Axion hot dark matter bounds
after Planck
Abstract
We use cosmological observations in the postPlanck era to derive limits on thermally produced cosmological axions. In the early universe such axions contribute to the radiation density and later to the hot dark matter fraction. We find an upper limit eV at 95% C.L. after marginalising over the unknown neutrino masses, using CMB temperature and polarisation data from Planck and WMAP respectively, the halo matter power spectrum extracted from SDSSDR7, and the local Hubble expansion rate released by the Carnegie Hubble Program based on a recalibration of the Hubble Space Telescope Key Project sample. Leaving out the local measurement relaxes the limit somewhat to 0.86 eV, while Planck+WMAP alone constrain the axion mass to 1.01 eV, the first time an upper limit on has been obtained from CMB data alone. Our axion limit is therefore not very sensitive to the tension between the Planckinferred and the locally measured value. This is in contrast with the upper limit on the neutrino mass sum, which we find here to range from eV at 95% C.L. combining all of the aforementioned observations, to eV from CMB data alone.
Preprint MPP2013113
a]Maria Archidiacono, a]Steen Hannestad, b]Alessandro Mirizzi, c]Georg Raffelt d]and Yvonne Y.Y. Wong
[a]Department of Physics and Astronomy, University of Aarhus
DK8000 Aarhus C, Denmark
[b]II. Institut für Theoretische Physik, Universität
Hamburg
Luruper Chaussee 149, D22761 Hamburg, Germany
[c]MaxPlanckInstitut für Physik
(WernerHeisenbergInstitut)
Föhringer Ring 6, D80805 München, Germany
[d]School of Physics, The University of New South Wales
Sydney NSW 2052, Australia
, , \emailAdd,
1 Introduction
Observations of the largescale structure in the universe, most notably of the cosmic microwave background (CMB) anisotropies and the galaxy distribution, are a powerful probe of both the expansion history and the energy content of the universe. The precision of these observations is such that in some cases cosmological constraints on particle physics models that could plausibly account for some of the energy content are more stringent than limits obtained from direct laboratory measurements. One notable example is the cosmological bound on the neutrino mass which is currently close to an order of magnitude more restrictive than existing laboratory bounds [1, 2, 3, 4]. Given this ability of cosmology to probe neutrino physics, it is of significant interest to investigate how well other hot dark matter (HDM) models with similar behaviours can be tested.
One example of particular interest is the case of hot dark matter axions. In reference [5] some of the present authors found that cosmological data do in fact provide an upper bound on the mass of hadronic axions comparable to that for neutrinos. The analysis was subsequently updated on several occasions to include progressively more recent data releases [6, 7, 8, 9], again with similar conclusions. Other authors also found comparable results [10]. Here, we provide another update incorporating the most recent measurement of the CMB temperature anisotropies by the Planck mission [11, 12], as well as the matter power spectrum extracted from the seventh data release of the Sloan Digital Sky Survey (SDSSDR7) [17], and the presentday Hubble parameter obtained from the most recent recalibration of the Hubble Space Telescope (HST) Key Project sample as part of the Carnegie Hubble Program [18].
The exercise of extracting information about lowmass particles from precision cosmological data has received a new twist in the past few years because combinations of different data sets are not necessarily consistent with the minimal (“vanilla”) CDM cosmology. In particular, many prePlanck observations prefer at the level a “dark radiation” component equivalent to an additional massless neutrino species (see, e.g., [13, 14, 15, 16] and references therein). The Planck data alone and in combination with baryon acoustic oscillations do not support the presence of a dark radiation component. However, the preferred value of the Hubble parameter is across the board significantly lower than that from direct measurements in our local neighbourhood [18]; if these measurements were to be brought in line with one another, then the presence of a dark radiation equivalent to could achieve the required result through its degeneracy with [12]. However, hovering around the level the case for dark radiation, prePlanck or postPlanck, is certainly not strong. Indeed, it could be as much a signature for new physics as it is a signature for poorly controlled systematics in some cosmological data sets. Nonetheless, since we are dealing here with lowmass particles which do contribute to the primordial radiation density, some additional care is required when dealing with local measurements.
The rest of the paper is organised as follows. In section 2 we briefly discuss the hot dark matter axion scenario. We describe the cosmological parameter space to be analysed in section 3, while section 4 summarises the data sets. Section 5 contains our results, notably, bounds on the axion mass and the neutrino mass sum from various data combinations, as well as a brief discussion of the impact of hot dark matter on the CMB observables and the corresponding degeneracies with . We conclude in section 6, followed by an appendix A in which we discuss the degeneracies of and in more detail.
2 Hot dark matter axions
The Peccei–Quinn solution of the CP problem of strong interactions predicts the existence of axions, lowmass pseudoscalars that are very similar to neutral pions, except that their mass and interaction strengths are suppressed relative to the pion case by a factor of order , where MeV is the pion decay constant, and is a large energy scale known as the axion decay constant or Peccei–Quinn scale [19, 20, 21]. Specifically, the axion mass is
(1) 
where is the mass ratio of the up and down quarks. We shall follow the previous axion literature and assume the canonical value . We note that variation in the range 0.38–0.58 is possible [22], although this uncertainty has no strong impact on our discussion. Henceforth we shall mostly use as our primary phenomenological axion parameter, instead of the more fundamental parameter .
Axions with masses in the region are well motivated as a cold dark matter (CDM) candidate, albeit with considerable uncertainty as regards the precise mass value [21, 23, 24, 25]. Such CDM axions would have been produced nonthermally by the realignment mechanism and, depending on the cosmological scenario, by the decay of axion strings and domain walls. In addition, a sizeable hot axion population can be produced by thermal processes if the Peccei–Quinn scale is sufficiently low [26, 27, 28, 29]. For
In principle, the parameter region
If axions do not couple to charged leptons, the main thermalisation process in the postQCD epoch is [26]. The axion–pion interaction is given by the Lagrangian [26]
(2) 
where, in hadronic axion models, the coupling constant is . Based on this interaction, the axion decoupling temperature was calculated earlier in [6], from which follows the presentday axion number density,
(3) 
where denotes the effective entropy degrees of freedom, and is the presentday photon number density. The top panel of figure 1 shows as a function of the axion mass . Before thermal axions become nonrelativistic by cosmological redshift, they contribute to the total radiation density an amount
(4) 
equivalent to additional light neutrino species, where is the presentday neutrino number density in one flavour, and is the photon energy density. We stress that equation (4) is strictly valid only when the axion temperature far exceeds . For order 0.1–1 eVmass axions this corresponds to a time well before photon decoupling; the phenomenology of such axions in the context of the CMB anisotropies and the matter power spectrum is closely linked to the axion mass scale, and cannot be reparameterised in terms of massless neutrino species. Indeed, as we shall show in section 5 and appendix A, the degeneracy of especially in the direction of the Hubble parameter is vastly different from that of massless neutrinos.
3 Cosmological model
We perform the analysis in the context of a CDM model extended to include hot dark matter neutrinos and axions, and vary a total of eight parameters:
(5) 
Here, and are the presentday physical CDM and baryon densities respectively, the angular the sound horizon, the optical depth to reionisation, and and denote respectively the amplitude and spectral index of the initial scalar fluctuations. We use the neutrino mass sum and the axion mass as fit parameters, noting that these can be easily converted to their corresponding presentday energy density via
(6)  
(7) 
where and are the neutrino and the axion presentday temperature respectively, and is shown in the bottom panel of figure 1 as a function of . Table 1 shows the priors adopted in the analysis. We stress that is always let free to vary in our analysis, because neutrino masses are an experimentally established fact, not some novel untested physics that can be discarded at will.
Parameter  Prior 

[eV]  
[eV] 
For convenience we assume the neutrino mass sum to be shared equally among three standard model neutrinos. Note however that implementing any other mass spectrum will return essentially the same conclusions, since current observations are primarily sensitive to , not to the exact mass splitting. In fact, discerning the neutrino mass spectrum—and thereby distinguishing between the normal and the inverted neutrino mass hierarchies—is unlikely to be possible even with the extremely precise measurements from the future Euclid mission [49].
4 Data and analysis
We consider three types of measurements: temperature and polarisation power spectra of the CMB anisotropies, the largescale matter power spectrum, and direct measurements of the local Hubble expansion rate. These are discussed in more detail below. To these data sets we apply a Bayesian statistical inference analysis using the publicly available Markov Chain Monte Carlo parameter estimation package CosmoMC [50] coupled to the CAMB [51] Boltzmann solver modified to accommodate two hot dark matter components. With the exception of the local Hubble parameter measurement, the likelihood routines and the associated window functions are supplied by the experimental collaborations.
4.1 CMB anisotropies
Our primary data set is the recent measurement of the CMB temperature (TT) power spectrum by the Planck mission [52], which we implement into our likelihood analysis following the procedure reported in [12]. This data is supplemented by measurements of the CMB polarisation from the WMAP nineyear data release [53], in the form of an autocorrelation (EE) power spectrum at and a crosscorrelation (TE) with the Planck temperature measurements in the same multipole range. We denote this supplement “WP”.
4.2 The matter power spectrum
We use the matter (halo) power spectrum (HPS) determined from the luminous red galaxy sample of the seventh data release of the SDSSDR7 [17]. The full data set consists of 45 data points, covering wavenumbers from to . We fit this data set following the procedure of reference [17], using an adequately smeared power spectrum to model nonlinear modecoupling constructed according to the method described in reference [54].
4.3 Local Hubble parameter measurements
We further impose a constraint on the presentday Hubble parameter based on measurements of the universal expansion rate in our local neighbourhood. The most recent value released as part of the Carnegie Hubble Program [18] is
(8) 
which uses observations from the Spitzer Space Telescope to calibrate the Cepheid distance scale in the HST Key Project sample. Following the analysis of reference [12], we add the statistical and systematic errors in quadrature, assuming both uncertainties to be Gaussiandistributed.
5 Results and discussions
The main results of our inference analysis are presented in table 2 and figure 2, which show, respectively, the onedimensional marginal limits on and derived from various data combinations, and the corresponding twodimensional marginal probability density contours in the subspace.
Data set  [eV]  [eV] 

Planck+WP  
Planck+WP+HPS  0.007–0.48  
Planck+WP+  
Planck+WP+HPS+ 
5.1 Axion mass limit from CMB alone
Most strikingly, while it was previously not possible to obtain an upper limit on the axion mass from the CMB anisotropies alone, we now find that the combination of Planck+WP is able to constrain quite stringently to eV (95% C.L.). This is in contrast with the CMBonly upper limit on the neutrino mass sum , which has been hovering around 1 eV since the WMAP fiveyear data release: here, we find eV (95% C.L.) after marginalising over the unknown axion mass, somewhat stronger than the prePlanck value of 1.19 eV [9].
The fundamental and irreducible effect of 0.1–1 eVmass hot dark matter neutrinos and axions on the CMB temperature anisotropies are similar in that both particle species transit from a ultrarelativistic state to a nonrelativistic one in close proximity to the photon decoupling era, as well as to the epoch of matter–radiation equality. This transition impacts strongly on the evolution of the radiationtomatter energy density ratio, which in turn affects the evolution of the spacetime metric perturbations and ultimately the photon temperature fluctuations. To quantify the precise effect of this transition on the CMB temperature power spectrum naturally necessitates solving the relevant Boltzmann equations. However, a general rule of thumb is that, assuming a fixed presentday total matter density, those perturbation modes that enter the horizon before the hot dark matter particle becomes fully nonrelativistic will suffer more strongly from decay of the metric perturbations in a mixed CDM+HDM cosmology than in a pure CDM scenario. This is translated firstly into an additional “uplift” of the primary CMB temperature fluctuations at last scattering on the same length scales. Secondly, the same decaying metric perturbations also alter the socalled early integrated Sachs–Wolfe (ISW) effect, which contributes to the CMB temperature anisotropies primarily around the first acoustic peak.
Following from this reasoning, what distinguishes between a hot dark matter neutrino population and an axion one of the same presentday (nonrelativistic) energy density is essentially twofold. Firstly, the axion population is always colder than the neutrino population by virtue of having decoupled from the cosmic plasma at an earlier time. Being a scalar degree of freedom the axion must also have a larger particle mass than the neutrino in order to make up the same presentday energy density [see equations (6)and (7)]. Together these push the axion’s relativistictononrelativistic transition to a higher redshift, so that its irreducible features on the CMB anisotropies occur on smaller scales (or higher multipoles). Secondly, while the axion is a new source of energy density in addition to the energy content of the CDM model, in endowing neutrinos with masses we are effectively “stealing” from the model’s radiation content. This effect combined with the axion’s relative coldness and heaviness means that a hot axion population will always cause less deviation to the radiationtomatter energy density ratio, and therefore leave a weaker signature on the CMB than a neutrino population of the same presentday energy density. Figure 3 shows the CMB temperature angular power spectra for a neutrino and an axion HDM scenario, juxtaposed against their CDM counterpart.
Thus what has enabled Planck to constrain the axion mass for the first time from CMB data alone can be traced to its significantly better measurement of the higher multipole moments from the third acoustic peak onwards (recall that WMAP is only cosmicvariance limited up to , roughly the second acoustic peak), which rules out much of the axion parameter space in the eV region manifesting on these scales. In contrast, the neutrino mass could already be constrained to order 1 eV by CMB data alone since the WMAP threeyear data release [55] via the early ISW effect on the thirdtofirst acoustic peak height ratio; new, high multipole measurements from Planck offer no significant improvement because there are no new unique signatures from subeVmass neutrinos on these scales. See also discussions in appendix A.
5.2 Correlation with
Combining CMB measurements with the halo power spectrum from SDSS DR7 tightens the axion mass bound to eV (95% C.L.), comparable to the prePlanck value of 0.82 eV [9]. The neutrino mass bound likewise improves to eV (95% C.L.). Note that formally the combination of Planck+WP+HPS yields also a 95% lower limit of eV. However, because neutrino oscillation experiments already limit the neutrino mass sum to eV, this cosmological lower limit is of little phenomenological consequence.
An interesting effect arises when we consider also local measurements of the Hubble parameter ; while the neutrino mass bound continues to tighten to eV (95% C.L.), inclusion of the local value leads only to a marginal 10% improvement on the axion mass bound. This effect can be understood from an inspection of figures 4 and 5, which show, respectively, the twodimensional marginal density contours in the  and subspaces. Clearly, a strong anticorrelation exists between and , so that adding to the analysis the local value—which is higher than the Planckinferred value—tends to tighten, perhaps artificially, the neutrino mass bound. In contrast, no such correlation exists between and , making the inference of independent of the value of . Thus we conclude that the axion hot dark matter bound is robust against the unresolved discrepancy between the Planckinferred value and that measured in our local neighbourhood. Figure 6 shows the twodimensional marginal probability density contours in the subspace with the inclusion of local data.
A deeper question now is, what causes and to exhibit different correlations with in the first place. We defer a more detailed discussion of the parameter degeneracies in the CMB temperature anisotropies to appendix A, but note here that at the most basic level, any degeneracy (or lack thereof) of and with can be traced to the parameter dependence of the angular sound horizon , where
(9)  
(10) 
are the sound horizon at decoupling and the angular diameter distance to the last scattering surface respectively, is the sound speed in the tightlycoupled photon–baryon fluid, and is the photon decoupling redshift. The angular sound horizon determines the location of the acoustic peaks in the CMB angular power spectrum, and is arguably the most wellmeasured CMB observable.
Clearly, the presence of the Hubble expansion rate in the expressions (9) and (10) indicates a priori a threeway degeneracy between the neutrino (or axion) energy density (and hence mass), the CDM energy density ,^{1}^{1}1The baryon density in principle also affects . However, can be pinned down by the CMB oddtoeven peak ratios. We therefore consider it to be a wellmeasured parameter. and the presentday Hubble parameter ,^{2}^{2}2In a flat spatial geometry, the dependence of on the vacuum energy can be reparameterised in terms of . meaning that for any choice of or , we can tweak and/or to maintain the same value. However, the amount of tweaking required for the axion scenario is generally less than that needed by massive neutrinos. This is because, as discussed in section 5.1, an axion population generally causes less deviation to the evolution of the radiation and matter energy densities than a neutrino population of the same presentday energy density. Indeed, assuming a fixed presentday total matter density, figure 3 shows that while the presence of a eV hot neutrino population causes a noticeable shift in the peak locations that can be restored by lowering the value to (compared to the reference CDM value of ), the shift due to a eV axion population is of a much lesser extent and requires only a small positive tuning of to . As we shall demonstrate in appendix A, the same negative correlation with remains true for the neutrino case even after accounting for uncertainties in the total matter density, while for the axion scenario the small tuning of may be negative or positive depending on the choice of so that overall no correlation exists between and .
6 Summary and conclusions
We have reexamined cosmological bounds on thermal axions based on the recent CMB temperature anisotropy measurement provided by the Planck mission, as well as other types of cosmological observations. For the first time CMB data alone provide a restrictive limit on the axion mass of eV (95% C.L.), which improves to 0.86 eV with the inclusion of the SDSS matter power spectrum, and to 0.78 eV when combined with local measurements of . The changes brought about by combining CMB data with other observations are obviously fairly minor, suggesting that the systematic uncertainties in our limits are small. In particular, the unsettled discrepancy between the Planckinferred value and that measured directly in our local neighbourhood does not strongly affect the axion limit. This is to be contrasted with the corresponding limit on the neutrino mass sum, which sees a dramatic reduction from eV (95% C.L.) from CMB alone to eV when combined with local measurements.
Note that when inferring the axion mass bound, we always marginalise over the unknown neutrino masses which contribute an unavoidable but unknown HDM fraction to the universe; in this sense, HDM contributions by neutrinos and axions are not alternatives, since the former must always be present. However, unless neutrinoless doublebeta decay eventually shows a signal, laboratory experiments are unlikely to provide neutrino mass information on the cosmologically relevant scale in the foreseeable future. Thus marginalisation over is arguably the only consistent approach to cosmological parameter estimation, and this situation is unlikely to change any time soon.
In comparison with the prePlanck state of affairs, although we now have an axion mass bound from postPlanck CMB data alone, the overall limit has not moved significantly away from order 1 eV. This situation is of interest for solar axion searches by the Tokyo axion helioscope [56, 57, 58] and by the CAST experiment at CERN [59, 60, 61, 62]. The axion–photon conversion efficiency in a dipole magnet directed towards the sun strongly degrades if the photon–axion mass difference is too large. This massmismatch can be overcome by filling the conversion pipe with a buffer gas at variable pressure that provides the photons with an adjustable refractive mass. The final CAST search range using He as buffer gas has reached a search mass of 1.17 eV, which is hard to push up any further [63]. In this sense, cosmological and helioscope axion mass limits are still largely complementary.
In the future, further improvements in HDM bounds on the cosmological front will likely come from the next generation of largescale galaxy and cluster surveys, notably the ESA Euclid mission [64]. Euclid has the potential to probe neutrino masses with a sensitivity of eV, sufficient to measure at the minimum mass sum of eV established by neutrino flavour oscillation experiments [49]. A similar sensitivity to HDM axions should likely be possible.
Acknowledgements
We thank Jan Hamann for help with the initial setup of the Planck likelihood code, and the anonymous referee for constructive suggestions that improve the clarity of the manuscript. We acknowledge use of computing resources from the Danish Center for Scientific Computing (DCSC), and partial support by the Deutsche Forschungsgemeinschaft through grant No. EXC 153 and by the European Union through the Initial Training Network “Invisibles,” grant No. PITNGA2011289442.
Appendix A Degeneracies of and in the CMB temperature anisotropies
Figures 4 and 5 show that while a strong anticorrelation exists between the neutrino mass sum and the Hubble parameter , no such correlation is present for the axion mass and . To understand these degeneracies, we note first of all that the most wellmeasured observable in the CMB temperature anisotropies is the angular sound horizon , defined in equations (9) and (10), which fixes the acoustic peak positions. Assuming a fixed baryon density (as we shall do for the rest of this section), depends on the dark matter density , the neutrino mass sum (or the axion mass as the case may be), and the Hubble parameter . Thus, already at this level it is clear that and (or ) must be correlated somehow. Indeed, as already shown in figure 3, holding (or ) fixed, the effect of on the positions of the acoustic peak can be countered by lowering the value of , while a finite can be offset by raising by a small amount.
The odd peak height ratios are likewise a wellmeasured set of quantities, and are usually described in the literature as a sensitive probe of the epoch of matter–radiation equality . A more accurate description, however, is that the ratios reflect the time evolution of the radiationtomatter energy density ratio , which in the vanilla CDM model (and in a few simple extensions of this model) is a oneparameter function subject to our choice of relative to the time of photon decoupling. A simple example where this is still the case is the extension of vanilla CDM with a nonstandard number of massless neutrinos, . Here, a larger can be offset by a larger matter density so as to restore and consequently all odd peak height ratios. The degeneracy between and is therefore exact as far as the function and its effects on the CMB are concerned, and when combined with the measurement of the angular sound horizon discussed above, gives rise to the wellknown positive correlation between and .
The case of a nonzero neutrino mass or axion mass is less straightforward, because no amount of adjustments could completely counter the effect of or on the radiationtomatter energy density ratio. To begin with, no unique definition of matter–radiation equality exists in this case, because for the 0.1–1 eVmass range under consideration there will always be some HDM particles that are neither ultrarelativistic nor completely nonrelativistic in the vicinity of photon decoupling. Furthermore, because HDM particles are continuously losing momentum through redshift, the additional redshift dependence they engender in cannot be mimicked over the whole timespan by simply tweaking , contrary to the case of . Therefore, any correlation between and or between and is necessarily approximate and subject to measurement errors (or cosmic variance).
This point is illustrated in figure 7, which shows the fractional change in for a neutrino HDM scenario and an axion scenario of the same presentday energy density, relative to the CDM case. To define we consider to be nonrelativistic all HDM particles with momenta less than a factor of the particle’s rest mass, where for we test a range of values from to . Clearly, independently of our choice of , varying only allows the matching of at one point in time, which we choose here to be , the scale factor at matter–radiation equality in the reference CDM model. Furthermore, the matching demands that we increase : by approximately 0.014 in the eV neutrino case, and by an amount 0.003–0.0205 in the eV axion case. When considered in conjunction with the threeway degeneracy [or the degeneracy] from discussed above and in section 5.2, this immediately leads us to the following conclusions:

For a fixed value of , raising necessitates that we lower even more in order to match the angular sound horizon of the reference CDM model. This reinforces the negative correlation already existing between and , even if no adjustments were made to in order to match ). Indeed, we see in the top panel of figure 8 that after incrementing by an amount 0.0146 and lowering to , the resulting CMB temperature power spectrum for the eV neutrino scenario matches the reference spectrum of the CDM model with to almost perfectly.

For a fixed , depending on the chosen value, the correlation between and induced by fixing the angular sound horizon scale can be mildly positive for no or a small increase in , or mildly negative at the upper end of the range. Two examples are shown in the bottom panel of figure 8 for the eV axion scenario: one case has increased by an amount 0.0028 and raised to , the other has a 0.01 increment in and . Both cases mimic the reference CDM spectrum reasonably well in the first two peaks. Therefore, marginalising over the uncertainty in leaves no apparent correlation between and .
Observe also that both adjusted axion scenarios in figure 8 begin to show similar signs of deviation, although in opposite directions, from the third acoustic peak onwards. These deviations were not previously measurable by WMAP alone because WMAP was cosmic variancelimited only up to . A eV axion scenario was therefore allowed by WMAP alone. Planck is able to rule out this scenario simply by virtue of its cosmic variancelimited measurement of the third acoustic peak.
Lastly, we briefly discuss the role of the angular diffusion scale measurement on the neutrino and the axion mass bounds. To this end, it is instructive to first consider the simpler case of a nonstandard ; here, a larger requires that we increase both and in order to fit the epoch of matter–radiation equality and the angular sound horizon. However, increased values of and also lead to a larger , defined as , where
(11) 
is the (squared) photon diffusion scale at decoupling, with the Thomson scattering crosssection, the free electron number density, and the baryontophoton energy density ratio. Thus, the net effect is that a larger causes more damping of the CMB temperature power spectrum at , and measurement of this damping tail can be a very effective means to pin down the value of (see, e.g., [65, 16]).
The same is not true however for the inference. This is because while we do need to increase to match the radiationtomatter energy density ratio, the value must be simultaneously lowered in order to fit the angular sound horizon scale; these two adjustments more or less cancel each other’s effect on . In comparison, the axion scenario has a marginally larger impact on the angular diffusion scale than the neutrino case because in order to fit , can be held more or less fixed. Physically, the difference can be traced to the fact that apart from the early ISW effect, the main impact of massive neutrinos that become nonrelativistic in bulk after photon decoupling on the CMB anisotropies is through the angular diameter distance to the last scattering surface . But since simply rescales the positions of the acoustic peaks and hence alters and in the same way, the ratio is independent of . In contrast, the majority of axions in the mass range considered become nonrelativistic before photon decoupling. Such axions therefore have a more direct impact on the physical sound horizon and the diffusion scale at decoupling, and remain discernible in the ratio . Nonetheless, the overall effect of axion masses on is still minor in comparison with ’s irreducible effects on the radiationtomatter energy density ratio and consequently the peak height ratios, and is therefore not a decisive factor in cosmological constraints on .
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