Gravi-Leptogenesis: Leptogenesis from Gravity Waves in Pseudo-scalar Driven Inflation Models

0.16 MB·English

by  S. H. Alexander

IPM Lepton and Hadron Physics School and Conference May 2006 1
Gravi-Leptogenesis:
Leptogenesis from Gravity Waves in Pseudo-scalar Driven Inflation Models
S. Alexander
Physics Department and the Institute for Gravitational Physics and geometry,
The Pennsylvania State University, 104 Davey Lab, University Park, PA 16802, U.S.A
M. Peskin
SLAC, Stanford University, 2575 Sand Hill Road, Menlo Park, CA 94025, U.S.A
M. M. Sheikh-Jabbari
Institute for studies in theoretical Physics and Mathematics (IPM),
P.O.Box 19395-5531, Tehran, IRAN
7
0 Inthistalkwepresentamechanismforleptogenesiswhichisbasedongravitywavesproduceddur-
0 inginflation. Weshowthatwheninflationisdrivenbyapseudo-scalarfieldthemetricperturbations
2 generatedduringinflationcanbecomebirefringent,thereforegivinganon-vanishingcontributionto
the gravitational triangle anomaly and sourcing lepton anti-lepton asymmetry. As this asymmetry
n
issourcedbythefieldswhichareactiveduringinflation,itisnotwashedoutordilutedbyinflation.
a
Theamountofmatterasymmetrygeneratedinourmodelcanbeofrealisticsizefortheparameters
J
within therangeofsomeinflationaryscenarios andgrandunifiedtheories. Thistalk isbasedon[1]
8
which has appeared on thearXiv as hep-th/0403069.
1

1
I. INTRODUCTION iii) CP and baryon number violating interactions
v
9 should be active at a time when the universe is out
3 One of the puzzles of astroparticlephysics and cos- of thermal equilibrium.
1 mology, which has been around for more than half A baryon excess this large cannot be produced in
1
a century and since the existence of antimatter and theearlyuniversewithintheStandardModel(SM)of
0
anti-particles established in collider experiments, has particlephysics[4]. Thisisduetothefactthatbaryon
7
0 been to explain, if our universe is mostly made out of number violating interactions in the Standard Model
/ matter, why and how this has happened during the are loop suppressed and the only source of CP viola-
h
courseofthe evolutionofthe universestartingfroma tioninthehadronicsectorisintheDiracphaseofthe
p
symmetric soup of matter and antimatter soon after CKMmixingmatrix,whichisnotenoughforexplain-
-
p the Big Bang. ingthe baryonasymmetryobservedtoday. Moreover,
e The fact that in the parts of the universe visible assumingthatthescaleofinflationislargerthanTeV,
h
to us there is an excess of matter over antimatter i.e. the SM is at work after inflation is ended, the
:
v has now been backed by the recent determinations out-of-equilibrium condition can be created at phase
i of the cosmological parameters from the cosmic mi- transitionsor throughlate decay ofmassive particles.
X
crowavebackgroundobservationsandtheWMAP ex- The most attractive choice for a phase transition is
r
a periment. Quantitatively this asymmetry is usually that associated with electroweak symmetry breaking.
giventhrough the ratio of excess of baryondensity to However, that phase transition is probably not suffi-
the photon density [2] ciently strongly first-order.
Although what is observed is a baryon asymme-
n
B =(6.5 0.4) 10−10 , (1) try, since the 1980’s it has been realized that the
n ± ×
γ standard weak interactions contain processes, medi-
pwhhoetroenns.BT=hinsbra−tino¯bhaansdthneγniiscethperonpuemrtbyetrhdaetnissittyimoef caotendvebrtybsaprhyaolenrsonasnd(SleUp(t2o)nsinasntadntaornest)h,ewrmhiaclhlyinatcetri--
independent, as the evolution of the n and n with vated at temperatures greater than 1 TeV. Thus, we
b γ
the cosmic Hubble expansion are the same. This is a canalsocreatethebaryonasymmetrybycreatingnet
smallnumber,butatthe sametime itislargeenough lepton number at high temperature through out-of-
to be a puzzle for models of particle physics. equilibrium and CP-asymmetricprocesses [5, 6]. Sce-
About forty years ago Sakharov stated the three narios of this type are known as leptogenesis.
necessary conditions to generate a matter-antimatter To use the possibility of lepton asymmetry and
asymmetrydynamicallyfromasymmetricinitialcon- sphalerons we, however, need to fulfill Sakharov con-
ditions [3]: ditions for leptons. Within the usual SM this does
i)ourparticlephysicsmodelshouldhavebaryonnum- not solve the issue and we are hence forced to asso-
ber violating vertices. ciate the observed baryon asymmetry of the universe
ii) CP should be violated. to physics beyond the SM.
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Most of the allowed parameter space of the mini- N = 3, and above that N = 0 [11, 12]. In any case,
malsupersymmetricStandardModelhasalreadybeen here we do not restrict ourselves to a specific parti-
excluded and large CP violating phases are strongly cle physics model, and keep an open mind on larger
constrainedinsupersymmetricmodels[7]thoughthey values of µ.
still could appear in the neutrino Yukawa couplings
that are used in the Fukugita-Yanagida scenario for
leptogenesis [6]. Models that can explain the baryon B. CP violation
excess typically involve exotic nonstandard physics,
CPviolatingcouplingsintheHiggsorsupersymmetry The need for CP violation manifests itself in our
sectors or in the couplings of the heavy neutral lep- model through the fact that a non-zero lepton num-
tons associated with neutrino mass [8]. In any event, ber generation can be achieved when RR˜ is non-
there is good reason to seek more effective sources of vanishing. As we will show explicitly,hRR˜ireceives
CP-violating out-of-equilibrium physics. a contribution with a definite sign from gravitational
Here we present a new mechanism for the creation fluctuationsproducedduringinflation,whichisdriven
of the matter-antimatter asymmetry, one associated bya pseudo-scalar field. In otherwords,CP-violation
with gravitationalfluctuations created during cosmo- in our model arises from the inflaton field with a CP-
logical inflation [1]. odd component. Note that during inflation the ex-
pectationvalue ofthe inflatonis non-vanishingandis
rolling in time.
II. OUTLINE OF THE MECHANISM
Such models of inflation can be naturally achieved
if the inflaton is a complex modulus field such as one
Let us first spell out how the three Sakharov con- finds in supergravity or superstring models. In or-
ditionsarerealizedinourmodelofmatter-antimatter der to use these models, however, we need to make
asymmetry, the gravi-leptogenesis. surethattheyhaveflatenoughpotentialsrequiredfor
(slow-roll) inflation. The simplest model of this kind
is when we have a single field inflation and a pseudo-
A. Lepton number violation scalar φ as the inflaton, known as natural inflation,
however it can be incorporated to have multiple ax-
The lepton number violation in our model comes ions such as in N-inflation models [16]. In fact, these
from triangle anomaly. As it is well-known [9], the models of inflation fit quite nicely into extensions of
lepton number current, and hence the total fermion the standard model and in string inspired inflation.
number current, has a gravitational anomaly in the Theimaginarypartφ ofthis field(whichwehence-
Standard Model. Explicitly, forthcallan‘axion’)cancoupletogravitythroughan
interaction
N
∂ Jµ = RR˜ (2)
µ ℓ 16π2 ∆L=F(φ)RR˜ , (4)
where
where F is odd in φ. Under P and CP
Jµ = ℓ¯γµℓ +ν¯γµν , (3a)
ℓ i i i i φ φ & F(φ) F(φ).
i=XL,R →− →−
1 Terms like (4) would generically appear once we inte-
RR˜ = ǫαβγδR R ρσ , (3b)
2 αβρσ γδ grate out heavy fermions axially coupled to φ or as a
result of the Green-Schwarz mechanism [13]. In the
and N = N N , which is three in the Standard
L − R Appendix we will explicitly show how a linear F of
Model. Theanomalyisaconsequenceofanimbalance
the form
between left- and right-handed leptons. In general
N
when heavy right-handed neutrinos are also added to
F(φ)= φ, (5)
the Standard Model, as is done in the seesaw mecha- (16π2M )
Pl
nismforexplainingthesmallnessoftheneutrinomass,
(2) will be correct in an effective theory valid below with N depending on the details of string compacti-
a scale µ, of order of the right-handedneutrino mass. fication, arises from heterotic string theory compacti-
More concretely N can in general be a function of fied to four dimensions [14].
energy. At low energies, below the right-handed neu-
trino mass scale N = 3. At higher energies, N could
be anywhere between zero to three, depending on the C. Out-of-equilibrium
details of the particle physics model invoked. In the
usual seesaw scenarios with three right handed neu- Out of equilibrium in our model is achieved noting
trinos, µ can be as large as 1014, for energies below µ thatweapplytheinteractionin(4)tothedynamicsof
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metric fluctuations during inflation where due to (ex- A. The equations of motion
ponential) growth of the background space-time, the
lepton number production is naturally out of equilib-
Plugging(8)intothe action(6),uptosecondorder
rium.
in h and h , we obtain
L R
We are now ready to compute the amount of the
leptonnumbergeneratedduringinflation. Inouranal- L= (h (cid:3)h +h (cid:3)h )
L R R L
ysis,weassumethatwehaveagivensuccessfulmodel −
∂2 ∂2 ∂2 ∂2
of inflation which involves an axion field and do not +16iF(φ) h h h h
elaborateonthedetailsoftheinflationarymodel. The (cid:20)(cid:18)∂z2 R∂t∂z L− ∂z2 L∂t∂z R(cid:19)
axiondrivenmodelsofinflation,thenaturalinflation, ∂2 ∂2 ∂2 ∂2
+a2 h h h h
arepreferredfromaparticlephysicsperspective. Nat- (cid:18)∂t2 R∂t∂z L− ∂t2 L∂t∂z R(cid:19)
uralinflationgivesasmallselfcouplingfortheinflaton
∂ ∂2 ∂ ∂2
field without fine tuning [10]. Throughout this note +Ha2 h h h h +O(h4)
R L L R
(cid:18)∂t ∂t∂z − ∂t ∂t∂z (cid:19)(cid:21)
weusenotationsandconventionsof[15],inparticular
M =2.44 1018 GeV is the reduced Planck mass. (10)
Pl
×
where
III. GRAVITY WAVE EVOLUTION
∂2 ∂ 1 ∂2
(cid:3)= +3H .
∂t2 ∂t − a2∂z2
Tobegin,wemustcomputethe productionofgrav-
itational waves during inflation under the influence As it is explicitly seen from (10), if h and h
L R
of the coupling (4). The action which describes the havethesamedispersionrelation,RR˜vanishes. Thus,
gravity waves is hence nonzero RR˜ requires “cosmological birefringence”
1 during inflation. Such an effect is induced by the
L= 2Mp2l −det g R+F(φ)RR˜. (6) addition of (4) to the gravitational equations. Lue,
p Wang,andKamionkowski(LWK) [18]andAlexander
IngeneralmetricperturbationsaboutanFRWuni- andMartin[19]havestudiedtheeffectsofsuchanin-
verse can be parameterized as teraction in generating observable parity-violation in
the cosmic microwave background. In a future work,
ds2 = (1+2ϕ)dt2+w dtdxi
− i (7) this coupling will be related to an observable to de-
+a2(t) ((1+2ψ)δ +h )dxidxj tect birefringence in binary systems for the LISA and
ij ij
(cid:2) (cid:3) Advanced LIGO gravitationalwave detectors [17].
where ϕ, ψ, w and h respectively parameterize the
i ij It is straightforward to obtain equations of motion
scalar,vector,andtensorfluctuationsofthemetric. It
for h and h :
L R
is straightforward to show that the scalar and vector
perturbations do not contribute to RR˜, and so we ig- Θ Θ
h = 2i h˙′ , h =+2i h˙′ , (11)
norethesefluctuationsinthefollowingdiscussion. We L − a L R a R
canalsofixagaugesothatthetensorfluctuationsare
parameterized by the two physical transverse trace- where
less elements of h . For such physical gravity waves
ij 4 d
which are moving in the z direction, the metric takes Θ= (F˙a2)/M2
the form a2dt Pl (12)
4(F′′φ˙2+2F′Hφ˙)/M2
ds2 = dt2+a2(t) (1 h )dx2 ≃ Pl
+
− −
+ (1+h+)dy2(cid:2)+2h×dxdy+dz2 (8) dotsdenotetimederivatives,andprimesdenotediffer-
(cid:3) entiationof F with respect to φ. To obtainthe above
where a(t) = eHt during inflation and h+, h× are equationswehaveusedthe factthatthe inflatonfield
functions of t, z. is only a function of time t. In the second line of the
To see the CP violationmore explicitly, it is conve- expression for Θ, assuming the slow-roll inflation, we
nient to use a helicity basis have dropped the terms proportional to φ¨ (explicitly
we have dropped 4F′φ¨/M2 ).
1 1 Pl
hL = (h+ ih×) , hR = (h++ih×) . (9) Note that with a constant φ, (4) is the Gauss-
√2 − √2
Bonnet term and being a total divergence (in four
Here h and h are complex conjugate scalar fields. dimensions), cannot affect the equations of motion;
L R
To be veryexplicit, the negativefrequency partof h thus, all terms in Θ involve derivatives of φ. These
L
is the conjugate of the positive frequency part of h , equations should be compared to those for evolution
R
andbotharebuiltfromwavefunctionsforleft-handed in flat space given by LWK [18]. The new term pro-
gravitons. portional to Hφ˙ leads to a substantial enhancement
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in the size of Θ. With this simplification, and the to the limit kη 1. In this region, we can ignore
≫
approximate form (5), the potential terms in (19) and take the solution to
be approximately a plane wave. More explicitly,
√2ǫ H 2
Θ= 2π2 (cid:18)M (cid:19) N , (13) g(η)=exp[ik(1 Θ2)1/2η(1+α(η))] , (20)
Pl −
where ǫ = 1(φ˙)2/(HM )2 is the slow-roll parameter where α(η) logη/η.
2 Pl ∼
of inflation [15].
IV. THE GREEN’S FUNCTION
B. Gravitational birefringence
We would now like to use (18) to compute the ex-
To see gravitational birefringence we need to solve pectation value of RR˜ in an inflationary space-time.
theequationsofmotionexplicitly. Letusfocusonthe It turns out that it will be dominated by the sub-
evolution of h and, more specifically, on its positive horizon, quantum part of the gravity-wave evolution.
L
frequency component. It is convenient to introduce Hence, to compute RR˜ we only need the two point
h i
conformal time (Green’s) function hLhR :
h i
1 1 ′ ′ ′ ′
η = = e−Ht . (14) G(x,t;x,t)= hL(x,t)hR(x,t)
h i
Ha H
= d3k eik·(x−x′)G (η,η′) . (21)
(Note that conformal time η runs in the opposite di- Z (2π)3 k
rection from t.) The evolution equation for h then
L
becomes Fork paralleltoz,theFouriercomponentGk satisfies
(15) with a delta-function source
d2 1 d d2 d2
h 2 h h = 2iΘ h . (15)
dη2 L− ηdη L− dz2 L − dηdz L d2 2(1 +kΘ) d +k2 G (η,η′)=i(Hη)2δ(η η′).
(cid:20)dη2 − η dη (cid:21) k M2 −
If we ignore Θ for the moment and let h eikz, Pl
L ∼ (22)
this becomes the equation of a spherical Bessel func-
For Θ=0, the solution of this equation is
tion:
ddη22hL−2η1ddηhL+k2hL =0 (16) Gk0(η,η′)=ℵh+L(k,η)h−R(−k,η′) η <η′
 h−(k,η)h+( k,η′) η′ <η ,
ℵ L R −
for which the positive frequency solution is (23)

where
h+(k,η)=e+ik(η+z)(1 ikη) . (17)
L − H 2 1
We now look for solutions to (15) with h eikz. (24)
L ∼ ℵ≡(cid:18)M (cid:19) 2k3
To do this, let Pl
hL =eikz ( ikη)ekΘηg(η) (18) and h−L is the complex conjugate of (17), and h+R, h−R
· − are the corresponding solutions of the h equation.
R
where g(η) is a Coulombwavefunction, we then have ForΘ=0,thesesolutionsarethesameasforh ,but
L
the structure of (23) will be preserved when we go to
d2 g+ k2(1 Θ2) 2 2kΘ g = 0 . (19) thecaseΘ=0. TheleadingeffectofΘistointroduce
dη2 (cid:20) − − η2 − η (cid:21) the expone6ntial dependence from (18),
This is the equation of a Schr¨odinger particle with G =e−kΘηe+kΘη′G (25)
k k0
ℓ = 1 in a weak Coulomb potential. When Θ = 0,
the Coulomb term vanishes and we find the spheri- for both η >η′ and η <η′. The prefactor is modified
cal Bessel function (17). For hL, the Coulomb term inorderΘ2,andthe wavefunctionsacquireadditional
is repulsive; for hR, with the opposite sign of the Θ corrections that are subleading for kη 1. Neither
term, the Coulomb potential is attractive. This leads of these effects will be important for ou≫r result.
to attenuation of hL and amplification of hR in the The Green’s function (25) can now be used to con-
early universe. This is the anticipated cosmological tracth andh toevaluatethe quantumexpectation
L R
birefringence which was also discussed by LWK [18]. value of RR˜. The result is
(For a more detailed treatment see [19].)
As we will see generation of the matter asymme- RR˜ = 16 d3k H2 (kη)2 k4Θ+O(Θ3)
try is dominated by modes at short distances (sub- h i a Z (2π)3 2k3M2 ·
Pl
horizon modes) and at early times. This corresponds (26)
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wherewehavepickeduponlytheleadingbehaviorfor Wecannowperformtheintegralstofindthelepton
kη 1. number density produced by the end of inflation
≫
We would like to emphasize that our expressionfor
hRR˜i is nonzero because of the effect of inflation in n= 1 H 2ΘH3 µ 6. (28)
producing a CP asymmetry out of equilibrium. The 72π4 (cid:18)MPl(cid:19) (cid:16)H(cid:17)
originalquantumstatefortheinflatonmighthavehad
Let us now analyze each factor in n:
nonzeroamplitudeforarangeofvaluesofφandmight
The factor (H/M )2 is the magnitude of the grav-
even have been CP-invariant. However, inflation col- Pl
•
ity wave power spectrum. We should stress that the
lapses the wavefunction onto a particular value of φ
usualgravitywavepowerspectrumiscomingfromthe
that is caught up in the local expansion of the uni-
super-horizon modes, while the main contribution to
verse. This value gives us a classicalbackgroundthat
n has come from the sub-horizon modes.
is CP-asymmetric.
The factor Θ is a measure of the effective CP viola-
Theaboveresultandcomputationsseemstobecru- •
tion caused by birefringent gravity waves.
cially dependent on the form of the Green’s function
ThefactorH3istheinversehorizonsizeatinflation;
or the vacuum state we have used. To resolve the •
this gives the density n appropriate units.
possible ambiguity in this regards, one may perform
Finally, the factor (µ/H)6 gives the enhancement
the above computation using a different method, the •
over one’s first guess due to our use of strongly
fermion level crossing, e.g. following [20]. This com-
quantum,shortdistance fluctuations to generateRR˜,
putation confirms the above results [21].
ratherthanthesuper-horizonmodeswhicheffectively
behave classically.
V. COMPUTING THE RATIO OF LEPTON
TO PHOTON DENSITIES
B. Photon number density
To complete our leptogenesis model, there are two
To understand the significance of the lepton num-
remaining steps. First, we need to compute the net
ber density (28), we should compareit to the entropy
leptonanti-leptonasymmetrygenerated. This, inour
density of the Universe just after reheating, or to the
model,shouldbe computedduringinflation. We then
photonnumber density. (Recallthat almostallofthe
needtohaveareheatingmodelthroughwhichwecan
entropy of the Universe is generated during the re-
compute the temperature and hence the entropy of
heating time and is carried in the massless degrees
the Universe after the inflation ended.
of freedom, i.e. photons.) In order this we need a
reheating model.
To illustrate the physical relevance of the result of
A. Lepton number density
our model, we take the simplest (and at the same
time very naive) reheating model, the instant reheat-
We are now ready to evaluate the lepton density
ingmodel. Thatis,weassumethatalloftheenergyof
thatarisesthroughthegravitationalanomaly(2). In-
the inflationary phase has been convertedto the heat
serting (26)into(2)andintegratingoverthe time pe-
of a gas of massless particles instantaneously. The
riod of inflation, we obtain
reheating converts the energy density of the inflaton
H−1 d3k 1 8H2k3η2Θ field φ
n= dη , (27)
Z Z (2π)3 16π2 M2 ρ=3H2M2 (29)
0 Pl Pl
where n is the lepton number density. The integral to radiation with
over k runs over all of momentum space, up to the
scale µ at which our effective Lagrangian description ρ=π2g∗T4/30 (30)
breaks down. The dominant contribution comes not and to the entropy density s, s = 2π2g∗T3/45,
from the modes which have left the horizon by the
where T is the reheating temperature, g∗ is the ef-
end of inflation (the super-horizon modes), k/H <
fective number of massless degrees of freedom and
1, but rather from very short distances compared to
s = 1.8g∗ nγ [15]. (Here we have assumed that the
thesescales. Theintegraloverη isdominatedatlarge
evolution of the Universe after the reheating era has
valuesofη,towardtheendofinflation(cf. (14)). The
been adiabatic). This gives
integral represents a compromise between two effects
of inflation, first, to blow up distances and thus carry nγ =1.28g∗−3/4(HMPl)3/2.
us to larger physical momenta and, second, to dilute
thegeneratedleptonnumberthroughexpansion. Itis Recallingthattheratioofthe presentbaryonnumber
nowclearthatthedominantcontributiontotheright- totheleptonnumberoriginallygeneratedinleptogen-
hand side comes from kη 1, as we had anticipated. esis is approximately n /n =4/11 [5], in our model
B L
≫
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we obtain range of validity of our effective field theory analy-
sis. Forexample,wecanaccommodateµ’s aslargeas
nB =4.05 10−5g∗3/4 H 7/2 Θ µ 6, (31) thestringscalewherequantumgravityeffectsbecome
nγ × (cid:18)MPl(cid:19) (cid:16)H(cid:17) important. Therefore,dependingonthedetailsofthe
model
If we are less naive, we might follow the dilution of n
andρwiththeexpansionoftheuniversetotheendof 1012GeV .µ.1016 1017GeV
−
reheating. The final result is the same (see, however,
or
[23]foracommentonthispoint). Withthe adiabatic
µ
expansionassumption, (31) can be compareddirectly 10−6 . .10−2 10−1 (35)
M −
to the present value of n /n given in (1). Pl
B γ
is a reasonable range for µ.
Insum,assumingthatscaleofinflationH saturates
C. Numerical results its current bound H 1014GeV, we have
≃
6
n µ
We should now estimate the numerical value of B 10+5 Θ . (36)
n ≃ (cid:18)M (cid:19)
n /n in our model to see if we have a viable lepto- γ Pl
B γ
genesismodel. Inourfinalresult(31)wehavefivedi-
Noting (34) and (35), it is readily seen that there
mensionless parameters, g∗, H/MPl, µ/MPl and the is big parameter space for obtaining the observed
slow-roll parameter ǫ and N (the latter two appear
baryon asymmetry of the Universe within our gravi-
throughtheCPviolationparameterΘ). Insertingthe
leptogenesis model.
expression for Θ we have
−1/2 6
nB 2.9 10−6g∗3/4√ǫ N H µ . VI. SUMMARY AND OUTLOOK
n ≃ × (cid:18)M (cid:19) (cid:18)M (cid:19)
γ Pl Pl
(32) Here we have constructed a leptogenesis model in
Within the usual supersymmetric particle physics which the lepton asymmetry is a result of gravita-
models g∗ 100 is a reasonable choice. The WMAP tional chiral anomaly. Indeed our model is a “mod-
∼
data,throughthedensityperturbationratioδρ/ρ(for ule”,whichusesminimalingredientsandcouldbe fit-
a single field inflation) leads to an upper bound on ted into a successful inflation model in which the in-
H/MPl ratio as [22] flaton field(s) has a pseudo-scalar component. From
the particle physics side,it is againlike a module and
H
.10−4, ǫ.0.01, (33) many different particle physics models could be in-
MPl voked. Here we did not discuss the details of neither
the inflationary nor the underlying particle physics
or H .1014 GeV.
models. It is of course veryimportant to explicitly fit
The factor N in (5) is inferred from the string the-
our module in viable physical models.
orycompactificationandisproportionaltothesquare
As our construction is a leptogenesis model, we
of the four dimensional M to ten dimensional (fun-
Pl needamechanismtoconverttheleptonasymmetryto
damental) Planck mass [14] (see also the Appendix).
baryon asymmetry. With the Standard Model this is
Within string theory,
usuallyperformedviathethermallyactivatedelectro-
N 102 1010 weaksphalerons,whichinourmodelshouldbe active
≃ − after inflation. This happens for reheat temperature
couldbeareasonablerange. Therefore,assumingthat Tr & 1 TeV. Within our naive “instant reheating”
the scale of inflation H saturates its current bound modelthat translatesto H &10−3 eV,which is nota
H/M 10−4, serious constraint on the inflationary models used.
Pl
≃ AccordingtorecentWMAPobservations,thescalar
Θ 10−8 100. (34) metricperturbationsgeneratedduringinflationhavea
≃ − size that gives density fluctuations with δρ/ρ 10−5.
∼
Thephysicallyviablerangefortheparameterµde- On the other hand, noting (1), one has
pendsonthedetailsoftheunderlyingparticlephysics
2
model on which our gravi-leptogenesis is based. For nB δρ
. (37)
example within the Standard Model + three heavy n ∼(cid:18) ρ (cid:19)
γ
righthandedneutrinos(andtheseesawmechanism)µ
The above could be a simple numerical accident or
could be of order of the right handed neutrino mass,
there might be some underlying deeper physics. In
in which case µ . 1012GeV. If we do not restrict
any case, our model which invokes gravity waves as
ourselves to the seesaw mechanism, µ can be larger.
the source for the baryon asymmetry and operates
The upper bound on µ is then only coming from the
duringinflation,couldbe aninterestingframeworkto
uncover the possible deeper physics behind (37).
IPM-LHP06-19May
IPM Lepton and Hadron Physics School and Conference May 2006 7
Acknowledgments In particular note that
∗(dΩ (ω))=RR˜.
We,andinparticularM.M.Sh-Jwhopresentedthe 3
talk, would like to thank organizers of IPM-LHP06
Upon compactification to four dimensions, the H2
school and conference.
term leads to
Appendix A: Realization of the model within
string theory [14] S M8 d6y d4x e−Φ(dB+α′Ω (ω))2
∼ 10Z Z 3
V
To see how this term appears in string theory, con- =M8 6 d4x[(dB)2+2α′(∗dB) Ω (ω)+ ]
sider e.g. the Heterotic SUGRA action: 10gs Z ∧ 3 ···
M2
1 = pl d4x((∂φ)2 2α′φRR˜)
S =M180Z d10x detg10 R+ 2(∂Φ)2 gs Z −
p (cid:0)
+ 1 e−ΦH2 + 1e−Φ/2(F )2 where the pseudoscalar φ (our inflaton field) is dual
12 ABC 4 AB to the two form B and the 4d Planck length, M is
(cid:1) 2 pl
defined as
where
1 ′ Mp2l =M180V6.
H =dB (Ω (A)+αΩ (ω))
3 2 3 3
− 4
And hence, finally:
(Ω (A) and Ω (ω) are the gauge and gravitational
Ch3ern-Simons 3three-forms, respectively. Explicitly, N=8π2Mp2lα′ =8π2 Mpl 2 1
gs (cid:18)M10(cid:19) √gs
2
Ω (ω)=Tr(ω dω+ ω ω ω)
3
∧ 3 ∧ ∧
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IPM-LHP06-19May