Microlensing Probes of Dark Matter Substructure

Microlensing Probes of Dark Matter Substructure

Fabian SchmidtPrinceton

DM on Galaxy Scales, Monterrey, 7/2013

Motivation

• Standard CDM predicts hierarchical DM substructure down to ~ MEarth

• Lower mass cutoff determined by details of DM production - but also tidal disruption...

• How can we detect / constrain this structure in the < 108 MSun range ?

• Key parameter: DM density within substructure

Basic picture

Basic picture

Earth/observer

DM subhalo

Source

Basic picture

Earth/observer

Source

DM subhalo

Lensing effect strongly exaggerated...

• Time (Shapiro) delay

• Doppler shift

• Astrometric perturbations

• Magnification

Lensing observables/

Zd

/ r

/Z

d

s(s )r2

:grav potential of lens in comoving frame (“moving screen” approx.)

/Z

ds

sr

Weak lensing approx (cf later)


• Doppler shift


• Magnification


Zd

/ r

/Z

d

s(s )r2


/Z

ds

sr


Light propagation


• Doppler shift


• Magnification


Zd

/ r

/Z

d

s(s )r2


/Z

ds

sr


Motion of source/observer

• Potential at scale radius:

• where

• Scale radius in terms of density:

Relevant scalings

s = (rs) ' 3 · 1012

Ms

106M/h

2/3

1/3s

Ms = M(< rs) s =(< rs)

rs = 14h1kpc

Ms

106M/h

1/3

1/3s

Note: no density profile assumed

subhalo mass subhalo density

• Potential at scale radius:

• where

• Scale radius in terms of density:

Relevant scalings

s = (rs) ' 3 · 1012

Ms

106M/h

2/3

1/3s

Ms = M(< rs) s =(< rs)

rs = 14h1kpc

Ms

106M/h

1/3

1/3s

Note: no density profile assumed

Relevant scalings

• For a moving lens, we have

• Order of magnitude estimates for individual lenses:

rsvrel

' 108 yr

Ms

106M/h

1/3

1/3s

vrel

200km/s

1

r ! 1

rs;

Zd ! rs; etc.

“Standard” CDM substructure

• Assume substructure made up of halos

• In reality, many subhalos will overlap

• To calculate lensing, we need halo mass and density

• Assume that , and obtain zf from spherical collapse

(< rs) ' (zf )

“Standard” CDM substructure

• zf from spherical collapse:

2

4

6

8

10

12

14

16

18

1e-06 0.0001 0.01 1 100 10000 1e+06 1e+08 1e+10

z f

Ms [Msun/h]

) s ' (1 + zf )3 . 104

D(zf )(M) = c

M = Mvir(Ms)

Substructure is very, very diffuse - subhalos are big

Observables

Time delay & Doppler shift

• Apply to “clocks”, e.g. ms pulsars

• Time delay: measure change in delay due to relative lens-source motion:

• Doppler shift: acceleration of source & observer due to gravitational pull of lens

t = 2vrel ·rZ

d =

: observed pulsar frequency

=

Zdt n · (r

src

robs

)

Observational prospects

• Time delay:

• Doppler shift:

• Doppler shift wins for close lenses and long observing times

T

obs

c

lens

0.03µs

yr

Tobs

yr

Ms

106M/h

2/3

1/3s

lens

kpc

1

vrel 0.1

µs

yr

Ms

106M/h

2/3

1/3s

vrel

300 km/s


• Time delay:

• Doppler shift:

• Doppler shift wins for close lenses and long observing times

T

obs

c

lens

0.03µs

yr

Tobs

yr

Ms

106M/h

2/3

1/3s

lens

kpc

1

vrel 0.1

µs

yr

Ms

106M/h

2/3

1/3s

vrel

300 km/s


• Time delay for ms pulsars - Siegel, Hertzberg, Fry (2007)

Probing Dark Matter with Pulsars 5

Figure 3. ∆Tp vs. time for a set of parameters corresponding tothe variables defined in Figure 2: l = 2kpc, x‖ = 1kpc, Mtot =104 M⊕, b = 10−4 pc, v⊥ = 200 km s−1, and Tp = 10−3s. ∆Tp

is illustrated in seconds and time in years. The solid curve is fora point mass, the dotted curve is for a microhalo of constantdensity with size rmax = 10−3 pc, and the dashed curve is for anextended microhalo with an NFW profile of size rmax = 10−3 pcand turnover radius r0 = 10−4.5 pc.

Figure 4. ∆t vs. time for a set of parameters corresponding tothe variables defined in Figure 2: l = 2kpc, x‖ = 1kpc, Mtot =104 M⊕, b = 10−4 pc, v⊥ = 200 km s−1, and Tp = 10−3s. ∆Tp

is illustrated in seconds and time in years. The solid curve is fora point mass, the dotted curve is for a microhalo of constantdensity with size rmax = 10−3 pc, and the dashed curve is for anextended microhalo with an NFW profile of size rmax = 10−3 pcand turnover radius r0 = 10−4.5 pc. The chosen set of parametersis very optimistic, but demonstrates the potential power of thismethod.

ear and quadratic terms from millisecond pulsar data beforeany further analysis is done. The majority of close dark mat-ter encounters (excepting actual observed transits) will havetheir signatures absorbed in this subtraction. We suggest,therefore, that it may be beneficial to develop non-standardanalysis techniques to explicitly search for this effect.

As can be seen from Figures 3 and 4, the time delaysignal is very sensitive to the density profile of the tran-siting source, and can be used to determine the densitiesof sufficiently massive transiting dark matter, if their im-pact parameters are small enough. Halos with steeper den-sity profiles and smaller cores are more easily detectable,whereas the more diffuse ones will have a more significantdeparture from the point-source template. Of course, at suf-ficently large impact parameters, all source behave as pointsources. Many of the data analysis techniques being pio-neered by groups searching for a cosmic gravitational wavebackground with pulsars (Jenet et al. 2006) should be ap-plicable to identifying the time delay signature of transitingdark matter substructure. As a caveat, it is worth point-ing out that unless the cusp of the Shapiro delay is clearlyvisible in the residuals, the fit is likely to become degener-ate in the presence of noise, as fast-transiting small masseswith large impact parameters will be difficult to discern fromslow-moving massive objects with small impact parameters.

4 DARK MATTER DISCRIMINATION

The small-scale structure produced in our universe is verysensitive to the type (thermal, non-thermal, or baryonic)and particle properties (masses and couplings) of dark mat-ter. While baryons cannot be all of the dark matter, theydo compose a significant amount of the nonluminous mat-ter in the universe (∼ 13 per cent). Baryons do not existin isolation in small mass clumps, but rather as a part ofvery large mass structures, due to the fact that their col-lapse is suppressed on all scales which enter the horizon priorto baryon-photon decoupling (at recombination). Therefore,even though baryons exist in the form of diffuse gas cloudsand compact halo objects today (in addition to stars), thesebaryons were never isolated from the non-baryonic dark mat-ter halos in which they are found. The signature that bary-onic dark matter (MACHOs) will leave in the pulsar timingmeasurements is that of a point mass source, as given bychanges over time in equation (8). For negligible changes inl and x‖, this would result in a pulse-arrival-time shift of

∆tf−∆ti # 2.95×10−11 s

(

Mtot

M⊕

)

ln

(

b2 + x2z

b2 + (xz + v⊥nTp)2

)

(12)

between n pulses. Additionally, MACHOs are compactenough that effects such as gravitational microlensing oughtto be observable if one knows where to look as well, allowingfor possible cross-correlation effects.

The vast majority (∼ 87 per cent) of dark matter isnon-baryonic, however (contained in ΩDM). In contrast tobaryons, the non-baryonic matter is practically collision-less, and thus collapses to form much more diffuse struc-tures than ordinary matter. Additionally, the suppressionof non-baryonic structure ceases at a much earlier epoch,as thermal dark matter kinetically decouples from the pri-mordial plasma at a typical temperature of O(10MeV), andnon-thermal dark matter is always decoupled both thermallyand kinetically from the plasma. The exact epoch of kineticdecoupling of thermally produced dark matter is determinedby the mass of the decoupled particles and its interactions;this in turn determines the mass function of small-scale darkmatter clumps (Bertschinger 2006). As a result, thermally

c© 2007 RAS, MNRAS 000, 000–000

Unfortunately, density assumed is ~1012 times higher than standard estimate...

Ms 102M


• Baghram, Afshordi, Zurek (2011)

7

100101

-21

-20

-19

-18

-17

-16

-15

-14

-13

-12

10

100 ns

1 s

π

ShapiroDoppler

White noiseV= 300 km/s M = 10 Mmin

-6µ =0.03z=1kpcsun

µ

FIG. 3: Pulsar residual power spectrum as a function of fre-quency (bottom x-axis) and the span of observation time inyears (top x-axis) for time delay caused by the Doppler effect(solid line) and time delay caused by Shapiro effect (dash-dotline). The long dashed lines represent levels of white noise for100 ns (bottom) and 1 µs (top) measured biweekly (see theAppendix) [18].

in the Milky Way halo), the typical distance of pulsarsto z0 = 1 kpc, the mean fraction of bound particles thatcan survive the tidal disruption period µ = 0.03 [17], theminimum mass of DM substructure Mmin = 10−6M"

and also the maximum Mmax = 1012M" (the total massof a galactic halo). Later we will show that hp is almostindependent of Mmax.The power spectra of Shapiro and Doppler effects in

Fig.(3) are well described by power-laws:

ωP δν

ν

(ω)|Shapiro

∝ ω−3, ωP δν

ν

(ω)|Doppler

∝ ω−4. (38)

These behaviors can be understood by noticing that the∆v integral (i.e. the last integral) in Eqs. (35-36) scalesas H2(ξs), if we use the spherical collapse relations ofSec. (III). Since most small structures with CDM initialconditions collapse around the same time, this is approx-imately constant. The contribution to the rest of theintegrals is dominated by k−1

y ∼ ∆r ∼ v/ω, so the inte-gral over distances scales as (∆r)3 ∝ ω−3. Plugging thisinto Eqs. (35-36) yields the scalings of Eq. (38).To physically understand the scaling for the Doppler

effect we can once more Fourier transform the power-spectrum in Eq. (36) to find that vDop. ∼ δν

ν is propor-tional to vt2 = (vt)×t, i.e. the magnitude of accelerationis proportional to distance traveled by the earth/pulsar.This is exactly what one expects for the gravitationalfield in a medium with roughly uniform density, and is

π

FIG. 4: Pulsar residual power spectrum as a function of fre-quency (bottom-x axis) and the span of time in years (topx-axis) for time delay caused by the Doppler effect (top-line)and time delay caused by Shapiro effect (bottom line) for amaximum mass of halo Mmax = 1012M! (dash-dot line) andMmax = 108M! (solid line).

due to the fact that most small substructure forms atroughly the same density ∝ H2(ξs). However, the direc-tion of acceleration is random, as different substructureswill dominate the local gravity on different scales.An important point to consider before examining the

effect of different parameters on pulsar timing is thestudy of the effect of maximum mass in the integrals.As we show in Fig. (4), the total dependence of hp

on maximum mass is small, where we plot the hp forMmax = 1012M", the total mass of a typical galaxy andMmax = 108M", for a more realistic tidal cut-off for sub-haloes at our position in the Milky Way. This confirmsthat, not surprisingly, most of the observable effects onpulsar timing comes from CDM small scale structure.Now we examine the dependence of the power spec-

trum on different parameters of the model. We plot thedimensionless amplitude hp for the Doppler effect for dif-ferent velocities of dark matter substructures and the µ-parameter of stable clustering in Fig. (5), which showsthat hp is proportional to velocity and the square root ofthe µ parameter.In Fig. (6), we plot the power spectrum for different

mass minima of DM substructures. As shown in Fig. (6),the ω−4 dependence of h2

p does not change by changingthe minimum of the mass. However, the amplitude ofthe signal increases when the interval of integration isincreased.In Figs. (7) and (8) we plot hp given different primor-

dial spectral index ns, for Doppler and Shapiro effectsrespectively. For ns < 1, the slope of hp does not change,

Doppler wins for long observing times (>~ 10 years)

using NL P(k) from stable clustering

Astrometric perturbations

• Measure proper motion of source -> time evolution of deflection angle

• In fact, need time dependence of this to disentangle from unknown proper motion

i iprop

= 2jlens

@

@j

Zd

s

s@i?

= 2vjlens,?

Zd

s

s@j?@

i?


• More sensitive to density due to additional spatial derivative

• Apply to stars targeted by GAIA

vrelrs

108µas

yr

Ms

106M/h

1/3

2/3s


• Erickcek & Law (2011):

The Astrophysical Journal, 729:49 (17pp), 2011 March 1 Erickcek & Law

0

5

-20 -10 0 10 20αx [µas]

100 yrsγ=1 (NFW)

0

550 yrs

γ=1.2

0

5

α y [µ

as]

5 yrsγ=1.5

0

5 1 yr γ=1.8

0

5

10

151 yr

γ=2.0 (SIS)

Figure 4. Deflection angle generated by a moving lens with dS = 5 kpc,dL = 50 pc, and vT = 200 km s−1. The virial mass of the lens is 5 × 105 M#and its concentration is Rvir/r−2 = 99. The inner density profile of the lens isgiven by ρ ∝ r−γ , and the different panels correspond to different values ofγ . The impact parameter is 1 arcsec, and only the portion of the image pathcorresponding to the time surrounding the moment of closest approach betweenthe image and lens is shown. Note that the image path becomes more linear andthe image motion slows down considerably as γ is decreased.(A color version of this figure is available in the online journal.)

(10 M# ! Mt ! 106 M#) subhalos that are capable of pro-ducing detectable astrometric lensing events. High-resolutionN-body simulations can probe the density profiles of only thelargest (Mt " 108 M#) subhalos, and even these profiles areunresolved at r ! 350 pc (Springel et al. 2008; Diemand et al.2008). For these large subhalos, Diemand et al. (2008) find thatρ ∝ r−1.2 in the innermost resolved regions, while Springelet al. (2008) see ρ ∝ r−(1.2–1.7) at their resolution limit for ninelarge subhalos, with no indication that the slope had reacheda fixed central value. Meanwhile, at the opposite end of themass spectrum, Diemand et al. (2005) find that the first Earth-mass dark matter microhalos have steeper density profiles withρ ∝ r−(1.5–2.0) at redshift z = 26, and higher-resolution simu-lations indicate that this steep profile extends to within 20 AUof the microhalo center (Ishiyama et al. 2010).

In light of this uncertainty, we consider a generic densityprofile

ρ(r) = ρ0

(r

r0

)−γ

(18)

with 1 < γ # 2. We assume that a constant-density core,if present, is significantly smaller than our typical impactparameters of 0.001 pc, and we assume that the subhalo does notcontain a black hole. Larger cores would decrease the lensingsignal while the presence of a black hole would enhance it byadding a point mass and steepening the density profile (Bertoneet al. 2005; Ricotti & Gould 2009). If we take this density profileas infinite when calculating the projected surface density Σ, wefind that

Σ(ξ ) =√

π ρ0r0Γ [0.5(γ − 1)]

Γ [0.5γ ]

(ξ

r0

)1−γ

, (19)

M2D(ξ ) = 2π3/2 (ρ0r

30

) Γ [0.5(γ − 1)](3 − γ )Γ [0.5γ ]

(ξ

r0

)3−γ

, (20)

where Γ[x] is the Euler gamma function.

Of course, this density profile does not extend to infinity; thesubhalo’s density profile will be truncated by tidal stripping,and it may also transition to a steeper power law, as in thecase of an NFW profile. If the density profile is truncated atr = Rt , then the surface density diverges from Equation (19)as ξ approaches Rt, but for ξ & Rt , Equations (19) and (20)are still good approximations. For instance, if γ = 1.5 (1.2),M2D(ξ ) for a subhalo truncated at Rt is greater than 80% (50%)the value given by Equation (20) if ξ # 0.1Rt . We will showin Appendix A that detectible astrometric signatures are onlyproduced if ξ < 0.03 pc, and Equation (20) is accurate towithin 20% for subhalos with γ $ 1.5, Mvir < 108 M#, andRt " 0.1 pc. Furthermore, the lower bound on Rt is significantlysmaller for subhalos with Mvir & 108 M#. We will thereforeuse Equation (20) and take Rt " 0.1 pc as a conservativelower bound, although we note that the resulting deflectionsmay be slightly overestimated, especially if γ ! 1.2. As shownin Figure 4, however, detecting a subhalo with γ ! 1.2 ischallenging, and we conclude that Equation (20) is accurate towithin ∼20% for subhalos of interest.

If a dark matter subhalo with a density profile given byEquation (18) passes in front of a star, Equation (1) tells usthat

(α = θα

(ξ

r0

)2−γ

ξ , (21)

where we have defined

θα ≡ 0.88 µas(

Γ [0.5(γ − 1)]2(3 − γ )Γ [0.5γ ]

) (1 − dL

dS

)

×(

pcr0

) (ρ0r

30

M#

). (22)

Like θSISE , θα depends on the distances to the lens and the source

only through the factor (1 − dL/dS). We also note that θα isrelated to the Einstein angle θE:

θα = θγ−1E

(r0

dL

)2−γ

. (23)

We will continue to assume that α & β so that ξ (see Figure 1)is approximately equal to dLβ.

Equation (22) gives the magnitude of the deflection anglein terms of the parameters of the density profile r0 and ρ0,but this is not the most useful description of the subhalo.Instead we characterize the subhalo by either its mass aftertidal stripping (Mt ≡ mbdMvir) or the mass contained within aradius of 0.1 pc from the subhalo center (M0.1 pc). Although Mtis a more standard and intuitive description of the subhalo mass,using M0.1 pc offers two advantages. First, M0.1 pc completelydetermines the deflection angle; without loss of generality, wecan set r0 = 0.1 pc, in which case

θα = 8.8 µas(

Γ [0.5(γ − 1)]2(3 − γ )Γ [0.5γ ]

) (1 − dL

dS

)

×(

3 − γ

4π

) (M0.1 pc

M#

). (24)

Second, M0.1 pc is the portion of the subhalo’s mass that isactually probed by astrometric microlensing because truncatingthe subhalo’s density profile at Rt = 0.1 pc does not affectits astrometric lensing signature. Therefore, using M0.1 pc tocharacterize the subhalo’s mass allows us to consider subhalos

6

Here, density ~105 times standard estimate - still not observable...


• Increasing density further,

• Can be used to constrain ultra-compact minihalos (UCMH) - Li, Erickcek, Law (2012)

The Astrophysical Journal, 729:49 (17pp), 2011 March 1 Erickcek & Law

0

5

-20 -10 0 10 20αx [µas]

100 yrsγ=1 (NFW)

0

550 yrs

γ=1.2

0

5

α y [µ

as]

5 yrsγ=1.5

0

5 1 yr γ=1.8

0

5

10

151 yr

γ=2.0 (SIS)

Figure 4. Deflection angle generated by a moving lens with dS = 5 kpc,dL = 50 pc, and vT = 200 km s−1. The virial mass of the lens is 5 × 105 M#and its concentration is Rvir/r−2 = 99. The inner density profile of the lens isgiven by ρ ∝ r−γ , and the different panels correspond to different values ofγ . The impact parameter is 1 arcsec, and only the portion of the image pathcorresponding to the time surrounding the moment of closest approach betweenthe image and lens is shown. Note that the image path becomes more linear andthe image motion slows down considerably as γ is decreased.(A color version of this figure is available in the online journal.)

(10 M# ! Mt ! 106 M#) subhalos that are capable of pro-ducing detectable astrometric lensing events. High-resolutionN-body simulations can probe the density profiles of only thelargest (Mt " 108 M#) subhalos, and even these profiles areunresolved at r ! 350 pc (Springel et al. 2008; Diemand et al.2008). For these large subhalos, Diemand et al. (2008) find thatρ ∝ r−1.2 in the innermost resolved regions, while Springelet al. (2008) see ρ ∝ r−(1.2–1.7) at their resolution limit for ninelarge subhalos, with no indication that the slope had reacheda fixed central value. Meanwhile, at the opposite end of themass spectrum, Diemand et al. (2005) find that the first Earth-mass dark matter microhalos have steeper density profiles withρ ∝ r−(1.5–2.0) at redshift z = 26, and higher-resolution simu-lations indicate that this steep profile extends to within 20 AUof the microhalo center (Ishiyama et al. 2010).

In light of this uncertainty, we consider a generic densityprofile

ρ(r) = ρ0

(r

r0

)−γ

(18)

with 1 < γ # 2. We assume that a constant-density core,if present, is significantly smaller than our typical impactparameters of 0.001 pc, and we assume that the subhalo does notcontain a black hole. Larger cores would decrease the lensingsignal while the presence of a black hole would enhance it byadding a point mass and steepening the density profile (Bertoneet al. 2005; Ricotti & Gould 2009). If we take this density profileas infinite when calculating the projected surface density Σ, wefind that

Σ(ξ ) =√

π ρ0r0Γ [0.5(γ − 1)]

Γ [0.5γ ]

(ξ

r0

)1−γ

, (19)

M2D(ξ ) = 2π3/2 (ρ0r

30

) Γ [0.5(γ − 1)](3 − γ )Γ [0.5γ ]

(ξ

r0

)3−γ

, (20)

where Γ[x] is the Euler gamma function.

Of course, this density profile does not extend to infinity; thesubhalo’s density profile will be truncated by tidal stripping,and it may also transition to a steeper power law, as in thecase of an NFW profile. If the density profile is truncated atr = Rt , then the surface density diverges from Equation (19)as ξ approaches Rt, but for ξ & Rt , Equations (19) and (20)are still good approximations. For instance, if γ = 1.5 (1.2),M2D(ξ ) for a subhalo truncated at Rt is greater than 80% (50%)the value given by Equation (20) if ξ # 0.1Rt . We will showin Appendix A that detectible astrometric signatures are onlyproduced if ξ < 0.03 pc, and Equation (20) is accurate towithin 20% for subhalos with γ $ 1.5, Mvir < 108 M#, andRt " 0.1 pc. Furthermore, the lower bound on Rt is significantlysmaller for subhalos with Mvir & 108 M#. We will thereforeuse Equation (20) and take Rt " 0.1 pc as a conservativelower bound, although we note that the resulting deflectionsmay be slightly overestimated, especially if γ ! 1.2. As shownin Figure 4, however, detecting a subhalo with γ ! 1.2 ischallenging, and we conclude that Equation (20) is accurate towithin ∼20% for subhalos of interest.

If a dark matter subhalo with a density profile given byEquation (18) passes in front of a star, Equation (1) tells usthat

(α = θα

(ξ

r0

)2−γ

ξ , (21)

where we have defined

θα ≡ 0.88 µas(

Γ [0.5(γ − 1)]2(3 − γ )Γ [0.5γ ]

) (1 − dL

dS

)

×(

pcr0

) (ρ0r

30

M#

). (22)

Like θSISE , θα depends on the distances to the lens and the source

only through the factor (1 − dL/dS). We also note that θα isrelated to the Einstein angle θE:

θα = θγ−1E

(r0

dL

)2−γ

. (23)

We will continue to assume that α & β so that ξ (see Figure 1)is approximately equal to dLβ.

Equation (22) gives the magnitude of the deflection anglein terms of the parameters of the density profile r0 and ρ0,but this is not the most useful description of the subhalo.Instead we characterize the subhalo by either its mass aftertidal stripping (Mt ≡ mbdMvir) or the mass contained within aradius of 0.1 pc from the subhalo center (M0.1 pc). Although Mtis a more standard and intuitive description of the subhalo mass,using M0.1 pc offers two advantages. First, M0.1 pc completelydetermines the deflection angle; without loss of generality, wecan set r0 = 0.1 pc, in which case

θα = 8.8 µas(

Γ [0.5(γ − 1)]2(3 − γ )Γ [0.5γ ]

) (1 − dL

dS

)

×(

3 − γ

4π

) (M0.1 pc

M#

). (24)

Second, M0.1 pc is the portion of the subhalo’s mass that isactually probed by astrometric microlensing because truncatingthe subhalo’s density profile at Rt = 0.1 pc does not affectits astrometric lensing signature. Therefore, using M0.1 pc tocharacterize the subhalo’s mass allows us to consider subhalos

6

Erickcek & Law (2011)

• We don’t know intrinsic flux (typically), thus only observe change in magnification:

• Very small, but mass-independent and proportional to density

• This is why we can look for planets with microlensing...

Flux Magnification

lens

rs 1013

Ms

106M/h

1/3

2/3s

lens

kpc

Fobs

vrel

r 1021yr1 s

lens

kpc


• Griest et al (2011): forecasted constraints on DM made of low-mass black holes, using Kepler photometric data

• Very low masses constrained - possible because BH are very dense

4

of expected events over the 5000 stars and then scaledthose results to 3.5 years of observation of 150,000 stars,assuming that 25% of these stars will be identified asvariable and not be useful[8]. That is we assume 390,000star-years. Our results are given in Figure 1. In orderto turn these results into the potential sensitivity of de-tecting PBH dark matter, we calculated the 95% C.L. foreach PBH mass, assuming that no events were detectedand that there was no background. These potential lim-its are shown in Figure 1, along with limits from earlierexperiments. We found very little difference between thetwo S/N requirements mentioned above, so show only thecase for 4 sequential 3-sigma event selection.We see from Figure 1 that we have the potential to de-

tect or rule out PBHs as the primary constituent of DMover the mass range 5 × 10−10M" < mPBH < 10−4M".Current limits from the MACHO/EROS experiments[11]are also shown as a dashed line. There are also limitsruling out halo fractions, f > 1, from femtolensing ofgamma ray bursts (GRB)[19], but they run from about10−16M" < mPBH < 10−13M", off to the left of Fig-ure 1, and other limits from picolensing of GRBs rulingout f < 4 from 4 × 10−13M" < mPBH < 8 × 10−10M",are in our mass range, but too weak to be seen in ourplot. We see that a microlensing search for PBHs throughKepler data has the potential to extend the mass sensi-tivity by almost two orders of magnitude below the MA-CHO/EROS limits which exclude DM masses down toaround 2 × 10−8M". Note that commonly quoted (e.g.[3]) limits from EROS alone[12] exclude masses down toaround 6 × 10−8M" and are not as strong as the earliercombined MACHO and EROS limits. There are no otherlimits in the mass range just below 2 × 10−8M", so thecapability of Kepler to search for these PBHs is unique.

DISCUSSION AND FUTURE WORK

In this theoretical paper we suggest a method to de-tect or rule out PBH DM over a large and unexploredmass range. If nothing is detected the method has thepotential to rule out around 40% of the total remainingmass range for PBH DM. While we emphasized PBHs,since microlensing depends only upon lens mass and size,such an experiment would detect or rule out any massivecompact halo object dark matter in the mass range de-scribed.The next step is clearly to perform the analysis sug-

gested here, and we have begun this task using pub-licly available Kepler data. This experiment requires adeep understanding of the lightcurve data including sys-tematic effects and possible backgrounds. It requires acareful selection of microlensing candidates and an accu-rate calculation of the efficiency of that selection method,as well as an understanding of false positives and back-ground events. Since Kepler transit searches have already

FIG. 1. Top panel: The expected number of events, Nexp (de-fined as 4 sequential measurements with flux 3-sigma aboveaverage) in 390,000 star-years of Kepler data. The thin hor-izontal line shows Nexp = 3, the limit for a 95% C.L. if noevents are detected. Bottom panel: Potential 95% C.L. ex-clusion of PBH dark matter. The area above the thick linewould be ruled out if no events were seen in 390,000 star-years.Also shown as a thin dashed line is the limit from combinedMACHO/EROS LMC microlensing searches[11]. The thinhorizontal line shows a halo consisting entirely of PBHs withlocal DM density ρ = 0.3GeVcm−3.

turned up many stellar flares on M and K dwarfs[20] us-ing selection criteria similar to those used above, a goodmethod of distinguishing microlensing from stellar flareswill be needed. However, there are some powerful dis-criminants against backgrounds. The lightcurves shouldmatch a limb-darkened finite source microlensing shape,and if enough events are found the rates should exhibitthe strong dependence on distance and source radius pre-dicted by the above formulas.Several effects in the data may mean that the actual

limits, supposing no PBHs are detected, will be differentfrom what we display in Figure 1. First, our simplis-tic removal of 25% of the stars as variable[8] needs tobe replaced by detailed analysis of all the lightcurves.Our calculation of thresholds using Gaussian statisticsneeds redoing in light of actual lightcurve data and othersources of noise that exist in real data. A more carefultreatment of stellar limb darkening may shorten the aver-

Conclusions

Conclusions

• It is natural to ask whether DM substructure within the Milky Way can be detected through lensing

• The main obstacle for standard CDM is that substructure is diffuse:

• Observables roughly scale as

• Time delay & Doppler:

• Astrometry:

• Magnification:

< 104

/ 1/3

/ 2/3

/

Conclusions• Thus, for standard CDM, time delay and Doppler

are most promising - but probably still out of reach even for futuristic experiments

• If DM substructure is much denser, then astrometry and magnification can become interesting

• Astrometry constrains ultra-compact minihalos

• Kepler constrains BH DM

• More observables still to be explored in this direction!

Documents

Microlensing Probes of Dark Matter Substructure