Phantom Energy and the Cosmic Horizon: Rh is still not a horizon!

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by  Geraint F. Lewis

Mon.Not.R.Astron.Soc.000,1–??(2002) Printed11December 2013 (MNLATEXstylefilev2.2)
Phantom Energy and the Cosmic Horizon:
R is still not a horizon!
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Geraint F. Lewis(cid:63)
Sydney Institute for Astronomy, School of Physics, A28, The University of Sydney, NSW 2006, Australia
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11December 2013
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n ABSTRACT
a There has been a recent spate of papers on the Cosmic Horizon, an apparently fun-
J damental, although unrecognised, property of the universe. The misunderstanding of

2 this horizon, it is claimed, demonstrates that our determination of the cosmological
makeup of the universe is incorrect, although several papers have pointed out key

] flaws in these arguments. Here, we identify additional flaws in the most recent claims
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of the properties of the Cosmic Horizon in the presence of phantom energy, simply
C demonstrating that it does not act as a horizon, and that its limiting of our view of
. the universe is a trivial statement.
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p Key words: cosmology: theory
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r
t
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a 1 INTRODUCTION 2 THE COSMIC HORIZON
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1 The presence of various horizons within our cosmological The concept of the Cosmic Horizon, Rh, was introduced by
models have greatly elucidated our understanding of the Melia (2007) who, in rewriting the standard Friedmann-
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workings of the universe, with both the particle and event Robertson-Walker (FRW) invariant interval in ‘observer-
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0 horizons limiting the connexions between past and future dependent coordinates’ found metric terms that appeared
3 cosmological events (Rindler 1956). The universe also pos- to be singular at a proper distance of Rh=1/H, where H
0 sesses a Hubble Sphere, which is not a horizon, and is the is the Hubble constant; it is important to remember that
. distance from an observer that comoving objects are mov- H evolves over cosmic time, and so R is a similarly evolv-
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ing, due to the cosmic expansion, at the speed of light (in ing proper distance from an observer. In this initial work,
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3 proper coordinates see Harrison 1991). Melia(2007)claimedthatthedivergenceofthemetriccom-
1 There has been the claim that the universe possess ponentsshowedthatRh representedaninfiniteredshiftsur-
: another, previously unidentified, horizon, dubbed the Cos- face,andhencealimittoourviewoftheuniverse.However,
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mic Horizon (R ), and the presence of this horizon signifi- this was shown as being incorrect by van Oirschot, Kwan,
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X cantlyinfluencesourobservationsofthecosmos(Melia2007, &Lewis(2010),whodemonstratedthattheinfiniteredshift
2009; Melia & Abdelqader 2009; Melia & Shevchuk 2012; wasduetoaunphysicalchoiceforthecoordinatevelocityof
r
a Bikwa, Melia, & Shevchuk 2012; Melia 2012b), although anemitteratadistanceofR ,andcorrectlyaccountingfor
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several authors have demonstrated that a number of the the coordinate transformation between FRW and observer
key claims about the Cosmic Horizon are incorrect (van coordinates results in the same redshift; hence we can see
Oirschot,Kwan,&Lewis2010;Lewis&vanOirschot2012; photons that have traveled through R .
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Bilicki & Seikel 2012). As noted in van Oirschot, Kwan, & Lewis (2010), the
Inthisbriefcontribution,wediscusstherecentclaimsof un-horizon-like properties come as no surprise as R is ex-
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R in the presence of phantom energy (Melia 2012a), again actlythesameastheHubbleSphere,averywellunderstood
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showingthemtobedemonstrablyincorrect.InSection2we conceptincosmology(Harrison1991).Whileinourcurrent
brieflyreviewthenatureofR ,whileinSection3wediscuss cosmology the Hubble Sphere will eventually become co-
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what R means for the path of a photon traveling though incident with our event horizon (see Figure 1 of Davis &
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anexpandinguniverse.Section4demonstratesthatR still Lineweaver 2004), it is not, in itself, a horizon.
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fails to behave as an unrecognised horizon in limiting our However, there have been continuing claims about the
view of the universe, and the conclusions are presented in fundamental nature of R , and recently, Bikwa, Melia, &
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Section 5. Shevchuk (2012) considered the paths of photons over cos-
mic history, examining the proper distance traversed since
the Big Bang. In considering several standard cosmological
models, they concluded that the fundamental property of
(cid:63) E-mail:[email protected] Rhisthatanyphotonwereceivetodaycannothavetraveled
(cid:13)c 2002RAS
2 Geraint F. Lewis
from a distance greater than R today. However, this was
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also shown to be incorrect by Lewis & van Oirschot (2012)
who demonstrated that R does not have to continuously
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groworasymptotetoaparticulardistance,andthatifdark
energy is actually of the form of phantom energy (with an
60
equation of state of ω<−1), then R can decrease. Hence,
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light rays arriving at an observer can have travelled from a yrs)50
l&arvgaenrpOriorpscehrodtis2t0a1n2c)e.thanRh today(seeFigure3ofLewis nce (GL40
ThequestionoftheinfluenceofphantomenergyonRh Dista
wasrevisitedinMelia(2012a),whoagainredefinedthecos- er 30
p
mologicalpropertiesofthisCosmicHorizon,concludingtwo o
Pr
key features of the photon paths should reassure us of its 20
fundamental importance. These are that;
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“The most important feature of these curves is that none of
those actually reaching us [today] ever attain a proper distance
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greaterthanthemaximumextentofourcosmichorizon.” 0 20 40 60 80 100 120
Time (Gyrs)
and
Figure1.ThebluecurvepresentsR forthecosmologicalmodel
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“every null geodesic that possesses a second turning point ...., outlinedinSection4,whiletheredlinespresentaseriesoflight
divergestoinfinity” rays emanating from the Big Bang (at the origin) and into the
future.Notethattheabscissarepresentcosmictime,whereasthe
Intheremainderofthispaper,wewilldemonstratethatthe
ordinateisproperdistancefromanobserver(ataproperdistance
first assertion is trivial, and the second is incorrect.
ofzero).Inthisrepresentation,fourlightraysarrivebackatthe
observer; the key light ray is the one arriving at ∼130 Gyrs, as
thishascrossedR threetimes.
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3 TO A PHOTON, JUST WHAT IS R ?
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a˙ (cid:16)1(cid:17)
As we have noted previously, Rh simply corresponds to the d˙= (ar)−a =Hd−1 (4)
a a
HubbleSphere(vanOirschot,Kwan,&Lewis2010;Lewis&
van Oirschot 2012), the distance from an observer at which so at proper distances of d > 0, the photons are traveling,
the universal expansion results in a proper velocity of the relativetotheobserver,atvelocitiesnotequaltothespeed
speedoflightforacomovingobject.Inthissection,wewill of light. At the distance of d=1/H, which is Rh (or, in its
discussthewhatpassingthroughtheHubbleSpheremeans properparlance,theHubbleSphere),thenitissimpletosee
toaphoton,althoughitshouldbenotedthatthishasbeen that d˙ = 0, and the photon is at rest with regards to the
discussed in detail previously (e.g. Ellis & Rothman 1993). observer (in proper coordinates).
StartingwiththeFRWinvariantintervalforaspatially What this means is that, in proper coordinates, a pho-
flat expanding space-time, ton crossing Rh represents an extremal or inflection point
ds2 =−dt2+a(t)2(cid:0)dr2+r2dΩ2(cid:1) (1) inthephoton’spath,and,consideringthatthephotonpath
began at r = 0 in the Big Bang, and returns to r = 0 at a
where r is the comoving radial coordinate, a(t) is the time- latertime(i.e.itisdetectedbyanobserver),theremustbe
dependent scale factor, and dΩ considers the angular coor- a ‘most-distant’ turn-around point in the photon’s journey,
dinates. With this, the proper distance to an object at a whereitstopsheadingoutwardandstartsheadinginwards,
comovingdistanceofr isd=a(t)r.Foranobserveratrest and this must coincide with the photon crossing Rh.
with regard to the comoving spatial coordinates, the expe- Insummary,theclaimbyMelia(2012a)thatRh repre-
rienced proper time is the same as the cosmic time, t, and sentsaboundtotheobservedphoton’spathisnothingmore
so we can talk of the relative velocity of the distant object than saying “The maximal distance from which we receive
with regards to the observer as being a photon is no more than the largest distance at which it
turns around in it journey”; this is a trivial statement.
d˙=a˙r+ar˙ (2)
wherethedotsdenotederivativeswithregardstot,andthe
r˙ represent motion relative to the comoving coordinates. 4 EVOLVING HORIZONS
If we consider a photon moving in the radial direction,
As a demonstration of the meaning of the Cosmic Hori-
so the angular terms in Equation 1 can be neglected, and
zon, we consider a universe in which R can be tuned to
remembering that a photon path follows ds = 0, then it is h
be increasing or decreasing by modulating the equation of
straight-forward to show
state of the dark energy component. We begin by adopting
1
r˙=± (3) the cosmological parameters of ΩM = 0.27, Ωω = 0.73 and
a H =72 km/s/Mpc. Unlike standard cosmological models,
o
and substituting the negative solution, as we are interested weallowtheequationofstateofthedarkenergycomponent,
in a photon approaching the observer, into Equation2, we ω,tochangeasafunctionoftime,adoptinganevolutionary
find form given by
(cid:13)c 2002RAS,MNRAS000,1–??
The Cosmic Horizon is still no horizon! 3
1.43 3
ω(t)=−1.1+ 1+exp(cid:0)6−th(cid:1) (5) R˙h = 2(1+ω) (6)
0.3
andhenceforphantomenergies,withω<−1,R wouldde-
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crease,foracosmologicalconstant(ω=−1)itisaconstant,
where th = t/(13.58 GYrs) is the scaled cosmic time; it andallotherfluids,withω>−1,Rhincreases.IfEquation5
should be noted that this evolution is not physically moti- ismodifiedsothattheultimateequationofstateofthedark
vated and simply provides a model in which we can exam- energycomponentisω=0,correspondingtomatter,R will
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ine the corresponding evolution of Rh. With this form, the increaseinthefuture,similartotheevolutionshowninFig-
transition in the form of the dark energy component takes ure 1. Again outward moving light rays encountering this
place at t ∼ 6th, and the duration of the change over is increasing Rh will change direction and will head back to
∼ 0.3th, although the values were chosen for purely illus- the observer.
trativepurposes.Essentially,intheearlyuniverse,thedark Anexaminationofthediscussionpresentedhereshould
energy component has an equation of state of ω ∼ −1.1, convincethereaderthatbymodifyingtheequationsofstate
correspondingtoaphantomenergy,whilearoundt∼5th ∼ of the energy components of the universe, we could ensure
65Gyrs the equation of state begins to transition to ω∼ 13, that Rh oscillates through a arbitrarily complex path, and
representing the equation of state of a relativistic mass-less similarly that photon journeys can be made to arbitrarily
fluid, such as photons (see Linder 1988). Note that given change their direction of motion toward or away from an
this evolution, expansion means that the universe will be- observer. Each change of direction is accompanied with the
comematterdominatedinthefuture,giventhattheenergy photon crossing in and out of R , which is extremely un-
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density in the now photon-like dark energy component di- horizon-likebehaviour(butpreciselywhatyouwouldexpect
minishes faster with cosmological expansion. as R being a turning point in a photon’s path).
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In Figure 1, we present the key properties of this uni-
verseintermsofthecosmictime(abscissa)andproperdis-
tance(ordinate).ThebluecurvedenotestheevolutionofR ,
h 5 CONCLUSIONS
whereas the red curves represent light rays which emanate
fromtheBigBang(attheorigin)andtravelintothefuture. We have examined the most recent claims of the funda-
Theleft-handhalfoftheplotcanbecompareddirectlywith mental nature of the Cosmic Horizon, Rh, made by Melia
Figure3ofLewis&vanOirschot(2012),withR increasing (2012a), especially its behaviour in the presence of phan-
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when the universe is matter dominated, and then decreas- tom energy. We have demonstrated that, by modifying the
ingasthephantomenergycomestodominate.However,as equation of state of the energy components in the universe,
the equation of state of the dark energy component evolves then Rh can be made to grown and shrink arbitrarily. Fur-
towards ω∼ 13, then Rh rapidly increases. thermore, as Rh represents the location of where a photon
AnexaminationofthelightraysinFigure1showthat, changes direction, or goes through an inflection point, (in
inthisrepresentationoftheuniverse,allphotonsheadout- proper coordinates) relative to an observer, an oscillating
wardsaftertheBigBang.Followingthisinitialmotion,three photon path can cross Rh a multitude of times before ar-
of the photon paths then encounter the evolving Rh only riving at an observer. The ’fundamental’ property of Rh in
once,andthenarrivebackattheobserver;thiscanbesimply termsofthefarthestproperdistancefromwhichanobserver
understoodas,inpropercoordinates,R markstheturning receives a photon is essentially a trivial statement.
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pointinaphoton’spath.OnephotonpathinFigure1does As we have stressed in other contributions (e.g. Lewis
not encounter R , so that there is no turning point in its &vanOirschot2012),theimportanceandevolutionoftrue
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path and it does not return to the observer. cosmic horizons is well understood, as is the mean of the
Hubble Sphere (Harrison 1991). Other than the trivial, the
TherearetwoadditionallightrayswhichcrossR more
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Cosmic Horizon of Melia (2012a) does not present a funda-
thanonce.Again,theseraysareinitiallyheadingawayfrom
mental limit to our view of the universe.
theoriginaftertheBigBang,andbothofthemcrossR and
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begin to head back towards the observer. However, due to
thepresenceofthephantomenergycomponent,bothphoton
pathsencounteradecreasingR andthemotionisreversed ACKNOWLEDGMENTS
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andthepathsagainmoveawayfromtheobserver;notethat
Luke Barnes and Krzysztof Bolejko are thanked for inter-
while one of these paths appear have a point of inflection,
esting discussions, and Brendon Brewer is thanked for his
just touching R , it actually does possess two crossings. As
h insights into Python.
theequationofstateofthedarkenergycomponentischang-
ingandinfluencingthecosmicexpansion,whileonephoton
path escapes to larger distance, the other is again reversed,
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