018 GeV for ωa ∼ 0.001, see [19, 72] formore details. An extensive treatment for these problemsis the main theme of our future studies.

Finally, the Higgs field can couple to the second ax-ion field with a higher mass [19], i.e. δv = (C1φ

21 +

C2φ22)/2M2

pl from theoretical perspectives. In that case,

4 LPA takes O(minutes) versus O(days) to finish one MCMC runof axi-Higgs on an average laptop.

the parameter (v/v0)ini used so far should be interpretedas the Higgs-VEV deviation at recombination (v/v0)rec

since the earlier Higgs-VEV deviation at BBN (v/v0)BBN

can be set to a different value depending on specific com-binations of C1, C2, φ1,ini, φ2,ini. The two-axion modelis necessary to archive the 2% Higgs-VEV uplift as re-ported in Tab. II at recombination and the 1% deviationat BBN as required by the produced abundance of lightelements [19]. This extended framework could simulta-neously resolve 7Li puzzle, Hubble tension and even S8

tension if the heavy axion mass is of order 10−26 eV, seeApp. A, or explain small-scale crisis if the heavy axion isFDM with ma ∼ 10−22 eV [73].

ACKNOWLEDGMENTS

I would like to express my appreciation to Prof. HenryTye and Prof. Tao Liu for the initial ideas leading to thispaper. I thank Leo Fung, Lingfeng Li and Yu-Cheng Qiufor useful discussions on the related topics. I am gratefulfor the encouragement from my girlfriend Shuting duringthe time of writing the paper. This work is supported bythe Area of Excellence under Grant No. AoE/P-404/18-3(6) issued by the Research Grants Council of Hong KongSAR.

Appendix A: The (non-)equivalenceof axi-Higgs and ΛCDM+me+ωa

The axi-Higgs model is equivalent but not identicalto the ΛCDM+me+ωa model. The latter is a simpleextension of ΛCDM which allows a non-standard elec-tron mass with the addition of an axion along withit. Thus, the me deviation in this model is alwaysconstant, independent of axion dynamics. On the otherhand, axi-Higgs has the electron mass modulated by theHiggs-VEV variation, hence the initial me deviation willdecay as soon as the axion density starts diluting at latetimes, see Fig. 4. Depending on the cosmic epoch beingconsidered, the main distinctive physics of axi-Higgswith ΛCDM+me+ωa can emerge at: (i) BBN whenthe Higgs-VEV uplift alters the abundance of severalprimordial elements such as D, 4He, 7Li; (ii) Reionzationwhen the electron mass are not as high as its initialvalue. We have also discussed that any models thatchange the electron mass without dismissing its devia-tion early enough would be in conflict with astrophysicaland laboratory observations at later times [19]. In termsof parameter limits, axi-Higgs can restore: ΛCDM whenω → 0; ΛCDM+ωa when (v/v0)ini → 1; ΛCDM+me

when ma → 0. The ΛCDM+me+ωa model is morestraightforward, it can reproduce: ΛCDM when ωa → 0and (me/me,0) → 1; ΛCDM+ωa when (me/me,0) → 1;ΛCDM+me when ωa → 0.

Despite some subtle difference, ΛCDM+me+ωa can

12

be perfectly treated as the phenomenological modelof axi-Higgs for ma . 10−29 eV within the regime ofthis work. We prove it in Tab. VI and Fig. 12 forma = 10−29 eV where the marginalized constraints ofcosmological parameters obtained from both modelsare completely matching. However, heavier axions willclearly differentiate axi-Higgs and ΛCDM+me+ωa. Forinstances, the results given in Tab. VI of the two modelsfor ma = 10−26 eV look drastically different, especiallywith (v/v0)ini unbounded for axi-Higgs. The reason isbecause the axion oscillation was triggered at very highredshifts, i.e. zosc ∼ O(105), which makes any reasonablevalue of the initial Higgs-VEV deviation disappearinglong before the moment when recombination kickedin, at roughly zrec ∼ 1100. We also notice that theaxion density is not restricted with this heavy axion,ωa ∼ 0.005, hence S8/σ8 is substantially suppressed.This is an important hint for S8 tension. The axi-Higgsmodel including another axion with mass ma ' 10−26

eV can resolve H0 and S8 tensions simultaneously.In the literature, the addition of this specific axionreconciles the exceed DM density preferred by EDE-typemodels [74, 75].

Data CMBbase+BAO+WLma 10−29 eV 10−26 eV

Model axi-Higgs ΛCDM+me+ωa axi-Higgs ΛCDM+me+ωa

ωb 0.02269+0.00018−0.00022 0.02269 ± 0.00020 0.02244 ± 0.00013 0.02254 ± 0.00016

ωc 0.1206+0.0023−0.0027 0.1206+0.0021

−0.0025 0.1145 ± 0.0016 0.1160 ± 0.0021100θMC 1.0501+0.0061

−0.0083 1.0501+0.0061−0.0075 1.04114 ± 0.00029 1.0463 ± 0.0049

τreio 0.0526 ± 0.0077 0.0522 ± 0.0079 0.0545 ± 0.0075 0.0533 ± 0.0078ln(1010As) 3.038 ± 0.016 3.037 ± 0.016 3.040 ± 0.015 3.040 ± 0.016

ns 0.9673 ± 0.0041 0.9669 ± 0.0043 0.9642 ± 0.0040 0.9617 ± 0.0048

(v/v0)ini 1.0131+0.0088−0.012 1.0128+0.0086

−0.011 unbounded 1.0074 ± 0.0070ωa < 0.00328 < 0.00304 0.0044 ± 0.0017 0.0049 ± 0.0019

H0 69.3+1.2−1.4 69.3 ± 1.3 67.75 ± 0.43 68.9 ± 1.1

S8 0.797 ± 0.012 0.796 ± 0.012 0.776 ± 0.015 0.775 ± 0.015σ8 0.793 ± 0.013 0.793 ± 0.013 0.764 ± 0.015 0.770 ± 0.016φini 2.6 ± 1.1 2.6 ± 1.0 1.99+0.49

−0.33 2.10+0.48−0.35

100θ∗ 1.04120 ± 0.00030 1.04120 ± 0.00031 1.04110 ± 0.00030 1.04098 ± 0.00032

TABLE VI. Marginalized distributions of cosmological pa-rameters in axi-Higgs and ΛCDM+me+ωa fitted with thebase data set CMBbase+BAO+WL for two axion masses:ma = 10−29 eV and ma = 10−26 eV. The unbounded con-straint of (v/v0)ini indicates the flat distribution spanning itswhole prior range. The parameter (v/v0)ini = me/me,0 in theΛCDM+me+ωa model.

Appendix B: The validity ofeffective fluid approximation

Integrating the exact axion equations is computa-tionally expensive due to the mismatch of the intrinsicCompton period and Hubble time scales. In practice,we can average over several periods of axion oscillationsto derive the effective fluid approximation (EFA) solu-tion after the background field becomes dynamical asdescribed in Sec. II. The natural question is: when wasthat transition triggered?

The axion equation of motion (5) tells us that theharmonic term, i.e. the last term, is dominated bythe Hubble friction until H ∼ ma, so this moment isalso a crude estimate of the transition redshift. Wethen parametrize this condition with the coefficient ξin Eq. (8), which is typically chosen to be ξ = 3 in theliterature [21, 40]. In principle, the later the transitionhappens, i.e. the larger ξ is, the more accurate dynamicswill be captured. Fig. 4 demonstrates how the physicalquantities of axi-Higgs evolve with the common choiceand two other alternatives: ξ = 1 corresponding tothe crude dynamics and ξ = 100 associated with thenearly-exact dynamics. Based on the fact that theapproximate curves closely trace the exact evolutionin each plot, we are reassured that the ξ = 3 choice issufficient up to the precision demanded by the currentdata. EFA implementation with the switch at ξ = 3may introduce biases on CMB spectra of more than 4σfor higher axion masses [76].

Appendix C: Supplementary materials

This appendix shows additional information quoted inthe main text for completeness and self-consistency.

Data CMBbase+BAOModel ΛCDM ΛCDM+ωa ΛCDM+me axi-Higgs

ωb 0.02241 ± 0.00013 0.02242 ± 0.00014 0.02246 ± 0.00017 0.02254 ± 0.00018ωc 0.11940 ± 0.00097 0.1192 ± 0.0010 0.1205+0.0018

−0.0020 0.1215 ± 0.0022100θMC 1.04097 ± 0.00028 1.04101 ± 0.00030 1.0438 ± 0.0046 1.0476+0.0053

−0.0060

τreio 0.0555+0.0070−0.0080 0.0560 ± 0.0082 0.0543 ± 0.0079 0.0554 ± 0.0078

ln(1010As) 3.045 ± 0.016 3.046 ± 0.017 3.044 ± 0.016 3.038 ± 0.016ns 0.9661 ± 0.0036 0.9665 ± 0.0038 0.9648 ± 0.0042 0.9643 ± 0.0042

(v/v0)ini 1 1 1.0040 ± 0.0065 1.0095+0.0077−0.0087

ωa 0 < 0.000860 0 < 0.00177H0 67.61 ± 0.43 67.41 ± 0.48 68.2 ± 1.1 68.7 ± 1.2S8 0.825 ± 0.012 0.821 ± 0.013 0.826 ± 0.013 0.822 ± 0.014σ8 0.8096 ± 0.0069 0.8028+0.0094

−0.0082 0.815 ± 0.011 0.811 ± 0.013φini 0 1.27+0.46

−0.72 0 1.74+0.62−1.1

100θ∗ 1.04118 ± 0.00028 1.04122 ± 0.00029 1.04114 ± 0.00031 1.04109 ± 0.00031

TABLE VII. Same as Tab. I with the base data set excludingWL data, i.e. CMBbase+BAO only.

Model ΛCDM+ωa ΛCDM+me

Code aHCAMB axionCAMB aHCAMB CAMB+CosmoRec

ωb 0.02220 ± 0.00016 −− 0.02256 ± 0.00017 0.02255 ± 0.00018ωc 0.1198 ± 0.0015 0.119 ± 0.002 0.1208 ± 0.0019 0.1208 ± 0.0018

me/me,0 1 −− 1.0077 ± 0.0069 1.0078 ± 0.0067ωa < 0.00349 < 0.003 0 0τreio 0.077 ± 0.016 −− 0.0552 ± 0.0074 0.0549 ± 0.0074

ln(1010As) 3.088 ± 0.031 −− 3.046 ± 0.014 3.045 ± 0.014ns 0.9645 ± 0.0048 −− 0.9656 ± 0.0041 0.9654 ± 0.0040

H0 65.9+1.6−1.0 66.61 ± 1.31 69.1 ± 1.2 69.1 ± 1.2

φini/Mpl 0.115 ± 0.045 0.11 ± 0.04 0 0

TABLE VIII. Marginalized posteriors of cosmological pa-rameters reproduced from aHCAMB compared with the previ-ous implementations in ΛCDM+ωa (see Tab. 3 of [22]) andΛCDM+me (see Tab. 1 of [14]) fitted with the data sets usedin the corresponding works, i.e. CMBP15 for ΛCDM+ωa;CMBfull+BAODR12 for ΛCDM+me. aHCAM independentlyyields consistent results with the other numerical codes.

13

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= 1= 3= 100

10 4 10 3 10 2

a

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|a|

= 1= 3= 100

10 4 10 3 10 2 10 1 100

a

10 12

10 10

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10 2

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102

104

|a|

= 1= 3= 100

FIG. 4. Axion dynamics with different transition coefficients in the axi-Higgs model. (Upper) axion background density on theleft and Higgs-VEV deviation on the left. (Lower) axion perturbations of k = 10−4 Mpc−1 mode on the left and k = 0.03 Mpc−1

mode on the right. The plots are generated with cosmological parameters fixed to their fiducial values.

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ah2

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CMBfull+BAOnoRSD+WL/SBA CMBfull+BAOnoRSD+WL/LPA CMBfull+BAOnew+WLnew/SBA CMBfull+BAOnew+WLnew/LPA

FIG. 10. LPA (empty-dashed) and SBA (filled-solid) posterior distributions in the axi-Higgs model fitted withCMBfull+BAOnoRSD+WL and CMBfull+BAOnew+WLnew for ma = 10−29 eV (blue) and ma = 2×10−30 eV (red), respectively

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SBALPA with SL

LPA with S8LPA with 8

FIG. 11. LPA posterior distributions for different choices of the CMB observable SL in the axi-Higgs model fitted withCMBfull+BAOnew+WLnew. The corresponding SBA posterior distributions are also shown for reference.

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axi-Higgs (10 26 eV)axi-Higgs (10 29 eV)

CDM+me + a (10 26 eV)CDM+me + a (10 29 eV)

FIG. 12. Posterior distributions in axi-Higgs and ΛCDM+me+ωa fitted with CMBbase+BAO+WL. The empty-solid contoursrepresent the distributions of ΛCDM+me+ωa.