Resonant Superstring Excitations during Inflation

0.38 MB·

by  G. J. Mathews

Resonant Superstring Excitations during Inflation
G. J. Mathews1,2, M. R. Gangopadhyay1, K. Ichiki3, T. Kajino2,4,5
1Center for Astrophysics, Department of Physics,
University of Notre Dame, Notre Dame, IN 46556
2National Astronomical Observatory, 2-21-1, Osawa, Mitaka, Tokyo 181-8588, Japan
3Department of Physics, Nagoya University, Nagoya 464-8602, Japan
4 Department of Astronomy, The University of Tokyo,
7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan and
5 International Research Center for Big-Bang Cosmology and Element Genesis,
and School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China
(Dated: January 4, 2017)
Weexplorethepossibilitythatboththesuppressionofthe(cid:96)=2multipolemomentofthepower
7 spectrumofthecosmicmicrowavebackgroundtemperaturefluctuationsandthepossibledipinthe
1 powerspectrumfor(cid:96)=10−30canbeexplainedastheresultoftheresonantcreationofsequential
0 excitationsofafermionic(orbosonic)open(orclosed)superstringthatcouplestotheinflatonfield.
2 Weshowthatasuperstringwith≈43or42oscillationscanfitthedipsintheCMBTTandEEpower
spectra at (cid:96)=2, and (cid:96)=20, respectively. We deduce degeneracy of N ≈1340 from which we infer
n
acouplingconstantbetweenthestringandtheinflatonfieldofλ=0.06±0.05. Thisimpliesmasses
a
J ofm≈540−750mpl forthesestates. Wealsoshowmarginalevidenceforthenextlowerexcitation
with n = 41 oscillations on the string at (cid:96) ≈ 60. Although the evidence of the dips at (cid:96) ≈ 20 and
3
(cid:96)≈60 are of marginal statistical significance, and there are other possible interpretations of these

] features, this could constitute the first observational evidence of the existence of a superstring in
O Nature.
C
PACSnumbers: 98.80.Cq,98.80.Es,98.70.Vc
.
h
p
- I. INTRODUCTION One such feature is the well known suppression of the
o (cid:96) = 2 moment of the CMB power spectrum observed
r both by Planck [5] and by the Wilkinson Microwave
t It is generally accepted that the energy scale of super-
s AnisotropyProbe(WMAP)[7]. Thereisalsoafeatureof
a strings is so high that it is impossible to ever observe
marginalstatisticalsignificance[6]intheobservedpower
[ a superstring in the laboratory. There is, however, one
spectrum of both Planck and WMAP near multipoles
epoch in which the energy scale of superstrings was ob-
1
(cid:96) = 10−30. Both of these deviations occur in an inter-
v tainablein Nature. That isinthe realmofthe earlymo-
esting region in the CMB power spectrum because they
7 ments of trans-Plankian [1] chaotic inflation out of the
correspond to angular scales that are not yet in causal
7 string theory landscape.
5 This paper explores the possibility that a specific se- contact when the CMB photons were emitted. Hence,
0 the observed power spectrum is close to the true primor-
quence of super-string excitations may have made itself
0 dial power spectrum for these features.
known via its coupling to the inflaton field of inflation.
.
1 This may have left an imprint of ”dips” [2] in the TT In the Planck inflation parameters paper [6], how-
0 power spectrum of the cosmic microwave background. ever, the deviation from a simple power law in the range
7
The identification of this particle as a superstring is pos- (cid:96)=10−30 was deduced to be of weak statistical signifi-
1
siblebecausetheremaybeevidenceforsequentialoscilla- cance due to the large cosmic variance at low (cid:96). In par-
:
v torstatesofthesamesuperstringthatappearondifferent ticular, a range of models was considered from the mini-
Xi scales of the sky. malcaseofakineticenergydominatedphaseprecedinga
short inflationary stage (with just one extra parameter),
The primordial power spectrum is believed to derive
r
a from quantum fluctuations generated during the infla- to a model with a step-like feature in the inflation gen-
erating potential and in the sound speed (with five extra
tionary epoch [3, 4]. The various observed power spec-
parameters). These modifications led to improved fits of
traofthecosmicmicrowavebackground(CMB)arethen
upto∆χ2 =12. However,neithertheBayesianevidence
modified by the dynamics of the cosmic radiation and
nor a frequentist simulation-based analysis showed any
matter fluids as various scales re-enter the horizon along
statistically significant preference over a simple power
witheffectsfromthetransportofphotonsfromtheepoch
law.
oflastscatteringtothepresenttime. Indeed, thePlanck
data[5,6]haveprovidedthehighestresolutionyetavail- Nevertheless, a number of mechanisms have been pro-
ableinthedeterminationCMBpowerspectra. Although posed [8] to deal with the suppression of the power spec-
theTTprimordialpowerspectrumiswellfitwithasim- trum on large scales and low multipoles. In addition
ple tilted power law [6], there remain at least two in- to being an artifact of cosmic variance [6, 9], large-scale
teresting features that may suggest deviations from the power suppression could arise from changes in the ef-
simplest inflation paradigm. fectiveinflation-generatingpotential[10],differinginitial
2
conditions at the beginning of inflation [2, 11, 12, 14– prediction in this model of a third possibly observable
19], the ISW effect [20], effects of spatial curvature [21], dip in the CMB power spectrum.
non-trivial topology [22], geometry [23, 24], a violation
of statistical anisotropies [25], effects of a cosmological-
constant type of dark energy during inflation [26], the II. RESONANT PARTICLE PRODUCTION
bounce due to a contracting phase to inflation [27, 28], DURING INFLATION
theproductionofprimordialmicroblack-holes[29],hemi-
spherical anisotropy and non-gaussianity [30, 31], the The details of the resonant particle creation paradigm
scattering of the inflationary trajectory in multiple field during inflation have been explained in Refs. [2, 37, 38].
inflationbyahiddenfeatureintheisocurvaturedirection Indeed,theideawasoriginallyintroduced[46]asameans
[32], brane symmetry breaking in string theory [33, 34], for reheating after inflation. Since Ref. [37], subsequent
quantum entanglement in the M-theory landscape [35], work [47–50] has elaborated on the basic scheme into a
or loop quantum cosmology [36], etc. model with coupling between two scalar fields. Here, we
In a previous work [2], we considered another possibil- summarize the essential features of a single fermion field
ity, i.e. that the suppression of the power spectrum in coupled to the inflaton as a means to clarify the physics
the range (cid:96) = 10−30 in particular could be due to the of the possible dips in the CMB power spectrum.
resonant creation [37, 38] of Planck-scale fermions that In this minimal extension from the basic picture, the
couple to the inflaton field. inflatonφispostulatedtocoupletoparticleswhosemass
The present paper is an extension of that work. Here, is of order the inflaton field value. These particles are
we show that both the suppression of the (cid:96)=2 moment then resonantly produced as the field obtains a critical
and the suppression of the power spectrum in the range value during inflation. If even a small fraction of the
(cid:96)=10−30couldbeexplainedfromtheresonantcoupling inflaton field is affected in this way, it can produce an
to successive numbers of oscillations of a single open or observable feature in the primordial power spectrum. In
closed fermionic superstring. Indeed, both the apparent particular, there can be either an excess in the power
amplitude and the location of these features arise natu- spectrumasnotedin[37,38],oradipinthepowerspec-
rallyinthispicture. Thereisalsoapredictionofanother trumasdescribedinRef.[2]. Suchadipoffersimportant
string excitation for (cid:96)≈60. newcluestothetrans-Planckianphysicsoftheearlyuni-
verse.
This result is significant in that accessing the mass
In the simplest slow roll approximation [3, 4], the gen-
scales of superstrings is impossible in the laboratory. In-
erationofprimordialdensityperturbationsofamplitude,
deed, the only place in Nature where such scales exist is
δ (k), when crossing the Hubble radius is just,
during the first instants of cosmic expansion in the in- H
flationary epoch. Here we examine the possibility that,
H2
of the myriads of string excitations present in the birth δ (k)≈ , (1)
H 5πφ˙
of the universe out of the M-theory landscape, it may be
that one string serendipitously made its presence known whereH istheexpansionrate,andφ˙ istherateofchange
viaanaturalcouplingtotheinflatonfieldduringthelast
of the inflaton field when the comoving wave number k
∼9 e-folds visible on the sky.
crosses the Hubble radius during inflation. The reso-
Weemphasize,however,thattheexistenceofsuchfea-
nant particle production could, however, affect the infla-
tures in the CMB power spectrum from string theory tonfieldsuchthattheconjugatemomentumφ˙ isaltered.
is not unique. In [33, 34] the suppression of the (cid:96) = 2
This could cause either an increase or a diminution in
and the dip for (cid:96) = 10−30 were simultaneously fit in a
δ (k) (the primordial power spectrum) for those wave
H
string-theory brane symmetry breaking mechanism. In
numberswhichexitthe horizonduringtheresonantpar-
this case, however, the source of the features is due to ticle production epoch. In particular, when φ˙ is accel-
the nature of the inflation-generating potential in string
erated due to particle production, it may deviate from
theory. This mechanism splits boson and fermion ex-
the slow-roll condition. In [37], however, this correction
citations, leaving behind an exponential potential that
was analyzed and found to be << 20%. Hence, for our
is too steep for the inflaton to emerge from the initial
purposes we ignore this correction.
singularity while descending it. As a result, the scalar
For the application here, we adopt a positive Yukawa
field generically ”bounces against an exponential wall.”
coupling of strength λ between an inflaton field φ and
Just as in [10], this steepening potential then introduces
a field ψ of N degenerate fermion species. This differs
an infrared depression and a preinflationary break in the
from [37, 38] who adopted a negative Yukawa coupling.
power spectrum of scalar perturbations, reproducing the
With our choice, the total Lagrangian density including
observed feature.
the inflaton scalar field φ, the Dirac fermion field, and
In the present work, however, rather than to address the Yukawa coupling term is then simply,
theimplicationsfortheinflation-generatingpotential,we
1
considerthepossibilityoftheresonantcreationofopenor L = ∂ φ∂µφ−V(φ)
closedfermionic(orbosonic)superstringswithsequential tot 2 µ
numbers of oscillations. We also note that there is a + iψ¯∂/ψ−mψ¯ψ+Nλφψ¯ψ . (2)
3
For this Lagrangian, it is obvious that the fermions have where the amplitude A and characteristic wave number
an effective mass of k can be approximately related to the observed power
∗
spectrum from the approximate relation:
M(φ)=m−Nλφ . (3)
(cid:96)
k ≈ ∗ , (9)
Thisvanishesforacriticalvalueoftheinflatonfield,φ = ∗ r
∗ lss
m/Nλ. Resonant fermion production will then occur in
wherer isthecomovingdistancetothelastscattering
lss
a narrow range of the inflaton field amplitude around
surface, taken here to be 13.8 Gpc [5]. For each reso-
φ=φ .
∗ nance the values of A and k determined from from the
∗
AsinRefs.[2,37,38]welabeltheepochatwhichparti-
CMB power spectrum relate to the inflaton coupling λ
clesarecreatedbyanasterisk. So,thecosmicscalefactor
and fermion masses m via Eqs. (7) and (8).
islabeleda atthetimet atwhichresonantparticlepro-
∗ ∗
duction occurs. Considering a small interval around this A=|φ˙∗|−1Nλn∗H∗−1 . (10)
epoch, one can treat H =H as approximately constant
∗ Theconnectionbetweenresonantparticlecreationand
(slow roll inflation). The number density n of particles
the CMB derives from the usual expansion in spherical
can be taken [2, 37, 38] as zero before t and afterwards (cid:80) (cid:80)
∗ harmonics, ∆T/T = a Y (θ,φ) (2 ≤ l < ∞
as n = n [a /a(t)]3. The fermion vacuum expectation l m lm lm
∗ ∗ and−l≤m≤l). Theanisotropiesarethendescribedby
value can then be written, theangularpowerspectrum, C =(cid:104)|a |2(cid:105), asafunction
l lm
(cid:104)ψ¯ψ(cid:105)=n Θ(t−t )exp[−3H (t−t )] . (4) of multipole number l. One then merely requires the
∗ ∗ ∗ ∗ conversionfromperturbationspectrumδ (k)toangular
H
where Θ is a step function. powerspectrumCl. Thisiseasilyaccomplishedusingthe
Then following the derivation in [37, 38], we can write CAMBcode[52]. Whenconvertingtotheangularpower
the modified equation of motion for the scalar field cou- spectrum, the amplitude of the narrow particle creation
pled to ψ: feature in δH(k) is spread over many values of (cid:96). Hence,
the particle creation features look like broad dips in the
φ¨+3Hφ˙ =−V(cid:48)(φ)+Nλ(cid:104)ψ¯ψ(cid:105) , (5) power spectrum.
where V(cid:48)(φ) = dV/dφ. The solution to this differential
equationafterparticlecreation(t>t )isthensimilarto
∗ III. FERMIONIC SUPERSTRINGS
that derived in Refs. [37, 38] but with a sign change for
the coupling term, i.e.
In string theory, the inclusion of fermions requires
φ˙(t>t ) = φ˙ exp[−3H(t−t )] supersymmetry. To incorporate supersymmetry into
∗ ∗ ∗
string theory, there are two different approaches:
− V(cid:48)(φ)∗(cid:2)1−exp[−3H(t−t )](cid:3) 1. The Ramond-Neveu-Schwarz (RNS) formalism which
3H ∗
∗ is supersymmetric on the string world-sheet.
+ Nλn∗(t−t∗)exp[−3H∗(t−t∗)] . (6) 2. The Green-Schwarz (GS) formalism which is
developed generally in a 10-dimensional Minkowski
Thephysicalinterpretationhereisthattherateofchange
background spacetime (or other spaetime).
of the amplitude of the scalar field rapidly increases due
to the coupling to particles created at the resonance φ=
These two formalisms are equivalent in Minkowski
φ .
∗ spacetime. In what follows we adopt the RNS formal-
Then, using Eq. (1) for the fluctuation as it exits the
ism.
horizon, and constant H ≈ H , one obtains the pertur-
∗
bation in the primordial power spectrum as it exits the
horizon:
A. RNS String Boundary Conditions and mode
[δ (a)] Expansions
δ = H Nλ=0 ,
H 1+Θ(a−a )(Nλn /|φ˙ |H )(a /a)3ln(a/a )
∗ ∗ ∗ ∗ ∗ ∗
The first task before us is to develop the RNS sector
(7)
and then deduce the mass spectrum both in the open
whereΘ(a−a )istheHeavisidestepfunction. Itisclear
∗
andclosedstringsector. IntheRNSsectortheintegrand
inEq.(7)thatthepowerinthefluctuationoftheinflaton
of the action is written as a function of the worldsheet
field will abruptly diminish when the universe grows to
light-cone coordinates τ,σ:
some critical scale factor a at which time particles are
∗
resonantly created.
Using k∗/k = a∗/a, then the perturbation spectrum S = 1 (cid:90) dτdσ∂ Xµ(σ−,σ+)∂ Xµ(σ−,σ+)
Eq. (7) can be reduced [38] to a simple two-parameter π + −
function. + i (cid:90) dτdσ(cid:2)ψµ(σ−,σ+)∂ ψ (σ−,σ+)
[δ (a)] π − + −µ
δH(k)= 1+Θ(k−kH∗)A(kN∗λ/=k)03ln(k/k∗) . (8) + ψ+µ(σ−,σ+)∂−ψ+µ(σ−,σ+)(cid:3) . (11)
4
Here Xµ represents bosonic strings. For fermionic fields to obey,
the action takes the form:
(cid:90)
SF = d2σ(ψ−∂+ψ−+ψ+∂−ψ+) , (12) ψ+µ|σ=π =−ψ−µ|σ=π . (16)
The NS sector describes bosons living on the back-
where d2σ ≡dτdσ. ground spacetime. In the NS sector then the mode
expansion of the fields are given by:
B. Open RNS Strings 1 (cid:88)
ψµ(τ,σ)= √ bµe−ir(τ−σ) ,
− r
2
n∈Z+1
For 10-dimensional superstring theories, open strings 2
are described by the variables ψαµ(τ,σ), with α = +,−. ψµ(τ,σ)= √1 (cid:88) bµe−ir(τ+σ) . (17)
For open strings in the light-cone gauge, the equations + 2 r
of motion imply ψl is right-moving, while ψi is left n∈Z+12
+ −
moving. In the full superstring theory, the state space
Here, we use the usual notation in string theory
breaksintotwosectors: 1)Theuppersignisadoptedfor
whereby n or m represent integer valued numbers,
statesintheRaymond(R)sector;2)whileintheNeuveu-
while r or s are half-integer value numbers.
Schwarz (NS) sector the lower choice of sign is used. A
periodic fermion corresponds to the R sector while an
anti-periodic fermion corresponds to the NS sector. C. Open RNS Mass Spectrum
One obtains the open superstring theory by applying
theGSOtruncationofthespace[13]. Thatis,intheNS
Havingidentifiedthebosonicandfermionicsectorsfor
sector one only keeps states that contain an odd number
both open and closed string sectors, one can deduce the
of fermions acting on the ground state. In the R sector
the mass spectrum in both cases. Key to the present
one keeps the set of states built upon the ground state
application is that in the R sector, the mass spectrum
|Ra(cid:105) (a=1,....,8) with an even number of fermions plus
1 has the simple form
thosestatesbuiltfromanoddnumberoffermionsonthe
ground state |Ra(cid:105). In this GSO truncation then bosonic (cid:112)
2 M = (n/α(cid:48)) , (18)
statesarisefromtheNS sector,whileallfermionicstates
arise in the R sector, even though world-sheet fermionic with n an integer eigenvalue of the number operator N⊥
states are used in both sectors. defined in such a way as to kill the the tachyon ground
Inthecaseofopenstrings,theboundariesareσ =0,π. states |Ra(cid:105) and |Ra(cid:105), while α(cid:48) is a normalization. In
1 2
particular, N⊥ counts the contributions of the fermionic
• Raymond Sector:
oscillators that appear in the states.
In this case one chooses the other end of the open
This means that, in our particle creation paradigm,
string to obey,
the presence of a single open string with a mass M =
(cid:112)
ψµ| =ψµ| . (13) (n0/α(cid:48)) should be accompanied by excitations with
+ σ=π − σ=π
n = n − 1, n = n − 2,... and also states with
−1 0 −2 0
The Raymond boundary condition induces n+1 =n0+1, n+2 =n0+2, etc.
fermions to the background spacetime. Thus, the So, one can entertain the possibility that the
field equations for the fermionic fields are: quadrupole suppression is due to a state with higher
(cid:112)
M = (n +1/)α(cid:48) thatresonatedduringinflationbe-
+1 0
fore the state M. We can also look for a state with
∂ ψµ =0 , M =(cid:112)(n −1/)α(cid:48).
− + −1 0
∂ ψµ =0 . (14)
+ −
Imposing the Raymond boundary conditions gives D. Closed RNS Strings
the mode expansion of the fields as,
For closed-strings the boundary conditions are peri-
odic:
1 (cid:88)
ψµ(τ,σ)= √ dµe−in(τ−σ) ,
− n
2
n∈Z
ψµ(τ,σ)= √1 (cid:88)dµe−in(τ+σ) . (15) ψ±µ(τ,σ)=±ψ±µ(τ,σ+π) , (19)
+ n
2
n∈Z where the positive sign describes the periodic or the
R sector boundary condition and the negative sign
• Neveu-Schwarz Sector: describes the anti-periodic or NS sector boundary
In this case one chooses the other end of the string condition. This leads to the following two choices for
5
the mode expansions of the left-going and right-going F. Number of oscillations on the string
states. The left-going states in the R sector and NS
sectors, respectively are:
In our previous paper [2] we related the mass of the
resonant particle to the scale k∗ and the number of e-
ψµ(τ,σ)= (cid:88)d˜µe−2in(τ+σ) , (20) folds N∗ of inflation associated with that scale, based
+ n upon an assumed a general monomial inflation effective
n∈Z potential. That is, the resonance condition relates the
ψµ(τ,σ)= (cid:88) ˜bµe−2ir(τ+σ) . (21) mass m to φ∗ via,
+ r
n∈Z+1
2 m=Nλφ , (25)
∗
For the right going mode one has the following two
choices, However, for a general monomial potential,
(cid:18) φ (cid:19)α
ψ−µ(τ,σ)= (cid:88)dµne−2in(τ+σ) , (22) V(φ)=Λφm4pl mpl , (26)
n∈Z
ψµ(τ,σ)= (cid:88) bµe−2ir(τ+σ) . (23) there is an analytic solution for φ∗ for a given scale in
− r terms of the number of e-folds of inflation N
∗
n∈Z+1
2
(cid:112)
Since there are two choices for the left-going and two φ∗ = 2αN∗mpl , (27)
choicesfortheright-goingstates,thereareatotaloffour
different sectors; R-R, R-NS, NS-R and NS-NS sectors. whereN∗isthenumberofe-foldsofinflationcorrespond-
The R-R and NS-NS modes correspond to background ing to a given scale k∗,
spacetime bosons, while the NS-R , R-NS sectors are the
background fermions. 1 (cid:90) φ∗ V(φ)
N = dφ =N −ln(k /k ) , (28)
∗ m2 V(cid:48)(φ) ∗ H
pl φend
E. Closed RNS Mass Spectrum where φ is the value of the scalar field at the end of
end
inflation,N isthetotalnumberofe-foldsofinflationand
Sinceclosedstringsareroughlyobtainedbymultiplica- the Hubble scale is k = h/2997.3 = 0.000227 Mpc−1
H
tively combining left moving and right moving copies of (for h=0.68) [5].
open strings, one expects a similar spectrum for closed So, for either open or closed superstrings we can write
strings. In the case of closed strings in the RNS sec-
√
(cid:112) (cid:112)
tor one must consider two possibilities: 1) R-R, NS-NS M = (n/α(cid:48))=Nλφ =Nλ 2α N −ln(k /k ) ,
∗ ∗ H
(bosonic); and 2) NS-R, R-NS (fermionic). (29)
NowusingtheGSOmechanismandimposingG-parity andwecanwritethemasscorrespondingtoagivenmul-
leads to type IIA and type IIB string theories. For the tipole on the sky
massspectrumofinteresthere,weexploretheNS-Rand
R-NS sectors which represent the fermionic states in the M((cid:96) )2 ∝(N −ln(k /k )) . (30)
∗ ∗ H
background spacetime. As in type IIA strings there are
56 gravitinos and 8 dilatons. Next, we make the simplifying assumption that the
In the NS-R, R-NS sectors one obtains a very similar resonantstatesinthespectrumdifferonlyinthenumber
mass spectrum to that of open strings, only in this case of oscillations on the string. Then the coupling to the
there is a zero-point offset ξ, inflaton field λ is the same, along with the number of
(cid:112) degenerate fermion states N at a given mass. We also
M = (n+ξ)/α(cid:48) , (24)
closed keep the same normalization of the mass scale α(cid:48).
wherethevalueofξ isdeterminedbythestringtheoryof Then if we take N = 50, we can write for the ratio
choice (i.e. type IIA or IIB). Also, the form changes for of the quadrupole ((cid:96)∗ = 2) suppression resonance to the
theGSsectorinotherbackgroundspace-times. However, (cid:96)∗ =20 resonance for open superstrings:
that is beyond the scope of this paper.
Thus, the only change for closed strings is that the M2((cid:96)∗ =2) = n+1 ≈ N −ln(k∗(2)/kH) . (31)
number of oscillations on the string becomes an effective M2((cid:96)=20) n N −ln(k (20)/k )
∗ H
numberofoscillationsn =n+ξ,withthevalueofξ ∼
eff
8 determined by the type of string theory. Henceforth, This relation can be used to deduce the number of oscil-
therefore we adopt n ≡ n, keeping in mind that the lations on the string.
eff
true number of oscillations is less for closed strings than
for open strings and also that n is not necessarily an (cid:20) N −ln(k (2)/k ) (cid:21)−1
n= ∗ H −1 . (32)
integer for closed strings. N −ln(k (20)/k )
∗ H
6
2500 1
2000
0.5
TTpC/2l1500 EEpC/2l
+1) +1)
l(l 1000 l(l
0
500
0 -0.5
0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 80 90
l l
FIG. 1: (Color online) The fit (res line) to (cid:96)≈2, (cid:96)≈20 and FIG. 2: Same as Figure 1 but in this case the lines are the
(cid:96) ≈ 60 suppression of the TT CMB power spectrum as de- derived EE CMB power spectrum based upon the fits to the
scribedinthetext. PointswitherrorbarsarefromthePlanck TT power spectrum shown in Figure 1.
Data Release [5]. The green line shows the best standard
ΛCDM power-law fit to the Planck CMB power spectrum
of freedom, i.e. the amplitude A and two independent
Once the number of oscillations is deduced, one can pre-
values for k .
dict k for the next state via: ∗
∗
(cid:18) (cid:19) Figure 2 similarly illustrates the derived EE CMB
n−1
ln(k /k )=N − [N −ln(k (20)/k )] (33) power spectrum based upon the fits to the TT power
∗ H n ∗ H
spectrum shown in Figure 1. Although this fit is not op-
timized, andtheuncertaintyinthedataislarge, thereis
We have made a straightforward χ2 minimization to
areductionintotalχ2 by∆χ2 =−5forthelinewithres-
fit the CMB Planck power spectrum [5] for the (cid:96)∗ ≈ 2
onant superstring creation. Hence, the EE spectrum is
and (cid:96)∗ = 20 resonances. We also searched via Eq. (33)
atleastconsistentwiththisparadigmandinfactslightly
forapossiblethirdresonantstringexcitationcorrespond-
favors it.
ing n−1 oscillations on the string. For simplicity and
speed we fixed all cosmological parameters at the values Under the assumption that the model errors are in-
deduced by Planck [5] and only searched over a single dependent and obey a normal distribution, then the
amplitude and two k∗ values, with the third k∗ value Bayesian information criterion (BIC) can be written [2]
predicted from Eq. (33). in terms of ∆χ2 as ∆BIC≈ ∆χ2 +(p·lnn), where p is
We deduce the following resonance parameters: the number of parameters in the test and n is the num-
ber of points in the observed data. When selecting the
best model, the lowest BIC is preferred since the BIC
(cid:96)≈2, A=1.7±1.5, k (2)=0.00048±0.00025hMpc−1
∗ is an increasing function of both the error variance and
thenumberofnewdegreesoffreedomp. Inotherwords,
(cid:96)≈20, A=1.7±1.5, k (20)=0.00149±0.00045hMpc−1 the unexplained variation in the dependent variable and
∗
the number of explanatory variables increase the value
of BIC. Hence, a negative ∆BIC implies either fewer ex-
(cid:96)≈60, A=1.7±1.5, k =0.00462±0.00035 h Mpc−1 planatory variables, a better fit, or both. For the ≈ 140
∗
data points in the range of the fits of Figure 1 plus 2,
Adoptingthisasthebestfitstringparametersgivesvia the inferred total improvement is ∆χ2 = −14 with the
Eq. (32), n=42.5 for the effective number of oscillation introduction of 3 new parameters. This corresponds to
on the string. a ∆BIC= +0.8. Generally, ∆BIC> 2 is required to be
Figure 1 illustrates the best fit to the TT CMB power considered evidence against a particular model. Hence,
spectrumthatincludesboththe(cid:96)≈2,(cid:96)≈20and(cid:96)≈60 one must conclude that although the fit including the
suppressionoftheCMB.ItisobviousfromFigure1that superstring resonances produces an improvement in χ2,
that the evidence for this fit is statistically weak due to it is statistically equivalent to the simple power-law fit.
the large errors in the data. Indeed, the total reduction Nevertheless, it is worthwhile to examine the possible
in χ2 is ∆χ2 =−9 for a fit with an addition of 3 degrees physical meaning of the deduced parameters.
7
IV. PHYSICAL PARAMETERS that the mass of the string excitation can be determined
once the coupling constant is known. To find the cou-
The coefficient A can be related directly to the cou- pling constant via Eq. (39), however, one must estimate
pling constant λ using the following approximation for the degeneracy of the string states.
theparticleproductionBogoliubovcoefficient[37,55–57] Wealsonotethattechnically,thedegeneracyispropor-
tional to the number of oscillations on the string. How-
(cid:18) −πk2 (cid:19) ever, since the number of oscillations n deduced here is
|β |2 ≈exp . (34)
k a2λ|φ˙ | large, the difference between the degeneracy of the n−1
∗ ∗ and n+1 states is small (a few percent) compared to
Then, the overall uncertainty in the mass. Hence our assump-
tionofaconstantdegeneracyforthestringexcitationsis
2 (cid:90) ∞ Nλ3/2 justified.
n = dk k2|β |2 = |φ˙ |3/2 . (35)
∗ π2 p p k 2π3 ∗
0
This gives, A. degeneracy of open superstrings
(cid:113)
Nλ5/2 |φ˙∗| What differs in the present application from that in
A = (36) our previous work [2] is that since we know the number
2π3 H
∗ of oscillations of the string we can approximately count
Nλ5/2 1
≈ √ . (37) thedegeneracyN ofthesuperstring. Forillustration,we
(cid:112)
2 5π7/2 δ (k )| simply describe the case of open superstrings for which
H ∗ λ=0
the R states in superstring theory can be expressed as,
where we have used the usual approximation for the pri-
mordial slow roll inflationary spectrum [3, 4]. 9 ∞ 9 ∞
(cid:89)(cid:89) (cid:89) (cid:89)
Now, given that the CMB normalization requires R sector = (αl )λn,l (dJ )ρm,J|RA(cid:105)
−n −m
δH(k)|λ=0 ∼10−5, we have l=2n=1 J=2m=1
⊗|p+,p(cid:126) (cid:105) . (41)
T
A∼1.3Nλ5/2. (38)
Here, |RA(cid:105) with A = 1,16 are the degenerate R sec-
Hence,forthemaximumlikelihoodvalueofA∼1.7±1.5, tor ground states, and |p+,p(cid:126) (cid:105) are string states. The
T
we have dJ ,dJ ,dJ ... are anticommuting creation operators
−1 −2 −3
acting on |RA(cid:105) and hence can only appear once in any
(1.1±1.0)
λ≈ . (39) given state. The αl are creation operators acting on
N2/5 −n
|p+,p(cid:126) (cid:105), and the quantities ρ are either zero or 1.
T m,J
The fermion particle mass m can then be deduced from Altogether then in the GSO truncation there are 16
the resonance condition, m=Nλφ . degenerategroundstatestimesadegeneracyof2foreach
∗
From Eq. (39) then we have m ≈ φ /λ3/2. For the ofthenoscillationsfromtheαl . operators. So,forour
∗ −n
(cid:96) ≈ 20 (k = 0.00149 ± 0.00045 h Mpc−1) resonance, state with n ≈ 42 the total degeneracy is N ≈ 1344.
∗
and k = a H = (h/2997.9) Mpc−1 ∼ 0.0002, we have This means that from Eq. (39) we deduce a coupling
H 0 0
N−N =ln(k /k )<1. TypicallyoneexpectsN(k )∼ constant of λ = 0.06±0.05. Inserting this into Eq. (40)
∗ H ∗ ∗
N ∼50−60. then implies a mass of the superstring of ∼ 540−750
We can then apply the resonance condition [Eq. (25)] m . Although this is a rather large number it is not
pl
todeducetheapproximaterangeofmassesforthestring inconsistent with the trans-Plankian physics of interest
excitations. Monomialpotentials[Eq.(26)]withα=2/3 here.
or α = 1 correspond to the lowest order approximation
tothestringtheoryaxionmonodromyinflationpotential
[63,64]. Moreover,thelimitsonthetensortoscalarratio V. CONCLUSION
from the Planck analysis [6] are more consistent with
α = 2/3 or 1. If we fix the value of A = 1.7, then from We have analyzed the possible dips at (cid:96) ≈ 2,20 and
therangeof50-60e-foldswewouldhaveφ∗ =(8−9)mpl (cid:96) ≈ 60 in the Planck CMB power spectrum in the con-
for α =2/3 or φ∗ =(10−11) mpl for α =1. Hence, we text of a model for the resonant creation of oscillations
have roughly the constraint, on a massive fermionic superstring during inflation. The
m best fit to the CMB power spectrum implies optimum
m∼(8−11) pl . (40) features with an amplitude of A ≈ 1.7 ± 1.5, located
λ3/2
at wave numbers of k = 0.00048±0.00025,0.00149±
∗
We note, however, that if the uncertainty in the normal- 0.00045,and0.00463 ± 0.00035 hMpc−1 . For string-
ization parameter A is taken into account, this range in- theory motivated axion monodromy inflation potentials
creases. Thisillustrationissimplymeanttodemonstrate consistent with the Planck tensor-to-scalar ratio, this
8
feature would correspond to the resonant creation of theory landscape. Perhaps, the presently observed CMB
superstring states with 41, 42 or 43 oscillations, re- power spectrum contains the first suggestion that one of
spectively with a degeneracy of N ≈ 1340 and with those many ambient superstrings may have coupled to
m∼540−750m andwithaYukawacouplingconstant the inflaton field leaving behind a relic signature of its
pl
λ between the fermion species and the inflaton field of existence in the CMB primordial power spectrum.
λ≈0.06±0.05)
Obviously there is a need for more precise determina-
tions of the CMB power spectrum for multipoles, par-
Acknowledgments
ticularly in the range of (cid:96) = 2−100, although this may
ultimately be limited by the cosmic variance. Neverthe-
less, in spite of these caveats, we conclude that if the WorkattheUniversityofNotreDameissupportedby
present analysis is correct, this may be the first hints the U.S. Department of Energy under Nuclear Theory
at observational evidence of a superstring present at the Grant DE-FG02-95-ER40934. Work at NAOJ was sup-
Planck scale. portedinpartbyGrants-in-AidforScientificResearchof
Indeed, one expects a plethora of superstring excita- JSPS (26105517, 24340060). Work at Nagoya University
tionstobepresentwhentheuniverseexitedfromtheM- supported by JSPS research grant number 24340048.
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