The multipole expansion of the local expansion rate Basheer Kalbouneh Christian Marinoni and Julien Bel Aix Marseille Univ Universit e de Toulon CNRS CPT Marseille France

2025-05-06 0 0 4.04MB 27 页 10玖币

侵权投诉

The multipole expansion of the local expansion rate

Basheer Kalbouneh, Christian Marinoni, and Julien Bel∗

Aix Marseille Univ, Universit´e de Toulon, CNRS, CPT, Marseille, France

(Dated: January 5, 2023)

We design a new observable, the expansion rate ﬂuctuation η, to characterize deviations from the linear

relation between redshift and dUniverseistance in the local Universe. We also show how to compress the result-

ing signal into spherical harmonic coeﬃcients in order to better decipher the structure and symmetries of the

anisotropies in the local expansion rate. We apply this analysis scheme to several public catalogs of redshift-

independent distances, the Cosmicﬂows-3 and Pantheon data sets, covering the redshift range 0.01 <z<0.05.

The leading anisotropic signal is stored in the dipole. Within the standard cosmological model, it is interpreted

as a bulk motion (307 ±23 km/s) of the entire local volume in a direction aligned at better than 4 degrees with

the bulk component of the Local Group velocity with respect to the CMB. This term alone, however, provides

an overly simplistic and inaccurate description of the angular anisotropies of the expansion rate. We ﬁnd that

the quadrupole contribution is non-negligible (∼50% of the anisotropic signal), in fact, statistically signiﬁcant,

and signaling a substantial shearing of gravity in the volume covered by the data. In addition, the 3D structure

of the quadrupole is axisymmetric, with the expansion axis aligned along the axis of the dipole.

Implications for the determination of the H0parameter are discussed. We ﬁnd that Hubble constant estimates

may show variation as high as ∆H0=(4.1±1.1) km/s/Mpc between antipodal directions along the dipole axis.

In the case of the Pantheon sample, this systematic diﬀerence is reduced to ∆H0=(2.4±1.1) km/s/Mpc once

model-dependent correction for peculiar velocity ﬂows are implemented. Notwithstanding, the axial anisotropy

in the general direction of the CMB dipole is still detected. We thus show how to optimally subtract redshift

anisotropies from Pantheon data in a fully model-independent way by exploiting the ηobservable. As a result,

the value of the best ﬁtting H0is systematically revised upwards by nearly 0.7 km/s/Mpc (about 2σ) compared

to the value deduced from the Hubble diagram using the uncorrected observed redshift. The goodness of ﬁt is

also improved.

PACS numbers:

I. INTRODUCTION

In accordance with the cosmological principle (CP), the

spatial sections of the Universe are maximally symmetric, that

is, rotationally and translationally invariant (e.g. [1]). This

statement about the symmetries of the Universe, sealed in the

Robertson & Walker line element, can only be interpreted in

a statistical sense, after convolving the spatial distribution of

matter with large smoothing kernels. This makes its empirical

conﬁrmation diﬃcult and subject to non-trivial systematicity.

Despite observational hurdles (e.g. [2]), convincing proofs

of isotropy are provided by the angular distribution of the tem-

perature ﬂuctuations of the cosmic microwave background

(CMB) [3, 4]. Also, 3D supporting evidence continues to

grow as spectroscopic studies reveal the structure of ever

larger and deeper regions of the Universe [5–14]. Analysis

of the spatial distribution of supernovae, i.e. objects whose

distances are estimated using redshift independent techniques,

also provides tentative conﬁrmation [15–22]. The nature of

the conﬁrmations remains preliminary, however, and there is

no shortage of evidence to the contrary [23–37]. If some of

these signals are not statistically signiﬁcant enough to reject

outright the CP, others appear as improbable in the framework

of the standard cosmological model.

However, it has long been known that in the local outskirts

∗Electronic address: basheer.kalbouneh@cpt.univ-mrs.fr,

christian.marinoni@cpt.univ-mrs.fr, julien.bel@cpt.univ-mrs.fr

of the Milky Way, at scales r<150h−1Mpc, the CP is violated

(e.g. [38, 39]). This region represents about half of the volume

used to determine the Hubble parameter H0, a fundamental

constant of the standard model and a consequence of the cos-

mological principle hypothesis. Traditionally, deviations from

CP predictions are treated perturbatively by expanding the

cosmological quantities into a smooth background component

and a ﬂuctuating part. Among the latter, a central role is occu-

pied by peculiar velocities. These super-Hubble motions con-

tain a lot of interesting cosmological information (e.g.[40])

and indeed their amplitude conﬁrms that the deviations from

the CP are in general agreement with the limits imposed by

the perturbation theory of the standard cosmological model

(e.g. [41–44]). However, it was soon realised that many sub-

tle systematic errors, if not properly subtracted, can compro-

mise their use as eﬃcient cosmological probes; non-Gaussian

issues, homogeneous and in-homogeneous Malmquist biases,

incompleteness of mass catalogs used to predict the amplitude

of peculiar velocities are among the pitfalls that most hamper

the analysis [45].

In order to free the investigation of local inhomogeneities

from certain statistical and observational complications, we

explore in this paper another direction. We develop a com-

pletely non-perturbative approach to inhomogeneities that fo-

cuses directly on the scale factor of the Universe as a rele-

vant variable to quantify deviations from uniformity (see also

[46, 47] for a similar approach). This is precisely the parame-

ter that is kept invariant in perturbative analyses, deﬁning the

reference background against which deformations in the spa-

tial sector of the metric are compared.

arXiv:2210.11333v3 [astro-ph.CO] 4 Jan 2023

In this spirit, we design an observable, the expansion rate

ﬂuctuation η, that provides information about ﬂuctuations in

the local expansion rate and is, at the same time, easily com-

parable with theoretical predictions. Indeed we will show that

it provides a model-independent means of analysing inhomo-

geneities, not even requiring the CP assumption as a prerequi-

site. This cosmographic approach (e.g. [48, 49]) allows the

results to be directly interpretable in alternative spacetimes

and can ideally guide the search for unconventional line el-

ements that capture the essential features of the local inhomo-

geneities.

From an observational point of view, the goal is to inves-

tigate the existence and signiﬁcance of anisotropies in the lo-

cal Universe through new methods of investigation. In this

speciﬁc case, by decomposing the angular ﬂuctuations of the

expansion rate into spherical harmonics and compressing in-

formation about anisotropies into a set of independent Fourier

coeﬃcients.

Multipolar expansion in spherical harmonics provides an

orthogonal insight into the nature of the local redshift-distance

relation and allows to go beyond the simple dipole model with

which anisotropies are traditionally described in the nearby

Universe. At the same time, it allows to deepen and extend

studies, such as those of [38, 47, 50–52] which attempt to

constrain the tidal ﬁeld component by analysing the shear

of the velocity ﬁeld generated by local gravitational ﬂuctua-

tions. In this respect, we focus on the study of the symmetries

and geometric structure of the harmonic multipoles, showing

how their analysis gives a simple and inexpensive description

of the structure of the anisotropies in the Hubble ﬂow. We

demonstrate that the three-parameter formula encoding such

information has predictive power comparable to that of much

more complex numerical studies of peculiar motions.

The paper is organized as follows: in Sec. II we intro-

duce the observable that optimally extract information about

the ﬂuctuation in the angular expansion rate, while in III we

present the method implemented to estimate the signal from

discrete datasets and to compress it into spherical harmonic

coeﬃcients. We also discuss how we estimate reconstruction

errors, both statistical and systematic. In Sec. IV, we describe

the data analyzed. Results are presented and interpreted in V.

Sec. VI provides summary and conclusion.

In the following, we present results in natural units (c=

1) and we refer to the standard ΛCDM model, as the ﬂat

Friedmann-Robertson-Walker (FRW) spacetime which best

ﬁts the Planck18 data [53]. Redshift is expressed with respect

to the CMB rest frame.

II. THE EXPANSION RATE FLUCTUATIONS η

We model the angular anisotropies in the redshift-distance

relation by directly exploiting the local expansion rate as a

target observable. In a perfectly uniform FRW universe, the

ratio z/dbetween the redshift and the proper distance of co-

moving particles is predicted to be constant, independent from

the particular line-of-sight along which it is estimated.

In any generic metric model describing the structure of lo-

cal space-time, i.e. the inhomogeneous distribution of mass at

the periphery of the Local Group of galaxies, it is possible, at

least in the limit of small separations, to relate the redshift z

and the proper distance as follows

z=˜

H0(l,b)d.(1)

In this expression, ˜

H0is a continuous function that depends

only on the angular coordinates (l,b) and can be constrained

experimentally. It is clear that the angular dependence is in

principle theoretically determinable as soon as a line element

is provided. Note that if the observer is only at rest relative

to the CMB, but not comoving with respect to the surround-

ing matter, then we expect a dependence of ˜

H0on the radial

distance even in very local regions of the Universe. As a mat-

ter of fact, in a generic spacetime, the characteristic distance

scale at which the linear limit of the equation (1) is reached is

not known apriori.

We actually characterize deviations from isotropy in the lo-

cal expansion rate via the (decimal) logarithmic relation

η≡log "˜

H0#.(2)

Here H0is a normalizing factor that, in the standard model of

cosmology, coincides with the value of the Hubble constant.

We ﬁx its amplitude by requiring that the average value of η

over the volume covered by data vanishes.

The justiﬁcation for the choice of this observable is statisti-

cal in nature. Errors are Gaussian only in the distance modulus

µand not in the redshift-independent distance dif these latter

are estimated as

d=10 µ−25

5.(3)

Therefore, given a sample of objects at spatial position r, the

discrete estimator of the continuous ﬁeld (2)

ˆη(r)=log "z

H0#+5−µ

5,(4)

is a random variable that follows a Gaussian distribution. In-

deed, we assume that the uncertainty δon ˆηis induced only

by the imprecision with which the redshift-independent dis-

tances are estimated (δ=σµ/5), i.e. we consider that any

error in the redshift estimate is negligible. As a consequence,

ˆηprovides an unbiased estimate of η, as can be easily veri-

ﬁed. As an added bonus, Eq. (2) also makes it possible to

quantify anisotropies in the expansion rate, regardless of the

value of the Hubble constant parameter used to normalize the

distance modules µ. A subtlety must be pointed out. It is

implicitly assumed, in the above argument, that in the limit

z<< 1, the range we are concerned in this paper, dis a fair

proxy for both the luminosity and angular diameter distance,

i.e. d ≈dL≈dA.

The expansion rate ﬂuctuations ηis not speciﬁcally tailored

to have only nice statistical properties. It also has a physical

content. Linear perturbation theory of the standard cosmolog-

ical model provides a framework for interpreting this observ-

able. According to it, the redshift observed in the CMB rest

frame is given by

z=zc+v(1 +zc),(5)

where zcis the cosmological redshift and vis the line of sight

component of the peculiar velocity of the source (assumed to

be non-relativistic) with respect to the CMB rest frame. By

inserting this last relation into 2 we get

η=log 1+v(1 +zc)

zc!≈v(1 +z)

zln 10 ,(6)

where we have assumed v << zc≈z. The ﬂuctuations in

the expansion rate are excited by radial peculiar velocity and

suppressed in inverse proportion to the object’s distance.

III. EXPANSION RATE FLUCTUATIONS: THE

SPHERICAL HARMONIC DECOMPOSITION

We can compress the information contained in the ηobserv-

able into a few coeﬃcients. To this end, we expand the expan-

sion rate ﬁeld ηin spherical harmonic (SH) components. We

orthogonally decompose ηon a sphere as follows

η=

∞

`=0

m=−`

a`mY`m(θ, φ)=

∞

`=0

η`,(7)

where

Y`m(θ, φ)=s(2`+1)(`−m)!

4π(`+m)! Pm

`(cos θ)eimφ,(8)

and Pm

`are associated Legendre polynomials. Thus, the

Fourier coeﬃcients a`mcan be expressed as

a`m≡Z2π

0Zπ

η(θ, φ)Y∗

`m(θ, φ) sin θdθdφ. (9)

Note that, due to the deﬁnition, the monopole (`=0) of η

vanishes.

In addition, one can deﬁne the angular power spectrum of

the ηanisotropy as

C`=h|a`m|2ie,(10)

where the expectation is intended to be over a statistical en-

semble of universes.

C`=1

2`+1

m=−`

|a`m|2,(11)

is an unbiased estimator for C`

The following section (§III A) describes in detail the pro-

cedure adopted to reconstruct the ﬁeld η(Ω) from discrete 3D

data with non-uniform sampling rates on the sky. In section

(III B) we describe how the SH coeﬃcients a`mare estimated.

We present the analytical formulas and numerical recipes for

evaluating measurement errors, both statistical and system-

atic, in the sections III C and III D, respectively.

A. Estimation of the angular ηﬁeld

The expansion rate ﬂuctuation estimator ˆη(r) is a discrete

random variable. The analysis of this observable can be sim-

pliﬁed, and the underlying theoretical model (2) can be better

traced if we convert it into a stochastic ﬁeld. We thus average

ˆη(r) over all the objects at position rwithin a given volume

V(Ω,R), where Ωis a solid angle centered on the observer and

Rthe depth of the catalog (i.e. its upper edge). The angular

anisotropies seen by the observer are thus piece-wise deﬁned

η(Ω)=PN

iˆη(ri)w(ri)W(ri|V(Ω,R))

iw(ri)W(ri|V(Ω,R)) (12)

where Nis the number of objects in the catalog, w(ri)=1/δ2

is a weight that takes into account the precision in the mea-

surement of the distance of the i-th object in the catalog.

W(r|V(Ω,R)) is a window function which evaluates to unity

if ri∈Vand is null otherwise.

It is clear that averaging has the advantage of reducing noise

at the cost of a lower angular resolution. The latter is essen-

tially controlled by the aperture of the solid angle Ω, although

it also depends, in principle, on the depth Rof the sample on

which the spatial averaging is performed.

In practice, we construct the η2D ﬁeld out of a discrete

point process ˆη(r), by ﬁrst partitioning the sky in Npix identi-

cal pixels (each subtending a solid angle Ωi) using HEALPix

[54] and then by applying eq. (12) to objects within the vol-

ume Vsubtended by each pixel Ωi. HEALPix is an algorithm

which tessellates a spherical surface into curvilinear quadri-

laterals, each covering the same area as every other. Although

characterized by a diﬀerent shape, the resulting pixels are lo-

cated on lines of constant latitude. This property is essential

for speeding up computation but is less than optimal for pix-

elating the discrete ηﬁeld. In fact, the counts can show large

variations from pixel to pixel. In addition, some of the pix-

els may end up containing no data at all. To tackle this issue,

we ﬁrst rigidly rotate the galaxy ﬁeld randomly, by looking

for conﬁgurations in which all the HEALPix pixels are ﬁlled

with objects and the least populated cell contains a maximum

number of objects. If as a result of diﬀerent rotations, the max-

imum number of galaxies in the least populated cells stays the

same, we pick up the conﬁguration for which the distribution

of the number of the galaxies in the pixels has the minimum

variance. This allows to minimize pixel-to-pixel ﬂuctuations

in the reconstructed value of ηand increase the signal-to-noise

ratio in the determination of the Fourier coeﬃcients. Note that

the rotation trick does not aﬀect the estimation of the angular

power spectrum ˆ

C`, which is, by deﬁnition, invariant under

rotation. However, once the Fourier coeﬃcients have been

estimated, we apply an inverse rotation to the pixels and η

maps so that, for the sake of clarity, the results are presented

in standard galactic (l,b) coordinates. The whole strategy is

graphically illustrated in Fig. 1.

The resolution of the HEALPix grid is calculated as Npix =

12N2

side where Nside =2t, and t∈N. The baseline grid, corre-

sponding to t=0, has 12 pixels. Our choice of the resolution

FIG. 1: Illustration of the rotation strategy to improve the estimation

of SH coeﬃcients. Upper: the standard HEALPix pixels (Nside =2

and Npix =48) tessellating the distribution of galaxies in the galactic

coordinates. Note the presence of an empty pixel. Center : rigid

rotation applied to the sample so that in each pixel falls at least one

galaxy (the minimum number is 5 in this example). Lower : the

inverse rotation is applied to both galaxies and pixels.

in the reconstruction of the angular ηmap, as explained in V,

is dictated by two criteria: the SH decomposition must result

in multipoles that have a suﬃciently high signal-to-noise ratio

and a suﬃciently low probability pto occur by chance in a

randomly ﬂuctuating ηﬁeld.

B. Estimation of the SH coeﬃcients

We estimate the Fourier coeﬃcients of the spherical har-

monic decomposition by slightly modifying the reconstruc-

tion scheme provided by the HEALPix algorithm. HEALPix

accomplishes that by means of an iteration scheme, the so-

called Jacobi iteration. The zeroth order estimator of the co-

eﬃcients of the expansion ﬁeld is

ˆa(0)

`m=4π

Npix

p=1

η(Ωp)Y∗

`m(θp, φp),(13)

where (θp,φp) are the angular coordinates of the center of each

pixel pand η(Ωp) is calculated by eq. (12). The Fourier coeﬃ-

cients a`mare then calculated up to the order `max =3Nside −1,

and the higher orders are

ˆa(k+1)

`m=ˆa(k)

`m+4π

Npix

p=1η(Ωp)−η(k)(θp, φp)Y∗

`m(θp, φp),(14)

where

η(k)(θp, φp)=

`max

`=0

m=−`

ˆa(k)

`mY`m(θp, φp).(15)

In matrix notation,

a(0) =Aη (16)

a(k+1) =a(k)+A(η−η(k)) (17)

η(k)=Sa(k),(18)

where ais the vector of the spherical harmonic coeﬃcients

(containing (`max +1)2elements), while ηand η(k)are vec-

tors representing the measured and estimated values of η

in each pixel (and thus contain Npix elements). Moreover,

A=4π

Npix Y∗

`m(θp, φp) and S=Npix

4πA∗T. The calculation of

a(k)is repeated until convergence i.e. until the residual has

zero Fourier coeﬃcients up to `max.

Instead of going through the iteration scheme, we proceed

in a diﬀerent way. We estimate analytically the asymptotic

limit a(∞)that should be ideally obtained in the limit of an

inﬁnite number of iterations. We ﬁrst write equation (17) as

a(k+1) =a(k)+Aη −ASa(k),(19)

a(k+1) =Aη +(I−AS)a(k),(20)

where Iis the identity matrix with size (`max +1)2×(`max +1)2.

By using a(0) =Aη we obtain

a(k+1) =a(0) +(I−AS)a(k).(21)

Under the assumption that this is convergent, for k→ ∞,

a(k+1) →a(k)we get

a(∞)=a(0) +(I−AS)a(∞),(22)

which results in

a(∞)=Ma(0),(23)

where M=(AS)−1. Note that we cannot take the inverse

of Aor Sindividually because they are not square matrices.

By this trick, we achieve two goals. First, we minimize the

computing time, moreover, and more importantly, we obtain a

closed form expression which simpliﬁes the estimation of the

error on the SH coeﬃcients, as we detail in the next section

and in Appendix B. The elements of the vector a(∞)represents

our best estimate (ˆa`m) of the coeﬃcients of the spherical har-

monic decomposition a`m.

C. Statistical measurement errors

In the following, we consider the SH coeﬃcients a`mas a

deterministic variable whose estimate

ˆa`m=a`m+`m,(24)

ﬂuctuates due to measurement errors `minduced by uncer-

tainties in the reconstruction of η. The expectation over dif-

ferent observational measurements made on the same sample

is thus E[ˆa`m]≡ hˆa`mi=a`m.i.e. we assume that the esti-

mator provides an unbiased estimate of the coeﬃcients of the

spherical harmonic expansion.

The variance of the estimator is deﬁned as

V[ˆa`m]=V[`m]≡σ2

`m,(25)

and, in general, depends on both modes `and m. An exact

analytical expression for σ`mis far from trivial and unenlight-

ening. It can be evaluated from the knowledge of the uncer-

tainties in the distance modulus measurements (see Appendix

B and cf. eq. B1 and 23).

An estimator of the power locked in each harmonic moment

`is the angular power spectrum (cf. eq. 11) estimator

C`=1

2`+1

m=−`

|a`m+`m|2.(26)

which can be expressed as

C`=1

2`+1

n=0

w(`)

n,(27)

where

w(`)

n=









ˆa2

`0n=0

2<[ˆa`n]2`≥n>0

2=[ˆa`(n−`)]22`≥n> `

(28)

This decomposition is conveniently chosen so to take into

account that al−mand a`mare conjugate variables (a`−m=

(−1)ma∗

`m). Thus, the variance of ˆ

C`reads

Vˆ

C`= 1

2`+1!22`

n=0

V[w(`)

n],(29)

where

V[w(`)

n]=









2σ4

`0+4σ2

`0a2

`0n=0

8σ(R)4

`n+16σ(R)2

`n<[a`n]2`≥n>0

8σ(I)4

`(n−`)+16σ(I)2

`(n−`)=[a`(n−`)]22l≥n> `

(30)

and where we have deﬁned σ(R)2

`m=V[<[ˆa`m]] and σ(I)2

`m=

V[=[ˆa`m]].

In appendix C we show that the analytical estimates ob-

tained using eq. (29) provides a fairly good approximation

to those obtained via a numerical Monte Carlo analysis (see

the comparison in TABLE III). Expression (29) can be further

simpliﬁed by making assumptions about the nature of the er-

rors. If the distribution of the galaxies is isotropic and all of

them have the same error in η(δi=δ), then, the real and

imaginary parts of ˆa`mare characterised by the same vari-

ance, and it will not depend on the considered harmonics (

σ2

`m=σ2≈4π

Nδ2), so,

V[ˆ

Cl]=2

2`+1h(σ2+ˆ

C`)2−ˆ

li.(31)

The above formula neatly isolates the two quantities contribut-

ing to the observed variance: the error in the estimation of the

distance modulus and the measured amplitude of the angular

power spectrum ˆ

C`.

Note, incidentally, that eq. (31) diﬀers from the expression

that would be obtained by averaging over a statistical ensem-

ble. Indeed, in this latter case Ee[ˆa`m]=0 (so ˆ

C`will be

replaced by zero) and also Ve[ˆa`m]=C`+σ2. It thus follows

that the theoretical expression factoring in the contributions of

cosmic variance is Ve[ˆ

C`]=2/(2`+1)(σ2+C`)2, if, again, it

is assumed that σ`mis isotropically distributed.

D. Systematic measurement errors

Eq. (26) provides a biased estimate of the local value of the

angular power spectrum ˆ

C`since its expectation over mea-

surements is

Eˆ

C`=ˆ

C`+1

2`+1

m=−`

σ2

`m.(32)

Measurements errors on the distance modulus lead to a sys-

tematic overestimation of the angular power spectrum ˆ

C`, and

the statistical bias term calculated in eq. (32) might not fully

remove the systematic shift. Indeed, expression (32) is strictly

valid if the estimator ˆa`m(cf. eq. (24)) is, as we assumed,

aﬀected only by statistical errors. It is true, however, that in-

completeness and anisotropies in the sky distribution, as well

as the pixelization and resolution strategy adopted to trans-

form the discrete ηobservable into a ﬁeld, could bias the ˆa`m

estimator. Although any constant systematic term added in 24

does not aﬀect the variance of the coeﬃcients of ˆa`m, it will

result in an additional (and analytically nontrivial) term in eq.

(32).

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ThemultipoleexpansionofthelocalexpansionrateBasheerKalbouneh,ChristianMarinoni,andJulienBelAixMarseilleUniv,Universit´edeToulon,CNRS,CPT,Marseille,France(Dated:January5,2023)Wedesignanewobservable,theexpansionrateuctuation,tocharacterizedeviationsfromthelinearrelationbetweenredshiftanddUniverseis...

展开>> 收起<<

The multipole expansion of the local expansion rate Basheer Kalbouneh Christian Marinoni and Julien Bel Aix Marseille Univ Universit e de Toulon CNRS CPT Marseille France.pdf

共27页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

The multipole expansion of the local expansion rate Basheer Kalbouneh Christian Marinoni and Julien Bel Aix Marseille Univ Universit e de Toulon CNRS CPT Marseille France

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: