Astrophysical Constraints on Dark Energy

0.12 MB·English

by  Chiu Man Ho

Astrophysical Constraints on Dark Energy
Chiu Man Ho∗ and Stephen D. H. Hsu†
Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
(Dated: October13, 2015)
Darkenergy(i.e.,acosmological constant)leads,intheNewtonianapproximation,toarepulsive
forcewhichgrowslinearlywithdistanceandwhichcanhaveastrophysicalconsequences. Forexam-
ple,thedarkenergyforceovercomesthegravitationalattractionfromanisolatedobject(e.g.,dwarf
galaxy) of mass 107M⊙ at a distance of 23 kpc. Observablevelocities of boundsatellites (rotation
curves) could be significantly affected, and therefore used to measure or constrain the dark energy
5 density. Here, isolated means that the gravitational effect of large nearby galaxies (specifically, of
1 their dark matter halos) is negligible; examples of isolated dwarf galaxies include Antlia or DDO
0 190.
2

ct I. INTRODUCTION darkmatter)energy-momentumtensor. Thiscanbesub-
O stituted in the original equation to obtain
The discovery of dark energy, which accounts for the
0 1
1 majorityofthe energyintheuniverse,is oneofthe most Rµν =8πG Tµν − gµνT −gµνΛ. (2)
(cid:18) 2 (cid:19)
significantofthe last20years. While therepulsiveprop-

] erties of dark energy are well known in the cosmolog-
O Inthe Newtonianlimit, one candecomposethe metric
ical context, they have not been as thoroughly under-
tensor as g = η +h with |h | ≪ 1. Specifically,
C stood on shorter, astrophysical, length scales. Previous µν µν µν µν
we are interested in the 00-component of the Einstein
. workhasconstrainedthe cosmologicalconstantonsolar-
h equation. We parameterize the 00th-component of the
system scales [1], but its effects are obviously too small
p metric tensor as
- to be directly observed.
o
In what follows, we discuss the repulsive dark energy
r g00 =1+2Φ, (3)
t force and its astrophysical effects on galactic scales. Be-
s
a cause this force grows linearly with distance, its effect is where Φ is the Newtonian gravitational potential. To
[ most likely to be significant for weakly-bound satellites leading order, one can show that [2]

withlargeorbits. Todetecttheeffectsofdarkenergy,we
3
mustfirstunderstandorbitsresultingfromordinarygrav- 1
2v itationaldynamics,withpotentialsmainlydeterminedby R00 ≈ 2∇~2g00 =∇~2Φ. (4)
5 the distribution of dark matter. We find that observa-
9 tions of distant satellites of isolateddwarfgalaxies could In the inertial frame of a perfect fluid, its 4-velocity is
5 be used to detect the effects of dark energy. Here, iso- given by uµ =(1,~0) and we have
0 latedmeanssufficientlyfarfromothersourcesofgravita-
. T =(ρ+p)u u −pg =diag(ρ, p), (5)
1 tionalpotential. Whendwarfgalaxysystemsarenotsuf- µν µ ν µν
0 ficiently isolated, the orbits of their satellites are subject
5 to tidal forces from nearby large galaxies. These tidal where ρ is the energy density and p is the pressure.
1 For a Newtonian (non-relativistic) fluid, the pressure is
forces can distort orbital shapes, and enforce an upper
: negligible compared to the energy density, and hence
v limit on orbital radii.
i T ≈ T00 = ρ. As a result, in the Newtonian limit, the
X 00-component of the Einstein equation reduces to
r
a II. NEWTONIAN GRAVITY AND ∇~2Φ=4πGρ−Λ, (6)
COSMOLOGICAL CONSTANT
which is just the modified Poisson equation for Newto-
The EinsteinequationwithcosmologicalconstantΛ is niangravity,includingcosmologicalconstant. Thisequa-
tion can also be derived from the Poisson equation of
1 Newtonian gravity, ∇~2Φ = 4πG(ρ +3p), with source
R − g R=8πGT +g Λ. (1)
µν µν µν µν
2 terms from matter and dark energy; p ≈ 0 for non-
relativistic matter, and p = −ρ for a cosmological con-
Contractingbothsideswithgµν,onegetsR=−8πGT−
stant.
4Λ where T ≡ Tµµ is the trace of the matter (including Assuming spherical symmetry, we have ∇~2Φ =
1 ∂ r2 ∂Φ and the Poisson equation is easily solved
r2 ∂r ∂r
to obt(cid:0)ain (cid:1)
∗Electronicaddress: [email protected] GM Λ
2
†Electronicaddress: [email protected] Φ=− − r , (7)
r 6
2
where M is the total massenclosedby the volume 4πr3. Galaxy Mass rc
3
Thecorrespondinggravitationalfieldstrengthisgivenby 106M⊙ 10.7 kpc
GM Λ 107M⊙ 23.1 kpc
~g =−∇~ Φ=(cid:18)− r2 + 3 r(cid:19) rˆ. (8) 108M⊙ 49.8 kpc
109M⊙ 107 kpc
Therefore,the cosmologicalconstantleads to a repulsive 1010M⊙ 231 kpc
force whose strength grows linearly with r. 1011M⊙ 498 kpc
One can also derive ~g by starting with the de Sitter- 1012M⊙ 1.07 Mpc
Schwarzschildmetric [3] 1013M⊙ 2.31 Mpc
2GM Λ 1014M⊙ 4.98 Mpc
2 2 2
ds = 1− − r dt (9)
(cid:18) r 3 (cid:19)
TABLE I: Galaxy masses (units of solar mass M⊙) and the
2GM Λ −1 corresponding rc.
2 2 2 2
− 1− − r dr − r dΩ
(cid:18) r 3 (cid:19)
thedarkforce: rotationalvelocitiesofstarsorgasclouds
whichdescribesthe spacetime outsidea sphericallysym- bound to these galaxiesshouldbe smaller thanthat pre-
metric mass distribution M in the presence of a cosmo- dictedbyordinaryNewtoniangravity. Thisinturncould
logical constant Λ. One then obtains Eq. (7) and hence provide a novel way to measure the cosmological con-
Eq. (8) by identifying Eq. (3) with the 00-componentof stant in the future. Rotation curves for many galaxies
the de Sitter-Schwarzschildmetric. havebeenmeasuredtoradiiof∼30kpcormore,andfor
somedwarfgalaxiesto∼10kpc[11]. Lowsurfacebright-
ness (LSB) galaxies may also be worthy of investigation
III. GALAXIES [12]. Some LSBs with total mass ∼ 1010M have disks
⊙
as large as 100kpc.
TheNavarro-Frenk-White(NFW)profile[13]isacom-
The results obtained in the previous section are rele-
monly used parametrization of dark matter halo energy
vanttogalaxies. Forinstance,inthe presenceofthe cos-
density:
mological constant Λ, Eq. (8) describes the Newtonian
gravitationalfieldstrengthoutsideagalaxywithaspher- ρ0
ically symmetric mass distribution M. From Eq. (8), it ρ= 2 , (11)
r/R (1+r/R )
is clear that when r is sufficiently large, the repulsive s s
dark force will dominate over the gravitational attrac- where ρ0 is a characteristic halo density and Rs is the
tion. The critical value of r beyond which this happens
scale radius. These two quantities vary from galaxy to
is given by
galaxy. While the detailedshapeofthe actualdarkmat-
ter density maydiffer fromthe NFW profile,the asymp-
3GM 1/3 3M 1/3 totic1/r3 behavioriswidelyaccepted. Ourresultsbelow
r = = , (10)
c
(cid:18) Λ (cid:19) (cid:18)8πρΛ (cid:19) will not be sensitive to the density profile at small r.
Consideradwarfgalaxy(DG)andalargergalaxy(LG)
where ρΛ = 8πΛG ≈ (2.3 × 10−3eV)4 is the observed (e.g.,theMilkyWay)whosecentersofmassareseparated
energy density of the cosmologicalconstant. Table I dis- by a distance R, and a satellite of the DG whose orbital
plays galactic masses in units of solar mass M⊙ and the radiusisroughlyr. IfthedistanceRissufficiently large,
corresponding r . we can neglect the gravitationalpotential of the LG and
c
Typical galaxies, including our Milky Way, have total treat the DG-satellite system as approximately isolated.
mass (including dark matter) & 1011−12M and sizes In that case, the values in Table I provide a roughguide
⊙
∼ 50kpc. According to Table I, r & 500kpc for these for distances r at which the dark energy force becomes
c
galaxies, so the dark force is not likely to affect internal significant. In the following section we will investigate
dynamics,butmayimpactgalaxy-galaxyinteractions[4], to what extent measurement of satellite velocities can
and limit the size of galaxy clusters (∼ 1014M , size ∼ constrain the dark energy density around the DG.
⊙
Mpc). Butfirstletus examineinmoredetailunder whatcir-
Some dwarf galaxies have total mass (including dark cumstances we can neglect the gravitationaleffects from
matter) ∼ 107M . These include Ursa Major II, Coma the (dark matter halos) of neighboring galaxies on the
⊙
Berenices, Leo T, Leo IV, Canes Venatici I, Canes Ve- DG. We will assume an NFW profile for both the DG
natici II, and Hercules (analyzed by [5]), and also Leo II halo and the larger galactic halo. The distance R from
[6]andLeoV[7]. TheirregulargalaxiesLeoA[8],Antlia the center ofthe DG to the center of the LGis generally
[9] and DDO 190 [10] also have masses around 107M . not equal to the distance from the satellite to the cen-
⊙
7
Forgalaxieswithmass∼10 M ,wehaver ∼23kpc. ter of the LG, which can vary from (R−r) to (R+r).
⊙ c
Thus, their galactic rotation curves could be affected by Therefore, the gravitational pull exerted on the satellite
3
Dwarf Galaxies distance to MW distance to M31 Dwarf Galaxies negligible tidal effect
Leo T 422 kpc 991 kpc Leo T r . 5.6+5.5 kpc
−2.8
Leo IV 155 kpc 899 kpc Leo IV r . 1.9+1.7 kpc
−0.9
Canes Venatici I 218 kpc 864 kpc Canes Venatici I r . 2.6+3.5 kpc
−1.5
Canes Venatici II 161 kpc 837 kpc Canes Venatici II r . 2.2+1.7 kpc
−0.9
Hercules 126 kpc 826 kpc Hercules r . 1.4+1.6 kpc
−0.8
Leo II 236 kpc 901 kpc Leo II r . 3.1+2.8 kpc
−1.5
Leo V 179 kpc 915 kpc Leo V r . 2.8+1.8 kpc
−1.1
Leo A 803 kpc 1200 kpc Leo A r . 10.6+12.6 kpc
−5.8
Antlia 1350 kpc 2039 kpc Antlia r . 18.8+23.8 kpc
−10.6
DDO190 2793 kpc 2917 kpc DDO190 r . 39.4+51.3 kpc
−22.4
TABLE II: Some dwarf galaxies with mass ∼ 107M⊙ and TABLEIV: Requiredorbital radii for thesatellites of some
their distances from the Milky Way (MW) and the An- dwarf galaxies with mass ∼107M⊙ to ensure negligible tidal
dromeda galaxy (M31). See Table 2 in [15]. effect. The upper limit on r is determined by requiring
FLtiGdal/FDG .0.1. Thecentralvaluecorrespondstotheexpo-
nent −1.6 in Eq. (16) while the ± values correspond to the
Dwarf Galaxies rhalf ρ0/GeV cm−3 exponents−1.6±0.4.
Leo T 178 ± 39 pc {0.028,0.22,1.74}
Leo IV 116 ± 30 pc {0.037,0.25,1.66}
gravitationalpull due to DG, FDG. This requires
Canes Venatici I 564 ± 36 pc {0.017,0.21,2.69}
Canes Venatici II 74 ± 12 pc {0.053,0.30,1.65} tidal GMLG(R) r GMDG(r)
Hercules 330 ± 63 pc {0.020,0.20,2.05} FLG ≈ R2 R ≪ FDG = r2 , (14)
Leo II 123 ± 27 pc {0.035,0.24,1.67} which implies
Leo V 42 ± 5 pc {0.086,0.38,1.72}
Leo A 354 ± 19 pc {0.019,0.20,2.11} MLG(R) 1/3 r
≪ 1. (15)
Antlia 471 ± 52 pc {0.018,0.21,2.42} (cid:18) MDG(r) (cid:19) R
DDO190 520 ± 49 pc {0.017,0.21,2.56}
Accordingto[15],manydwarfgalaxiesareatleast100
TABLE III: Some dwarf galaxies with mass ∼ 107M⊙ and
kpcawayfromtheMilkyWayandmuchfartherfromthe
their half-light radii rhalf and ρ0. For each galaxy, the three
Andromeda galaxy (M31). Some of these dwarf galaxies
0d.i4ff,er−e1n.t6,va−lu1.e6s+of0.ρ40}cinorEreqs.po(n16d).to the exponents {−1.6− with mass ∼ 107M⊙ include Leo T, Leo IV, Canes Ve-
natici I, Canes Venatici II, Hercules, Leo II, Leo V, Leo
A,Antlia andDDO 190. Theirdistances fromthe Milky
Way and M31 are shown in Table II.
by the LG is different from the pull on the DG, leading FortheMilkyWay(MW),wehaveρ0 ∼0.2GeVcm−3
to a tidal effect. (See [14] for previous work regarding
and R ∼25 kpc (see Fig. 1 in [16]). For dwarf galaxies
s
the tidal effects on orbiting satellites around their host with mass ∼ 107M , the analysis in [17] suggests that
⊙
galaxies.) This tidaleffect is repulsive: it pulls apartthe
R ∼0.795kpc(see theirTable3whichgivesthe bestfit
s
DG-satellitesystem. Perhapssurprisingly,formanyDGs
parameters for some dwarf galaxies assuming the NFW
(i.e.,neartheMilkyWay),thetidaleffectislargeenough
profile). Intermsofthe half-lightradiusrhalf (the radius
to distort and even destabilize the satellite orbits.
atwhich half of the totallight is emitted), [17]obtains a
Let the total mass of the DG enclosed within r be relation for the mean density hρi interior to rhalf:
MDG(r) and that of the LG enclosed within R±r be
MLG(R±r). Then we have hρi ∼ 2600 rhalf −1.6±0.4 GeV cm−3. (16)
(cid:18) pc (cid:19)
r
MDG(r) = 4πr′2ρDG(r′)dr′, (12) (Note that M pc−3 ≈ 40 GeV cm−3). In the second
Z0 ⊙
column of Table III, the half-light radii of some dwarf
R±r
MLG(R±r) = 4πr′2ρLG(r′)dr′. (13) galaxies with mass ∼107M⊙ are listed. We adopt these
Z0 values of rhalf from [15, 17, 18]. For each of the galaxies
listedinTableIII,byusingr =rhalf andthecorrespond-
Thecircularityandstabilityofthe satelliteorbitscanbe ing value of hρi obtained from Eq. (16) (together with
guaranteed by requiring that the magnitude of the tidal R ∼0.795kpc)inthe NFWdensity profile,weestimate
s
force due to the LG, FLtiGdal, is much smaller than the the value of ρ0. The three different values of ρ0 listed in
4
N orbital radii 95% CI of c/10−84GeV2 at r is given by
5 1-10 kpc {1.31, 1.78}
2 GMDG(r) 1 2
5 5-15 kpc {1.49, 1.66} v (r)= − Λr , (17)
r 3
5 10-20 kpc {1.54, 1.61}
5 15-25 kpc {1.55, 1.60} where MDG(r) can be obtained by a simple integration:
5 20-30 kpc {1.56, 1.59}
r+R r
3 s
10 1-10 kpc {1.43, 1.70} MDG(r)=4πρ0Rs (cid:20)ln(cid:18) R (cid:19)− r+R (cid:21) . (18)
s s
10 5-15 kpc {1.53, 1.64}
10 10-20 kpc {1.56, 1.61} Thus, for a set of measurements on v2(r) at some level
10 15-25 kpc {1.56, 1.60} of sensitivity, we fit v2(r) with
10 20-30 kpc {1.57, 1.59}
a r+b r
2 2
v (r)= ln − −cr , (19)
TABLE V: 95% confidence interval (CI) of c, assuming 1% r (cid:20) (cid:18) b (cid:19) r+b(cid:21)
error in v2. N is numberof satellites.
where a and b are some constants. A (statistically sig-
N orbital radii 95% CI of c/10−84GeV2 nificant) positive fit value for c suggests the existence of
a cosmological constant. The cosmological value for c is
5 1-10 kpc {0.40, 2.31} c=1.58×10−84GeV2. (Notethatinarealisticsituation
5 5-15 kpc {1.11, 1.81} wecannotrelyontheNFWparametrizationbeingexact,
10 1-10 kpc {0.68, 2.07} andthefittingfunctionshouldprobablybeslightlymore
10 5-15 kpc {1.25, 1.78} general than the one used above.)
In Table V, we simulate the results of measurements
TABLEVI: 95% confidenceinterval(CI)ofc,assuming5% on v2(r) with corresponding error of 1%. We take
error in v2. N is numberof satellites. ρ0 ∼ 0.2GeV cm−3 and Rs ∼ 0.795kpc for the dwarf
galaxies. We vary the number of satellites N and their
(randomlygenerated)orbitalradii. Forexample,at95%
thethirdcolumnofTableIIIcorrespondtotheexponents confidence level, one could bound c to be positive using
{−1.6−0.4, −1.6, −1.6+0.4} in Eq. (16). 5 satellites at r ∼ 1−10 kpc. In order to bound c close
Using the normalization factors (ρ0 and Rs) for both to its cosmologicalvalue, one wouldneed, e.g.,at least 5
the Milky Way and the dwarf galaxies with mass ∼ satellites atr ∼10−20kpc or10 satellites atr ∼5−15
107M , we can determine the satellite radii for which kpc.
⊙
Eq. (15) is satisfied. In Table IV, we display the re- In the event that dark energy is dynamical [19], as
quiredorbitalradiiforthesatellitesofsomedwarfgalax- opposed to a rigid cosmological constant, it might form
ies with mass ∼ 107M , assuming negligible tidal ef- inhomogeneous clumps on galactic length scales. This
⊙
fect. The upper limit on r is determined by requiring behaviorcould,inprinciple,bedetectablethroughtheef-
FLtiGdal/FDG . 0.1. The central value corresponds to the fects discussedhere: Λ fromastrophysicalmeasurements
exponent−1.6inEq. (16)whilethe±valuescorrespond would be larger than the known cosmological value. In
to the exponents −1.6±0.4. Table VI, we simulate the results from measurements
on v2(r), assuming that the corresponding error is 5%.
Again, we take ρ0 ∼ 0.2GeV cm−3 and Rs ∼ 0.795kpc
for the dwarf galaxies. The table indicates that even
IV. CONSTRAINTS ON DARK ENERGY
at the sensitivity of 5%, one could rule out (at 95%
DENSITY FROM ROTATION CURVES
confidence level) any Λ that is significantly larger than
1.58×10−84GeV2 byusing,e.g.,5satellitesatr∼1−10
Forgalaxieswithmass∼107M ,wehaver ∼23kpc, kpc. The very existence of satellites of dwarf galaxies
⊙ c
so the effect of the dark energy force becomes observa- (even those close to the Milky Way, and hence subject
tionally significant for r ∼ 20kpc. For such radii, Table to significanttidal forces that limit r) provides an upper
IV indicates that the tidal effect due to the Milky Way limit onthe localdark energy density, probably no more
might have already distorted and even destabilized the than anorder of magnitude largerthan the cosmological
satelliteorbitsinthedwarfgalaxiesLeoT,LeoIV,Canes value.
VenaticiI,CanesVenaticiII,Hercules,LeoII,LeoVand
Leo A. On the other hand, satellite orbits of Antlia and
DDO 190 could be strongly affected by the dark energy V. MISSING SATELLITE PROBLEM
force at radii for which the tidal effect due to the Milky
Way is negligible. Observationsindicatefewersatellitegalaxiesthanpre-
For isolated dwarf galaxies such as Antlia and DDO dictedbynumericalsimulationsinvolvingcolddarkmat-
190, the rotational velocity-squared v2 of their satellites ter. This is known as the missing satellite problem [20].
5
For instance, simulations predict a few hundred satellite force discussed here as a possible solutionto the missing
galaxieswithinafewMpcradiusoftheLocalGroup,but satellite problem. We are instead suggesting a possible
we have observed at least five times fewer. alternative method for measuring the local dark energy
FortheLocalGroupwithmass∼1013−14M ,wehave density through rotation curves of dwarf galaxies.
⊙
r ∼ a few Mpc, which naively suggests that the dark
c
forcemightplayalimiting roleinthe bindingofsatellite
galaxies. However,ifsimulationsofgalaxyformationare Acknowledgements. WethankRobertScherrer,James
performed with sufficient resolution and sufficiently late Schombert, Brian O’Shea, Megan Donahue, Jay Strader
end-times in an expanding ΛCDM universe,the effect of andMatthewWalkerforusefulconversations. Thiswork
the dark energy has already been accounted for (barring was supported by the Office of the Vice-President for
more exotic scenarios, like clumping of dark energy). Research and Graduate Studies at Michigan State Uni-
In other words, we are not suggesting the dark energy versity.
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