Energy conditions in the epoch of galaxy formation

0.12 MB·English

by  Matt Visser

Energy conditions in the epoch of galaxy formation
Matt Visser
Physics Department, Washington University, St. Louis, Missouri 63130-4899
(Dated: Originally submitted November1996; Revised February 1997)
The energy conditions of Einstein gravity (classical general relativity) do not require one to fix a
specific equation of state. In a Friedmann–Robertson–Walker universe where the equation of state
for the cosmological fluid is uncertain, the energy conditions provide simple, model-independent,
and robust bounds on the behaviour of the density and look-back time as a function of red-shift.
Current observations suggest that the “strong energy condition” is violated sometime between the
epoch of galaxy formation and the present. This implies that no possible combination of “normal”
matter is capable of fittingtheobservational data.
Published in: Science 276 (1997) 88-90; doi: 10.1126/science.276.5309.88
5
1
Currently at: School of Mathematics, Statistics, and Operations Research,
0
Victoria University of Wellington, PO Box 600, Wellington 6140, New Zealand.
2

n
a The energy conditions of Einstein gravity (classical cosmological density are automatically included as spe-
J
general relativity) place restrictions on the stress-energy cial cases of this analysis.

7 tensorT (energy-momentumtensor)[1–3]. Thistensor The bounds I derive from the SEC are independent
µν

is a 4 by 4 matrix built up out of the energy density, of whether or not the universe is open, flat, or closed—
]
c momentum density, and the 3 by 3 stress tensor (pres- which means that the density parameter (Ω parameter,
q sure and anistoropic stresses). The energy conditions the ratio of the actual density to the critical density
- force various linear combinations of these quantities to needed to close the universe) does not have to be speci-
r
g be positive and have been used, for instance, to derive fied. Thusmyapproachisindependentoftheexistenceor
[ many theorems of classical general relativity—such as nonexistenceofanyofthestandardvariantsofcosmolog-

1 the singularity theorems; the area increase theorem for ical inflation [12], which typically predict Ω=1 [13–15].
v blackholes,andthepositivemasstheorem—withoutthe If current observations are correct, then the “strong
9 need to assume a specific equation of state [4–6]. Gen- energy condition” (SEC) must be violated sometime be-
1 eralrelativistsandparticlephysicistswouldbesurprised tween the epoch of galaxy formation and the present.
6
if large violations of the classical energy conditions oc- This implies that no possible combination of “normal”
1
cur at temperatures significantly below the Planck scale matter is capable of fitting the observational data, and
0
kT < E ≈ 1019 GeV, T ≈ 1032 K [7]. (Above one needs to do something drastic to the cosmological
. Planck
1 the Planck scale quantum gravity takes over, the whole fluid, either introduce a cosmologicalconstant Λ [16], or
0
framework of classical cosmology seems to break down, have a very non-standard weak form of cosmological in-
5
and the question is moot [8–10].) flation that persists right up to galaxy formation.
1
ThespacetimegeometryofthestandardFRWcosmol-
: Current observations seem to indicate that the strong
v ogyis describedby specifying the geometryofspaceasa
energy condition (SEC) is violated rather late in the life
i
X of the universe—somewhere between galaxy formation function of time, using the spacetime metric [17, 18]
r and the present time, in an epoch where the cosmolog-
a
ical temperature never exceeds 60 K. I shall show this dr2
2 2 2 2 2 2 2
by using the energy conditions to developsimple and ro- ds =−dt +a(t) +r (dθ +sin θdφ ) .
1−kr2
bust bounds for the density and look-back time [11] as a (cid:20) (cid:21)
(1)
function of red-shift in a Friedmann–Robertson–Walker
Here ds is the invariant interval between two events, t is
(FRW)cosmology. TheexperimentalobservationsIneed
comoving time (time as measured by an observerfollow-
arethepresentdayvalueoftheHubbleparameterH0,an
ing the averageHubble flow), a(t) is the scale parameter
age estimate for the age of the oldest stars in the galac-
describing the size of the universe as a function of time,
tic halo, and an estimate for the red-shift at which these
and(r,θ,φ) aresphericalpolarcoordinatesusedtocover
oldest stars formed. From the theoretical side, I only
all of space (with space at time t being defined as the
need to use a FRW cosmology subject to the Einstein
constant-t slice through the spacetime). The parameter
equations and classical energy conditions. Using the en-
k depends on the overall geometry of space, and only
ergy conditions to place bounds on physical parameters
takes on the values
ofthe universeallowsme to avoidthe needto separately
analyzecold,hot,lukewarm,ormixeddarkmatter. Simi-
larly,MACHOS(massivecompacthaloobjects),WIMPS +1 closed (if Ω>1),
(weakly interacting massive particles), axions, massive k = 0 flat (if Ω=1), (2)
neutrinos, and other hypothetical contributions to the (−1 open (if Ω<1).
2
The two non-trivial components of the Einstein equa- very seriously wrong. The WEC additionally requires
tions yield the totaldensity ρ andtotalpressurep ofthe that the density is positive [20].
cosmologicalfluid as a function ofthe scale factor a [19]. To understandwhat the SEC requires for the physical
universe, consider the quantity d(ρa2)/dt, and use the
Einstein equations to deduce
3 a˙2 k
ρ= + . (3)
8πG a2 a2
(cid:20) (cid:21) d
2
(ρa )=−aa˙(ρ+3p). (12)
dt
1 a¨ a˙2 k
p=−8πG 2a + a2 + a2 . (4) Thus
(cid:20) (cid:21)
They can be combined to deduce the conservation of
d
2
stress-energy SEC=⇒ sign (ρa ) =−sign(a˙). (13)
dt
(cid:20) (cid:21)
a˙ This implies that
ρ˙ =−3 (ρ+p). (5)
a
Here a˙ is the (time dependent) velocity of expansion of SEC=⇒ ρ(a)≥ρ0(a0/a)2 for a<a0. (14)
the universe. Combined with the scale factor a(t) it de-
fines the (time dependent) Hubble parameter In terms of the red-shift (1+z =a0/a):
H(t)= a˙(t). (6) SEC=⇒ ρ(z)≥ρ0(1+z)2. (15)
a(t)
The subscript zero denotes present day values, and the
There are several different types of energy condition SEC provides a model-independent lower bound on the
in general relativity, the two main classes being aver- density of the universe extrapolated back to the time of
aged energy conditions (that depend on some averageof thebigbang. AnotherviewpointontheSECcomesfrom
the stress-energy tensor along a suitable curve), and the considering the quantity
point-wise energy conditions (that depend only on the
stress-energy tensor at a given point in spacetime). The
standardpoint-wiseenergyconditionsarethenullenergy 3 a¨
ρ+3p=− . (16)
condition (NEC), weak energy condition (WEC), strong 4πG a
(cid:20) (cid:21)
energy condition (SEC), and dominant energy condition
That is
(DEC). Basic definitions are given in [1–3] and for the
special case of a FRW spacetime the general formulae
simplify.
SEC=⇒ a¨<0. (17)
The SEC implies that the expansion of the universe is
NEC ⇐⇒ (ρ+p≥0). (7)
decelerating—and this conclusion holds independent of
whether the universe is open, flat, or closed.
For the DEC, use the Einstein equations to compute
WEC ⇐⇒ (ρ≥0) and (ρ+p≥0). (8)
d
6 5
(ρa )=+3a a˙(ρ−p). (18)
dt
SEC ⇐⇒ (ρ+3p≥0) and (ρ+p≥0). (9)
Thus
d
DEC ⇐⇒ (ρ≥0) and (ρ±p≥0). (10) 6
DEC=⇒ sign (ρa ) =+sign(a˙). (19)
dt
(cid:20) (cid:21)
The NEC is enough to guarantee that the density of the
universe goes down as its size increases. The DEC therefore provides an upper bound on the en-
ergy density.
NEC ⇐⇒ sign(ρ˙)=−sign(a˙). (11)
6
DEC=⇒ ρ(a)≤ρ0(a0/a) for a<a0. (20)
If the NEC is violated the density of the universe must
increaseastheuniverseexpands—sosomethinghasgone In terms of the red-shift
3
DEC=⇒ ρ(z)≤ρ0(1+z)6. (21) H0 ≤τf−1 ≤62±8 km s−1 Mpc−1 . (28)
When we look into the sky and see some object, the When we actually look into the night sky, we infer that
look-backtimetothatobject(τ =t0−t)isdefinedasthe the oldest stars seem to have formed somewhat earlier
difference betweent0 (the ageofthe universenow)andt than the development of galactic spiralstructure [24]. A
(the age of the universe when the light that we are now canonical first estimate is [25]
receivingwasemitted). Ifweknowthevelocityofexpan-
sion of the universe a˙ as a function of scale parameter a
Redshift at formation of oldest stars≡z ≈15. (29)
we simply have f
This now bounds the Hubble parameter
a0 da
τ(a;a0)=t0−t= . (22)
Za a˙(a) SEC=⇒ H0 ≤58±7 km s−1 Mpc−1 . (30)
Byputtingalowerboundona˙ wededuceanupperbound Recent estimates of the present day value of the Hubble
on look-back time. In particular since the SEC implies parameter are[26]
that the expansion is decelerating then
H0 ∈(65,85) km s−1 Mpc−1 . (31)
1 a0−a
SEC=⇒ τ(a;a0)=t0−t≤ , (23)
H0 a0 (I have chosen to use a range of H0 values on which
thereiswidespreadthoughnotuniversalconsensus[26].)
independent of whether the universe is open, flat, or But even the lowest reasonable value, H0 = 65 km s−1
closed. Expressed in terms of the red-shift: Mpc−1,isonlyjustbarelycompatiblewiththeSEC,and
thatonlybytakingtheyoungestreasonablevalueforthe
1 z ageof the globularclusters. For currently favoredvalues
SEC=⇒ τ(z)=t0−t≤ H01+z. (24) of H0 we deduce that the SEC must be violated some-
where between the formation of the oldest stars and the
Thisprovidesuswitharobustupper boundontheHub- present time.
ble parameter Note the qualificationsthatshouldbe attachedto this
claim: We have to rely on both stellar structure calcula-
tions for τ and an estimate for z . Decreasing z to be
f f f
1 z
SEC=⇒ ∀z :H0 ≤ . (25) moreinlinewiththeformationoftherestofthegalactic
τ(z)1+z structure (z ≈ 7) makes the problem worse, not bet-
f
This is enough to illustrate the age-of-the-oldest- ter (H0 ≤ 54 ± 7 km s−1 Mpc−1). Increasing zf out
to its maximum conceivable value, z ≈ 20 [24], does
stars problem (often mischaracterized as the age-of-the- f
not greatly improve the fit to the SEC since the bound
universe problem). Suppose there is some class of stan- becomes H0 ≤ 59±8 km s−1 Mpc−1. All of these dif-
dard candles whose age of formation, τ , can be esti-
f ficulties are occurring at low cosmological temperatures
mated [21]. Suppose further that if we look out far
(T ≤ 60 K), and late times, in a region where the basic
enough we can see some of these standard candles form-
equation of state of the cosmological fluid is supposedly
ing at red-shift z (or can estimate the red-shift at for-
f understood [27].
mation). Then
Incontrast,the NECdoesnotprovideanystrongcon-
straintonH0. Foraspatiallyflatuniverse(k =0,Ω=1,
1 z 1 as preferred by inflation advocates [13–15])
f
SEC=⇒ H0 ≤ ≤ . (26)
τ 1+z τ
f f f
ln(1+z)
The standard candles currently of most interest (simply NEC+(k =0)=⇒ τ =t0−t≤ . (32)
because they have the best available data and provide H0
the strongestlimit) are the globular clusters inthe halos Somewhatmorecomplicatedformulaecanbederivedfor
ofspiralgalaxies: stellarevolutionmodelsestimate(they k =±1, (open or closed universes).
donotmeasure)the ageoftheoldeststarsstillextantto This implies a (very weak) bound on H0. In order
be 16±2×109 yr[22]. Thatis,atanabsoluteminimum for cosmologicalexpansion to be compatible with stellar
evolution and the NEC
9
Age of oldest stars≡τ ≥16±2×10 yr. (27)
f
ln(1+z )
f
NEC+(k =0)=⇒ H0 ≤ . (33)
Using zf <∞, this implies that [23] τf
4
The best value for τ (16×109 yr), and best guess for IftheSECisviolatedbetweentheepochofgalaxyfor-
f
zf (zf ≈15),givesH0 ≤170kms−1 Mpc−1. Decreasing mation and the present, then how does this affect our
zf to about 7 reduces this bound slightly to H0 ≤ 129 ideas concerning the evolution of the universe? The two
km s−1 Mpc−1. Both of these values are consistent with favorite ways of allowing SEC violations in a classical
the observational bounds on H0. Even for the highest field theory are by using a massive (or self-interacting)
Hubble parameter (H0 =85 km s−1 Mpc−1), and oldest scalar field [30], or by using a positive cosmological con-
age for the oldest stars (t =18×109 yr), z ≥3.6 well stant [31]. A classical scalar field ϕ, that interacts with
f f
withintheobservationalboundsonz . Thepresentdata itself via some scalar potential V(ϕ), can violate the
f
is therefore not in conflict with the NEC. SEC [30], but not the NEC, WEC, and DEC [31]. In-
The DEC provides us with a upper bound on the en- deed
ergydensityρ,andthereforeanupperboundontherate
of expansion. This translates to a lower bound on the
look-backtime andalowerboundonthe Hubble param- (ρ+3p)| =ϕ˙2−V(ϕ). (38)
ϕ
eter. For a spatially flat universe
ItisthispotentialviolationoftheSEC(dependingonthe
3 3 detailsofthetimerateofchangeofthescalarfieldandits
1 a −a
0
DEC+(k =0)=⇒ τ =t0−t≥ 3H0 a30 . (34) sfieellfdisntseoraactttiroanctpivoetetnotiaald)vtohcaattemsaokfesincfloastmioonlo[g1i3c–a1l5s]c.alIanr
the present context, using a massive scalar field to deal
Somewhatmorecomplicatedformulaecanbe derivedfor
with the age-of-the-oldest-stars problem is tantamount
k =±1. In terms of the red-shift
to assertingthata lastdyinggaspofinflationtook place
as the galaxies were being formed. This is viewed as an
1 1 unlikely scenario [32].
DEC+(k =0)=⇒ τ =t0−t≥ 3H0 1− (1+z)3 . In contrast, the current favorite fix for the age-of-the-
(cid:18) (cid:19)
(35) oldest-stars problem is to introduce a positive cosmolog-
So the ages of the oldest stars provide the constraint ical constant Λ [16], in which case
1 1
DEC+(k =0)=⇒ H0 ≥ 3τ 1− (1+z )3 . (36) (ρ+3p)total =(ρ+3p)normal−2ρΛ. (39)
f (cid:18) f (cid:19)
This also is a relatively weak constraint, The observed SEC violations then imply
DEC+(k=0)=⇒ H0 ≥20±3 km s−1 Mpc−1 . (37) ρΛ ≥ 1(ρ+3p)normal. (40)
2
Thepresentobservationaldataisalsonotinconflictwith
the DEC. Under the mild constraint that the pressure due to nor-
The estimated value of H0 has historically exhibited malmatter in the presentepochbe positive, this implies
considerable flexibility: While it is clear that the rela- that more that 33% of the present-day energy density is
tionship between the distance and red-shift is essentially due to a cosmologicalconstant.
linear,theabsolutecalibrationoftheslopeoftheHubble I have shown that high values of H0 imply that the
diagram (velocity of recession versus distance) has var- SEC must be violated sometime between the epoch of
ied by more than an order of magnitude over the course galaxy formation and the present. This implies that the
of this century. Hubble parameter estimates from 500 age-of-the-oldest-starsproblemcannotsimplybefixedby
km s−1 Mpc−1 to 25 km s−1 Mpc−1 can be found in adjusting the equation of state of the cosmologicalfluid.
the published literature [28, 29]. Current measurements Since allnormalmatter satisfiesthe SEC,fixingthe age-
give credence to the range 65—85 km s−1 Mpc−1 [26]. of-the-oldest-stars problem will inescapably require the
Thereliabilityofthedataonτ andz ishardertoquan- introduction of “abnormal” matter—indeed large quan-
f f
tify, but there appears to be broad consensus within the tities of abnormal matter, sufficient to overwhelm the
community on these values [22, 24, 25]. gravitationaleffects of the normal matter, are needed.
5
[1] S. W. Hawking and G. F. R. Ellis, The large scale [18] S. Weinberg, Gravitation and Cosmology, (Wiley, New
structureofspace-time,(Cambridge,England,1973),see York, 1972), pp.412–415.
pp.88-91 and 95–96. [19] A cosmological constant, if present, is absorbed into the
[2] R. M. Wald, General Relativity, (Chicago University definition of the total density and total pressure.
Press, Chicago, 1984), see pp.218–220. [20] Negative energy densities are extremely rare in physics.
[3] M. Visser, Lorentzian wormholes—from Einstein to The only known examples are from small quantum ef-
Hawking,(AIPPress,NewYork,1995),seepp.115–118. fects (such as the experimentally verified Casimir effect,
[4] See [1] pp. 263, 266, 271, 272, 292–293, 311, 318, 320, see[3]pp.121-125), orfrom ahypotheticalnegativecos-
354–357. mological constant, see [3] pp. 129-130. Negative energy
[5] See[2] pp.226–227, 232, 233, 237–241. does not mean antimatter. Antimatter has positive en-
[6] See[3] pp.118–119. ergy. Negative energy means an energy less than that of
[7] These classical energy conditions are violated by quan- the normal undisturbed vacuum.
tum effects of order ¯h, with typical quantum violations [21] A standard candle is simply any class of astrophysi-
beingapproximatelyhTµνiviolation≈¯hc9/(GM)4.See[3] cal objects that is sufficiently well understood, suffi-
pp.128–129.Hereh¯ isPlanck’sconstant,cisthespeedof cientlywellcharacterized,andhassufficientlyniceobser-
light, G is Newton’s constant, and M is the mass of the vationalfeatures,tobewidelyacceptedbyobservational
bodyunderconsideration.Thesequantumeffectsarenot astronomersasausefuldiagnostictool.Themostfamous
expected to be significant for large classical systems— standard candles are the Cephid variables, whose abso-
particularly in cosmological settings. For a general dis- lute luminosity is a known function of their period [17]
cussion of quantum effects in semiclassical gravity see pp.20,106 and[18]pp.433-438. HereIwant asimilarly
N. D. Birrell and P. C. W. Davies, Quantum fields in well-behaved class of objects to trace out galaxy forma-
curved spacetime, (Cambridge, England, 1982), and S. tion.
A. Fulling, Aspects of quantum field theory in curved [22] See [17] p. 106.
space–time, (Cambridge, England, 1989). [23] Here I haveexpressed theHubbleparameter in terms of
[8] General Relativity: An Einstein Centenary Survey, thestandardastrophysicalunitsofkilometerspersecond
edited by S. W. Hawking and W. Israel, (Cambridge, per Megaparsec, with a parsec being approximately 3×
England, 1979). 1016 m.
[9] Three Hundred Years of Gravitation, edited by S. W. [24] See [17] pp.610–611. Note thelarge uncertainties.
Hawking and W. Israel, (Cambridge, England, 1987) [25] See [17] p. 614. This number is a model-dependent esti-
[10] See[3] pp.53–73. mate,notanobservation,fortunatelytheanalysisofthis
[11] Look-backtimetoanobject issimplydefinedasthedif- report is relatively insensitive to theprecise value of z .
f
ferencebetween theage of theuniversenowand theage [26] Particle Data Group, Review of Particle Properties,
of theuniversewhen thelight that we are now receiving Phys.Rev.D54(1996)1–720,seethemini-reviewonpp.
from that object was emitted. 112–114, and references therein. Slightly different num-
[12] Cosmological inflation is a brief period of anomalously bers are given on p.66.
rapid expansion in the early universe during which the [27] Thestandardpictureisthattheuniverseismatterdom-
universeinflatesbyanenormousfactor.Inflationiscom- inated (i.e. dust) out to z ≈ 1000, so that one expects
monly invoked as a hypothesis to explain the horizon the equation of state to bep=0. See[17] p. 100.
problem, the flatness problem, and the monopole prob- [28] See [17] pp.106–108.
lem as discussed in [13–15]. [29] See [18] pp.441–451.
[13] E. W. Kolb and M. S. Turner, The Early Universe, [30] See [1] p.95.
(Addison–Wesley,Redwood City, 1990). [31] See [3] p.120.
[14] A. D. Linde, Inflation and Quantum Cosmology, (Aca- [32] Standard variants of inflation are driven by GUT-scale
demic, Boston, 1990). (grandunifiedtheory)phasetransitionsintheearlyuni-
[15] I. Moss, Quantum Theory, Black Holes, and Inflation, verseandtakeplacewhenenergiesareoforderkT ≈1014
(Wiley,Chichester, 1996). GeV, (see [13]) with temperatures of order T ≈ 1027 K,
[16] S.LeonardandK.Lake,TheTinsleyDiagramRevisited, whereas galaxy formation takesplace for T ≤60 K.
The Astrophysical Journal, 441, (1995) L55–L56. [33] Acknowledgement: This research was supported by the
[17] P. J. E. Peebles, Principles of Physical Cosmology, U.S. Department of Energy. Correspondence to Matt
(Princeton UniversityPress, 1993). Visser; [email protected]
(Current e-mail: [email protected])