Table Of ContentCosmological scalar field perturbations can grow
Miguel Alcubierre,1 Axel de la Macorra,2 Alberto Diez-Tejedor,3,4 and Jos´e M. Torres1
1Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma de M´exico,
Circuito Exterior C.U., A.P. 70-543, M´exico D.F. 04510, M´exico
2Instituto de Fisica, Universidad Nacional Auto´noma de M´exico,
Circuito Exterior C.U., A.P. 20-364, M´exico D.F. 04510, M´exico
3Santa Cruz Institute for Particle Physics and Department of Physics,
University of California, Santa Cruz, CA, 95064, USA
4Departamento de F´ısica, Divisio´n de Ciencias e Ingenier´ıas,
Campus Le´on, Universidad de Guanajuato, Le´on 37150, M´exico
(Dated: September17, 2015)
Ithasbeenargued thatthesmallperturbationstothehomogeneous andisotropic configurations
5
of a canonical scalar field in an expanding universe do not grow. We show that this is not true
1
in general, and clarify the root of the misunderstanding. We revisit a simple model in which the
0
zero-modeofafreescalarfieldoscillates withhighfrequencyaroundtheminimumofthepotential.
2
Under this assumption the linear perturbations grow like those in the standard cold dark matter
p scenario, but with a Jeans length at the scale of the Compton wavelength of the scalar particle.
e
Contrarytopreviousanalysesintheliteratureourresultsdonotrelyontime-averagesand/orfluid
S
identifications, and instead we solve both analytically (in terms of a well-defined series expansion)
5 and numerically the linearized Einstein-Klein-Gordon system. Also, we use gauge-invariant fields,
1 which makes the physical analysis more transparent and simplifies the comparison with previous
works carried out in different gauges. As a byproduct of this study we identify a time-dependent
] modulation of the different physical quantities associated to the background as well as the pertur-
c
bations with potential observational consequences in dark matter models.
q
-
r PACSnumbers: 98.80.-k,98.80.Jk,95.35.+d,03.50.-z
g
[
I. INTRODUCTION ing of the problemshows that the contrastin the energy
2
density freezes.1
v
8 Perfectfluidstructureformationthendemandsmatter
ItisdifficulttooverestimatetherelevanceoftheJeans
1 with a low speed of sound in order to have a window,
9 instabilityinmodernphysicalcosmology. Inordertoun- c /H <λ<1/H, where the Jeans mode is released and
s
6 derstand the emergence of cosmic structure, we need a
the perturbations can grow. Cold dark matter (CDM)
0 mechanismthattransformsthe nearlyhomogeneousand
represents the simplest realization of this scenario, for
1. isotropicearlyuniverseweinferfrome.g. the cosmicmi- which one assumes c2 0 for all relevant modes.2
0 crowavebackgroundobservations,to the highly clumped For a canonical scsal≈ar field we have c2 = 1, see e.g.
5 one we can see today at the scale of galaxy clusters and s
Ref. [3], and it is usually argued that this implies that
1 below. Inthe standardcosmologicalscenariothis transi-
: smallperturbationsintheenergydensitydonotgrow[4].
v tionis possiblethanks to the instability ofascalarmode
This would seem to be a very serious argument against
i thatappearswhenwe coupledarkmatter (DM) to grav-
X the whole scalar field DM program [5], so we find it
ity, the gravitational Jeans instability.
mandatory to clarify the issue.
r
a In a universe dominated by a barotropic perfect fluid In this paper we will consider the simplest situation:
with equation of state p = p(ε) ε, the behavior of a universe dominated by a real massive scalar field ϕ
≪
the small perturbations in the energy density depends satisfying the Klein-Gordon equation, (✷+m2)ϕ = 0.
crucially on the speed of sound, c2 = ∂p/∂ε, and the
s
Hubble radius, H−1 (throughout this paper we use nat-
ural units such that c = ~ = 1). Roughly speak-
ing, we can distinguish three different regions in Fourier 1 Actually all these statements are gauge dependent. We can al-
space [1]: i) On scales smaller than the Jeans length, wayschoosetoworkine.g. theuniformdensitygauge,wherethe
λ = c (π/Gε )1/2 c H−1, the contrast in the energy contrastintheenergydensityvanishesidenticallyatallscales,or
J s 0 s
∼ ine.g. thesynchronous gauge, wherethecontrastintheenergy
density oscillates with damping amplitude, due to the
density grows even for those modes larger than the Hubble ra-
stabilizingeffectofpressureandtheexpansionoftheuni-
dius. Itisonlyintermsoftheconformal-Newtoniangaugethat
verse,respectively. HereGistheNewtonconstantandε0 thebehavioroutlinedinthepreviousparagraphmakessense,and
is the background matter density. ii) Above this length it is only in this gauge that we can easily compare our results
scalebutstill belowthe Hubble radius,self-gravitydom- withthosesketchedinthethreepointsabove.
2 IfDMconsistsofcollisionlessparticlesitismoreappropriateto
inates and the contrast in the energy density grows: the
talkaboutafree-streaming,ratherthanaJeans,length;however,
Jeans instability comes into play. iii) Finally, at scales
theideaissimilar,seee.g. Sections 10.2and10.3inRef.[2]for
larger than the Hubble radius a relativistic understand- adiscussion.
2
damped oscillations growing mode freeze down We will follow the notation in Chapters 7 and 8 of
−−−−−−−−−−−−−−−−−|−−−−−−−−−−−−−−−|−−−−−−−−−−→ λ
λJ H−1 Ref. [1]; in particular we use the signature (+,−,−,−)
forthespacetimemetric. We highlyrecommendthis ref-
FIG. 1: Behavior of the linear perturbations in the contrast erence to the reader interested in the details about some
to the energy density for different wavelengths λ. The Jeans ofthedefinitionsandconventionsbelow. Oneshouldalso
lengthλJ dependsonthemattercontentintheuniverse. For mentionthatrecentlytheproblemofstructureformation
aperfectfluidλJ ∼csH−1,andstructureformationdemands with a massive scalar field has been studied by perform-
c2s ≪1. However,whenacanonicalscalarfieldoscillateswith ing full nonlinear numerical simulations of the Einstein-
highfrequencyaroundtheminimumofthepotentialwehave
Klein-Gordonsystem, both in the relativistic [9] and the
λ ∼m−1, even though c2 =1. As usual this picture should
J s nonrelativistic [10]regimes. However,we believe thatby
be understood in theconformal-Newtonian gauge.
studying the problem from the point of view of a mode
analysisinperturbationtheoryonecanmoreclearlysep-
arate the relevant physical mechanisms that come into
Here the box is the d’Alembert operator in four space-
play at different scales.
time dimensions, and m is the mass of the scalar par-
ticle [this corresponds to a scalar field potential of the
form V(ϕ) = m2ϕ2/2, so that the mass is defined as
II. THE HOMOGENEOUS AND ISOTROPIC
m2 ∂2V/∂ϕ2]. The more interesting case with a com-
BACKGROUND
≡
plexfieldincludingthe self-interactionsandthepresence
ofothermattercomponents(e.g. radiation,baryons,etc)
At very large scales the universe is (nearly) homoge-
will be presentedelsewhere. We willshow that whenthe
neous and isotropic; that makes it possible to introduce
scalar field is slowly rolling down the potential, linear
the idea of a homogeneous and isotropic background.
perturbations cannot grow, in accordance with common
According to the current cosmological observations this
wisdom. However, when the scalar field oscillates with
backgroundcanbedescribedintermsofaflatRobertson-
high frequency m H around the minimum of the po-
≫ Walker (RW) metric of the form
tential term, the Jeans length is not given by the naive
value csH−1 one would guess from a perfect fluid anal- ds2 =a2 dη2 dx2 dy2 dz2 . (1)
ogy. InsteaditisdeterminedbytheComptonwavelength − − −
of the scalar particle, c m−1, and the evolution of per- Here η is the confo(cid:0)rmal cosmological tim(cid:1)e, (x,y,z) a
s
turbations larger than this scale almost mimics that of spatial coordinate system comoving with the expansion,
thestandardCDMscenarioeventhoughc2 =1(seeFig- and a(η) the scale factor. Conformal time η is related
s
ure1fordetails). Themaindifferencewithrespecttothe to the standard comoving cosmological time t through
standard CDM evolution (apart from the appearance of t = adη. The expansion rate is codified in the Hubble
a nonvanishing Jeans length), is the presence of a time- parameter, H = /a = a′/a2, with the prime denot-
R H
dependent modulation of the different physical quanti- ing the derivative with respect to conformal time. The
ties. We will present below both an analytical argument spacetimesymmetriesinEq.(1)guarantythattheback-
based on expansions of the solution of the relevant cos- ground field cannot depend on the spatial coordinates,
mological equations, and results from simple numerical so that ϕ(η,~x) = ϕ0(η). Under these assumptions the
simulations. Klein-Gordon equation simplifies to
Similarresultshavebeenpresentedbeforeinthelitera- ϕ′′+2 ϕ′ +a2m2ϕ =0. (2)
ture,seeforinstanceRefs.[6](asfarasweknowthegrav- 0 H 0 0
itational instability of a canonical scalar field —in the There are two different timescales in Eq. (2): on the
context of a static universe— was reported for the first one hand that associated to the cosmologicalexpansion,
time by Khlopov, Malomed and Zel’dovich in Ref. [7]). −1 = (aH)−1, and on the other that defined by the
However,to our knowledge, we present for the first time Hmass of the scalar field, (am)−1. In this paper we will
a description of the problem in terms of gauge-invariant concentrate on the case when the expansion rate of the
fields. Furthermore, note that our analysis does not rely universeisveryslowwhencomparedtothecharacteristic
on time averages and/or fluid identifications. Instead time of oscillation of the scalar field, that is (aH)−1
we present a well-defined series expansion that makes it (am)−1,whichimpliesH m. Thisinequalityisalwa≫ys
≪
possible to find analytic solutions order by order in the satisfied at late times. We will briefly discuss the regime
expansion parameter: it is well known that the time av- H m later on in this section.
≫
erageoftheproductoftwofunctionsisnotingeneralthe In order to proceedwe propose a solution of the form
productof the time averagesofthose functions, and also
ϕ (η) = ϕH(η) ϕm(η)+ 2(H/m) , (3a)
that a scalar field is not a perfect fluid [8], so one needs 0 0 0 O
to take some care with the standard procedure. With (η) = H(η)(cid:2)[1+ (H/m)] , (cid:3) (3b)
H H O
this formalismthe results emergemorenaturallythanin a(η) = aH(η) 1+ 2(H/m) . (3c)
previous works, and it is convenient in order to identify O
the root of the misunderstanding; see Eq. (15) and the Here functions with a(cid:2)superscript H(cid:3) vary on cos-
paragraphs below. mological timescales, (fH)′ aHfH, whereas those
∼
3
with a superscript m vary on timescales given by the
mass of the scalar field, (fm)′ amfm. In princi- CDM
∼ Analytic
ple, higher-order terms oscillate in time with high fre- 1
Numerical
quency. Consequently, the derivative of one of those
terms is not necessarily suppressed in the series expan-
0.1
sion, e.g. d[ (H/m)]/dη aH. That is the reason for
O ∼
which the series expansion of the Hubble parameter m 1
has a linear term in H/m, even though such a termHis H / 0.01
not present in the expression for the scale factor a. 0.8
Introducingtheansatz(3)intotheKlein-Gordonequa-
0.001 0.6
tion we obtain
0.4
AM
ϕ (η)= Pl sin(mt)+ 2(H/m) , (4) 0.0001 2.5 3 3.5 4
0 (mη)3 O
(cid:2) (cid:3) mη 10
with A an integration constant, M = 1/√8πG the re-
Pl
ducedPlanckmass,andwherewehavefixedanarbitrary
FIG. 2: Evolution of the conformal Hubble factor H/m as
phase to zero. For later convenience, and with no loss of
a function of the conformal time mη for a free scalar field
generality, from now on we will also choose A2 = 24.
oscillating around the minimum of the potential. We show
From the above expression we see that the scalar field
for comparison the solution in standard CDM (solid line),
oscillates around ϕ0 = 0 with constant frequency m in and both the analytical approximation in Eq. (6) (dashed
comoving time t (so that the frequency will increase in line),andthefullnumericalsolutionoftheFriedmann-Klein-
conformal time η as the universe expands), and an am- Gordon system, Eqs. (2) and (5), (dotted line). In the figure
plitude that decays as 1/η3. Introducing this expression the integration begins at mη = 101/3, with a = 102/3 and
for ϕ (η) into the Friedmann equations, H/m=2×10−1/3.
0
8πG
2 = a2ε , (5a)
0
H 3 Eq. (4) above, we obtain ϕ M . That guaran-
0 Pl
H′−H2 = −4πGa2(ε0+p0), (5b) rteoellsplaarrgaemveatleuress[|1ǫ1sr]|,=wi|tηhsr|ǫ|=|2M≪(P2Ml/2ϕ20/2≫)(∂1 oVf/tVhe)2slaonwd-
one finds η M2(∂2V/V), so we csarn≡safelyPlconcluϕde that the
sr ≡ Pl ϕ
universe is not in a period of slow-roll inflation, as was
2 3 H
(η) = 1 sin(2mt)+ 2(H/m) , (6) evident from Eq. (6).
H η − 4 m O
(cid:20) (cid:18) (cid:19) (cid:21) At this point one should mention that the dominant
a(η) = (mη)2 1+ 2(H/m) . (7) termsinthesolutionoftheFriedmann-Klein-Gordonsys-
O
tem givenby Eqs.(4), (6) and (7) coincide with those of
Forcompletenesswe(cid:2)canalsointeg(cid:3)ratetheexpressionfor
the exact solution found in Ref. [12] for a scalar field
the comoving time to obtain
evolving in a universe dominated by a barotropic fluid
such that H = 2/(3γt), with γ constant (when γ = 1
(mη)3
mt= madη = 1+ 3(H/m) . (8) this background barotropic fluid can be associated with
3 O
Z theaverageenergydensityandpressureofthescalarfield
(cid:2) (cid:3)
itself). Something similar happens with the solution re-
In the above equations we have used the fact that the
ported in Ref. [13], obtained in terms of time averages.
background energy density and pressure are given by
Themaindifferencewithourresultsisthe factthathere
1 ϕ′2 1 ϕ′2 wehavemadeexplicittheexistenceofhigher-orderterms
ε = 0 +m2ϕ2 , p = 0 m2ϕ2 . (9)
0 2 a2 0 0 2 a2 − 0 in H/m, and in particular we have shown the first oscil-
(cid:18) (cid:19) (cid:18) (cid:19) lating subdominant contribution to the Hubble parame-
In particular we find that, to lowest order in the ter. Notehoweverthatthisisnotonlyapurelyacademic
series expansion, the background energy density red- question: the inclusion of the higher-order terms will be
shifts with the inverse of the comoving volume, crucialnextfortheunderstandingoftheevolutionofthe
ε 1/a3, whereas for the background pressure we ob- small perturbations.
0
tain∼p (1/a3)cos(2mt). Note that to this same order Figure 2 shows the evolution of the conformal Hubble
0
∼
we cannot distinguish the expansion rate from that in a factor /m as a function of the conformal time mη for
H
CDM universe, = 2/η, even tough p ε during a free scalar field oscillating around the minimum of the
0 0
H | | ∼
the evolution. Interestingly, this does not depend on the potential. The solid line corresponds to standard CDM,
mass of the scalar particle, as long as it is large enough while the dashed and dotted lines show the case of the
when compared to the expansion rate of the universe. scalar field using both the analytical approximation in
Since H/m 2/(mη)3, the condition H m Eq.(6),andafullnumericalsolutionoftheKlein-Gordon
∼ ≪
demands mη 1. In terms of the scalar field, andFriedmannsystem,Eqs.(2)and(5),respectively[for
≫
4
the numericalcasewesolvefor fromEq.(5b),anduse two different series expansions: one in the small pertur-
H
Eq.(5a)tomonitorthenumericalerror]. Noticethatthe bations of the spacetime metric and the scalar field, and
evolution for the case of the scalar field follows closely the other in the expansion rate of the universe. We will
that of CDM with small oscillations around it. Initially soon introduce a new one in terms of the Jeans length.)
we haveH/m=0.2andthe oscillationsare stillevident, Inordertomoveforwardwefinditconvenienttodefine
but their amplitude decreases as time goes on. the new quantities
Nextwewillconsiderthebehaviorofthesmallpertur-
ϕ′ aϕ′
bations around the homogeneous and isotropic solution v =a δϕ+ 0 Ψ , z = 0 . (13)
in Eqs. (4), (6), and (7). In particular, we find that an
(cid:18) H (cid:19) H
unstable Jeansmode growsat scalesbetween the Comp-
The gauge-invariantfield v(η,~x) is usually known as the
ton length of the scalar particle and the Hubble radius
Mukhanov-Sasaki variable, and the function z depends
of the universe; see Eq. (21) below for details.
only on the spacetime background, z =z(η). Note that,
leavingthescalefactoraside,the Mukhanov-Sasakivari-
III. THE LINEAR PERTURBATIONS able represents the scalar field perturbation evaluated
in the spatially flat gauge ψ = E = 0 [14]. Whenever
the scalar field dominates the evolution of the universe,
In this paper we will not consider vector and tensor
Eqs. (12a) and (12b) simplify to
modes, and will concentrate on the scalar sector of the
perturbations. After all this is the sector that contains a2Ψ v ′
the Jeans mode of the theory. With this assumption, ∆ = 4πGz2 , (14a)
z
the most general expression for the spacetime metric of (cid:18) H (cid:19) (cid:16) (cid:17)
a universe close to a flat homogeneous and isotropic RW a2Ψ ′ v
= 4πGz2 . (14b)
one takes the form z
(cid:18) H (cid:19) (cid:16) (cid:17)
ds2 = a2 (1+2φ)dη2+2∂ Bdxidη These two equations can be combined to obtain
i
[(1(cid:8) 2ψ)δij 2∂i∂jE]dxidxj . (10) z′′
− − − v′′ c2∆v v =0, (15)
For the perturbed scalar field we will (cid:9)also write − s − z
ϕ=ϕ0+δϕ. Here φ, B, ψ, E and δϕ are functions of withc2 =1analogoustoa“speedofsound”foracanon-
s
the spacetime coordinates η and ~x, with δϕ ϕ and
0 ical scalar field [3]. The main reason for this identifica-
≪
φ, B, ψ, E 1.
tion is the similarity of Eq. (15) with that obtained for
≪
All of these fields depend on the choice of coordi-
a barotropic perfect fluid with sound speed c ; see e.g.
s
nates used to write the line-element in Eq. (10), and
Eq. (7.65) in Ref. [1]. Notice that even though here we
they could in principle be describing fictitious inhomo-
are only interested in the case of a scalar field with no
geneities. For this reason it is sometimes convenient to
self-interactions, so far the analysis of the perturbations
work with gauge-invariantfields, such as [1]
is general and valid for an arbitrary potential.
Equation (15) above fixes a characteristic length scale
Φ = φ−(1/a)[a(B−E′)]′ , (11a) given by csaz′′/z −1/2 (in this paper we will work with
Ψ = ψ+ (B E′), (11b) comoving wa|venum| bers k, but we will talk about phys-
H −
δϕ = δϕ ϕ′(B E′). (11c) ical wavelengths λ, with λ = 2πa/k). This length pro-
− 0 −
vides us with a ruler to discriminate between large and
Note that the above fields coincide with the amplitude smallscalesinthescalarfieldperturbations. Thecorrect
of the metric and the scalar field perturbations in the estimation of this length scale for the different physical
conformal-Newtonian(alsoknownintheliteratureasthe situations —and in particular the relative size it takes
longitudinal) coordinate system, for which B =E =0. when compared to the Hubble radius— lies precisely at
Tolinearorderinthenewfieldvariables,the00and0i the heart of the usual misunderstanding.
Einstein field equations, essentially the Hamiltonian and As we willfindsoon,see Eqs.(20)and(21) below,the
momentum constraints, take the following form: scale c az′′/z −1/2 determines the Jeans length of the
s
| |
scalar field. However, this quantity is not always related
∆Ψ 3 (Ψ′+ Φ)=4πGa2δε to the naive guess c H−1 one would expect from a per-
− H H s
=4πG ϕ′0δϕ′+a2m2ϕ0δϕ−ϕ′02Φ , (12a) fect fluid analogy. In order to clarify this point, let us
Ψ′+HΦ=4π(cid:2)Gϕ′0δϕ, (cid:3) (12b) cisonroslildinergtdwoowrneptoretsheentmatiinviemcuamseso.f Wthehepnottehnetisaclawlaerhfiaevlde
where ∆is the Laplaceoperatorinflatspace,andwhere ϕ′0 ∼aHϕ0,whichimplies|z′′/z|∼(aH)2. Thisfixesthe
δε=δε ε′(B E′)isthegauge-invariantdensitypertur- characteristic Jeans length to the Hubble radius. Short-
bation.−Fo0r a −scalar field, and to this order in the series wavelengthperturbations(whencomparedtothislength
expansion,therearenoanisotropicstressesandthei=j scale)oscillateinspaceandtimeasv sin(c kη+~k ~x),
s
field equations fix Φ=Ψ. (Note that so far we are us6ing with c2 = 1. The Jeans mode is then∼stabilized and·the
s
5
perfect fluid analogy seems possible, i.e. small pertur- Eq. (15) can be written in the form
bations to the homogeneous and isotropic solutions do
not grow. Consider, for instance, the case of the infla- v(η,~x)= 1 C v (η)ei~k·~x+c.c. . (17)
ton during a slow-roll regime, or a quintessence field in L3/2 ~k ~k
thepresentuniverse,asparticularrealizationsofthissce- ~kX6=0h i
nario.
Here C are some dimensionless integration constants
However,whenthezero-modeofthescalarfieldisoscil- ~k
that label the different possible solutions, c.c. denotes
lating with highfrequency m H aroundthe minimum
osufltthseinpotze′′n/tzial, w(eamha)v2e. inTsht≫eeadneϕw′0l∼enagtmhϕs0c,awlehiischwreel-l ceoqmuaptlieoxnconjugate, and the functions v~k(η) satisfy the
| | ∼
inside the Hubble radius, making possible the growth of
apecratunrobnaictaiolnsscawliatrhfiaeHld<c2k=<1c)−s.1aNmow(rtehmeenmabiveertehsatitmfoar- v~k′′+ω~k2(η)v~k =0, with ω~k2(η)=k2− zz′′ . (18)
s
tion c H−1 for the Jeans length has nothing to do with
s
the correct one, c m−1. Notethattheboundaryconditionsdemandki =2πni/L,
s
with n = 1, 2,... and i = 1,2,3. The zero-mode
Let us consider in more detail this second scenario. i ± ±
k =0 is already included in the descriptionof the space-
Notice that, using the solutions in Eqs.(4), (6), and (7),
time background,and for that reasonit does not appear
the “mass” term in Eq. (15) takes the form
in Eq. (17).
z′′ H According to the Eq. (18) the behavior of the mode-
= a2m2 1+6 sin(2mt)+ 2(H/m) . (16)
functions v (η) depends crucially on the relative value
z − m O ~k
(cid:20) (cid:18) (cid:19) (cid:21) betweenthe squareofthewavenumberk2,andtheback-
As mentioned before, this is a purely background quan- groundfunction z′′/z. For those modes smaller than the
tity. For practical reasons we will work with periodic Compton wavelength, k am, we obtain (remember
≫
boundary conditions over a box of comoving size L. We that in the fast expansion regime the Compton wave-
canalwaystakethelimitL attheendofthecalcu- length is always well inside the Hubble radius, and then
→∞
lations. Under this assumption, the general solution to am aH)
≫
1
v (η)= 1+ 2(H/m,am/k) exp i 1+ 2(H/m,am/k) kη , (19a)
~k √2k O − O
(cid:0) (cid:1) (cid:2) (cid:0) (cid:1) (cid:3)
whereas for modes larger than this quantity, k am, we find
≪
A(k) 1 k k 3 H
v (η) = B(k)z¯ B(k)a sin(mt) cos3(mt)+ 2(H/m)
~k √2m − 5 am aH − 2 m O
(cid:26) (cid:18) (cid:19)(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:21)
d(mη) i 1 k k 3 H
iB−1(k)z¯ + B−1(k) cos(mt) sin3(mt)+ 2(H/m)
− z¯2 7 a2 am aH − 2 m O
Z (cid:18) (cid:19)(cid:18) (cid:19)(cid:20) (cid:18) (cid:19) (cid:21)
+ 2(H/m,k/am) , (19b)
O
(cid:27)
(orv =vr+ivi ifweprefertothinkintermsoftworeal, Eq.(19a),this isnotnecessarilytrueforfactorsofk/aH
~k ~k ~k
linearly independent mode-functions vr and vi.) Here in the case of Eq. (19b), where they can even dominate
~k ~k formodesinsidetheHubbleradius,i.e. thosemodeswith
we havedefined z¯=z/√6M . Also,the functions v (η)
Pl ~k aH <k <am. This implies that, for modes larger than
have been normalized so that v (v∗)′ v′(v∗) = i, with
~k ~k − ~k ~k the Compton wavelength, the lowest nonvanishing con-
the convention that in the asymptotic limit H/m 0,
tribution to the Mukhanov-Sasaki variable is not given
→
k/(am) ,werecoverstandardmasslessplanewaves, by lim v (η) z¯ iz¯ d(mη)/z¯2, as one could have
i.e. v~k(η→) =∞e−ikη/√2k in Eq. (19a). In the above ex- naivelyk→ex0pe~kcted∼. In−cidentally the extra terms will be
pression A(k) is a phase factor, A(k)2 = 1, and B(k) crucial to determine the bRehavior of the modes relevant
| |
is a realnumber necessaryto connect the two regimes in for structure formation. Since aH η−1 and am η2,
∼ ∼
Eqs.(19a)and(19b)atk am. Forourpurposesinthis eventually all modes reach this regime. This completes
∼
paper we will not need to determine these quantities. our discussion of the Mukhanov-Sasakivariable.
Notice that, while factors of aH/k are small in However,theMukhanov-Sasakivariableisonlyanaux-
6
iliary field (remember that this quantity is related to the Newtonian potential Ψ, or to the contrast in the en-
the perturbation in the scalar field evaluated in the spa- ergy density, δε/ε . Introducing the expressions for the
0
tially flat gauge). In order to make contact with obser- Mukhanov-Sasakivariable,Eqs.(19),intoe.g. Eq.(14a),
vations we need to move our attention, for instance, to we obtain for the Newtonian potential4
i aH 1
cos(mt)e−ikη , if k am,
3 1 √2k k a ≫
Ψ~k(η)=r2 MPl × √A(2km)(cid:18)15B((cid:19)k)+iB−1(k) akH akm a13 , if k ≪am. (20)
(cid:20) (cid:18) (cid:19)(cid:16) (cid:17) (cid:21)
whereas from the Hamiltonian constraint (12a), we find for the contrast in the energy density5
i k 1
− cos(mt)e−ikη , if k am,
√2k aH a ≫
(cid:18) (cid:19)
2
δε0ε(cid:12)(cid:12)~k =r23M1Pl × √AA((2kkm)) "−615B(k)(cid:18)akH(cid:19) −iB−a1H(k)(cid:16)Hamm(cid:17)a131#, if aH ≪k ≪am, (21)
(cid:12) B(k)+9iB−1(k) cos2(mt), if k aH.
(cid:12) √2m(cid:20)−5 (cid:18) k (cid:19)(cid:16) k (cid:17)a3(cid:21) ≪
Note that the two expressions above have been re- which is again the same behavior as in the standard
ported only to the lowest non-vanishing order in the CDM scenario. On the other hand, those modes larger
series expansions. A pattern of small-amplitude, high- than the Hubble radius, k < aH, freeze with a time-
frequency oscillations are expected around the expres- dependentmodulationincos2(mt). Itispossibletotrace
sions in Eqs. (20) and (21), but this will be enough for back this oscillatory dependency to a background term,
the purposes of this paper. ′ 2 cos2(mt)/a,thatappearsintheexpressionfor
H −H ∼
thecontrastinthe energydensity,seeEqs.(23)and(24)
For modes larger than the Compton wavelength,
in footnote 5. These oscillations are therefore not ex-
k <am, the Newtonian potential mimics the behavior
pected to appear in the case ofa complex scalar field for
ofstandardCDM:oneofthe solutionsremainsconstant,
whiletheotherdecreasesintimewithH/a η−5;seefor which H′−H2 ∼1/a.
∼ In particular, the growing mode in the energy density
instance Eq. (7.53) in Ref. [1] for details. On the other
contrast δε/ε 1/(aH)2 η2 t2/3 (i.e. δε/ε a to
hand, for modes smaller than the Compton wavelength, 0 0
∼ ∼ ∼ ∼
the lowest order in the expansion series) at those scales
k >am, the solution oscillates in time with a decreasing
amplitude, H 1/η3. This decay in the amplitude of larger than the Compton wavelength but smaller than
∼ the Hubble radius, 1/m<λ<1/H, is usually identified
the Newtonian potential is characteristic of a barotropic
withstructureformationintheuniverse(seeforinstance
perfect fluid p = p(ε) ε with a nonvanishing Jeans
scale, c2 =0. ≪ Eq. (7.56)in Ref. [1]). Note howeverthat the (large)os-
s 6 cillations for the contrastin the energy density of modes
Something similar happens for the contrast in the en-
largerthanthe Hubble radiusarenotpresentinthe case
ergy density, where high-frequency modes with k > am
of CDM.
oscillate in time with damped amplitude, 1/a2H η−1.
At this point it is interesting to stress once again the
∼
In contrast, there appear two different regimes for the
modes larger than the Compton wavelength. Perturba-
tions larger than the Compton length but still smaller
than the Hubble radius, aH < k < am, can grow in 5 Herewehaveusedtheidentity
time as 1/(aH)2 η2, or decrease as 1/(Ha3) η−3,
∼ ∼ δε =8πG z2 1 + H′−H2 1 v~k ′− v~k , (23)
ε0(cid:12)(cid:12)~k a2 (cid:20)(cid:18)3 k2 (cid:19)H(cid:16) z (cid:17) (cid:16) z (cid:17)(cid:21)
(cid:12)
that(cid:12)relates, in Fourier space, the Mukhanov-Sasaki variable to
4 Herewehaveusedtheidentity the c(cid:12)ontrast in the energy density. Using the evolution for the
Ψ (η)=−8πG z2 H v~k ′ , (22) background universewecanwrite
~k a2 (cid:20)2k2 (cid:16) z (cid:17)(cid:21) 1 + H′−H2 = 1 1−9 aH 2cos2(mt)+O(H/m) . (24)
that relates, in Fourier space, the Mukhanov-Sasaki variable to 3 k2 3" (cid:18) k (cid:19) #
theNewtonianpotential.
7
0.0012 and then in this gauge the contrast in the energy den-
0.0009
k/m= 464 sity grows with time as the scale factor, even for those
0.0006
0.0003 modes that are well outside the Hubble horizon. (It is
0 in this gauge that the matter power spectrum is usually
-0.0003
presentedinthe literature.) Note that inthis coordinate
-0.0006
-0.0009 system the large oscillations in the energy density dis-
-0.0012
appear, and then it is not clear for us if they could be
3 4 5 6 7 8 9 10
observableinpracticeoriftheyareonlyagaugeartifact.
0.0002
Figure 3shows the evolutionofthe contrastin the en-
k/m= 2.32
0.00015 ergy density, δε/ε , for three individual Fourier modes
0|~k
that are representative of the different regimes. For this
0.0001
figure we have considered the same background evolu-
5e-05 tion of Figure 2, so that initially we have mη =101/3,
a = 102/3 and /m = 2 10−1/3. In order to calculate
0
H ×
3 4 5 6 7 8 9 10 the contrast in the energy density we integrate numeri-
2e-05
callyEqs.(14b)and(18)forthefunctionv andtheNew-
-3 ~k
1.5e-05 k/m= 4.64 x 10 tonianpotentialΨ~k,simultaneouslywiththebackground
evolution. WethenusetheHamiltonianconstraint(12a)
1e-05
to find the perturbation in the energy density, δε .
~k
5e-06 The top panel in Figure 3 corresponds to a mode
with k/m=108/3 464, which is clearly in the regime
0 ≈
k > am for the time interval mη (2,11). We can see
3 4 5 6 7 8 9 10
mη thatthe contrastinthe energyden∈sity oscillateswithan
amplitude that decays like 1/(a2H), as expected. One
can also clearly see from the figure that we have oscil-
FIG. 3: Evolution of the contrast in the energy density,
lations with two quite different frequencies: a high fre-
fδeεr/eεn0t|~kF,ouasriearfmuondcteiso,nk/omf c=on(f1o0r2m,a1l/2ti,m1e0−m3)η×fo1r02t/h3r,eiendthife- quency oscillation coming from the term e−ikη, modu-
lated by a lower frequency oscillation that corresponds
universedepictedinFigure2. Thelowerandhigherwavenum-
berswerechoseninordertolieoutsideoftheregion[aH,am] tothetermcos(mt)inEq.(21). Themiddlepanelcorre-
during the evolution. Solid lines represent the results of the spondstoamodewithk/m=1/2 102/3 2.32,sothat
× ≈
numerical evolution, whereas dashed lines are obtained from aH < k < am, also in the same time interval. In this
an appropriate combination of the envelopes of the linearly caseweseethatthe contrastinthe energydensitygrows
independent solutions in Eq. (21). Modes shorter than the as 1/(a2H2), with small oscillations. These modes are
Compton wavelength (top panel) oscillate with damped am- interesting for structure formation since they grow as if
plitude. Onthecontrary,modeslargerthanthecosmological
they were made of CDM. Finally, the bottom panel cor-
horizon (bottom panel) freeze, showing an oscillatory behav- responds to a mode with k/m = 10−7/3 4.64 10−3,
ior associated to the inherent oscillation of the scalar field. ≈ ×
for which we have k < aH in the evolution. In this
Finally,thosemodesthatarelargerthantheComptonwave-
case the contrast in the energy density rapidly reaches a
length but smaller than the Hubble radius (middle panel)
grow at thesame rate as thescale factor, as it is expectedin regime where it oscillates with constant amplitude. No-
the case of CDM. These modes can give rise to structures in tice that in all these three scenarios the frequency of the
thelateuniverse. Notethatsincetheequationsarelinearwe oscillationsincreasesastime goesby. This isagainto be
can rescale thevertical axesin thefiguresarbitrarily, as long expected since the frequency should be constant in cos-
as the amplitudes remain always less than unity. mologicaltimet,sothatitincreasesinconformaltimeη.
factthatthesestatementsareonlyvalidintheconformal- IV. DISCUSSION
Newtoniancoordinatesystem,wherethegauge-invariant
fields Ψ and δε/ε take the same values as the Newto- Foraperfectfluidwithequationofstatep=p(ε) ε,
0
≪
nianpotentialandcontrastintheenergydensity,respec- there is a close relation between the Hubble radius and
tively. If we move to e.g. the synchronous coordinate the Jeans length, λ c H−1. This is no longer true in
J s
∼
system φ = B = 0, the contrast in the energy den- the case of a canonical scalar field, where the character-
sity is related to the gauge-invariant field δε/ε through istic Jeans lengthis fixedinsteadby λ c az′′/z −1/2.
0 J s
δε /ε = δε/ε ε′/(aε ) aΨdη, the constant of inte- When this length scale is much smaller t∼han t|he H|ubble
s 0 0− 0 0
gration in this formula corresponding to an unphysical radius, perturbations can grow. For a perfect fluid this
fictitious mode. As in theRstandard CDM scenario, the is possible only if c2 1. For a scalar field we have
expression for intermediate wavelengths aH k am c2 = 1, but we cousld≪still satisfy az′′/z −1/2 H−1.
≪ ≪ s | | ≪
in Eq. (21) describes now all the modes larger than the This is what happens, for instance, when the zero-mode
Compton wavelength of the scalar particle, k am, ofthescalarfieldisoscillatingwithhighfrequency(when
≪
8
compared to the expansion rate of the universe) around Ifthemassofthescalarparticleliesatthescaleof10µeV
a minimum of the potential, where az′′/z −1/2 m−1. or above, as it is expected for the QCD axion [16], the
| | ∼
Note that this value for the Jeans length at the scale of Jeans length would be smaller than a centimeter, and
the Comptonwavelengthofthe scalarparticlecannotbe the growth of cosmic structures would be probably in-
resolved in terms of a nonrelativistic analysis, as it was distinguishable to that in the standard CDM scenario,
previously done by Hu et al. in Ref. [5]. Indeed, the at least while in the linear regime (see Ref. [9] for the
value of the Jeans length we identify in this paper does case when nonlinearities become important). However,
not coincide with the naive estimation they reported in if we consider ultralight scalar particles of masses as low
Eq. (4) of that reference, and which has been frequently as 10−22eV [5, 17], the Jeans length grows to the scale
used after that (see e.g. Ref. [10]). of parsecs. This could have observable physical conse-
ForthosemodeslargerthantheJeanslengththescalar quences in cosmology [10, 18], alleviating, for instance,
field follows the evolution in the standard CDM sce- the missing satellite discrepancy [19].
nario, except for the large oscillations of the contrast in In order to determine properly the new expression for
the energy density for modes larger than the Hubble ra- themasspowerspectrumwewouldneedamoreelaborate
dius when evaluated in e.g. the conformal-Newtonian analysis that includes, on the one hand, a knowledge of
gauge. Note however that these (large) oscillations are the initial conditions for the scalar field after inflation,
not present in the behavior of the Newtonian potential, as well as the evolution of the perturbations during the
for which the scalar field behaves like CDM, or even in radiationdominatedera. Weleavethisstudyforafuture
the super-Hubble modes of the contrast to the energy paper.
density when evaluated in e.g. the synchronous gauge. Note added.—After the first version of this paper was
As far as we know these oscillations of the contrast in submitted for publication we became aware of Ref. [20],
the energy density for long wavelength modes in certain where the linearized Einstein-Klein-Gordon system is
coordinate systems had not been previously reported in considered in the context of cosmological reheating. Al-
the literature, and it will be very interesting to explore thoughtherearesomesimilaritiesintheanalysis,themo-
if they are only a gauge artifact or if on the contrary tivationofourpaperis different. We thanksProf.James
they could have imprinted some signatures on the cos- P. Zibin for pointing this reference out.
mological observables. We expect these oscillations will
not affect the large scale structure of the universe. Af-
ter all, the distribution of galaxies is only sensitive to
Acknowledgments
the subhorizon modes, and these modes follow the stan-
dardCDMevolution. However,theycouldaffecte.g. the
cosmic microwave background photons at large angular We are grateful to Juan Carlos Hidalgo, David Marsh
scales, which traced super-Hubble scales for a long pe- and Dar´ıo Nu´n˜ez for useful comments and discussions
riodof time during the cosmologicalevolution. We leave about a first version of this paper. This work was par-
a more detailed analysis of this point for a future work. tially supported by PIFI, PROMEP, DAIP-UG, CAIP-
ForthosemodessmallerthantheJeanslengththeevo- UG, the “Instituto Avanzadode Cosmolog´ıa”(IAC) col-
lution cannot bring the small perturbations in the early laboration,DGAPA-UNAM under Grants No. IN115311
universetothenonlinearregime,andtheinhomogeneities and No. IN103514,CONACyT M´exico under grants No.
areerased. Thiswillintroduceacutoffinthemasspower 182445andNo. 167335,and PAPIIT-IN101415. ADT is
spectrum for the distribution of galaxies in the universe. alsosupportedinpartbyGrantNo.FQXi-1301fromthe
Something similar happens in warm DM scenarios [15]. Foundational Questions Institute (FQXi).
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