Effects of Inhomogeneity on the Causal Entropic prediction of Lambda

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by  Daniel Phillips

Effects of Inhomogeneity on the Causal Entropic prediction of Λ
Daniel Phillips and Andreas Albrecht
Department of Physics, UC Davis
(Dated: March 9, 2009)
The Causal Entropic Principle aims to predict the unexpectedly small value of the cosmological
constant Λ using a weighting by entropy increase on causal diamonds. The original work assumed
a purely isotropic and homogeneous cosmology. But even the level of inhomogeneity observed in
our universe forces reconsideration of certain arguments about entropy production. In particular,
we must consider an ensemble of causal diamonds associated with one cosmology, and we can no
longer immediately discard entropy production in thefar futureof theuniverse. Dependingon our
choices fora probability measure andourtreatment of black holeevaporation, theprediction for Λ
may be left intact or dramatically altered.
9
0
I. INTRODUCTION may interpret this as an assumption that the number of
0
observers is proportional to the entropy increase within
2
Abroadlineofargumentintendedtoresolveoramelio- the causal diamond, but in the spirit of [1] we may sim-
r
a rate the notorious problem of the apparent smallness of plytakethisweightasahypothesisandremarkthat∆S
M the cosmologicalconstant (ρ ≈1.25×10 123 in Planck has severaladvantages over some other weightings: 1) It
Λ −
is hearteningly generic, allowing at least the theoretical
units)istorejectthenotionofafundamentalvalueforΛ
9 possibilityofapplicationtouniverseswithmuchdifferent
altogether. In this approach,well-knownfrom the string
low energy physics from ours. 2) It seems less contrived
theory landscape as well as other “multiverse” notions,
] than typical “anthropic”reasoning;though we may con-
c the problem is transformed to the search for a selection
q principle that may explain why a value as small as ob- template observers in a universe with no galaxies, it is
- difficult to imagine them without significant entropy in-
r served is probable. In order to make this formulation
g two broad decisions must be made, both of which can crease. 3)As shownin [1], it can actually reproduceand
[ improve upon previous anthropic results.
be controversial: the choice of selection principle, and

1 the probability measure. The first tends to be contro- Even after accepting the program to calculate likeli-
v versial because a choice of selection principle is a choice hoodsofphysicalparametersfromsomeaprioritheoret-
2 abouthowtocategorizeourimaginedexperimentalsam- ical distribution and after fixing a probability measure,
2 ple of universes in which measurements occur, and thus a full calculation of the probability distribution for Λ
6 leads to difficult questions about observers. The choice is a formidable task. In an ideal case we would have
1
ofprobabilitymeasurehasitsownwell-knowndifficulties a background theory giving us some set of cosmological
.
3 relating to defining probabilities across different infinite parameters and their prior distribution. We would then
0 spaces. Different choices for either selection principle or allowallparametersto varyandmakea predictionforΛ
9 probabilitymeasurecanleadtowildlydifferentprobabil- by marginalizing over the other parameters,in a scheme
0 ity predictions, easily changinga predictionof likelihood such as that in [3]. As a first step, Bousso et al. [1] fol-
:
v to an exponentially disfavoredone. low the usual simplification of holding all other physical
Xi Boussoet al. [1]suggestedanovelcombinedapproach, parametersfixedwhile modifying only the positive value
of Λ in a flat FRW universe. Other work has discussed
theso-called“CausalEntropicPrinciple”(CEP).Forflat
r
a universes with a positive fundamental cosmological con- aspects of the CEP for Λ≤0 [4, 5], but in this work we
stant, one can define the causal diamond for a particu- keepthe same Λ>0 assumption used in the first papers
on this subject.
lar world line λ(τ) as the intersection of interiors of the
future cone at earliest times and the past cone at late A common drawbackof varying only Λ in such an ap-
times1. Theresultingregionisfiniteincomovingvolume proach is the possibility that variation in other param-
inthisflatpositive-lambdaFRWuniverse,anddiamond- eters could significantly affect the prediction for Λ it-
shapedwhendrawnincomovingcoordinatesandconfor- self. TheclassicinstanceisthatWeinberg’spredictionof
maltime. (SeeFig. 1). Ifwerestrictourprobabilitymea- ρΛ <10−121[6]undertheselectionprinciplethatgalaxies
sure to the finite interior of this diamond, we can avoid must form is softened by allowing the density contrast
thedifficultyindefiningaprobabilitymeasureoninfinite Q or the baryon-to-photon ratio to increase from that
spaces. Moreover, the proposed selection principle is a observed in our universe. Greater early anisotropies or
simple weighting proportionalto the entropy production matterdensitiesandreducedradiationpressurecouldal-
∆S occurring within the causal diamond. Loosely one low structure to form earlier and thus significantly push
up the allowable value of Λ [7, 8]. Cline et al. [9] have
shown that the entropic approach, at least for Λ, is re-
silientwhen varyingQ. More recentlyit has been shown
1 The CEP has been recently extended to curved FRW universes that allowingthe curvatureof the universeto varyalong
in[2] with Λ can dramatically change the CEP predictions for
2
Λ,dependingonexactlywhatpriorsonetakeonthecos- II. THE CAUSAL ENTROPIC PREDICTION
mic curvature[2]. Other authors have suggested poten- FOR Λ
tiallimitationsoftheCEPalongwithrelatedapproaches
[4, 10].
Simply stated, the CEP [1] assumes that the proba-
bilistic weighting for cosmological parameters is propor-
Thispaperexaminesadifferentsimplificationthathas
tional to the increase in entropy ∆S within a causal di-
been made so far in all work on the CEP: that of an
amond associated with that cosmology. Given a mul-
isotropic, homogeneous universe. Our own universe’s
tiverse populated with different cosmologies, the CEP
small primordial fluctuations allow us to make these ap-
thusbecomesatooltocalculateprobabilitydistributions
proximations to great effect for the overall evolution of
for measurements of the cosmological parameters them-
theuniverse. Butastimeprogressesweknowthatstruc-
selves. Although in principle one could ask the CEP to
ture formation proceeds apace and entropy production,
thus give predictions for a full range of cosmological pa-
if it is associated with structure, becomes less spatially
rameters,followingBoussoet al. wewillleaveallcosmo-
homogeneous. Because the causal entropic approach ex-
logical parameters fixed at their observed values except
amines only the causal region surrounding a particular
the cosmologicalconstant Λ.
worldline,wemusttrytoformulatehowdeparturesfrom
Thecausaldiamondisdefinedasthevolumecontained
homogeneitymayaffecttheentropicweight,andwhether
within the future cone of an early event (taken to be re-
those variations can affect the prediction for Λ. (In the
heating following inflation) as well as within the past
processwe estimate entropyproductionwellinto the era
cone of a late event on the same world line. The causal
of cosmological constant domination, but we note that
diamondisthus the regionofspaceinfullcausalcontact
our approach is not directly related to the arguments in
with a particular world line. Following the original ar-
[4] about the cosmologicalheat death of observers.)
gument we will also restrict ourselves to purely positive
In section II we briefly review the CEP method. In Λ, so that all cosmologies will eventually be dominated
section III we discuss the the resulting prediction for Λ by the cosmological constant. In every case a de Sitter
andcommentontheincreasinginhomogeneityofentropy horizon will thus form and define the past light cone for
producedatlatetimes,illustratedbyblackholeevapora- the causal diamond.
tion. In sectionIV we describe the necessity of replacing The CEP choice to restrict entropy increase to that
a single causal diamond with an ensemble given an in- within a causal diamond originated from a holography
homogeneous cosmology. Section V discusses the nature argument: the universe simply does not consist of a re-
of long-term entropy sources that might compete with gion larger than a single causal diamond. We will not
stellar entropy production for causal diamonds contain- try and argue the pros and cons of this point here, but
ing collapsed structures. In section VI we discuss effects simply take this restriction as one of our input assump-
on the predicted probability distribution for ρ , and in tions. Thereishoweveranimportantextrastepwhichwe
Λ
section VII we summarize our conclusions. Throughout will talk about in greater detail. If an entire cosmology
we use Planck units with h¯ =c=G=1. is represented only by a single causal diamond, we need
some way to choose this causaldiamond, or equivalently
defineaparticularworldlineassociatedwithaparticular
setofcosmologicalparameters. Thereisnodifficulty do-
ing so in a homogeneous,isotropicuniverse,as allcausal
diamonds are identical. Such is clearly not the case for
an inhomogeneous universe, and we will thus introduce
the statistical notion of an ensemble of causal diamonds
associatedwith aparticularcosmology. Itshouldbe em-
phasized that this complication is required even with a
very strictly holographic interpretation of the causal di-
amond.
Black holes immediately come to mind in calculations
ofcosmologicalentropy. Theentropyassociatedwiththe
formationofablackholehorizonisexplicitlyexcludedin
the CEP, as is de Sitter horizon entropy. This exclusion
isimportantasasinglesupermassive(107M )blackhole
canhave anentropy of1091[11], exceeding a⊙ll other non-
FIG.1: Acausaldiamond(depictedschematicallyhere)isthe
regionwhichcancausallyimpactandbecausallyimpactedby horizon entropy sources. One might object, as noted in
aworldlineλ(τ). Thefiniteentropyproducedintheresulting [1],thatthisentropycannotbehiddenforever,asonvery
spacetimevolumeisusedintheCausal EntropicPrincipleas long time scales the black hole will evaporate and return
a cosmological weighting factor its entropy to the rest of the causal diamond. Bousso et
al. [1] argued that a typical late-time causal diamond is
empty and thus we may discount this entropy. We will
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revisit this issue below. During the current cosmological era for cosmologies
Itisalsoimportanttonotethattheweightingw(ρ )∝ similar to ours, calculations in [1] revealed stars to be
Λ
∆S includes only entropy increase occurring within the the greatest contributor to ∆S due to photons absorbed
causal diamond. Therefore various processes which one andreemitted by cooldust. This largecontributionmay
might imagine to be strong contributors to entropy in- beseenfromestimatingentropyincreaseforaprocessby
creaseturn outnot to be significant. For example,CMB ∆S = ∆E where ∆E is the energy released and T is the
T
photonsrepresentalargeamountofcurrententropy,but typicaltemperature (k =1). Forstars,typicalenergies
B
not of entropy increase within the causal diamond sur- released in fusion are about 7 MeV/nucleon. While typ-
rounding our world line. The causal diamond at recom- ical stars produce visible light with an effective T ∼eV,
binationenclosedamuchsmalleramountofmatter(and perhapshalfofthephotonsareabsorbedandreemmitted
photons)thanaHubbleradiustodaydoes,somostCMB bycooldustwithaT ∼20meV. Itisthecombinationof
photons within our horizon must have entered through highenergyper nucleon,ubiquity ofstellarburning,and
the bottom cone of the causal diamond; these photons the low effective temperature of muchof the reprocessed
do not contribute to ∆S. Other events in the early uni- starlight which gives stellar entropy the edge over other
verse suchas nucleosynthesis likewise contribute little to processes.
this measure of ∆S owing to the small size of the causal
diamond. So Bousso et al. [1] restricted themselves to
processesactiveduring the eraofrelativelylargecomov- III. COMOVING VOLUME OF UNIVERSE
ing scale for the causal diamond. One of the purposes
of this paper is to examine whether the very long times Stellar entropy production per comoving volume
available for entropy production in the future of a Λ- reaches a maximum of 2.7 × 1063/Mpc3/yr comoving,
dominateduniversecancompensateforthesmallvolume as shown in Fig. 2. The causal diamond gets as large as
of matter in causal contact with an observer once a de ∼1013Mpc3. With∼1010 yearsthatgivesanintegrated
Sitter horizon forms. stellar entropy production ∆S ≈1086 reachedby about
0
Varying the cosmological constant directly affects the 10 billion years.
size of the causal diamond, with the comoving 4-volume Under the CEP with stars as the major source of en-
containedproportionaltoΛ−1. Thereforeevenbeforeac- tropy production, one obtains a weighting and hence a
counting for the effects of entropy production, the CEP predicted probability distribution for ρ . With several
Λ
rewards smaller values of Λ with greater weight, owing different star formation models [1, 9] the predicted 1-σ
to their larger causaldiamonds, at least measuredin co- probability band of roughly 10−124 <∼ρΛ <∼10−122 easily
moving volume. If we found entropy production to be contains our universe’s observed value.
dominated by a process producing a constant entropy
rate per comoving volume, such a process would trans- 73 Total ∆S per comoving volume
x 10
late an a priori flat distribution of ρ 5
Λ
dP
=const 4
dρ
Λ
3
into a flat distribution in log(ρ )
Λ
S
∆
dP 2
dρ ∝w(ρΛ)=ρ−Λ1
Λ
1
dP
∝w(ρ )ρ =const 0
Λ Λ
dlog(ρ ) 7 8 9 10 11 12 13
Λ
log(t) (years)
The reduction from a flat distribution to one flat in log
space is an indication of how much work the causal dia- FIG. 2: Integrated stellar entropy production per comoving
mond portion of the CEP is doing on its own. For real- Mpc3, calculated using the Nagamine et al. star formation
istic entropy sources, the total entropy production (and model [12] considered in [1]. The long tail is produced by
thus probabilistic weight) was calculated via low-mass white dwarfs with lifetimes up to 1013 years, but
by 1010 years we have already seen a large fraction of stellar
w(ρ )∝∆S(ρ )= ∞dtV (ρ ,t)∂2S(ρΛ,t) entropyproduction.
Λ Λ Z c Λ ∂V ∂t
0 c
A comoving volume of a particular scale contains a
Here V is the directly calculable comoving 3-volume of fixedamountofmattersolongastheuniverseishomoge-
c
the causaldiamondas a functionof Λ andt and ∂S˙/∂V neousoverthe scaleof consideration. Buta flatuniverse
c
is the entropy production rate per comoving volume. with a cosmologicalconstantwill forma horizonof fixed
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tropy produced by stars in the first 1010 years of cosmic
evolution. The time scale is enormous: log (τ ) =
10 BH
0.4 83 + 3log [M /106M ], or about 1086 years in this
r Vc case, but1w0e caBnHnot igno⊙re the situation out of hand as
e
p 0.2 any worldline which tracks matter has a high chance of
R ultimately ending up near (or even in) a black hole.
F
S
0
7 8 9 10 11 12 13
Vc
r x 1063 IV. WHY WE CARE ABOUT PARTICULAR
pe 4 WORLD LINES
n
o
ucti 2 In the the formulation of Bousso et al. [1] the uni-
d
o verseis consideredto be exactly FRW homogeneousand
r
p 0 isotropic, which makes a distinction between comoving
nt. 7 8 9 10 11 12 13 volumeandmassunnecessary. Indeedoverthesizeofthe
E 12
x 10 causal diamond this assumption is quite accurate at the
10
beginning of our universe and well through the current
time, as the universe is homogeneous well below scales
5
c approachingthe Hubble length or the current size of the
V
causal diamond. As mentioned above it is in the future
0 that differences among causal diamonds may arise.
7 8 9 10 11 12 13
Because of the assumption of homogeneity in [1], a
log(t) (years) particular set of cosmological parameters resulted in a
unique, representative causal diamond. Thus the proba-
FIG. 3: A peak in star formation (top plot) is followed by bility is given by
a peak in entropy production (middle plot) per comoving
volume. In our universe the peak in comoving 3-volume of dP dp dN
∝w(ρ )
the causal diamond (bottom plot) is near the time of max- dρ Λ dN dρ
Λ Λ
imal stellar entropy production per Vc. The 3-volume Vc(t)
of the causal diamond is determined by Λ; a much earlier where dN representsthedensityofvacuapervalueofΛ.
peak (larger Λ) would not allow the diamond to capture as dρΛ
much entropy production. Universes with smaller Λ would We may take dN to be flat if the landscape has values
dρΛ
give larger causal diamonds in late times, but would capture spaced tightly in the region of interest, and if 0 is not a
little more stellar entropy production, and are less likely ow- special value. With these assumptions, the spacings of
ing to our flat prior. All plots are assuming a homogeneous vacua can be assumed to be uniform for Λ near 10 123.
−
universe. The quantity dp is the term representing the theory’s
dN
prior probability for Λ. Following previous work, we as-
sume prior probability is flat; in other words, the back-
physical size. Eventually the the comoving radius corre-
ground theory is indifferent to vacua, choosing among
sponding to the horizon length will drop below the scale
them with equal probability.
of matter inhomogeneity. In physical terms, for a world
Critically, in [1], the weighting w(ρ ) is the weight of
line near a gravitationally collapsed halo, the amount of Λ
asinglerepresentativecausaldiamondwithcosmological
mass enclosed by a causal diamond will eventually ap-
constant density ρ . If a particular set of cosmological
proximate a constant value, rather than exponentially Λ
parameters does not yield a single causal diamond, we
emptying out. Comoving coordinates are no longer a
mustreplaceoursinglecalculationofw(ρ )withaprob-
particularly good choice within a collapsed halo. Λ
ability distribution
In our universe a large halo might have mass 1015M .
Today’s ρ ≈ 3.3×1010 M⊙ gives a corresponding c⊙o-
m Mpc3
w(ρ )= w(ρ ,λ)dλ
moving volume of about 3×104Mpc3, which is nearly Λ Z Λ
λ
a factor of a billion smaller than the maximum comov-
ing size. Any late-time entropy source must therefore where the integral over λ is one over all possible world
compensate for effectively having a causal diamond 3- lines (and hence causal diamonds) given a particular set
volume approximately 10 9 of that during peak stel- of cosmologicalparameters from our backgroundtheory.
−
lar entropy production. Whether this is possible de- This discussion may seem counter to the spirit of the
pends upon details: the lengths of time available and causal diamond approach in [1]. Yet unless our back-
the scale of entropy production. The most dramatic ex- ground theory is itself phrased in terms of causal di-
ample would be the inclusion of Hawking radiation from amonds, we cannot skip smoothly from a distribution
a black hole. Release of a 107M black hole’s 1091 en- of cosmological parameters to a distribution of results
tropyasHawkingradiationwould⊙completelyswampen- for causal diamonds. Our prior distribution of Λ or the
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spacing of vacua is phrased in terms of cosmological pa- world lines piercing each spacelike surface with constant
rameters, not particular world lines. Assuming perfect cosmic time. In the homogeneous limit for comoving co-
homogeneity simply means taking the weight function ordinatesthis gridsimply remainsfixedintime, yielding
w(ρ ,λ)tobeproportionaltoadeltafunctionpeakedat a fixed world line density, and corresponding to the sim-
Λ
a particular world line λ that is “typical” of a perfect ple choice made in Bousso et al. [1]
0
FRW universe. Given the tremendous variety of world Things are more complicated as the universe becomes
lines for any structure-forming cosmology, this assump- lesshomogeneous. A startingpointis toimagine placing
tionseemsunrealistic: anextremecounterexamplewould test particles in a fixed, constant spatial density at an
be a world line that runs directly into a black hole hori- early cosmic time, and watching the particles trace out
zon at an early era. Nonetheless it remains to be seen geodesicsastheuniverseevolves. Ofcourse,ouruniverse
whether consideringanensembleofworldlinesfora cos- seems to have performed this very experiment, and as Λ
mology rather than a single one makes a difference in dominates we have a picture of most matter eventually
predictions for ρ . residingwithinisolatedgravitationallyboundhalos,with
Λ
In order to calculate the entropy production probabil- exponentiallyemptyingspaceinbetween. Inthispicture,
itydistributionoveranensembleofworldlinesλ,weneed atlate times the spatialdensity distributionofgeodesics
to describe how the density of a bundle of world lines parallelsthatofmatteritself,soatleastroughly,aproba-
behaves over time relative to the coordinates in which bilitydistributionforentropyproductionoverworldlines
we wish to measure entropy production. We argue that would be equivalent to integration over the matter dis-
for an inhomogeneous universe there are multiple ways tribution.
to parametrize these world lines and that the choice of There are other choices that yield dramatically differ-
parametrization directly affects the results of CEP cal- ent answers, however. If we choose a slice at late cosmic
culation. timeandparametrizeworldlinestohaveaconstantden-
It should be noted that even in the case of a per- sity in physical coordinates, the vast majority of world
fectly homogeneous FRW universe not all world lines linesatlatetimeswillbelocatedinnearlyemptyregions
(and hence causal diamonds) are created identically, as withalmostnoentropyproduction. Whenwetraceback
one could imagine arbitrary boosts or even accelerated theworldlinestothebeginningoftheuniverse,theywill
paths relative to a comoving observer. Even with mod- notbehomogeneouslydistributedrelativetomatter,but
est boosts, observers on these paths would have a dif- for the purposes of calculating entropy increase at early
ferent experience of the universe owing for example to times there is no significant difference since the entropy
a strong CMB dipole. Since the group of boosts is not production itself is homogeneous in space.
compact, one might expect a “typical” boost to be ar- On the other hand, with this second choice, any en-
bitrarily far from the comoving rest frame, with corre- tropyproductionatlate times will be exponentiallysup-
spondingly anisotropic physics. Given a homogeneous, pressedby the rarity of worldlines that are located near
isotropicuniverse,thepreservationofsymmetryafforded matter,andsogiventhischoiceitisjustifiabletodiscard
by the choice of a comoving observer seems an enticing late-time entropy sources. It is important to observe,
motivation for picking a comoving causal diamond. But however, that the second choice seems at best no better
it must be emphasized that this is indeed a choice, and motivatedthanthefirst,andindeedthatonecouldimag-
any appeal that comovingcoordinates are natural in the ine many other intermediate choices for parameterizing
sense that they follow typical matter distributions (and worldlines. Forthe remainderofthe paperwe treatthis
perhapsthusobservers)hasimplicationsfortheinhomo- choiceasanopenquestion,andwillestimatetheeffectof
geneous case. late-time entropy production where it seems to matter:
that is, under the assumption that typical world lines
When we move to an inhomogeneousuniverse we can-
follow matter distribution from an early time. Therefore
notevenappeal to a notionofpreservingsymmetry. For
webeginbyaskingwhatastrophysicalprocessesmaypro-
the purposes of simplification we will leave out acceler-
duce substantial entropy well into the future.
atedworldlinesanddescribeourcollectionofworldlines
as a congruence of timelike geodesics, with each space-
time point lying on a single geodesic. One can construct
V. LONG-TERM ENTROPY PRODUCTION
such a congruence by specifying a spacelike slice and
examining geodesics orthogonal to this slice. Different
slices, however, typically result in different inherited pa- A. Black holes
rameterizationsfor the worldlines. We will describe two
suchchoicesinwhatfollows,butthereareofcoursemany Black holes contain much more entropy than all other
others. astrophysical sources. In [1], black hole horizon entropy
Foraslicepickedataconstantcosmictimeinthevery as well as that associated with the formation of a de
homogeneousearlystagesofauniverselikeours,there is Sitter horizon were explicitly excluded from the tally of
a natural parametrization: our entropy production can entropy increase. Maor et al. [10] raise the possibility
be measured on a per-mass or, equivalently, comoving thatgravitonsproducedduring blackholemergerscould
coordinate basis, and so we can simply imagine a gridof by themselves exceed stellar entropy increase. But even
6
if one does not count a significant early-time increase in a process with a combination of high energy released,
entropy from black holes, on the very long time scale loweffectivetemperature,andnearuniversaloccurrence.
of black hole evaporation, this entropy increase can no One possibility is annihilations of dark matter. Dark
longer be avoided. Hawking radiation returns entropy matter masses perhaps 6 times baryonic matter, so the
to the matter sector, and it will typically dominate the totalavailable energyis ≈ 6 GeV per baryon. If WIMPs
early-timestellarentropyproductionasestimatedinsec- have weak scale masses, the typical handful of annihi-
tion (III). lation products by themselves cannot produce anywhere
near enough entropy. So the interesting case is if the
annihilation happens in a low-temperature context so
B. Stellar entropy that many low-energy products (typically photons) can
be produced by a single annihilation.
Adams et al. [13] explore WIMP capture by white
Low-mass white dwarfs may continue burning for as
dwarfs. Over the long term white dwarfs make up the
longas1013years. Moreover,eventhoughstarformation
bulk of collapsed stellar objects, and they have densities
is already dropping dramatically in our universe due to
greatenoughto capturemassive WIMPs overtime. Due
depletionofcoolgas,somesmallbutfinitestarformation
toDMannihilationsthedwarfshaveaveryextendedpe-
ratewilllikelyexistfarintothefutureowingtocollisions
riod of low luminosity and low temperature. Adams et
amongsub-stellarmassesandwhitedwarfs. Further,one
al. give typical T ≈ 63 K, or about 5 meV for DM an-
might wonder about the time behavior of star formation
nihilations, which with 6 GeV/baryon energy gives only
in universes with very different values of Λ.
1012 entropy from annihilating all DM.
Canstars in a collapsedregionfar into the future ever
exceedthe1086entropyproducedbythestarsinthefirst
1010 years? We can calculate an upper bound by simply
D. Proton decay
imagining all baryons within a halo are converted into
stars and burned. Consider a massive halo (1015M ).
Baryons make up about 1/6 of the matter content,⊙or One can also ask about proton decay within white
1o.r6p×er1h0a1p4sM1⊙071=h2y×d1r0o3−g×2e71nk0g4a4/kbtagormyosn. ≈ 1.5×1071 baryons, mddweacataeryflyssur(8ecl8he%aasseodfp.fi→Tnyapel+iscta+ellllyπa0ra,bm1ouaGstse1)V./3FpioesrrlonasuttcytloepoincneaulistGruiUnltoTis-
Each instance of fusion releases about 7 MeV per whichfreelystreamoutofevenwhite dwarfsratherthan
baryon. At a temperature of 20 meV for dust- thermalizing. Thus we need a temperature of T ≈ 10 6
−
reprocessing, that is about 3 × 108 entropy per dust- ev or about 10 2 K. For proton decay in white dwarfs,
−
processed baryon. Even if over very long times 100% T ≈ .06 K with proton decay lifetime Γ = 1037 years.
of baryons are burned to hydrogen, and half are repro- Using the same bounds on proton decay as Adams et
cessed by dust (an overestimate as dust is depleted over al.[13], 32 < logΓ < 41, but since T4 ∝ Γe Γt, we can
−
time), that allows only ≈1079 entropy, 7 orders of mag- only push that temperature down another order of mag-
nitudelessthanisproducedbystellarentropy∆S0 upto nitude with the simplest proton decay models. But a
1010 years. It would seem that for the observed cosmo- protondecay mechanismoriginatingfrom a higher order
logical parameters future stellar entropy production can operator could produce much longer lifetimes and cor-
not compete with that in the past. respondingly lower temperatures, perhaps allowing this
Varyingthe cosmologicalconstantaffects the estimate process to compete with early stellar evolution.
in two ways: increasing Λ leads to earlier vacuum domi-
nation and a smaller value of ∆S . However, it simulta-
0
neously leads to a smaller typical halo size as discussed E. Dynamical effects
later. Eventually large Λ will lead to a severe drop in
star formation rates at both early and later times. Sim- Given the approximations involved, either proton de-
ilarly, small values of Λ will push vacuum domination cay or WIMP annihilation might be considered reason-
later and later, eventually leaving less stellar entropy to ablecompetitorstostellarentropyproduction∆S inthe
0
be produced inthe vacuum-dominatedera. Thus it does matter-dominated era. In order to calculate the maxi-
not appear that stellar entropy in late eras is a strong mum entropy for each we have simply given each pro-
competitorto∆S0,evenwhenthecosmologicalconstant cess a maximal value assuming complete conversion of
is varied. a certain large halo. But halo masses themselves may
not be stable on the time scales considered (τ ≈ 1024
years for WIMP annihilation and ≈ 1037 years for pro-
C. Dark Matter annihilation ton decay). There are two competing dynamic processes
within halos over the very long term ([13]). Interactions
To compete with∆S weneedapproximately1015 en- between starslead to dynamic relaxationandejection of
0 individual stars on a time scale of τ ≈ 100τ ≈
tropy per baryon. With the possible exception of Hawk- evap relax
ing radiation, this appears to be a tall order. We need 100Rv 12lnN(N2) ≈1019−1020 years for typical galactic ra-
7
dius R, random velocity v, and number of stars N. At tle different except that they generally avoid the partic-
the same time, gravitational radiation should cause or- ular dynamics that over time seem to eject matter from
bits to decay and eventually drop matter into a central the halo or collapse it into black holes. Such an ap-
black hole, on a time scale of ≈1024 years. Adams et al. proachmightseemartificial,butitwouldstillbejustone
estimate perhaps 1-10% of matter remains bound to the of many parameterizations, and would immediately re-
central black hole while the remainder is lost from the storetoimportanceHawkingentropyorthatfromcertain
galaxy. models of proton decay. Subtle changes in parametriza-
Matter ejected from the gravitational bounds of a tion clearly can yield very different ideas of what consti-
galaxywillingeneralbelostfromthedeSitterhorizonas tutesatypicalworldline,andwenowmaketheargument
well. Taking the point of view of a world line following that if there is a significant late-time entropy source in
an example white dwarf ejected this way, within a few our parametrization, it can drastically affect the CEP
Hubble times the former host halo will have redshifted prediction for a cosmologicalparameter.
beyondthehorizonandtheonlycontinuingsourceofen-
tropy increase within an observer’s horizon and causal
diamond would be that produced from the single white VI. EFFECTS ON PREDICTION FOR ρΛ
dwarfstar. Eventhecompleteprotondecayofsuchastar
wouldproduceacompletelynegligibleamountofentropy All of our late-time effects are at approximately fixed
compared to ∆S given the small matter content within mass within a horizon. Assuming we are examining an
0
the horizon. On the other hand, we may still wonder astrophysical process which is independent of halo scale
about a single large black hole ejected in this fashion, (whichiscertainlytrueforprotondecayitself,butshould
since Hawkingradiationoverextremely longtimes could be considered a simplification for black holes and white
compete with early entropy. dwarfprocesses,since largerhalosmayhavedifferentas-
If this picture of dynamical effects is correct, for a trophysics), the only determinant of entropy production
world line near the leftover central black hole in a halo, isthemassofthehalo. Thuswecanframeourlong-term
of the processes considered again it is only Hawking ra- entropy weight as a function of the mass of the halo,
diation that could compete with ∆S0, as on time scales w(ρΛ,λ) → w(ρΛ,Mhalo). If we assume matter fairly
much shorter than proton decay, essentially all matter tracesworldlines(i.e.,splitupphasespaceevenlyatearly
will have either been ejected from the halo or have al- times),thenwecantakeadvantageofthePress-Schechter
ready collapsedinto the centralblack hole. WIMP anni- mass function[14] to estimate the probability for each
hilation within white dwarfs has a time scale of ≈ 1025 worldline(andthuscausaldiamond)tobe withinahalo
years, so the story is relatively similar: rather than con- of mass M and hence to have a weight w(ρΛ,M). If
tribute appreciably to entropy gain within a single halo, we are calculating total entropy, to a first approxima-
this process will take place mostly within isolated white tion the weight for a worldline within a mass M halo is
dwarfs. proportional to M in the scale invariant approximation.
Dynamical effects may also have important implica- If f(M,t)∝ exp(− δc ) is the P-S differential mass
√2σ(M,t)
tions for counting entropy from Hawking radiation. We function, then the integrated weight function
intentionally made the choice to parametrize world lines
so that they essentially followedtypical paths of matter. w(ρ )≡ w(ρ ,λ)dλ
We claimed that this choice was in essence arbitrary, if Λ Z Λ
straightforward. But having made this choice, we must
accept that the long-term dynamical behavior of matter
would also be that for a typical world line. Our picture = MP(M)w(ρ ,M)
Z Λ
has two basic fates for matter on time-scales well before
blackholeevaporation: eitheritiswithinablackhole,or
insomesmallchunkofmatterwithnoblackholeswithin
the de Sitter horizon. = Mf(M,∞)dM
Z
With our parametrization choice it then seems that it
is quite common for world lines themselves to intersect where f(M,∞) is the P-S mass function evaluated at a
black holes, but that it is not common for world lines to late time. In a Λ-dominated universe, these mass frac-
staywithinaHubbleradiusofablackholelongenoughto tions approach a fixed value on the timescale of t ≈
Λ
observeblackholeevaporation. Classicalworldlinesmay 16.7 billion years for our universe, as the cosmological
end at the singularity of a black hole, but our approach constant freezes structure formation[3]. The final halo
ofignoringhorizonentropyis not nearlyso obviousonce mass fraction and differential P-S function are given in
thehorizonitselfiscrossed. InthispicturetheCEPmay Fig. 4.
be safe from the need to count Hawking radiation, but Thus the weighting for this kind of constant late-time
the details are far from immediate. entropy source turns out to be proportional to the mass
Itshouldalsobementionedthatwemighttrytotweak of the halo a typical piece of matter finds itself within.
our matter-following geodesic parametrization to be lit- One expects this typical halo mass to decrease for larger
8
Final halo mass fraction F(M) ρΛ in logarithmic space:
1
F(M)0.5 10 ρΛρΛ 0.1 ρΛ0.01 ρΛ dlodgP(ρΛ) ∝w(ρΛ)ρΛ ∝const
100 ρ
Λ Inthecaseoflate-timeentropyitisnottheactualsize
0 ofthe causaldiamondvolumebutratherthemassofthe
0 5 10 15 20
typical halo that controls the entropic weight. Smaller
15 cosmologicalconstants result in larger structure, so that
the mass-weighted“typical”halo(andhence the weight)
F/dmu) 105 scaTlehseaopvperroaxllimwaetigelhytiansgρf−Λo1r.a cosmology will be ∆S +
d 0
0( 0 ∆S . Wherethe latetimeentropyincreasedominates,
log1 −5 100 ρ1Λ0 ρρΛΛ0.1 ρ0Λ.01 ρΛ thelflataetpriorin ρΛ is thus transformedto aflatdistribu-
−10 tion for log(ρΛ). While this result addresses the cosmo-
0 5 10 15 20
log10(M /M ) logicalconstantproblemtosomedegree(asweneedonly
halo sun
explain the smallness of log Λ), compared with earlier
work we have lost the peak in the probability distribu-
FIG. 4: Top: Fraction of total mass at late times in halos
tionassociatedwithapredictionoftheactualvalueofΛ.
smallerthanmassM,plottedforarangeofcosmological con-
Unless we can definitively rule out significant late-time
stants relative to the observed value ρΛ. Bottom: the late-
entropy sources, such a result would undermine some of
timedifferentialPress-Schechterhalofractionf(M).Notethat
asΛincreases,thehalofractionshiftsapproximatelyinversely thesuccessoftheCEP.NonethelesstheCEPstillbenefits
to smaller masses. to an extentfrom the suppressionof structure formation
for large values of ρ , which shows up in the weighting
Λ
2 dropping below the ρ−Λ1 form (as can be seen in Fig. 5).
0
VII. CONCLUSIONS
−2
))Λ Standard treatments of the Causal Entropic principle
ρ
w( −4 considera one-to-onemapping betweencosmologicalpa-
og( ∝ ρ−1 rameters and causal diamonds. The inhomogeneity of
l Λ
−6 a realistic universe introduces additional complexity be-
cause different observers can experience very different
−8 causal diamonds, even with the same cosmological pa-
rameters. One must have some method of picking a typ-
−10 ical causal diamond, or of characterizing an ensemble of
−126 −125 −124 −123 −122 −121 −120 −119
log(ρΛ) causal diamonds for a given cosmology. We have shown
that with one reasonable choice of parameterizations for
FIG. 5: Unnormalized weight w(ρΛ) ∝ Mf(M,∞)dM as- the ensemble of causal diamonds, we are forced to con-
signed to a constant entropy productionRafter Λ domination sider very slow entropy sources in the far future. Dy-
for cosmologies near ours. Over this range of Λ our typical namical effects on the typical halo over long times may
halo size, and hence entropic weight, is approximately pro- prevent these slow entropy sources from being impor-
portional to ρ−Λ1 (dashed curve). tant contributors to the overall measure, but it is easy
toimagineparticularparameterizationswherethisisnot
the case. The entropy associated with black hole evap-
Λ. Indeed, from Fig. 4, we can see that for Λ within a
oration or certain models of particle decay could then
few orders of magnitude of the observed value, f(M,t)
ruin CEP predictions for the value of the cosmological
scales approximately inversely in mass with increasing
constant.
ρ . Thus at least in this vicinity,
Λ It shouldalsobe noted that ina universe with enough
inhomogeneity and with smaller causal diamond sizes,
Z Mf(M,t)dM ∝ρ−Λ1, the effect of the inhomogeneity would be pushed to ear-
lier time scales and we would need to worry about the
as plotted in Fig. 5. clumping of stellar entropy production itself rather than
Recalling from section II, a flat prior on ρ combined merelylate-timeevents. Anexamplewouldbeauniverse
Λ
withaconstant-rateentropysourceandacausaldiamond with much larger Λ and also much larger initial fluctua-
volumeV ∝ρ−Λ1 leadsto aflatpredicteddistributionfor tions.
9
There are methods to parametrize causal diamonds CEP-specificissue. Anditisonethatmustbeaddressed
that seem to avoid the late-time entropy production is- to be confident of CEP predictions for nonidealized cos-
sue discussed here for universes similar to ours. But this mologies.
ambiguity seems to point at least to an incompleteness
inthe CEPascurrentlyformulated. Onecouldofcourse
simply make a felicitous choice of parameterizations and
add it to the CEP. But for a wide range of cosmological
Acknowledgments
parameters it may be still be difficult to be sure of cap-
turing a “typical” causal diamond in this fashion. The
reliance on entropy associated with a single causal dia- We wish to thank Steve Carlip, Damien Martin and
mondmakesthis issue muchmore difficultthanit would Brandon Bozek for many fruitful discussions. This work
befor(e.g.) aper-baryonmeasure,andinthatsenseisa was supported by DOE grant DE-FG03-91ER40674.
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