On Asymptotically De Sitter Space: Entropy & Moduli Dynamics

On Asymptotically de Sitter Space: Entropy & Moduli Dynamics To Appear... [1206.nnnn]

Mahdi Torabian International Centre for Theoretical Physics, Trieste

String Phenomenology Workshop 2012 Isaac Newton Institute , CAMBRIDGE

Tuesday, June 26, 12 WMAP 7-year Cosmological Interpretation 3

TABLE 1 The state of Summarythe ofart the cosmological of Observational parameters of ΛCDM modela Cosmology

b c Class Parameter WMAP 7-year ML WMAP+BAO+H0 ML WMAP 7-year Mean WMAP+BAO+H0 Mean

2 +0.056 Primary 100Ωbh 2.227 2.253 2.249−0.057 2.255 ± 0.054 2 Ωch 0.1116 0.1122 0.1120 ± 0.0056 0.1126 ± 0.0036 +0.030 ΩΛ 0.729 0.728 0.727−0.029 0.725 ± 0.016 ns 0.966 0.967 0.967 ± 0.014 0.968 ± 0.012 τ 0.085 0.085 0.088 ± 0.015 0.088 ± 0.014 2 d −9 −9 −9 −9 ∆R(k0) 2.42 × 10 2.42 × 10 (2.43 ± 0.11) × 10 (2.430 ± 0.091) × 10 +0.030 Derived σ8 0.809 0.810 0.811−0.031 0.816 ± 0.024 H0 70.3km/s/Mpc 70.4km/s/Mpc 70.4 ± 2.5km/s/Mpc 70.2 ± 1.4km/s/Mpc Ωb 0.0451 0.0455 0.0455 ± 0.0028 0.0458 ± 0.0016 Ωc 0.226 0.226 0.228 ± 0.027 0.229 ± 0.015 2 +0.0056 Ωmh 0.1338 0.1347 0.1345−0.0055 0.1352 ± 0.0036 e zreion 10.4 10.3 10.6 ± 1.210.6 ± 1.2 f t0 13.79 Gyr 13.76 Gyr 13.77 ± 0.13 Gyr 13.76 ± 0.11 Gyr a The parameters listed here are derived using the RECFAST 1.5 and version 4.1 of the WMAP[WMAP-7likelihood 1001.4538] code. All the other parameters in the other tables are derived using theRECFAST1.4.2andversion4.0oftheWMAP likelihood code, unless stated otherwise. The difference is small. See Appendix A for comparison. Theb Larson expansion et al. (2010). rate “ML” of refers the to theUniverse Maximum Likelihood is increasing. parameters. c Larson et al. (2010). “Mean” refers to the mean of the posterior distribution of each parameter. The quoted errors show the 68% confidence levels (CL). Darkd 2 energy3 is introduced2 to account−1 for this accelerated expansion. ∆R(k)=k PR(k)/(2π )andk0 =0.002 Mpc . e “Redshift of reionization,” if the universe was reionized instantaneously from the neutral state to the fully ionized state at zreion. Note that these values are somewhat different from those in Table 1 of Komatsu et al. (2009a), largely because of the changes in the treatment of reionization history in the Boltzmann code CAMB (Lewis 2008). A fsmallThe present-day positive age of cosmological the universe. constant (vacuum energy) is phenomenologically the simplest source of DE. [See recently Bousso 1203.0307] TABLE 2 Summary ofIts the presence 95% confidence indicates limits on deviations that we from live the simple in an (flat, Asymptotically Gaussian, adiabatic, power-law)de SitterΛ CDMUniverse model except. for dark energy parameters 2 a Section Name Case WMAP 7-year WMAP+BAO+SN WMAP+BAO+H0 Tuesday, June 26, 12 Section 4.1 Grav. Waveb No Running Ind. r<0.36c r<0.20 r<0.24 c Section 4.2 Running Index No Grav. Wave −0.084 2.7 N/A 4.34−0.88 (68% CL) j local k Section 6 Gaussianity Local −10

To study the consequences of the fact that the Universe asymptotes to a de Sitter space.

Of particular interest for string phenomenology/cosmology, the implications it might have for the moduli dynamics and inflation in the early Universe.

Turned-around stand point: what should be the initial conditions for a universe to asymptote to de Sitter space.

Tuesday, June 26, 12 29/54 Hinf mPl 1/3 eNefold N (43) ! m H Hubble Pl ! 0 "

1/2 1/3 1/2 1/6 Hinf Heq Hinf H0 eNefold ∼ H H ∼ H H ! eq " ! 0 " ! 0 " ! eq " (44)

1/3 2/27 Hinf 2/3 H0 H0 6 2/3 N − 10− N − m " Hubble H m ∼ · Hubble Pl ! eq " ! Pl " (45)

25 mPl 1 Heq 1 1 Nefold ! ln ln ln NHubble 64 ln NHubble 54 H0 −3 H0 −3 ∼ −3 ! " ! " (46)

τϕ ≪ τuniverse (47)

2 ϕ¨ +(3H + Γϕ)˙ϕ + mϕϕ =0 (48)

Treheat T0 1/3 (HreheatLreheat) NHubble (49) Hreheat ∼ H0

N 1/3 (1) (50) Hubble ∼ O

reheat 27 NHubble " 10 De(51) Sitter Space in a Nutshell H m (52) osc ∼ ϕ

De Sitterm space2 is the maximally symmetric solution of the vacuum Einstein H Γ m ϕ (53) decay ∼ ϕ ∼ ϕ m equations! Pl with" a positive cosmological constant (it is positively curved).

T T 10MeV (54) reheat " BBN ∼

mϕ " 50 TeV (55)

R =(3/Λ)1/2 (56)

An observer in de Sitter space is surrounded by a cosmological horizon. The static coordinates cover the causal patch (operationally meaningful portion of de Sitter space, a region that can be probed by an observer). The horizon is observer dependent.

Tuesday, June 26, 12 De Sitter Space in a Nutshell

An object, if released, will accelerate towards the horizon. Once it crosses the horizon, it can no longer be retrieved. It is said that the entropy of ordinary matter is lost when it crosses the horizon. 2 2 mPl mPl The de Sitter Horizon has finite non-zeroSdS (30) entropy. [Gibbons and Hawking PRD∼ (1977)]Λ ∼ H ! " The total entropy in de Sitter space S S S + S (31) total ∼ dS ∼ bulk horizon The holography argument: the horizon entropy is proportional to area. 2 2 mPl mPl 1 The entropy of empty de Sitter SdSrSchwarzschild ! H0− (30)(32) ∼ Λ ∼ H It also implies that there is a bound on the entropy! of" any matter system can be put in a causal domain of an asymptotically2 de Sitter universe. Stotal SbulkT + SmhoriznH SdS (31)(33) ∼5 0 ! Pl[See0 recently∼ Conlon 1203.4576] Tuesday, June 26, 12

3 3/12 2 TrSchwarzschildm! H0− m (32) S 0 Pl Pl (34) 0 ∼ H ! H " H # 0 $ # 0 $ # 0 $ 2 T0 ! mPlH0 (33) 3 3/2 3 3 T0 mPl (TreheatLreheat) (T0L0) NHubble ! NHubble ∼ ∼ H0 H0 3 #3/2 $ 2 # $ T m m (35) S 0 Pl Pl (34) 0 ∼ H ! H " H # 0 $ # 0 $ # 0 $ 3 NHubble (L0H0) (36) ∼ 3 T (T L )3 (T L )3 0 N (35) reheat reheat ∼ 0 0 ∼ H Hubble # 0 $ eNefold L ζ (37) osc H N ∼(L Hinf)3 (36) Hubble ∼ 0 0

m 2/3 m 4/3 ϕ H t Pl Lreheat Losc e inf Losc (38) ∼ LoscΓϕ ζ ∼ mϕ (37) # ∼$ Hinf # $

3/2 2/3 m 4/3 m ϕ m L L Treheatϕ mPl L Pl (38)(39) reheat ∼ osc Γ ∼ ∼ oscmPl m # ϕ $ # #$ ϕ $

3/2 1/2 m H 3/2 1 Pl mϕ 0 1/3 Lreheat ! HT0−reheat mPl NHubble(39)(40) m∼ϕ m mPl # $ # #Pl $ $

31//26 1/12/2 1 mmPlPl HH0 0 1/31/3 LreheatLosc !! HH0− NHubbleNHubble (40)(41) mmϕ mmPl ## ϕ$$ ## Pl$$

1/16/6 1/21/2 H 1 mmPl HH0 1/3 NLefold H−inf Pl 0 N 1 1/3 (41) e osc !! 0 Hubbleζ− N (42) H mmϕ mmPl Hubble 0 ## ϕ$$ ## Pl$ $

31/6 31/2 2 N Hinf mPl H0 1 1/3 e efold Treheat T0 ζ− mNPl (42) ! H m m ! Hubble (43) 0 H# ϕ $ − # HPl $ H # 0 $ # 0 $ # 0 $

3 3 2 Treheat T0 2/m9 Pl mϕ H0 (43) H0 − H0 ∼ H0 (44) # $m #! $m # $ Pl # Pl $ mϕ 3/2 H 1/3 0 (44) m ∼ m # Pl $ # Pl $ 4

4 De Sitter Space in a Nutshell

We understand that, the growth of the de Sitter horizon to its asymptotic value at late times is a response to the matter that crosses the horizon. However, the horizon has no knowledge of the entropy of matter that was already beyond it to start with. [Bousso 0205177] Morals: The universe with a cosmological constant will asymptotically approach a de Sitter space (i.e. an observer’s causal domain agrees with an empty de Sitter space at late times) and any observer will eventually be surrounded by an event horizon.

For a universe to asymptote to de Sitter space, there is a bound on the amount of stored entropy that a local observer can ever measure.

Tuesday, June 26, 12 Moduli Dynamics in a Nutshell [see Acharya and Choi talks] Moduli are ubiquitous in M/String Theory compactifications. They parametrize the geometric properties (the moduli space) of the compact manifold.

The low energy phenomenological parameters in 4 dimensions are functions of them.

They need to get vev’s and gain mass through some dynamical stabilization and supersymmetry breaking mechanism.

Phenomenological and observational data greatly constrain the mechanism. In string phenomenological models, when the supersymmetry breaking mediation mechanism is via gravity, the masses of moduli and all the other scalars of the SSM are about the graviton mass. Not very far from few/tens of TeV. [Many authors, see recently Acharya,Kane,Kuflik 1006.3272] 7

Tuesday, June 26, 12 29/54 Hinf m29Pl/54 1/3 N Nefold Hinf mPl 1/3 1 e efolde ! ! N ζ− NHubble(43) (45) m mPlH H0 Hubble Pl ! !0 " " 1/2 1/3 1/2 1/6 29/H54inf Heq Hinf H0 H Nefoldm 1/2 1/3 1/2 1/6 Nefold infe PlHinf 1Heq1/3 29/54 Hinf H0 Nefold Hinf mPl 1/3 e ! e ∼NefoldHeqζ− NHubbleH0 (45)∼ H0 Heq mPl ∼H0eH ! ! " H ! ∼" NHHubble! H"(43) ! " ! ! "eq " m!Pl 0 H"0 ! 0 " ! eq " (46) ! " (44) H 1/2 H 1/3 H 1/2 H 1/6 Nefold inf eq 1/2 inf1/2 1/3 01/3 1/2 11//26 1/2 1/6 e Hinf Hinf Heq Heq Hinf HinfH0 mPl H0 ∼ H eNefoldNHefold ∼ H 1/3 H 2/27 eq Hinfe 0 2/3 H0 0 H0 eq 6 2/3 ! " ! ∼N"−∼Heq H! H0" H!∼ H"0 ∼ 10m−Heq N − H H m " !Hubble!" Heq!" "! m0 !" (46)"∼! ! Pl· "" Hubble! 0 " ! eq " Pl ! eq " ! Pl " (44) (47) (45) 1/2 1/3 1/2 1/2 1/6 H H H 1/3 m2/27 H Nefold inf eq 1inf/3 2Pl/27 0 e Hinf 2/3 H0 H0 6 2/3 HinfN − H0 H0 10 2/3N − 2/3 ∼ Heq H"0 25Hubble∼mPlmPl 1 HHeq0 12 − Heq Hubble 6 1 2 ! " Nm!Pl ""ln !Heq "lnmPl! "∼ζ Nln!Hubble−N· " 1064− ζ lnNNHubble− efoldm!Pl H!eq " !mPl " (47) Hubble∼ · Hubble 54 ! H0" −3! H"0 −3 (45)∼ −3 ! " ! " (46) (48) 1/3 2/27 Hinf H0 H0 25 2mPl 2/3 1 Heq6 21 2/3 1 " Nefold ! lnζ25NHubble− mPlln 10−1 ζ NHlnHubble−eqNHubble 64 1/3ln NHubble 1/3 mPl Heq mNPl 54 Hln0 τ−3 ∼ τ H0ln· −3 +ln(∼ζN −−(47)3 ) 64+ln(ζN − ) ! " ! efold" ! ! " ϕ ≪!universe" Hubble Hubble 54 H0 −3 (48)H0 (46) ∼ ! " ! " (49)

25 mPl 1 Heq τϕ τuniverse1/3 2 (47)1/3 ϕ¨ +(3H +≪Γϕ)˙ϕ + m ϕ =0 (48) Nefold ! ln ln +ln(ζNHubble− ) ϕ 64+ln(ζNHubble− ) 54 H0 −3 H0 τϕ ≪ τuniverse∼ (50) ! " ! " (49) 2 ϕ¨ +(3H + Γϕ)˙ϕ + m ϕ =0 (48) Treheat ϕ T0 12/3 (Hϕ¨reheat+(3LHreheat+ Γ) ϕ)˙ϕ +NmHubbleϕϕ =0 (49) (51) τϕ ≪Hτuniversereheat ∼ H(50)0

Treheat T0 1/3 Treheat(HreheatLreheat) NHubbleT0 1(49)/3 Hreheat 2 (1/H3 L∼ H0 ) N (52) ϕ¨ +(3H + Γϕ)˙ϕ + m ϕN=0reheat reheat(1) (51) Hubble(50) Hreheatϕ Hubble ∼ O ∼ H0

N 1/3 1/3(1) (50) Treheat T0Hubble1/N3 (1) (53) reheat∼HubbleO 27 (HreheatLreheat) NNHubble 10 ∼(52)O (51) Hreheat ∼ H0 Hubble "

Moduli Dynamicsreheat inreheat 27a Nutshell27 1/3 NHubbleN"Hubble10 " 10 (51) (54) NHubble (1) (53) Moduli commence oscillations∼ O when Hosc mϕ (52) The scalar condensates behave as pressureless∼ dust and quickly dominate the dynamics of the reheatUniverse 27 H Hmosc mϕ (52) (55) NHubble " 10 osc ∼ ϕ ∼ (54) Moduli are relatively light and gravitationallyHdecay coupledΓϕ to all the other(53) species. They are unstable but long-lived. ∼ 2 H m m (55)2 mϕ osc ϕ Hdecay Γϕ ϕmϕ (56) ∼ Hdecay Γϕ mϕ (53) ∼ ∼∼ ∼m mPl ! Pl " ! " 2 The decay of moduli releases a hugemϕ amount of entropy in the Universe Hdecay Γϕ mϕ Treheat " TBBN(56)1MeV (57) and dilutes its content. The entropyTreheat releaseTBBN must10MeV necessarily∼ be followed(54) ∼ ∼ mPl" ∼ by the observable (the last) nucleosynthesis.! "

T T 1MeV mϕ " 10(57) TeV (58) reheat " BBN ∼

After they decay, the standard hot big-bang cosmology1/ follows.2 mϕ " 10 TeV R =(3/Λ(58)) (59)

8 Tuesday, June 26, 12 1/2 33 R =(3/Λ) H 10−(59)eV (60) 0 ∼ 5 5 33 H0 10− eV m 50(60) TeV (61) ∼ ϕ ∼

m 50 TeV (61) ϕ ∼ 5

5 29/54 1/2 N Hinf mPl 1 1/3 R =(3/Λ) (59) e efold ζ− N (45) ! m H Hubble Pl ! 0 " The Entropy of the CMB in

Asymptotically De33 Sitter Space 1/2 1/3 1/2 1/6 H0 10− eV (60) Hinf Heq Hinf H0 ∼ eNefold ∼ H H ∼ H IfH the Universe is dominated by a positive cosmological constant at late ! eq " ! 0 " ! 0 " ! eq " time,(46) then the entropy of the CMB radiation is bounded. mϕ 50 TeV [Fischler(61) et.al 0307031] ∼ 1/2 1/3 1/2 1/2 1/6 Hinf Heq Hinf mPl H0 eNefold In the presence of a cosmological constant, any local observer eventually ∼ Heq H0 ∼ mPl H0 Heq 1 ! " ! " ! " !perceives" ! the space" as a box of size H − . (62) (47) 0 This box is the region that is causally accessible to the local observer living in asymptotic de Sitter space. 1/3 2/27 Hinf H0 H0 2 2/3 6 2 2/3 ζ N − 10− ζ N − m " H m Hubble ∼ This· boxHubble stores a limited amount of entropy. Any attempt to squeeze more Pl ! eq " ! Pl " entropy(48) meets with black hole formation and absorbed by the horizon.

Therefore, there is also a threshold on the amount of entropy of the CMB. 25 mPl 1 Heq 1/3 1/3 Nefold ! ln ln +ln(ζNHubble− ) 64+ln(ζNHubble− ) 54 H0 −3 H0 ∼ ! " ! " (49) 9

Tuesday, June 26, 12 τϕ ≪ τuniverse (50)

2 ϕ¨ +(3H + Γϕ)˙ϕ + mϕϕ =0 (51)

Treheat T0 1/3 (HreheatLreheat) NHubble (52) Hreheat ∼ H0

N 1/3 (1) (53) Hubble ∼ O

reheat 27 NHubble " 10 (54)

H m (55) osc ∼ ϕ m 2 H Γ m ϕ (56) decay ∼ ϕ ∼ ϕ m ! Pl "

T T 1MeV (57) reheat " BBN ∼

mϕ " 10 TeV (58)

5 The Entropy of the CMB in Asymptotically De Sitter Space 1 1/2 1/3 1/2 1/6 rSchwarzschild H− (30) Hinf Heq Hinf H0 Pack the CMB photons in a box of the size of the horizon,! 0 the causal patch eNefold 1 1 ∼ 1H/2eq 1H/30 ∼ 1H/20 1H/6eq r rSchwarzschildH− H− (30) (30) Hinf! "Heq ! " Hinf ! "H0 ! " Schwarzschild ! 0! 0 eNefold (44) ∼ Heq H0 ∼ H0 Heq 2 ! " ! " ! " ! " At the verge of black hole formations T0 ! mPlH0 (31) (44) 1/3 2/27 2 2 Hinf 2/3 H0 H0 T0 Tm0 PlHm0PlH0 (31) (31) ! ! " NHubble− (45) A local observer in the Universe with a positive cosmological constant will mPl 1H/3eq 2m/27Pl Hinf 2/3 H0 H0 3 3/2 ! " ! " never see more entropy in the CMB than " NHubble− (45) T0 mPl mPl Heq mPl S0 (32) 3 3 3/2 ! 3/2 2 ! " ! " T0 Tm∼Pl H0 m mPlH0 S 0 ! " Pl! " (32) 25 mPl 1 Heq 1 1 0 S0 ! ! (32) Nefold ! ln ln ln NHubble 64 ln NHubble ∼ H0 ∼ HH00 "H0 H0 54 H −3 H −3 ∼ −3 ! " !! "" ! ! " " 25 mPl ! 10 " Heq ! 10 " 1 3 Nefold ! ln ln ln NHubble 64 (46)ln NHubble T 54 H0 −3 H0 −3 ∼ −3 (T L )3 (T L )3 0 N (33) ! " ! " reheat reheat 0 0 3 3 Hubble (46) ∼ T0 ∼ H0 (T L )3 (3T L )3 3 TN0 ! " (33) reheat(TreheatreheatLreheat) 0 (0T0L0) HubbleNHubble (33) Treheat T0 1/3 ∼ ∼ ∼ H∼0 H (H L ) N (47) ! !" 0 " reheat reheat Hubble TreheatHreheat T0 ∼1/H30 3 (HreheatLreheat) NHubble (47) N10 Hubble (L0H0) (34) Hreheat ∼ H0 3∼ Tuesday, June 26, 12 NHubble (L0H0) 3 (34) NHubble∼ (L0H0) (34) 1/3 ∼ NHubble (1) (48) 1/3 ∼ O H t N (1) (48) e inf Hubble ∼ O Hinf t e LoscH t (35) L ζ e inf∼ Hinf (35) osc reheat 27 ∼LoscHinf (35) N 10 (49) ∼ Hinf Hubble " reheat 27 NHubble " 10 (49) 2/3 4/3 2/3 mϕ 4/3 mPl Lreheatmϕ Losc 2/3 mPl Losc 4/3 (36) Lreheat Losc ∼ mϕ ΓLϕosc ∼ mPl mϕ(36) L ∼ L Γϕ !∼ "L mϕ ! " (36) reheat ∼ !osc "Γ ∼ !osc "m ! ϕ " ! ϕ " 3/2 m 3/2 mϕ T mTreheat ϕ mPl 3/2 (37) (37) reheat Pl ∼ mϕ mPl T ∼ mmPl ! " (37) reheat ∼ ! Pl "m ! Pl "

3/2 31//22 1/2 mPl 1 mHPl0 H1/03 1/3 L 1 H− 3/2 1/2 N (38) Lreheat ! Hreheat0− ! 0 NHubble (38)Hubble m1 mPl mmϕ H0 mPl 1/3 Lreheat H− ϕ ! Pl" ! N" (38) ! !0 "m ! "m Hubble ! ϕ " ! Pl " 1/6 11//62 1/2 1 mPl 1 mHPl0 H1/03 1/3 L H− 1/6 N1/2 (39) osc ! 0Losc ! Hm0− H Hubble NHubble (39) m1 ϕ Pl mmϕPl 0 mPl 1/3 Losc H!− " ! " N (39) ! 0 m ! "m ! "Hubble ! ϕ " ! Pl "

1/6 1/12/6 1/2 Hinf mPl H0 1/3 Nefold N Hinf mPl 1H0 1/3 e ! e efold 1/6 ζ− 1N/2Hubble N(40) (40) H Hminf! mPl m H0 1/3 Hubble eNefold 0 ! ϕ "H0 ! mPlϕ" mPlN (40) ! H m ! "m ! "Hubble 0 ! ϕ " ! Pl " 3 3 2 Treheat T0 3 mPl 3 2 Treheat3 3 T0 m2 Pl(41) H0Treheat− H0 T0∼ H0 mPl (41) ! " H!0 " − !H0 " ∼ H0 (41) H! − "H ! ∼ " H ! " ! 0 " ! 0 " ! 0 " 3/2 1/3 mϕ H03/2 1/3 3m/2ϕ 1H/30 (42) mPl mϕ ∼ mPl H0 (42) ! " m!Pl "∼ mPl (42) m ! ∼" m ! " ! Pl " ! Pl "

29/54 Nefold Hinf m29/Pl54 1/3 e Hinf mPl 1/3 N (43) eNefold ! m H N Hubble (43) ! m HPl ! 0 " Hubble Pl ! 0 " 4 4 4 29/54 N Hinf mPl 1 1/3 Treheat " TBBN 1MeV (59) e efold ζ− N (45) ∼ ! m H Hubble Pl ! 0 "

1/2 1/3 1/2 1/6 mϕ " 10 TeV (60) Hinf Heq Hinf H0 eNefold 29/54∼ Heq H0 ∼ H0 Heq N Hinf mPl !1 1/3" ! " ! " ! "mϕ " 10 TeV (59) e efold ζ− N (45) (46) ! m H Hubble 1/2 Pl ! 0 " R =(3/Λ) (61) 1/2 1/3 1/2 1/2 1/6 N Hinf Heq Hinf mPl H0 1/2 1/2 1e/3efold 1/2 1/6 R =(3/Λ) (60) Nefold Hinf Heq ∼HinfH H0 H ∼ m H H e ! eq " ! 0 " ! Pl " ! 0 " ! eq " 33 ∼ H H ∼ H H (47) H0 10− eV (62) ! eq " ! 0 " ! 0 " ! eq " ∼ (46) 33 H0 10− eV (61) 1/3 2/27 ∼ Hinf H0 H0 2 2/3 6 2 2/3 1/2 1/3 1/2 1/2 ζ N 1−/6 10− ζ N − mϕ 50 TeV (63) Hinf Heq "Hinf mPl H0 Hubble Hubble ! eNefold mPl Heq mPl ∼ · ∼ H H ∼ m! " !H " H ! eq " ! 0 " ! Pl " ! 0 " ! eq " (48) (47) mϕ ! 50 TeV (62) 1 H0− (64) 25 mPl 1 Heq 1/3 1/3 1/3 2/27Nefold ln ln +ln(ζN − ) 64+ln(ζN − ) Hinf H0 H0 !2/3 2/3 Hubble 1 Hubble 2 54 H60 2−3 H0 ∼ H− (63) " ζ NHubble− !10− "ζ NHubble− ! " 0 mPl Heq mPl ∼ · (49) ! " ! " 2 (48) NHubble > 10 (65)

2 τϕ ≪ τuniverse N(50)Hubble > 10 (64) 25 mPl 1 Heq 1/3 1/3 Nefold ! ln ln +ln(ζNHubble− ) 64+ln(ζNHubble− ) 54 H0 −3 H0 ∼ ζ( 1) (66) ! " ! " ≥ 2 2 (49) mPl mPl 2 S ϕ¨ +(3H(30)+ Γϕ)˙ϕ + m ϕ =0 (51)ζ( 1) (65) dS ∼ Λ ∼ H ϕ ≥ ! " Moduli τDynamicsϕ ≪ τuniverse in (50) 1 1/2 1/3 1/2 1/6 rSchwarzschild ! H0− Stotal S(30)dS SbulkN+efoldShoriznTreheatHinf (31) Heq T0 1H/3inf H0 Asymptotically∼ ∼ Dee Sitter Space(HreheatLreheat) NHubble (52) ∼HreheatHeq H0 ∼ H∼0 H0 Heq 2 ! " ! " ! " ! " ϕ¨ +(3H + Γϕ)˙ϕ + mϕϕ =0 (51) (44) 29/54 In the moduli dominated cosmology, the reheating1 after1/2 the moduli decay Hinf mPl 1/3 r HR−=(3/Λ) (32) (59) eNefold ζ 1N 2 (45) Schwarzschild ! 0 ! triggers− THubble0 ! them PlstandardH0 hot big-bang(31) cosmology. Treheat T0 mPl H0 (HreheatLreheat) (53) ! " The CMB radiation has also resulted from the moduliH decay. ∼1/3H 2/27 Treheat T0 H1/inf3 reheat 2/3 H0 0 H0 (HreheatLreheat) NHubble N −(52) (45) So the entropy bound shouldHreheat tell us2 about the∼ moduliH0 dynamics." Hubble T0 ! mPlH0 m33Pl (33) Heq mPl 1/2 1/3 1/2 1/6 H0 10− eV ! "(60)! " 3 3/2 1 N Hinf Heq Hinf T H0 m ∼ e efold The observed0 homogeneityPl and isotropy are explained if Lreheat Hreheat− (54) S0 ! T (32) T # ∼ Heq H0 ∼ ∼H0 H Heq H reheat3 3/2 0 2 ! " ! " ! " 0 ! " 0 (HreheatLreheat) (53) ! " ! (46)" HT0 mPl ∼ mH25Pl mPl 1 Heq 1 1 S0 reheat! Nefold ! 0 ln (34) ln ln NHubble 64 ln NHubble ∼ H0 H0 mϕ" 50H540 TeV H0 −3 H(61)0 −3 ∼ −3 # $ # $ ∼# $ ! N 1/"3 !(1) " (55) 3 Hubble ∼ O (46) 1/2 1/3 The3 total1/2 entropy3 1/T 2released0 1in/6 the Universe1/3 N (1)3 (54)3/2 Nefold Hinf (HTreheateq LreheatH) inf (T0L0)mPl HN0 Hubble (33)Hubble e ∼ ∼ H0 3 3 ∼TO0 mPl ∼ Heq H0 ∼ mPl H0 (T!reheat"LHreheateq ) (T0L0) NHubble ! NHubble ∼ ∼ H 1Treheat H T0 ! " ! " ! " ! " ! " # 0 $ H0− # N reheat0 $ 1027 (62) 1/3 (56) (47) Adiabatic expansion thereafter (H(35)Hubblereheat"Lreheat) NHubble (47) reheat 27 Hreheat ∼ H0 3 NHubble " 10 (55) NHubble (L0H0) The number (34)of Hubble spheres in the present Universe 1/3 2/27 ∼ 3 Hinf H0 H0 2/3 2/3 NHubble (L0H0) 2 (36) 2 6 2Observational lower bound∼ NHubble > 10 [SilkH et.al1/3 m1101.5476](63) (57) " ζ NHubble− 10− ζ NHubble− Nosc ∼ ϕ (1) (48) mPl Heq mPl ∼ · 11 Hubble ! " ! " Hosc mϕ (56) ∼ O Tuesday, June 26, 12 Hinf(48)t ∼ e eHinf t Losc L(35)osc ζ (37) 2 ∼ Hinf ∼ Hinf mϕ 2 Hdecay ΓNϕ reheatmϕ 1027 (58) (49) 25 mPl 1 Heq 1/3 1/3 mϕ ∼ Hubble∼ "mPl Nefold ! ln ln +ln(ζNHubble− ) 64+ln(ζNHubble− Hdecay) Γϕ mϕ (57) ! " 54 H0 −3 H0 ∼ ∼ 2/3∼ mPl 4/3 ! " ! " 2/3 4/3 mϕ ! mPl" mϕ (49)LreheatmPl Losc Losc (38) 5 Lreheat Losc Losc ∼ Γ(36)ϕ ∼ mϕ ∼ Γϕ ∼ mϕ # $ # $ ! " ! " Treheat " TBBN 1MeV (58) ∼ 3/2 τϕ ≪ τuniverse (50) m T m ϕ (39) 3/2 reheat ∼ Pl m 5 mϕ # Pl $ Treheat mPl (37) ∼ mPl 2 ! " 3/2 1/2 ϕ¨ +(3H + Γϕ)˙ϕ + mϕϕ =0 (51) 1 mPl H0 1/3 Lreheat ! H0− NHubble (40) mϕ mPl 3/2 1/2 # $ # $ 1 mPl H0 1/3 Lreheat ! H0− N (38) Hubble 1/6 1/2 Treheat T0 m1ϕ/3 mPl (H L ) !N " ! (52)" 1 mPl H0 1/3 reheat reheat Hubble Losc ! H0− N (41) Hreheat ∼ H0 m m Hubble # ϕ $ # Pl $ 1/6 1/2 1 mPl H0 1/3 L H− N (39)1/6 1/2 1/3osc ! 0 N HubbleHinf mPl H0 1 1/3 N (1) mϕ mPl(53)e efold ζ− N (42) Hubble ∼ O ! " ! " ! H m m Hubble 0 # ϕ $ # Pl $

1/6 1/2 3 3 2 Hinf mPl H0 1/T3reheat T0 mPl reheatNefold 27 (43) NHubblee "!10 (54) NHubble (40) ! H m m H0 − H0 H0 0 ! ϕ " ! Pl " # $ # $ # $

mϕ H 2/9 3 3 2 0 (44) H mTreheat T0 (55) mPl ! osc ∼ ϕ mPl(41) mPl H − H ∼ H # $ ! 0 " ! 0 " ! 0 "

2 4 mϕ 3/2 1/3 Hdecay Γϕ mϕ mϕ H(56)0 ∼ ∼ mPl (42) ! m " ∼ m ! Pl " ! Pl "

Treheat " TBBN 1MeV 29(57)/54 ∼ Hinf mPl 1/3 eNefold N (43) ! m H Hubble Pl ! 0 " mϕ " 10 TeV (58) 4

5 1 1/2 1/3 1/2 1/6 rSchwarzschild H− (30) Hinf Heq Hinf H0 ! 0 eNefold ∼ H H ∼ H H ! eq " ! 0 " ! 0 " ! eq " 1 1/2 1/3 1/2 1/6(44) rSchwarzschild ! H0− (30) N Hinf Heq Hinf H0 1 e efold 1/2 1/3 1/2 1/6 2 − H H H H rSchwarzschild ! H0 (30) Nefold∼ H inf H eq ∼ H inf H 0 T0 ! mPlH0 (31) e ! eq " ! 0 " ! 0 " ! eq " ∼ Heq H0 1∼/3 H0 2/27 H(44)eq H!inf " 2/!3 H"0 ! H0 " ! " 2 " NHubble− (45)(44) T0 ! mPlH0 (31) mPl Heq mPl 2 ! 1"/3 ! 2/"27 T 3 m H 3/2 (31) T 0 ! Plm 0 Hinf 2/3 H0 H0 0 Pl " N − 1/3 2/27 (45) S0 ! (32) mH Hubble H H m H ∼ H H Plinf 2/3! eq "0 ! Pl "0 0 3 0 3/2 25 "mN − 1 H 1 (45) 1 ! "T ! m" m PlHubble H eq m 20 Pl Nefold lnPl ln eq lnPlNHubble 64 ln NHubble S0 m m 2 (32) ! ! " ! " HPl 3 ! PlH 3/2 54 H0 −3 H0 −3 ∼ −3 SdS∼ T00 mPl0 (30) 25 !m " 1 !H "1 1 S ∼! Λ "∼ H! "3 (32) Pl eq (46) 0 ! T Nefold ! ln ln ln NHubble 64 ln NHubble 3 ∼ H0 3 ! H"00 54 H0 −3 H0 −3 ∼ −3 (TreheatLreheat) (!T0L0") ! " NHubble (33) 25 ! m"Pl 1! H"eq 1 1 ∼ ∼ H0 3 Nefold ! ln ln ln NHubble(46) 64 ln NHubble ! T"0 54 H0 −3 H0 −3 ∼ −3 (T LStotal )3 SdS(T LSbulk)3 + Shorizn N (31)1 (33) Treheat! " 1!/2 "T0 1/31/3 1/2 1/6 reheat reheat∼ ∼0 0 rSchwarzschildHubble! H0− (30) Nefold Hinf Heq Hinf H0 ∼ ∼ H0 3 e (HreheatLreheat) NHubble (47)(46) ! T0 " Hreheat ∼ H0 3 3 Treheat ∼ Heq TH00 1/3 ∼ H0 Heq (TreheatLreheat) (T0L0) 3 NHubble (33) ! " ! " ! " ! " N ∼ (L H∼ ) H (34) (HreheatLreheat) NHubble (47) (44) Hubble 0 0 1 0 Hreheat ∼ H0 rSchwarzschild∼ ! H0−! " (32) 32 Treheat T0 1/3 NHubble (L0H0T)0 ! mPlH0 (34) (31) (Hreheat1/3 Lreheat) NHubble (47) ∼ Hreheat NHubble (1)∼ H0 1/3 (48)2/27 3 1H/3inf ∼ O2/3 H0 H0 N − (45) NHubble2 Hinf(Lt0H0) (34) NHubble" Hubble(1) (48) T0 ! me∼PlH0 (33) mPl Heq mPl L (35) ∼ O ! " ! " osc Hinf t 3 3/2 1/3 ∼ Hinfe T0 mPl L (35) NHubble (1) (48) osc S0 ! (32) N reheat ∼10O27 (49) 3 ∼ H3/∼2inf H0 2 H0 Hubble " Hinf t! " ! " 25reheat mPl 27 1 Heq 1 1 ModuliT0 DynamicsmPl e inmPl N NHubbleln " 10 ln (49)ln N 64 ln N S0 !L (34) (35) efold ! Hubble Hubble ∼ H osc2/H3 " H 4/3 54 H0 −3 H0 −3 ∼ −3 # 0 $m # ∼0 $Hinf #m 0 $ ! " ! " AsymptoticallyL L ϕ De2/ 3SitterL PlSpace4/3 (36)3 reheat 27 (46) reheat osc mϕ osc3 mPl3 T0 NHubble " 10 (49) L ∼ L (ΓTϕ L ∼ ) L (mT ϕL ) (36)N (33) Adiabatic expansion reheat osc reheat reheat 3osc 0 0 3/2 Hubble ∼ ! "Γϕ ∼ ∼! m"ϕ ∼ H0 3 ! 3 " T0 ! " !mPl " (TreheatLreheat) (T0L0) 2/3 NHubble ! 4/3 NHubble Treheat T0 1/3 ∼ mϕ ∼ H0 mPl H0 (HreheatLreheat) N (47) Lreheat Losc # $Losc # $ (36) Hubble ∼ Γ ∼ 3/2 m (35) Hreheat ∼ H0 ! ϕ " mϕ !3/2 ϕ " 3 The reheating temperature Treheat mPl mNϕHubble (L0H0) (37) (34) Treheat∼ mPlmPl ∼ (37) ∼ ! m"3Pl 1/3 NHubble (L0!H0) " (36) ∼ 3/2 NHubble (1) (48) mϕ ∼ O The physical size of the universeT reheatat the timemPl of reheating eHinf t (37) ∼3/2 3/2 m 1/21/2 m H PlLosc (35) 1 1 PlmPl H!inf t0H0 " 1/13/3 − e ∼ Hinf LreheatLreheat! H!0 H0− NHubbleNHubble (38)(38) reheat 27 mLϕoscmϕ ζ mPlmPl (37) N 10 (49) ! ! "∼"!Hinf! " " Hubble " 3/2 1/2 Energy conservation in expanding1 universemPl H0 2/3 1/3 4/3 L H− m N m(38) reheat ! 0 2/13/6 1ϕ/2 4/Hubble3 Pl mLmmreheatϕ1ϕ/6 LoscHmPl1/m2 Pl Losc (36) L 1L m1 !Pl Pl "∼H!L0 0 Γ"ϕ 1/31/3∼ (38)mϕ reheatLosc Hosc0− osc N (39) Losc ! H!0∼− Γϕ ∼ ! mNϕ" Hubble !(39) " m# mϕ$ m mPl# Hubble$ The physical size of the universe! at!ϕ the" beginning"! !Pl " "of oscillations 1/6 1/2 mPl H0 3/2 1/3 3/2 1 mϕ Losc ! H0− 1/6mTϕ 1/m2NHubble (39) (37) HTreheatmm mPl mreheatH Pl (39) Nefold inf ϕ Pl1/6 Pl01/2 1/3m e H !m ∼ " !mHPl " ∼ N Pl (40) Nefold !inf Pl 12 # 0 $ 1/Hubble3! " e ! H0 mϕ mPl NHubble (40) Tuesday, June 26, 12 H m! " m! " 0 ! ϕ " ! Pl " 3/21/6 1/2 1/23/2 1/2 mPl H0 1/3 N Hinf1 mPl 1H0mPl 1/3H0 1/3 eLreheatefold ! H0− L 3 H− 3 NHubbleN 2 (40) (40)N (38) ! T mreheatϕ ! Tm0 Pl m Hubble Hubble Hreheat0 # 3 mϕ$ # 03 m$Plmϕ Pl2 mPl (41) TreheatH! −"T0 H! !∼"m"PlH ! " ! 0 " ! 0 " ! 0 " (41) H0 − 1/6H0 ∼1/2 H0 mPl H0 1/3 ! 1 " 3 ! " 3 ! 1/6" 2 1/2 Losc ! H0− mPlNHubble H0 (41) 1/3 Treheatmϕ 3/2 Tm0Pl1 1m/3Pl # Losc$ !#H0− $ N(41)Hubble (39) mϕ Hm0ϕ mPl H0 3/−2 H0 ! ∼1/"3 H0! " (42) ! mϕm"Pl ! ∼H"mPl ! " ! 1"/6 !01/2 " (42) Nefold Hinf mPl H0 1 1/3 e ! mPl ∼ mPl ζ− N1Hubble/6 (42)1/2 H0! mϕ" 3/2 m!HPlinf "mPl1/3 H0 1/3 # N$efold # $ me ϕ ! H290/54 NHubble (40) Nefold Hinf mHPl0 mϕ 1/3 mPl (42) e m!Pl ∼ m!Pl N"Hubble! " (43) ! 3 m"Pl H0!329/54 " 2 TreheatHinf m!TPl0 " mPl1/3 Nefold ! (43) e H! − H NHHubble (43) # 0 m$Pl #H00 $ # 3 0 $ 3 2 ! T"reheat T0 mPl H m 29/54 4 (41) Nefold inf PlH0 − 1/H30 ∼ H0 e ! ! 2/9" N!Hubble" ! (43)" mϕm HH0 Pl ! 0 " (44) 4 m ! m Pl # Pl $ mϕ 3/2 H 1/3 0 4 (42) m ∼ m ! Pl " ! Pl " 4

29/54 Hinf mPl 1/3 eNefold N (43) ! m H Hubble Pl ! 0 "

4 The Entropy of the Horizon in Asymptotically De Sitter Space

Objects passing through the horizon to the outside do not experience anything dramatic from the point of view of the static coordinates.

In fact, nothing is intrinsic with the horizon, not even its existence.

It is defined relative to the an observer it can communicate with in the future.

The growth of the de Sitter horizon to its asymptotic value at late times is a response to the matter that crosses the horizon. The horizon encodes information about matter that has passed to the outside of the cosmological horizon of the local observer.

A detector localized at the position of the horizon can accumulate and send data to a local observer within the horizon distance. However, it has no knowledge of the entropy of matter that was already beyond it to start with. 13

Tuesday, June 26, 12 R =(3/Λ)1/2 (61)

33 H 10− eV (62) 0 ∼

mϕ ! 50 TeV (63)

1 H0− 29/54 (64) Hinf mPl 1/3 eNefold N (43) ! 2 2 Hubble mmPlPl Hm0Pl SdS ! " (30) ∼ Λ 2∼ H NHubble > 10 ! " (65) 1/2 1/3 1/2 1/6 2 2 Nefold Hinf mPlHeq mPl Hinf H0 e SdS (30) Stotal ∼SdSΛ S∼bulkH+ Shorizn (31) ∼ Heq ∼ ∼H0 ∼ H0 Heq ! ζ"( 1)! !" " ! (66)" ! " ≥ (44) Stotal SdS Sbulk + Shorizn1 (31) rSchwarzschild∼ ∼ ! H0− (32) 1/3 2/27 Hinf0.0133 <2/Ω3 < H0.00840 (95%H CL)0 (67) 6 2/3 − k − − " NHubble 1 10− NHubble mPl rSchwarzschildHeq ! Hm0−Pl ∼ (32) · ! 2 " ! " T0 ! mPlH0 (33) (45) Nefold = Hinf t (68) 2 T0 ! mPlH0 (33) 25 m3Pl 1 3/2Heq 1 2 1 Nefold ! lnT0 mPlln mPl ln NHubble 64 ln NHubble S250/5454 H! −3 H −1/23 (34)1/9∼ −3 ∼ H0 0 H0 "0 H0 27 H0 # !$ 3Losc"# $3/!2 H0#" $2 mPl 28 10− T0 mPl mPl (46)10− S0 ! 1/3 ! (34) ∼ mPl 1 ! mPl TBBN ∼ ∼ HH00− N H0 " H0 ! " # $ Hubble# $ ! # 3" $ ! " 3 3 T0 (69) (TreheatLreheat) (T0L0) NHubble (35) ∼ τϕ ≪∼τuniverseH0 3 (47) 3 3 #T0 $ (TreheatLreheat) (T0L0) NHubble (35) 2/3 ∼ ∼ H02/9 TBBN mϕ H0# $ ! ! 3 (70) m NmHubble (mL0H0) (36) Pl Pl ∼ Pl 2 ! " ϕ¨ +(3N H +!(ΓLϕH)˙ϕ")3+ mϕϕ =0(36) (48) Hubble ∼ 0 0

10 TeV ! mϕ ! 50 TeVeHinf t (71) L ζ Hinf t (37) Treheat osc e T0 1/3 (HLosc ∼ ζ LHinf ) N (37) (49) reheat∼ Hreheatinf Hubble Hreheat ∼ H0 1/3 2/9 2/27 H0 TBBM Hinf 2/3 2 H0 (N2/3 ζ− ) 4/3 ! mmϕ 2/Hubble3 mm!Pl 4/3 Heq LreheatmPl Losc mPlϕ 1/3 Losc Pl mPl (38) ! " ! Lreheat"∼ Losc Γ ∼ Losc m ! "(38) ∼ ΓϕϕN ∼ m(1)ϕϕ (72) (50) ## $$Hubble ∼ #O# $$

3/2 11 1/3 16 3/2 1/3 10 GeV (N − ζ) H 10 mmGeVϕϕ (N − ζ) Hubble T!Treheatreheatinf !mmreheatPlPl 27Hubble (39)(39) ∼∼NHubblemmPl" 10 (51) ## Pl$$ (73)

3/2 1/2 1 mPl 3/2 H0 1/2 1/3 1 mPl 1 Heq m1Pl mPlH0 1/31/3 25 mPl 1 Heq 1/3 Lreheat ! H0−1 NHubble− (40) − ln Lreheatln ! H0− mlnϕ HoscmPl +ln(mϕ ζNNHubbleHubble) !(40)Nefold (52)! ln ln +ln(ζNHubble) 2 H0 −3 H0 −#m9 ϕ $ TBBN# m∼Pl$ 54 H0 −6 H0 ! " ! "# $! # " $ (74) ! " ! " The Entropy1 /6of the1/ 2Horizon in 1 mPl H0 1/3 L H− 1/6 1/2N (41) osc ! 0 mPl H0 Hubble1/3 2 1/3 1 mϕ mPl 1/3mϕ AsymptoticallyL−osc ! H0− # $ # De$ Sitter− NHubble Space(41) 41 + ln(ζNHubble) ! NHefolddecaymϕ! 63Γ +ϕm ln(PlζmNϕHubble) (75) (53) # $∼ # ∼ $ mPl The amount of entropy that has passed1/6 through1/2 ! the horizon" (hereafter the Nefold Hinf mPl H0 1 1/3 horizon entropy)e as! can be seen by1/6 a local observer1/2ζ− NHubble in a universe(42) dominated HinfH0 mmPlϕ2 mHPl0 1/3 by a cosmologicaleNefold constant# H $ # $ ζ 1N (76) (42) ! Treheat0 " TBBN 10MeV− Hubble (54) H0 mϕ mPl ∼ # 3$ # 3 $ 2 Treheat T0 mPl ! (43) 29/54 H 3 H 3 H 2 1/2 Hinf mPl 1/3 0 − 0 0 R =(3/Λ) (59) Nefold 1 #Treheat $ # T0 $ # mPl$ 6 e ! ζ− NHubble (45) mϕ " 50! TeV (43) (55) mPl H0 H0 − H0 H0 ! " # $ 3/#2 $ #1/3 $ mϕ H0 If it is caused by moduli dynamics,! then (44) mPl mPl 33 # $ # 2/$9 H 10− eV (60) 1/2 1/3 1/2 m1ϕ/6 H0 1/2 0 Hinf Heq Hinf H0 R =(3/Λ) (44)∼ (56) eNefold m ! m ∼ H H ∼ H H Pl # Pl $ ! eq " ! 0 " ! 0 " ! eq " (46) 4 33 m 50 TeV (61) Immediate observation H0 10− eV ϕ ! (57) ∼ 4 1/2 1/3 1/2 1/2 1/6 N Hinf Heq Hinf mPl H0 e efold 14 ∼ Heq H0 ∼ mPl H0 Heq 1 ! " ! Tuesday," June 26, 12! " ! " !m "50 TeV H− (58) (62) (47) ϕ ∼ 0

1/3 2/27 Hinf H0 H0 2 2/3 6 2 2/3 2 ζ N − 10− ζ N − NHubble > 10 5 (63) m " H m Hubble ∼ · Hubble Pl ! eq " ! Pl " (48)

25 mPl 1 Heq 1/3 1/3 Nefold ! ln ln +ln(ζNHubble− ) 64+ln(ζNHubble− ) 54 H0 −3 H0 ∼ ! " ! " (49)

τϕ ≪ τuniverse (50)

2 ϕ¨ +(3H + Γϕ)˙ϕ + mϕϕ =0 (51)

Treheat T0 1/3 (HreheatLreheat) NHubble (52) Hreheat ∼ H0

N 1/3 (1) (53) Hubble ∼ O

reheat 27 NHubble " 10 (54)

H m (55) osc ∼ ϕ

m 2 H Γ m ϕ (56) decay ∼ ϕ ∼ ϕ m ! Pl "

T T 1MeV (57) reheat " BBN ∼

mϕ " 10 TeV (58)

5 R =(3/Λ)1/2 (61) R =(3/Λ)1/2 (61)

33 H0 1033− eV (62) H 10∼− eV (62) 0 ∼

mϕ ! 50 TeV (63) mϕ ! 50 TeV (63)

1 H1 0− (64) H0− (64)

N > 102 (65) Hubble 2 NHubble > 10 (65)

R =(3/Λ)1/2 (61) ζ( 1) (66) ζ( 1)≥ (66) ≥

33 H 10− eV (62) 0 ∼ 0.0133 < Ωk < 0.0084 (95% CL) (67) 0−.0133 < Ω < 0.0084 (95% CL) (67) − k

mϕ ! 50 TeV (63)Nefold = Hinf t (68) Nefold = Hinf t (68)

1 25/54 1/2 1/9 H0− 27 H0 (64) Losc H0 mPl 28 Moduli Dynamics10− 25 in/54 1/2 1/9 10− H0 ! Losc1 1/3 !H0 mPl 27 ∼ mPl H− N mPl TBBN ∼ 28 10− ! " ! 0 1/3Hubble! ! " ! " 10− ∼ mPl 1 mPl TBBN ∼ Asymptotically De! Sitter" SpaceH0− NHubble ! " ! (69)" 2 (69) NHubble > 10 (65) 2/3 2/9 TBBN mϕ H0 2/3 ! ! 2/9 (70) TBBNm mϕm H0m ! Pl "! !Pl ! Pl " (70) ζ( 1) mPl (66) mPl mPl ≥ ! " ! " 10 TeV ! mϕ ! 50 TeV (71) 10 TeV ! mϕ ! 50 TeV (71) 0.0133 < Ω < 0.0084 (95% CL) (67) − k

Nefold = Hinf t (68)

25/54 1/2 1/9 27 H0 Losc H0 mPl 28 10− ! 1/3 ! 10− ∼ mPl 1 mPl TBBN ∼ ! " H0− NHubble ! " ! " (69) 15

Tuesday, June 26, 12

6 6

6 Eternal Inflation and Asymptotically De Sitter Space False vacuum eternal inflation is generic in the sting theory landscape of metastable de Sitter vacua. The tiny cosmological constant is also explainable in this framework. [See Recently, Bousso 1203.0307] The inflation takes place in a false minimum, it ends locally through a first-order phase transition (decay by bubble nucleation).

Inside each bubble there is an open homogenous and isotropic RW universe with negative spatial curvature.

This curvature has to be dilute away by a subsequent inflationary (non-eternal, not necessarily slow-roll) era. 16

Tuesday, June 26, 12 29/54 N Hinf mPl 1 1/3 Treheat " TBBN 1MeV (59) e efold ζ− N (45) ∼ ! m H Hubble Pl ! 0 "

1/2 1/3 1/2 1/6 mϕ " 10 TeV (60) H29inf/54 Heq H29inf/54 H0 Nefold H m T T 1MeV (59) N He inf mPl Nefold1 1/3 inf Pl 1 1/3 Treheat TBBN reheat1MeV" BBN (59) efold e ζ− N (45) " ∼ e ! ∼ Heq ζ− NH!0 ∼ (45)H0 HubbleHeq ∼ m H! " ! Hubblem"Pl H!0 " ! " Pl ! 0 " ! " (46) R =(3/Λ)1/2 (61) m 10 TeV (60) H 1/2 H 1/3 H 1/2 H 1/6 ϕ " Nefold 1/2 inf 1/3 eq 1/inf2 1/02 1/6 m 10 TeV (60) 1/2 e1H/3 H 1/2 H1/6 m H ϕ " N Hinf NefoldHeq inf ∼ HHinf eq HH0 inf∼ H Pl H 0 e efold e ! eq " ! 0 " ! 0 " ! eq " ∼ Heq H0∼ Heq∼ H0 H0 H∼eq mPl H0 (46)Heq 33 ! " ! "! "! ! " " ! !" " ! " ! " H0 10− 1/eV2 (62) (46) (47) R∼=(3/Λ) (61) 1/2 1/2 1/3 1/2 2 1/2 21R/6 =(3/Λ) (61) Hinf Heq Hinf mmPlPl mHPl0 Nefold S (30) e 1/3 2/27 dS ∼ Heq H0 ∼ mPl ∼ HΛ0 ∼ Heq 1/2Hinf 1H/30 ! H0"1/2! 2 " 21//32 ! "6 1/!26 2"/3 ! " H 10 33 eV (62) N Hinf Heq Hinf ζmNPl− H100− ζ N − (47)! " mϕ0 ! 50− TeV (63) e efold m " H m Hubble ∼ · Hubble ∼ Pl ! eq " ! Pl " ∼ Heq H0 ∼ mPl H0 Heq (48) 33 ! " ! " ! " ! " !S " S S + S H 10− eV(31) (62) 1/3 2/27(47) total dS bulk horizn0 Hinf H0 H0 2 2/3 ∼ 6 ∼2 2/3 ∼ " ζ NHubble− 10− ζ NHubble− mϕ ! 501 TeV (63) m H m ∼ · H− (64) Pl ! eq " ! Pl " 0 25 mPl 1 Heq 1/3 (48) 1/3 1/3 Nefold !2/27ln ln +ln(ζNHubble− ) 64+ln(ζNHubble− 1 ) Hinf H0 H0 54 2 H02/3 −3 H6 0 2 2/3 rSchwarzschild∼ ! H0− (32) " ζ !NHubble− " 10!− ζ"NHubble− mϕ ! 50 TeV 1 (63) m H m ∼ · (49) H0− (64) 29/54 Pl eq Pl 25 1/m2 Pl 1 Heq 1/3 1/3 2 H m ! " ! " N R =(3/Λln) ln +ln((59)ζN − ) 64+ln(ζN − ) N > 10 (65) Nefold inf Pl 1 1/3 efold ! (48) Hubble Hubble Hubble e ζ N (45) 54 H0 −3 H0 ∼ ! − Hubble ! " ! " 2 mPl H0 T0 ! m(49)PlH0 (33) ! " τϕ τuniverse (50) ≪ 1 N > 102 (65) Eternal Inflation Hand0− Hubble (64) 25 mPl 1 Heq 33 1/3 1/3 Nefold ln ln H 10+ln(ζeVN − ) 64+ln((60)ζN − ) ζ( 1) (66) 1/2 1/3 1/2 1/6 ! 0 − Hubbleτϕ ≪ τuniverse Hubble3 (50) 3/2 2 ≥ N Hinf Heq Hinf H0 54 H0 −3 H0∼ ∼ T0 mPl mPl e efold ! " ! " Asymptotically2 S De Sitter Space(34) ϕ¨ +(3H + Γ )˙ϕ + m (49)ϕ 0=0 !(51)29/54 ∼ Heq H0 ∼ H0 Heq ϕ ϕ ∼H H0 m H0 " H0 ζ( 1) (66) ! " ! " ! " ! " Nefold #inf $ Pl # $1 1/3 # $ 2 ≥ (46) e ! ζ− NHubbleNHubble(43)> 10 (65) ϕ¨ +(3H + Γ )˙ϕ + m2mϕPl=0 H0 (51) m Number50 TeV of e-foldsϕ to ϕreach(61)! to observable" limit 0.0133 < Ωk < 0.0084 (95% CL) (67) ϕ ! 3 − 3/2 τϕ ≪ τuniverse (50) 3 3 T0 mPl 1/2 1/3 1/2 1/2 1/6 Treheat (TreheatLT0reheat11//2)3 (T0L1/03) 1/2NHubble !10/.20133 < Ω1k/<6N0Hubble.0084 (95% CL) (67) (HreheatLreheat) H N H (52) H m − H N Hinf Heq Hinf mPl H0 Nefold inf Hubble∼ eq ∼ Hinf0 Pl H00 efold Hreheat e ∼ H0 # $ # $ e Treheat T0 1/3 ζ( 1) (35) (66) ∼ Heq H0 ∼ mPl H0 Heq 1 (Hreheat∼LreheatHeq) NHHubble0 ∼ (52)mPl H0 Heq ! " ! " ! " ! " ! " HH−reheat ! "∼ H0!(62) " ! " ! "≥ ! " (47) 0 (44) 2 N = H t (68) ϕ¨ +(3H + Γϕ)˙ϕ + mϕϕ =0 (51) efold inf Treheat T0 3 (HreheatLreheat) NHubble(53) (L0H0) (36) Treheat 1/3 T0 2/27 ∼ 1/3 2/27 Hreheat Hinf (H HL0 ∼ H)0 H0 2 (53)2/3 6 2 2/3 reheat reheat − 0.0133 < Ωk < −0.0084 (95% CL) (67) Hinf H0 H0 2/3 2/3 The physicalHreheat2 " size of the∼ universeH0 ζ atN Hubblethe beginning10− ζ N ofHubble oscillations 2 6 2 NHubble > 10m H (63)m − ∼ · " ζ NHubble− 10− ζ NHubble− Pl ! eq " ! Pl " mPl Heq mPl ∼ · Treheat T0 1/3 eNefold (45) ! " ! " 1 (48) (HreheatLreheat) NHubble (52) Losc ζ (37) Hreheat ∼ HL0reheat Hreheat− 1 (54) #Lreheat H− ∼ (54)Hinf # reheat 25 mPl 1 Heq 1 Nefold = H1inf t (68) ζ ( 1) dependsNefold onln the detailed(64) ln history ofln theNHubble Universe64 rightln NHubble after inflation ≥ ! 54 H −3 H −3 ∼ −3 25 mPl 1 Heq 1/3 1/3 ! 0 " ! 20/3" 4/3 Nefold ! ln ln +ln(ζNHubble− ) 64+ln(ζNHubble−Treheat) until moduli1T/03 1 /oscillations.3 mϕ mPl (46) 54 H0 −3 H0 ∼ (HreheatLreheat) NHubbleNHubble(1)Lreheat(53)(1) Losc (55)(55) Losc (38) ! " ! " ∼ O ∼ O ∼ Γϕ ∼ mϕ (49) Hreheat ∼ H0 # $25/54 # $ 1/2 1/9 H0 Losc H0 mPl τϕ ≪ τuniverse (47) Previously we found ! 1 1/3 ! reheat 27 mPl H−3/2N mPl TBBN reheat NHubble27" 10 ! " (56) mϕ 0 Hubble ! " ! " 1 NHubble " 10 (56) τϕ ≪ τuniverse (50) Lreheat H− (54) Treheat mPl 17 (39) (69) reheat ∼ mPl # 2# $ Tuesday, June 26, 12 ϕ¨ +(3H + Γϕ)˙ϕ + mϕϕ =0 (48)

Hosc mϕ 3/(57)2 1/2 2 Hosc mϕ ∼ 1 mPl(57) H0 1/3 1/3 L H− N (40) ϕ¨ +(3H + Γϕ)˙ϕ + mϕϕ =0 (51) N (1) ∼ Treheatreheat(55)! 0 T0 1/3 Hubble Hubble (H Lmϕ ) mNPl (49) ∼ O H reheat# reheat$ #H Hubble$ reheat 2 ∼ 0 mϕ Hdecay Γϕ mϕ 2 1/6(58) 1/2 ∼ ∼mϕ mPl m H Treheat T0 1/3 H Γ m ! 1"1/3 Pl (58) 0 1/3 reheat 27decay ϕ ϕ Losc ! H0−NHubble (1) NHubble(50) (41) (HreheatLreheat) NHubble (52) NHubble " 10 ∼ ∼ m(56)Pl m ∼ O m Hreheat ∼ H0 ! " ϕ Pl # $ # 5 $

reheat 1/6 275 1/2 H NHubblem " 10 H (51) 1/3 Nefold inf Pl 0 1 1/3 N (1) (53) e ! ζ− NHubble (42) Hubble Hosc mϕ (57) H m m ∼ O ∼ 0 # ϕ $ # Pl $

Hosc mϕ (52) 3∼ 3 2 Treheat T0 mPl N reheat 1027 (54) 2 ! (43) Hubble " mϕ H0 − H0 H0 Hdecay Γϕ mϕ (58)# $ # $ 2 # $ ∼ ∼ m mϕ ! Pl " Hdecay Γϕ mϕ (53) ∼ ∼ mPl 2/9 mϕ !H0 " Hosc mϕ (55) ! (44) 5 mPl mPl ∼ T T #10MeV$ (54) reheat " BBN ∼

2 mϕ 4 H Γ m (56) mϕ " 50 TeV (55) decay ∼ ϕ ∼ ϕ m ! Pl "

R =(3/Λ)1/2 (56) T T 1MeV (57) reheat " BBN ∼ 33 H 10− eV (57) 0 ∼ mϕ " 10 TeV (58) m 50 TeV (58) ϕ ∼ 5

5 1/2 R =(3R =(3/Λ)/1Λ/2) (61)(61)

R =(3/Λ)1/2 (61) 33 33 H H 10−10−eV eV (62)(62) 0 ∼0 ∼

33 H 10− eV (62) 0 ∼ m 50 TeV (63) ϕm!ϕ ! 50 TeV (63)

mϕ ! 50 TeV (63) 1 H0− 1 (64) H0− (64)

1 H0− (64) 2 1/2 NHubble > 10 2 (65) R =(3/Λ) NHubble(61) > 10 (65)

2 NHubble > 10 (65) 33 H 10− eV (62) 0 ∼ ζ( 1) (66) ≥ζ( 1) (66) ≥ ζ( 1) (66) mϕ ! 50 TeV ≥ (63) 0.0133 < Ω < 0.0084 (95% CL) (67) − k 0.0133 < Ωk < 0.0084 (95% CL) (67) 1 − H0− 0.0133 < Ω < 0.(64)0084 (95% CL) (67) − k Nefold = Hinf t (68) N > 102 (65) Hubble Nefold = Hinf t (68) Nefold = Hinf t (68) 25/54 1/2 1/9 27 ζ( H1)0 L(66)osc H0 mPl 28 10− ≥ 10− 25/!54 1 1/3 ! 1/2 1/9 ∼ mPlH H− N L mPlH TBBNm ∼ 10 27! "250 /54 0 Hubbleosc ! 1"/20 ! 1/"9Pl 10 28 27− H0 ! Losc 1/3 H!0 m(69)Pl 28 − 10− ∼ mPl 1 mPl TBBN 10− ∼ 0.0133 < Ωk < 0.0084 (95% CL)! 1H (67)01−/3NHubble! ∼ m!Pl " H− N mPl! "TBBN! ∼" − ! " 0 Hubble ! " ! (69)" (69) 2/3 2/9 TBBN mϕ H0 Nefold = Hinf t ! (68)! (70) m 2/3m m 2/9 PlT 2/3 Plm PlH2/9 !TBBNBBN" mϕ ϕ !H0 "0 ! ! ! ! (70) (70) 25/54 EternalmPlmPl 1m/Inflation2 Pl mPl m1/9Pl m Pland 27 H0 L!osc ! " H"0 mPl! !" 28 " 10− 10− ! 1 1/3 ! ∼ mPl H− N 10 TeVmPl! mϕT!BBN50 TeV∼ (71) ! " Asymptotically0 Hubble ! " ! De" Sitter Space (69) 10 TeV10 TeV! m!ϕ !m50ϕ ! TeV50 TeV (71) (71) 2/3 2/9 The HubbleTBBN scale1/ during3mϕ Hthe0 2 non-eternal/9 inflation 2/27 H0 TBBM H(70)inf H0 m ! m ! m 2/3 2 ! Pl " Pl ! Pl " ! (NHubbleζ− ) ! Heq 1/3 mPl 2/9 mPl mPl2/27 ! H0 " 1!/T3 BBM " 2/9Hinf 2/3 !H0 " 2/27 H0 TBBM Hinf 2/32 (72)H0 ! (NHubble(N ζ− ) !ζ 2) 10H TeVeq ! mϕ !m50Pl TeV m!Pl(71) Hubble − m!Pl ! H"eq ! m"Pl mPl ! " mPl ! " ! " (72)! " (72) 11 1/3 16 1/3 − − 1/3 10 2/GeV9 (NHubbleζ) ! Hinf !2/2710 GeV (NHubbleζ) H0 TBBM Hinf 2/3 2 H0 (N ζ− ) (73) H m ! m Hubble ! m ! eq " ! Pl " Pl ! Pl " 1 mPl 1 Heq (72)1 mPl 1/3 25 mPl 1 Heq 1/3 ln ln ln +ln(ζNHubble− ) ! Nefold ! ln ln +ln(ζNHubble− ) The number2 of e-foldingsH0 −3 H0 −9 TBBN 54 H0 −6 H0 1 m!Pl "1 H!eq "1 m!Pl " 1/3 25 m!Pl "1 H!eq " 1/3 1 mPl 1 ln Heq 1 lnmPl 1/3ln 25+ln(mζNPl − 1 )(73)H!eqNefold ! 1/3 ln ln +ln(ζN − ) ln ln ln +ln(ζN − ) N ln Hubbleln +ln(ζN − ) Hubble 2 H −23 HH0−9 −3T H0 −Hubble9 ! TefoldBBN! 54 H −6 H Hubble54 H0 −6 H0 ! 0 " !! 0 " " ! BBN!" " ! " ! 0 " ! 0 " ! " ! " (73) (74) 1/3 1/3 41 + ln(ζNHubble− ) ! Nefold ! 63 + ln(ζNHubble− ) (74) 1/3 1/3 41 + ln(ζNHubble− ) ! Nefold ! 63 + ln(ζNHubble− ) (75)

18 6 Tuesday, June 26, 12 6 6 6 Conclusions

Moduli dynamics is studied in a universe which is asymptotically de Sitter spacetime in the far future.

Holographic bounds have been used to constrain the mass of the moduli and the physical size of the Universe.

Eternal inflation has also been studied in this framework. Bounds have been put on the scale and duration of non-eternal inflation following bubble nucleation.

The study is based on reheating after moduli decay. There are other contributions to the entropy of the Universe: e.g. field theory configurations, structures formed from density perturbations (can be generated from different mechanisms), etc; they need to be studied in future.

Tuesday, June 26, 12