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Curvature Constraints from the Causal Entropic Principle

Brandon Bozek, Andreas Albrecht, and Daniel PhillipsPhysics Department, University of California, Davis.

(Dated: February 6, 2009)

Current cosmological observations indicate a preference for a cosmological constant that is dras-tically smaller than what can be explained by conventional particle physics. The Causal EntropicPrinciple (Bousso, et al.) provides an alternative approach to anthropic attempts to predict ourobserved value of the cosmological constant by calculating the entropy created within a causal di-amond. We have extended this work to use the Causal Entropic Principle to predict the preferredcurvature within the multiverse. We have found that values larger than k = 40m are disfavoredby more than 99.99% and a peak value at = 7.9 10

123 and k = 4.3m for open universes.For universes that allow only positive curvature or both positive and negative curvature, we find acorrelation between curvature and dark energy that leads to an extended region of preferred values.Our universe is found to be disfavored to an extent depending the priors on curvature. We also pro-vide a comparison to previous anthropic constraints on open universes and discuss future directionsfor this work.

PACS numbers:

I. INTRODUCTION

The simplest explanation for the observed accelera-tion of the universe is Einsteins cosmological constant,. However, the value that explains the acceleration ismany orders of magnitude smaller than that expectedfrom quantum field theory. We are then left either todetermine a method to set the cosmological constant toa small value or to consider an environmental variablevarying from place to place in the multiverse.

Following the environmental approach numerous au-thors (many inspired by the pioneering work ofWeinberg[1]), have sought to explain the observed valueof by postulating that the most likely universe to beobserved would be that which contains the largest poten-tial to contain observers. However, such anthropic ap-proaches can become burdened by complicated assump-tions on the nature of observers. In their Causal En-tropic Principle (CEP) Bousso et al. [2] took this rea-soning in a simple and elegant direction by associatingobservers with entropy increase. Initial applications ofthis approach have successfully predicted our value of[2, 3]. The CEP has added appeal because there hasbeen long standing (if not universal[4]) acceptance of theidea that entropy increase would need to be imposed asa condition specific to observers rather than a global andeternal property of the Universe[5, 6, 7, 8]. Specifically,the CEP gives a weight to each set of cosmological pa-rameters proportional to the entropy produced within acausal diamond in the corresponding cosmology. In ad-dition to the original work[2] which found our value of to be within one sigma of the peak of their predictedprobability distribution, the CEP was further developedby Cline et al. [3], exploring constraints on other cosmo-logical values such as density contrast, baryon fraction,matter abundance, and dark matter annihilation rate.

In this paper we develop this method further by usingCEP to jointly predict the values of curvature and

most likely to be observed. Since Cline et al. [3] didnot find significant features in their extended parameterspace, we chose to vary only k and for this work andhold all other parameters fixed. However, given the tailin the probability distribution for positive curvature, aninteresting extension to this work would be to also varyadditional parameters such as the density contrast.

Anthropic constraints on curvature are interestingin their own right, in the context of the flatnessproblem[9, 10] which suggests that in the absence ofsomething like cosmic inflation[10, 11, 12] the most nat-ural realization of big bang cosmology would be highlydominated by curvature. A number of authors have al-ready considered anthropic bounds on curvature[13, 14,15], but we believe this is the first work to apply theCEP to curvature. We find that our results place an up-per limit on the allowed negative curvature as expected,but one that is looser than those works mentioned abovedue to a more lenient tolerance on the sizes of structurethat are allowed to form. Further, we find that our peakprobability for open universes to be away from the upperedge of our probability distribution, not dictated by it,as would be the case in the other work. We also considersolutions that allow for just positive curvature and forboth positive and negative curvature. In these cases wefind a tail in the probability distribution that allows for awide range of allowed and k, a large fraction of whichare significantly larger than our measured values. Wefind that our universe is not ruled out in any scenario,but, depending on ones choice of priors on curvature,disfavored to a certain degree.

In section II we review the CEP and the cosmology wewill be considering. We then review the star formationmodel we will use in section III. We discuss our resultsin section IV and our conclusions in section V.

210126 10124 10122 101200

0.1

0.2

0.3

0.4

0.5

Prob

. Den

sity

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3

3.5

4

4.5

t (Gyr)

V c

0 5 10 15 20 25 300

0.02

0.04

0.06

0.08

0.1

0.12

t (Gyr)

SFR

10122 10120 10118 101160

1

2

3

4

5

6

7

8

Prob

. Den

sity

0 0.2 0.4 0.6 0.8 10

0.005

0.01

0.015

0.02

0.025

0.03

0.035

t (Gyr)

V c

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

1.4

t (Gyr)

SFR

FIG. 1: Top Left Panel: Probability density for with fixed curvature of k = 10m (dashed) and k = 0 (solid). TopCenter Panel: The comoving volume (in units of 1012Mpc3) for = 10

123 (larger) and = 10122 (smaller) are shown for

k = 10m (dashed) and k = 0 (solid). Top Right Panel: The star formation rate in units ofM

Mpc3yr. The upper curve for each

value of curvature represented in solid and dashed respectively is = 10122 and the lower is = 10

123 . The bottom rowis the probability density, comoving volume, and star formation rate for k = 50m. The blue/dashed curve is = 10

119

and the black/solid curve is = 10120 .

II. THE CAUSAL ENTROPIC PRINCIPLE

According to the CEP the probability distribution for is given by the equation:

d2P

ddk= P0 w(, k)

d2p

ddk(1)

where w(, k) is a weighting factor, P is the totalprobability, P0 is a normalization factor, and p is thetotal prior. We will assume the joint prior probabil-ity of p(, k) to be independent giving p(, k) =p() p(k). The prior for is an expression of howthe multiverse is populated by physics with differentvalues of . Here we use the standard form (sometimesmotivated by the string theory landscape) taken by pre-

vious authors[1, 2]: dpd = constant. For simplicity we

also take dpdk = constant, which will enable a discussion

of the flatness problem later in the paper. These flatpriors mean the largest allowed cutoff values of k and set the typical values for the prior. The value of thecutoff turns out to be unimportant because for flat pri-ors w(, k) dictates the shape of the final probabilitydistribution.In the CEP framework we set w(, k) = S, where

S is the total entropy produced within a causal dia-mond. After considering numerous astrophysical sourcesfor entropy production, Bousso et al. [2] find that thedominant form of entropy production is star light reradi-

ated by dust. As in [2], S is given by.

S =

ti

d2S

dVcdtVcdt (2)

where Vc is the total comoving volume of an observerscausal patch. A causal patch is defined by a future lightcone taken at an initial point, such as reheating followinginflation, intersected by a past light cone at a late timepoint, which in the case of a universe dominated by a cos-mological constant is bounded by a de Sitter horizon andin the case of a universe dominated by positive curvaturethe late time event is the crunch.We use the metric:

ds2 = dt2 + a(t)2R20[d2 + Sk()

2d2] (3)

where Sk() = sin() for positive curvature, Sk() =sinh() for negative curvature, and Sk() = for nocurvature. The causal diamond is then given by R0 =2 |

2 + |, where =

dta(t) . The comoving volume

is then:

Vc =

2piR30[12 sin(2)] for k = +1

43 R

30

3 for k = 02piR30[

12 sinh(2) ] for k = -1

(4)

The scale factor a(t) can be found by solving the Fried-mann equation:

H2 =8pi

3(ma3

+ +ka2

) (5)

3where = /8pi, k = 3k/8piR20, and k = {1, 0, 1}

for a negative, flat, and positively curved universe respec-tively. The value of the matter density today (a = 1) isset at m = 5.2 10

124 in Planck units which we usethroughout unless otherwise noted. Following previouswork in the topic we neglect radiation. In this work wehold m fixed and allow the curvature today, k, and to vary.

The other part of Eqn. 2, d2S

dVcdt, is the entropy pro-

duced per comoving volume per time, which is calculatedby the convolution

d2S

dVcdt(t) =

t0

d2S

dMdt(t t)(t

)dt (6)

where d2S

dMdt (t t) is the entropy production rate per

stellar mass at time t due to stars born at an earliertime, t, and (t

) is the star formation rate at t. Theentropy rate per stellar mass is found by calculating

d2S

dMdt(t t) =

1

M

Mmax(tt)0.08M

d2s

dNdtIMF (M)dM

(7)

where IMF (M) is the initial mass function andd2s

dNdtis

the entropy production rate for a single star. The latteris given by the stellar luminosity divided by the effectivetemperature. The number of photons emitted by a staris dominated by the half that are reprocessed by dust atan effective temperature of 20mev. This is given by:

d2s

dNdt=

LTeff

=1

2(M

M)3.5 3.7 1054yr1. (8)

The prefactor in Eqn. 7 is the average initial mass,M = 0.48M. The lower limit of Eqn. 7 is the min-imum mass of a star that can support nuclear burning.Following Bousso, we take the upper limit to be

Mmax(t t) =

{100M for t t

< 105 yr

(1010yrtt )

0.4M for t t > 105 yr

(9)

III. STAR FORMATION

A key aspect to this work is how well the star formationrate is modeled. While Bousso et al. [2] use star forma-tion rates of Nagamine [16] et al. and Hopkins and Bea-com [17] with some simple modifications to extend thesemodels to include different cosmological constant values,we will follow Cline et al. [3] who use a model proposedby Hernquist and Springel (HS) [18]. The HS model wasfound to produce similar results to those found in Boussoet al. but was more straightforward to extrapolate to thecase were a larger number cosmological parameters arevaried. The HS star formation model is given by thisequation:

= ms0q(t)(1 erf(

a

2

c4

)) (10)

k /

m

10126 10124 10122 10120103

102

101

100

101

102

FIG. 2: The 68.27% (dark grey), 95.44% (light grey), 99.73%(inner white), and 99.99% (outer white) contours for a mul-tiverse that only admits negative curvature universes. Thesolid blue line that cuts through the other 4 contours is theanthropic bound from [15] for smaller sized galaxies.

where c = 1.6868, a = 0.707, s0 = 3.79951063 (taken

from [18]), and m (same as above) are constants and 4is the root-mean-square density fluctuation that corre-sponds to the mass scale that virializes at a temperatureof T = 104K.The star formation efficiency, q(t), encompasses the

rate and efficiency of radiative cooling within a collaps-ing object that leads to star formation. HS model thisprocess with:

q(t) = ((t)

((t)m + m)1

m

)p (11)

where (t) = ( HH0 )2/3. = 4.6, m = 6, and p = 2.72

are constants fit from numerical simulations and H0 =

70km/sMpc . For universes with positive curvature that end in

a crunch, the star formation rate of Eqn. 10 continues upuntil the crunch. We therefore needed to place a boundon late time star formation when we no longer trust ourmodel. We set = 0 when r = m in the collapsingphase, where r = 1.5 10

127. This choice allows foran exploration of the CEP properties without a stronglimiting effect put in by hand.Star formation rates for several different values of cur-

vature and are shown in the right panel of Fig. 1.Their corresponding probability curves are shown in theleft panel of Fig. 1.

IV. RESULTS AND DISCUSSION

Using the equations of the previous section we solve forthe probability distribution over a wide range of open andclosed universes. To enable the most general discussionwe consider both open and closed cosmologies togetherand separately. (It is commonly [15] but not universally[19] thought that the string theory landscape only leadsto the open case.)

4FIG. 3: The 68.27% (dark grey), 95.44% (light grey), 99.73%(inner white), and 99.99% (outer white) contours for a multi-verse that only admits positive curvature universes.

k/

m

10125 10120 101155000

4000

3000

2000

1000

0

FIG. 4: The 68.27% (dark grey), 95.44% (light grey), 99.73%(inner white), and 99.99% (outer white) contours for a mul-tiverse that admits both positive and negative curvature uni-verses.

If we were to only consider open universes, then Fig.2 depicts the resulting probability density distribution inlog -log k space. Values of curvature of k > 40mfall outside of the 99.99% CL. Smaller values of andcurvature lead to larger causal diamonds and thereforehave the most total entropy production, as depicted inFig. 1. This balances the majority of vacua havinglarger values of both and curvature, giving a peakvalue at = 7.9 10

123 and k = 4.3m. Usingthe upper bound (95% CL) on negative curvature fromWMAP+HST [20], our universe of = 1.25 10

123

and k = 0.016m is in the 99.73% CL. Fig. 1 illus-trates that for a fixed value of negative curvature thedistribution for remains roughly unchanged from thedistribution for a flat universe.

Now considering only positively curved universes, thereis a clear correlation between and curvature which

comes from competition between positive curvature and. This can cause the universe to loiter[21] in astate with little cosmic expansion but plenty of struc-ture growth. These conditions conspire to create both alarger causal diamond and enhanced linear growth. Thisresults in the tail on the bottom right of Fig. 3 wherethere is ridge between collapsing regions to the left andnon-collapsing regions to the right in the 68% CL. For afixed value of positive curvature, small values of leadto a universe that will recollapse before there is signif-icant star formation. As is increased the recollapseis delayed allowing for more star formation and a largercausal diamond, giving more total entropy produced andtherefore a more likely universe within the CEP frame-work. This continues until is large enough to allow fora non-collapsing universe, at which point larger values of begin to suppress growth. The narrowing of the tailcomes from deviations from the ridge having large energydensities that lead to either a rapid recollapse or early domination. Using the upper bound (95% CL) on pos-itive curvature from WMAP+HST [20], our universe of = 1.25 10

123 and k = 0.06m is in the 99.99%CL.Fig. 4 shows the 2 dimensional probability density dis-

tribution, d2P

d log dk, in log -k space for both positive

and negative curvature. The full span range of curva-ture allowed by WMAP+HST [20] (0.06m k 0.016m) is in the 95.44% CL. A significant fraction ofthe values within the 68.27% CL are positively curveduniverses of both large amounts of curvature and darkenergy compared with our universe due to the compet-ing effects mentioned above leading to a similar tail onthe lower right.A recent paper by Bousso and Leichenauer [22] has

argued that the asymptotic behavior of the star forma-tion model shown in upper right panel of Fig. 1 maybe unphysical. Since the CEP framework depends on anaccurate accounting of star formation in universes far dif-ferent than ours, a careful study of different models is animportant aspect of developing this work further. How-ever, we suspect that the asymptotic behavior leads to asubdominant effect on the final probability distributionssince it coincides with a decreasing comoving volume thatwill diminish the contribution it will make to the totalentropy contribution.On the other hand, the bottom right portion of the tail

in Figs. 3 and 4 is an area where we have little confidencein our star formation model as the duration of matterdomination is increasingly smaller as we move furtherout onto the tip of the distribution. A different cut onlate time star formation from the one we chose above oranother star formation model may find the bottom righttip less favored or ruled out.We have extended the CEP to include curvature and

found that regardless of whether one considers both posi-tive and negative curvature or just one of the two options,a non-zero value of curvature appears to be preferred. Wehave also found that our universe is not ruled out in any

5scenario considered here, but is somewhat disfavored insome scenarios. The favored values for an open universeare just a few orders of magnitude larger than values fa-vored by modern data, so to the extent that the flatnesspuzzle is about why the curvature is not given by thePlanck scale, the CEP seems to put a significant dent inthe flatness puzzle. This is not dissimilar to anthropicarguments of curvature, where structure formation is cutoff by excessive curvature, however the CEP offers a lessrestrictive initial assumption. In Fig. 2 we also plotthe bound on negative curvature calculated by Freivo-gel et al. [15] (which are similar to those of Vilenkinand Winitzki [13] and Garriga et al. [14]) by demandingthat structures at least as large as a small sized galaxyform. Our plot allows for somewhat more curvature thanis allowed by these methods. Setting a structure forma-tion limit based on smaller galactic masses brings thecurvature limit closer to ours. Ultimately our rough re-production of the anthropic cutoff is unsurprising as ourmain entropy source, star formation, cuts off along withstructure formation. However, our actual prediction forcurvature is not against a cutoff for structure formationas would be the case for a simple bound. The causal en-tropic weighting provides additional rewards for smallercurvatures in the form of increased star formation andentropy production.

V. CONCLUSIONS

Anthropic constraints on observable parameters areinteresting when considering implications for the multi-

verse. The CEP has appealing advantages over previousanthropic attempts. We find that the CEP places upperlimits on the amount of curvature that is observable andwhile not ruling out our universe, the CEP finds largercurvature preferable to our measured value. We also findan intriguing feature in the probability space for posi-tive curvature of an elongated tail stretching into regionsof large curvature. Our results for negatively curveduniverses are broadly consistent with previous anthropicbounds on curvature but less constraining due to a morelenient tolerance for the minimum mass of structure al-lowed. Still, like the previous work we find that anthropicconsiderations seem to offer cosmology considerable relieffrom the flatness problem.

Acknowledgments

We thank Lloyd Knox, Damien Martin, and espe-cially James Cline for very helpful discussions. We alsothank Tony Tyson for computing resources as well asPerry Gee and Jim Bosch for technical computing sup-port. This work was supported by DOE grant DE-FG03-91ER40674.

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