Alexander Kusenko (UCLA)IPMU ’08

Echoes of supersymmetry: BAU, relic Q-balls, and gravity waves• Introduction: SUSY Q-balls• Inflation+SUSY⇒ Q-balls• stable Q-balls as dark matter• constrains• gravitational waves

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Alexander Kusenko (UCLA)IPMU ’08

Echoes of supersymmetry: BAU, relic Q-balls, and gravity waves

Affleck−Dine condensate
baryons

gravitywaves
baryonic Q−balls

unstablestabledark matter
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Alexander Kusenko (UCLA)IPMU ’08

SUSY and Q-balls
nucleus

Why would one suspect thatSUSY ⇒ Q-balls?

SUSY

SUSY
SUSY

Bose−Einstein

Q−ball3

Alexander Kusenko (UCLA)IPMU ’08

Q-balls

Let us consider a complex scalar field φ(x, t) in a potential that respects a U(1) symmetry:φ → eiθ φ.
vacuum: φ = 0R
1

conserved charge: Q = 2i„«↔
φ† ∂0 φ d3x

Q 6= 0 ⇒ φ 6= 0 in some finite domain⇒ Q-ball [Rosen; Friedberg, Lee, Sirlin; Coleman]

4Alexander Kusenko (UCLA)IPMU ’08

R3h221
12 |φ̇| + 2 |∇φ| + U (φ)

Minimize energy E = d x
Q = const. Introduce Lagrange multiplier:hR
1iunder the constrainti↔
φ∗ ∂ t φ d3x

E = E + ω Q − 2ii
R 3 1 ∂R 3 h1
2= d x 2 ∂ φ − iωφ + d x 2 |∇φ|2 + Ûω (φ) + ωQ,
where Ûω (φ) = U (φ) − 12 ω 2 φ2.

• Minimize blue by setting φ = eiωtφ̄(x)• Minimize red by choosing φ̄(x) to be
the bounce for tunneling in Ûω (φ) = U (φ) − 12 ω 2φ2.
• Finally, minimize E with respect to ω.

5Alexander Kusenko (UCLA)IPMU ’08Q-balls exist whenever Ûω (φ) = U (φ) − 12 ω 2φ2 is not positive definite
for some value of ω.
2 21U(ϕ) − −2 ωϕsmall ω , large Q

ω=0ϕ(0)0ϕlarge ω, small Q6

Alexander Kusenko (UCLA)IPMU ’08

Q-balls exist if
 2U (φ) φ = min, for φ = φ0 > 0[Coleman]

Finite φ0: M (Q) ∝ Q

Flat potential (U (φ) ∼ φp, p < 2); φ0 = ∞:






M (Q) ∝ Qα, α < 17Alexander Kusenko (UCLA)IPMU ’08

Q-balls exist in (softly broken) SUSY because• the theory has scalar fields• the scalar fields carry conserved global charge (baryon and leptonnumbers)• attractive
 2scalar interactions (tri-linear terms, flat directions) force(U (φ) φ ) = min for non-vacuum values.

8Alexander Kusenko (UCLA)IPMU ’08

MSSM, gauge mediated SUSY breaking

Baryonic Q-balls (B-balls) are entirely stable if their mass per unit baryon
charge is less than the proton mass.
M (Q) = MS Q3/4 ⇒M (QB )− 1/ 4∼MQS
QBfor QB ≫MS1GeV4< 1GeV>12∼ 10Such B-balls are entirely stable.



9Alexander Kusenko (UCLA)IPMU ’08

Baryon asymmetry


n


+0.3

η ≡ nBγ = 6.1 −0.2 × 10−10(WMAP)






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Alexander Kusenko (UCLA)IPMU ’08

What happened right after the Big Bang?

• Inflation probably took place• Baryogenesis – definitely after inflation
Standard Model is not consistent
with the observed baryon asymmetry (assuming inflation)

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Alexander Kusenko (UCLA)IPMU ’08

Affleck–Dine baryogenesis• Natural if SUSY+Inflation• Can explain matter• Can explain dark matter• Predictions can be tested soon

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Alexander Kusenko (UCLA)IPMU ’08

Inflation

All matter is produced during reheating after inflation.
SUSY ⇒ flat directions.
During inflation, scalar fields
are displaced from their minima.

13Alexander Kusenko (UCLA)IPMU ’08

Affleck – Dine baryogenesis

at the end of inflationa scalar condensate
develops a large VEValong a flat direction

CP violation is due totime-dependent background.
Baryon asymmetry: φ = |φ|eiωt
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Alexander Kusenko (UCLA)IPMU ’08

Affleck – Dine baryogenesis: an example
Suppose the flat direction is lifted by a higher dimension operator Wn =1n+3. The expansion of the universe breaks SUSY and introduces massΦnMterms m2 ∼ ±H 2.
The scalar potential:

22V = −H |Φ| +1
M2n+4|Φ|2nAssume the inflation scale E ∼ 1015 GeV The Hubble constantHI ≈ E 2/Mp ≈ 1012 GeV. TR ∼ 109 GeV

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Alexander Kusenko (UCLA)IPMU ’08

In this example, the final baryon asymmetry isnBnγ∼nB(ρI /TR)∼ 10−10∼nB TR ρΦnΦ mΦ ρITR109GeVMpm 3/ 2! (n−1)(n+1)

Correct baryon asymmetry for n = 1. (For n > 1, too big.)

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Alexander Kusenko (UCLA)IPMU ’08

Fragmentation of the Affleck-Dine condensate[AK, Shaposhnikov]
small inhomogeneities can growunstable modes:
p0 < k < kmax = ω 2 − U ′′(φ)⇒ Lumps of baryon condensate⇒ Q-ballstx

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Alexander Kusenko (UCLA)IPMU ’08

Fragmentation ≈ pattern formation

Familiar example:

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Alexander Kusenko (UCLA)IPMU ’08

Numerical simulations of the fragmentation

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40
20060

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20002040
60[Kasuya, Kawasaki]

19Alexander Kusenko (UCLA)IPMU ’08

Two-dimensional charge density plots [Multamaki].
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3030

20202020

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1010

10102030

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10(a) mt = 02030

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10(b) mt = 752030

4010(c) mt = 150

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2020

1010

10102030(e) mt = 525

40102030(f) mt = 675

40102030(g) mt = 8253040(d) mt = 375

4010204010203040(h) mt = 900

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Alexander Kusenko (UCLA)IPMU ’08

Three-dimensional charge density plots [Multamaki].

(i) mt = 900(j) mt = 1050(k) mt = 1200(l) mt = 1350

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Alexander Kusenko (UCLA)IPMU ’08

Fragmentation of AD condensate can produce Q-balls

SUSY Q-balls may be stable or unstable
if stable ⇒ dark matterAffleck−Dine condensate
baryonst

baryonic Q−balls
unstablestablex
dark matter[AK, Shaposhnikov; Enqvist, McDonald]22

Alexander Kusenko (UCLA)IPMU ’08

Stable Q-balls as dark matter

Q-balls can accommodate baryon number at lower energy than a nucleon⇒ B-Balls catalyze proton decay Signal:ρdEGeV∼100
3dlcm
1 g/cmHeavy ⇒ low flux⇒ experimental limits from Super-Kamiokande and other large
detectors

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Alexander Kusenko (UCLA)IPMU ’08

Present experimental limits[Arafune et al.];24

Alexander Kusenko (UCLA)IPMU ’08

A “candidate event”[Lattes, Fujimoto and Hasegawa, Phys.Rept. 65, 151 (1980)]

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Alexander Kusenko (UCLA)IPMU ’08

Unstable B-balls

Gravity mediated SUSY breaking typically produces potentials which growas ∼ φ2 up to the Planck scale.
Hence, Q-balls are unstable.

Decay of Q-balls results in late non-thermal production of LSP.
Ordinary and dark matter arise from the same process. Hence, one may
be able to explain why Ωmatter and Ωdark are not very different.
[Fijii,Yanagida; Enqvist, McDonald; Laine, Shaposhnikov]

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Alexander Kusenko (UCLA)IPMU ’08Ωdark/ Ωmatter ∼ 10

• Dark matter is stable Q-balls [Laine, Shaposhnikov]• Dark matter is LSP produced non-thermally from decay of unstable
Q-balls [Enqvist, McDonald; Fujii, Hamaguchi; Fujii, Yanagida]• Dark matter is gravitino produced non-thermally from decay of unstable
Q-balls [Fujii, Yanagida]

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Alexander Kusenko (UCLA)IPMU ’08ΩB−ball/ Ωmatter ∼ 10

[Laine, Shaposhnikov]• Gauge-mediated SUSY breaking• QB ∼ 1026±2 (in agreement with numerical simulations)
More specifically, ΩB−ball/Ωmatter ∼ 10 implies
− 1/ 2
MQBSUSYηB ∼ 10−10 TeV102628

Alexander Kusenko (UCLA)IPMU ’08ΩB−ball/ Ωmatter ∼ 1029

Alexander Kusenko (UCLA)IPMU ’08ΩB−ball/ Ωmatter ∼ 1030

Alexander Kusenko (UCLA)IPMU ’08“We’ve established a clear link”

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Alexander Kusenko (UCLA)IPMU ’08“We’ve established a clear link”

32Alexander Kusenko (UCLA)IPMU ’08

Astrophysical constraints• Q-balls pass through ordinary stars and planets• SUSY Q-balls accumulate inside white dwarfs and neutron stars• SUSY Q-balls can convert nuclear matter into squark condensate– old estimates underestimated the rates– new rates too high, unless the flat direction is lifted by baryon
number violating operators.

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Alexander Kusenko (UCLA)IPMU ’08

Interactions of SUSY Q-balls with matter (old picture)Q−ballxquark
χquark∝ m12 , slow
χThis process was thought to limit
the rate at which the Q-balls
could process baryonic matter.
Lifetimes of neutron stars were
though to be greater than the ageof the universex

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Alexander Kusenko (UCLA)IPMU ’08

Interactions of SUSY Q-balls with matter (correct picture)Q−ballquarkx
antiquark

There is a Majorana mass termfor quarks inside coming fromthe quark-squark-gluino vertex.
Probability ∼ 1 for a quark toreflect as an antiquark. Veryfast!
[AK, Loveridge, Shaposhnikov].

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Alexander Kusenko (UCLA)IPMU ’08

Interactions of SUSY Q-balls with matter

The MSSM Lagrangian contains terms describing interactions of quarks ψwith squarks φ and gluinos λ:√a
L = −g 2Tij(λaσ 2ψj φ∗i ) + C.C. + ...

and also the Majorana mass terms for gluinos:

LM = M λaλa.
Of course, the quarks also have dirac mass terms.

36Alexander Kusenko (UCLA)IPMU ’08

In the basis {ψL, ψR, λ}, the mass matrix has a (simplified) form:
0m ϕL m0 ϕR ϕL ϕR MThe squark fields φ grow large inside the Q-ball. This mass term causes aquark to scatter off a Q-ball as an antiquark, with probability of order 1.
[AK, Loveridge, Shaposhnikov]

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Alexander Kusenko (UCLA)IPMU ’08

Interaction rates are not limited by weak-scale cross section.
Signatures in detectors do not change significantly
Neutron stars: can they survive long enough?
Pulsars ages: oldest pulsars have (Ṗ /P ) ∼ (0.3 − 3) × 10−10yr−1

Some pulsars are also known to be (at least) as old as 10 Gyr based on thecooling ages of their white dwarf companions
Inside a neutron star Q-ball VEV grows fast and reaches vealues at which
the flat direction is lifted by higher-dimension operators

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Alexander Kusenko (UCLA)IPMU ’08

Generally, the lifting terms can be written in the formnV (φ)lifting ≈ λnM4

φMn−1+m ∗ n−1−mφM• If m 6= 0, the baryon number is broken. Q-balls inside a neutron starreach some maximal size and stop growing in size. The rate of conversionof matter into condensate stabilizes at a small value. This is allowed.
• If m = 0, Q-balls change the way they grow after reaching a certainsize Qc.

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Alexander Kusenko (UCLA)IPMU ’08

Q-balls along ”Flat” and ”Curved” directionsϕω
MRFD√1 ΛQ1/4√2π √2ΛQ−1/4
4π 32 ΛQ3/4√1 Q1/42Λ

CDϕmax√−1/4
Λ2ϕ−12ΛQ=π= ωccmax
ωQ
1/3
1/123
3 1/3
1Q
=()
(Q/Q)
RF Dc8π Λ2 ϕmax
2The change from FD to CD makes the Q-ball grow faster.

40Alexander Kusenko (UCLA)IPMU ’08

As soon as this happens, the neutron star is consumed very quickly:t

FD Q-balls
1010 years

CD Q-balls
1500 years

Q-balls that go from FD to CD for Q < 1057 are ruled out, unless thelifting terms can break the baryon number.

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Alexander Kusenko (UCLA)IPMU ’08

White Dwarfs

White dwarfs can also accumulate SUSY Q-balls. The rate of consumption
is lower because of the lower density. Nevertheless, one should consider
a possible limit coming from the fact that some very old (10 Gyr) whitedwarfs are known to have cooled down to very low temperatures; they emit
Lwd = 3 × 10−5L⊙ = 7 × 1028 erg/s.
Q-balls must not produce more heat than this.
No new limits arise. For m = 0, Q-balls are ruled out by stability of neutronstars. For m 6= 0, the rate of heat release is much than Lwd.

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Alexander Kusenko (UCLA)IPMU ’08

Gravitational radiation from the fragmentation process

One can expect gravitational waves if• large masses move around• relativistic velocities• no spherical symmetry
All of these conditions can be satisfied for some flat directions.

43Alexander Kusenko (UCLA)IPMU ’08

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101020304010(m) mt = 020304010(n) mt = 752030

4010(o) mt = 150

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2020

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1010

10

102030(q) mt = 525

40102030(r) mt = 675

40102030(s) mt = 8253040(p) mt = 375

4010204010203040(t) mt = 900

Two-dimensional charge density plots.

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Alexander Kusenko (UCLA)(u) mt = 900(v) mt = 1050

IPMU ’08(w) mt = 1200(x) mt = 1350

Three-dimensional charge density plots.
The lack of spherical symmetry in the early steps of fragmentation means
gravity waves can be produced.

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Alexander Kusenko (UCLA)IPMU ’08

Analytical estimates [AK, Mazumdar]

The mass density of the condensate undergoing fragmentation can bewritten as ρ(x, t) = ρ0 + ρ1(x, t) , whereρ1(x, t) = ǫρ0Z

d3k eαkt cos(ωt − ~k·~x) .

The quadrupole moment that generates gravity waves:
Dij =Zd3x xixj T 00(x, t) ,where the energy-momentum tensor T 00(x, t) ≈ ρ(x, t).
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Alexander Kusenko (UCLA)IPMU ’08

Based on the analytical and numerical calculations of the condensate
fragmentation [Kawasaki et al.],k ∼ ξk × 102H∗, ωk ∼ vk ∼ ξk × 102 vH∗,
where H∗ is the Hubble constant at the time of the condensate.

For ω ∼ 102vH∗, the power in gravitational waves in a Hubble volume:
2 6
ρ
−20v
4P ∼ 10 ξk G 4 .
H∗

For mode φ(x, t) ≈ R(t) exp{αk t} cos(ωk t − kx), where R(t) is a slowly
changing function of time,47

Alexander Kusenko (UCLA)IPMU ’08

At the time of production [AK, Mazumdar],ΩGW ∗ ∼ 10−3ξk−3ξv6

ρ20(H∗MPl)4

The energy density depends on the type of SUSY breaking and the type offlat direction.
Strong gravitational waves:• gravity mediated SUSY breaking (more mass per scalar)• not the flat direction of AD baryogenesis: ηB = nB /nγ ∼ 10−10 toosmall• (B + L) flat directions OK: sphalerons destroy (B + L), so there is noconstraint on the initial density carried by the (B + L) flat directions.
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Alexander Kusenko (UCLA)IPMU ’08Predictions:

Peak frequency of the gravitational radiation observed today, f∗ = ωk /2π:
f = f∗

a∗a0= f∗≈ 0.6 mHz ξkξv



a∗arh

gs,rh
100

gs,0gs,rh1 / 6 

1 / 3 Trh1 TeVT0Trhf∗10H∗

,Trh ∼ 1 TeV ⇒ mHz frequency, accessible to LISA Trh ∼ 100 TeV ⇒10–100 Hz frequency, accessible to LIGOIII and BBO
Spectral signature: signal is peaked at the longest wavelength, determinedby the size of the Q-balls, and it falls off as 1/f 3 for larger frequencies.
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Alexander Kusenko (UCLA)IPMU ’08

The fraction of the critical energy density ρc stored in the gravity waves
today isΩGW4 2H∗= ΩGW∗
H01/3
−51.67 × 10100−8 −3 6 −2∗ ≈ 10≈ξk ξv hΩGW2hgs,∗a∗a0

LISA sensitivity: ΩGW h2 ∼ 10−11 at mHz frequencies

LIGO III sensitivity: ΩGW h2 ∼ (10−5 −10−11) in the (5−103) Hz frequency
band.
Numerical simulations under way [AK, Mazumdar, Multamäki]

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Alexander Kusenko (UCLA)IPMU ’08Conclusion• SUSY + Inflation ⇒ Q-balls, some may be stable, may be dark matter• Typical size large ⇒ typical density small ⇒ need large detectors tosearch for relic Q-balls• Gravitational waves from the fragmentation of (B + L) flat directions
may be observed by LIGO III and LISA, as well as BBO.

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