”Thermal axion constraints in non-standard thermal histories
There is no direct evidence for radiation domination prior to big-bang nucleosynthesis, and so it is useful to consider how constraints to thermally-produced axions change in non-standard thermal histories. In the low-temperature-reheating scenario, radiation domination begins as late as MeV, and is preceded by significant entropy generation. Axion abundances are then suppressed, and cosmological limits to axions are significantly loosened. In a kination scenario, a more modest change to axion constraints occurs. Future possible constraints to axions and low-temperature reheating are discussed.
address=California Institute of Technology, Mail Code 130-33, Pasadena, CA 91125 address=California Institute of Technology, Mail Code 130-33, Pasadena, CA 91125 address=California Institute of Technology, Mail Code 130-33, Pasadena, CA 91125
If the axion has mass , it will be produced thermally, with cosmological abundance where is the effective number of relativistic degrees of freedom when axions freeze out [Crewther et al.(1979), Baluni(1979), Peccei and Quinn(1977), Chang and Choi(1993), Turner(1987), Kolb and Turner(1990), Moroi and Murayama(1998)]. If , axions free-stream to erase density perturbations while they are relativistic, and thus suppress the matter power spectrum on small scales, much like neutrinos [Hu et al.(1998), Eisenstein and Hu(1997), Kolb and Turner(1990), Hannestad(2006), Chang and Choi(1993), Crotty et al.(2004), Tegmark et al.(2004a), Barger et al.(2004), Seljak et al.(2005), Fukugita et al.(2006), Spergel et al.(2007), Pierpaoli(2003)]. Data from large-scale structure (LSS) surveys and cosmic microwave-background (CMB) observations impose the constraint to light hadronic axions [Hannestad et al.(2005), Hannestad et al.(2007), Melchiorri et al.(2007)]. We restrict our attention to hadronic axions in this work.
These constraints were determined in the standard radiation-dominated scenario. The transition to radiation domination after inflation might be gradual [Kamionkowski and Turner(1990)]. In a modified thermal history, relic abundances may change, due to modified freeze-out temperatures and suppression from entropy generation.
The universe could have reheated to a temperature as low as [Kawasaki et al.(1999), Ichikawa et al.(2005), Giudice et al.(2001a), Hannestad(2004), Kolb et al.(2003)]. This low-temperature reheating (LTR) scenario may be modeled simply through the decay of a massive particle into radiation, with fixed rate . This decay softens the scaling of temperature with cosmological scale factor , increasing the Hubble parameter and leading to earlier freeze-out for certain relics. Entropy generation then highly suppresses these relic abundances. Kination models offer another alternative to the standard thermal history, without entropy production, and cause more modest changes in abundances [Salati(2003)].
Past work has determined the relaxation in constraints to neutrinos, weakly interacting massive particles, and non-thermally produced axions are relaxed in LTR [Giudice et al.(2001b), Giudice et al.(2001a), Yaguna(2007)]. Here, we present new constraints to thermally-produced axions in the LTR scenario. We point the reader to Ref. [Grin et al.(2008)] for a discussion of the more modest changes to axion constraints in the kination scenario, and for additional details relevant to the following discussion. We conclude by discussing the impact of future LSS surveys and CMB measurements of the primordial helium abundance on the allowed parameter space for axions.
2 Low-temperature reheating (LTR)
where and denote the energy densities in the scalar field and radiation, and is the cosmological scale factor, whose evolution is given by the Friedmann equation. The reheating temperature is defined by [Giudice et al.(2001b), Chung et al.(1999), Kolb and Turner(1990)], where is the Planck mass and is the value of when .
At the beginning of reheating, dominates the energy density. The temperature is related to the radiation energy density by [Kolb and Turner(1990)] . We integrate Eqs. (1) to obtain the dependence of on [Grin et al.(2008)]. When the scalar begins to decay, the temperature rises quickly to a maximum and then falls as . This shallow scaling of temperature with scale factor results from the transfer of scalar-field energy into radiation. When overtakes near , the epoch of radiation domination begins, with the usual scaling [Grin et al.(2008)].
During reheating, [Giudice et al.(2001b), Giudice et al.(2001a)], the universe thus expands faster than during radiation domination, and the equilibrium condition is harder to meet. Relics with freeze-out temperature will thus have suppressed abundances because they never come into chemical equilibrium. Relics with come into chemical equilibrium, but their abundances are reduced by entropy production.
3 Axion production
Standard hadronic axions with are produced by the channels , , and [Hannestad et al.(2005), Chang and Choi(1993), Kolb and Turner(1990), Berezhiani et al.(1992)]. Numerically evaluating the expression from Ref. [Chang and Choi(1993)] for the axion-production rate and solving Eq. (1) for , we estimate the axion freeze-out temperature using the condition . As is lowered, axions freeze out earlier due to the higher value of , as shown in Fig. 1. As increases, the epoch becomes less relevant, and asymptotes to its standard value. Now, since [Chang and Choi(1993)], higher-mass axions keep up with the Hubble expansion for longer and generally decouple at lower temperatures.
The resulting axion abundance is [Grin et al.(2008)]
where is the axion mass in units of .
When , the present mass density in axions is severely suppressed, because of entropy generation. Using the numerical solution for , we obtain . In the right panel of Fig. 1, we show normalized by its standard value, , as a function of . For , the axion abundance asymptotes to .
4 Constraints to axions
Most constraints to the axion mass come from its two-photon coupling [Raffelt(1996), Kolb and Turner(1990), Kaplan(1985), Srednicki(1985), Kephart and Weiler(1987), Bershady et al.(1991), Turner(1987), Ressell(1991), Grin et al.(2007), Gnedin et al.(1999), Andriamonje et al.(2007), Asztalos et al.(2002)]. This coupling depends on the up-down quark mass ratio , for which there are experimentally allowed such that vanishes, and so constraints to axions from star clusters, helioscope, RF cavity, and telescope searches may all be lifted [Buckley and Murayama(2007), Moroi and Murayama(1998)]. In contrast, the hadronic couplings do not vanish for any experimentally allowed values. Axion searches based on these couplings are underway, and have already imposed the range [Ljubicic et al.(2004)]. These couplings also determine the relic abundance of axions, and so constraints may be obtained from cosmology.
Mass constraints to thermal axions from cosmology are considerably relaxed because of entropy generation. A conservative constraint is obtained by requiring that axions not exceed the matter density of [Spergel et al.(2007)] and is shown by the dot-dashed hashed region in Fig. 2. If , constraints are considerably relaxed. When , we obtain , equal to the standard result.
Axions will free stream at early times, decreasing the matter power spectrum on length scales smaller than the comoving free-streaming scale, evaluated at matter-radiation equality:
This suppression is given by if [Hannestad et al.(2005), Hannestad et al.(2007), Hannestad and Raffelt(2004)] and imposes a constraint to . Including entropy generation, the relationship between the effective axion temperature and the neutrino temperature is
Using Sloan Digital Sky Survey (SDSS) measurements of the galaxy power spectrum [Tegmark et al.(2004b)] and Wilkinson Microwave Anisotropy Probe (WMAP) [Spergel et al.(2003)] 1-year measurements of the CMB angular power spectrum, Refs. [Hannestad et al.(2005), Hannestad et al.(2007), Hannestad and Raffelt(2004)] derived limits of . We map these results into the plane.
We calculate and for axions in LTR, and thus obtain the upper limit to the axion mass as a function of , shown in Fig. 2. For this data set, the smallest length scale for which the galaxy correlation function can be reliably probed is [Grin et al.(2008), Hannestad et al.(2005), Hannestad et al.(2007)]. For , , and this axion mass constraint is lifted. At high , the constraint from LSS/CMB data () supercedes the constraint .
Future instruments, such as the Large Synoptic Survey Telescope (LSST), will measure the matter power-spectrum with unprecedented precision ( [Zhan et al.(2006), Tyson(2006)]. This order of magnitude improvement over past work [Tegmark et al.(2004), Tegmark et al.(2006)] leads to the improved sensitivity shown by the dotted line in Fig. 2. To estimate possible constraints to axions from LSST measurements of the power spectrum, we recalculated our limits using the approximate scaling , assuming for .
We also estimate the possible improvement offered by including information on smaller scales (), as probed by measurements of the Lyman- flux power spectrum [Viel et al.(2004)], also shown in Fig. 2. This is indicated by the dashed line in Fig. 2. We can see that higher and lower values are probed because of information on smaller length scales.
5 Axions, LTR, and BBN
Future limits to axions may follow from constraints to the total density in relativistic particles at . This is parameterized by the axionic contribution to the total effective neutrino number [Grin et al.(2008), Chang and Choi(1993)]:
For sufficiently high masses, the axionic contribution saturates to at high [Chang and Choi(1993)]. In Fig. 3, we show , evaluated at whichever which saturates the best cosmological bound for a given .
A comparison between the abundance of He () and the predicted abundance from BBN places constraints at MeV [Cardall and Fuller(1996)]; thus constraints to He abundances are also constraints on and . Here we apply the scaling relation [Steigman(2007)]:
Direct measurements of , including a determination of from CMB observations, lead to the 68% confidence level upper limit of [Cyburt et al.(2005), Ichikawa and Takahashi(2006), Ichikawa et al.(2008)]. From Fig. 3, we see that this bound cannot constrain or . If future measurements reduce systematic errors, constraints to will be obtained for the lighter-mass axions.
Constraints to and may follow from CMB measurements of . He affects CMB anisotropies by changing the ionization history of the universe [Trotta and Hansen(2004)]. CMBPol (a proposed future CMB polarization experiment) is expected to approach , leading to the sensitivity limit [Trotta and Hansen(2004), Kaplinghat et al.(2003), Ichikawa et al.(2008), Eisenstein et al.(1999), Baumann(2008)]. As shown in Fig. 3, for , such measurements of may impose stringent limits on the axion mass. Also, if axions with mass in the range are directly detected, might impose an upper limit to .
LTR suppresses the abundance of thermally-produced axions, once , as a result of dramatic entropy production. The cosmologically allowed window for is extended as a result. Future probes of the matter power spectrum or the primordial helium abundance may definitively explore some of this parameter space.
Acknowledgements.D.G. was supported by the Gordon and Betty Moore Foundation and thanks the organizers of DM08. T.L.S. and M.K. were supported by DoE DE-FG03-92-ER40701 and the Gordon and Betty Moore Foundation.
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