Tensor Detection Severely Constrains Axion Dark Matter
David J. E. Marsh, Daniel Grin, Renee Hlozek, Pedro G. Ferreira
.. Tensor Detection Severely Constrains Axion Dark Matter
David J. E. Marsh ∗ , Daniel Grin , Ren´ee Hlozek , and Pedro G. Ferreira Perimeter Institute, 31 Caroline St N, Waterloo, ON, N2L 6B9, Canada Department of Astronomy and Astrophysics, University of Chicago, Illinois, 60637, U.S.A. Department of Astronomy, Princeton University, Princeton, NJ 08544, USA and Astrophysics, University of Oxford, DWB, Keble Road, Oxford, OX1 3RH, UK (Dated: June 20, 2014)The recent detection of B-modes by BICEP2 has non-trivial implications for axion dark matterimplied by combining the tensor interpretation with isocurvature constraints from Planck. In thispaper the measurement is taken as fact, and its implications considered, though further experimentalverification is required. In the simplest inflation models r = 0 . H I = 1 . × GeV. Ifthe axion decay constant f a < H I / π constraints on the dark matter (DM) abundance alone ruleout the QCD axion as DM for m a (cid:46) χ / µ eV (where χ > f a > H I / π 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 backreaction of fluctuations is included. High f a 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 m a (cid:38) − eV we find Ω a / Ω d (cid:46) − , with stronger constraints on heavier axions. Lighter axions,however, are allowed and (inflationary) model-independent constraints from the CMB temperaturepower spectrum and large scale structure are stronger than those implied by tensor modes. PACS numbers: 14.80.Va,98.70.Vc,95.85.Sz,98.80.Cq
Introduction:
The recent measurement of large angleCMB B-mode polarisation by BICEP2 [1], implying atensor-to-scalar ratio r = 0 . +0 . − . has profound implica-tions for our understanding of the initial conditions ofthe universe [2], and points to an inflationary origin forthe primordial fluctuations [3–5]. The inflaton also drivesfluctuations in any other fields present in the primordialepoch and so the measurement of r , which fixes the in-flationary energy scale, can powerfully constrain diversephysics. In this work we will discuss the implicationsfor axion dark matter (DM) in the case that the tensormodes are generated during single-field slow-roll infla-tion (from now on we simply refer to this as ‘inflation’)by zero-point fluctuations of the graviton. In this workwe assume that the measured value of r both holds up tocloser scrutiny experimentally, and is taken to be of pri-mordial origin. We relax these assumptions in our closingdiscussion. We stress that our conclusions are one con-sequence of taking this measurement at face value, butalso that they apply to any detection of r .The scalar amplitude of perturbations generated dur-ing inflation is given by [7] A s = 12 (cid:15) (cid:18) H I πM pl (cid:19) = 2 . × − (1)where H I is the Hubble rate during inflation, (cid:15) = − ˙ H/H is a slow-roll parameter, and M pl = 1 / √ πG = 2 . × GeV is the reduced Planck mass. The zero-point fluctua- ∗ [email protected] tions of the graviton give rise to tensor fluctuations withamplitude A T = 8 (cid:18) H I πM pl (cid:19) , (2)so that the tensor to scalar ratio is r = A T /A s = 16 (cid:15) .The measured values of r and A s give: H I = 1 . × GeV . (3)It is this high scale of inflation that will give us strongconstraints on axion DM.Axions [8–10] were introduced as an extension to thestandard model of particle physics in an attempt todynamically solve the so-called ‘Strong- CP problem’ ofQCD. The relevant term in the action is the CP -violatingtopological term S θ = θ π (cid:90) d x(cid:15) µναβ Tr G µν G αβ , (4)where G µν is the gluon field strength tensor. The θ termimplies the existence of a neutron electric dipole moment, d n . Experimental bounds limit d n < . × − e cm [11]and imply that θ (cid:46) − . The Peccei-Quinn [8] (PQ)solution to this is to promote θ to a dynamical field, the The value of r = 0 . r atthe percent level and do not substantially alter our conclusions. a r X i v : . [ a s t r o - ph . C O ] J un axion [9, 10], which is the Goldstone boson of a sponta-neously broken global U (1) symmetry. At temperaturesbelow the QCD phase transition, QCD instantons lead toa potential and stabilise the axion at the CP -conservingvalue of θ = 0. The potential takes the form [12] V ( φ ) = Λ (1 − cos φ/f a ) . (5)The canonically normalised field is φ = f a θ , where f a isthe axion decay constant and gives the scale at which thePQ symmetry is broken. Oscillations about this potentialminimum lead to the production of axion DM [13–19] .Axions are also generic to string theory [24–26], wherethey and similar particles come under the heading ‘axion-like particles’ (e.g. Ref. [27]). Along with the QCD axionwe will also consider constraints on other axions comingfrom a measurement of r .Just as the graviton is massless during inflation, lead-ing to the production of the tensor modes, if the axion ismassless during inflation (and the PQ symmetry is bro-ken) it acquires isocurvature perturbations [28, 29] (cid:112) (cid:104) δφ (cid:105) = H I π . (6)Thus high-scale inflation as required in the simplest sce-nario giving rise to r implies large amplitude isocurvatureperturbations [30, 31].The spectrum of initial axion isocurvature density per-turbations generated by Eq. (6) is (cid:104) δ a (cid:105) = 4 (cid:42)(cid:18) δφφ (cid:19) (cid:43) = ( H I /M pl ) π ( φ i /M pl ) . (7)Given that axions may comprise but a fraction Ω a / Ω d ofthe total DM, the isocurvature amplitude is given by A I = (cid:18) Ω a Ω d (cid:19) ( H I /M pl ) π ( φ i /M pl ) . (8)The ratio of power in isocurvature to adiabatic modes isgiven by: A I A s = (cid:18) Ω a Ω d (cid:19) (cid:15) ( φ i /M pl ) . (9)These isocurvature modes are uncorrelated with theadiabatic mode. The QCD axion is indistinguishablefrom CDM on cosmological scales, and the Planck col-laboration [6] constrains uncorrelated CDM isocurvatureto contribute a fraction A I A s < . . (10) For more details see e.g. Refs. [20–23].
Given certain assumptions, in particular that the PQsymmetry is broken during inflation and that the QCDaxion makes up all of the DM, this implies the limit H I ≤ . × GeV (cid:18) f a GeV (cid:19) . , (11)which is clearly inconsistent by many orders of magnitudewith the value of Eq. (3) implied by the detection of r . The QCD Axion:
We now discuss the well known im-plications of a measurement of r as applied to the QCDaxion (e.g. [31–34]). For the QCD axion the decay con-stant is known to be in the window10 GeV (cid:46) f a (cid:46) GeV , (12)where the lower bound comes from stellar cooling [35] andthe lesser known upper bound from the spins of stellarmass black holes [36].The homogeneous component of the field φ evolves ac-cording to the Klein-Gordon equation in the expandinguniverse ¨ φ + 3 H ˙ φ + V ,φ = 0 . (13)Once Hubble friction is overcome, the field oscillates inits potential minimum, with the energy density scalingas matter, and provides a source of DM in this ‘vacuumrealignment’ production. There are various possibilitiesto set the axion relic density, depending on whether thePQ symmetry is broken or not during inflation.The relic density due to vacuum realignment is givenby Ω a h ∼ × (cid:18) f a GeV (cid:19) / (cid:104) θ i (cid:105) γ , (14)where angle brackets denote spatial averaging of the shortwavelength fluctuations [39], 0 < γ < , and we have droppedthe factor f ( θ i ) accounting for anharmonic effects forsimplicity.The PQ symmetry is broken during inflation if f a >H I / π and then the homogeneous component of θ is a We note that for 10 GeV (cid:46) f a (cid:46) GeV there is no ex-actly known expression for Ω a when oscillations begin duringthe QCD phase transition (e.g. [31, 40]). Also, in order for largeentropy production to be possible oscillations must begin in amatter dominated era, giving another slightly different expres-sion (which can be absorbed into γ ) [41]. More rigorously the condition is [32] f a > Max { T GH , T max } where T GH is the Gibbons-Hawking temperature of de Sitterspace during inflation, T GH = H I / π [37, 38] and T max is themaximum thermalisation temperature after inflation, T max = γ eff E I ( γ eff is an efficiency parameter and E I = 3 / (cid:112) M pl H I ). free parameter in each horizon volume. Even in the sim-plest case where (cid:104) θ i (cid:105) ∼ ¯ θ i , then for large f a ∼ GeVEq. (14) already implies a modest level of fine tuning to θ i ∼ − if the axion is not to overclose the universe, ρ a > ρ crit , where ρ crit is the critical density for flatness.However, this fine tuning is easy to accommodate in theso-called ‘anthropic axion window’ [32].Combining Eqs. (10), and (14) with the measured valueof r and setting Ω d h = 0 .
119 [7], the tensor and isocur-vature constraints put an upper limit on the axion DMfraction ofΩ a, QCD Ω d (cid:46) × − γ (cid:18) f a GeV (cid:19) / (cid:18) . r (cid:19) (cid:18) Ω d h . (cid:19) . (15)This constraint essentially rules out the high- f a QCD ax-ion as a DM candidate, showing the far reaching impli-cations of the measurement of r . Barring an impossiblyhuge [31] dilution of axion energy density, γ (cid:28)
1, thissmall abundance gives an upper limit on the QCD axioneffective initial misalignment angle (cid:104) θ i (cid:105) (cid:46) × − γ (cid:18) f a GeV (cid:19) − / (cid:18) Ω d h . (cid:19) (cid:18) . r (cid:19) . (16)In low f a models the axion does not acquire isocur-vature perturbations since the field is not establishedwhen the PQ symmetry is unbroken. Therefore with low- f a there is no additional constraint on axions derivedfrom combining the measurement of r with the boundon A I /A s , other than setting the scale for this scenario.When the PQ symmetry is broken after inflation, theaxion field varies on cosmologically small scales with av-erage (cid:104) θ (cid:105) = π /
3, which should be used in Eq. (14) tocompute the relic abundance. The requirement of notoverproducing DM, Ω a h < . f a to f a < . × χ − / GeV [32] where χ can vary by an order of magnitude or more and ac-counts for theoretical uncertainties (including productionfrom string decay) . For low f a there are relics of the PQtransition no longer diluted by inflation [21]. While do-main walls are problematic, string decay can be the dom-inant source of axion DM in this scenario. The case oflow f a axions has been discussed extensively elsewhere,and we discuss them no further here. Ultra-light Axions:
In this section we further developthe ideas presented in Ref. [44] and show an estimate of the combined constraints on axion parameter spacefrom isocurvature, a confirmed detection of r , and othercosmological constraints of Ref. [45].Ultra-light axions are motivated by string theory con-siderations, with the mass scaling exponentially with the See e.g. Ref. [40] where it is argued that the value of f a assumingno string contribution, χ = 1, still gives a useful benchmark forthe excluded masses. CMB+LSS
CMB (WMAP1) log ( m a / eV) ⌦ a ⌦ d - - - FIG. 1: Constraints in axion parameter space: regions be-low curves are allowed. The solid red line shows the re-sult of the present work which constrains axions using themeasured value of r = 0 . +0 . − . shown in thin lines) andthe Planck constraint on axion isocurvature, A I /A s < . m a < H eq . We also show the 95% exclusion contours ofRef. [45] from CMB (WMAP1) and CMB+Lyman-alpha for-est power spectra, which are significantly stronger than thetensor/isocurvature constraint for intermediate mass axions,and are independent of the inflationary model. moduli [26], or simply by a Jeffreys prior on this un-known parameter. They differ from the QCD axion inthat they need not couple to QCD, or indeed the stan-dard model. For such a generic axion the temperaturedependence of the mass cannot be known, as the massesarise from non-perturbative effects in hidden sectors. Aslong as the mass has reached its zero-temperature valueby the time oscillations begin, the relic abundance dueto vacuum realignment is given byΩ a ≈ a H m a (cid:42)(cid:18) φ i M pl (cid:19) (cid:43) , (17)where a osc is the scale factor defined by 3 H ( a osc ) = m a when oscillations begin: it can be approximated by usingthe Friedmann equation and assuming an instantaneoustransition in the axion equation of state from w a = − w a = 0 at a osc . When m a (cid:46) − eV the relic abun-dance cannot be significant unless f a (cid:38) GeV > H I and therefore in what follows we consider only the casewhere the PQ symmetry is broken during inflation . For a single axion this is true, but for many axions, as in the
Pressure perturbations in axions can be described us-ing a scale-dependent sound speed, leading to a Jeansscale below which density perturbations are suppressed[26, 45–49]. When the mass is in the range 10 − eV (cid:46) m a (cid:46) − eV this scale can be astrophysical or cos-mological in size and therefore can be constrained us-ing the CMB power spectrum and large-scale structure(LSS) measurements [45, 50, 51]. The size of the effectis fixed by the fraction of DM in axions, Ω a / Ω d , andso constraints are presented in the ( m a , Ω a / Ω d ) plane.Constraints from the CMB are particularly strong for m a (cid:46) H eq ∼ − eV where the axions roll in their po-tential after equality, shifting equality and giving rise toan Integrated Sachs-Wolfe (SW) effect from the evolvinggravitational potential [50].Light axions also carry their own isocurvature pertur-bations [44], with the spectrum Eq. (7). Fixing the ini-tial field displacement in terms of the DM contributionfrom Eq. (17) allows us to place a constraint across the( m a , Ω a / Ω d ) plane given by the measured value of r andthe Planck constraint on A I /A s . The measured valueof r restricts the allowed values of Ω a to be small. Weshow this constraint with the solid red line on Fig. 1,along with the CMB (WMAP1) and LSS (Lyman-alphaforest) constraints of Ref. [45]. Regions below curves areallowed.The Planck constraints on axion isocurvature applyonly to the case where the axions are indistinguishablefrom CDM, however the suppression of power due to ax-ion pressure shows up also in the isocurvature powerfor low masses [44] and the Planck constraints cannotbe applied. Work on constraining this mode is ongo-ing [51]. The CMB isocurvature constraint is drivenby the SW plateau. As the axionic Jeans scale crossesinto the SW plateau at low mass and suppresses theisocurvature transfer function [44], the signal-to-noise SN R ∝ /l max , where l max ∼ l Jeans ∼ √ m a . There-fore we estimate that the isocurvature limit is given by( A I /A s ) max ∝ ( A I /A s ) maxold × (cid:112) − eV /m a . This esti-mate is used to obtain the dashed line in Fig. 1.Fig. 1 shows the huge power of the measurement of r to constrain axions, giving Ω a / Ω d < − for m a (cid:38) − eV, far beyond the reach even of the Lyman-alphaforest constraints. For m a (cid:46) − eV, however, theconstraints from the CMB temperature and E-mode po-larisation and LSS (WMAP1 and SDSS [45], Planck andWiggleZ in preparation [51]) are stronger than the ten-sor/isocurvature constraint, and are independent of theinflationary interpretation of BICEP2. Ruling out axions:
Spatial averaging of short wave-length modes gives rise to an irreducible back-reactioncontribution to (cid:104) φ (cid:105) and thus Ω a . If the required smallvalues cannot be obtained, the corresponding axion is axiverse [26], an N-flation type scenario for DM could be relevant. ruled out. Specifically (cid:104) φ (cid:105) = ¯ φ + σ φ = ¯ φ + (cid:104) δφ (cid:105) . (18)The mean homogeneous value, ¯ φ , can be tuned or dy-namically made arbitrarily small (e.g. via coupling toa tracking field [42, 43]); fixing ¯ φ = 0 gives the ir-reducible contribution to Ω a from fluctuations. Plug-ging the variance into Eq. (16) we find that the QCDaxion with f a > H I / π is totally ruled out [31] (un-less also f a (cid:29) M pl ), further taking the low f a valueabove this rules out m a (cid:46) χ / µ eV. Applying thisto the ultra-light axion abundance in Eq. (17) we findthat Ω a / Ω d < − over the entire range of masses weconsider, which is always below the amount necessary tosatisfy the tensor plus isocurvature constraint, and thusno ultra-light axions are completely excluded. This is be-cause order Planckian field displacements are necessaryfor non-negligible abundance in ultra-light axions, while H I < M pl sources the fluctuation contribution. Discussion:
We have considered the implications ofthe BICEP2 detection of r on axion DM. In the simplestinflation models r = 0 . H I = 1 . × GeV.Axions with f a > H I / π acquire isocurvature perturba-tions and are constrained strongly by the Planck bound A I /A s < .
04. All such high f a QCD axions are ruledout. Even if they can exist (by somehow suppressingthe fluctuation contribution to the abundance), evadingisocurvature bounds will require searches for them to beindependent of the DM abundance [52]. In the general,non-QCD, case low f a < H I / π axions [53] are unaf-fected by the tensor bound. High f a axions [26, 41] arestrongly constrained, although for m a (cid:46) − eV sup-pression of power in the isocurvature mode can loosenconstraints [44]. One may consider the high- f a ultra-light axions ‘guilty by association’ to the QCD axion,but this is a model-dependent statement and axion hi-erarchies are certainly possible [54] and indeed desirableif the inflaton is also an axion, as many high H I modelsdemand.There are in principle (at least) five ways around theisocurvature bounds. The first is to produce gravita-tional waves during inflation giving r = 0 . H I low [55, 56]. Secondly, entropy production af-ter the QCD phase transition can dilute the QCD axionabundance. This is possible in models with light mod-uli and low temperature reheating (e.g. [57] and refer-ences therein). Light axions oscillate after nucleosyn-thesis and cannot be diluted by such effects. Thirdly,if the axions are massive during inflation they acquireno isocurvature, although a shift symmetry protects ax-ion masses. Fourthly, non-trivial axion dynamics duringinflation suppressing isocurvature are possible e.g. vianon-minimal coupling to gravity [58] or coupling the in-flation directly to the sector providing non-perturbativeeffects, e.g. the QCD coupling [59, 60]. Such couplingsmay alter the adiabatic spectrum and produce observ-able signatures through production of primordial blackholes. Finally coupling a light ( m a (cid:46) − eV) axionto (cid:126)E · (cid:126)B of electromagnetism could induce ‘cosmologicalbirefringence’ [61] leading to production of B-modes thatare not sourced by gravitational waves [26, 62]. This pos-sibility will be easy to distinguish from tensor and lensingB-modes by its distinctive oscillatory character at high (cid:96) , measurable for example by SPTPol and ACTPol.Other cosmological constraints on axions are morepowerful than the tensor/isocurvature bound for lightmasses m a (cid:46) − [45, 51]. We are exploring thismass range with a careful search of parameter space us-ing nested sampling [44]. Isocurvature constraints willimprove in the future [63], as will constraints on Ω a / Ω d [50], both of which could allow for a detection consistentwith the tensor bound [44]. In the regime m a (cid:38) − eVthe tensor bound is stronger than current cosmologicalbounds on Ω a . However, in this regime axions can playa role in resolving issues with galaxy formation if theyare dominant in DM [49]. Future weak lensing surveyswill cut into this regime [64] and surpass the indirecttensor bound. If these axions are necessary/detected inlarge scale structure this would imply either contradic- tion with the tensor bound, or other new physics duringinflation. The same is true for direct detection of a high f a QCD axion DM [65].
Note added in proof:
The related paper Ref. [66] refer-ring to the QCD axion has also recently appeared.
Acknowledgments
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