2. Dark matter properties from cosmology & astrophysics200.145.112.249 › webcast › files ›...

2. Dark matter properties from cosmology & astrophysics

Pasquale Dario Serpico (Annecy-le-Vieux, France) School on Dark Matter ICTP-SAIFR (São Paulo, Brazil) - 27/06/2016

DM EVIDENCE @ MANY SCALES

“Astrophysical”“Cosmological” (growing effect of non-linearities, baryonic gas dynamics, feedbacks...)

CMBanis.

(Growth & Pattern of)Large Scale Structures

Clusters(X-rays, lensing)

Galaxies Dwarfs(rotation curves, fits...)

‣ Exact solutions or linear perturbation theory applied to simple physical systems: credible and robust!

‣ Many would say: Suggests “cold” collisionless additional species, rather than a modification of GR (IMHO: academic debate mostly influenced by “classical” thinking… the need for new d.o.f. is key observation!)

‣ Tells that its majority is non-baryonic, rather than e.g. brown dwarf stars, planets...

Especially cosmological evidence of paramount importance for Particle Physics!

Beyond SM explanation needed, but gravity is universal: no particle identification! discovery via other channels is needed to clarify particle physics framework

But what to look for depends on“theoretical prejudice” (curse of DM searches)

FIG. 1: The power spectrum of matter. Red points with error bars are the data from the Sloan

Digital Sky Survey [9]; heavy black curve is the ΛCDM model, which assumes standard general

relativity and contains 6 times more dark matter than ordinary baryons. The dashed blue curve is

a “No Dark Matter” model in which all matter consists of baryons (with density equal to 20% of

the critical density), and the baryons and a cosmological constant combine to form a flat Universe

with the critical density. This model predicts that inhomogenities on all scales are less than unity

(horizontal black line), so the Universe never went nonlinear, and no structure could have formed.

TeVeS (solid blue curve) solves the no structure problem by modifying gravity to enhance the

perturbations (amplitude enhancement shown by arrows). While the amplitude can now exceed

unity, the spectrum has pronounced Baryon Acoustic Oscillations, in violent disagreement with

the data.

matter model, on the other hand, the oscillations should be just as apparent in matter as

they are in the radiation. Indeed, Fig. 1 illustrates that – even if a generalization such

as TeVeS fixes the amplitude problem – the shape of the predicted spectrum is in violent

5

TWO COMMENTS

! Don’t be (too) fooled by the whole debate dark matter vs. modified gravity

! Do we really need new degrees of freedom? Among fundamental particles of the SM, only neutrinos are at most weakly charged (i.e. “dark”, or invisible if you prefer) massive and cosmologically stable, we’ll see if they work.

“DM OR MODIFIED GRAVITY”?The difference between the two may be less sharply defined than you naively think!

Ex.: take one of the most frequently considered models of modified gravity, f(R)

arX

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FTPI-MINN-08/35, UMN-TH-2716/08, arXiv:0809.1653

Dark Matter from R2-gravity

Jose A. R. CembranosWilliam I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, 55455, USA

The modification of Einstein gravity at high energies is mandatory from a quantum approach. Inthis work, we point out that this modification will necessarily introduce new degrees of freedom.We analyze the possibility that these new gravitational states can provide the main contribution tothe non-baryonic dark matter of the Universe. Unfortunately, the right ultraviolet completion ofgravity is still unresolved. For this reason, we will illustrate this idea with the simplest high energymodification of the Einstein-Hilbert action: R2-gravity.

PACS numbers: 04.50.-h, 95.35.+d, 98.80.-k

INTRODUCTION

Different astrophysical observations agree that themain amount of the matter content of our Universe isin form of unknown particles that are not included in theStandard Model (SM). Typical candidates to account forthe missing matter can be found in well motivated ex-tensions of the electro-weak sector. However, there is afundamental sector in our model of particles and interac-tions, where the introduction of new degrees of freedomis not only well motivated, but absolutely necessary.

The non-unitarity and non-renormalizability of thegravitational interaction described by the Einstein-Hilbert action (EHA) demands its modification at highenergies. The main idea of this work is to realize that thiscorrection cannot be accomplished without the introduc-tion of new states; these states will typically interact withSM fields through Planck scale suppressed couplings andpotentially work as dark matter (DM).

R2-GRAVITY

In spite of many and continuous efforts, the ultraviolet(UV) completion of the gravitational interaction is stillan open question. In these conditions, it is difficult tomake general statements about its phenomenology. Wewill adopt a very conservative and minimal approach inorder to capture the fundamental physics of the prob-lem. The simplest correction to the EHA at high energiesis provided by the inclusion of four-derivative terms inthe metric that preserve the general covariance principle.The most general four-derivative action supports, in ad-dition to the usual massless spin-two graviton, a massivespin-two and a massive scalar mode, with a total of eightdegrees of freedom (in the physical or transverse gauge[1, 2]). In fact, four-derivative gravity is renormalizable.However, the massive spin-two gravitons are ghost-likeparticles that generate new unitarity violations, break-ing of causality, and inadmissible instabilities [3].

In any case, in four dimensions, there is a non-trivialfour-derivative extension of Einstein gravity that is free of

ghosts and phenomenologically viable. It is the so calledR2-gravity since it is defined by the only addition of aterm proportional to the square of the scalar curvature tothe EHA. This term does not improve the UV problemsof Einstein gravity but illustrates our idea in a minimalway. In fact, R2-gravity only introduces one additionalscalar degree of freedom, whose mass m0 is given by thecorresponding new constant in the action, as one can seein Eq. (1):

SG =

!√

g

"

−Λ4 −M2

Pl

2R +

M2Pl

12 m20

R2 + ...

#

(1)

$%&'

DE

$ %& '

EHA

$ %& '

DM

where MPl ≡ (8πGN )−1/2 ≃ 2.4 × 1018 GeV, Λ ≃2.3×10−3 eV, and the dots refer to higher energy correc-tions that must be present in the model to complete theUV behaviour. In this work, we will show that just theAction (1) can explain the late time cosmology since thefirst term can account for the dark energy (DE) content,while the third term is able to explain the dark matter(DM) one. The first term is just the standard cosmolog-ical constant, that we will neglect along this work. Wewill focus on the new phenomenology that introduces thethird term when it can be identified with the observedDM. R2-gravity modifies Einstein’s Equations (EEs) as[4, 5] (following notation from [6]):

(

1 −1

3 m20

R

)

Rµν −1

2

(

R −1

6 m20

R2

)

gµν

− Iαβµν∇α∇β

(1

3 m20

R

)

=Tµν

M2Pl

, (2)

where Iαβµν ≡ (gαβgµν − gαµgβν). The new terms donot modify the standard EEs at low energies except forthe mentioned introduction of a new mode. In fact, if weimpose to preserve standard gravity up to nuclear den-sities or Big Bang Nucleosynthesis (BBN) temperatures,the constraints on m0 are just m0

>∼ 10−12 eV. In thiscase, the new terms are expected to be negligible at den-sities lower than ρ ∼ TBBN

4 ∼ (100 MeV)4 or curvatureslower than HBBN

2 ∼ TBBN4/MPl

2 ∼ (10−12 eV)2. Inthis paper, we will discuss in detail the restrictions and

T. P. Sotiriou & V. Faraoni, “f(R) Theories Of Gravity,” Rev. Mod. Phys. 82, 451 (2010) [0805.1726]

comes quite naturally, since it is actually introduced byparticles with spin !excluding gauge fields". Remarkably,the theory reduces to GR in vacuo or for conformallyinvariant types of matter, such as the electromagneticfield, and departs from GR in the same way that Palatinif!R" gravity does for most matter fields that are usuallystudied as sources of gravity. However, at the same time,it exhibits new phenomenology in less-studied cases,such as in the presence of Dirac fields, which includetorsion and nonmetricity. Finally, we repeat once morethat Palatini f!R" gravity, despite appearances, is really ametric theory according to the definition of Will !1981"!and the geometry is a priori pseudo-Riemannian".11 Onthe contrary, metric-affine f!R" gravity is not a metrictheory !hence the name". Consequently, it should also beclear that T!" is not divergence-free with respect to thecovariant derivative defined with the Levi-Civita con-nection !nor with !! actually". However, the physicalmeaning of this last statement is questionable and de-serves further analysis since in metric-affine gravity T!"

does not really carry the usual meaning of a stress-energy tensor !for instance, it does not reduce to thespecial relativistic tensor at an appropriate limit and atthe same time there is also another quantity, the hyper-momentum, which describes matter characteristics".

III. EQUIVALENCE WITH BRANS-DICKE THEORY ANDCLASSIFICATION OF THEORIES

In the same way that one can make variable redefini-tions in classical mechanics in order to bring an equationdescribing a system to a more attractive, or easy tohandle, form !and in a similar way to changing coordi-nate systems", one can also perform field redefinitions ina field theory in order to rewrite the action or the fieldequations.

There is no unique prescription for redefining thefields of a theory. One can introduce auxiliary fields, per-form renormalizations or conformal transformations, oreven simply redefine fields to one’s convenience.

It is important to mention that, at least within a clas-sical perspective such as the one followed here, twotheories are considered to be dynamically equivalent if,under a suitable redefinition of the gravitational andmatter fields, one can make their field equations coin-cide. The same statement can be made at the level of theaction. Dynamically equivalent theories give exactly thesame results when describing a dynamical system thatfalls within the purview of these theories. There areclear advantages in exploring the dynamical equivalencebetween theories: we can use results already derived forone theory in the study of another, equivalent, theory.

The term “dynamical equivalence” can be consideredmisleading in classical gravity. Within a classical perspec-

tive, a theory is fully described by a set of field equa-tions. When we are referring to gravitation theories,these equations describe the dynamics of gravitating sys-tems. Therefore, two dynamically equivalent theoriesare actually just different representations of the sametheory !which also makes it clear that all allowed repre-sentations can be used on an equal footing".

The issue of distinguishing between truly differenttheories and different representations of the sametheory !or dynamically equivalent theories" is an intri-cate one. It has serious implications and has been thecause of many misconceptions in the past, especiallywhen conformal transformations are used in order toredefine the fields !e.g., the Jordan and Einstein framesin scalar-tensor theory". It goes beyond the scope of thisreview to present a detailed analysis of this issue. Werefer the interested reader to the literature and specifi-cally to Sotiriou et al. !2008" and references therein for adetailed discussion. Here we simply mention that, giventhat they are handled carefully, field redefinitions anddifferent representations of the same theory are per-fectly legitimate and constitute useful tools for under-standing gravitational theories.

In what follows, we review the equivalence betweenmetric and Palatini f!R" gravity with specific theorieswithin the Brans-Dicke class with a potential. It is shownthat these versions of f!R" gravity are nothing but differ-ent representations of Brans-Dicke theory with Brans-Dicke parameter #0=0 and −3/2, respectively. We com-ment on this equivalence and on whether preference toa specific representation should be an issue. Finally, weuse this equivalence to perform a classification of f!R"gravity.

A. Metric formalism

It was noticed quite early that metric quadratic gravitycan be cast into the form of a Brans-Dicke theory, and itdid not take long for these results to be extended tomore general actions that are functions of the Ricci sca-lar of the metric !Teyssandier and Tourrenc, 1983; Bar-row, 1988; Barrow and Cotsakis, 1988; Wands, 1994" #seealso Cecotti !1987", Wands !1994", and Flanagan !2003"for the extension to theories of the type f!R ,!kR" withk$1 of interest in supergravity$. This equivalence hasbeen reexamined recently due to the increased interestin metric f!R" gravity !Chiba, 2003; Flanagan, 2004a;Sotiriou, 2006b". We now present this equivalence insome detail.

We work at the level of the action but the same ap-proach can be used to work directly at the level of thefield equations. We begin with metric f!R" gravity. Forconvenience we rewrite here the action !5",

Smet =1

2%% d4x&− gf!R" + SM!g!",&" . !50"

One can introduce a new field ' and write the dynami-cally equivalent action,

11As mentioned in Sec. II.B, although the metric postulatesare manifestly satisfied, there are ambiguities regarding thephysical interpretation of this property and its relation with theEinstein equivalence principle !see Sec. VI.C.1".

461Thomas P. Sotiriou and Valerio Faraoni: f!R" theories of gravity

Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010

equivalent to

Smet =1

2!! d4x"− g#f$"% + f!$"%$R − "%& + SM$g#$,%% .

$51%

Variation with respect to " leads to

f"$"%$R − "% = 0. $52%

Therefore, "=R if f"$"%!0, which reproduces the action$5%.12 Redefining the field " by &= f!$"% and setting

V$&% = "$&%& − f„"$&%… , $53%

the action takes the form

Smet =1

2!! d4x"− g#&R − V$&%& + SM$g#$,%% . $54%

This is the Jordan-frame representation of the action ofa Brans-Dicke theory with Brans-Dicke parameter '0=0. An '0=0 Brans-Dicke theory #sometimes called“massive dilaton gravity” $Wands, 1994%& was originallyproposed by O’Hanlon $1972a% in order to generate aYukawa term in the Newtonian limit and has occasion-ally been considered in the literature $Deser, 1970;Anderson, 1971; O’Hanlon, 1972a; O’Hanlon and Tup-per, 1972; Fujii, 1982; Barber, 2003; Davidson, 2005;Dabrowski et al., 2007%. It should be stressed that thescalar degree of freedom &= f!$"% is quite different froma matter field; for example, like all nonminimallycoupled scalars, it can violate all of the energy condi-tions $Faraoni, 2004a%.

The field equations corresponding to the action $54%are

G#$ =!

&T#$ −

12&

g#$V$&% +1&

$"#"$& − g#$!&% ,

$55%

R = V!$&% . $56%

These field equations could have been derived directlyfrom Eq. $6% using the same field redefinitions that werementioned above for the action. By taking the trace ofEq. $55% in order to replace R in Eq. $56%, one gets

3!& + 2V$&% − &dVd&

= !T . $57%

This last equation determines the dynamics of & forgiven matter sources.

The condition f"!0 for the scalar-tensor theory to beequivalent to the original f$R% gravity theory can be seenas the condition that the change of variable &= f!$R%needed to express the theory as a Brans-Dicke one#Eq. $54%& be invertible, i.e., d& /dR= f"!0. This is a suf-ficient but not necessary condition for invertibility: it isonly necessary that f!$R% be continuous and one to one$Olmo, 2007%. By looking at Eq. $52%, it is seen that

f"!0 implies &= f!$R% and the equivalence of the actions$3% and $51%. When f" is not defined, or it vanishes, theequality &= f!$R% and the equivalence between the twotheories cannot be guaranteed $although this it is nota priori excluded by f"=0%.

Finally, we mention that, as usual in Brans-Dicketheory and more general scalar-tensor theories, one canperform a conformal transformation and rewrite the ac-tion $54% in what is called the Einstein frame $as opposedto the Jordan frame%. Specifically, with the conformaltransformation

g#$ → g#$ = f!$R%g#$ ' &g#$ $58%

and the scalar field redefinition &= f!$R%→ & with

d& ="2'0 + 32!

d&

&, $59%

a scalar-tensor theory is mapped into the Einstein framein which the “new” scalar field & couples minimally tothe Ricci curvature and has canonical kinetic energy, asdescribed by the gravitational action,

S$g% =! d4x"− g( R2!

−12

#(&#(& − U$&%) . $60%

For the '0=0 equivalent of metric f$R% gravity we have

& ' f!$R% = e"2!/3&, $61%

U$&% =Rf!$R% − f$R%

2!„f!$R%…2 , $62%

where R=R$&%, and the complete action is

Smet! =! d4x"− g( R2!

−12

#(&#(& − U$&%)+ SM$e−"2!/3&g#$,%% . $63%

A direct transformation to the Einstein frame, withoutthe intermediate passage from the Jordan frame, hasbeen discovered by Whitt $1984% and Barrow and Cot-sakis $1988%.

We stress once more that the actions $5%, $54%, and $63%are nothing but different representations of the sametheory.13 Additionally, there is nothing exceptionalabout the Jordan or the Einstein frame of the Brans-Dicke representation, and one can actually find infinitelymany conformal frames $Flanagan, 2004a; Sotiriou et al.,2008%.

B. Palatini formalism

Palatini f$R% gravity can also be cast in the form of aBrans-Dicke theory with a potential $Flanagan, 2004b;

12The action is sometimes called “R regular” by mathematicalphysicists if f"$R%!0 #e.g., Magnano and Sokolowski $1994%&.

13This has been an issue of debate and confusion #see, forexample, Faraoni and Nadeau $2007%&.

462 Thomas P. Sotiriou and Valerio Faraoni: f$R% theories of gravity

Rev. Mod. Phys., Vol. 82, No. 1, January–March 2010

= f 0(R) V () = R() f(R())

In particular, the simplest such theory adds a quadratic term in Ricci scalar (R2…)

Equivalent to a theory with a scalar of mass O(m0)

where

J. A. R. Cembranos, “Dark Matter from R2-gravity,” PRL 102, 141301 (2009) [0809.1653]

In the low curvature scenario R<< m02 this is standard GR + massive scalar minimally coupled to matter, with ! linearly coupled to the trace of the energy momentum tensor

2

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1

Dark Energy Dominated

MatterDominated

a

t/t0

a

0.1

-32

0.1+7x10

0.2

-32

0.2

+7x10

t/t0

FIG. 1: Evolution of the scale factor of the Universe: a(t)as function of time t (time is normalized to the age of theUniverse t0 ≃ 4.3 × 1017 s [24], and a(t0) = 1). The standardevolution is modulated by a coherent oscillation. Althoughthis oscillation has a very small amplitude, it has associateda high frequency (determined by the mass of the scalar mode,m0 ≃ 1 eV in the figure) and it can constitute the observedamount of DM (we do not take into account the inhomo-geneities coming from the clusterization process).

possible signatures of the model. We will see that theones already discussed are not dominant.

It is straight forward to check that the metric gµν =[1 + c1 sin(m0t)]ηµν is solution of the linearized Eq. (2),i.e. for c1 ≪ 1, without any kind of energy source. Inthis work we will argue that the energy stored in suchoscillations behaves exactly as cold DM and can explainthe missing matter problem of the Universe (see Fig. 1).We want to emphasize that this new mode of the metricis an independent degree of freedom that eventually willcluster and generate a successful structure formation if itis produced in the proper amount.

INTERACTION WITH STANDARD MODELPARTICLES

In order to be quantitative, we need to write the actionfor the new scalar degree of freedom of the metric in acanonical way. This work can be done directly [2] (what itis called inside the Jordan frame) or through a conformaltransformation of the metric [7] (what it is known as theEinstein frame). As we will work in the limit in whichR ≪ m2

0, in both cases, the metric can be expanded

perturbatively as

gµν = gµν +2

MPlhµν −

!

2

3

1

MPlφ gµν , (3)

where gµν is its classical background solution, hµν takesinto account the standard two degrees of freedom associ-ated with the spin-two (traceless) graviton, and φ corre-sponds to the new mode. This scalar field has associateda canonical kinetic term with the mass m0 as we havealready commented.

We will deduce the couplings of this scalar gravitonwith the SM fields by supposing that gravity is minimallycoupled to matter (in the Jordan frame). In such a case,φ is linearly coupled to matter through the trace of thestandard energy-momentum tensor:

Lφ−Tµν=

1

MPl

√6φT µ

µ . (4)

It implies that the couplings with massive SM particlesare given a tree level. In particular, the three body in-teractions are given by:

Ltree-levelφ−SM =

1

MPl

√6φ"

2 m2ΦΦ2 −∇µΦ∇µΦ (5)

+#

ψ

mψ ψψ −2 m2W W+

µ W−µ − m2Z ZµZµ

$

with the Higgs boson (Φ), (Dirac) fermions (ψ), and elec-troweak gauge bosons, respectively. In contrast withwhat has been claimed in previous studies, this fielddoes couple to photons and gluons due to the confor-mal anomaly induced at one loop by charged fermionsand gauge bosons. We find (following notation from [8]):

Lone-loopφ−SM =

1

MPl

√6φ%αEMcEM

8πFµνF

µν

+αscG

8πGa

µνGµνa

&

, (6)

The particular value of the couplings (cEM and cG) de-pends on the energy. We will be particularly interestedon the coupling with photons, which leads to potentialobservational decays of φ. We will perform all the calcu-lations restricting ourselves to the content of the SM butthe exact values of the couplings depend also on heavierparticles, charged with respect to these gauge interac-tions, that may extend the SM at higher energies.

ABUNDANCE

In principle, the above interactions of the scalar modewith the SM could produce a thermal abundance of φ at avery early stage of the Universe. However, it is expectedthat higher order corrections to Action (1) will be im-portant at this point. In fact, it will typically take place

“DM OR MODIFIED GRAVITY”?It is possible for ! to be the DM (a “light one”, ~ meV to MeV) produced via the misalignment mechanism (see last lecture), and decaying into γγ via a Planck-mass & loop suppressed coupling

J. A. R. Cembranos, “Dark Matter from R2-gravity,” PRL 102, 141301 (2009) [arXiv:0809.1653]

2

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


MatterDominated

a

t/t0

a

0.1

-32

0.1+7x10

0.2

-32

0.2

+7x10

t/t0







perturbatively as

gµν = gµν +2

MPlhµν −

!

2

3

1

MPlφ gµν , (3)



Lφ−Tµν=

1

MPl

√6φT µ

µ . (4)



1

MPl

√6φ"

2 m2ΦΦ2 −∇µΦ∇µΦ (5)

+#

ψ



$


Lone-loopφ−SM =

1

MPl

√6φ%αEMcEM

8πFµνF

µν

+αscG

8πGa

µνGµνa

&

, (6)


ABUNDANCE


2

0 0.2 0.4 0.6 0.8 1

0

0.2

0.4

0.6

0.8

1


MatterDominated

a

t/t0

a

0.1

-32

0.1+7x10

0.2

-32

0.2

+7x10

t/t0







perturbatively as

gµν = gµν +2

MPlhµν −

!

2

3

1

MPlφ gµν , (3)



Lφ−Tµν=

1

MPl

√6φT µ

µ . (4)



1

MPl

√6φ"

2 m2ΦΦ2 −∇µΦ∇µΦ (5)

+#

ψ



$


Lone-loopφ−SM =

1

MPl

√6φ%αEMcEM

8πFµνF

µν

+αscG

8πGa

µνGµνa

&

, (6)


ABUNDANCE


4

is φ1 ∼ 109 GeV, i.e. with a lower abundance (See Fig.2). If m0

>∼ 10 MeV, the gamma ray spectrum originatedby inflight annihilation of the positrons with interstellarelectrons is even more constraining than the 511 keV pho-tons [13].

On the contrary, if m0 < 2 me, the only decay channelthat may be observable is the decay in two photons. Wefind:

Γφ→γγ =α2

EMm30

1536π3M2Pl

|cEM |2 . (15)

By taking into account all SM charged particles and as-suming φ to be much lighter than all of them: cEM =11/3, and

Γφ→γγ ≃

!

2.5 × 1029s

"1 MeV

m0

#3$−1

. (16)

As it has been discussed in detail in [18, 19], if m0<∼

1 MeV, it is difficult to detect these gravitational decaysin the isotropic diffuse photon background (iDPB). Thegamma-ray spectrum at high Galactic latitudes can havecontributions from Galactic and extragalactic sources,but it seems well fitted at Eγ <∼ 1 MeV by assuming Ac-tive Galactic Nuclei (AGN) as main sources. The spec-trum observed by COMPTEL (-the Compton ImagingTelescope- over the energy ranges 0.8 − 30 MeV [20]),SMM (-the Solar Maximum Mission- for 0.3−7 MeV [21])and INTEGRAL ( 5−100 keV [22]), fall like a power law,with dN/dE ∼ E−2.4 [20], and it will dominate any pos-sible signal from R2-gravity if m0

<∼ 1 MeV.However, a most promising analysis is associated with

the search of gamma-ray lines at Eγ = m0/2 from local-ized sources, as the GC. The iDPB is continuum sinceit suffers the cosmological redshift. However, the mono-energetic photons originated by local sources may givea clear signal of R2-gravity. INTEGRAL has performeda search for gamma-ray lines originated within 13 fromthe GC over the energy ranges 0.08− 8 MeV. It has notobserved any line below 511 keV up to upper flux limitsof 10−5-10−2 cm−2 s−1, depending on line width, energy,and exposure [23]. Unfortunately, these flux limits are,at least, one order of magnitude over the expected fluxesfrom φ decays with m0

<∼ 1 MeV, even for cuspy ha-los. The photon flux originated by R2-gravity dependson m0 as ΦEγ=m0/2 ∝ m2

0. This strong dependence im-plies that only the heavier allowed region could be de-tected with reasonable improvements of present experi-ments [19].

CONCLUSIONS

In conclusion, we have studied the possibility that theDM origin resides in UV modifications of gravity. Al-though our results may seem particular of R2-gravity, the

Ex

clud

edY

uk

awa

Fo

rce

Ex

clud

edG

amm

aR

aysExcluded

Overproduction

!h

=0.104 -0.116

" 2

#cms-2

-1

$=(1.05-0.06)x10

511

-3

+9

10

11

12

13

10

10

10

10

10

-4 -2 0 2 4 6 8

10 10 10 10 10 10 10

"(G

eV)

1

m(eV)0

FIG. 2: Parameter space of the model: m0 is the mass of thenew scalar mode and φ1 is its misalignment when 3H ∼ m0

(we assume ge 1 = gs1 ≃ 106.75, and γs1 ≃ 1). The left sideis excluded by modifications of Newton’s law. The right oneis excluded by cosmic-ray observations. In the limit of thisregion, R2-gravity can account for the positron productionin order to explain the 511 keV line coming from the GCconfirmed by INTEGRAL [12] (up to m0 ∼ 10 MeV). Theupper area is ruled out by DM overproduction. The diagonalline corresponds to the NBDM abundance fitted with WMAPdata [24].

low energy phenomenology of the studied scalar mode isubiquitous in high energy corrections of the EHA com-ing from string theory, supersymmetry or extra dimen-sions (reproduced in form of dilatons, radions, gravis-calars or other moduli fields). The instability of the de-duced DM predicts a deviation from EEs at an energyscale: 1 TeV <∼ ΛG

<∼ 105 TeV. Consequences for hier-archy interpretations, baryogenesis or inflation deservefurther investigations.

Acknowledgments

I am grateful to K. Olive, M. Peloso, and M. Voloshinfor useful discussions. This work is supported inpart by DOE grant DOE/DE-FG02-94ER40823, FPA2005-02327 project (DGICYT, Spain), and CAM/UCM910309 project.

[1] K. S. Stelle, Phys. Rev. D 16, 953 (1977).

4




Γφ→γγ =α2

EMm30

1536π3M2Pl

|cEM |2 . (15)


Γφ→γγ ≃

!

2.5 × 1029s

"1 MeV

m0

#3$−1

. (16)







CONCLUSIONS


Ex

clud

edY

uk

awa

Fo

rce

Ex

clud

edG

amm

aR

aysExcluded

Overproduction

!h

=0.104 -0.116

" 2

#cms-2 -1

$=(1.05-0.06)x10

511

-3

+9

10

11

12

13

10

10

10

10

10

-4 -2 0 2 4 6 8

10 10 10 10 10 10 10

"(G

eV)

1

m(eV)0





Acknowledgments



How would you call that? A DM candidate? A modified gravity scenario? Note that the best you could hope to find is that DM decay is somewhat related to the Planck scale…

4




Γφ→γγ =α2

EMm30

1536π3M2Pl

|cEM |2 . (15)


Γφ→γγ ≃

!

2.5 × 1029s

"1 MeV

m0

#3$−1

. (16)







CONCLUSIONS


Excl

uded

Yukaw

aF

orc

e

Excl

uded

Gam

ma

Ray

sExcludedOverproduction

!h

=0.104 -0.116

" 2

#cms-2 -1

$=(1.05-0.06)x10

511

-3

+9

10

11

12

13

10

10

10

10

10

-4 -2 0 2 4 6 8

10 10 10 10 10 10 10

"(G

eV)

1m(eV)0





Acknowledgments



SO, MUCH ADO ABOUT NOTHING? Well, not really:

you may still ask yourself if there are modifications to the known long-range force (IR phenomenon!) directly coupled to the energy momentum tensor (~“responding to mass”)

All the evidence from O(10 Gpc) down to at least O(Mpc)-linear or quasi-linear regime-consistent with the latter hypothesis & no solid alternative of the first type has emerged,

while at small scales it is not conclusive, and in the solar system one certainly needs neither

You may think of it as a quantitative rather than qualitative difference wrt DM:

e.g. is the mass scale associated to the new degrees of freedoms <<10-20 eV scale we have directly probed in the Solar System (maybe of the level of 10-33 eV associated to the Hubble scale) or way above it?

another way to look at it: is the dynamics we see the manifestation of the exchange of light virtual particles “universally coupled to all matter”(note: even in this case we expect that these particles can be produced!) or to the standard universal gravitational coupling of real (relatively heavier) new particles to the matter we see?

For its minimality, henceforth we stick to the latter hypothesis (DM), i.e. new physics responsible for DM only in the UV rather than also in the IR

NEUTRINOS AS DARK MATTER?

m2atm ' 2.4 103 eV2

=c

'P

i mi

45 eV

Failed! This is a powerful argument excluding general classes of candidates (relativistic relics as DM, or so-called hot DM)

ΩDM≈0.3(Planck)⇒Σmi ≈ 15 eV

Condition 1. Must be massive (which is already a departure from SM...)

Fulfilled! Oscillations established, at least 2 massive states, measured splitting implies at least one state heavier than 0.05 eV

Condition 2. Must match cosmological abundance

Failed! Direct mass limits combined with splittings from oscillation experiments impose upper limit of about 7 eV to the sum (After KATRIN, potentially improved to ~0.7 eV)

Condition 3. Must allow for structure formation (of the right kind)

we will perform this computation in lecture 3

DM IS NOT “HOT” (IT IS NOT RELATIVISTIC)!dark matter is not “hot”: cannot have a relativistic velocity distribution(at least from matter-radiation equality for perturbation to grow)

This is the more profound reason why neutrinos would not work as DM, even if they had the correct mass: they were born with relativistic velocity distribution which prevents structures below O(100 Mpc) to grow till late!

Cartoon Picture:ν’s “do not settle” in potential wells that they can overcome by their typical velocity: compared

with CDM, they suppress power at small-scales

Neutrino free streaming

baryons, cdmΦ

ν

More quantitative picture:see R. Sheth on perturbation growth in radiation era (but some more notions in Lec. 4)

THE NUMERICAL PROOFΛCDM run vs. cosmology including neutrinos (total mass of 6.9 eV)

simulation by Troels Haugbølle, see

http://users-phys.au.dk/haugboel/projects.shtml

http://users-phys.au.dk/haugboel/projects.shtml

“MACRO” CANDIDATES

• QCD strangelets (E. Witten, PRD 30, 272 (1984), See also J. Madsen, astro-ph/9809032), require- that the most stable form of hadronic matter is one containing strange quarks (u-d-s

“nuggets”), if the opening of a third “Fermi well” is not (over)compensated by the larger mass penalty for s quarkanswer to this conjecture still unknown

- a first order QCD phase transitionLattice studies show that this is not realized Y. Aoki et al. Nature 443, 675 (2006) [hep-lat/0611014] Even if strangelets could exist, no mechanism present in the SM to make them the DM(might still make DM if BSM degrees of freedom alter early universe…)

Phenomenologically, we only ask that DM interaction rate Γ=σ n v is “small. But we only measure DM energy density ρ, not n. Hence Γ=(σ/m) ρ v small only requires (σ/m) small!

Composite/macroscopic object may make the DM, if σ/m small enough. Also, must be done before BBN. 2 possibilities in the SM I am aware of

• Primordial Black Holes (PBHs)Highly constrained & require new physics, anyway (more in a moment)

PRIMORDIAL BLACK HOLES (PBH)Need a mechanism seeding the small scale primordial perturbation allowing for direct collapse to BHs before BBN. This only “shifts” the need for new physics to the dynamics of this mechanism, usually non-minimal inflationary scenarios, e.g. S. Clesse and J. García-Bellido, “Massive Primordial Black Holes from Hybrid Inflation as Dark Matter and the seeds of Galaxies,'' PRD 92, 023524 (2015) [1501.07565]

8

It is maximal at the critical point of instability. The mild-waterfall therefore induces a broad peak in the scalarpower spectrum for modes leaving the horizon in phase-1and just before the critical point. The maximal ampli-tude for the scalar power spectrum is given by

Pζ(kφc) ≃ΛM2µ1φc

192π2M6Pχ2ψ2

0

. (34)

Depending of the model parameters, the curvature per-turbations can exceed the threshold value for leading tothe formation of PBH.This calculation was performed assuming that ψc =

ψ0. It is important to remark that for values of ψ0

and Λ given by Eqs. (11) and (10), one gets that N1,N2 and the amplitude of the scalar power spectrumdepend on a concrete combination of the parameters,Π ≡ M(φcµ1)1/2/M2

P, plus some dependence in χ2. Butχ2 itself depends only logarithmically on Π. As a result,χ2 varies by no more than 10% for relevant values of Π2.The parameters φc, µ1 and M appear to be degenerateand all the model predictions only depend on the valuegiven to Π. Nevertheless, Eq. (34) implicitly assumesthat field values are strongly sub-Planckian. In the op-posite case, when φc ∼ M ∼ MP, we find importantdeviations and the numerical results indicate that the wa-terfall is longer by about two e-folds and that the powerspectrum is enhanced by typically one order of magni-tude, compared to what is expected for sub-Planckianfields and for identical values of Π2.As a comparison between numerical and analytical

methods, we have plotted in Fig. 2 the power spectrum ofcurvature perturbations for Π2 = 300 and sub-Planckianfields, by using the analytical approximation given byEq. (31), by using the δN formalism including all terms(i.e. the contributions from phase-1 and phase-2) in N,φ

and N,ψ, and by integrating numerically the exact dy-namics of multi-field perturbations. As expected we finda good qualitative agreement between the different meth-ods. Nevertheless, one can observe about 20% differenceswhen using the analytical approximation, which actuallyis mostly due to the fact that N2

,ψ has been neglected. Inthe rest of the paper, we use the numerical results for abetter accuracy.In Figure 3 the power spectrum of curvature perturba-

tions has been plotted for different values of the param-eters. This shows the strong enhancement of power notonly for the modes exiting the Hubble radius in phase-1,but also for modes becoming super-horizon before fieldtrajectories have crossed the critical point. One can ob-serve that if the waterfall lasts for about 35 e-folds thenthe modes corresponding to 35 <∼ Nk

<∼ 50 are also af-fected. As expected one can see also that the combi-nation of parameters Π drives the modifications of thepower spectrum. We find that it is hard to modify inde-pendently the width, the height and the position of thepeak in the scalar power spectrum.Finally, for comparison, the power spectra assuming

ψc = ψ0 and averaging over a distribution of ψc values

!70 !60 !50 !40 !30 !20 !10 010! 10

10! 8

10! 6

10! 4

0.01

1

N k

PΖ!k"

FIG. 2. Power spectrum of curvature perturbations for pa-rameters M = φc = 0.1MP, µ1 = 3× 105MP. The solid curveis obtained by integrating numerically the exact multi-fieldbackground and linear perturbation dynamics. The dashedblue line is obtained by using the δN formalism. The dottedblue line uses the δN formalism with the approximation ofEq. (31).

are displayed. They nearly coincide for Π2 <∼ 300 but wefind significant deviations for larger values.

FIG. 3. Power spectrum of curvature perturbations for pa-rameters values M = 0.1MP, µ1 = 3 × 105MP and φc =0.125MP (red), φc = 0.1MP (blue) and φc = 0.075MP (green),φc = 0.1MP (blue) and φc = 0.05MP (cyan). Those pa-rameters correspond respectively to Π2 = 375/300/225/150.The power spectrum is degenerate for lower values of M,φand larger values of µ1, keeping the combination Π2 con-stant. For larger values of M, φc the degeneracy is broken:power spectra in orange and brown are obtained respectivelyfor M = φc = MP and µ1 = 300MP/225MP. Dashed linesassume ψc = ψ0 whereas solid lines are obtained after av-eraging over 200 power spectra obtained from initial con-ditions on ψc distributed according to a Gaussian of widthψ0. The power spectra corresponding to these realizations areplotted in dashed light gray for illustration. The Λ parame-ter has been fixed so that the spectrum amplitude on CMBanisotropy scales is in agreement with Planck data. The pa-rameter µ2 = 10MP so that the scalar spectral index on thosescales is given by ns = 0.96.

9

IV. FORMATION OF PRIMORDIAL BLACKHOLES

In this section, we calculate the mass spectrum of PBHthat are formed when O(1) curvature perturbations orig-inating from a mild-waterfall phase re-enter inside thehorizon and collapse. Furthermore we show that theabundance of those PBH can coincide with dark matterand can evade the observational constraints mentionedin Sec. II.The mass of a PBH whose formation is associated to a

density fluctuation with wavenumber k, exiting the Hub-ble radius |Nk| e-folds before the end of inflation, is givenby [16]

Mk =M2

P

Hke−2Nk , (35)

where Hk is the Hubble rate during inflation at time tk.In our model, Hk ≃

!

Λ/3M2P.

Assuming that the probability distribution of densityperturbations are Gaussian, one can evaluate the fractionβ of the Universe collapsing into primordial black holesof mass M at the time of formation tM as [7]

βform(M) ≡ ρPBH(M)

ρtot

"

"

"

"

t=tM

(36)

= 2

# ∞

ζc

1√2πσ

e−ζ2

2σ2 dζ (37)

= erfc

$

ζc√2σ

%

. (38)

In the limit where σ ≪ ζc, one gets

βform(M) =

√2σ√πζc

e−ζ2c2σ2 . (39)

The variance σ of curvature perturbations is related tothe power spectrum through ⟨ζ2⟩ = σ2 = Pζ(kM ), wherekM is the wavelength mode re-entering inside the Hubbleradius at time tM .For the study of PBH formation in our model, the

range of masses has been discretized and the value ofβform has been calculated for mass bins ∆M . This corre-sponds to PBH formed by the density perturbations reen-tering inside the horizon between tM and tM+∆M . Wehave considered mass bins whose width corresponds toone e-fold of expansion between these times, i.e. ∆Nk =∆ ln k = (∆ lnM)/2 = 1. This is sufficiently small for thepower spectrum to be considered roughly as constant oneach bin. Thus one gets σ(Mk) =

!

Pζ(k)∆ ln k.In the following, we use Eq. (38) instead of Eq. (39), so

that the results are accurate when σ >∼ ζc. Calculatingζc has been the subject of intensive studies [7–15], usingboth analytical and numerical methods. An analyticalestimate has been determined recently using a three-zonemodel to describe the PBH formation process [7]. For

a given equation of state parameter w at the time offormation, it is given by

ζc =1

3ln

3(χa − sinχa cosχa)

2 sin3 χa, (40)

with χa = π√w/(1 + 3w). In the present scenario, PBH

are formed in the radiation dominated era and one canset w = 1/3, leading to ζc ≃ 0.086. Different values havebeen obtained with the use of numerical methods, butit seems a general agreement that ζc lies in the range0.01 <∼ ζc <∼ 1 (see Fig. 3 of [7] for a comparison betweendifferent methods). For the sake of generality, ζc will bekept as a free parameter.

10!20 10!15 10!10 10!5 1 10510!16

10!14

10!12

10!10

10!8

10!6

M PBH! M!

Βform

10!20 10!15 10!10 10!5 1 1050.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

M PBH! M!

Βeq

FIG. 4. PBH abundances at the time of formation βform(M)(top panel) and at matter-radiation equality β(M,Neq) (bot-tom panel). The color scheme corresponds to the parame-ters given in Fig. 3. The blue dashed curve is obtained forΠ2 = 300 but with M = φc = 0.01MP and µ1 = 3 × 108MP,illustrating that PBH masses can be made arbitrarily largefor a given value of Π2. The critical curvature ζc has beenset so that the right amount of dark matter has been pro-duced at matter-radiation equality. Values of ζc are reportedin Table IV

In our scenario of mild waterfall, the peak in the powerspectrum of curvature perturbations is broad and cov-ers several order of magnitudes in wavenumbers. There-

9

IV. FORMATION OF PRIMORDIAL BLACKHOLES

In this section, we calculate the mass spectrum of PBHthat are formed when O(1) curvature perturbations orig-inating from a mild-waterfall phase re-enter inside thehorizon and collapse. Furthermore we show that theabundance of those PBH can coincide with dark matterand can evade the observational constraints mentionedin Sec. II.The mass of a PBH whose formation is associated to a

density fluctuation with wavenumber k, exiting the Hub-ble radius |Nk| e-folds before the end of inflation, is givenby [16]

Mk =M2

P

Hke−2Nk , (35)

where Hk is the Hubble rate during inflation at time tk.In our model, Hk ≃

!

Λ/3M2P.

Assuming that the probability distribution of densityperturbations are Gaussian, one can evaluate the fractionβ of the Universe collapsing into primordial black holesof mass M at the time of formation tM as [7]

βform(M) ≡ ρPBH(M)

ρtot

"

"

"

"

t=tM

(36)

= 2

# ∞

ζc

1√2πσ

e−ζ2

2σ2 dζ (37)

= erfc

$

ζc√2σ

%

. (38)

In the limit where σ ≪ ζc, one gets

βform(M) =

√2σ√πζc

e−ζ2c2σ2 . (39)

The variance σ of curvature perturbations is related tothe power spectrum through ⟨ζ2⟩ = σ2 = Pζ(kM ), wherekM is the wavelength mode re-entering inside the Hubbleradius at time tM .For the study of PBH formation in our model, the

range of masses has been discretized and the value ofβform has been calculated for mass bins ∆M . This corre-sponds to PBH formed by the density perturbations reen-tering inside the horizon between tM and tM+∆M . Wehave considered mass bins whose width corresponds toone e-fold of expansion between these times, i.e. ∆Nk =∆ ln k = (∆ lnM)/2 = 1. This is sufficiently small for thepower spectrum to be considered roughly as constant oneach bin. Thus one gets σ(Mk) =

!

Pζ(k)∆ ln k.In the following, we use Eq. (38) instead of Eq. (39), so

that the results are accurate when σ >∼ ζc. Calculatingζc has been the subject of intensive studies [7–15], usingboth analytical and numerical methods. An analyticalestimate has been determined recently using a three-zonemodel to describe the PBH formation process [7]. For

a given equation of state parameter w at the time offormation, it is given by

ζc =1

3ln

3(χa − sinχa cosχa)

2 sin3 χa, (40)

with χa = π√w/(1 + 3w). In the present scenario, PBH

are formed in the radiation dominated era and one canset w = 1/3, leading to ζc ≃ 0.086. Different values havebeen obtained with the use of numerical methods, butit seems a general agreement that ζc lies in the range0.01 <∼ ζc <∼ 1 (see Fig. 3 of [7] for a comparison betweendifferent methods). For the sake of generality, ζc will bekept as a free parameter.

10!20 10!15 10!10 10!5 1 10510!16

10!14

10!12

10!10

10!8

10!6

M PBH! M!

Βform

10!20 10!15 10!10 10!5 1 1050.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

M PBH! M!

Βeq

FIG. 4. PBH abundances at the time of formation βform(M)(top panel) and at matter-radiation equality β(M,Neq) (bot-tom panel). The color scheme corresponds to the parame-ters given in Fig. 3. The blue dashed curve is obtained forΠ2 = 300 but with M = φc = 0.01MP and µ1 = 3 × 108MP,illustrating that PBH masses can be made arbitrarily largefor a given value of Π2. The critical curvature ζc has beenset so that the right amount of dark matter has been pro-duced at matter-radiation equality. Values of ζc are reportedin Table IV

In our scenario of mild waterfall, the peak in the powerspectrum of curvature perturbations is broad and cov-ers several order of magnitudes in wavenumbers. There-

@ matter/radiation eq.

PS @ CMBscales

features responsible for direct collapse to BH @ horizon crossingBH mass function at formation

14

k (Mpc1)

P (k)

WIMP kinetic decoupling

P R(k)

109

108

107

106

105

104

103

102

101

103

102

101 1 10 10

210

310

410

510

610

710

810

910

1010

1110

1210

1310

1410

1510

1610

1710

1810

19

1010

109

108

107

106

105

104

103

102

Allowed regions

Ultracompact minihalos (gamma rays, Fermi -LAT)

Ultracompact minihalos (reionisation, WMAP5 e)

Primordial black holes

CMB, Lyman-↵, LSS and other cosmological probes

FIG. 6: Constraints on the allowed amplitude of primordial density (curvature) perturbations P (PR) at all scales. Here wegive the combined best measurements of the power spectrum on large scales from the CMB, large scale structure, Lyman-↵observations and other cosmological probes [152, 153, 156]. We also plot upper limits from gamma-ray and reionisation/CMBsearches for UCMHs, and primordial black holes [43]. For ease of reference, we also show the range of possible DM kineticdecoupling scales for some indicative WIMPs [74]; for a particle model with a kinetic decoupling scale kKD, limits do not applyat k > kKD. Note that for modes entering the horizon during matter domination, P (but not PR) should be multiplied by afurther factor of 0.81.

to be n . 1.17. Since large-scale observations actuallyput much stronger limits on the spectral index, we havealso considered the case of n = 0.968 ± 0.012, as ob-tained by WMAP observations, and constrained the al-lowed additional power below some small scale ks to be atmost a factor of 10–12 (assuming a step-like enhance-ment in the spectrum). As a third example, we haveobtained quasi-model-independent limits, of the order ofPR . 106, on perturbation spectra that can at leastlocally be well described by a power law. We would liketo stress, however, that it is intrinsically impossible toconstrain primordial density fluctuations in a completelymodel-independent way; one thus has to re-derive suchlimits for any particular model of, e.g., inflation whichproduces a spectrum that does not fall into one of theseclasses. Here, we have provided all the necessary tools todo so.

We have mentioned that present gravitational lens-ing data cannot be used to constrain the abundance ofUCMHs – essentially because they are simply not point-like enough, even in view of their highly dense and con-centrated cores. Future missions making use of the light-curve shape in lensing events, however, are likely to probeor constrain their existence. This would be quite remark-able as it would allow us to put limits on the power spec-trum without relying on the WIMP hypothesis for DM.Most of our formalism is readily extended, or can in factbe directly applied to, such constraints arising from grav-itational microlensing.

Finally, we have compiled an extensive list of the most

stringent limits on PR(k) that currently exist in the lit-erature for the whole range of accessible scales, from thehorizon size today down to scales some 23 orders of mag-nitude smaller. Direct and indirect observations of thematter distribution on large scales – in particular galaxysurveys and CMB observations – constrain the powerspectrum to be PR(k) 2 109 on scales larger thanabout 1Mpc. On sub-Mpc scales, on the other hand, onlyupper limits exist. From the non-observation of PBH-related e↵ects, one can infer PR . 102 101 on allscales that we consider here. UCMHs are much moreabundant and thus result in considerably stronger con-straints, PR . 106, down to the smallest scale at whichDM is expected to cluster (this depends on the nature ofthe DM; for typical WIMPs like neutralino DM, e.g., itfalls into the range kmax 8 104 3 107 Mpc1).

It is worth recalling that the observational evidencefor a simple, nearly Harrison-Zel’dovich spectrum of den-sity fluctuations is obtained by probing a relatively smallrange of rather large scales. The limits we have providedhere will thus be very useful in constraining any model ofe.g. inflation, or phase transitions in the early Universe,that predicts deviations from the most simple case andwhich would result in more power on small scales.

T. Bringmann, P. Scott ,Y. Akrami, PRD 85, 125027 (2012) [1110.2484]

poorly constrained

scales!

BOUNDS ON PBH6

1015 1020 1025 1030 1035 1040

BH mass, g

108

106

104

102

100

PB

H/

DM

Femto EROS+MACHO

Hawking+-rays

FIRAS

WMAP3

Star Formation

Superradiance

104 GeV

cm3

103 GeV

cm3

102 GeV

cm3

FIG. 3: Constraints on the fraction of PBHs as DM. Shadedregions are excluded. The blue shaded regions correspond tothe revised constraints derived in this paper assuming the DMdensities of (104,103,102) GeV/cm3 and the velocity disper-sion of 7 km/s. The red shaded regions represent the existingconstraints coming from various observations as explained inSect. III A.

obtain these constraints we have considered stars withmasses M M 7M, the progenitors of WDs, andstars with 8M M 15M which become NSs. WDslead to better constraints for low PBHs masses, whileNSs are more competitive at high masses. The transi-tion between the two regimes is around mBH 1020 g.For comparison, the constraints of Ref. [13] are also pre-sented in Fig. 2. Both constraints are derived assumingthe velocity dispersion of v = 7 km/s. Such velocities arecharacteristic of globular clusters and dwarf spheroidalgalaxies. The constraints for other velocity dispersionscan be obtained by a simple rescaling as described above.

At large PBH masses, the revised constraints are sub-stantially more stringent than those of Ref. [13]. In thismass range the energy loss mechanism is so ecient thatall the gravitationally bound PBH that ever cross the starhave time to be captured and transferred to a compactremnant. As pointed out in Sect. II, the number of suchtrajectories turned out to be much larger than it was as-sumed in Ref. [13]. In this range of masses the gain inthe total captured mass (and therefore, the improvementin the constraints) is the ratio between the two curvesof Fig. 1 calculated at r = R R, which is close to1.8 103. For stellar masses M > 8M, this number isreduced by a factor two.

At small PBH masses, the revised constraints are,on the contrary, somewhat worse than the estimate ofRef. [13]. The reason for this is that some of the PBHthat are inside the star by the end of the adiabatic con-traction, are in fact on elongated orbits and spend mostof the time outside the star. When the energy losses arenot ecient, these PBH do not have enough time to losetheir energy and get captured by the star. Such orbitswere, but should not have been, included in the estimatesin Ref. [13], hence the di↵erence.

Clearly, most stringent constraints come from observa-

tions of compact stars in regions with a high DM densityand a low DM velocity dispersion. As a benchmark, weconsider the values DM = (104, 103, 102) GeV cm3 andv = 7 km s1. The constraints that would result inthese cases are shown in Fig. 3 together with the otherexisting constraints. The strongest constraints shown inlight blue correspond to the highest value of DM den-sity considered, i.e DM 104 GeV cm3 and decreaselinearly for lower values of DM density. The correspond-ing conditions may be present in the cores of metal-poorglobular clusters at the epoch of star formation, if theyare proved to be of a primordial origin [38] (see detaileddiscussions in [12, 13]). Another place where similar con-ditions, albeit with somewhat lower densities, could existare dwarf spheroidal galaxies that are considered to beDM dominated [39, 40] and have very low velocity dis-persions [39]. However, at present compact objects suchas NS or WD have not been observed in dwarf spheroidalgalaxies. Nonetheless, surveys for pulsars and X-ray bi-naries have already revealed some glimpses of NSs ex-istence in dSph galaxies [41, 42], even though a clearobservation is still lacking.

IV. CONCLUSIONS

In this paper we have considered the adiabatic contrac-tion of DM during the formation of a star. By simulatingthe behavior of 30 million particles, we reconstructedthe phase space distribution of the DM at the end of thestar formation. In particular, the number of particlesn(r) within a given radius r was found to be propor-tional to r1.5, which corresponds to the DM density pro-file (r) / r1.5, in agreement with previous calculationsand the Liouville theorem.

At the same time, we have found that the adiabaticcontraction creates a rather special distribution of par-ticle orbits. Namely, if one considers the particles thatcontribute to n(r) for a small r, a substantial (O(1)) frac-tion of them have very elongated orbits with periastrasmaller than r. In fact, the number of particles (r) thathave periastra smaller than r scales as (r) / r. Suchparticles spend only a small fraction of time close to thecenter, so their individual contributions to the densityat small r are suppressed. However, they are numerousenough to contribute non-negligibly to the density. Atr = R, there are about 1.8 103 more particles thathave periastra smaller than r than there are particleswithin r.

This has implications for the DM capture by stars aftertheir formation. A large number of particles that con-stitute the DM cusp around the newly-formed star haveorbits that cross the star, which potentially leads to theircapture. This factor has not been taken into account inthe previous estimates.

As an application, we have considered the capture ofPBH by stars, which leads to the constraints on thePBH abundance. We have recalculated the constraints of

see e.g. F. Capela, M. Pshirkov, P. Tinyakov, PRD 90, 083507 (2014) [arXiv:1403.7098] & refs. therein

Even so, a number of constraints exist

which exclude a dominant “monochromatic” PBH contribution to DM at any mass but 10-14-10-9 M⦿ (potentially excluded as well)

BH evaporate (emitting gamma-rays) on times comparable or shorter than lifetime of Universe

BH would induce “interferometry” pattern in the energy spectrum of lensed GRBs

PBH capture in stars catalyze fast conversion in BH, while “old” evolved objects like WD or NS are observed (DM-density dependent bound)

bounds from CMB spectral distortions, secondary anisotropies induced (e.g. reionization) either via accretion byproducts or by explosive runaway instability (if rotating, due to “plasma mass”)

direct searches via micro-lensing, plus other arguments (do not strictly require them to be BHs)

SOME NEWSAlthough limits can be weakened a bit for PBH masses distributed over many decades, the

best hope to evade them is to assume that a sizable evolution of the mass function takes place (e.g. born sub-solar, thus evading CMB constraints, merging to tens of M⦿ , to evade lensing)

Recently, this way out has been shown to be on shaky grounds!

CMB excludes that more than 3-4% of DM (of any mass) has converted into any invisible radiation (like GW), over the whole history from recombination to now!

Figure 1. Comparison of the lensed TT (top) and EE (bottom) power spectra for several de-caying DM lifetimes and a fixed abundance f

dcdm

= 0.2. Boxes show the (binned) cosmic varianceuncertainty.

– 8 –

Figure 4. Constraints on the decaying dark matter fraction fdcdm

as a function of the lifetime

dcdm

in the long-lived and intermediate regime. All datasets also include CMB low-` data from eachspectrum and the lensing reconstruction. Blue (red) lines and contours refer to the case without (with)high-` polarization data. Inner and outer coloured regions denote 1 and 2 contours, respectively.

• We argued in section 3.1, second bullet, that there is an intermediate regime givenroughly by

dcdm

2 [10

1, 10

3

] Gyr

1, for which the DM decay starts after recombina-tion and decaying DM has totally disappeared by now. Results for this case are shownin the bottom-right panel of Fig. 4. In that case, the CMB is mostly insensitive tothe time of the decay. Our runs show that in this regime, the CMB can tolerate up to4.2% of dcdm at 95% CL. This is an important number, standing for the fraction ofdark matter that can be converted entirely into a dark radiation after recombination,without causing tensions with the data. Although we do not show the full parameterspace up to

dcdm

' 10

3

Gyr

1, we have checked that the behaviour stays the same(this can also be inferred from the smallest values of

dcdm

plotted in Fig. 5.)

• Finally, we show in Fig. 5 the constraints applicable to the short-lived regime,

dcdm

>10

3

Gyr

1, for which the decay happens before recombination. To accelerate the explo-ration of the parameter space, we scan over log

10

(

dcdm

) with a flat prior. We howevercut at 10

6

Gyr

1 for obvious convergence issues, and consider chains as converged whenR 1 < 0.1. Changing the upper bound would not change at all our conclusions.Note that with such a bound, we are also covering the region of parameter space for

– 17 –

either PBH do not make a sizable fraction of the DM or their mass function evolution should be negligible: in the two merger events seen by LIGO ~5% conversion of mass into GW!

V. Poulin, PDS, J. Lesgourgues, arXiv:1606.02073

MICROLENSING

u = /E

e.g. L. Wyrzykowski et al.,arXiv:1106.2925 & refs. therein

Several searches (EROS, OGLE...) for μlensing events towards Magellanic Cloud exclude dominant MACHOs component as halo DM from 10-7 to 10 M⦿

idea: Compact object crossing along the los acts as a gravitational lens, inducing a time-dependent magnification pattern depending on the geometry and mass. From

limits to the rate of such events, DM fraction limits to the associated mass

ang. distance source-lens

depends on lens mass and geometry

tE =time to cross Einstein angular size

MICROLENSING CONSTRAINTS

2 1034g' 1026g

some events expecteddue to stellar BH

← goes to

EXAMPLE OF LIMITS ON “MACHOS” Too large DM “particle” would disrupt bound systems of different orbital sizes

(function of mass) via associated time-dependent gravitational potentials

• thickness of disks: MX < 106 M⦿

satellites, globular clusters: MX < 103 M⦿

• Halo-wide binaries: Method proposed in J. Yoo, J. Chaname, A. Gould, Astrophys. J. 601, 311 (2004)

H-W.Rix and G. Lake, astro-ph/9308022 & refs. therein

M. A. Monroy-Rodríguez and C. Allen, ApJ J. 790 159 (2014) [1406.5169]

Latest bound MX ≾ 100 M⦿ to ≾ 5 M⦿

(for higher and higher “binary quality” cuts)7

FIG. 7.— Exclusion contour plot at 95% confidence level. The dashed,the dotted and the long dashed lines represent the microlensing-based limitsfrom EROS (Afonso et al. 2003) and MACHO (Alcock et al. 2001), and thelimit based on disk stability (Lacey & Ostriker 1985), respectively. Our limitsfrom the 3.′′5 < ∆θ < 900′′ sample are represented as a solid line.

provided that the combined limits from the three curves ruledout each mass separately.Hence, the only remaining window open for MACHOs

would be black holes with a mass function strongly peakedatM ∼ 35 M⊙. Even this model is more strongly constrainedthan shown Figure 7 because the MACHO (Alcock et al.2001) results and our results each separately place weak limitson this model, which when combined comes close to ruling itout.

7. CONCLUSIONSIn this paper, we have investigated the evolution of halo

wide binaries in the presence of MACHOs and estimatedupper limits of MACHO density as a function of their as-sumed mass by comparing our simulations to the sample ofhalo wide binaries of CG. We exclude MACHOs with massesM > 43 M⊙ at the standard local halo density ρH at the 95%confidence level.MACHOs have been a major dark-matter candidate ever

since observations first established that this mysterious sub-stance dominates the mass of galaxies. Prodigious efforts overseveral decades have gradually whittled down the mass rangeallowed to this dark-matter candidate. However, the windowfor MACHOs with 30 M⊙ ! M ! 103 M⊙ remained com-pletely open while constraints in the range 103 M⊙ ! M !106 M⊙ were somewhat model dependent. Our new limits onMACHOsM > 43 M⊙ all but close this window.

We are grateful to Éric Aubourg and Kim Griest for pro-viding data for Figure 7. We thank John Bahcall, BohdanPaczynski, and especially Scott Tremaine for valuable com-ments that significantly improved the paper. A detailed cri-tique by referee Terry Oswalt also greatly improved the pa-per. This work was supported by grant AST 02-01266 fromthe NSF.

APPENDIX

IONIZED BINARIESAfter disruption of a binary system, the two ionized members remain in similar Galactic orbit, so it is only the separation along

the direction of the orbital motion that can keep increasing, while the perpendicular separation oscillates.In the Coulomb regime, the average post-ionization gain in the relative velocity of binaries along the orbital direction is (see

eq.[5]),!

"

v2∥#

=!

32πG2ρM∆t3v

lnΛ, (A1)

where ∆t is the remaining time to 10 Gyr after disruption. For a diffusive process that is uniform over time ∆t, the root-mean-square separation parallel to the orbital motion is

!

"

d2∥#

=!

"

v2∥#

∆t2

3= 2000 pc

$

ρ

ρH

%1/2$ M30M⊙

%1/2& v300 km s−1

'−1/2$

lnΛ

5.1

%1/2$∆t

10 Gyr

%3/2

, (A2)

where the Coulomb logarithm is calculated at the scaled quantities.For a conservative limit of the tidal radius, at = 3 pc, the time required for ionized binaries to separate farther than at is 0.13 Gyr,

so that only binaries ionizing within the last ∼ 1% of the age of the Galaxy have a significant chance to be confused with boundsystems, even for the lowest mass perturbers that we can effectively probe. In the tidal limit, binaries escape with characteristicvelocities of the transition separation∼ 104 AU, i.e., 300 m s−1. Hence, they drift one tidal radius in only 10 Myr, so their impactis even smaller. Nevertheless, since there are only a handful of binaries in the widest-separation bins, it is important to make acareful estimate of the contribution from ionizing binaries. We take account of ionized binaries in the simulations as follows.Binaries are considered ionized either when they have positive energy or they have a > at . They are then assigned a relativevelocity equal to their escape velocity in the former case, or zero in the latter. A random orbital direction is chosen. The ionizedbinaries in the Coulomb regime then continue to suffer perturbations to the end of the simulations whose effect we calculateusing equation (A1) and (A2). At the end of the simulation, the binary is assigned a transverse separation drawn randomly froma sinusoidal distribution of amplitude,

d⊥,MAX =v⊥Ω= 26 pc

& v⊥km s−1

'

, (A3)

where v⊥ is the final transverse velocity, Ω =√2vc/R0 is the epicyclic frequency, and R0 is the Galactocentric distance. Finally,

we “observe” the ionized binary from a random orientation and record the projected separation. We find that the ionized binaries

4

FIG. 2.— Binary distributions as a function of semi-major axis. 100,000binaries are generated following an arbitrarily chosen flat (α=1) distributionrepresented as a thick solid line. The halo density is set to be ρH . The squares,triangles and circles represent binary distributions for three different massesof perturber, after T = 10 Gyrs evolution. The fitting curves for each modelare shown as dashed lines.

est impacts in the tidal regime, even though as we discuss in§ 3, the single closest encounter dominates. The mass of eachbinary component is set to be 0.5 M⊙. We evolve 100,000binary systems in each simulation. To illustrate the depen-dence on the mass of perturbers, we show their effects on anartificial initial binary distribution that is independent of semi-major axis (see Fig. 2). As expected, for the binary systemsthat are initially tightly bound, the final distributions are al-most the same as the initial ones regardless of the mass of theperturbers. However, at wide separations, the distributionsare driven to a new power law, which becomes steeper withincreasing mass.This approach works well for any individual initial power-

law distribution, but is too time-consuming to process the verylarge number of distributions required for the comparison ofdata and models. In § 5.5, we introduce a scattering-matrixformalism that substantially improves the efficiency of mass-production Monte Carlo simulations.

4.7. Fitting FormulaMotivated by the fact that the final distribution is well-

approximated by power laws at each extreme, we use a two-line (five-parameter) fitting formula given by,

H(x) =!

f (x)−n +g(x)−n"−1/n

, (10)where f (x) and g(x) are (two-parameter) straight lines in theirargument, x = log(a/AU), each corresponding to the respec-tive asymptotic behaviors of H(x). The fifth parameter n per-mits a smooth transition between f (x) and g(x) in the inter-mediate region. We calculate the five parameters of equa-tion (10) by minimizing χ2 for a given data set, and the fittingcurves for four different masses of perturber are representedas dashed lines in Figure 2.

5. RESULTSHere, we present our main calculations on the evolution of

binary distributions with various initial slopes under the influ-ence of various perturber masses and halo densities, and weevaluate the transition separations. Although the semi-majoraxis of a binary system is a direct indicator of the bindingenergy of the system and is the theoretically most tractablequantity, it is not observable. It is the angular separations onthe sky that we can directly measure from observations. Tocompare our results with the data, we calculate physical sep-arations projected on the sky plane and convolve these withan adopted distance distribution to predict the binary distribu-tions as a function of angular separation.In principle, one could compare models directly to the ob-

served projected separations since CG give individual dis-tance estimates to each binary. However, while the observa-tional selection function is quite simple for angular separa-tions (essentially just a pair of Θ-functions), it is rather com-plex and would be extremely difficult to model for projectedphysical separations. Hence, we compare our models to themost directly observed quantity: angular separations.

5.1. Semi-Major Axis, aWe begin the simulation with a flat distribution,

dN/d loga ∝ const for calculation of scattering matri-ces. We then investigate the dependence of the resultingfinal binary distribution on perturber mass and halo density.Figure 3 shows a sample of our results with the initialpower-law distribution, dN/d loga ∝ a−0.567, in agreementwith the observations as summarized in § 2. Two models withthe same perturber mass but widely different halo densities,

FIG. 3.— Evolution of binary distributions with the initial power law, α =1.567 obtained in § 2, represented as a thick solid line. 50,000 binaries areevolved in the presence of perturbers of mass 1000 M⊙ The circles show thebinary distributions as functions of semi-major axis while the triangles showthe distributions of projected physical separations of the binary componentsonto the sky plane. The solid lines represent fitting curves for semi-majoraxis, and the dashed lines for projected physical separation. Two verticalarrows indicate transitions, at , from the unperturbed to the perturbed regimes.

A RECENT ONE…

T. D. Brandt,“Constraints on MACHO Dark Matter from Compact Stellar Systems in Ultra-Faint Dwarf Galaxies,” ApJ Letters, in press [1605.03665]

Presence/stability of stellar systems in ultra-faint dwarf Galaxies close the 20-100 M⦿ window

(notably a stellar cluster in Eridanus II could push the limit to ~3 M⦿, barring ~1% chance alignments)

5

Fig. 4.— Constraints on MACHO dark matter from microlens-ing (blue and purple, Alcock et al. 2001; Tisserand et al. 2007) andwide Galactic binaries (green, Quinn et al. 2009), shown togetherwith the constraints from the survival of compact ultra-faint dwarfgalaxies and the star cluster in Eridanus II. I conservatively adopt adark matter density of 0.02 M⊙ pc−3 in Eri II and 0.3 M⊙ pc−3 inthe ultra-faint dwarfs, assume a three-dimensional velocity disper-sion σ = 8 kms−1, and use two definitions of the heating timescale.A low-density halo and initially compact cluster weaken the con-straints from Eri II. Even in this case, assuming dark matter halosto have the properties that are currently inferred, MACHO darkmatter is excluded for all MACHO masses !10−7 M⊙.

portional to the cluster mass (Binney & Tremaine 2008),and the cluster in Eri II is 1.5–2 orders of magnitude lessmassive than Fornax 4 (Mackey & Gilmore 2003), theFornax globular cluster nearest the center of that dwarf(at 240 pc in projected separation). This scenario there-fore requires very different dark matter halos in the twogalaxies or severe mass loss during Eri II’s inspiral, andalso luck to catch the cluster on the point of disruption.This problem of coincidence is generic to any scenario inwhich Eri II’s cluster was initially compact. The proba-bility of observing the system in such a transient state issignificantly higher if the cluster’s age is ∼3 Gyr ratherthan ∼12 Gyr.Other possibilities to evade the constraints include

an intermediate-mass black hole (!104 M⊙) to provide

binding energy, or a chance alignment such that the clus-ter only appears to reside in the center of Eri II. Bothwould be surprising. Such a black hole would have a masscomparable to the total stellar mass of its host galaxy. Amassive black hole would also be expected to host a re-laxed MACHO cluster of comparable mass, in which caseit may not avoid the problem of dynamical heating at all.A chance alignment of a cluster physically located at thegalaxy’s half-light radius is possible; the most naıve esti-mate, the fraction of solid angle lying within a few rh inprojection, gives a chance alignment probability of ∼1%at a physical distance of ∼300 pc from the galaxy core.While many scenarios could, in principle, account for

the survival of the star cluster in Eri II, it is harder toappeal to coincidence for the entire sample of compactultra-faint dwarfs. Assuming the measured velocity dis-persions to reflect the properties of their dark matterhalos, these dwarfs should have much larger half-lightradii if their dark matter is all in the form of MACHOs!10 M⊙. The strongest constraints, however, may comefrom the cluster in Eri II, and could be improved withbetter data. Precise photometry with the Hubble SpaceTelescope could resolve the question of whether the clus-ter is intermediate-age or old, while spectroscopy of clus-ter members and nonmembers would give another probeof Eri II’s dark matter content. While future observa-tions will determine the strength of the constraints fromEri II, existing data from Eri II and from the sample ofcompact ultra-faint dwarfs appear sufficient to rule outdark matter composed exclusively of MACHOs for allmasses above ∼10−7 M⊙.

I thank Ben Bar-Or, Juna Kollmeier, Kris Sigurdson,and especially Scott Tremaine for helpful conversationsand suggestions, and an anonymous referee for helpfulcomments. This work was performed under contract withthe Jet Propulsion Laboratory (JPL) funded by NASAthrough the Sagan Fellowship Program executed by theNASA Exoplanet Science Institute.

REFERENCES

Ackermann, M., Albert, A., Anderson, B., et al. 2015, PhysicalReview Letters, 115, 231301

Alcock, C., Allsman, R. A., Alves, D. R., et al. 2001, ApJ, 550,L169

Baade, W., & Hubble, E. 1939, PASP, 51, 40Bechtol, K., Drlica-Wagner, A., Balbinot, E., et al. 2015, ApJ,

807, 50Belokurov, V., Walker, M. G., Evans, N. W., et al. 2009,

MNRAS, 397, 1748Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second

Edition (Princeton University Press)Bird, S., Cholis, I., Munoz, J. B., et al. 2016, ArXiv e-prints,

1603.00464Brooks, A. M., & Zolotov, A. 2014, ApJ, 786, 87Chaname, J., & Gould, A. 2004, ApJ, 601, 289Chandrasekhar, S. 1943, Reviews of Modern Physics, 15, 1Clowe, D., Bradac, M., Gonzalez, A. H., et al. 2006, ApJ, 648,

L109Cole, D. R., Dehnen, W., Read, J. I., & Wilkinson, M. I. 2012,

MNRAS, 426, 601Crnojevic, D., Sand, D. J., Zaritsky, D., et al. 2016, ArXiv

e-prints, 1604.08590Ferrer, F., & Hunter, D. R. 2013, JCAP, 9, 005Georgiev, I. Y., Hilker, M., Puzia, T. H., Goudfrooij, P., &

Baumgardt, H. 2009, MNRAS, 396, 1075

Goerdt, T., Moore, B., Read, J. I., Stadel, J., & Zemp, M. 2006,MNRAS, 368, 1073

Griest, K. 1991, ApJ, 366, 412Harris, W. E. 1996, AJ, 112, 1487Hernandez, X., Matos, T., Sussman, R. A., & Verbin, Y. 2004,

PhRvD, 70, 043537Hodge, P. W. 1961, AJ, 66, 83Inoue, S. 2011, MNRAS, 416, 1181Kharchenko, N. V., Piskunov, A. E., Roser, S., Schilbach, E., &

Scholz, R.-D. 2005, A&A, 438, 1163King, I. R. 1966, AJ, 71, 64Klypin, A., Kravtsov, A. V., Valenzuela, O., & Prada, F. 1999,

ApJ, 522, 82Koposov, S. E., Belokurov, V., Torrealba, G., & Evans, N. W.

2015a, ApJ, 805, 130Koposov, S. E., Casey, A. R., Belokurov, V., et al. 2015b, ApJ,

811, 62Lacey, C. G., & Ostriker, J. P. 1985, ApJ, 299, 633Laevens, B. P. M., Martin, N. F., Bernard, E. J., et al. 2015, ApJ,

813, 44Lovell, M. R., Eke, V., Frenk, C. S., et al. 2012, MNRAS, 420,

2318Mackey, A. D., & Gilmore, G. F. 2003, MNRAS, 340, 175Maraston, C. 2005, MNRAS, 362, 799

LESSON LEARNEDDark Matter requires “new physics”, beyond known theories, in order to

be produced, and most likely is made of new degrees of freedom itself

! How much DM is out there! DM is not “hot” (non-relativistic velocity distribution… as for the neutrinos) ! Must be stable or long-lived ! DM must be sufficiently heavy! DM... is dark, and dissipationless! DM is collisionless (or not very collisional)! DM has small interactions with ordinary matter

Cosmology and astrophysics also give us some “particle physics” constraint

Only a handful of similar indications for BSM: explains the interest of particle physicists!

let us review these constraints. We assume everywhere that there is a single species accounting for (most of) DM. Multi-component is certain (at least ν’s!) but one expect one to dominates (in the SM sector, for instance, it happens!), to prevent further “fine tuning”

AN IMPORTANT NUMBER...

Recent determination (Planck 2015, 68% CL)

Ωch2=0.1188±0.0010, i.e. Ωc~0.26

s0 = 2889

T,0

2.725

3

cm3

c =3H2

0

8GN= 1.054 105h2GeV cm3

X,0 = MX nX,0 = MX s0 Y0

where heff ~ 2+3×2(4/11)×7/8~3.91 comes from accounting for γ’s & ν’s

[Main] Goal (Lec. 3-4-5) compute value of number to entropy density ratio, Y0

Xh2 = 2.74 108MX

GeV

Y0

LIFETIME

Figure 1. CMB temperature power spectrum for a variety of models, all with the same parameters100 s,!ini

dcdm

,!b

, ln(1010As), ns, reio = 1.04119, 0.12038, 0.022032, 3.0980, 0.9619, 0.0925 takenfrom the Planck+WP best fit [26]. For all models except the “Decaying CDM” one, the decayrate

dcdm

is set to zero, implying that the “dcdm” species is equivalent to standard cold DM with apresent density !

cdm

= !ini

dcdm

= 0.12038. The “Decaying CDM” model has dcdm

= 20 km s1Mpc1,the “Tensors” model has r = 0.2, and the “Open” (“Closed”) models have k = 0.02 (0.2). Themain di↵erences occur at low multiples and comes from either di↵erent late ISW contributions ornon-zero tensor fluctuations.

To check (ii), we plot in Figure 1 the unlensed temperature spectrum of models with dcdm

set either to 0 or 20 km s1Mpc1

3. To keep the early cosmological evolution fixed, we stickto constant values of the density parameters (!ini

dcdm

, !b

), of primordial spectrum parameters(As, ns) and of the reionization optical depth

reio

. Of course, for dcdm

= 0, the dcdmspecies is equivalent to standard cold DM with a current density !

cdm

= !ini

dcdm

. We need tofix one more background parameter in order to fully specify the late cosmological evolution.Possible choices allowed by class include h, or the angular scale of the sound horizon atdecoupling, s = rs(t

dec

)/ds(tdec

). We choose to stick to a constant value of s, in order toeliminate the e↵ect (i) described above, and observe only (ii). We see indeed in Figure 1 thatwith such a choice, the spectra of the stable and decaying DM models overlap everywhereexcept at small multipoles. To check that this is indeed due to a di↵erent late ISW e↵ect, weshow in Figure 2 the decomposition of the total spectrum in individual contribution, for thestable model and a dcdm model in which the decay rate was pushed to 100 km s1Mpc1.

Since the dominant e↵ect of decaying DM is a modification of the small-` part of theCMB temperature spectrum, in the rest of the analysis, it will be relevant to investigate de-

3It is useful to bear in mind the conversion factor 1 km s

1Mpc

1= 1.02 10

3Gyr

1.

– 7 –

Figure 3. Matter power spectrum P (k) (computed in the Newtonian gauge) for the same modelsconsidered in Figure 1. The black curve (Stable CDM) is hidden behind the red one (Tensors).

When introducing the curvature parameter, one gets a combination of the first two e↵ectsonly. Moreover, variations of

dcdm

and k leading to an e↵ect in the CMB of the sameamplitude give e↵ects on the P (k) with very di↵erent amplitudes. This comparison showsthat, at least in principle, CMB lensing e↵ects and direct constraints on P (k) may help tobreak degeneracies, and to measure

dcdm

independently of k and r. This can only beconfirmed by a global fit to current observations.

3.2 The data

The parameter extraction is done using a Metropolis Hastings algorithm, with a Choleskydecomposition to better handle the large number of nuisance parameters [27]. We investi-gate two combinations of experiments which we denote by A and B. Both share the Plancklikelihoods, consisting of the low-`, high-`, lensing reconstruction and low-` WMAP polari-sation, as well as the WiggleZ data [28], and the BOSS measurement of the Baryon AcousticOscillation scale at z = 0.57 [29]. The set B adds the BICEP2 public likelihood code [16].We used the publicly available Monte Python4 code [30] for the analysis.

We performed the analysis selecting flat priors for the following set of parameters

!b, H0

, As, ns, reio,!dcdm+dr

,dcdm

, r,k ,

in addition to the other nuisance parameters for the Planck likelihood, omitted here forbrevity. The first five cosmological parameters stand respectively for the baryon density, the

4https://github.com/baudren/montepython_public

– 9 –

Even, if decaying into invisible channels, bounds still exist from CMB & Large Scale structures

(under the sole assumption of relativistic byproducts)

τ≿160 Gyr (CMB only)

τ≿170 Gyr (with other consistent data)

V. Poulin, PDS, J. Lesgourgues, arXiv:1606.02073

B. Audren et al. JCAP 1412, 028 (2014) [1407.2418]

at very least one order of magnitude longer than age of the universe

CMB mostly affected by late integrated Sachs-Wolfe effect (modification of homogeneous & perturbed

DM density at late times affects evolution of metric fluctuation) LSS helps in breaking partial degeneracy

with curvature & tensor modes

Stringent bounds from astrophysics if decaying into “visible” species (e±,γ) ≿1027 s (Calore’s lectures)

LOWER LIMIT ON DM MASS

DeBroglie

=h

mv. kpc =) m & 1022 eV (v ' 100 km/s)

f g

h3

m > O(10 100) eV

dark matter is confined/detected at least at astrophysical scales, hence must be “localized” and behave classically there.

For fermions a much stronger bound holds, due to the fact that they obey Fermi-Dirac statistics (“Pauli principle”)

Conservation of phase space density of a non-interacting fluid (Liouville Eq.) + condition that any observable, coarse grained p.s. density must be lower than the real one, in turn lower than the above maximum, one derives

S. Tremaine and J. E. Gunn, Phys. Rev. Lett. 42, 407 (1979)

A. Boyarsky, O. Ruchayskiy and D. Iakubovskyi, JCAP 0903, 005 (2009)

updated lower limit around ~400 eV

EXERCISEConsider a spherical, uniform system, gravitationally bound, of mass M and radius R, made of “cold” fermionic particles of mass m, with g spin dof. Compute the lower limit on m

Hints/Guidelines

gravitationally bound: all particles should have a velocity lower than the gravitational escape velocity (compute the latter for such a system)

cold fermions: the system contains N=M/m particles. But the “coldest” fermion system still settles in “Fermi levels”… compute the Fermi velocity as a function of M,R, m, g.

estimate: Given the dependence on M, R, which systems are optimal to set bounds on m? Search the literature for typical values of M, R of astrophysical systems (clusters of Galaxies, Milky Way-like Galaxy, dwarf spheroidals) and to derive a(n order of magnitude) bound

Bonus: Direct (terrestrial) experiments tell us that neutrinos have a maximum allowed mass of about 2.2 eV. What’s the maximum overall mass they could contribute to a Milky Way-sized halo (assuming that they can be “maximally packed” into it)? What if the forthcoming Katrin experiment pushes the upper limit to 0.2 eV, as expected?

DM IS... DARK AND DISSIPATIONLESS DM must not couple much to γ’s (perturbation shape & amplitude argument, not seen in e.m. channels…). Neither it can couple to its own “invisible photons” exactly like baryons!

Why? For instance, because DM forms extended, triaxial halos, while baryons “sink” in inner halo parts, form disks, etc. since they can dissipate energy by e.m. emission. At Galactic scale, evidence from tidal streams of satellite galaxies (some diagnostic also via lensing…)

e.g. D. R. Law, S. R. Majewski, K. V. Johnston, “Evidence for a Triaxial Milky Way Dark Matter Halo from the Sagittarius Stellar Tidal Stream” Astrophys. J. 703, L67 (2009) & refs. to it

SOME COMPLICATIONS To make the argument quantitative,

simulations needed. Details do matter (plenty/all of them) e.g. the baryonic disk does make the DM one “more spherical”…

e.g. S. Kazantzidis, M. G. Abadi and J. F. Navarro, “The Sphericalization of Dark Matter Halos by Galaxy

Disks,” Astrophys. J. 720, L62 (2010) [1006.0537]

Don’t get confused about claims of a possible “dark disk”, maybe related to mass extinctions!

The bulk of DM must be dissipationless! Some room for sub-leading component (<5% according to

J. Fan, A. Katz, L. Randall and M. Reece, “Dark-Disk Universe,” PRL 110, 211302 (2013) [1303.3271])

which is dissipative and may form ‘disks’. To call it the DM is as accurate as saying that missing baryons or

standard neutrinos are the DM.

e.g. L. Randall and M. Reece,“Dark Matter as a Trigger for Periodic Comet Impacts,”

PRL 112, 161301 (2014) [1403.0576]

F. Y. Cyr-Racine, R. de Putter, A. Raccanelli, K. Sigurdson,“Constraints on Large-Scale Dark Acoustic Oscillations from

Cosmology,” PRD 9 063517 (2014)[1310.3278]

Further constraints from “dark oscillations”

DM IS COLLISIONLESS (WRT BARYONIC GAS)

cm2

g= 1.78

barn

GeV

! if DM-DM interaction too strong, spherical structures would be obtained rather than triaxial. From actual clusters, one can derive σ/m<0.02 cm2/g

7

System v0[km/s] /m

cm2/g

References

Bullet Cluster 1000 1.25 [41, 43]

Galactic Evaporation 1000 0.3 [45]

Elliptic Cluster 1000 0.02 [46]

Dwarf Evaporation 100 0.1? [45]

Black Hole 100 0.02? [59]

Mean Free Path 44 2400 0.01 0.6 [57]

Dwarf Galaxies 10 0.1 [56]

TABLE I: The systems we consider and the observational bound they place on DM self-scattering cross section. Entries markedwith an asterisk ? are velocity dependent bounds. For more details, see text.

also consider in detail the bounds placed by elliptic clusters, as those represent the tightest bounds on a system withhigh velocity DM.

In Fig. (5) we show tr/m as a function of m, assuming a Maxwellian distribution with characteristic velocityv0 = 1000 km/s, which is approximately the value found in galaxy clusters. For values of m greater than 0.5 GeVwe include only modes of zero and one, while for m < 0.5 GeV, we include 5. As can be seen in Fig. 2, for ourchoice of , m and v, this is an acceptable trade-o between computational speed and accuracy. We also display theapproximate solutions for the cross section and transfer cross section, as given by Eqs. (9) and (10), again integratingover a Maxwellian distribution for both incoming particles (the upper line is the approximate cross section, while thelower is tr). We can clearly see that for systems with velocity distributions centered around 1000 km/s no boundson MeV-scale dark forces can be placed.

0 1 2 3 4

1106

2106

5106

1105

2105

5105

1104

mGeV

trm c

m2 g

0 1 2 3 4 5

11062106

510611052105

510511042104

mGeV

trm c

m2 g

FIG. 5: tr/m as a function of m, assuming that m = 500 GeV, and = 0.01 (left) and = 0.1 (right). A thermal velocitydistribution Eq. (14) with dispersion v0 = 1000 km/s = 3.3 103c, characteristic of galaxy clusters, was used. Contributionsfrom modes up to = 5 are included in the exact numerical cross section for m < 0.2 GeV, while only 1 are includedabove this mass. The approximate solutions from Eqs. (9) and (10) are also shown (dashed red lines).

However, dwarf galaxies, with velocity dispersions of 10 km/s [61], provide a non-trivial constraint. In Fig. 6we show the velocity-averaged tr/m as a function of m, this time for a dwarf galaxy-appropriate value ofv0 = 10 km/s. Again, both the exact numerical solution (with 5 for all values of m) and the approximatesolutions are shown. Taking the upper bound on /m to be the 0.1 cm2/g derived from dwarf galaxies, we can placea bound requiring

m 40 MeV (15)

for the larger value of considered and slightly weaker (m > 30 MeV) for smaller . Although clusters present atighter bound on the scattering cross section, the characteristic velocity in these systems is far higher (Table I) andthe stronger constraint comes from dwarf galaxies.

A full simulation for the case of velocity dependent cross sections, as expected in models with a Sommerfeldenhancement, would improve on our estimate and we advocate strongly for it to be carried out.

From M. R. Buckley and P. J. Fox, Phys. Rev. D 81, 083522 (2010) (*=v-dependent)

! From Bullet cluster, σ/m<0.7-1.3 cm2/g,

! similar bounds from different arguments, for a compilation see e.g.

! Very loose from particle physics standard (barn/GeV level), but much less than atomic or molecular cross sections!

Jordi Miralda-Escudé ApJ 564 60 (2002)

S. W. Randall et al. ApJ 679, 1173 (2008)

but different levelsof robustness...

Exercise: estimate the geometric cross section for hydrogen atomic scattering… & compare!

ACTUAL SIMULATIONS NEEDED…& COMPLICATIONSA. H. G. Peter, M. Rocha, J. S. Bullock and M. Kaplinghat,B. MNRAS 430, 105 (2013) [1208.3026]

• For instance, ref.

found that, while interactions among dark-matter particles will drive galaxy and cluster halos to become spherical in their centers, DM self-interaction cross sections at least as large as σ/m = 0.1 cm2/g are allowed

• Typically, such large σ require a light mediator (at least if one wants to stay perturbative), sort of “dark photon” (can be massive, though) velocity dependent σ implied (smaller in clusters!)

L. G. van den Aarssen, T. Bringmann and C. Pfrommer, PRL 109, 231301 (2012)[1205.5809]

S. Tulin, H. B. Yu and K. M. Zurek, “Resonant Dark Forces and Small Scale Structure,'' PRL 110, 111301 (2013) [1210.0900]…

Results.—We show the complete parameter space wherea dark force can account for DM small scale structure andrelic density. For scattering, to compare with astrophysicalbounds, we consider the velocity-averaged cross sectionh!Ti ¼

Rd3v!Te

"ð1=2Þv2=v20=ð2"v2

0Þ3=2, where v0 is themost probable velocity for a DM particle. Figure 2 shows

contour plots of h!Ti for two cases, symmetric and asym-metric DM, in the mX-m# parameter space.For symmetric DM (Fig. 2, left), we take the average of

attractive and repulsive cross sections, !T ¼ ð!attT þ

!repT Þ=2, with $X chosen to reproduce the observed DM

relic density at each point. The blue shaded contour regionsshow h!Ti=mX on dwarf scales (v0 ¼ 10 km=s) in theranges 0:1–1 cm2=g (light) and 1–10 cm2=g (dark) to solvesmall scale structure problems. The lower range is preferredfor a constant cross section; Ref. [11] found 0:1 cm2=gmatched small scale structure observations, whereas1 cm2=g caused too low central densities in dwarf spher-oidals. Simulationswith av-dependent classical (attractive-only) force preferred the upper range (or larger) [10].The red (green) contour lines show h!Ti=mX ¼ 0:1and 1 cm2=g on MW (cluster) scales with v0 ¼200 ð1000Þ km=s, showing the approximate upper limitsfrom observations. Reference [11] found that 1 cm2=g pro-duced a too-small central DMdensity in galaxy clusters andis only marginally consistent with MW-scale halo shapeellipticity constraints, whereas 0:1 cm2=g is consistent withthese constraints [11]. In the resonant regime, we havecomputed !T numerically. This region shows a pattern ofresonances formX & 10 GeV–TeV, where!att

T is enhanced,allowing for greater mX values for fixed h!Ti=mX. Thedashed lines indicate where we use analytic formulas toextrapolate our results into the Born (mX ' m#=$X) andclassical (mX ( m#=v) regimes. Our numerical calcula-tionmaps smoothly into these regions, again confirming ouragreement with the analytic formulas [24]. The crosses

dw0.1

dw1dw

10

MW 0.1MW 1cl 0.1

cl 1

10 4 0.001 0.01 0.1 1

1

10

100

1000

104

m GeV

mX

GeV

Asymmetric dark matter X 10 2

0.1

dw0.1

dw1

dw10

MW 0.1

MW 1

cl 0.1

cl 1

10 4 0.001 0.01 0.1 10.1

1

10

100

1000

104

m (GeV)

mX

(GeV

)

Symmetric dark matter

FIG. 2 (color online). Symmetric (left) and asymmetric (right) DM parameter space in mX-m# plane. Blue shaded regions showwhere DM self-scattering solves dwarf-scale structure anomalies, whereas red (green) lines show bounds from Milky Way (cluster)scales. Numerical values indicate h!Ti=mX in cm2=g on dwarf (‘‘dw’’), Milky Way (‘‘MW’’), and cluster (‘‘cl’’) scales. For symmetricDM, $X is fixed to obtain the observed relic density; for asymmetric DM, $X ¼ 10"2 is fixed to deplete X, !X density for mX &300 GeV (dotted line). Dashed lines show extrapolation using analytic formulas, whereas the ‘‘x’’ marks indicate the parameter pointsutilized in Fig. 1.

dwarf Milky Way cluster

Bornresonant

classical

10 100 100010 5

10 4

10 3

10 2

0.1

1

10

100

v km s

Tm

Xcm

2g

attr

activ

eon

ly

FIG. 1 (color online). Velocity-dependence of !T for sampleparameters within different regimes. Blue line shows Bornformula (A1), in agreement with numerical results (blue dots),formX ¼ 4 GeV,m# ¼ 7:2 MeV, $X ¼ 1:8) 10"4. Green lineshows classical formula (A2), in agreement with numericalresults (stars), for mX ¼ 2 TeV, m# ¼ 1 MeV, $X ¼ 0:05.Red lines show !T in the resonant regime for mX ¼ 100 GeV,$X ¼ 3:4) 10"3, illustrating s-wave resonance (solid, m# ¼205 MeV), p-wave resonance (dot-dashed, m# ¼ 20 MeV), ands-wave antiresonance (dashed, m# ¼ 77 MeV).

PRL 110, 111301 (2013) P HY S I CA L R EV I EW LE T T E R Sweek ending

15 MARCH 2013

111301-3

SOLVING PROBLEMS?… AND MORE SIMULATIONS

M. Kaplinghat, S. Tulin and H. B. Yu,“Dark Matter Halos as Particle Colliders: Unified Solution to Small-Scale Structure Puzzles from Dwarfs to Clusters,” PRL 116, 041302 (2016) [1508.03339]

• some collisional aspects may be beneficial for possible issues of CDM paradigm at small scales, on core sizes and central densities of dwarfs, LSBs, and galaxy clusters e.g.

T. Sepp et. al. “Simulations of Galaxy Cluster Collisions with a Dark Plasma Component” arXiv:1603.07324

non-interacting DM distribution displayed in blue (80%), dark plasma (20%) in red.

• New simulations with dark radiation currently being developed, see e.g.

Bottom line: DM should not have self-interactions exceeding the barn/GeV level; slightly smaller σ could also be beneficial… however be aware that the relevant

scales are affected by baryonic effects, feedbacks, etc. Unclear to isolate smoking guns (and simulations with both baryons and collisional DM eventually needed)

DM COUPLING TO NORMAL MATTER

D. M. Jacobs, G. D. Starkman and B. W. Lynn, “Macro Dark Matter,” MNRAS 450, 4, 3418 (2015) [1410.2236]

arX

iv:0

705.

4298

v2 [

astro

-ph]

12

Oct

200

7

Towards Closing the Window on Strongly Interacting Dark Matter:Far-Reaching Constraints from Earth’s Heat Flow

Gregory D. Mack,1, 2 John F. Beacom,1, 2, 3 and Gianfranco Bertone4

1Department of Physics, Ohio State University, Columbus, Ohio 432102Center for Cosmology and Astro-Particle Physics, Ohio State University, Columbus, Ohio 43210

3Department of Astronomy, Ohio State University, Columbus, Ohio 432104Institut d’Astrophysique de Paris, UMR 7095-CNRS,

Universite Pierre et Marie Curie, 98bis boulevard Arago, 75014 Paris, [email protected], [email protected], [email protected]

(Dated: 30 May 2007)

We point out a new and largely model-independent constraint on the dark matter scattering crosssection with nucleons, applying when this quantity is larger than for typical weakly interactingdark matter candidates. When the dark matter capture rate in Earth is efficient, the rate ofenergy deposition by dark matter self-annihilation products would grossly exceed the measuredheat flow of Earth. This improves the spin-independent cross section constraints by many orders ofmagnitude, and closes the window between astrophysical constraints (at very large cross sections)and underground detector constraints (at small cross sections). In the applicable mass range, from∼ 1 to ∼ 1010 GeV, the scattering cross section of dark matter with nucleons is then bounded fromabove by the latter constraints, and hence must be truly weak, as usually assumed.

PACS numbers: 95.35.+d, 95.30.Cq, 91.35.Dc

I. INTRODUCTION

There is a large body of evidence for the existence ofdark matter, but its basic properties – especially its massand scattering cross section with nucleons – remain un-known. Assuming dark matter is a thermal relic of theearly universe, weakly interacting massive particles areprime candidates, suggested by constraints on the darkmatter mass and self-annihilation cross section from thepresent average mass density [1]. However, as this re-mains unproven, it is important to systematically testthe properties of dark matter particles using only late-universe constraints. In 1990, Starkman, Gould, Es-mailzadeh, and Dimopoulos [2] examined the possibilityof strongly interacting dark matter, noting that it indeedhad not been ruled out. Many authors since have ex-plored further constraints and candidates. In this litera-ture, “strongly interacting” denotes cross sections signif-icantly larger than those of the weak interactions; it doesnot necessarily mean via the usual strong interactions be-tween hadrons. We generally consider the constraints inthe plane of dark matter mass mχ and spin-independentscattering cross section with nucleons σχN .

Figure 1 summarizes astrophysical, high-altitude bal-loon/rocket/satellite detector, and underground detectorconstraints in the σχN–mχ plane. Astrophysical limitssuch as the stability of the Milky Way disk constrainvery large cross sections [2, 3]. Accompanying and com-parable limits include those from cosmic rays and thecosmic microwave background [4, 5]. Small cross sec-tions are probed by CDMS and other underground de-tectors [6, 7, 8, 9]. A dark matter (DM) particle can bedirectly detected if σχN is strong enough to cause a nu-clear recoil in the detector, but only if it is weak enoughto allow the DM to pass through Earth to the detector.

0 5 10 15 20log m

χ[GeV]

-30

-25

-20

-15

-10

-35

-40

-45

log σχN

[cm

2 ]

Underground Detectors

Cosmic Rays

MW redone

MW Disk Disruption

XQC

IMP 7/8 IMAX

SKYLAB

RRS

FIG. 1: Excluded regions in the σχN–mχ plane, not yet in-cluding the results of this paper. From top to bottom, thesecome from astrophysical constraints (dark-shaded) [2, 3, 4, 5],re-analyses of high-altitude detectors (medium-shaded) [2, 10,11, 12], and underground direct dark matter detectors (light-shaded) [6, 7, 8, 9]. The dark matter number density scales as1/mχ, and the scattering rates as σχN/mχ; for a fixed scat-tering rate, the required cross section then scales as mχ. Wewill develop a constraint from Earth heating by dark matterannihilation to more definitively exclude the window betweenthe astrophysical and underground constraints.

In between the astrophysical and underground limitsis the window in which σχN can be relatively large [2].High-altitude detectors in and above the atmosphere

6 D. M. Jacobs, G. D. Starkman, and B. W. Lynn

Figure 1. Constraints on Macro cross section and mass, assum-ing the Macros have a single mass, inferred from elastic scatteringwith baryons and other Macros. In dark grey are those inferredfrom comparison of observed to numerically-simulated galaxiesand clusters with self-interacting dark matter (Randall et al.(2008); Rocha et al. (2013)); overlapping, in light grey are thelarge-scale structure constraints taken from Dvorkin et al. (2014);the orange region correspond to the mica constraints (Price &Salamon (1986)); the green region is the Skylab constraint (Shirk& Price (1978); Starkman et al. (1990)). The black and greenlines correspond to objects of constant density 1 g cm3 and3.6 1014 g cm3, respectively.

Figure 2. Constraints on Macro cross section and mass, assumingthe Macros have a single mass, inferred from inelastic scatteringwith baryons. Here, the grey region is the constraint applied fromself-interacting dark matter (which the Macro-baryon interactionshould not exceed) and the blue region corresponds to the inelas-tic mica constraints.

of the impact parameter of a wave packet as it travels pastthe lensing mass is given by the Einstein radius,

RE

=

r4GM

DOLDLS

DOS. (19)

Two electromagnetic waves passing in the vicinity ofthe lens with characteristic photon energy, E, and arrivingat the location of the observer have amplitudes that addand interfere, resulting in an intensity (or flux) amplificationgiven by

A =2 + u2 + 2 cos

up4 + u2

, (20)

where u r0

RE

, and the phase di↵erence = Er be-tween the two waves is determined by both the energy ofthe photons and di↵erence in their path lengths, r.

One way to detect the presence of an intervening lensis to look for a modulation of an observed photon flux asa function of E – this is the basic idea behind femtolens-ing (Gould (1992)). When considering sources for which r0changes substantially in an observing time, the amplifica-tion would vary as a function of time – this is the basisof microlensing (Paczynski (1986)). Historically, the abovetechniques have been used to place constraints on the abun-dance of primordial black holes, brown dwarfs, or other com-pact astrophysical objects comprising a fraction of the darkmatter density. To constrain the abundance of Macros, wewill (conservatively) insist that the Einstein lensing angle,E =

p4GM

X

/d exceed the angular size of the Macro,X = R

X

/d, for a characteristic distance, d. This amountsto the requirement

X

/MX

. 3

d

Gpc

cm2 g1 . (21)

2.4.1 Femtolensing

Femtolensing refers to gravitational lensing where the an-gular separation between two lensed images from the samesource is of order 1015 arcseconds. At such scales, imagescannot be resolved, however an interference pattern in theenergy spectrum of background sources would be observable.For a gamma-ray burst (GRB), for example, the magnifica-tion is energy-dependent and this results in an intensity pat-tern in the energy spectrum (Gould (1992); Barnacka et al.(2012)).

The lensing probability is determined by the opticaldepth to the source, and a “lensing cross section” (Bar-nacka et al. (2012)). First, the optical depth may be calcu-lated by noting that the cross section for “strong” lensingevents is characteristically given by (see e.g. Fukugita et al.(1992))

= R2E

= 4GMX

DOLDLS

DOS. (22)

The di↵erential probability of a beam of light being a lensedby a cosmological distribution of Macros is then

d = nX

(zL)dt

=32

X

H20 (1 + zL)

3 DOLDLS

DOS

dt

dzL

dzL , (23)

c - RAS, MNRAS 000, 1–??

8 D. M. Jacobs, G. D. Starkman, and B. W. Lynn

Figure 3. Constraints on the Macro cross section and mass (as-suming the Macros have a single mass), including the Macro-photon constraints and applicable for both elastically- andinelastically-scattering candidates. In red are the femto- andmicro-lensing constraints, while in grey are the CMB-inferredconstraints (see text for references). The black and green linescorrespond to objects of constant density 1 g cm3 and 3.6 1014 g cm3, respectively. Black hole candidates lie on the ma-genta line, however, these may be ruled out for other reasons(see Capela et al. (2013b,a); Pani & Loeb (2014); Defillon et al.(2014)); objects within the hatched region in the bottom-rightcorner should not exist as they would simply be denser than blackholes of the same mass.

3.1.1 Model III

Since a detailed description of the core charge distributionwithin the Macro is unknown, we take it to be uniform.In the places/eras of interest, the Macros will be immersedin a fluid of protons, electrons and, depending on the era,positrons. We will assume the overall distribution of the fluidto be determined by the hydrostatic equilibrium between thefluid pressure and the electrostatic force.

Recall that the number densities and pressures of afermion species are given by

ni =12

Z 1

mi

d eEeEq

eE2 m2i

e(eE+Viµi)/T + 1

(33)

Pi =1

32

Z 1

mi

d eE

eE2 m2

i

3/2

e(eE+Viµi)/T + 1

, (34)

where here eE is only the relativistic part of the energy, i.e.E = eE + Vi, where E is the full energy eigenvalue of theHamiltonian. Using this definition, the chemical potentials,µi are the same as their background values in the absenceof the Macro. In the classical limit, applicable in the casesof interest, the exponential dominates in the denominatorsand one finds

ni = eVi/T ni (35)

Pi = eVi/T niT , (36)

where the barred values are the background values, and wehave defined Ve V so that Ve+ = Vp = V .

Since the system is taken to be spherically symmetric,the condition for electric hydrostatic equilibrium is

dP

dr=

Qint(r)↵r2

(ne + ne+ + np) , (37)

where Qint (r) is the (integer) charge within a sphere of ra-dius, r, and is given by

Qint (r) =

8>><

>>:

QX

r

RX

3

+ 4

Z r

0

der er2 (np + ne+ ne)

QX

+ 4

Z r

0

der er2 (np + ne+ ne) ,

(38)where the top and bottom lines apply for r < R

X

andr > R

X

, respectively. In the limit r ! 0, one can showthat dP/dr ! 0 for a reasonably well-behaved potential8,indicating

V 0(0) = 0 . (39)

The other boundary condition on V comes from the require-ment

limr!1

V (r) ! 0 (40)

The value of V (0) will generally not be known, except forwhen a full analytic solution for V (r) is available, hence nu-merical “shooting” will be required in order to numericallyevolve V (r) from r = 0. The di↵erential equation to solve isfound by taking a derivate of (37) with respect to r, whichsimply results in a version of the Poisson equation9

V 00(r)+2rV 0(r) = 4↵

±n(R

X

r)+np +ne+ ne

,

(41)where n 3 |Q

X

| /(4R3X

), and upper or lower signs referto Q

X

> 0 or QX

< 0, respectively. After dividing by T ,defining q2 4↵n/T , y qr, and v(y) V/T , we arriveat the dimensionless equation

v(y) +2yv(y) = ± (y

X

y) +np + ne+ ne

n, (42)

where a dot is a derivative with respect to y and yX

qRX

.Generally, this equation must be solved numerically; how-ever, in the case |v(y)| 1 it is easy to show that

np + ne+ ne ' (np + ne+ + ne) v(y) (43)

due to overall charge neutrality. Therefore the solution to(42) is approximately

v(y) =

8>><

>>:

1a2

c1eay eay

y

(y < y

X

)

±c2eay

y(y > y

X

) ,

(44)

where a2 n/n and n np + ne+ + ne . By the matchingof v(y

X

) and v(yX

) at yX

we learn

c1 =1

2a3eay

X (1 + ayX

) (45)

c2 =1a3

(sinh ayX

ayX

cosh ayX

) . (46)

8 This is true if V (r) is finite at r = 0, for example.9 This is consistent with our convention that V = e, where is the electrostatic potential; it satisfies r2 = , where isthe charge density.

c - RAS, MNRAS 000, 1–??

Same bounds as before also apply (order of magnitude!), plus a number of additional ones

G. D. Mack, J. F. Beacom and G. Bertone, “Towards Closing the Window on Strongly Interacting Dark Matter: Far-Reaching Constraints from Earth's Heat Flow,'' PRD 76, 043523 (2007) [0705.4298]

By the way, if DM is a fundamental particle, M< MPl~1019 GeV (higher, would be a BH!) But could be an exotic composite state! similar bounds as for BH apply down to ~4x1024 g

(or 10-8 M⦿), and sometimes below that. Extra bounds exist. For details see

Too strong coupling with photons damp CMB anisotropy tail σDMγ/m = 8 x10-31 cm2/GeVand with neutrinos, suppressed Ly α power spectrum σDMν/m = 8 x10-33 cm2/GeV

R. J. Wilkinson, C. Boehm and J. Lesgourgues, JCAP 1404, 026 (2014) [1309.7588], JCAP 1405, 011 (2014) [1401.7597]

SUMMARY OF WHAT WE LEARNED

Hopefully, some clarification on the (partially) misleading dichotomy DM vs Modified Gravity: the key is the need for some new degree of freedom.

It turns out that this requires new physics (even for-the highly constrained-BHs, need some exotic production mechanism) with some specific properties.

Astrophysical and cosmological arguments put some constraints on DM. Unfortunately, cannot get “too far” since “gravity is universal” it does not tell us what the underlying new physics is.

We need some “strategy” to identify what DM is. For that, first we need some extra input/constraint must necessarily come from theory, i.e. let’s try to define some classes of candidate: We Shall classify them via production mechanisms, the leitmotif of Lectures 3-4-5.

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