Can Non-standard Recombination Resolve the Hubble Tension?

Miaoxin Liu, Zhiqi Huang,∗ Xiaolin Luo, Haitao Miao, Naveen K. Singh, and Lu Huang

School of Physics and Astronomy, Sun Yat-sen University, 2 Daxue Road, Tangjia, Zhuhai, China

The inconsistent Hubble constant values derived from cosmic microwave background

(CMB) observations and from local distance-ladder measurements may suggest new physics

beyond the standard ΛCDM paradigm. It has been found in earlier works that, at least phe-

nomenologically, non-standard recombination histories can reduce the & 4σ Hubble tension

to ∼ 2σ. Following this path, we vary physical and phenomenological parameters in REC-

FAST, the standard code to compute ionization history of the universe, to explore possible

physics beyond standard recombination. We find that the CMB constraint on the Hubble

constant is sensitive to the Hydrogen ionization energy and 2s→ 1s two-photon decay rate,

both of which are atomic constants, and is insensitive to other details of recombination.

Thus, the Hubble tension is very robust against perturbations of recombination history,

unless exotic physics modifies the atomic constants during the recombination epoch.

∗ huangzhq25@mail.sysu.edu.cn

arX

iv:1

912.

0019

0v1

[as

tro-

ph.C

O]

30

Nov

201

9

2

I. INTRODUCTION

The tremendous efforts made by cosmic microwave background (CMB) experiments facilitate

us to explore the early as well as the current universe. The analysis of the temperature and

the polarization anisotropy spectra hints for a spatially flat universe filled with ∼ 5% known

matter in the standard chart of particle physics, ∼ 26% cold dark matter (CDM) whose particle-

physics nature is yet to be determined, and ∼ 69% dark energy with repulsive gravity, whose

microscopic nature is usually interpreted as the cosmological constant Λ or equivalently the vacuum

energy [7]. Within this ΛCDM paradigm and using the latest CMB data from the Planck satellite,

the Planck collaboration was able to precisely determine the cosmological parameters [3], among

which the current expansion rate of the universe, namely the Hubble constant is constrained to be

H0 = 67.4 ± 0.5 km s−1Mpc−1.

The remarkable success of the ΛCDM paradigm, however, is challenged by the recent mea-

surements of the local expansion rate of the universe. The local Hubble constant inferred from

low-redshift distance ladder (SH0ES), H0 = 74.03± 1.42 km s−1Mpc−1, is in 4.4σ tension with the

CMB+ΛCDM result [33–36]. More recently, another independent class of H0 measurements using

the time-delay of strong-lensing quasar images (H0LiCow, STRIDES) starts to join the high-H0

camp [41, 43]. The combination of SH0ES and H0LiCow results gives H0 = 73.8±1.1 km s−1Mpc−1,

which raises the low-z (redshift z . 1) and high-z (redshift z ∼ 1100) Hubble tension to 5.3σ [43].

The ever increasing Hubble tension, if not due to some unknown observational biases and sys-

tematics [1, 20], may suggest a fundamental flaw in the standard ΛCDM paradigm. The possibility

of discovering new physics beyond ΛCDM has inspired discussion of many theoretical models, most

of which are related to the dark components [5, 6, 8, 11, 17–19, 24, 26, 27, 29, 32, 37, 42, 45, 46].

Other possibilities are also intensively investigated [2, 10, 12, 13, 16, 23, 47]. Ref. [28] studied

the impact of non-standard primordial fluctuations from inflation, and found that the CMB H0

constraint is insensitive to the primordial conditions, thus excluding early-universe origin of the

Hubble tension for a broad class of models. Following a series of attempts to explore non-standard

recombination [21, 22, 31], Ref. [14] found that phenomenological modification to the timing and

width of the recombination process can significantly reduce the Hubble tension to ∼ 2σ. However,

unlike the primordial fluctuations that are related to unknown physics at very high energy scales

& 1014GeV, the recombination process depends on well-tested physics at energy scales . eV. The

phenomenological perturbations to the ionization fraction function Xe(z) in Ref. [14] as well as in

earlier works [21, 22, 31] may be too “nonphysical” to comply with the basic physical pictures of

3

TABLE I. Parameters in RECFAST

parameter definition precision

EH,energy Hydrogen ionization energy theoretically and experimentally determined constant

AH,2γ Hydrogen 2s→ 1s two-photon rate theoretically determined constant

fH Hydrogen fudge factor phenomenological, has some uncertainties

AHe,2γ Helium 2s→ 1s two-photon rate theoretically determined constant

fHe Helium fudge factor phenomenological, has some uncertainties

recombination. To tackle this problem, in this paper we use a very different approach to perturb

recombination. We vary the parameters in the system of equations that govern the evolution of

Xe(z). This method automatically retains the major physical structure of Helium and Hydrogen

recombination processes, and directly connects the perturbations in Xe(z) to underlying physical

parameters. We can then identify the physical degrees of freedom that may relieve the H0 tension.

II. METHOD

The standard RECFAST code based on a series of work [30, 39] contains a fast and approximate

algorithm to compute the helium and hydrogen recombination processes. Despite the presence of

a few phenomenological “fudge factors” and a few fitting parameters in RECFAST, it is believed

that sub-percent level accuracy can be achieved after careful calibration against more detailed

calculations [15, 38, 40, 44].

In Table I we list the major (non-cosmological) parameters involved in RECFAST algorithm.

In principle the HeI and HeII ionization energies should also be included in the list. However, the

prediction of CMB observable is not sensitive to the helium parameters. Including too many helium

parameters would be redundant for CMB analysis. In the latest version RECFAST V1.5.2, there

are many more parameters to capture details of Helium recombination. These parameters only

lead to tiny ( percent) corrections to CMB observables and have little impact on cosmological

parameters. Thus, hereafter we will only discuss the parameters listed in Table I.

If we admit the precision and robustness of RECFAST code in the ΛCDM paradigm, none of the

parameters in Table I has much room to vary. Physics beyond ΛCDM may lead to re-calibration of

the phenomenological fudge factors fH and fHe, or even make them time-dependent. The variation

of atomic constants EH,ion, AH,2γ etc., however, would be much more difficult to achieve and is

considered to be rather “controversial”. To fully explore the impact of non-standard recombination

on H0, we nevertheless relax the atomic constants in part of our analysis, too. More concretely,

4

we rescale the parameters in Table I with scaling factors sH,energy, sH,2γ , sH,fudge +nH,fudge(a− a∗),

sHe,2γ , sHe,fudge, respectively, where a is the cosmological scale factor normalized to unity today,

and a∗ = 1/1090 is its value at recombination in standard scenario. We are allowing some time-

dependence of the hydrogen fudge factor, because it governs the evolution of Xe for a much longer

time (till the late universe). In total we have six recombination parameters. We assume a uniform

prior in [0.8, 1.2] for each scaling parameter s..., and uniform prior in [−0.2, 0.2] for the running

parameter nH,fudge. These parameters are added to the standard CosmoMC [25] for a full MCMC

analysis with all the parameters varying. We use the Planck final release TTTEEE + lowE +

lensing likelihood [4] and apply a lower-bound to the reionization redshift: zreion > 6, which is

independently inferred from GunnPeterson trough [9]. The reionization prior can effectively control

the degeneracy between reionization and the non-standard RECFAST Xe.

We modify the publicly available CosmoMC package to add the aforementioned six extra re-

combination parameters to the standard six cosmological parameters: the baryon density Ωbh2,

the CDM density Ωch2, the angular extension of sound horizon on the last scattering surface θ, the

reionization optical depth τ , the amplitude As and tilt ns of the primordial scalar power spectrum.

We do a MCMC run with all the parameters varying, as well as a “less controversial” run with the

fudge factors varying but with the atomic constants fixed (sH,energy = sH,2γ = sHe,2γ = 1). We dub

the two runs “vary-all” and “vary-fudge”, respectively.

III. RESULTS

We take the Planck best-fit ΛCDM [3] as a reference model, and show the variations of lnXe

trajectories in the left panel of Figure 1. Unlike the parametrization used in Ref. [14] where

the Xe variations are mainly due to the shift of hydrogen recombination redshift z?, defined by

Xe(z?) = 0.5, our parametrization explores many more degrees of freedom. To more explicitly

demonstrate this, in the right panel of Figure 1 we show, by selecting sub-samples with z? almost

frozen, the rich structures in lnXe(z) due to the degrees of freedom beyond a z? shift.

The hydrogen recombination redshift z? is indeed a key quantity connected to the Hubble

tension. In Figure 2 we demonstrate the strong degeneracy between z? and H0 for the “vary-all”

run. We find that the Hubble tension can be significantly relieved when z? is set free, in agreement

with Ref. [14]. However, we find that significant relaxation of z? can only be achieved by variation

of the hydrogen ionization energy or the hydrogen 2s→ 1s two-photon decay rate, both of which

are well-determined atomic constants by quantum mechanics.

5

500 1000 1500 2000 2500z

0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

0.4ln

X e1 samplesmean

500 1000 1500 2000 2500z

0.4

0.3

0.2

0.1

0.0

0.1

0.2

0.3

0.4

lnX e

1 samplesmean

FIG. 1. Left panel: trajectories of lnXe(z) randomly selected from the top 68.3% likelihood-ranked MCMC

samples from the “vary-all” run. For better visibility we have subtracted lnXe(z) of the reference model -

Planck best-fit ΛCDM [3]. The dark- and light-gray areas are marginalized 68.3% and 95.4% confidence-

level regions, respectively. Right panel: the same as the left panel, except that only samples with restricted

hydrogen recombination redshift 1089.156 < z? < 1090.335 (Planck + ΛCDM ∼ 3σ bounds) are used.

60 63 66 69 72 75H0

1050

1065

1080

1095

1110

1125

1140

z *

0.96

0.98

1.00

1.02

1.04

sH,energy

60.0 62.5 65.0 67.5 70.0 72.5 75.0H0

1050

1065

1080

1095

1110

1125

z *

0.85

0.90

0.95

1.00

1.05

1.10

1.15

sH,2

FIG. 2. The marginalized posterior distribution of z? and H0 for the “vary-all” run. The left and right

panels show the dependence of the distribution on the hydrogen ionization energy parameter sH,energy and

on the hydrogen 2s→ 1s two-photon rate parameter sH,2γ , respectively.

The final marginalized constraints on H0 are shown in figure 3, for both “vary-all” and “vary-

fudge” runs. The relaxation of atomic constants in “vary-all” run frees the hydrogen recombination

redshift and hence significantly worsens the constraints on H0. This can be understood because the

CMB H0 constraint is mainly determined by the angular diameter distance to the last scattering

surface, i.e., the z = z? surface. In the less controversial “vary-fudge” run with the atomic constants

6

vary-fudge vary-all

ΛCDM SH0ES+H0LiCow

60 70H0

00.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11.1

1.2

P/P

max

FIG. 3. The marginalized posterior distribution of H0.

frozen, the constraint on H0 does not significantly differ from the standard ΛCDM. The Hubble

tension with SH0ES + H0LiCow measurement persists at a 5.2σ level.

IV. CONCLUSION

By connecting the perturbations in ionization history to physical parameters, we found that

exotic variations of atomic constants must be claimed in order to have a significant relaxation of the

CMB constraint on the Hubble constant. If one admits the basic physical properties of hydrogen,

there is no obvious room to vary the recombination history to relieve the Hubble tension. Our

result does not contradict with previous findings in Ref. [14]. Rather, we point out how “non-

standard” the recombination process has to be in order to serve as a buffer belt between the low-

and high-redshift Hubble drivers.

[1] Addison, G. E., Huang, Y., Watts, D. J., et al. 2016, ApJ, 818, 132

[2] Adhikari, S., & Huterer, D. 2019, arXiv:1905.02278

[3] Aghanim, N., et al. 2018, arXiv:1807.06209

[4] —. 2019, arXiv:1907.12875

[5] Agrawal, P., Cyr-Racine, F.-Y., Pinner, D., & Randall, L. 2019, arXiv e-prints, arXiv:1904.01016

[6] Agrawal, P., Obied, G., & Vafa, C. 2019, arXiv:1906.08261

[7] Akrami, Y., et al. 2018, arXiv:1807.06205

[8] Alexander, S., & McDonough, E. 2019, Physics Letters B, 797, 134830

7

[9] Becker, R. H., et al. 2001, Astron. J., 122, 2850

[10] Benetti, M., Graef, L. L., & Alcaniz, J. 2018, Journal of Cosmology and Astroparticle Physics, 2018,

066066

[11] Bhattacharyya, A., Alam, U., Pandey, K. L., Das, S., & Pal, S. 2019, The Astrophysical Journal, 876,

143

[12] Bolejko, K. 2018, Physical Review D, 97, doi:10.1103/physrevd.97.103529

[13] Carneiro, S., de Holanda, P., Pigozzo, C., & Sobreira, F. 2019, Physical Review D, 100,

doi:10.1103/physrevd.100.023505

[14] Chiang, C.-T., & Slosar, A. 2018, arXiv e-prints, arXiv:1811.03624

[15] Chluba, J., & Thomas, R. M. 2011, Mon. Not. Roy. Astron. Soc., 412, 748

[16] D’Eramo, F., Ferreira, R. Z., Notari, A., & Bernal, J. L. 2018, Journal of Cosmology and Astroparticle

Physics, 2018, 014014

[17] Di Valentino, E., Melchiorri, A., & Mena, O. 2017, Physical Review D, 96,

doi:10.1103/physrevd.96.043503

[18] Di Valentino, E., Melchiorri, A., Mena, O., & Vagnozzi, S. 2019, arXiv:1908.04281

[19] Di Valentino, E., et al. 2018, Physical Review D, 97, doi:10.1103/physrevd.97.043513

[20] Efstathiou, G. 2014, MNRAS, 440, 1138

[21] Farhang, M., Bond, J. R., & Chluba, J. 2012, ApJ, 752, 88

[22] Farhang, M., Bond, J. R., Chluba, J., & Switzer, E. R. 2013, ApJ, 764, 137

[23] Graef, L. L., Benetti, M., & Alcaniz, J. S. 2019, Physical Review D, 99, doi:10.1103/physrevd.99.043519

[24] Karwal, T., & Kamionkowski, M. 2016, Physical Review D, 94, doi:10.1103/physrevd.94.103523

[25] Lewis, A., & Bridle, S. 2002, Physical Review D, 66, 568

[26] Lin, M.-X., Benevento, G., Hu, W., & Raveri, M. 2019, Phys. Rev., D100, 063542

[27] Lin, M.-X., Raveri, M., & Hu, W. 2019, Phys. Rev., D99, 043514

[28] Liu, M., & Huang, Z. 2019, arXiv e-prints, arXiv:1910.05670

[29] Miao, H., & Huang, Z. 2018, The Astrophysical Journal, 868, 20

[30] Peebles, P. J. E. 1968, ApJ, 153, 1

[31] Planck Collaboration, Ade, P. A. R., Aghanim, N., et al. 2016, A&A, 594, A13

[32] Poulin, V., Smith, T. L., Karwal, T., & Kamionkowski, M. 2019, Physical Review Letters, 122,

doi:10.1103/physrevlett.122.221301

[33] Riess, A. G., Casertano, S., Yuan, W., Macri, L. M., & Scolnic, D. 2019, The Astrophysical Journal,

876, 85

[34] Riess, A. G., et al. 2016, ApJ, 826, 56

[35] —. 2018, ApJ, 861, 126

[36] —. 2018, ApJ, 855, 136

[37] Rossi, M., Ballardini, M., Braglia, M., et al. 2019, arXiv:1906.10218

[38] Scott, D., & Moss, A. 2009, MNRAS, 397, 445

8

[39] Seager, S., Sasselov, D. D., & Scott, D. 1999, ApJ Lett, 523, L1

[40] —. 2000, ApJS, 128, 407

[41] Shajib, A. J., et al. 2019, arXiv:1910.06306

[42] Sola, J., Gomez-Valent, A., Perez, J. d. C., & Moreno-Pulido, C. 2019, arXiv:1909.02554

[43] Wong, K. C., et al. 2019, arXiv:1907.04869

[44] Wong, W. Y., Moss, A., & Scott, D. 2008, MNRAS, 386, 1023

[45] Yang, W., Mukherjee, A., Di Valentino, E., & Pan, S. 2018, Physical Review D, 98,

doi:10.1103/physrevd.98.123527

[46] Yang, W., Pan, S., Di Valentino, E., et al. 2018, JCAP, 1809, 019

[47] Zhang, X., & Huang, Q.-G. 2019, arXiv e-prints, arXiv:1911.09439