Thank you for the invitation.

DISCUSSION OF

COSMIC ACCELERATION

Paul H. Frampton and Kevin Ludwick

MIAMI 2010 CONFERENCE

DECEMBER 15, 2010

OUTLINE

Accelerating Expansion

Dark Energy Problem

Emergent Gravity

Holographic Principle

Observable Universe

Evolution

2

References

PHFSolution to the Dark Energy Problem

arXiv:1004.1285

PHFPossible Solution of Dark Matter, The So-

lution of Dark Energy, and Gell-Mann as

Great Theoretician

arXiv:1004.1910 [hep-ph].

D.A. Easson, PHF and G.F. SmootEntropic Accelerating Universe

arXiv:1002.4278 [hep-th]

Entropic Inflation

arXiv:1003.1528

PHF and Kevin LudwickSeeking Evolution of Dark Energy

In preparation

3

Accelerating expansion

In fundamental theoretical physics, there was,at the beginning of the twenty-first century, animpossible seeming problem which might not besolved for a hundred years. The problem wasthe dark energy in cosmology, comprising someseventy percent of the universe.

Another cosmological problem, closely entwinedwith the dark energy problem, is the questionwell posed, now almost eight decades ago, byTolman whether one can construct a consistentcyclic model, given the seemingly contradictoryconstraint imposed by the second law of ther-modynamics. The most developed solution, ofthis Tolman conundrum, is that suggested byBaum and myself in Phys. Rev. Lett. 2007.

4

The most important observational advance incosmology, since the early studies of cosmic ex-pansion in the 1920’s ,was the dramatic and,at that time, surprising discovery, in the wan-ing years of the twentieth century, that the ex-pansion rate is accelerating. This was first an-nounced in February 1998, based on the concor-dance of two groups’ data.

Many subsequent experiments concerning theCosmic Microwave Background (CMB), LargeScale Structure (LSS), and other measurementshave all confirmed the 1998 claim. We maytherefore adopt the position, that the acceler-ated expansion rate is an observed fact.

5

Assuming general relativity, together withthe cosmological principle of homogeneity andisotropy, the scale factor a(t) in the FRW met-ric satisfies the Friedmann-Lemaıtre equation

H(t)2 =

a

a

2=

8πG

3

ρ (1)

where I shall normalize a(t0) = 1 at the present,time t = t0, and ρ is an energy density sourcewhich drives the expansion of the universe.

6

Two established contributions to ρ are ρm frommatter (including dark matter) and ργ radia-tion, so that

ρ ⊇ ρm + ργ (2)

with ρm(t) = ρm(t0)a(t)−3 and ργ(t) =

ργ(t0)a(t)−4.

7

For the observed accelerated expansion, themost popular approach is to add to the sources,in Eq.(1), a dark energy term ρDE(t) with

ρDE(t) = ρDE(t0)a(t)−3(1+ω) (3)

where ω = p/ρ is the equation of state. Forthe case ω = −1, as for a cosmological con-stant, Λ, and discarding the matter and radia-tion terms which are smaller I can easily inte-grate the Friedmann-Lemaıtre equation to find

a(t) = a(t0) eHt (4)

where√3H =

√Λ =

√8πGρDE.

8

By differentiation with respect to time p times,we obtain for the pth derivative

dp

dtpa(t)|t=0 = (

√

Λ/3)p (5)

Therefore, if Λ > 0 is positive, as in a De Sittergeometry, not only is the acceleration (p = 2)positive and non-zero, but so are the jerk (p =3), the snap (p = 4), the crackle (p = 5), thepop (p = 6) and all p ≥ 7.

The insertion of the dark energy term worksvery well as a part of the ΛCDM model. How-ever, it is an ad hoc procedure which gives noinsight into what dark energy is.

9

Dark Energy Problem

With this background, I shall now move to mydifferent explanation for the accelerated expan-sion which obviates any dark energy, includingany need for a cosmological constant.

I now adopt a different approach, with no darkenergy, where instead the central role is playedby the assumption of the holographic principle,and by the overiding concept of entropy.

10

The essential assumption is the aforementionedholographic principle, by which I understandthat all the information about the universe is en-coded on its two-dimensional surface. What thisimplies is, however unlikely it seems and how-ever contrary to everyday experience, that thethree-dimensional world I apparently observe, issomehow an illusion. This can lead to a reinter-pretation of the cosmic acceleration, and possi-bly the most dramatic new insight into gravityin over three centuries.

11

Consider the Schwarzschild radius (rs), and thephysical radius (r), of the Sun (⊙). They are(rs)⊙ = 3km and r⊙ = 800, 000km. Theirratio is (ǫ)⊙ ≡ (rs/r)⊙ = 3 × 10−6. Onecan readily check that, for the Earth or for theMilky Way, that the ratio ǫ = (rs/r) is like-wise much smaller than one: ǫ << 1. Suchobjects are nowhere close to being black hole.Now consider the visible universe (VU), withmass MV U = afewtimes1023M⊙. It has(rs)V U ∼ 45Gly, and (r)V U ∼ 48Gly, hence(ǫ)V U ∼ 1. The visible universe, within whichwe all live, is close to being a black hole. The so-lution to the dark energy problem follows, pro-viding I so approximate the visible universe.

12

At this horizon, there is PBH temperature, Tβ,which I can estimate as

Tβ =h

kB

H

2π∼ 3× 10−30K. (6)

This temperature of the horizon informationscreen leads to a concomitant FDU accelerationaHorizon, outward, of the horizon given by therelationship

aHorizon =

2πckBTβh

= cH ∼ 10−9m/s2 .

(7)

When Tβ is used in Eq. (7), I arrive at a cos-mic acceleration which is essentially in agree-ment with the observations.

13

From this viewpoint, the dark energy is non-existent. Instead there is a consequence of thesecond law of thermodynamics, acting to createthe appearance of a dark energy component ofthe driving density on the right-hand-side of theFriedman-Lemaıtre equation, Eq.(1).

I have discussed a theory underlying the acceler-ated expansion of the universe based on entropy.This approach provides a physical understand-ing of the acceleration phenomenon which waslacking in the description as dark energy.

14

The entropy of the universe has received somerecent attention, in part because it relates tothe feasibility of constructing a consistent cyclicmodel. For example, the cyclic model, assum-ing its internal consistency will indeed be fullyconfirmed, provides the solution to a difficultentropy question originally posed, seventy-fiveyears earlier, by Tolman. The accelerated ex-pansion rate is no longer surprising. It is the in-evitable consequence of information storage onthe surface of the visible universe.

15

Emergent Gravity

This solution of the dark energy problem notonly solves a cosmological problem, it casts acompletely new light on the nature of the grav-itational force. Since the expansion of the uni-verse, including the acceleration thereof, canonly be a gravitational phenomenon, I arrive atthe viewpoint that gravity is a classical result ofthe second law of thermodynamics. This meansthat gravity cannot be regarded as, on a footingwith, the electroweak and strong interactions.

Although this can be the most radical change,in gravity theory, for over three centuries, it isworth emphasizing that general relativity, andits classical tests, remain unscathed, as does theprediction of gravitational waves.

16

My result calls into question almost all of thework done on quantum gravity, since the discov-ery of quantum mechanics. For gravity, thereis no longer necessity for a graviton. In thecase of string theory, the principal motivationfor the profound and historical suggestion byScherk and Schwarz that string theory be rein-terpreted, not as a theory of the strong interac-tion, but instead as a theory of the gravitationalinteraction, came from the natural appearanceof a massless graviton in the closed string sector.

I am not saying that string theory is dead.What I am saying is, that string theory can-not be a theory of the fundamental gravitationalinteraction, since there is no fundamental grav-itational interaction.

17

The way this new insight emerged, and the solu-tion of the dark energy problem itself, was as anatural line of thought, following the discoveryof a cyclic model and the subsequent investiga-tions of the entropy of the universe, including apossible candidate for dark matter.

18

Another ramification, of my solution of the darkenergy, problem is the status, fundamental ver-sus emergent, of the three spatial dimensions,that we all observe every day. Because the solu-tion assumes the holographic principle, at leastone spatial dimension appears as emergent. Re-garding the visible universe as a sphere, withradius of about 48 Gly, the emergent space di-mension is then, in spherical polar coordinates,the radial coordinate, while the other two co-ordinates, the polar and azimuthal angles, re-main fundamental. Physical intuition, relatedto the isotropy of space, may suggest that, ifone space dimension is emergent, then so mustbe all three. This merits further investigation,and may require a generalization of the holo-graphic principle. On the other hand, a funda-mental time coordinate is useful in dynamics.

19

Of course, this present discussion of cosmic ac-celeration, is merely one small step towards theultimate goal, of a cyclic model, in which timenever begins or ends.

20

Once we accept that gravity is emergent, and aresult of the 2nd law of thermodynamics, thenin the law F = Gm1m2/r

2 at least one of thesystems 1 and 2 must be large, meaning that ithas at least e3 ∼ 20 particles. The gravitationalforce between two elementary particles e.g. twoprotons, is zero. So, at LHC, gravity is not onlyirrelevant, it is zero!

21

Holographic principle

For the case of a sphere, with mass M , of ra-dius R, where R will be the co-moving radiusfor the expanding universe, a simplified, andnon-covariant form, of the holographic princi-ple, states that the entropy, S/k, has an upperlimit equal to that of a black hole, i.e.

S

k

≤

S

k

BH=

1

4

4πR2S

l2Planck

(8)

where G is Newton’s constant, RS = 2GMis the Schwarzschild radius and lPlanck is thePlanck length. It is interesting, from theviewpoint of the physical understanding of thevisible universe, to use accurate observationaldata to check, whether the simplied, and non-covariant, Eq.(8) is satisfied at the present time,t = t0, and in the past, cognizant that, withdark energy, if R sufficiently increases, Eq.(8)will, in any case, eventually be violated.

22

The holographic principle is supported, bystring theory. The AdS/CFT correspondenceis an explicit realization of Eq.(8), and so, apartfrom the non-trivial subtlety that our universeis dS, not AdS, from the viewpoint of string the-ory, there is every reason to believe the covari-ant holographic principle, and to wish to checkEq.(8). It is related to recent considerations ofthe entropy of the universe.

However, physics is an empirical science, andtherefore the scientific method dictates that weshould find a physical example, in which Eq.(8)can be calculated. The result, reported here, isthat a detailed and accurate check of Eq.(8),as applied to the visible universe, fails, by astatistically-significant amount, although in thepast, a few billion years ago, it was satisfied.

23

I should define, precisely, what is meant by thevisible universe. It is the sphere, centered forconvenience at the Earth, and with a radiusdA(Z

∗) = 14.0± 0.1Gpc. The value of dA(Z∗)

is the particle horizon corresponding to the re-combination red shift Z∗ = 1090 ± 1, i and ismeasured directly by WMAP7, without need-ing the details of the expansion history. Thus,”visible” means with respect to electromagneticradiation.

24

Observable Universe

The motion is that the visible universe, so de-fined, is a physical object which should be sub-ject to the holographic principle. It is an ex-panding, rather than a static, object, yet myunderstanding is that the principle, at least inits covariant form, is still expected to be valid. Ishall use the notation employed by the WMAP7paper, from which all observational data aretaken.

The present age, t0, of the universe is measuredto be

t0 = 13.75± 0.13Gy (9)

25

The comoving radius, dA(Z∗), of the visible uni-

verse, is, likewise, measured to one percent ac-curacy, as

dA(Z∗) ≡ (1 + Z∗)DA(Z

∗) = c∫ t0t∗

dt

a(t)= 14.0± 0.1Gpc (10)

where it is noted that the measurement, ofdA(Z

∗), does not require knowledge, of the ex-pansion history, a(t), for t∗ ≤ t ≤ t0.

26

The critical density, ρc, is provided by the for-mula

ρc =

3H20

8πG

(11)

whose value depends on H0, as does the total,baryonic plus dark, matter density, ρm

ρm ≡ Ωmρc (12)

Because the error on the Hubble parameter,H0,is several per cent, it is best to avoid H0, inchecking the holographic principle.

27

The mass of the matter, M(Z*), contained inthe visible universe, is

M(Z∗) =4π

3dA(Z

∗)3ρm (13)

and the Schwarzschild radius, RS(Z∗), is given

by

RS(Z∗) ≡ 2GM(Z∗) (14)

Collecting results enables the desired accuratecheck of the holographic principle. The mass isM(Z∗) = 5.5× 1023M⊙.

28

The holographic pronciple requires that

Sk

V.U.

Sk

BH

≤ 1 (15)

A shift parameter, R, was defined by Bond, Ef-stathiou and Tegmark (BET) as

R =

√

ΩmH20

c(1 + Z∗)DA(Z

∗) (16)

which was, with great prescience, introduced byBET, as a dimensionless quantity, to be mea-sured, accurately, by CMB observations.

29

This BET shift parameter, R, of Eq. (16), isgiven in WMAP7, as

R = 1.725± 0.018 (17)

A little algebra shows that the BET shift pa-rameter R provides the most accurate availablecheck, of the holographic principle, by virtue ofthe result

Sk

V.U.

Sk

BH

≡ R4 = 8.85± 0.37 (18)

showing a violation, by 21σ.

30

To my knowledge, the visible universe is, atpresent, the only physical object, for which it ispossible to calculate, and compare with experi-ment, or observation, the simplified holographicprinciple.

From Eq. (15), the radius dA(ZHP ) = (1 +ZHP )DA(ZHP ), at which the violation ofEq.(8), begins, is dA(ZHP ) = 8.4 ± 0.1Gpc,at a time, comparable to when the cosmic de-celeration ends, and becomes acceleration. Thisis strongly supportive of the idea of an entropicaccelerating universe.

The original aim, of the present work, was toconfirm, at t = t0, the inequality, Eq.(15). Itwas, therefore, surprising to learn that it is vio-lated, with high statistical significance, and hasbeen so, for five billion years.

31

0.6

0.7

0.8

0.9

z* =0.5

Ω1z

Ω0z

-4 -3 -2 -1 1 2

-20

-15

-10

-5

Figure 1: Equation of state linear in z: ωz(z) = ωz

0 + ωz

1z. Plot of ωz

1-ωz

0 for 0.5 < z⋆ < 0.9in increments of z⋆ = 0.05.

32

-0.2

-2

Ω1z

z*

Ω0z= -2 Ω0

z= -0.2

0.5 1.0 1.5 2.0

-60

-40

-20

20

40

Figure 2: Equation of state linear in z: ωz(z) = ωz

0 +ωz

1z. Plot of ωz

1-z⋆ for −2 < ωz

0 < −0.2in increments of ωz

0 = 0.2.

33

Ω1z= 2

Ω1z= -2

Ω0z

z*

0.5 1.0 1.5 2.0

-8

-6

-4

-2

2

Figure 3: Equation of state linear in z: ωz(z) = ωz

0 + ωz

1z. Plot of ωz

0-z⋆ for −2 < ωz

1 < 2 inincrements of ωz

1 = 0.5.

34

z* = 0.5

Ω1CPL

Ω0CPL

0.9

-4 -3 -2 -1 1 2

-80

-60

-40

-20

Figure 4: Equation of state linear in a: ωCPL(z) = ωCPL

0 +ωCPL

1z

1+z. Plot of ωCPL

1 -ωCPL

0 for0.5 < z⋆ < 0.9 in increments of z⋆ = 0.05.

35

Ω0CPL= -0.2Ω0

CPL= -2

-0.2

-2

Ω1CPL

z*

0.5 1.0 1.5 2.0

-300

-200

-100

100

200

Figure 5: Equation of state linear in a: ωCPL(z) = ωCPL

0 + ωCPL

1z

1+z. Plot of ωCPL

1 -z⋆ for

−2 < ωCPL

0 < −0.2 in increments of ωCPL

0 = 0.2.

36

Ω1CPL= 2

Ω1CPL= -2

Ω0CPL

z*0.5 1.0 1.5 2.0

-8

-6

-4

-2

Figure 6: Equation of state linear in a: ωCPL(z) = ωCPL

0 + ωCPL

1

z

1+z. Plot of ωCPL

0 -z⋆ for

−2 < ωCPL

1 < 2 in increments of ωCPL

1 = 0.5.

37

As we can see, unexpected degeneracies inarise with these models. In particular, forω0 = −1, z⋆ = 0.74 (the value of z⋆ for theω = −1 model), there are 2 solutions. Dis-covery of evolution will use low redshift data.Analysis has to be made independent of the DEequation of state.

Thank you for your attention.

38