Recognising Axionic Dark Matter by Compton and de-Broglie Scale ...

Recognising Axionic Dark Matter by Compton and de-Broglie Scale Modulation of Pulsar Timing. Ivan De Martino,1, ∗ Tom Broadhurst,1, 2, † S.-H. Henry Tye,3, ‡ Tzihong Chiueh,4, 5, § Hsi-Yu Schive,6, ¶ and Ruth Lazkoz1, ∗∗

arXiv:1705.04367v1 [astro-ph.CO] 11 May 2017

1

Department of Theoretical Physics, University of the Basque Country UPV/EHU, E-48080 Bilbao, Spain 2 Ikerbasque, Basque Foundation for Science, E-48011 Bilbao, Spain 3 Institute for Advanced Study and Department of Physics, Hong Kong University of Science and Technology, Hong Kong 4 Department of Physics, National Taiwan University, Taipei 10617, Taiwan 5 National Center for Theoretical Sciences, National Taiwan University, Taipei 10617, Taiwan 6 National Center for Supercomputing Applications, Urbana, IL 61801, USA (Dated: May 15, 2017) Light Axionic Dark Matter, motivated by string theory, is increasingly favored for the “no-WIMP era”. Galaxy formation is suppressed below a Jeans scale, of ≃ 108 M⊙ by setting the axion mass to, mB ∼ 10−22 eV, and the large dark cores of dwarf galaxies are explained as solitons on the de-Broglie scale. This is persuasive, but detection of the inherent scalar field oscillation at the Compton frequency, ωB = (2.5 months)−1 (mB /10−22 eV ), would be definitive. By evolving the coupled Schr¨ odinger-Poisson equation for a Bose-Einstein condensate, we predict the dark matter is fully modulated by de-Broglie interference, with a dense soliton core of size ≃ 150pc, at the Galactic center. The oscillating field pressure induces General Relativistic time dilation in proportion to the local dark matter density and pulsars within this dense core have detectably large timing residuals, of ≃ 400nsec/(mB /10−22 eV ). This is encouraging as many new pulsars should be discovered near the Galactic center with planned radio surveys. More generally, over the whole Galaxy, differences in dark matter density between pairs of pulsars imprints a pairwise Galactocentric signature that can be distinguished from an isotropic gravitational wave background. PACS numbers: 03.75.Lm, 95.35.+d, 98.56.Wm, 98.62.Gq

INTRODUCTION

Axion is a compelling choice for extending the standard model in particle physics, naturally generating oscillating dark matter with symmetry broken by the simple misalignment mechanism [1–3]. In particular, such fields are a generic feature of string theory arising in the dynamical compactification to 4 space-time dimensions for describing our Universe, so that multiple stabilized, complex scalar fields naturally appear, many of them contain an axionic mode (e. g. Ref [4]). Such an axion is effectively massless during the hot big bang epoch and as the Universe cools below some critical temperature it rolls down a small non-pertubatively generated potential, oscillating about the minimum, corresponding to a zeromomentum axion that shares the same wave function. It is argued that very light axions can be very natural in this context [5] and furthermore, in a ”string theory landscape”, the cosmological constant (dark energy) may also be naturally small and accompanied by correspondingly very light scalar bosons [6]. Coherent axion Compton oscillation at the frequency mB results in oscillating field pressure at a 2mB frequency but a static energy density to the leading order. Over many periods, the pressure averages to zero resembling standard cold dark matter. To the next order, the axion pressure is scale dependent, being higher at small scale, and therefore axions mimic cold dark matter on

large scales but as hot dark matter on small scale. At a certain scale, self-gravity is balanced by the field pressure, yielding a static, centrally located and highly nonlinear density peak, or soliton. The soliton scale depends on the gravitational potential depth that is 150 pc for the Milky Way [15], much smaller than the size of the Galaxy. Throughout the Galaxy, gravity is balanced by the axion pressure in form of density “granules” of similar scale to the soliton and these density fluctuations are the interference of many large-amplitude de Broglie waves and are unstable, regenerating themselves with a lifetime of about 1 Myr for our Galaxy [17].The Compton oscillation leads to an oscillating gravitational potential which can affect pulsar timing measurements [7] and the coherence of this field is limited by the de-Broglie scale. In this paper, we calculate how that these modulations affect the relative timing between pulsars, and the Earth clock with an approximately Galactocentric density dependence which is fully modulated by the interference and very large within the central soliton predicted within all galaxies in this context. This is an important feature that is absent in Ref [7], which if observed would directly support this very light bosonic dark matter scenario. The parameters used are as follows: Hubble constant H0 = 70 km/sec/Mpc and the critical density ρc = 6 GeV m−3 , the present dark matter density ρ0 = 1.5 GeV m−3 .

2 COMPTON SCALE PRESSURE OSCILLATION AND PULSAR TIMING

For a single Scalar field with inherent harmonic oscillation of the potential we have: φ(x, t) = A(x) cos(mB t + α(x)), where mB is the mass of the associated scalar boson. From the energy-momentum tensor one obtains an oscillating pressure p(x, t) = − 21 m2B A2 cos (ω t + 2α) with frequency ω = 2πν = 2mB . These oscillations are usually neglected as the average pressure over the period is zero. However, at the Compton scale of interest here, the oscillating scalar field generates an oscillating gravitational potential that is imprinted on pulsar-timing residuals in proportion to the local boson mass density [7]. It is important to appreciate that all clocks are modulated by an oscillating scalar field, include Earth clocks, so cancellation of measured pulsar timing residuals is feasible, and likewise when pairs of clocks/pulsars are maximally out of phase then their combined relative timing residual is enhanced. It is customary to write the time dependent part of the time residuals as the relative frequency shift of the pulse δt(t) = −

Z

t

0

ν(t′ ) − ν0 ′ dt , ν0

(1)

where ν(t) is the frequency of the pulse at the detector at the time t, and ν0 is the pulse emission frequency at the pulsar. In order to rewrite the eq. (1) in term of the oscillating gravitational potential one needs to consider the linearized Einstein equations in the Newtonian gauge with the two potentials h00 = 2Φ and hij = −2Ψδij . Then, the frequency shift can be recast as [8] ν(t) − ν0 ≈ Ψ(xp , t0 ) − Ψ(x, t). ν0

(2)

Thus, to compute the frequency shift one can consider only the oscillating contribution of the potential Ψ. Therefore, from the spatial components of the linearized Einstein equations one get the amplitude of the oscillating part to be Ψc (x) = π

GρDM (x) . m2B

FIG. 1: The left panel shows light axionic density galaxy profiles of for a range of boson mass, mB , for the massive simulated halo shown on the right (visualized with the yt package [10]), where the granular de-Broglie scale structure is visible on 100 pc scales, including the dense central soliton - a stable standing wave centered on the potential minimum. Also indicated is an NFW profile as a blue dashed curve, which fits well the azimuthally averaged density profile.

(3)

We stress here that the amplitude of the oscillation will vary on the sky as the density of dark matter in the Galaxy has a modulated variation with a mean radial decline that is Navarro-Frenk-White (NFW) like [9] (see Figure 1) which locally affects the strength of pulsar timing as above. Plugging eqs. (2) and (3) into (1), one obtain the timedependent part of the time residuals for the i − th pulsar by taking the fluctuation with respect to the average sig-

nal (∆ti = δti − < δti >)    1 ωDi ∆ti (t) = Ψ(xi ) sin ωt − + 2αi − ω c  − Ψ(xe ) sin (ωt + 2αe ) ,

(4)

where αi ≡ α(xi ) and Di are the phase and the distance to the ith pulsar, and αe ≡ α(xe ) is the phase of the Earth clock. In order to maximize the effect of the oscillating gravitational potential and avoid the same dependence of Earth clocks on the local density of axionic matter, we may study the difference of the time residuals between two pulsars. The differences in such parities timing residuals will be enhanced by the modulation of the axion density field on the de-Broglie scale throughout the Galaxy, as shown in Figures 1& 2, as our simulations have demonstrated that the density structure of virialized halos is predicted to be fully modulated on the de-Broglie scale, representing the macroscopic interference patterns of a Bose-Einstein Condensate, as we discuss below.

de-BROGLIE SCALE GALACTIC STRUCTURE

Bosonic Dark Matter, such as Axions, if sufficiently light can satisfy the ground state condition, where the deBroglie wavelength exceeds the mean particle separation set by the observed density of dark matter. This state can be simply described by a coupled Schrdinger-Poisson

3 equation, by analogy with the Gross-Pitaeviski coupled equations for a Bose-Einstein condensate. Specifically, its functional form in comoving coordinates can be expressed as   ∂ ∇2 i + − aV ψ = 0 , (5) ∂τ 2 ∇2 V = 4π(|ψ|2 − 1) ,

(6)

where ψ is the wave function, V is the gravitation potential and a is the cosmological scale factor. The system is normalized to the time scale dτ = χ1/2 a−2 dt, and to the scale length ξ = χ1/4 (mB /~)1/2 x, where χ = 23 H02 Ω0 and mB indicates the particle mass, while H0 and Ω0 represent the present Hubble and dark matter density parameters, respectively [11]. The simplest case of no self-interaction was first advocated by Hu et al. [12] for which the boson mass is the only free parameter, termed “Fuzzy Dark Matter” with further analytical work in relation to dwarf galaxies [13, 14]. The first cosmological simulations in this context, dubbed ψDM , by [15] have uncovered a rich nonlinear structure by accurately solving the above equation evolved with an initial standard power spectrum truncated at the Jeans scale. These pioneering simulations have revealed unpredicted small scale structure on the de-Broglie scale and established that a stable, solitonic core forms around the potential minimum within each virialized halo, providing a natural explanation for the dark matter dominated cores of dwarf Spheroidal galaxies [15]. The central soliton is predicted to be surrounded by an extended halo with “ granular” texture on the deBroglie scale [16] and shown in Figure 1, which when azimuthally averaged follows the form derived from N-body simulations of purely collisionless CDM parameterized by the well known NFW profile [15]. The identification of a centrally stable soliton with the large cores of dSph galaxies has allowed the boson mass to be estimated with little model dependence, using the inverse relation between the width of the soliton and the boson mass [15, 17, 18]. The best constraint comes form the well studied Fornax dwarf spheroidal (dSph), with an estimated halo mass is 4 × 109 M⊙ , yielding the soliton peak density is 2 GeV cm−3 and core width 1 kpc [15] and mB = 0.8 × 10−22 eV. Similar but somewhat larger values of mB are derived using dSph galaxies from the SDSS survey [19]. This can then be used to obtain the soliton scale expected for the Milky Way by using the scaling between Halo mass and Soliton mass scal4/3 −1/3 ing law: ρpeak ∝ Mhalo and rc ∝ Mhalo derived from simulations [15, 17], which for our Galaxy with a mass MW = 2 × 1012 M⊙ [20], predicts a MW taken to be Mhalo 3 −3 soliton peak density ρMW and solipeak = 8 × 10 GeV cm MW ton core width rc = 120 pc (Figure 1). The Milky Way halo is generally taken to have a dark matter density 0.3 GeV cm−3 in the solar neighborhood, with re-

cent careful dynamical study by Portail et al. [20] revising upward this figure to 0.6 GeV cm−3 . The predicted core density within the soliton for our Galaxy ranges over 4 − 8 × 103 and from Eq.(5) above, we have −2 ∆t = πGρDM /2m3B = (0.7 − 1.4) × 10−27 m−3 . B sec −7 −1 Since mB = 1.5×10 sec , we arrive at ∆t = 200−400 nsec; this ∆t is within the reach of current pulsar time arrays. The central soliton then provides approximately two orders of magnitude enhancement of the DM density within r . 100pc, dominating over any pulsar lying elsewhere in the galaxy, (or the relatively small Earth clock modulation), so that such favorably located pulsars can be regarded as standard clocks, having effectively no pulse time variation when making comparisons of timing residuals between two such pulsars, as shown in Figure 2 (top left). For pairs of pulsars within this soliton region, separated by more than the Compton wavelength of (> pc scale), the amplitude of the sum of the two pulsar timing residuals that may be compared for detecting relative timing can partially or fully cancel, or add constructively by up to a factor of two, as shown in Figure 2, bottom left panel. The above assumes a boson mass equal to mB = 10−22 eV, but allowing this to vary we have ρ ∝ m2B and rc ∝ −1 m−1 B , and thus ∆t ∝ mB , the soliton density becomes higher, but the core radius becomes smaller for a higher particle mass compensating to some extent.

PAIRWISE TIMING AMPLITUDES

It is important to appreciate that all clocks are modulated within an oscillating scalar field including Earth clocks and so in practice we can work only with relative timing residuals, either between pairs of pulsars or between any pulsar and a time standard based on precise Earth clocks. The ticks of an individual clock are cyclically slowed and increased at the Compton frequency, with a magnitude that is proportional to the mass density of the scalar field local to each pulsar, where the ”amplitude” of this effect is the time difference induced by Eqn 3. In practice, pulsar timing measurements are typically averaged over a sizeable number of pulses on a relatively short timescale of hours and this rate is then compared on longer timescales, a practice that is well suited to the larger than monthly Compton frequency modulation that we seek. The timing amplitude of any such modulation is largest for a pulsars within the central soliton and also when the pulsars are spatially located such that they are out of phase relative to the Compton frequency. In general this phase difference can mean that for any given pair of pulsars, for which the local Axion density is equal, can range in relative amplitude from zero to double the amplitude of each separately, as shown by blue shaded area in Figure 2. Furthermore, for well separated pulsars

4 (on a scale greater then the de-Broglie scale) the timing amplitude range can be enhanced by another factor of two because the de-Broglie scale interference that fully modulates the local density about the mean level, shown in Figure 2. Ideally, the relative pulsar timing may not have to rely on being referred to any Earth clock for simultaneous observations of different pulsars through the same telescope, or for telescopes that are highly synchronous, with the advantage that any vagaries and the precision limit of Earth clocks may then be canceled out. So here we calculate the relative timing amplitudes for pairs of pulsars S(t):  1 Ψ(x1 ) sin (ωt + α′1 ) − S(t) = ∆t1 (t) − ∆t2 (t) = ω  − Ψ(x2 ) sin (ωt + α′2 ) , (7) i where we have defined α′i = 2αi − ωD c . To illustrate this difference we calculate the predicted relative timing signal in Fig. 2 between local pulsars, and pulsars close to the Galactic center at a radius of 50 and 500 pc, fixing the boson mass to mB ∼ 0.8 × 10−22 eV. In this case, the relative timing amplitude is dominated by the pulsar closer to the Galactic center and it can reach an amplitude of the order of 600 ns. While taking the pulsars at approximately the same distance from the Galactic center, the relative timing amplitude signal strongly depends on the phase of the pulse, canceling when the signals as seen from Earth are in phase. For sake of convenience, we define ratio of the dark matter density at the location of the two pulsars as   ρDM (x1 ) Ψ(x1 ) = . (8) δρDM = Ψ(x2 ) ρDM (x2 )

In order to estimate the sensitivity of the current and forthcoming pulsar timing array detectors to the oscillation of the Dark matter scalar field, we compute the average square signal over all the phases √ p 2p 2 hS (t)i = Ψ(x1 )2 + Ψ(x2 )2 , (9) 2ω and relate it to the GW strain, and since the Ψ(x) amplitude of the oscillating scalar field only depends on the dark matter density distribution, one can always re-write Ψ(x2 ) in term of Ψ(x1 ) √ q 6 hc = (10) Ψ(x2 ) 1 + δρ2DM . 2 We compute the characteristic amplitude shown in Fig. 3, highlighting the relatively strong signal expected for pulsars within 0.5Kpc of the Galactic center, We also over-plot the confidence regions for the current results

FIG. 2: Predicted timing signal between pulsar pairs, for a light scalar field of mB ∼ 0.8 × 10−22 eV. In the top panels the relative timing signal between local pulsars at 8 kpc, and pulsars close to the Galactic center at a radius of 50 and 500 pc. In the bottom left and right panels both members of the pair are located at the same Galactocentric radius of 0.5 and 8 kpc, respectively, with relative phases chosen as indicated in the inset box, bottom right, illustrating the signal can cancel in such cases. The shaded regions indicate how the density modulation on the de-Broglie scale can enhance or diminish the pairwise relative timing signal.

from PTA, PPTA and SKA experiments to allow a direct comparison with expected sensitivities of these new surveys [21], where even one such central pulsar can provide a sufficient timing residual to test for the presence of light axionic dark matter.

DISCUSSION AND CONCLUSIONS

We have distinguished between Compton and deBroglie scale modulations within the axionic interpretation of dark matter and examined their effects on Pulsar timing. For our Galaxy the de-Broglie scale is approximately 100 times larger than the Compton scale, corresponding to ≃ 150pc for the Milky Way, with the favored axion mass of mB ≃ 10−22 eV . Within virialized halos our simulations reveal that the density distribution is fully modulated on the de-Broglie scale, as shown in Figure 1, with a dense soliton at the center of radius ≃ 150pc where the pulsar timing effect is strong. The coherence of this Compton oscillation will also vary on the de-Broglie scale, being coherent within de-Broglie sized patches, and becoming unrelated between patches separated by a larger scale. We have made self consistent predictions for pulsar timing residuals by including the spatial dependence of light bosonic dark matter revealed in our ψDM simulations [15]. The de-Broglie interference is most conspicuous by the formation of central soliton on the de-Broglie scale, rep-

5 feature is expected dynamically on a scale of . 150/mB pc, which lies between the scale length of the stellar bulge (1kpc) and the smaller pc scale region of influence of the central Milky Way black hole. Indeed careful dynamical modeling by Portail et al. [20] has recently uncovered a central shortfall of 2 × 109M⊙ of ”missing matter” which may help account for the 100 km/s motion of stars within the central ≃ 120 pc of the galaxy [31–33] and which we aim to examine in the context of the near spherical soliton potential that we predict here.

FIG. 3: Characteristic strain measured between pairs of pulsars, with Galactocentric radii chosen as in Figure 2 and compared with the expected sensitivities from the current and forthcoming PTA experiments (adapted from [7, 21]), with the corresponding oscillation frequency shown above. We highlight the relatively high signal strength we calculate for pulsars within the Galactic soliton region as a function of Axion mass in a series of curves, demonstrating that this signal is already detectable for central Galactic pulsars. For comparison, the diagonal dotted line is the prediction obtained by [7] for local pulsars, assuming a smooth density distribution, with the blue shaded region representing the wider range we predict that includes our de-Broglie DM density modulation. The local upper limit obtained by [34] is also shown (black square).

resenting a stable, time-independent ground state, where the pulsar timing residuals are expected to be nearly two orders of magnitude higher than those imprinted on local pulsars, due to the relatively high central density of the soliton, that much exceeds in density a corresponding NFW profile. Such central millisecond pulsars are expected to be formed in large numbers within the bulge and near the galactic center [22, 23], and can account for the GeV gamma-ray excess [24, 25] and are being searched with some success [26, 27]. Detection will be compromised within the inner 100 pc has a high plasma density and the ISM contains small scale irregularities, causing dispersion of pulse arrival time and pulse smearing, although “corridors” of lower scattering may be evident [28, 29]. The smearing is a low-pass filter making millisecond pulsars undetectable at low frequency, but decreases rapidly at high frequency (