Neutrino Masses, Dark Energy and the Gravitational Lensing of Pregalactic HI

0.52 MB·English

by  R. Benton Metcalf

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed22January2009 (MNLATEXstylefilev2.2)
Neutrino Masses, Dark Energy and the Gravitational
Lensing of Pregalactic HI
9 R. Benton Metcalf
0
Max Plank Institut fu¨rAstrophysics, Karl-Schwarzchild-Str. 1, 85741 Garching, Germany
0
2

n
22January2009
a
J

3
ABSTRACT

We study the constraintswhichthe nextgenerationofradiotelescopescouldplace on
]
O the massandnumberofneutrinospeciesbystudying the gravitationallensingofhigh
redshift21 cmemissionincombinationwithwide-angle surveysofgalaxylensing.We
C
use simple characterizationsof reionization history and of proposed telescope designs
.
h toforecasttheconstraintsanddetectabilitythresholdforneutrinos.Itisfoundthatthe
p degeneracy between neutrino parameters and dark energy parameters is significantly
- reducedbyincorporating21cmlensing.Thecombinationofgalaxyand21cmlensing
o
could constrain the sum of the neutrino masses to within ∼0.04 eV and the number
r
t of species to within ∼ 0.1. This is an improvement of a factor of 2.6 in mass and
s
a 1.3 in number over a galaxy lensing survey alone. This includes marginalizing over
[ an 11 parametercosmologicalmodel with a two parametermodel for the dark energy
equationofstate.Ifthedarkenergyequationofstateisheldfixedatw≡p/ρ=−1the
2
constraintsimproveto∼0.03eVand0.04.Theseforecastederrorsdependcriticallyon
v
thefractionofskythatcanbesurveyedinredshifted21cmemission(25%isassumed
5
4 here) and the redshift of reionization (z = 7 is assumed here). It is also found that
2 neutrinos with masses too small to be detected in the data could none the less cause
0 a significant bias in the measured dark energy equation of state.
.
1 Key words: large-scale structure of Universe – dark matter – gravitational lensing
0
– intergalactic medium – low frequency radio astronomy
9
0
:
v
i
X 1 INTRODUCTION limit is considerably looser if the density of dark energy is
r allowed to evolve with time (Seljak et al. 2005; Hannestad
a One of the most tantalizing questions in experimental par- 2005; Spergel et al. 2007).
ticle physics – the nature of neutrino mass – is connected
Massiveneutrinosintheappropriatemassrangearerel-
to one of the most tantalizing questions in observational
ativisticwhentheydecouplefromthephotons,electronsand
cosmology –thenatureofdarkenergy.Cosmological obser-
baryonswhenthetemperatureoftheuniverseisafewMeV.
vationspresentlyputthemoststringentupperlimitsonthe
They have the effect of suppressing the power spectrum
absolutemassofneutrinos(Seljak et al.2005;Spergel et al.
of density fluctuations on small scales (scales smaller than
2007) and might continue to do so for some time to come.
the horizon when the neutrinos become non-relativistic,
However,apartialdegeneracybetweenneutrinomassesand
k > k 0.018√m Ω h Mpc−1 eV−1/2 where k is
thedarkenergy’sequationofstatelimitshowwelleitherone nr ≃ ν m
the Fourier wave number, m is the neutrino’s mass and
can be determined from cosmological measurements based ν
h is the Hubble parameter in units of 100 km s−1Mpc−1)
on theevolution of structureformation.
due to free-steaming. This suppression begins at the time
Atmosphericandsolarneutrinooscillation experiments of decoupling and continues until today because the ther-
stronglyindicatethatneutrinoshavemassandthatthesum mal velocity of the neutrinos is still significant (v
therm
of their masses is larger than the measured mass splittings 150(1+z)m kms−1eV−1whennon-relativistic).Thefree≃-
ν
mtPheeνnmetlseνcfrt∼>orompn∆βn-emduet2acrtmianyo+emx√pa∆sesrmimo2sfuen2n.t≃5se0gV.i0v,5etheaVenr.euDpairpreeecrntobmocuoeanmsdpureoetn-- sktnrreaamsiknfgs s≃ca0le.8c2o√ntΩinmu((11e++szzt))2o3+dΩeΛcreamesVνe fhromMpitcs−m1.aNxiemuutrminoosf
will resist falling into dark matter halos with velocity dis-
ingmeasurementsoftheotherflavors(seeFogli et al.(2006) ` ´
persions < v . For a review of neutrinos in cosmology
for a review of experimental results). Combinations of cos- therm
see Lesgou∼rgues & Pastor (2006).
mologicaldatagiveaconstraintof m <0.66eVassum-
ν ν
ing a flat cosmology with a cosmological constant, but this It has been proposed that neutrino masses could be
P
2 R.B. Metcalf
measured by the next generation of weak gravitational ionized bubbles (Zaldarriaga et al. 2004) and make thedis-
lensing surveys (EUCLID1,LSST2,PanSTARRS3, DES4) tribution non-Gaussian.
(Hannestad et al. 2006; Kitching et al. 2008) and that the
existenceofmassiveneutrinosmaylimittheabilityofthese
surveystomeasurepropertiesofthedarkenergy(Hannestad
2005; Kiakotou et al. 2007). Dark energy causes structure 2.2 gravitational lensing
formation to evolve differently than it would otherwise at
Itisconvenienttoexpressthelensingformalism intermsof
low redshift (z < 1). A degeneracy arises between this and the convergence, κ(~θ,z ), at a position θ~ on the sky which
the late-time fr∼ee-streaming of massive neutrinos since at s
to an excellent approximation is related directly to thedis-
scales significantly smaller than k the suppression is scale
fs tribution of matter through which thelight passes
independent and the lensing surveys will be sensitive to a
limited range in k. However, at z > 1 dark energy is ex- 3 ∞ (1+z)
κ θ~,z = H Ω dz g(z,z )δ θ~,z (2)
pected to have negligible effects on structure formation so s 4 o m E(z) s
if higher redshifts can be probed the degeneracy can be re- “ ” Z0 “ ”
3 zi+δz (1+z)
moved.TheCMB (cosmicmicrowave background)provides ≃ 4HoΩm δ ~θ,zi dz E(z) g(z,zs)
information on the power spectrum at z ∼ 1100, but Silk Xi “ ”Zzi−δz
damping limits its sensitivity to scales below knr. Gravita- = G(z ,z )δ θ~,z (3)
i s i
tional lensing of pregalactic 21-cm radiation can provide a
probe of structure formation on the scales needed, at the Xi “ ”
redshifts needed. with
Theprospectsformeasuringgravitationallensingofthe ∞ D(z,0)D(z′,z)
pregalactic 21 cm radiation have been studied by a num- g(z,zs)= dz′ η z′,zs D(z′,0) . (4)
ber of authors (Zahn & Zaldarriaga 2006; Metcalf & White Zz
` ´
2007b;Hilbert et al.2007;Lu & Pen2007;Metcalf & White H is the Hubble parameter. The convergence can be
o
2007a). The methods used in this paper are described in thoughtofasaprojecteddimensionlesssurfacedensity.The
more detail in Metcalf & White(2007a). weightingfunctionforthesourcedistancedistribution,η(z),
is normalized to unity. D(z′,z) is the angular size distance
between the two redshifts and δ(~x,z) is the fractional den-
2 FORMALISM & MODELING sity fluctuation at redshift z and perpendicular position ~x.
The function
2.1 pregalactic 21 cm radiation
A thorough review of pregalactic 21 cm radiation can be E(z)= Ωm(1+z)3+ΩΛ(1+z)3f(z), (5)
foundinFurlanettoet al.(2006).Forthispaperwewillmen- q
where Ω and Ω , are the present day densities of matter
tion only the barest essentials needed to specify our model m Λ
and dark energy measured in units of the critical density.
for 21 cm emission. The fluctuations in the 21 cm bright-
It is assumed that the universe is flat – Ω +Ω =1. The
ness temperature depend on the spin temperature, T , the m Λ
s
function describing the evolution of dark energy with red-
ionization fraction, x and thedensity of HI through
H
shift can bewritten
T T Ω h2
δTb≃24(1+δb)xH„ s−TsCMB«„0.b02 « f(z)= ln(1−+1 z) 0 [1+w(a)] dlna (6)
0.15 1+z 1/2 Z−ln(1+z)
mK (1)
ׄΩmh2 10 « where w(a) is the equation of state parameter for the dark
energy – theratio of the of its pressure to its density – and
(Field 1959; Madau et al. 1997). As is commonly done, we
a=(1+z)−1 is thescale parameter.
will assumethatthespintemperatureismuchgreaterthan
Forourpurposesitisconvenienttoexpressequation(3)
the CMB temperature. This leaves fluctuations in x , and
H
as a matrix equation,
the baryon density δ = (ρ ρ )/ρ as the sources of
b b − b b
brightness fluctuations. We will make the simplifying as- K=Gδ, (7)
sumption that x = 1 until the universe is very rapidly
H
and uniformly reionized at a redshift of zreion. We will where the the components of K are the convergences run-
take δb to be distributed in the same way as dark matter ningover all position angles ~θ and source redshifts, zs. The
according to the CDM model. Nonlinear structure forma- componentsofthevectorδ runoverall position angles and
tion (Peacock & Dodds 1996) and linear redshift distortion foreground redshifts z .Thematrix Gis afunction of most
i
(Kaiser 1987) are included. Realistically, the reionization of the global cosmological parameters – Ω , Ω , w, etc –
m Λ
process will be inhomogeneous and may extend over a sig- and is independent of position on the sky. This equation
nificant redshift range. This will increase Cν(ℓ) by perhaps holds equally well if κ and δ are expressed spherical har-
afactor of10 onscales larger than characteristic sizeof the monicspaceorintheu-v planewhereinterferometerobser-
vations are carried out.
Within frequency bands of size 1 MHz we use an esti-
1 www.dune-mission.net
2 www.lsst.org matorfortheFouriertransformoftheconvergence,κ(ℓ,zs),
3 pan-stars.ifa.hawaii.edu denotedκˆ(ℓ,zs),basedontheZahn & Zaldarriaga(2006)es-
4 https://www.darkenergysurvey.org/ timatorandthencombinethebandsasinMetcalf & White
Neutrino Masses, Dark Energy and the Gravitational Lensing of Pregalactic HI 3
(2007a). Thenoise in κˆ(ℓ,ν) within onefrequency band is average mass of the neutrino species and through this the
free-streaming scale while Ω h2 affects the degree of sup-
(2π)2 ν
Nκˆ(ℓ,ν)= pression to thepower spectrum.
2
Themeasurementsweconsiderareactuallysensitiveto
[ℓ ℓ′C (ℓ′,k)+ℓ (ℓ ℓ′)C (ℓ′ ℓ,k)]2 −1 anylightparticlespecieswithsignificantcosmologicaldensi-
d2ℓ′ · ν · − ν | − |
×"Xk Z CνT(ℓ′,k)CνT(|ℓ′−ℓ|,k) # twieosulndotbejudsitffneeruentrtinfoors.reTlihcepianrtteircplersettahtaiotnwoefrethperopdauracmedetineras
(8)
different way, for example axions, but the late-time physics
where CT(ℓ,k) is the power spectrum of the actual bright- would bethe same.
ν
ness temperature, while C (ℓ,k) = CT(ℓ,k)+CN(ℓ,k) is
ν ν ν
the observed power spectrum which includes noise. ℓ is the
angular Fourier mode number. Because the estimator is a 2.4 model observations
sumoveralltheobservedpairsof visibilities, it will(bythe
2.4.1 21 cm observations
centrallimittheorem)beclosetoGaussiandistributedeven
though it is quadratic in the visibilities. We will concentrate on the planned SKA (the Square Kilo-
The κˆ(ℓ,z)s can begrouped into a data vector meterArray)5 becauseit is theonly plannedtelescope that
willhavealargeenoughsizetoberelevanttoneutrinocon-
D=Kˆ K (9)
− straints.Itisonlythecoreofthetelescopethatwillbeused
=Kˆ Gδ (10) forobservingpregalactic21cmradiation.PlansfortheSKA
−
corehavenotbeenfinalized,butitisexpectedtohaveadi-
where the components run over all the combinations of zi ameter of D 6 km (ℓ 104), an aperture covering
tel max
and ℓ that are measured. To a good approximation modes ∼ ∼
fraction of f 0.02 (the total collecting area of the
cover
from different bands and ℓ’s separated by more than the telescopesdivided∼byπ(D /2)2)andafrequencyrangeex-
tel
resolution of the telescope are statistically independent so
tending down to 100 MHz which corresponds to z 13.
thecovariance matrix for D is diagonal ∼ ∼
It is expected that the SKA will be able to map the 21 cm
N δ Nκˆ(ℓ ,ν ). (11) emission with a resolution of ∆θ 1 arcmin. For reference,
ij ≃ ij i i onearcminute(fwhm)correspond∼stobaselinesof5.8kmat
with|ℓi−ℓj|largerthantheresolution.Thelikelihoodfunc- z = 7 and 11 km at z = 15. We will take ℓmin = 10 to be
tion for the correlations in κ(ℓ,zi) is given by thelowest mode to bemeasured.
1 1 The noise in each visibility measurement will have a
ln = Kˆ†C−1Kˆ C, (12)
L −2 − 2| | thermalcomponent andacomponentresultingfrom imper-
fect foreground subtraction. Here we model only the ther-
where
malcomponent.Ifthetelescopesin thearray areuniformly
C=N+Cκ (13) distributed on the ground, the average integration time for
each baseline will be the same and the power spectrum of
(Metcalf & White2007a).C hereisthe(cross-)powerspec-
κ thenoise will be
trum of theconvergence for two different source redshifts,
2π T λ 2 (2π)3T2
[Cκ]ij = κ(~ℓ,zi)κ(~ℓ,zj) . (14) CℓN = ∆νto „fcovseyrsDtel« = ∆νtofc2overℓmsyasx(ν)2, (15)
D E
Thiscanbecalculated usingexpression(2)andamodelfor (Zaldarriaga et al. 2004; Morales 2005; McQuinn et al.
thematter power spectrum. 2006) where T is the system temperature, ∆ν is the
sys
bandwidth, t is the total observation time, and ℓ (λ) =
o max
2πD /λ is the highest multipole that can be measured by
tel
2.3 structure formation with massive neutrinos
the array, as set by the largest baselines. At the relevant
To calculate the linear matter power spectrum we use the frequencies, the overall system temperature is expected to
analytic formulae of Eisenstein & Hu (1999) with the mod- bedominatedbygalacticsynchrotronradiation.Wewillap-
ification of Kiakotou et al. (2007) that improves accuracy proximate thebrightness temperature of this foreground as
when the neutrino masses are small. We use the method Tsky = 180 K(ν/180 MHz)−2.6, as appropriate for regions
of Peacock & Dodds (1996) to transform this linear power well away from the Galactic Plane (Furlanettoet al. 2006).
spectrum into a power spectrum with nonlinear structure Thisresultsinlargereffectivenoiseforhigherredshiftmea-
formation. This method has not been tested thoroughly surements of the 21 cm emission. We will consider an ob-
against simulations of nonlinear structure formation with servation time of 90 dayswhich might be achievable within
massive neutrinos, but we expect that therelation between three seasons of observation and we will assume that the
thelinearandnonlinearpowerspectrumwillbeonlyweakly surveycovers 25% of thesky.
affected by this. There is no baryon acoustic oscillations in
thepower spectrum.
2.4.2 galaxy weak lensing survey
We assume that there are N species of neutrinoswith
ν
the same mass. The total density of neutrinos is fixed For comparison and for combining with the 21 cm lensing
by the physics at the time of decoupling to be Ωνh2 = wewillconsideramodelgalaxylensingsurvey.Thenoisein
m /93.14eV.Ifthemassesarenotdegeneratethemea-
ν ν
sured N will not be an integer. These are the parameters
ν
P
we will try to constrain. Roughly speaking Nν controls the 5 www.skatelescope.org/
4 R.B. Metcalf
powerspectrumestimatesfromsuchasurveycanbewritten
asN (ℓ)=σ2/n wheren istheangularnumberdensityof
κ ǫ g g
backgroundgalaxiesandσ istheroot-mean-squareintrinsic
ǫ
ellipticity of those galaxies. This neglects all systematic er-
rorsaswellasphotometricredshiftuncertainties.Following
standardassumptions,wemodeltheredshiftdistributionof
usable galaxies as η(z) z2e−(z/zo)1.5, where z is set by
o
∝
the desired median redshift, and we adopt σ = 0.25. The
ǫ
EUCLIDsatellite(theimagingpartofwhichwaspreviously
knownasDUNE6)proposestosurvey20,000squaredegrees
on the sky to a usable galaxy density of n 35 arcmin−2
g
≃
with a median redshift of z 0.9. Several planned ground-
∼
basedsurveys–LSST,PanSTARRS–willcovercomparable
areas to EUCLID at a similar depth. In order to use tomo-
graphic information, we divide the galaxies into 10 redshift
binseach containing thesame numberof galaxies.
Figure1.Theforecastedconstraintsonthecombinedmassand
number of neutrinos with the fiducial model {Pνmν,Nν} =
{0.66eV,3}marginalizedoveralltheothercosmological param-
2.4.3 CMB observations
eters. The dotted curves are for a EUCLID-like galaxy lensing
The Planck Surveyor7 will doa full skysurvey of theCMB survey,thedashedcurvesarefora21cmlensingsurvey,thesolid
curves are for the tomographic combination of the two surveys
radiation with higherresolution andmoresensitivitytopo-
andthedot-dashed curveisforPlanckbyitself.
larizationthanisnowavailable.Weincorporatethesefuture
measurementsintoourforecastsbycalculatingtheexpected
Fisher matrix. To do this we use thesame techniqueas de- 3 RESULTS
scribedintheappendixofRassat et al.(2008).Thisincludes
thecontributionsfromboththescalarandtensorperturba- Toassesshowwelltheproposedobservationscouldmeasure
tions. neutrino properties we adopt two statistical methods that
are popular in the literature – Fisher matrix forecasts and
theBayesian evidence method.
2.5 the cosmology
3.1 Fisher matrix forecast
Weusea11parametercosmologicalmodel.Theenergyden-
sities in dark energy and baryons are Ω and Ω in units Themaximumlikelihoodestimateforanyparametercanbe
Λ b
of the critical density. The density of matter is fixed to found by maximizing (12) with respect to that parameter.
Ω = 1 Ω to make the geometry flat. The primordial TheerrorinthisestimatorisoftenforecastusingtheFisher
m Λ
−
power spectrum is matrix defined as
Po(k)=As Hkc−1 ns+ddlnnsklnk (16) Fij =−fi∂∂p2il∂npLjfl. (17)
„ o «
The expected error in the parameter p , marginalized over
a
The dns parameter is included because if the primordial all other parameters, is σ2 F−1 . The unmarginalized
dlnk a ≃ aa
power spectrum is not a pure power-law it might partially errorestimate(theerrorwhenallotherparametersareheld
mocktheeffectofearlyneutrinofree-streamingonthepower fixed)is (F )−1. ` ´
aa
spectrum.Thetimedependentdarkenergyequationofstate Sinceℓ-modesseparatedbymorethantheresolutionof
has two parameters w(a) = w +w (1 a). There are two the telescope will not be correlated, we can break the like-
o a
neutrino parameter N and m . In−cluding dns and a lihood function up into factors representing each resolved
ν ν ν dlnk
non-constantwmakesthisamoregeneralsetofparameters region in ℓ-space (Metcalf & White 2007b). The result is
P
withmorepotentialdegeneraciesthanhasusuallybeenused that there are (2ℓ+1)f independent measured modes
sky
∼
when predicting constraints on massive neutrinos. The op- for each value of ℓ, where f is the fraction of sky sur-
sky
ticaldepthtoCMBlastscatteringsurfaceisτ.Thefiducial veyed. The Fisher matrix can then be further simplified to
model is set to τ,h,Ω h2,Ω h2,n , dns ,w ,w ,N = thewidely used form,
{ Λ b s dlnk o a ν}
0.09,0.7,0.343,0.0223,1,0, 1,0,3 . The normalization,
{As, is set so that the fluctua−tions w}ithin a sphere of radius F = 1 ℓmax (2ℓ+1)f tr C−1C, C−1C, . (18)
8 Mpc is σ = 0.75 in the fiducial model. Two values for ab 2 sky a b
Σ m areu8sed,0.66 eVand0.09 eV.Allcalculated param- ℓ=Xℓmin ˆ ˜
ν ν
eterconstraintsaremarginalizedovertheotherparameters. The error estimates on the neutrino parameters are
shown in figures 1 through 3 for combinations of different
dataandmodelassumptions.Figure1showstheconstraints
usingthethreekindsofdatabythemselvesandthecombina-
6 www.dune-mission.net tion of galaxy and 21 cm lensing. These lensing surveysare
7 www.rssd.esa.int/index.php?project=Planck combined tomographically not just by adding their Fisher
Neutrino Masses, Dark Energy and the Gravitational Lensing of Pregalactic HI 5
matricesaswouldbethecaseiftheywereindependentmea-
surements.Tenredshiftbinswithequalnumbersofgalaxies
are used and ten redshift bins of fixed z-width are used for
the 21 cm lensing. The CMB does a relatively good job of
constraining the N , but not as good a job of constraining
ν
thesum of themasses.
Infigure2theCMBconstraintsarecombinedwith the
lensingconstraints.Thereisadrasticimprovementbecause
the parameter degeneracies are significantly reduced (Note
thechangeinscalefrom figure1).Theforecasted errorsare
0.04 eVand0.1for thecombination oflensingsurveyswith
m = 0.66. This is an improvement of 2.6 in mass and
ν ν
1.3 in N over either of the lensing surveys by themselves.
ν
P
In figure 3 the dark energy equation of state is kept fixed
(w = 1). The improvement in the constraints illustrates
−
the degeneracy between neutrinos and dark energy param-
eters. The effect on the galaxy lensing constraints are par-
Figure 2. The same as figure 1 except the Planck constraints
ticularlystrong.Ifoneonlyconsiderscosmological constant
are combined with the lensing constraints and the scale has
been changed. Therearetwofiducial modelsat {Pνmν,Nν}= models the constraints improve to 0.03 eV and 0.04 for the
{0.66eV,3}and{Pνmν,Nν}={0.09eV,3}. combined lensing case.
Figure4showstheconstraintsinthew - m plane.
o ν ν
It can be seen more directly here how the degeneracy be-
P
tween dark energy and massive neutrinos is reduced by in-
cluding 21 cm lensing, especially when the neutrino masses
are small. Note also that if it were assumed that neutrinos
are massless, the maximum likelihood would not give the
correct valuefor w .
o
These calculations are in agreement with previous cal-
culations of the constraints from future galaxy lensing sur-
veys (Kitchinget al. 2008; Hannestad et al. 2006) where
more restricted cosmological models were used and thus
stronger constraints were found.
3.2 Bayesian evidence
Anotherstatistical question that could be asked is whether
thedatarequiresmassiveneutrinos.Onewaytoanswerthis
questionis usingtheaverage ratio oftheBayesian evidence
Figure 3. Same as figure 2 except the dark matter equation of
(Jeffreys1961)formodelswithandwithoutmassiveneutri-
stateisheldfixedat{wo,wa}={−1,0}.
nos. This technique has been used extensively in cosmolog-
ical parameter estimation (Liddle et al. (2006); Saini et al.
(2004); Heavens et al. (2007) for example) and galaxy lens-
ing surveysspecifically (Kitching et al. 2008).
The Bayesian evidence is the probability of the data,
D, given the model, M, marginalized over the parameters
of that model, θ ,
{ }
E(DM)= dθ p(Dθ,M)p(θM). (19)
| | |
Z
Theintegralisoveralloftheparameterspace.Bayesfactor
is theratio of the evidencesfor two competing models
E(DM )
B | o . (20)
≡ E(DM1)
|
Here model M is the simpler model with n parameters
o o
and model M is more complex and has more parameters,
1
n >n .
1 o
Figure4.Theconstraintsinthewo–Pνmν plane.Planckcon- Tomakeaforecast,Bayesfactormustbeaveragedover
straintsareincluded.Notetheimprovementinthewaconstraints expecteddatasets.ItwillbeassumedthatmodelMo isthe
especiallywhentheneutrinomassesaresmall. real case so that the averaging is according to this model.
WeareaskinghowwellwecanexpecttoroleoutmodelM .
1
6 R.B. Metcalf
wesee that theconstraints from thelensing of galaxies and
21 cm put similar constraints on the neutrino parameters.
By combining these data sets the constraints are improved
bymorethanafactoroftwoineachdimension.Acombined
mass as small as 0.2 eV should be detected, 0.19 eV if N
ν
is fixed to 3. Figure 6 shows how these constraints change
if the dark energy equation of state is fixed to the cosmo-
logical constant value, w ,w = 1,0 . The constraints
o a
{ } {− }
drastically improve which again demonstrates the degener-
acybetweendarkenergyandmassiveneutrinos.Inthiscase,
a combined mass of 0.09 eV would bedetectable.
4 DISCUSSION
If it is assumed that neutrinos are massless, or have too
Figure5.Thelogoftheaverageevidenceratiobetweenamodel smallamasstobeofsignificance,thedarkenergyequation
with and without massive neutrinos. The axis are the deviation ofstate,w=p/ρ,couldbeunderestimatedfrom cosmologi-
from the fiducial model {Pνmν,Nν} = {0,3}. The three con- calconstraints.Thiswouldbethecaseevenifthedatadoes
tourcurvesofeachtypearefor1,2.5and5.AccordingtoJeffreys notallowforadetectionofneutrinomass.Itisalsotruethat
(1961) |lnB|<1is“inconclusive”, 1<|lnB|<2.5is“substan- anyindependentconstraintontheneutrinomasswouldim-
tial”, 2.5 < |lnB| < 5 is “strong” and |lnB| > 5 is “decisive” prove future cosmological constraints on dark energy. The
evidencefortheadditionalparameters.Theexpectedconstraints KATRIN8beta-decayexperimentexpectstoreachalevelof
fromPlanckhavebeenincorporated.Thedottedcurvesarefora
0.2eVfortheelectronneutrinomassandthusmighthavean
EUCLID-like galaxy lensing survey, the dashed curves are for a
impactondarkenergyconstraints.Insection3itwasshown
21cmlensingsurveyandthesolidcurvesareforthetomographic
thataneutrinomassassmallas m /N =0.03eVcould
combinationofthetwosurveys. ν ν ν
bias w high by 1 σ in future galaxy lensing surveys if it
o
∼ P
isnotaccounted forandif itis accountedfortheerrorbars
increase by a factor of 3.4 for w and 2 for w . Lensing
o a
of high redshift 21 cm is a way to reduce this degeneracy
by adding constraints on the growth of structure at higher
redshifts where dark energy is presumed to contribute very
little. Type Ia Supernovae surveys and surveys aimed at
measuring the baryon acoustic oscillations at z 1 will
∼
put constraints on dark energy that are based on the lumi-
nosity distance redshift relation and so are independent of
thegrowthofstructure.Itisimportantthatmethodsbased
bothonstructureformation andonluminosity distanceare
fullyrealizedtoavoidsystematicerrors.Thetwoprobesare
alsonecessarytotestalternativetheoriesofgravityonlarge
scales which might also provide an explanation for the ap-
parent acceleration of thecosmological expansion.
Acknowledgments
IwouldliketothankS.Whiteforhelpfuldiscussionand
Figure 6. Same as figure 5 except the dark matter equation of
criticism.IwouldalsoliketothankJ.Wellerforloaningme
stateisheldfixedat{wo,wa}={−1,0}.
hiscodeforcalculatingthePlanckpriorsonthecosmological
parameters.
Incalculating thisweusetheapproximation to B derived
h i
by Heavenset al. (2007) theSavage-Dickey ratio
F(1) 1 REFERENCES
hBi≃(2π)−(n1−no)/2s||F(o)||exp»−2δθνFν(µ1)δθµ– (21) Eisenstein, D. J. & Hu, W.1999, ApJ, 511, 5
where Field, G. B. 1959, ApJ,129, 536
Fogli, G. L., Lisi, E., Marrone, A., & Palazzo, A. 2006,
−1
δθi = F(o) Fj(α1)δθα (22) Progress in Particle and Nuclear Physics, 57, 742
»“ ” –ij Furlanetto, S. R., Oh, S. P., & Briggs, F. H. 2006,
and the range of the indexes are i=1...n , α=n +1...n Phys.Rep.,433, 181
o o 1
and µ,ν = 1...n . We have assumed that there are no a Hannestad, S.2005, Physical Review Letters, 95, 221301
1
priori limits on therange of theneutrinoparameters.
Figure5showsevidenceratioasafunctionofthedevi-
ationfromthefiducialmodel,{ νmν,Nν}={0,3}.Again 8 www-ik.fzk.de/katrin/
P
Neutrino Masses, Dark Energy and the Gravitational Lensing of Pregalactic HI 7
Hannestad, S., Tu, H., & Wong, Y. Y. 2006, Journal of
Cosmology and Astro-Particle Physics, 6, 25
Heavens,A.F.,Kitching,T.D.,&Verde,L.2007,MNRAS,
380, 1029
Hilbert, S., Metcalf, R. B., & White, S. D. M. 2007, MN-
RAS,382, 1494
Jeffreys,H.1961,TheoryofProbability(OxfordUniversity
Press)
Kaiser, N. 1987, MNRAS,227, 1
Kiakotou, A., Elgaroy, O., & Lahav, O. 2007, ArXiv e-
prints,709
Kitching, T. D., Heavens, A. F., Verde, L., Serra, P., &
Melchiorri, A. 2008, Phys.Rev.D,77, 103008
Lesgourgues, J. & Pastor, S. 2006, Phys.Rep.,429, 307
Liddle, A. R., Mukherjee, P., Parkinson, D., & Wang, Y.
2006, Phys.Rev.D,74, 123506
Lu, T. & Pen, U.-L.2007, ArXiv e-prints, 710
Madau,P.,Meiksin,A.,&Rees,M.J.1997,ApJ,475,429
McQuinn,M.,Zahn,O.,Zaldarriaga, M., Hernquist,L.,&
Furlanetto, S.R. 2006, ApJ, 653, 815
Metcalf, R.B. & White, S. D.M. 2007a, in preparation
—. 2007b, MNRAS,381, 447
Morales, M. F. 2005, ApJ, 619, 678
Peacock, J. A.& Dodds, S. J. 1996, MNRAS,280, L19
Rassat, A., Amara, A., Amendola, L., Castander, F. J.,
Kitching,T.,Kunz,M.,Refregier,A.,Wang,Y.,&Weller,
J. 2008, ArXive-prints
Saini,T.D.,Weller,J.,&Bridle,S.L.2004,MNRAS,348,
603
Seljak, U., Makarov, A., McDonald, P., Anderson, S. F.,
Bahcall, N. A., Brinkmann, J., Burles, S., Cen, R., Doi,
M.,Gunn,J.E.,Ivezi´c,Zˇ.,Kent,S.,Loveday,J.,Lupton,
R.H.,Munn,J.A.,Nichol,R.C.,Ostriker,J.P.,Schlegel,
D. J., Schneider, D. P., Tegmark, M., Berk, D. E., Wein-
berg,D.H.,&York,D.G.2005, Phys.Rev.D,71,103515
Spergel, D.N.,Bean,R.,Dor´e,O.,Nolta,M. R.,Bennett,
C.L.,Dunkley,J.,Hinshaw,G.,Jarosik,N.,Komatsu,E.,
Page,L.,Peiris,H.V.,Verde,L.,Halpern,M.,Hill,R.S.,
Kogut,A.,Limon,M.,Meyer,S.S.,Odegard,N.,Tucker,
G. S., Weiland, J. L., Wollack, E., & Wright, E. L. 2007,
ApJSup.,170, 377
Zahn, O. & Zaldarriaga, M. 2006, ApJ, 653, 922
Zaldarriaga, M., Furlanetto, S. R., & Hernquist, L. 2004,
ApJ,608, 622