Cosmic Holography

Cosmic HolographyCosmic Holography

Bin Wang( 王斌 )Department of Physics,

Fudan University

2.7K cosmic microwave background (CMB)

The universe started in a low-entropy state and hasnot yet reached its maximal attainable entropy.

Questions:

1. Which is this maximal possible value of entropy?2. Why has it not already been reached after so many billion years of cosmic evolution?

900 10S =

Entropy of our universe

Bekenstein Entropy Bound (BEB)For Isolated Objects

Isolated physical system of energy and size (J.D. Bekenstein, PRD23(1981)287)

Charged system with energy , radius and charge

(Bekenstein and Mayo, PRD61(2000)024022; S. Hod,PRD61(2000)024023; B. Linet, GRG31(1999)1609)

Rotating system

(S. Hod, PRD61(2000)024012; B. Wang and E. Abdalla,PRD62(2000)044030)

• Charged rotating system (W. Qiu, B. Wang, R-K Su and E. Abdalla, PRD 64 (2001) 027503 )

E R

E R e/BEBS S ER£ = h

2( / )(1 / (2 ))S ER e ER£ -h

2 2 2 1/ 2/ (1 / ( ))S ER s E R£ -h

+

2 2 2 1/ 2 2( / )[(1 / ( ) / (2 )]S ER s E R e ER£ - -h

“Entropy bounds for isolated system depend neither

On background spacetime nor on spacetimedimensions.”

Universal

Holographic Entropy Bound (HEB)Holographic Principle

Entropy cannot exceed one unit per Planckian area of itsboundary surface

(Hooft, gr-qc/9310026; L. Susskind,J. Math. Phys. 36(1995)6337)

AdS/CFT Correspondence“Real conceptual change in our thinking aboutGravity.”(Witten, Science 285(1999)512)

2HEB pS S Al-£ =

Comparison of BEB and HEBIsolated System

ForFor

Cosmological ConsiderationCosmological entropy

HEB violated for largeBEB too loose for large

Two bounds cannot naively be used in cosmology.Both of them need revision in a cosmological context.

22 2 2

/ / ( )

,,

BEB s p s

HEB p ps BEB HEBs BEB HEB

S ER GER G R Rl R GES Al R lR R S SR R S S

-- -

= = = == =

> <= =

h h

3S R:RR

24

HEB

BEB

S RS ER R

:: :

Problem: In a general cosmological setting,

Natural Boundary?Particle Horizon

BEB (J.D. Bekenstein, Inter. J. Theor. Phys. 28(1989)967)FS(W. Fischler and L. Susskind, hep-th/9806039)

Comparison of BEB and FS-HEB Around the Plank time, they appear to be saturated,

which could justify the initial “low” entropy value.

+

4 2 2 22

/ / / ( ) /( )BEB p p p p p

p FS HEB

S ER Md d Hd d lHd S

r-

= = = = =h h h

FS HEB BEBS S- <

Questions still exit

For collapsing universe, FS entropy bound fails

For universes with negative cosmological constants,FS bound fails

(W. Fischler and L. Susskind, hep-th/9806039; N. Kaloper andA. Linde, PRD60(1999)103509; B. Wang and E. Abdalla,PLB471(2000)346)

Hubble Entropy Bound (HB)

- Hubble radius- number of Hubble-size regions within the volume - maximum entropy of each Hubble-size region

Relation among HB, FS, BEB

A possible relation between the FS, HB and a generalizedsecond law of thermodynamics (GSL) has been discussed.(R. Brustein, PRL84(2000)2072; B. Wang and E. Abdalla,PLB466(1999)122, PLB471(2000)346)

1H -

Hn2 2

H pS H l- -=V

3 2 2 2( ) ( / )HB H H p pSV S n S V H H l VHl- - - -< = = =

1/ 2 1/ 2HB BEB SFS S S=

1H -

Validity of BEB and HB limited self-gravity,

strong self-gravity,

Friedmann equation

For

Holographic Bekenstein-Hawking entropy of a universe-sizedblack hole

2 / ,BEBS ER np=( 1) / (4 ),HBS n HV G= -

1, 1R H HR-< <1, 1R H HR-> >

22

16 1 (1)( 1)GEH Rn n Vp= --

( 1) (2)4BHVS n GR= -

1: (3)1: (4)

BEB BH

BEB BH

HR S SHR S S

> >< <

Relation among

SubstitutingThe relation can be written as

This is very similar to the 2D Cardy formula

At the turning point between the limited self-gravity andstrong self-gravity BEB and HB have been unified Friedmann equation corresponds to the generalized Cardy

formula(E. Verlinde, hep-th/0008140; B. Wang, E. Abdalla,PLB503(2001)394)

, ,BEB HB BHS S S2 2 2 2( ) (5)HB BEB BH BEBS S S S+ - =

2 / , 2 /BH BH BEBS E R n S ER np p= =

2 (2 ) (6)HB BH BHRS E E Enp= -

02 ( / 24)/ 6 (7)S c L cp= -

Inhomogeneous Cosmologies Pietronero’s (1987) case that luminous large-scale matter

distribution follows a fractal pattern has started a sharpcontroversy in the literature.

CfA1 redshift survey (de Lapparent, Geller & Huchra, 1986,1988) was the first to reveal structures such as filamentsand voids on scales where a random distribution of matterwas expected.

Do observations of large-scale galaxy distribution supportor dismiss a fractal pattern? How inhomogeneous ismatter distribution?

Relativistic aspects of cosmological models. Any relativisticeffect on observations?

(L. Pietronero and col.)

The real reason, though, for our adherence here tothe Cosmological Principle is not that it is surelycorrect, but rather, that it allows us to make use ofthe extremely limited data provided to cosmology byobservational astronomy. …If the data will not fit into this framework, we shallbe able to conclude that either the CosmologicalPrinciple or the Principle of Equivalence is wrong.Nothing could be more interesting.

Weinberg, 1972

Large Scale StructureLarge Scale Structure

Relativistic Model

Let us start with the inhomogeneous spherically symmetricmetric as:

where

where

(E. Abdalla, R. Mohayaee, PRD59(1999)084014)

2 2 2 2 2 2 2( , )[ / ( ) ] (8)ds dt R r t dr f r r dw= - + +

2 2 2 2sin (9)d d dw q q y= +

( , ) ( )FRWR t r R t perturbation= +

Description of the Inhomogeneous UniverseDescription of the Inhomogeneous Universe

In normalized comoving coordinates the metric of theparabolic LTB model is

where , and

Characteristics of the realistic model: Spherical symmetry Describing a fractal distribution of galaxies, not refering to

either initial of final moment

2 2 2 2 2 2 2 22 2 2 2

( sin ) (10)( )( sin )a bab

ds dt R dr R d dh dx dx r x d d

q q yq q y

¢= - + + += + +%

2[ 1, ]abh diag R¢= -1/ 3 2/ 31(9 ) ( ) (11)2R F t b= +

Lemaitre-Tolman-Bondi modelLemaitre-Tolman-Bondi model

Our proposal of a holographic principle ininhomogeneous cosmology

The entropy inside the apparent horizon can never exceed thearea of the apparent horizon in Plank units.

22/ 4 AHp rS A l bp£ = %

Defining Apparent Horizonto the aerial radius, with the result

is the physical apparent horizon, denotes the proper apparent horizon.

Fractal behavior in parabolic models have been found byRibeiro (APJ1992). They are

Where and around 0.65 and around 50 are required to obtain fractalsolutions.

2 0aba br h r rÑ º ¶ ¶ =% % %

[ ]( ) 3 ( ) (12)AH AHF r t rb= +( )[ ]3

2AH AHr t rb= +%AHr

0 0(13)

p

qF ar

rb b h=ìïïí = +ïïî

( )0 1(14)ln

pF are rbb h

=ìïïí = +ïïî[ ]5 410 ,10 ,pa - -Î [ ] [ ]0 10.5,4 , 1000,1300b hÎ Î

q 0h

Model 1:

Model 2:

Define the local entropy density

Standard big-bang cosmology: When a particle becomesnonrelativistic and disappears, its entropy is transferred toother relativistic particle species still present in the thermalplasma.

Photons and neutrinos share the entropy of the universe.

Reasonable suppose: Entropy of the universe is mainlyproduced before the dust-filled era.

+

First Law of thermodynamics, .Considering that in the expansion of the universe, The radiation always has the property of black body Conservation of the number density of the photon

We have: , the expression for the redshift is

We obtain the relation .

4/ /s T aT Tr= =

00

h hkT kTn n=

0

110

0 01 (15)dt dt dr dr Rz R Rd d d d Rl l l ll l l l

--

=

æ ö ¢æ ö ÷ç÷ ¢ ¢ç+ = = =÷ç÷ç ÷ç ÷ç ¢è ø è ø

0 0TR T R Const¢ ¢= =

The local entropy density in the inhomogeneous case canbe expressed as

The total entropy measured in the comoving space inside theapparent horizon is

For homogeneous dust universe the local entropy density isonly a function of proportional to , the consistenttotal entropy value

(B. Wang, E. Abdalla and T. Osada, PRL85(2000)5507)

( ) ( ) ( ) ( )3

0 0 3 31 1, (16), ,s t r a T R C

R t r R t r¢= =¢ ¢

( ) 20

, 4 . (17)AHrS s t r R R drp ¢= ò

t ( )3a t-

34 .3 AHS rps=

Fig. 1: Relation between and with different at thebeginning of the dust-filled universe when .

S A p5

0 0.97 10t -= ´

We now face the question:

the holographic principle has to be challenged

it can be used to select a physically acceptablemodel

We prefer the second, more constructive, alternative.

Fig. 2: Inhomogeneous models which can accommodatereasonable entropy to meet the present observable value.

Fig. 3: Choosing parameters in order to meet the entropyvalue in the present observable universe.

Conclusions

Possible to modelize highly inhomogeneousstructures

Entropy constraints can give valuable informationthrough the holographic principle

I. Upper bound on the numbI. Upper bound on the number of e-foldings from hologrer of e-foldings from hologr

aphyaphy

The number of e-foldings during inflationThe number of e-foldings during inflation

Number of e-folds:Number of e-folds: Horizon problem, flatness problem & entropy problemHorizon problem, flatness problem & entropy problem Relate to the slow roll parameters and fluctuations prediction of Relate to the slow roll parameters and fluctuations prediction of

inflationinflation


The existence of an upper bound for the number of e-folThe existence of an upper bound for the number of e-foldings has been discussed.dings has been discussed.

In general it is model dependent. The bound has been oIn general it is model dependent. The bound has been obtained in some very simple cosmological settings, while btained in some very simple cosmological settings, while it is still difficult to be obtained in nonstandard models. it is still difficult to be obtained in nonstandard models.

Using the holographic principle, the consideration ofUsing the holographic principle, the consideration of physical details connected to the universe evolution can physical details connected to the universe evolution can

be avoided. We have obtained the upper bound for the nbe avoided. We have obtained the upper bound for the number of e-foldings for a standard FRW universe as well umber of e-foldings for a standard FRW universe as well as non-standard cosmology based on the brane inspired as non-standard cosmology based on the brane inspired idea of Randall and Sundrum models.idea of Randall and Sundrum models.

Holographic PrincipleHolographic Principle

Motivated by the well-known example of black hole entroMotivated by the well-known example of black hole entropy, an influential holographic principle has put forward, spy, an influential holographic principle has put forward, suggesting that microscopic degrees of freedom that build uggesting that microscopic degrees of freedom that build up the gravitational dynamics actually reside on the bounup the gravitational dynamics actually reside on the boundary of space-time. dary of space-time.

This principle developed to the Maldacena's conjecture oThis principle developed to the Maldacena's conjecture on AdS/CFT correspondence and further very important cn AdS/CFT correspondence and further very important consequences, such as Witten's identification of the entroonsequences, such as Witten's identification of the entropy, energy and temperature of CFT at high temperaturepy, energy and temperature of CFT at high temperatures with the entropy, mass and Hawking temperature of ths with the entropy, mass and Hawking temperature of the AdS black hole.e AdS black hole.

Cosmic HolographyCosmic Holography

We thus seek at a description of the We thus seek at a description of the powerful holographic principle in powerful holographic principle in cosmological settings, where its testing is cosmological settings, where its testing is subtle.subtle.

The question of holography therein: for flat The question of holography therein: for flat and open FLRW universes the area of the and open FLRW universes the area of the particle horizon should bound the entropy particle horizon should bound the entropy on the backward-looking light cone. on the backward-looking light cone.

Verlinde-Cardy formulaVerlinde-Cardy formula

FLRW universe filled with CFT with a dual AdS description has FLRW universe filled with CFT with a dual AdS description has been done by Verlinde, revealing that when a universe-sized blbeen done by Verlinde, revealing that when a universe-sized black hole can be formed, an interesting and surprising corresponack hole can be formed, an interesting and surprising correspondence appears between entropy of CFT and Friedmann equatiodence appears between entropy of CFT and Friedmann equation governing the radiation dominated closed FLRW universes.n governing the radiation dominated closed FLRW universes.

Generalizing Verlinde's discussion to a broader class of univerGeneralizing Verlinde's discussion to a broader class of universes including a cosmological constant: matching of Friedmann ses including a cosmological constant: matching of Friedmann equation to Cardy formula holds for de Sitter closed and AdS flaequation to Cardy formula holds for de Sitter closed and AdS flat universes. t universes.

However for the remaining de Sitter and AdS universes, the argHowever for the remaining de Sitter and AdS universes, the argument fails due to breaking down of the general philosophy of tument fails due to breaking down of the general philosophy of the holographic principle. In high dimensions, various other aspehe holographic principle. In high dimensions, various other aspects of Verlinde's proposal have also been investigated in a numcts of Verlinde's proposal have also been investigated in a number of works. ber of works.

Verlinde-Cardy formula in Brane CosmologyVerlinde-Cardy formula in Brane Cosmology Further light on the correspondence between Friedmann equation aFurther light on the correspondence between Friedmann equation a

nd Cardy formula has been shed from a Randall-Sundrum.nd Cardy formula has been shed from a Randall-Sundrum.

CFT dominated universe as a co-dimension one brane with fine-tunCFT dominated universe as a co-dimension one brane with fine-tuned tension in a background of an AdS black hole, Savonije and Verlied tension in a background of an AdS black hole, Savonije and Verlinde found the correspondence between Friedmann equation and Cnde found the correspondence between Friedmann equation and Cardy formula for the entropy of CFT when the brane crosses the blaardy formula for the entropy of CFT when the brane crosses the black hole horizon.ck hole horizon.

Confirmed by studying a brane-universe filled with radiation and stifConfirmed by studying a brane-universe filled with radiation and stiff-matter, quantum-induced brane worlds and radially infalling brane.f-matter, quantum-induced brane worlds and radially infalling brane.

The discovered relation between Friedmann equation and Cardy forThe discovered relation between Friedmann equation and Cardy formula for the entropy shed significant light on the meaning of the holmula for the entropy shed significant light on the meaning of the holographic principle in a cosmological setting.ographic principle in a cosmological setting.

The general proof for this correspondence for all CFTs is still difficult The general proof for this correspondence for all CFTs is still difficult at the moment.at the moment.

The number of e-foldings from holographyThe number of e-foldings from holography

Our motivation here is the use of the correspondOur motivation here is the use of the correspondence between the CFT gas and the Friedmann eence between the CFT gas and the Friedmann equation establishing an upper bound for the numquation establishing an upper bound for the number of e-foldings during inflation. ber of e-foldings during inflation.

Recently, Banks and Fischler have considered tRecently, Banks and Fischler have considered the problem of the number of e-foldings in a univhe problem of the number of e-foldings in a universe displaying an asymptotic de Sitter phase, aerse displaying an asymptotic de Sitter phase, as our own. As a result the number of e-foldings is our own. As a result the number of e-foldings is not larger than 65/85 depending on the type of s not larger than 65/85 depending on the type of matter considered.matter considered.

The number of e-foldings from holographyThe number of e-foldings from holography

Here we reconsider the problem from the Here we reconsider the problem from the point of view of the entropy content of the point of view of the entropy content of the Universe, and considering the correspondUniverse, and considering the correspondence between the Friedmann equation anence between the Friedmann equation and Cardy formula in Brane Universes.d Cardy formula in Brane Universes.

Brane cosmologyBrane cosmology

Metric:Metric: We consider a bulk metric defined by We consider a bulk metric defined by

and and L L is the curvature radiusis the curvature radius of AdS spacetime.of AdS spacetime. k k takes the values 0, -1, +1 takes the values 0, -1, +1

corresponding to flat, open and closed geometrcorresponding to flat, open and closed geometrics, and is the corresponding ics, and is the corresponding

metric on the unit three dimensional sections.metric on the unit three dimensional sections.


Black hole horizon:Black hole horizon:

The relation between the parameter The relation between the parameter mm and the and the Arnowitt-Deser-Misner (ADM) mass of the five Arnowitt-Deser-Misner (ADM) mass of the five dimensional black hole dimensional black hole M M isis

is the volume of the unit 3 sphere.is the volume of the unit 3 sphere.


Metric on the brane:Metric on the brane: Here, the location and the metric on the boundarHere, the location and the metric on the boundar

y are time dependent. We can choose the brane y are time dependent. We can choose the brane time such thattime such that

The metric on the brane is given byThe metric on the brane is given by

CFT on the braneCFT on the brane

The Conformal Field Theory lives on the brane, whiThe Conformal Field Theory lives on the brane, which is the boundary of the AdS hole. The energy for ch is the boundary of the AdS hole. The energy for a CFT on a sphere with volume is given a CFT on a sphere with volume is given by The density of the CFT energy can be by The density of the CFT energy can be expressed as expressed as

EntropyEntropy

The entropy of the CFT on the brane is equal to tThe entropy of the CFT on the brane is equal to the Bekenstein-Hawking entropy of the AdS black he Bekenstein-Hawking entropy of the AdS black holehole

The entropy density of the CFT on the brane isThe entropy density of the CFT on the brane is

Friedmann EquationFriedmann Equation From the matching conditions we find now the cosmoFrom the matching conditions we find now the cosmo

logical equations in the brane,logical equations in the brane,

is the critical brane tension. Takingis the critical brane tension. Taking the Friedmann Eq. reduces to the Friedmann equation the Friedmann Eq. reduces to the Friedmann equation

of CFT radiation dominated brane universe without cof CFT radiation dominated brane universe without cosmological constant. osmological constant.

If the brane-world is a de Sitter uIf the brane-world is a de Sitter universe or AdS universe, respectively.niverse or AdS universe, respectively.

Friedmann EquationFriedmann Equation UsingUsing

the Friedmann equation can be written in the formthe Friedmann equation can be written in the form

is the effective positive cosmological constant in four diis the effective positive cosmological constant in four di

mensions.mensions. Using Friedmann equation becomes Using Friedmann equation becomes

which corresponds to the movement of a mechanical nonrwhich corresponds to the movement of a mechanical nonrelativistic particle in a given potential.elativistic particle in a given potential.

Entropy BoundEntropy Bound

For a closed universe there is a critical value for For a closed universe there is a critical value for which the solution extends to infinity (no big crunwhich the solution extends to infinity (no big crunch)ch)

The entropy in such a universe can be rewriten asThe entropy in such a universe can be rewriten as

at the end of inflation. We take to be the energy at the end of inflation. We take to be the energy density during inflation, that is, density during inflation, that is,

Upper bound on the number of e-foldingsUpper bound on the number of e-foldings

Scale factor at the exit of inflation leads to the value Scale factor at the exit of inflation leads to the value , where , where corresponds to the apparent horizon corresponds to the apparent horizon duringduring inflation, and we obtaininflation, and we obtain

We get We get

where we used the usual values where we used the usual values

Brane corrections to the Friedmann equationBrane corrections to the Friedmann equation Let us consider now very high energy brane corrections tLet us consider now very high energy brane corrections t

o the Friedmann equation. From the Darmois-Israel condo the Friedmann equation. From the Darmois-Israel conditions we finditions we find

wherewhere l l is the brane tension and in the very high energy l is the brane tension and in the very high energy limit the term dominates.imit the term dominates.

Within the high-energy regime, the expansion laws correWithin the high-energy regime, the expansion laws corresponding to matter and radiation domination are slower tsponding to matter and radiation domination are slower than in the standard cosmology. han in the standard cosmology.

Slower expansion rates lead to a larger value of the numSlower expansion rates lead to a larger value of the number of e-foldings. However, the full calculation has not beber of e-foldings. However, the full calculation has not been obtained due to the lack of knowledge of this high-enen obtained due to the lack of knowledge of this high-energy regime. ergy regime.

CFT energy density and entropy density relationCFT energy density and entropy density relation

The energy density of the CFT and the entropy dThe energy density of the CFT and the entropy density are related as follows,ensity are related as follows,

Substitute in the Friedmann equation as before, lSubstitute in the Friedmann equation as before, l

eading to a bound for the entropy, as well as a beading to a bound for the entropy, as well as a bound for the scale factor,ound for the scale factor,


The era when the quadratic energy density is important. The era when the quadratic energy density is important. The brane tension is required to be bounded byThe brane tension is required to be bounded by

and then the number of e-foldings is and then the number of e-foldings is where where

is taken. is taken. The number of e-foldings obtained is bigger than the vaThe number of e-foldings obtained is bigger than the va

lue in standard FRW cosmology, which is consistent wilue in standard FRW cosmology, which is consistent with the argument of Liddle et al.th the argument of Liddle et al.


In summary:In summary:

we have derived the upper limit for the number of e-foldings based uwe have derived the upper limit for the number of e-foldings based upon the arguments relating Friedmann equation and Cardy formula. pon the arguments relating Friedmann equation and Cardy formula.

For the standard FRW universe our result is in good agreement with For the standard FRW universe our result is in good agreement with Literatures. Literatures.

For the brane inspired cosmology in four dimensions we obtained a lFor the brane inspired cosmology in four dimensions we obtained a larger bound. Considering such a high energy context, the expansioarger bound. Considering such a high energy context, the expansion laws are slower than in the standard cosmology, and our result can laws are slower than in the standard cosmology, and our result can again be considered to be consistent with the known argument. n again be considered to be consistent with the known argument.

The interesting point here is that using the holographic point of view, The interesting point here is that using the holographic point of view, we can avoid a complicated physics during the universe evolution awe can avoid a complicated physics during the universe evolution and give a reasonable value for the upper bound of the number of e-fnd give a reasonable value for the upper bound of the number of e-foldings.oldings.

II. WMAP constraint on II. WMAP constraint on P-term inflationary P-term inflationary

modelmodel

Supersymmetric inflationary modelSupersymmetric inflationary model

Besides the standard model, supersymmetry has been conBesides the standard model, supersymmetry has been considered both as a blessing and as a curse for inflationary sidered both as a blessing and as a curse for inflationary model building. model building.

It is a blessing, primarily because it allows one to have vIt is a blessing, primarily because it allows one to have very flat potential, as well as to fine-tune any parameters ery flat potential, as well as to fine-tune any parameters at the tree level. Moreover it seems more natural than that the tree level. Moreover it seems more natural than the non-symmetric theories.e non-symmetric theories.

It is a curse, because during inflation one needs to consiIt is a curse, because during inflation one needs to consider supergravity, where usually all scalar fields have too der supergravity, where usually all scalar fields have too big masses to support inflation.big masses to support inflation.

Exceptions:Exceptions: The The N=1N=1 generic D-term inflation generic D-term inflation The The N=1 N=1 supersymmetric F-term inflationsupersymmetric F-term inflationavoids the general problem of inflation in supergravity.avoids the general problem of inflation in supergravity.

P-term inflationary modelP-term inflationary model

A new version of hybrid inflation, the ``P-term inflatA new version of hybrid inflation, the ``P-term inflation'' has been introduced in the context of N=2 sion'' has been introduced in the context of N=2 supersymmetry. [Kallosh & Linde]upersymmetry. [Kallosh & Linde]

It is intriguing that once one breaks N=2 supersymIt is intriguing that once one breaks N=2 supersymmetry and implements the P-term inflation in N=metry and implements the P-term inflation in N=1 supergravity, this scenario simultaneously lead1 supergravity, this scenario simultaneously leads to a new class of inflationary models, which ints to a new class of inflationary models, which interpolates between D-term and F-term models. erpolates between D-term and F-term models.

P-term inflationary modelP-term inflationary modelThe effective potential in units $M_p=1$ isThe effective potential in units $M_p=1$ is

A general P-term inflation model has 0<f<1 with the speA general P-term inflation model has 0<f<1 with the spe

cial case f=0 corresponding to the D-term inflation, while cial case f=0 corresponding to the D-term inflation, while f=1 corresponds to the F-term inflation. f=1 corresponds to the F-term inflation.

Above, s_e is the bifurcation point indicating the end of iAbove, s_e is the bifurcation point indicating the end of inflation. nflation.

The second term in the potential is due to the one-loop cThe second term in the potential is due to the one-loop correction and the third term to the supergravity correctioorrection and the third term to the supergravity correction. n.

The inflationary spaceThe inflationary space

In a single field slow-roll inflation model with a pIn a single field slow-roll inflation model with a potential V(s) the amplitude of curvature perturbaotential V(s) the amplitude of curvature perturbation is given by tion is given by

The spectral index is defined byThe spectral index is defined by

The logarithmic derivative of the spectral index iThe logarithmic derivative of the spectral index iss

The WMAP resultsThe WMAP results

WMAP result favors purely adiabatic fluctuations WMAP result favors purely adiabatic fluctuations with a remarkable feature that the spectral index with a remarkable feature that the spectral index runs from n>1 on a large scale to n<1 on a small runs from n>1 on a large scale to n<1 on a small scale. More specifically on the scale scale. More specifically on the scale

k=k=

It is of interest to investigate whether the P-term It is of interest to investigate whether the P-term inflation can accommodate these observational inflation can accommodate these observational result.result.

Number of e-foldsNumber of e-foldsFrom the potential form we learnt that inflation consists of tFrom the potential form we learnt that inflation consists of t

wo long stages, one of them is determined by the one-lowo long stages, one of them is determined by the one-loop effect and the other is determined by the supergravity op effect and the other is determined by the supergravity corrections. corrections.

The total duration of inflation can be estimated byThe total duration of inflation can be estimated by

Where N_k is supposed to be a reasonable number of e-Where N_k is supposed to be a reasonable number of e-

foldings.foldings.

Thus we require . Thus we require . For the F-term inflation For the F-term inflation f=1f=1 and and N_k=60, g<0.15N_k=60, g<0.15, which i, which i

s exactly the argument given by Linde and Riotto (97)s exactly the argument given by Linde and Riotto (97)

Number of e-foldsNumber of e-folds


Slow-roll parametersSlow-roll parameters

For the P-term inflationFor the P-term inflation

The end of inflation is determined byThe end of inflation is determined by

Slow-roll parametersSlow-roll parameters

Behavior of slow-roll parametersBehavior of slow-roll parameters

The value of $s_{end}$ obtained from $\epsilon=1$ for small $s$ ($s_{eThe value of $s_{end}$ obtained from $\epsilon=1$ for small $s$ ($s_{end}<s_0$) is the real end point of inflation. nd}<s_0$) is the real end point of inflation.

Comparison with WMAP resultComparison with WMAP result

Strategy: Strategy:

Express Express s (s=s_k)s (s=s_k) as a function of as a function of s_{end}s_{end} and and N_kN_k for for different values ofdifferent values of f f and and g. N_kg. N_k is the number of e-foldin is the number of e-foldings between the time the scales of interest leave the horigs between the time the scales of interest leave the horizon and the end of inflation. Inserting such an zon and the end of inflation. Inserting such an s s into slointo slow-roll parameters and we can obtain the spectral index w-roll parameters and we can obtain the spectral index and its logarithmic derivative.and its logarithmic derivative.

Comparison with WMAP resultComparison with WMAP result

Dependence of the spectral index and its logarithmDependence of the spectral index and its logarithmic derivative on ic derivative on f f for different values of for different values of g g when thwhen the number of e-foldings are 60 and 70, respectivee number of e-foldings are 60 and 70, respectively.ly.

Comparison with WMAP resultComparison with WMAP result There is a threshold value of There is a threshold value of g(min)g(min) to force the spectral to force the spectral

index index nn to meet the minimum observational result to meet the minimum observational result 1.04.1.04.

With the increase of With the increase of NN, the threshold value , the threshold value g(min)g(min) can be can be smallersmaller. . However due to the existence of the upper bounHowever due to the existence of the upper bound of the number of e-foldings, this threshold value cannot d of the number of e-foldings, this threshold value cannot be reduced arbitrarily.be reduced arbitrarily.

For fixed For fixed ff, both the values of the spectral index and its l, both the values of the spectral index and its logarithmic derivative increase with the increase of ogarithmic derivative increase with the increase of gg. For . For fixed fixed gg, they increase with , they increase with ff as well. as well.

We cannot enforce both spectral index and its running to We cannot enforce both spectral index and its running to meet the WMAP observational result at the same time fomeet the WMAP observational result at the same time for the common range of r the common range of ff and and g.g.

Comparison with WMAP resultComparison with WMAP result For fixed For fixed gg, the dependence of the spectral index and its , the dependence of the spectral index and its

logarithmic derivative on logarithmic derivative on ff for different values of the num for different values of the number of e-foldings is shownber of e-foldings is shown

with the increase of the number of e-foldings botwith the increase of the number of e-foldings both h nn and its running increase. and its running increase.

Comparison with WMAP resultComparison with WMAP result The spectral index and its logarithmic derivative depenThe spectral index and its logarithmic derivative depen

d on the number of e-foldings for different fixed values d on the number of e-foldings for different fixed values of of gg and and f f..

nn decreases from decreases from n>1n>1 to to n<1n<1 as as kk increases (N ) increases (N ) Again, n and its running cannot comply with Again, n and its running cannot comply with observation at the same time.observation at the same time.

WMAP constraint on P-term inflationary modelWMAP constraint on P-term inflationary modelIn summary:In summary: The P-term inflation model with a running parameter The P-term inflation model with a running parameter 0<f<10<f<1 displays displays

a richer physics.a richer physics.

In addition to the upper bound on In addition to the upper bound on gg determined by a reasonable nu determined by a reasonable number of e-foldings to solve the horizon problem as required by the inmber of e-foldings to solve the horizon problem as required by the inflation, the observational data of the spectral index together with the flation, the observational data of the spectral index together with the upper limit of the number of e-foldings puts the lower bound on the cupper limit of the number of e-foldings puts the lower bound on the choice of hoice of gg..

Obtaining a logarithmic derivative spectral index such that n>1 on laObtaining a logarithmic derivative spectral index such that n>1 on large scales while n<1 on small scale for P-term inflation. (Not for D-, rge scales while n<1 on small scale for P-term inflation. (Not for D-, F-term inflations)F-term inflations)

It is not possible to accommodate both observational ranges of It is not possible to accommodate both observational ranges of n n anand its running at the same time.d its running at the same time.

The larger values of the logarithmic derivative of the spectral index cThe larger values of the logarithmic derivative of the spectral index can be around an be around -0.011-0.011 for values of for values of ff and and gg keeping the spectral index keeping the spectral index within the WMAP range.within the WMAP range.

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Cosmic Holography