Lecture 4 on String Cosmology: Brane Inflation and Cosmic Superstrings

Home , Cosmic string, Henry Tye, String cosmology

Henry Tye

Cornell University and HKUST

Asian Winter School, Japan, January 2012 BraneBrane W worldorld Brane world in Type IIB

" -30 10 m

+ W µ! G Inflationary Scenario

Quantum Fluctuations

V( ! ) Slow!Roll Region

Damped Oscillations, Reheating

! How is inflation realized in brane world ?

Brane inflation Dvali and H.T. hep-ph/9812483

Inflaton is an open string mode

Inflaton potential comes from the closed string exchange D3-anti-D3 brane inflation

D3 Relatively flat potential ?

anti-D3

C.P. Burgess, M. Majumdar, D. Nolte, F. Quevedo, G. Rajesh, R. Zhang, hep-th/0105204 G. Dvali, Q. Shaﬁ and S. Solganik, hep-th/ 0105203 Flux compactiﬁcation where all moduli of the 6-dim. manifold are stabilized

D7−branes D7−branes

warped throat

Figure 1: A schematic picture of the Calabi-Yau manifold is presented here. The large Giddings,circle given by Kachru,dashed line Polchinskirepresent the 3-cycle where NS-NS three form H3 is turned on. The smaller circle in the throat stands for the 3-cycle where the R-R three form F3 Kachru,is turned onKallosh,. Also shown Linde,are D7-br Trivedianes wraping 4-cycKKLTles. There mavacuumy exists a number of throats likande the manyone shown othershere. There is a mirror image of the entire picture due to the IIB/Z2 orientifold operation.

From the zero mode we obtain the usual relation between the gravity strength in four dimensions and the fundamental mass scale of the higher dimensional theory

2 D 2 D 4 2A 2 M = M d y g e . (2.21) P s | mn| (0) ⇥ One may choose (0) = 1 as a convention, but in order to compare its magnitude to the excited modes magnitude we keep it as (0) which is of course a constant. For the excited mode we impose the following normalization condition

D 2 D 4 2A 2 M d y g e (y) (y) = M (2.22) s | ab| (m) (m0) P mm0 ⇥ After this general discussions we would like to ﬁnd the KK spectrum of the gravitons and other closed string modes in the KS background. We postpone the spectrum analysis until section 6 after some introduction of the KS background.

3. A Throat in the Calabi-Yau Manifold

A KKLT vacuum involves a Calabi-Yau (CY) manifold with ﬂuxes [1]. Consider F- theory compactiﬁed on an elliptic CY 4-fold X. The F-theory 4-fold is a useful way

7 The KKLMMT scenario

D7−branes D7−branes

D3-brane

anti-D3-brane Figure 1: A schematic picture of the Calabi-Yau manifold is presented here. The large circle given by dashed line represent the 3-cycle where NS-NS three form H3 is turned on. The smaller circle in the throat stands for the 3-cycle where the R-R three form F3 is turnedKachru,on. Also sho Kallosh,wn are D7-br Linde,anes wrapi Maldacena,ng 4-cycles. There MacAllister,may exists a num bTrivedi,er of throats like the one shown here. There ishep-th/0308055a mirror image of the entire picture due to the IIB/Z2 orientifold operation.

From the zero mode we obtain the usual relation between the gravity strength in four dimensions and the fundamental mass scale of the higher dimensional theory

3. A Throat in the Calabi-Yau Manifold

A KKLT vacuum involves a Calabi-Yau (CY) manifold with ﬂuxes [1]. Consider F- theory compactiﬁed on an elliptic CY 4-fold X. The F-theory 4-fold is a useful way

7 2THEAUTHOR

What is eternal inﬂation ?

a(t) eHt ⇠ H2 1/G⇤ ⇠ Consider a patch of size 1/H. Suppose T 3/H. ⇠ 3 3 9 The universe would have grown by a factor of e · = e . If inflation ends in one patch, there are still many other (causally disconnected) patches which continue to inflate. So inflation never ends.

Ln g e ⇠ The inflationary proper(tiesg) ndependsln(g/gc) maximum dimension max = 6, 7, and 7.8, and we take Nf =1, 2, 6oftheD3-branecoordinates⇠ to be dynamicalsensitiv fields. The coeelcientsyc onhave rmsthe size Q, andpr the rescaledoper quantitiestiesc ˆ = of the throat LM LM c /Q are drawn from a distribution that has unit variance. We begin at x = x 0.9, LM M 0 8⌘ 14 = 0 ✓1 = ✓2 = 1 = 2 = =1.0 , with arbitrary radial velocityx ˙ 0,arbitraryangular10 >Gµ>10 ⌘˙ { and} compactification. velocity 0 in the direction, and all otherProbability angular velocities vanishing. with We would N now likee-folds: to understand how the observables depend on the input parameters Q, max, Nf ,and ,andon ˙ M the initial data x0, x˙ 0, 0. 3 P (N ) N e ⇠ e

Radial mode + 5 angular modes Figure 1: Examples of downward-spiralingN. Agarwal, trajectories R. for Bean, a particular L. McAllister, realization of theShiu poten-G. Xu,and 1103.2775 Underwood, tial. The black dotsin mark 60 and 120 e-folds before the end of inflation (7 of the 8 curves shown achieve Ne > 120); inflation occurs along an inflection point that is not necessarilyBean, parallel to Shandera, HT, Xu, the radial direction. Red curves have nonvanishing initial angular velocities ˙ 0, while blue curves have S˙ 2=0. x S3 Baumann, Klebanov, Maldacena, Steinhardt, 0 McAllister, Dymarsky, Seiberg, Murugan, . . . Burgess, Cline, Dasgupta, Firouzjahi, Stoica, . . . 4 Results for the Homogeneous Background

As a first step, we study the evolution of the homogeneous background. In 4.1, we show that § for fixed initial conditions, the probability of Ne e-folds of inflation is a power law, and we show that the exponent is robust against changes in the input parameters , N ,and . In max f M 4.2 we present a simple analytic model that reproduces this power law. We study the e↵ect § of varying the initial conditions in 4.3, and we discuss DBI inflation in 4.4. § §

9 Outline Introduction Cosmic String Network Brane Inﬂation Microlensing Other SummaryPossibilities

Some simple scenarios

Features:

• Mobile D3s D7 • warped throats D3 • anti-D3s in throats D3 D3 • Wrapped D7s • DBI D3

Henry Tye Brane Inﬂation : an Overview 7/32 Testing Brane Inflation

• Compare power spectrum and its running • Tensor mode (B mode polarization) • Non-Gaussianity • Steps (from Gauge-gravity duality) AData blip inat CMBhigh k power (l) will spectrum improve

ACBAR Kuo etc., astro-ph/0611198 A smallPow steper spectrum in the potential can generate such a blip

Adam, Cresswell, Easther, astro-ph/0102236 As the D3 brane moves down the throat: The inﬂationarCascadey properties depends sensitively on the properties of the throat Klebanov-Strasslerand thr compactiﬁcation.oat r = r SU((K + 1)M) × SU(KM) 0 l = 1 r = r SU((K − 1)M) × SU(KM) 1 l = 2 D3 SU((K − 1)M) × SU((K − 2)M) r = r2

...... Radial...... mode +

SU 5M angular× SU M modes (2 ) ( ) Shiu and Underwood, in Bean, Shandera, HT, Xu, S2 x S3 Baumann, Klebanov, Maldacena, Steinhardt, McAllister, Dymarsky, Seiberg, Murugan, . . . Burgess, Cline, Dasgupta, Firouzjahi, Stoica, . . . RG ﬂow and Seiberg duality transition RG Flow and Seiberg Duality

8π2 T = g2

ˆ S b = 2 U T2 T (N (2 S 2 ( − ) U 1) M (N ) ) ) (2) M T1 ) + Cascade (N N SU U( S ) (1 T 1 ˆb = 0

ln(r/r0)

ln(r2/r0) ln(r1/r0) 0 • The anomalous mass dimension has a correction that depends on which step the RG flow is at. This means that the coupling flows depend on which step the flow is at. • Using gauge/gravity duality, we see that the dilaton runs and it has a kink at the position where Seiberg duality transition takes place. • this leads to steps in the warp factor, which then leads to steps in the inflaton potential. φ˙2 h4(φ), it is most sensitive to the step in the warp factor. φ˙2 increasing by a factor ∼ of (1 + 2b)acrossastepdecreasesPR by a factor of (1 + 2b). The CMB data shows that around l 20, there is a dip in the power spectrum by about 20% in k-space, which gives ∼ 2b 0.2. Using (2.9), we have ∼ 3g M 1 s 0.2 . (5.11) 8π (p +1)3 ∼ ! " i Given that g M 33 to ﬁt the spacing of the steps, we immediately get s ≈

pi =2. (5.12)

With input of 4 quantities : the position l 20, the fractional height (size) ∆T/T 0.2 Predictions ∼ # and the width ∆l 5ofthe2ndstepaswellastheposition(atl 2) of the ﬁrst step, p ∼ ∼ • Afterwe can fitting use ∆l thel and feature∆T/T at pl −~3 to20 get in aWMAP complete data, set of predictions: ∝ ∝

pl∆T/T ∆lp it predicts additional 2 2 0.7 1 • ∼ ∼ ∼ 3 20 0.2 5 steps : their positions, ∼ ∼ 4 170 0.08 40 their heights and their ∼ ∼ ∼ 5 1300 0.04 260 widths. ∼ ∼ ∼ Note that ∆T/T 0.7and∆T/T =0.2fortheﬁrsttwostepsareprobablytoobigtobe ∼ it alsotreated predicts as a perturbation. non- We shall take this to mean that the size of the ﬁrst step can be • very big. GaussianityWe should features point out due that the warp factor Eq.(2.9) is a good approximation only when to thep steps.1. This condition is also necessary to ignore the running of the dilaton. So p =3is % already too small. Moreover, as we will see in Sec. 5.4, the glitch in the power spectrum around X.l Chen,20 may R. Easther, be too large E. Lim, to be explained by a step feature in the warp factor due ∼ to the dualityastro-ph/0611645 cascades.

5.2 QualitativeGirma Hailu Analyses and HT, Hep-th/0611353 around a Single Step R. Bean, X. Chen, G. Hailu, HT and J. Xu, 0802.0491 We consider a sharp step in the warped geometry T (r). We parametrize the size of the step as 2b ∆T/T and the width as ∆φ 2d.Weusethetanhfunctioninterpolation ≡ ≡ between the two sides of the step, i.e.

r4 r r T (r) T 1+b tanh − s . (5.13) ≡ 3 R4 d # ! "$ 5.2.1 The Evolution of the Sound Speed We use the case of positive b to illustrate the evolution of the sound speed, the formulae are the same for negative b.Thedirectionofthestepissuchthat,forpositiveb, T steps up as r increases, i.e. the warp factor h T 1/4 (or the speed-limit h2)increasesasr increases. ∝ The ﬁrst stage is when branes move across the step, where the speed-limit increases suddenly. Since the time that the branes spend to across the step is very short comparing to the Hubble time (see Appendix C), during which we can ignoretheHubbleexpansion

–18– At the end of brane inflation:

B rane i nflatio n, 2003

D3-Brane and anti-D3-braneTye annihilate:

All energy released goes to strings:

fundamental strings and D1-branes

Callan and Ma ldacena Sa , hep-t vvidy, h h/9708 ep-th/ 147 Hashim 970814 oto, he 7 p-th/0 204203 Well-known important cosmological properties:

−3 • Monopoles : density ~ a Disastrous • Domain walls : density ~ 1/a Dangerous −2 cosmic strings : density ~ a • −4 interaction cuts it down to a during radiation Safe 10−11

N. Jones, H. Stoica, H.T., hep-th/0203163 S. Sarangi , H.T., hep-th/0204074 Cosmic strings

• Cosmic string interactions produce a scaling cosmic string network. History of cosmic strings

• Early 1980s : proposed to generate density perturbation as seed for structure formation; as an alternative to inflation; Gµ > 10 −6 Kibble, Zeldovich, Vilenkin, Turok, Shellard, ...... • In 1985, Witten attempted to identify the cosmic strings as fundamental strings in superstring (heterotic) theory. He pointed out a number of problems with this picture: tension too big, no production and the stability issue. • In early 1990s, COBE data disfavors cosmic strings. • By late 1990s, CMB data supports inflation and ruled out cosmic string as an explanation to the density perturbation. • In 1995, Polchinski and others pointed out the presence of D- branes in string theory. This led to the brane world/brane inflation scenarios, which led to a revival of cosmic strings, which can have much lower tensions and can be quite stable. • These cosmic strings were produced cosmologically. (p,q) Superstrings • In contrast to vortices in Abelian Higgs model, cosmic strings from brane inflation should have a spectrum in tension. • This is the (p,q) strings, where p and q are coprime. (1,0) strings are fundamental strings while (0,1) strings are D1-strings. • The spectrum depends on the particular brane 1 2 inflationary scenario.1 1 2 1!2 1+2 or

2 2 1 2 1 2 2 2 Gµp,q = !p gs + q Gµ They have non-trivial interactions.

E. Copeland, R. Myers and J. Polchinski, hep-th/0312067 G. Dvali and A. Vilenkin, hep-th/0312004 mode that is swallowed by the gauge ﬁeld inside the D3-brane via the Higgs (or Green- Schwarz) mechanism. Since the axion remains massless, the domain wall tension is exactly zero, i.e., there is no domain wall. At the same time, the D1-string becomes a vortex (a D1-vortex) in the Abelian Higgs model, with localized energy density (see Fig. 2), though it is no longer BPS. As a consequence, the D1-strings survive inside D3-branes and they will evolve as a network of cosmic strings in our universe. The actual phenomenology of the cosmic string network does depend on the details of the inﬂationary scenario.

As measured by C2, the net RR charge of this D1-vortex is zero. Besides a positive contribution to the RR charge from the winding number of the axion, there is a negative contribution to the RR charge coming from the magnetic flux. This is screening. (However, as shown in Figure 1, these two contributions do not cancel locally, so there is a non-trivial RR charge density.) That is, as we move a D1-string inside a D3-brane, it loses its RR charge, but retains its winding number. (One can define this same charge in the Abelian Higgs model, and likewise, a vortex there also has a net zero charge.) Note that the winding number is identified with the RR charge outside the D3-brane. Since it is conserved when a D1-string moves inside the D3-brane (it becomes the winding number of the D1-vortex), it is a more useful quantum number to keep track.

3. The Stability of a D1-String Inside a D3-Brane

To set up the problem, we first consider a D1-string outside a D3-brane, i.e., a D1-string as a BPS D1-brane with coupling to the 2-form RR field C2. Next we put it inside a D3-brane. The key new ingredient is the coupling of the abelian gauge field A1 to C2. The Green- Schwarz mechanism takes place and A1 becomes massive. This model can be exactly solved where the D1-string becomes a vortex in the Abelian Higgs model. This D1-vortex remains topologically stable with localized energy density, though they are no longer BPS. There is a very rich literature [31] on Chern-Simons vortices and relation between Chern-Simons we have terms and the Abelian Higgs model, though in different contexts.

2 ad dφ = d " dC2 = 2πnaδ (x ), (3.2) 3.1∧ D1-String Outsind⊥ e a D3-brane "dC = adφ = a , 2 r Let us dimensionally reduce the 10-dimensional theory to an effective 4-dimensional theory where we have chosen φ to be dimensionless and " = "4 unless we specify otherwise. Note that d dφ is not identicallytzoergoestinctehφe iksinnoettsiicngtle rvmaluefod;rφC(x2 )frinocmreatshesebby u2πlkn ascittion. Consider n D1-strings coupled to C2 in ∧ µ circles the z-axis once. the four uncompactified dimensions, all sitting along the z-axis at r = x = x2 + y2 = 0. | ⊥| After rescaling C to obtain a canonical kinetic term, we have (with constant dilaton 3.2 D1-String InsideD1-stringa D3-brane inside D3-brane2 ! background): Now consider the same D1-string inside a D3-brane, with the D3-brane world volume action that involves the C and the Abelian gauge field A . 2 1 1 S = dC 2 + 2πnaδ2(x ) C . (3.1) 1 2 1 2 2 2 2 S = G2 + dC2 + ξC2 G2 + 2πnaδ (x−) MC24 2| |(3.3) ⊥ ∧ − 2| | 2| | ∧ ⊥ ∧ !M4 " 2 where G2 = dA1, ξ = √2τ3κ4w, hτ3eirsethae=D3√-br2aτn1eκte4nmsioena. sNuorteesthtahteCR2,RA1c,hξaarngdeaohfaaveD1-string whose coordinate is δ (x )dx dy. ⊥ dimension of mass. For ξ = 0 (e.g., when V6 ), both A1 and C2 are massless. For ∧ τ1D3i-sbrantehe D1-st→rin∞g tension and κ4 is the 4-dimensional effective gravitational coupling. It finite ξ, spontaneous symmetry breaking via the Green-Schwarz mechanism takes place. 2 2 2 2 The equations of motionisforetlhaisteadctiotno atrhe:e 10-dimensional coupling κ4V6 = κ = κ10gs where V6 is the 6-dimensional compactified volume and gs is the string coupling. Introducing the dual of C2, adφ = &dC2, dD1"-vdorAtex1lo=op ξdC2, (3.4) 2 d " dC2 = ξG2 + 2πnaδ (x ). (3.5) D1-string ⊥ D1-vortex There are two ways to solve these equations. First let uL.s sLeblondolve for C and2. T HThe solution of (3.5) is (in non-compact space): hep-th/0402072 – 5 –

Figure 3: A D1-string intersecting a D3-brane an"ddfoCrm2ing=a Dξ1-Avor1tex+insaidde tφhe.D3-brane. The (3.6) picture also shows a D1-vortex loop existing on its own. where, by analogy with the solution outside the D3-brane, we get the δ function from the as measured by C2. This screening of C2 charge happens because C2 becomes massive. multi-vaNlauiveeldy, tφhe.y lPoouk tlitkeinagD1t-shtriisngbscareceknedinby Eanqa.n(ti3-D.41-)strwinge. Hgoewte:ver, they cannot annihilate, since the origin of the RR charge for the D1-component is different from that for the anti-D1-component. Furthermodre,"thdeAtwo c=har"ge(sadξodnφot c+ancξel2lAocal)ly.. As we (3.7) move this vortex outside a D3-brane, the magnet1ic flux around the core disap1pears since the gauge field is absent outside the D3-brane. So this vortex recovers its RR charge and becomes a D1-string. In this sense, we believe the conserved winding number is the For the ground state (i.e., n = 0), where φ = 0 (or is a pure gauge), we see that A1 has appropriate quantity to follow as we move a D1-string inside/outside a D3-brane. In the mass ξ.taTchyhonecontdhenesartiown apiyctutreo, asBoPlSvDe1-tsthriinsg iss eatvoortfex,csoouheprelewde siemqpluy aseteitohantsthe(3.4,3.5) is to first solve for vortex changes its thickness, tension, and magnetic flux as it moves in/out a D3-brane, but A1, it is always topologically stable, due to the winding number measured by the axion φ. 1/2 As ξ gs goes to zero, the vortex grows (1/ξ) to infinite size and its energy density ∼ spread throughout the D3-brane. The D1-string tension (ξ 2) also goes to infinity while the "dA1 = ξC−2 + dη1 ξC2 (3.8) D1-vortex tension grows only logarithmically. So, as a D1-string moves→inside a D3-brane, it goes to a lower tension vortex with infinite size, i.e., just like dissolution. since we are considering vortices without electric flux, so we expect dη1 = 0. Putting this 4.3 Low Energy Effective Theory in String Theory back into the equation for C2, we have The δ-function is the origin of the singular behavior of the tension. Including the partner of C (i.e., Φ ) should get rid of the δ-function singularity. In string theory, this massive 2 | | 2 2 Higgs mode is always present. Todget"hedr Cwit2h φ=, th(ey fo)rm"aξcomCp2lex+sca2laπr mnoadeδ. C(oxnsid)er (3.9) ⊥ the low energy effective theory of a generic Type II−B orientifold model. One may write the effective Lagrangian for all the string modes as where C2 has mass ξ. We can view this system as

L = L0,2 + L0,I + LM (4.9)

– 13 – – 6 – Strings and axions • A point particle can be charged under a gauge field, a one-form field. • A string is charged under a two-form field. • In 4-dim., a two-form field (NS-NS or RR) is dual to an axion. • In a typical realistic stringy vacuum, there are a number of axions. • So we expect a variety of cosmic string types. Scaling of the Cosmic Superstring Network

100 independent of initial conditions 80 Insensitive to the details of the 60 interactions

10ΓGµ Ω = 20 cs 2 gs

0 1 10 100 1000

M. Jackson, N. Jones and J. Polchinski, hep-th/0405229 H.T., I. Wasserman, M. Wyman, astro-ph/0503506 Relative density of (p,q) strings

∼ −8 1 np,q µp,q

0.1

0.01

0.001

0.0001

246 Cosmic string tension spectrum in a warped deformed conifold (Klebanov-Strassler)

One may view the strings as D3-branes wrapping a 2- to have a simple (expected)cycle insideform: the S3 at the bottom of the throat.

2 2 hA q bM 2 2 ⇥p Tp,q 2 + ( ) sin ( ), (1.3) ⇥ 2⇥ ⇥gs ⇥ M but the way it comesba=0bout.93266is interesting. Indeed, the tension is obtained by min- imizing the HamiltoniaMn ofis thethe RRD3-brane flux wrappingworld S3v. olume action after integrating out the extra dimensions. Care must be taken with the Hamiltonian when one has an S. Gubser, C. Herzog, I. Klebanov, hep-th/0405282, electric ﬁeld on a D-brane.H.F Firouzjahi,or example L. Leblond,the H.T.,Chern-Simons hep-th/0603161. terms which do not con- tribute to the stress energy tensor due to their topological nature nevertheless aect the Hamiltonian (hence the tension and energy) by coming into play via the con- jugate momentum. This contribution turns out to be crucial here and leads to the above simple formula (1.3). This formula has the right limits. Setting either p = 0 or q = 0 reproduces Eq.(1.2). For M and b = h = 1, it reduces to Eq.(1.1). Because p is ⇤ A ZM -charged with non-zero binding energy, binding can take place even if (p, q) are not coprime. Also, M fundamental strings can terminate to a point-like baryon, irrespective of the number of D-strings around. The paper is divided as follows. Sec. 2 reviews the properties of the KS throat we need. Sec. 3 contains the calculation and the main result. Sec. 4 includes general discussions and some comments on the issues remaining.

2. A Throat in the Calabi-Yau Manifold

2.1 The Conifold A cone is deﬁned by the following equation in C4

4 2 wi = 0 (2.1) i=1

Here Eq.(2.1) describes a smooth surface apart from the point wi = 0. The geometry around the conifold is studied in [30]. The base of the cone is a manifold X given by the intersection of the space of solutions of Eq.(2.1) with a sphere of radius r in C4 = R8, w 2 = r2 | i| i We are interested in Ricci-ﬂat metrics on the cone which in turn imply that the base of the conifold is a Sasaki-Einstein manifold. The simplest ﬁve dimensional Sasaki-Einstein manifold for N = 1 supersymmetry is T 1,1 and it is the only manifold for which the deformation is explicitly known [12].

4 Example : M=5

A baryon with mass M 3/2h /√α! ∼ A (M-p,0)

(p,0)

X. Siemens, X. Martin and K. Olum, astro-ph/0005411, T. Matsuda, hep-th/0509061, . . . . Search for Cosmic Strings

• Lensing • Cosmic Microwave Background Radiation • Gravitational Wave Burst • ∆T/T (Doppler effect) • Pulsar Timing • Stochastic Gravitation Radiation Background Possible CMB B-mode detection

^ ^ cosmic string lensing cosmic string introduces a deﬁcit angle

☀ cosmic ↑ string ➘ earth identify ●| ● ↓ ➚ ☀↓ CSL-1 Sazhin etc. astro-ph/0302547

z=0.46 ± 0.008 identical spectra with conﬁdence level above 99.9% 1.9 arc sec ⇓ -7 Gµ~ 4 x 10 Radio telescope ?

Recall Cowen and Hu.

National Radio Astronomy Observatory Shami Chatterjee, Jim Cordes, H.T., Ira Wasserman Unfortunately not (higher resolution Hubble pictures):

January 2006 If it is cosmic string lensing Bound on cosmic string tension

log(Gµ) ns =0.95 U. Seljak and A. Slosar, astro-ph/0604143 WMAP

0 ≤ β ≤ 0.05

ns ∼ 0.98 + β → ns ≤ 1.03 log r ∼−8.8+60β Gµ < 10−8 log Gµ ∼−9.4+30β S. Shandera and H.T., 0601099 cusps and kinks are quite common in string evolution

CUSP

h(t) ~ |t|1/3

wave form of gravitational wave bursts KINK

h(t) ~ |t| 2/3

Damour and Vilenkin A cusp

Blanco-Padillo and Olum Gravitational wave radiation from cusps Damour and Vilenkin

Log ← prediction → 10 More recent analysis −5 0 10 10 1/year

−5 10 −10 10 1/year

−10 10 [Hz] [Hz] γ γ

−15 10 −20 −1/3 −3 −21 −1/3 −3 Λ Cosmology A =10 s , p=10 −15 50% 10 Λ Cosmology A =10 s , p=10 −20 −1/3 50% −21 −1/3 Λ Cosmology A =10 s , n=3/2 Λ Cosmology A =10 s , n=3/2 50% −20 −1/3 −3 50% Λ Cosmology A =10 s , n=3/2, p=10 Cosmology A =10−21 s−1/3, n=3/2, p=10−3 50% Λ 50% −20 −20 10 10 9 −12 −11 −10 −9 −8 −7 −6 −5 −4 −12 −11 −10 −9 −8 −7 −6 −5 −4 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 Gµ Gµ and dashed curves of Fig. 1 show γ and η computed us-

−6 10 ing the Damour-Vilenkin cosmological functions namely, 1/year Eqs. (62), (63) and (64). For the dashed-dot curves we −8 10 −21 −1/3 have used an amplitude estimate of A50% = 10 s . 0 −100 TX.his Siemens,amplitude estim aJ.te Creighton,can be obtained u sing the Ini- 10 10 tial LIGO sensitivity curve, setting the SNR threshold to −12 10 I. Maor, S. Majumder, [Hz] 1, and assuming all cusp events are optimally oriented γ

−2 −14 −20 −1/3 10 Λ Cosmology A =10 s (as used for the dashed horizontal lines of Fig. 1 in [13]). 10 −2 50% −21 −1/3 Λ Cosmology A =10 s K. Cannon and J. Reed, 10 50% This is also our estimate for the amplitude in the case of −16 −20 −1/3 10 DV Cosmology A =10 s 50% −21 −1/3 Advanced LIGO. The dashed curves show γ and η com- DV Cosmology A =10 s 50% gr-qc/0603115 −20 −1/3 −4 −18 10 puted with the amplitude A50% = 10 s , which we 10 −12 −11 −10 −9 −8 −7 −6 −5 −4 10−4 10 10 10 10 10 10 10 10 feel is more appropriate for Initial LIGO. The thick and Gµ 10 thin solid curves show γ and η computed by evaluating −6 the cosmological functions (Eqs. (A4), (A6) and (A8))

η 10 η 0 10 numerically for the Λ universe (see Appendix A). The −6 10−2 thick solid curves correspond to our amplitude estimate −8 10 10 for Initial LIGO, and the thin solid curves to our estimate −20 −1/3 −3 −21 −1/3 −3 Cosmology A =10 s , p=10 −4 for Advanced LIGO. Λ 50% 10 Λ Cosmology A =10 s , p=10 −8 50% −20 −1/3 η The fu−n21ction−a1/3l dependence of the rate of gravitational −10 Λ Cosmology A =10 s , n=3/2 10−6 10 Λ Cosmology A =10 s , n=3/2 10 50% −20 −1/3 wave bursts on Gµ is discussed in detail in Appendix B. −20 −1/3 −3 Cosmology A =10 s 50% Λ 50% −21 −1/3 −3 Λ Cosmology A =10 s , n=3/2, p=10 Cosmology A =10 Cosmology−21 s−1/3 AHere=10we sum smari,s en=3/2,those fip=10ndings. From left to right, −8 Λ Λ 50% 10 50% −20 −1/3 50% DV Cosmology A =10 s the first steep rise in the rates as a function of Gµ of −12 −10 50% −21 −1/3 DV Cosmology A =10 s 50% 10 10−10 the dashed and dashed-dot curves of Fig. 1 comes from −12 −11 −10 −9 −8 −7 −6 −5 −4 10 −12 −11 −10 −9 −8 −7 −6 −5 −4 −12 −11 −10 −9 −8 −7 −6 −5 −4 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 1010 10 10 10 10 1010 ev10ents produ10ced at smal10l redshifts (10z 1). The peak and Gµ Gµ Gµ subsequent decrease in the rate star"ting around Gµ 10−9 comes from events produced at larger redshifts bu∼t still in the matter era (1 z zeq). The final rise comes FIG. 1: Plot of the rate of gravitational wave bursts, γ from events produced in"the "radiation era (z z ). $ eq (top panel), and the probability η of having at least least For “classic” cosmic strings (p = ε = n = 1), the mat- one event in our data set with amplitude larger than A50% ter era maximum in our estimate for the rate of events in a year of observation (bottom panel), as a function of Gµ. −4 For all curves we have set α = ΓGµ, Γ = 50, f∗ = 75 Hz, at Initial LIGO sensitivity is about 7 10 events per FIG. 2: Comparison of γ (top panel) and η with a year of FIG. 3: Same as Fig. 2 but with thyeearA, wdhivchaisnscubestdantLialIlyGlowOer tahamn×tphelira-te 1 per c = p = 1, and the ignorance constants g1 = g2−=211. Th−e 1/3 ∼ observation (bottom panel), as a function of Gµ for several tuddaesh-deost tanidmdaashteed cuorvfesAsho5w0%γ an=d η 1co0mputed wsith yea.r suggested by the results of Damour and Vilenkin the Damour-Vilenkin cosmological functions Eqs. (62), (63) [11, 12, 13]. The bulk of the difference arises from our −21 −1/3 −20 −1/3 string models. For all three curves an Initial LIGO ampli- and (64), with A50% = 10 s , and A50% = 10 s estimate of a detectable amplitude. This is illustrated by −20 −1/3 respectively. The thick and thin solid curves show γ and η the dashed-dot and dashed curves of Fig. 1, which use the tude estimate of A50% = 10 s has been used, and the computed in a universe with a cosmological constant with same cosmological functions, and two estimates for the −20 −1/3 −21 −1/3 − − − − amplitudes A50% = 10 s , and A50% = 10 s 21 1/3 20 1/3 amplitude, A50% = 10 s and A50% = 10 s cosmological functions have been evaluated in the Λ universe. respectively. respectively. Our amplitude −est3imate results in a de- puted with a reconnection probacbreialseitinytheobfurspt ra=te b1y a0bout a.facTtorhoef 100 at the The model parameters are identical to those of Fig. 1 except matter era peak. A more detailed discussion of the effect of events in our data set with amplitudes greater than dashed curves show γ and η comofpthue atmepdlitudfeoonrthleoraotepcasn bewfoiutndhin Aappendix B. where indicated. The solid curves show γ and η computed A . In an observation time T the probability of not 50% The remaining discrepancy arises from differences in the −3 having such an event is exp( γT ). Hence, the odds of size given by Eq. (57) with ε = co1sm,oloagyn, ads wenll as=facto3rs/of2O.(1) Tthaht wiesre dropped with a reconnection probability of p = 10 . The dashed having at least least one even−t in our data set with am- in the previous estimates, which account for a further plitude larger than our minimum detectable amplitude curves show γ and η computed with a size of loops given by is the value of n we expect whednecretahse eby sfapctoer coftarbuoumt 10. oTfhispiseirllu-strated by is, the difference between the dashed-dot and thin solid − Eq. (57) with ε = 1, and n = 3/2. The dashed-dot curves turbations oηn= 1loenγgT . strings i(s65)invcuervressoef Flyig. 1,pwrhiochpusoe rthteisoamneaamlplittuode estimate − −21 −1/3 show the combined effect of a low reconnection probability, the mode number, and the resultA50w% e= 1e0 xpsect, anidfthtehDeamosupr-Veilcen-kin cosmo- −3 Figure 1 shows the rate of burst events, γ, as well logical interpolating functions (Eqs. (62), (63) and (64)) p = 10 , as well as a size of loops given by Eq. (57) with as the probability η of having at least least one event and the Λ universe functions (Eqs. (A4), (A6) and (A8)) truinmour odaftapseet rwtituh armbpalitutdieolanrgser othnan lAo50n% gforsatrirnespgecstiviesly.dWohemn zi

• 10 (Damour and Vilenkin, 2001) • 100 or more for cosmic superstrings (2004) • down by a factor of 100 (2006) • (p,q) string spectrum raises this by a factor of about 5 • lots of loops raises it more • tension is getting smaller ? • effect of beads ? C. J. Hogan, astro-ph/0605567

FIG. 1: Predicted gravitational wave backgrounds from strings, and noise sources. Broad band energyR.de nCaldwellsity is shown iandn unit sB.of tAllen,he critica lPRD45,density for h34470 = 1, a(1992)s a function of frequency, for α = 0.1. Noise levels are shown for current millisecond pulsar data (MSP), and projected LISA sensitivity in maximum resolution and Sagnac modes. Confusion noise is shown for massive black hole binaries (MBHB), the summed Galactic binary population including binary white dwarfs (UB+WUMa+GCWDB+CV), and extragalactic populations of white dwarfs (XGCWDB) and neutron stars (XGNSB). Radiation from loop populations at high redshift (H) and present-day (P) is shown, labled by the value of Gµ. Dotted curves show the contributions of z > 1 loops where they are subdominant to the P contributions. Current (MSP) sensitivity is at about Gµ 10−10, ≈ and LISA will reach to around Gµ 10−15. ≈ population. The fraction of horizon-size loops that needs to stabilize in order for radiation- era loops to dominate near f is only (Gµ)1/2(Ω /√30Ω ) 600(Gµ)1/2 (times some peak ≈ M R ≈ numerical factors), which is a small number for the light strings we are contemplating. Unless α is relatively large, accurate estimates of the radiation background will require estimates of loop spectra with very large dynamic range.

9 Search for Cosmic Strings with low tension

• LensingX • Cosmic Microwave BackgroundX Radiation • Gravitational Wave Burst • ∆T/T (DopplerX effect) • Pulsar Timing ? • Stochastic Gravitation Radiation Background • Micro-lensing • Cusp Doppler effect ? 2THEAUTHOR

What is eternal inﬂation ?

Ln g e ⇠ (g) n ln(g/g ) Low tension⇠ stringsc 8 14 10 >Gµ>10

• Warped geometry can provide very low tension cosmic strings. • They radiate gravitational waves much slower, so they live much longer, in particular the small loops. • The small loops can cluster like dark matter; thus their local density is 5 orders of magnitude larger than that from the scaling cosmic string network.

David Chernoff and HT, 0708.4282 Chernoff 0908.4077 was a numerical error of 1/32 in the pc to cm conversion and finally the horizon size 2 4 changed from 3 to 4 Gpc or (3/4) . All in all its a reduction2of 10 . was a numerical error of 1/3 in⇥ the pc to cm conversion and finally the horizon size We have estimated the rate RL assuming that the loops are2constantly traversing 4 changed from 3 to 4 Gpc or (3/4) . All in all its a reduction of 10 . new areas of the sky. That assumption should be fine as long as the total observing ⇥ time is less than the characWterise thaic pverioe edstimaof mottedion theof therateloopR(i.e.L assumingif its oscillatingthat the loops are constantly traversing and begins to lense thenewsameareapartsstheooffobservthetheer-sourcebaskyckgroline. ofundTsighhat.moThetrecaharasthasumptioctenristiconce).EinsteinnaThengleshouldis shortesbte fine as long as the total observing timescale is for the shortest string to move across its ⇥oEwn= 8⌅loGµop size: time is less than the characteris3tic Gµperiod of motion of the loop (i.e. if its oscillating = 12.04 10 10 . (3.1) was a numerical error of 1/3 in ⇥the pc to2 cm10conversion and finally the horizon size 2 ⇥ ⇥ 4 and beginslcghangtoedlensfrom e3 tothe4GRpGµcsameor (3/4) .paAll rtsin all itsofa reductionthe baofckgro10 . und more than once). The shortest tosc The character135yristics angular size of a stellar source at distance R is ⇥ (3⇥= R.11)/R. The We have estimated the8 rate R assuming that the loops ⇥are consta⇥ ntly traversing ⇥ relatc iv⇥e size is 10 L timescale is for the shortest string to10 move across its own loop size: new areas of thesky⇥. That⇥assumptio5 2 n10should 10be0kpfinec as long as the total observing the observer-source line of sigh⇥ =t.4.6The10charac⇥teristic Einstein angle is (3.2) ⇥ ⇥ Gµ R time is less than theEcharacteristicperiod of⇥mot ion of⇥the loop (i.e. if its oscillating This means that any experiment withwhicdurah impliestionthat⇥theT stellar< toscsourcewillwillbgenerallye in ablineare well describregedime.as a pToinhet source. the observander-sourcebegins lineto lensof sighe the⇥t.E Thesame= 8⌅charaGpaµrtscteorfisttheic Einsteinbackgroaundnglemois re than once). The shortest Thetimescalerelativisticallyis for themovingshortestanMicro-lensingd oscillatingstring to mostringve acrosswilllgcreatitseobrighwn lotnesopssize:fluctuatioRns Gµin expected number of transient lensed sources from a background3 populatioGµ n of size NS the background star tha⇥t c=an8=b⌅eG1se.µ04archedt10oscfor in a microlensing. 13exp5yrerimenst. (3.1) (3.11) E 2 ⇥ 2 10 10 8 observed in an experiment of durwasatioa nnumericalThe⇥TactualiserroNsituaLr tion=of 1Ris/3Lso⇥mewinThatheNtSmorep.lc tocomplica⇥cm cocted.nv⇥ersionFGµor a loaondp, afinas oppllyosedthe10tohoa rizon size 3 g Gµ⇥ ⇥R = 1.04 t2osc10 135yrs . 4 (3.1) ⇥(3.11) changstraed ighfrotmstring,3 to one4 Gexppcectsor (3lensing/4) .likAlle thatcin allof aitsp10oina treductionmass10 8for photoof ns10with .impact The characteristic duration Theof lechansingracterisevticenantgularwillsizebeof a⇥ste⇥llar source2⇥ 10at distance R is ⇥ = R /R. The parameter large compared to the size of the loop⇥ and le⇥nsing like⇥that of ⇥a ⇥straight⇥string relatWive sizehaveisestimated the rate RL assuming that the loops are constantly traversing This meansThe chaforThisracterpathsthatmeansisthatic tanapathatngularssycloansesizeyexteoxpaopferimenseerimenagmenstellattowithrf thesourcetdurastring.withattion10distanceWe⇥willduraT

– 14 – Outline Lensing Introduction Microlensing of Stars TypicalCosmic String Network scenario Schematic Lensing Microlensing LSST Rates Summary

t 300s(R/10 kpc )(Gµ/10 10)andt 70 yr (Gµ/10 10). ⇠ osc ⇠ Repetition : 103 ⇠ Flux 3.0

2.5

2.0

1.5

1.0

0.5

Time 0 10 20 30 40

Henry Tye Brane Inﬂation : an Overview 29/32 Outline Lensing Introduction Microlensing of Stars Cosmic String Network Schematic Lensing Microlensing LSST Rates Rate for LSSTSummary (Large Synoptic Survey RateTelescope)... for LSST (Large similar Synoptic to Survey European Telescope) GAIA - - - - similar for European GAIA

LSST : 4 108 stars observed 103 times each with a 15 s ⇥ exposure over a 10 year period. Red for 10 kpc and blue for 100 kpc. Figure with = 1; expect 103. G G ⇠ Log NL 10, 100 kpc 2

! " 1

0 ! ! Log Μ !14 !13 !12 !11 !10 !9 !8

Henry Tye Brane Inﬂation : an Overview 30/32 Superstring theory may be tested

• Instead of searching for tiny particles or signatures in accelerators, we can search for distinctive features suggested by string theory. • Steps in the CMB power spectrum suggested by gauge- gravity duality in a warped throat. • Production of cosmic superstrings that stretch across the universe. The string tensions have the right values so these cosmic superstrings are compatible with all present day observational bounds and yet can be detected in the near future. • Micro-lensing detection offers the best hope to reach to very low tensions and provide very distinctive signatures. Thank you !