Tensor Detection Severely Constrains Axion Dark Matter
The recent detection of B-modes by BICEP2 has non-trivial implications for axion dark matter implied by combining the tensor interpretation with isocurvature constraints from Planck. In this paper the measurement is taken as fact, and its implications considered, though further experimental verification is required. In the simplest inflation models implies . If the axion decay constant constraints on the dark matter (DM) abundance alone rule out the QCD axion as DM for (where accounts for theoretical uncertainty). If then vacuum fluctuations of the axion field place conflicting demands on axion DM: isocurvature constraints require a DM abundance which is too small to be reached when the back reaction of fluctuations is included. High QCD axions are thus ruled out. Constraints on axion-like particles, as a function of their mass and DM fraction, are also considered. For heavy axions with we find , with stronger constraints on heavier axions. Lighter axions, however, are allowed and (inflationary) model-independent constraints from the CMB temperature power spectrum and large scale structure are stronger than those implied by tensor modes.
Introduction: The recent measurement of large angle CMB B-mode polarisation by BICEP2 bicep (), implying a tensor-to-scalar ratio has profound implications for our understanding of the initial conditions of the universe lyth1997 (), and points to an inflationary origin for the primordial fluctuations guth1981 (); linde1982 (); albrecht1982 (). The inflaton also drives fluctuations in any other fields present in the primordial epoch and so the measurement of , which fixes the inflationary energy scale, can powerfully constrain diverse physics. In this work we will discuss the implications for axion dark matter (DM) in the case that the tensor modes are generated during single-field slow-roll inflation (from now on we simply refer to this as ‘inflation’) by zero-point fluctuations of the graviton. In this work we assume that the measured value of both holds up to closer scrutiny experimentally, and is taken to be of primordial origin. We relax these assumptions in our closing discussion. We stress that our conclusions are one consequence of taking this measurement at face value, but also that they apply to any detection of .
The scalar amplitude of perturbations generated during inflation is given by planck2013cosmo ()
where is the Hubble rate during inflation, is a slow-roll parameter, and GeV is the reduced Planck mass. The zero-point fluctuations of the graviton give rise to tensor fluctuations with amplitude
so that the tensor to scalar ratio is 222The value of is in slight tension with current temperature measurements. Increasing the damping in the tail, or violating slow roll helps reduce the tension, albeit in an ad hoc fashion. bicep (); planck2013inflation (). The corrections affect isocurvature amplitudes and at the percent level and do not substantially alter our conclusions.. The measured values of and give:
It is this high scale of inflation that will give us strong constraints on axion DM.
Axions pecceiquinn1977 (); wilczek1978 (); weinberg1978 () were introduced as an extension to the standard model of particle physics in an attempt to dynamically solve the so-called ‘Strong-CP problem’ of QCD. The relevant term in the action is the CP-violating topological term
where is the gluon field strength tensor. The term implies the existence of a neutron electric dipole moment, . Experimental bounds limit cm baker2006 () and imply that . The Peccei-Quinn pecceiquinn1977 () (PQ) solution to this is to promote to a dynamical field, the axion wilczek1978 (); weinberg1978 (), which is the Goldstone boson of a spontaneously broken global symmetry. At temperatures below the QCD phase transition, QCD instantons lead to a potential and stabilise the axion at the CP-conserving value of . The potential takes the form gross1981 ()
The canonically normalised field is , where is the axion decay constant and gives the scale at which the PQ symmetry is broken. Oscillations about this potential minimum lead to the production of axion DM turner1983b (); turner1983 (); dine1983 (); abbott1983 (); preskill1983 (); turner1986 (); berezhiani1992 ()333For more details see e.g. Refs. Kim:1986ax (); book:kolb_and_turner (); raffelt2001 (); sikivie2008 ().. Axions are also generic to string theory witten1984 (); witten2006 (); axiverse2009 (), where they and similar particles come under the heading ‘axion-like particles’ (e.g. Ref. ringwald2012b ()). Along with the QCD axion we will also consider constraints on other axions coming from a measurement of .
Just as the graviton is massless during inflation, leading to the production of the tensor modes, if the axion is massless during inflation (and the PQ symmetry is broken) it acquires isocurvature perturbations axenides1983 (); seckel1985 ()
The spectrum of initial axion isocurvature density perturbations generated by Eq. (6) is
Given that axions may comprise but a fraction of the total DM, the isocurvature amplitude is given by
The ratio of power in isocurvature to adiabatic modes is given by:
These isocurvature modes are uncorrelated with the adiabatic mode. The QCD axion is indistinguishable from CDM on cosmological scales, and the Planck collaboration planck2013inflation () constrains uncorrelated CDM isocurvature to contribute a fraction
Given certain assumptions, in particular that the PQ symmetry is broken during inflation and that the QCD axion makes up all of the DM, this implies the limit
which is clearly inconsistent by many orders of magnitude with the value of Eq. (3) implied by the detection of .
The QCD Axion: We now discuss the well known implications of a measurement of as applied to the QCD axion (e.g. fox2004 (); hertzberg2008 (); 5yearWMAP (); mack2009a ()). For the QCD axion the decay constant is known to be in the window
The homogeneous component of the field evolves according to the Klein-Gordon equation in the expanding universe
Once Hubble friction is overcome, the field oscillates in its potential minimum, with the energy density scaling as matter, and provides a source of DM in this ‘vacuum realignment’ production. There are various possibilities to set the axion relic density, depending on whether the PQ symmetry is broken or not during inflation.
The relic density due to vacuum realignment is given by
where angle brackets denote spatial averaging of the short wavelength fluctuations lyth1992 (), is a dilution factor if entropy is produced sometime after the QCD phase transition and before nucleosynthesis (for example by decay of a weakly coupled modulus)444We note that for there is no exactly known expression for when oscillations begin during the QCD phase transition (e.g. fox2004 (); wantz2009 ()). Also, in order for large entropy production to be possible oscillations must begin in a matter dominated era, giving another slightly different expression (which can be absorbed into ) acharya2010a ()., and we have dropped the factor accounting for anharmonic effects for simplicity.
The PQ symmetry is broken during inflation555More rigorously the condition is hertzberg2008 () where is the Gibbons-Hawking temperature of de Sitter space during inflation, gibbons1977 (); bunch1978 () and is the maximum thermalisation temperature after inflation, ( is an efficiency parameter and ). if and then the homogeneous component of is a free parameter in each horizon volume. Even in the simplest case where , then for large Eq. (14) already implies a modest level of fine tuning to if the axion is not to overclose the universe, , where is the critical density for flatness. However, this fine tuning is easy to accommodate in the so-called ‘anthropic axion window’ hertzberg2008 ().
This constraint essentially rules out the high- QCD axion as a DM candidate, showing the far reaching implications of the measurement of . Barring an impossibly huge fox2004 () dilution of axion energy density, , this small abundance gives an upper limit on the QCD axion effective initial misalignment angle
In low models the axion does not acquire isocurvature perturbations since the field is not established when the PQ symmetry is unbroken. Therefore with low- there is no additional constraint on axions derived from combining the measurement of with the bound on , other than setting the scale for this scenario. When the PQ symmetry is broken after inflation, the axion field varies on cosmologically small scales with average , which should be used in Eq. (14) to compute the relic abundance. The requirement of not overproducing DM, , then limits the maximum value of to hertzberg2008 () where can vary by an order of magnitude or more and accounts for theoretical uncertainties (including production from string decay)666See e.g. Ref. wantz2009 () where it is argued that the value of assuming no string contribution, , still gives a useful benchmark for the excluded masses.. For low there are relics of the PQ transition no longer diluted by inflation book:kolb_and_turner (). While domain walls are problematic, string decay can be the dominant source of axion DM in this scenario. The case of low axions has been discussed extensively elsewhere, and we discuss them no further here.
Ultra-light Axions: In this section we further develop the ideas presented in Ref. marsh2013 () and show an estimate of the combined constraints on axion parameter space from isocurvature, a confirmed detection of , and other cosmological constraints of Ref. amendola2005 ().
Ultra-light axions are motivated by string theory considerations, with the mass scaling exponentially with the moduli axiverse2009 (), or simply by a Jeffreys prior on this unknown parameter. They differ from the QCD axion in that they need not couple to QCD, or indeed the standard model. For such a generic axion the temperature dependence of the mass cannot be known, as the masses arise from non-perturbative effects in hidden sectors. As long as the mass has reached its zero-temperature value by the time oscillations begin, the relic abundance due to vacuum realignment is given by
where is the scale factor defined by when oscillations begin: it can be approximated by using the Friedmann equation and assuming an instantaneous transition in the axion equation of state from to at . When the relic abundance cannot be significant unless and therefore in what follows we consider only the case where the PQ symmetry is broken during inflation777For a single axion this is true, but for many axions, as in the axiverse axiverse2009 (), an N-flation type scenario for DM could be relevant..
Pressure perturbations in axions can be described using a scale-dependent sound speed, leading to a Jeans scale below which density perturbations are suppressed hu2000 (); amendola2005 (); axiverse2009 (); marsh2010 (); park2012 (); marsh2013b (). When the mass is in the range this scale can be astrophysical or cosmological in size and therefore can be constrained using the CMB power spectrum and large-scale structure (LSS) measurements amendola2005 (); marsh2011b (); marsh_etal_inprep (). The size of the effect is fixed by the fraction of DM in axions, , and so constraints are presented in the plane. Constraints from the CMB are particularly strong for where the axions roll in their potential after equality, shifting equality and giving rise to an Integrated Sachs-Wolfe (SW) effect from the evolving gravitational potential marsh2011b ().
Light axions also carry their own isocurvature perturbations marsh2013 (), with the spectrum Eq. (7). Fixing the initial field displacement in terms of the DM contribution from Eq. (17) allows us to place a constraint across the plane given by the measured value of and the Planck constraint on . The measured value of restricts the allowed values of to be small. We show this constraint with the solid red line on Fig. 1, along with the CMB (WMAP1) and LSS (Lyman-alpha forest) constraints of Ref. amendola2005 (). Regions below curves are allowed.
The Planck constraints on axion isocurvature apply only to the case where the axions are indistinguishable from CDM, however the suppression of power due to axion pressure shows up also in the isocurvature power for low masses marsh2013 () and the Planck constraints cannot be applied. Work on constraining this mode is ongoing marsh_etal_inprep (). The CMB isocurvature constraint is driven by the SW plateau. As the axionic Jeans scale crosses into the SW plateau at low mass and suppresses the isocurvature transfer function marsh2013 (), the signal-to-noise , where . Therefore we estimate that the isocurvature limit is given by . This estimate is used to obtain the dashed line in Fig. 1.
Fig. 1 shows the huge power of the measurement of to constrain axions, giving for , far beyond the reach even of the Lyman-alpha forest constraints. For , however, the constraints from the CMB temperature and E-mode polarisation and LSS (WMAP1 and SDSS amendola2005 (), Planck and WiggleZ in preparation marsh_etal_inprep ()) are stronger than the tensor/isocurvature constraint, and are independent of the inflationary interpretation of BICEP2.
Ruling out axions: Spatial averaging of short wavelength modes gives rise to an irreducible back-reaction contribution to and thus . If the required small values cannot be obtained, the corresponding axion is ruled out. Specifically
The mean homogeneous value, , can be tuned or dynamically made arbitrarily small (e.g. via coupling to a tracking field marsh2011 (); marsh2012 ()); fixing gives the irreducible contribution to from fluctuations. Plugging the variance into Eq. (16) we find that the QCD axion with is totally ruled out fox2004 () (unless also ), further taking the low value above this rules out . Applying this to the ultra-light axion abundance in Eq. (17) we find that over the entire range of masses we consider, which is always below the amount necessary to satisfy the tensor plus isocurvature constraint, and thus no ultra-light axions are completely excluded. This is because order Planckian field displacements are necessary for non-negligible abundance in ultra-light axions, while sources the fluctuation contribution.
Discussion: We have considered the implications of the BICEP2 detection of on axion DM. In the simplest inflation models bicep () implies . Axions with acquire isocurvature perturbations and are constrained strongly by the Planck bound . All such high QCD axions are ruled out. Even if they can exist (by somehow suppressing the fluctuation contribution to the abundance), evading isocurvature bounds will require searches for them to be independent of the DM abundance arvanitaki2014 (). In the general, non-QCD, case low axions cicoli2012c () are unaffected by the tensor bound. High axions axiverse2009 (); acharya2010a () are strongly constrained, although for suppression of power in the isocurvature mode can loosen constraints marsh2013 (). One may consider the high- ultra-light axions ‘guilty by association’ to the QCD axion, but this is a model-dependent statement and axion hierarchies are certainly possible Kim:2004rp () and indeed desirable if the inflaton is also an axion, as many high models demand.
There are in principle (at least) five ways around the isocurvature bounds. The first is to produce gravitational waves during inflation giving while keeping low senatore2011 (); cook2011 (). Secondly, entropy production after the QCD phase transition can dilute the QCD axion abundance. This is possible in models with light moduli and low temperature reheating (e.g. iliesiu2013 () and references therein). Light axions oscillate after nucleosynthesis and cannot be diluted by such effects. Thirdly, if the axions are massive during inflation they acquire no isocurvature, although a shift symmetry protects axion masses. Fourthly, non-trivial axion dynamics during inflation suppressing isocurvature are possible e.g. via non-minimal coupling to gravity folkerts2013 () or coupling the inflation directly to the sector providing non-perturbative effects, e.g. the QCD coupling dvali1995 (); jeong2013b (). Such couplings may alter the adiabatic spectrum and produce observable signatures through production of primordial black holes. Finally coupling a light () axion to of electromagnetism could induce ‘cosmological birefringence’ carroll1990 () leading to production of B-modes that are not sourced by gravitational waves pospelov2008 (); axiverse2009 (). This possibility will be easy to distinguish from tensor and lensing B-modes by its distinctive oscillatory character at high , measurable for example by SPTPol and ACTPol.
Other cosmological constraints on axions are more powerful than the tensor/isocurvature bound for light masses amendola2005 (); marsh_etal_inprep (). We are exploring this mass range with a careful search of parameter space using nested sampling marsh2013 (). Isocurvature constraints will improve in the future hamann2009 (), as will constraints on marsh2011b (), both of which could allow for a detection consistent with the tensor bound marsh2013 (). In the regime the tensor bound is stronger than current cosmological bounds on . However, in this regime axions can play a role in resolving issues with galaxy formation if they are dominant in DM marsh2013b (). Future weak lensing surveys will cut into this regime lensing_inprep () and surpass the indirect tensor bound. If these axions are necessary/detected in large scale structure this would imply either contradiction with the tensor bound, or other new physics during inflation. The same is true for direct detection of a high QCD axion DM budker2013 ().
Note added in proof: The related paper Ref. visinelli2014 () referring to the QCD axion has also recently appeared.
Acknowledgements.We are especially grateful to the anonymous referee, whose suggestions greatly improved the manuscript. We are grateful to Luca Amendola for providing us with the contour constraints of Ref. amendola2005 (), and to Asimina Arvanitaki, Piyush Kumar and Maxim Pospelov for discussions. PGF acknowledges support from STFC, BIPAC and the Oxford Martin School. DG is funded at the University of Chicago by a National Science Foundation Astronomy and Astrophysics Postdoctoral Fellowship under Award NO. AST-1302856. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
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