Small-Scale Cosmic Microwave Background Anisotropies as a Probe of the Geometry of the Universe

7 Pages·0.17 MB·English

by  Marc Kamionkowski

IASSNS-HEP-93/91
POP-548
CfPA-94-01
UTAP-173
December 1993
Small-Scale Cosmic Microwave Background Anisotropies
as a Probe of the Geometry of the Universe
1z 2(cid:3) 3;4(cid:14)
Marc Kamionkowski , David N. Spergel , and Naoshi Sugiyama
1
School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540
2
Princeton University Observatory, Princeton, NJ 08544.
3
Department of Astronomy, University of California, Berkeley, CA 94720
4
Department of Physics, Faculty of Science, University of Tokyo, Tokyo 113, Japan
ABSTRACT
Weperformdetailedcalculationsofcosmicmicrowavebackground(CMB)anisotropiesinaCDM-dominated
open universe with primordial adiabatic density perturbations for a variety of reionization histories. We show
that to a great extent, the CMB anisotropies depend only on the geometry of the Universe, which in a matter
dominated universe is determined by (cid:10), and the optical depth to the surface of last scattering. In particular,
the location of the primary Doppler peak depends primarily on (cid:10) and is fairly insensitive to the other un-
known parameters, such as (cid:10)b, h, (cid:3), and the shape of the power spectrum. Therefore, measurements of CMB
anisotropies on small scales may be used to determine (cid:10).
Submitted to The Astrophysical Journal Letters
z
[email protected]
(cid:3)
[email protected]
(cid:14)
[email protected]
49 naJ 3 3001049/hp-ortsa
1. INTRODUCTION 2. REIONIZATION
The recent discovery of anisotropies in the cosmic microwave background In the standard big-bang scenario, most of the plasma in the Universe
(CMB) (Smootetal. 1992)hasushered inanewera incosmology(forarecent
recombinedtoformneutralhydrogenandheliumatz (cid:24)1300. Yetobservations
review,seeWhite,Scott,&Silk1993). Althoughthecurrent detectionsarestill
of highredshift quasars at z (cid:24)4 implythat the intergalactic mediumhas been
accompanied by signi(cid:12)cant uncertainties, the CMB anisotropies on all angular
mostly ionized since at least that redshift (Jenkins & Ostriker 1991). This
scales willsoon be mapped out with great precision.
impliesthat sometimebetween z (cid:24)4 and z (cid:24)1300 the Universe was reionized
The detailed shape of the CMB spectrum can depend quite sensitively on
perhaps by a burst of star formation or by early active galactic nuclei. This
theuncertainparametersinvolved,anditispossiblethatdetailedmeasurements
reionizationatredshiftzRhasasigni(cid:12)cante(cid:11)ectontheCMBspectrum(see,e.g.
ofCMBanisotropiescanbeusedtodeterminesomeoftheuncertainparameters
Sugiyama,Silk,& Vittorio1993)as it suppresses (cid:13)uctuations at angularscales
such as the Hubble constant, the baryon density, and spectral index (Bond et
1=2
much smaller than the horizon size at that redshift, (cid:18)H (cid:24) 2((cid:10)=zR) , and
al. 1993). In thisLetter,we pointout that CMB anisotropies mayalsoprovide
produces additional (cid:13)uctuations both through Doppler scattering at angular
informationon the value of (cid:10).
In a previous paper (Kamionkowski& Spergel 1993; hereafter referred to scales comparable to (cid:18)H and at arcsecond to arcminute scales through second
as paper I), we computed the large-angle anisotropies in a CDM-dominated order e(cid:11)ects (Ostriker & Vishniac 1986, Vishniac 1987).
open universe with primordialadiabatic density perturbations. We found that If we assume that the (cid:12)rst stars are similarto those observed today, then
modelswith0:35<(cid:10)<0:8and 0:4<h<0:6(where his the Hubble constant we can estimate the number of ionizing photons produced per hydrogen atom
(cid:24) (cid:24) (cid:24) (cid:24)
(cid:0)1 (cid:0)1
in units of 100 km sec Mpc ) are compatible with the observed amplitude turnedintostars. Mostoftheionizingradiationisproducedbymassivestarson
oflarge-angleCMB(cid:13)uctuations,the amplitudeandspectrum of(cid:13)uctuations of the mainsequence. A 30M(cid:12) star burns 7M(cid:12) of hydrogenreleasing 0:05M(cid:12)c2
galaxycountsintheAPM,CfA,andQDOTsurveys,andanageoftheUniverse of energy. Roughly 1/4 of this energy is emitted as ionizing radiation (Kurucz
greater than 13 Gyr.
1979). Since these massive stars account for approximately 1/4 of the initial
The analytic treatment in Paper I is valid only for angular scales which
mass function, star formation releases (cid:24) 100 keV in ionizing radiation per
probe comoving distance scales well outside the horizon at the surface of last
processed baryon. For an O star spectrum, for a typical ionization, roughly
scattering. On angular scales smaller than those probed by COBE, peculiar
13.6eV goes to ionize hydrogen and an additional4 eV goes into photoelectric
motions of the plasma at the surface of last scattering must be taken into
heating. Thus,thereare(cid:24)5000ionizingphotonsemittedperprocessed baryon.
account. An accurate treatment requires numerical solution of the Boltzmann
(cid:0)4
This suggests that a minimumof 2(cid:2)10 of the Universe must be in bound
equations.
objects for the Universe to be mostlyreionized.
Inthe next Section,wediscuss thereionizationofthe Universe byanearly
These ionizing photons will create a Stromgren sphere of ionized material
generation of star formation and argue that reasonable reionization histories
around each collapsed region. The mass of this Stromgren sphere will be de-
imply an optical depth to Thompson scattering of (cid:24) 0:5(cid:0)1 between us and
(2)
the redshift of recombination. In Section 3, we brie(cid:13)y describe our numerical termined by the balance between ionization and (cid:11) (T), the temperature (T)
results, and compare with our previous analytic calculations. We also show dependent recombinationrate to energy levels above the ground state (Spitzer
that the CMB spectrum depends primarilyon (cid:10) and the optical depth to the 1978). The number of ionizing photons needed per hydrogen atomto keep the
(2)
surface of last scattering. In the (cid:12)nal Section, we make concluding remarks. gas mostlyionized at redshift z is therefore, (cid:24)t(z)=ne(z)(cid:11) (T), where t(z) is
1 2
Here, (cid:27)(MJ;0) is the amplitude of mass (cid:13)uctuations today at the Jeans mass
scale and D is the linear-theory growth factor. The Jeans mass after recombi-
6
nation is (cid:24) 10 M(cid:12). Fig. 1 shows the fraction of mass in collapsed objects as
(cid:0)1=2 3=2
a function of (cid:28)(z) ' 0:04(cid:10)bh(cid:10) xez for di(cid:11)erent COBE-normalized cos-
mologies. Using our estimate of required mass fraction for ionization implies
that the Universe is reionized at a redshift of 40{ 50 creating an opticaldepth
toThompsonscattering between us andlargeredshifts of(cid:24)0:5(cid:0)1:Bardeen et
al. (1987) argue that unreionized open models violate limits from the OVRO
experiment on the amplitude of (cid:13)uctuations on small angular size. However,
the opticaldepth estimated here is su(cid:14)cient to smoothout the (cid:13)uctuations on
these scales and evade this limit. As we will see in the next Section, ongoing
experiments will enable us to constrain (cid:28) for this familyof models.
3. RESULTS
The CMB spectra are computed using the numericaltechniques described
by Sugiyama& Gouda (1992). We assume a scale-invariant Harrison-Peebles-
FIG.1. This (cid:12)gure shows the fractionofthe massof the Universe incollapsed
Zel'dovichprimordialspectrum of density perturbations. On scales larger than
objects as a function ofthe opticaldepth between the present and a givenred-
2 the curvature scale, the de(cid:12)nition of scale invariance is somewhat ambiguous,
shift. These calculations assumed h = 0:6, (cid:10)bh = 0:015 and a scale-invariant
butinPaperIitwasshownthatthisambiguitya(cid:11)ectsonlythelowestmultipole
spectrumofdensity(cid:13)uctuations. Theshadedregioncorresponds tothefraction
moments(l <9for(cid:10)'0:1andl<5for(cid:10)'0:3),andthattheCMBspectrum
of baryons in stars required to ionize the Universe. (cid:24) (cid:24)
isrelativelyinsensitivetotheexactshapeofthepowerspectrumonscaleslarger
the Hubble time,andne(z) is the electron density. Forreasonable assumptions than the curvature scale.
about the electron temperature, this implies that (cid:24) 5(cid:0)20 photons must be In Fig. 2 we plot the COBE-normalized CMB spectrum as a function of
producedperhydrogenatomtokeeptheUniversereionizedbackto(cid:28) (cid:24)1. This multipole moment l for several values of (cid:10) and for optical depths (cid:28) = 0 (no
(cid:0)1=2 3=2
suggests that the Universe should become reionized when the fraction of mass reionization) and (cid:28) = 1, where (cid:28) ' 0:04(cid:10)bh(cid:10) xe(zls) . Here, (cid:10)b is the
(cid:0)4 (cid:0)3
in bound objects reaches a level of (cid:24)2(cid:2)10 (cid:0)4(cid:2)10 . mass density of baryons, zls is the redshift of the surface of last scattering,
We can estimate the fractional mass of the Universe that has collapsed and xe is the ionization fraction. Note that the location of the (cid:12)rst Doppler
into bound objects of mass greater than the Jeans mass, fcoll(MJ), by using peak increases as (cid:10) is decreased, and is insensitive to the ionization history.
the Press-Schechter formalism(Press & Schechter 1974): Reionization decreases the amplitude of the Doppler peak by roughly a factor
(cid:18) (cid:19) (cid:0)2(cid:28)
1 1:68 e , and leads to another smaller peak at lower l which arises from peculiar
fcoll(MJ;(cid:10);z)= Erfc p : (2:1)
2 2(cid:27)(MJ;0)D((cid:10);z) motions of scatterers at low redshift in a reionized model. At large l, the
3 4
results are consistent with Bond et al. (1993).
(cid:10) (cid:28) SP Sask MAX MSAM2 MSAM3 ARGO WD
0.2 0.0 1.1 1.1 2.1 2.3 1.6 1.3 2.0
0.5 0.0 1.4 1.4 2.8 3.1 2.1 1.7 1.9
1.0 0.0 1.4 1.4 2.6 2.8 1.7 1.6 1.5
0.2 0.5 0.8 0.8 1.3 1.5 0.9 0.8 1.1
0.5 0.5 1.0 1.0 1.8 1.9 1.3 1.1 1.1
1.0 0.5 1.0 1.0 1.7 1.9 1.1 1.2 1.0
0.2 1.0 0.8 0.8 1.0 1.2 0.6 0.8 0.7
0.5 1.0 1.0 1.0 1.4 1.5 0.8 1.0 0.8
1.0 1.0 1.0 1.0 1.4 1.6 0.8 1.0 0.7
0.2 2.0 0.9 0.8 0.9 1.1 0.4 0.8 0.4
0.5 2.0 1.0 1.0 1.1 1.3 0.5 0.9 0.5
1.0 2.0 1.0 1.0 1.1 1.3 0.5 0.9 0.5
Observed 0:86(cid:6)0:37 1:2(cid:6)0:4 4:7(cid:6)0:8 3:8(cid:6)1 1:8(cid:6)0:5 2:1(cid:6)0:5 <2:2
FIG. 2. The COBE-normalized CMB spectrum as a function of multipole
1.6 1:6(cid:6)0:5 1:5(cid:6)0:4
momentl for several values of (cid:10) and for opticaldepths (cid:28) =0 (no reionization)
and (cid:28) =1. Here we have taken (cid:10)b =0:06and h=0:5. Throughout the paper,
(cid:3)=0. Table 1. Predictions for ((cid:1)T=T)rms(cid:2)105 for several experiments for various
values of (cid:10) and (cid:28) and the reported level of observed CMB (cid:13)uctuations. The
multipole spectrum drops sharply at the angular scale subtended by the Silk two values quoted for the MSAM experiment represent the r.m.s. tempera-
ture (cid:13)uctuations with and without \sources" (Cheng et al. 1993, Page 1993).
dampingscale at recombination. The location of this fall-o(cid:11)is sensitive to (cid:10).
The two values for the MAX experiment are from two di(cid:11)erent regions of the
Interestingly enough, in all these models the spectra on COBE scales,
sky. The higher value (Gundersen et al. 1993) is reported as a detection. The
l <(cid:24) 13, are very close to (cid:13)at, especially when we consider the e(cid:11)ect of cosmic lower value (Meinhold et al. 1993) may be (cid:13)uctuation due to free-free emis-
variance; at l = 10 this introduces a 1(cid:27) error of about 30% in the prediction. sion,synchrotron emission,or mayrepresent CMB (cid:13)uctuations. For the White
Dish experiment (Tucker et al. 1993), we use the \Method I" 2-beam di(cid:11)er-
The numerical results for the shape and amplitude of the spectrum on these
encing scheme rather than the \Method II" quadrupole di(cid:11)erencing scheme.
scales agrees well with the analytical results in Paper I, especially for the case
The ARGO (de Bernardis et al. 1993) and SP (Schuster et al. 1993) results
of no reionization. On scales l >15, the tail end of the Doppler peak becomes
(cid:24) are the amplitudes of Gaussian correlation function; the r.m.s. temperature
important and a numerical calculation is needed. For (cid:10) = 1, the numerical (cid:13)uctuations will di(cid:11)er slightly fromthese numbers.
5 6
scaling with (cid:28). Variation of any of the parameters with (cid:12)xed (cid:28) has relatively
little e(cid:11)ect on the CMB spectrum.
Fig. 2 and Fig.3 show that variations in the location of the Doppler peak
are highly insensitive to all the uncertain parameters except for the geometry
of the Universe. Although we have used only scale-invariant spectra, the re-
sults of Bond et al. (1993) show that the location of the Doppler peak is also
relatively insensitive to changes in the primordial spectral index, the addition
ofacosmologicalconstant,or thee(cid:11)ects ofgravitationalwaves. Ifthe Universe
is (cid:13)at but dominated by a cosmological constant, then the total mass density
(non-relativistic matter plus vacuum-energy density) of the Universe is (cid:10)= 1,
and the Doppler peak is still located at l ' 200 (Bond et al. 1993). The ro-
bustness of the prediction for the location ofthe Doppler peak as a function of
(cid:10) should comeas no surprise: This simplyre(cid:13)ects the angularscale subtended
by the horizon at the surface of last scattering.
Informationon the value of (cid:10) should soon be availableeven without fully
mapping out the angular CMB spectrum around the Doppler peak. If (cid:10) = 1,
FIG. 3. The COBE-normalized CMB spectrum as a function of multipole
then the amplitudeat l (cid:24)200is greater than that at l(cid:24)800,but if (cid:10)<1,the
momentl for several values of (cid:10)b and h. In all models,(cid:10)=0:3 and (cid:28) =1.
ratio is reversed. In Figure 4, we plot the predictions of scale-invariant models
withdi(cid:11)erentionizationhistoriesandvaluesof(cid:10). Figure4showsthatallofthe
In Table 1, we list the predictions for ((cid:1)T=T)rms for the various models
(cid:13)at models tend to along a curve in this temperature ratio plot. The various
for several di(cid:11)erent CMB experiments. All of the models are computed for
vacuum-dominated and tilted models studied by Bond et al. (1993) also lie
(cid:10)b = 0:06 and h = 0:5. Note that the detection of CMB (cid:13)uctuations at the alongthiscurve. Thepositionalongthe curvedepends mostlyuponthe optical
(cid:0)5
level of greater than 10 by the MAX (Meinhold et al. 1993; Gundersen
depth of the Universe. Open models with (cid:10) < 0:5 lie in upper portion of the
et al. 1993), MSAM (Cheng et al. 1993; Page 1993), SP (Schuster et al.
plot. This suggests a combinationof experiments probing l (cid:24)200 and l (cid:24) 700
1993), Saskatoon (Wollacket al. 1993), and ARGO (de Bernardis et al. 1993) could potentiallydetermine the geometry of the Universe.
experiments suggest that the optical depth of the Universe due to reionization
is less than 2. This is consistent withour theoretical expectations for Gaussian
4. CONCLUSIONS
theories.
In Fig. 3 we plot the CMB spectrum for several values of (cid:10)b and h for Inpaper I,weshowed that COBE-normalizedlow-(cid:10)modelsare consistent
(cid:10)=0:3,and for (cid:12)xed optical depth. The similarityof the curves demonstrates with galaxy counts in large-scale surveys and with the age of the Universe.
thatthe primarye(cid:11)ect ofvarying(cid:10)b,h,zls, andxe canbe determinedby their Due to cosmic variance, as well as the uncertain contribution of tensor modes,
7 8
We have also estimated when the (cid:12)rst generation of stars mayhave reion-
ized the intergalactic medium in these models. Because of the strong depen-
dence of the number of collapse objects on redshift, the predicted epoch of
reionization is relatively insensitive to assumption about the details of primor-
dialstars(aslongassomemassivestarsarepartofthis(cid:12)rststellargeneration).
In the low (cid:10) models considered here, we expect that the Universe should have
an opticaldepth of(cid:24)0:5(cid:0)1 between us and the epoch ofrecombination. This
additionalopacitytophotonscatteringreduces thepredicted amplitudeof(cid:13)uc-
tuations on smallangular scales but does not shift the location of the Doppler
peak.
1=2
The location of the Doppler peak, l (cid:24) 200=(cid:10) re(cid:13)ects the angular scale
subtended by the horizon at the surface of last scattering, so it should be
no surprise that it is relatively insensitive to other cosmological parameters.
Similarly, the location of the sharp drop-o(cid:11) in the multipole spectrum at l (cid:24)
3 (cid:0)3=4 1=2
9(cid:2)10 (cid:10) ((cid:10)bh) , the angle subtended by the Silk damping scale at the
surface of last scatter, also depends on (cid:10). Although the amplitude of the
FIG. 4. The points in this plot represent the predicted ratios of temperature
Doppler peak may be suppressed by reionization, it is quite likely that it will
(cid:13)uctuations fordi(cid:11)erent cosmologicalparameters. The open symbolsrepresent
still be distinguishable for reasonable ionization histories.
(cid:28) =0models,thestar-shaped symbolsrepresent (cid:28) =0:5and(cid:28) =1models,the
Current measurements are still too uncertain to discriminate between the
(cid:12)lled symbols represent the (cid:28) = 2 models. The triangular and three-pointed
various values of (cid:10). Accurate results fromthe White Dish and MSAM experi-
symbolsrepresent (cid:10)=0:2models. Thesquare andfour-pointedsymbolsrepre-
mentsshouldbe abletodistinguishbetween a(cid:13)atuniverse andauniverse with
sent (cid:10)=0:5models. The pentagons and (cid:12)ve-pointed symbolsrepresent (cid:10)=1
(cid:10) (cid:24) 0:3. Experiments with higher resolution which probe multipole moments
models. The solid, dashed and dotted lines connect the (cid:10) = 0:2, (cid:10) = 0:5 and
100 < l < 1000 (angular scales 0:1(cid:14) < (cid:18) < 1(cid:14)) will be able to make the deter-
(cid:24) (cid:24) (cid:24) (cid:24)
(cid:10)=1 models. In the source-free MSAM dataset, MSAM2/COBE = 1:6(cid:6)0:5:
minationmore precise. We hope that the ideas presented here will give added
However, if the sources are included MSAM2/COBE = 3:5(cid:6)1:0. The current
impetus to the development of such detectors.
White Dish upper limitis not yet strong enough to constrain models.
discriminationbetween thevariousvaluesof(cid:10)basedonmeasurementsoflarge-
5. ACKNOWLEDGMENTS
angleCMBanisotropiesalonewascurrently impossibleandlikelytoremainso.
InthisLetter, wehaveperformeddetailednumericalcomputationsofthe CMB We thank T. Herbig, W. Hu, N. Gouda, L. Page, J. Silk, G. Tucker, and
spectrum on smallerangular scales and found that informationon the value of M. White for fruitful discussions. We would like to thank L. Page, G. Tucker,
(cid:10) is most likely encoded in the angular spectrum of CMB anisotropies. and M. White for providing us with appropriate window functions. MK was
9 10
supported in part by the U.S. Department ofEnergy under contract DEFG02- Smoot,G. F. et al. 1992,ApJ (Letters), 396, L1
90-ER40542. DNS was supported in part by NSF grant AST88-58145 (PYI)
Spitzer, L. 1978, Physical Processes in the Interstellar Medium (Wiley Inter-
and NASA grant NAGW-2448. NS was supported by a JSPS (Japan Society
science: New York), p. 107
of the Promotionof Sciences) Postdoctoral Fellowshipfor Research Abroad.
Sugiyama,N. & Gouda, N. 1992, Prog. Theor. Phys., 88,803
Sugiyama,N., Silk,J., & Vittorio, N. 1993,ApJ (Letters), 419,L1
Tucker, G. S., Gri(cid:14)n, G. S., Nguyen, H. T., & Peterson, J. B. 1993, ApJ
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