The standard Friedmann model of cosmology is based on the Copernican
Principle, i.e. the assumption of a homogeneous background on which structure
forms via perturbations. Homogeneity underpins both general relativistic and
modified gravity models and is central to the way in which we interpret
observations of the CMB and the galaxy distribution. It is therefore important
to probe homogeneity via observations. We describe a test based on the fossil
record of distant galaxies: if we can reconstruct key intrinsic properties of
galaxies as functions of proper time along their worldlines, we can compare
such properties at the same proper time for our galaxy and others. We achieve
this by computing the lookback time using radial Baryon Acoustic Oscillations,
and the time along galaxy world line using stellar physics, allowing us to
probe homogeneity, in principle anywhere inside the past light cone. Agreement
in the results would be an important consistency test -- although it would not
in itself prove homogeneity. Any significant deviation in the results however
would signal a breakdown of homogeneity.Comment: Accepted for publication in JCAP. Matches published version. Minor
  changes: ref. added and longer discussion on performing the test
  observationally. Results unchange

Heavens, AF

Jimenez, R

Maartens, R

Journal of Cosmology and Astroparticle Physics

arXiv

13.02.13 KB. Accepted version ok to add to Spiral. IO

Spiral - Imperial College Digital Repository

Prepared for submission to JCAPTesting homogeneity with the fossilrecord of galaxiesAlan F. Heavens,a Raul Jimenezb and Roy Maartensc,daSUPA, Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh, EH9 3HJ, UKbICREA & ICC, University of Barcelona (IEEC-UB), Barcelona 08024, SpaincDepartment of Physics, University of Western Cape, Cape Town 7535, South AfricadInstitute of Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UKE-mail: afh@roe.ac.uk, raul.jimenez@icc.ub.edu, roy.maartens@port.ac.ukAbstract. The standard Friedmann model of cosmology is based on the Copernican Principle, i.e.the assumption of a homogeneous background on which structure forms via perturbations. Homo-geneity underpins both general relativistic and modified gravity models and is central to the way inwhich we interpret observations of the CMB and the galaxy distribution. It is therefore important toprobe homogeneity via observations. We describe a test based on the fossil record of distant galaxies:if we can reconstruct key intrinsic properties of galaxies as functions of proper time along their world-lines, we can compare such properties at the same proper time for our galaxy and others. We achievethis by computing the lookback time using radial Baryon Acoustic Oscillations, and the time alonggalaxy world line using stellar physics, allowing us to probe homogeneity, in principle anywhere insidethe past light cone. Agreement in the results would be an important consistency test – although itwould not in itself prove homogeneity. Any significant deviation in the results however would signala breakdown of homogeneity.Keywords: Early Universe; GravityarXiv:1107.5910v2  [astro-ph.CO]  15 Sep 2011Contents1 Introduction 12 Lookback time in a general universe 23 Probing homogeneity with the fossil record 44 Conclusions 51 IntroductionThe standard model of the Universe is homogeneous, with structure formation described via pertur-bations on a fixed background. Within general relativity (GR), it then follows that the accelerationof the Universe is driven by dark energy. The homogeneous standard model is a simple, predictivemodel that successfully accommodates all observations up to now. However, we should probe thefoundations of this model as far as possible. This is further motivated by the fact that there is stillno satisfactory description of the dark energy that is central to the model. We can probe the assump-tion that GR holds on cosmological scales, by investigating modified gravity theories and by devisingconsistency tests of GR. This probe is only effective if we assume homogeneity. Alternatively, we canprobe the assumption of homogeneity itself [1–3].Homogeneity is not established by observations of the CMB and the galaxy distribution – wecannot directly observe homogeneity, since we observe down the past lightcone, recording propertieson 2-spheres of constant redshift and not on spatial surfaces that intersect that lightcone. Whatthese observations can directly probe is isotropy. In order to link isotropy to homogeneity, we haveto assume the Copernican Principle, i.e. that we are not at a special position in the Universe. TheCopernican Principle is not observationally based; it is an expression of the intrinsic limitation ofobservations from one spacetime location.We cannot observationally prove homogeneity, but we can develop consistency tests that woulduncover deviations from homogeneity if these deviations existed. In other words, we can test forconsistency of the Copernican Principle and for violations of the Copernican Principle. If we find noviolation, then the indirect evidence for homogeneity is strengthened. However, any single significantviolation of the Copernican Principle would disprove homogeneity.Here we propose a new test of homogeneity, based on intrinsic properties of galaxy evolution.Homogeneity means that fundamental properties, such as star formation histories, should be uniform– this is the ‘fossil record’ implicitly carried by galaxies via their evolution. By observing distantgalaxies, and then tracing our own galactic history back to the proper time of emission, we can checkfor consistency between properties of our galaxy and distant galaxies.This is closely related to the ‘cosmic chronometer’ project [4–6], which uses galaxy spectral prop-erties to find H(z) in a Friedmann model (see also [7]). Here we do not assume Friedmann geometrybut rather test for it. It is also related to earlier work which attempted to prove homogeneity fromuniformity of the thermal history of galaxies [8]. However it turns out that there are inhomogeneousmodels with uniform thermal histories [8], and the proposal fails as a proof of homogeneity. It doessurvive however as a probe of homogeneity – i.e., violations of uniformity would indicate a break-down of the Copernican Principle and thus of homogeneity. The question is how to apply the idea inpractice.Before describing our proposal, we briefly review other tests of the Copernican Principle andhomogeneity.• The geometric null test of [9] is based on the tight relation between curvature and expansionhistory in a Friedmann spacetime. This relation is independent of dark energy and other con-tributions to the energy-momentum tensor (and also independent of the field equations). The– 1 –quantityK(z) := H20 (1 + z)4 +H2(z)[(1 + z)2{DL(z)D′′L(z)−D′2L (z)}+D2L(z)]+ (1 + z)H(z)H ′(z)DL(z) [(1 + z)D′L(z)−DL(z)] (1.1)is identically zero at all redshifts in a Friedmann spacetime with arbitrary content and curvature.It follows thatK(z) significantly different from 0 ⇒ violation of homogeneity. (1.2)(How to implement the test is discussed in [10].)• The baryon acoustic oscillation (BAO) feature in the galaxy distribution provides a furthergeometric test of homogeneity [1, 9, 11]. Future large-volume surveys will allow the detection ofthe BAO scale in both radial and transverse directions. These scales are equal in a Friedmannbackground, so thatradial BAOtransverse BAO− 1 significantly different from 0 ⇒ violation of homogeneity. (1.3)• Galaxy clusters with their hot ionized intra-cluster gas, act via scattering of CMB photons likegiant mirrors that carry information about the last scattering surface seen by the cluster – andtherefore serve as indirect probes inside our past lightcone. The Sunyaev-Zeldovich (SZ) effecton the CMB temperature anisotropies probes the monopole (thermal SZ) and dipole (kineticSZ) seen by the cluster in a perturbed Friedmann model. Thus [1, 12]Non-perturbative thermal or kinetic SZ temperature effect ⇒ violation of homogeneity.(1.4)(In related work, the SZ effect has been used to place constraints on inhomogeneous void models[13–15].)The SZ effect on CMB polarization provides a new and different probe of the monopole anddipole, and in addition probes the quadrupole and octupole seen by the cluster, in a perturbedFriedmann model. Thus [1]:Non-perturbative SZ polarization effects ⇒ violation of homogeneity. (1.5)Other tests that may become possible with future observations include: a geometric null test based onthe time drift of the cosmological redshift [16]; massive neutrinos propagate inside our past lightconeand provide in principle a probe of deviations from homogeneity [17], if the cosmic neutrino backgroundcan be observed.We now turn to our proposed probe of the Copernican Principle, based on the fossil recordcarried by galaxies, which is also an indirect probe of conditions inside our past lightcone. First wemust deal with the problem of lookback time.2 Lookback time in a general universeA distant galaxy, with worldline E , emits photons at event E that we observe with redshift zE atevent O on our galaxy worldline O (see Fig. 1). In order to compare the intrinsic properties of E andO at the same proper time, we need to compute the lookback time tLB = tO − tE , where t denotesproper time along galaxy worldlines. This is straightforward in a Friedmann model – but we cannotassume the geometry of spacetime if our aim is to test for homogeneity. So we need to compute thelookback time in a covariant way, valid in a general spacetime.The galaxy 4-velocity field is uµ = dxµ/dt. The past-pointing photon 4-momentum is kµ =dxµ/dv, where v is the null affine parameter with v = 0 at O. Then1 + z = uµkµ, kµ = (1 + z)(−uµ + nµ), uµnµ = 0, nµnµ = 1, (2.1)– 2 –Figure 1. Lookback time in a general spacetime.where nµ is a unit vector along the line of sight. For observers comoving with the matter, an incrementdv in null affine parameter corresponds to a time increment dt, wheredt = −uµkµdv = −(1 + z)dv. (2.2)We need to relate v to z: by (2.1),dzdv= kν∇ν(uµkµ) = kµkν∇µuν , (2.3)where the last equality follows since kµ is geodesic. The covariant derivative is split as∇µuν =13Θhµν + σµν + ωµν − uµu̇ν , (2.4)where hµν = gµν +uµuν projects into the galaxy instantaneous rest space, Θ is the volume expansionrate (Θ = 3H in a Friedmann model), σµν is the shear, ωµν is the vorticity and u̇µ is the acceleration.Since matter is pressure-free, u̇µ = 0. Putting everything together, we getdzdv= (1 + z)2[13Θ + σµνnµnν]. (2.5)Now we integrate along the lightray from O to E, using (2.2) and (2.5):tO − tE =∫ zE0dz(1 + z)[Θ(z)/3 + σµν(z)nµnν] . (2.6)This will give us the lookback time – provided that we can uniquely relate the time intervals alonggalaxy worldlines that cross the light ray to a time interval along our worldline O. In order to do this,we need the existence of spatial 3-surfaces that are everywhere orthogonal to uµ; these will then be– 3 –surfaces of constant proper time. The necessary and sufficient condition for these surfaces to exists isan irrotational flow:ωµν = 0. (2.7)Then we can uniquely identify the event E′ where the constant proper time surface t = tE throughE intersects O. (For rotating matter, it is not clear whether we can consistently define a lookbacktime.)3 Probing homogeneity with the fossil recordIn order to compare the two galaxies at the same proper time, we need a method to evaluate thelookback time (2.6) from some observable. The observed expansion rate along the line of sight is [2]Hobs,los(z) = nµnν∇µuν =13Θ(z) + σµν(z)nµnν . (3.1)This leads to a covariant expression for the radial part of the Alcock-Paczynski formula, valid in anyspacetime; in particular, we get a covariant expression for the comoving radial BAO scale, set by themaximum comoving distance a sound wave travels, rbao,los, without assuming Friedmann geometry[1]:rbao,los =δz(z)Hobs,los(z)=δz(z)Θ(z)/3 + σµν(z)nµnν. (3.2)In principle, Hobs,los can be found from observations of the extent of the baryon ruler in redshift,δz(z), for a range of redshifts along the line of sight to the emitting galaxy. Then the lookback timeis simplytO − tE =∫ zE0dz(1 + z)Hobs,los(z). (3.3)We can estimate how many independent patches there are in the sky where the BAO featurecan be measured, using the Friedmann formalism in [18]. Note that radial BAO observations giveus Hobs,los times the rbao,los; we assume the latter is computed to sufficient accuracy (∼ 1% in thestandard model) that the main error source is from Hobs,los. The angular size of a box with redshiftdepth of δz is δθ = (1 + z)DA/rbao,trans, where for this estimate we assume rbao,los = rbao,trans. Thentaking redshift bins of, for example, width 0.2, a 10 (20)% measurement of Hobs,los(z) requires boxeswith size ∼ 650(450)Mpch−1. We ignore complications from bias and assume that the survey will belarge enough such that shot noise is subdominant. This estimate is supported by looking directly intoa suite of dark matter simulations and determining the BAO feature [19]. Therefore, there are ∼ 50independent patches in the sky where the homogeneity test can be performed.For the galaxy fossil record, we assume that from the integrated spectrum of a galaxy E observedat redshift zE , we can obtain its specific star formation rate S at the event E of emission, correspondingto E ’s proper time tE (see Fig. 1). Then we use a model of star formation to compute S for ourgalaxy in the past, at the event E′, corresponding to proper time t = tO − tE , which we determinefrom the radial BAO data via (3.3). If the universe is homogeneous, then these rates should agreewithin the errors – and this should be true for galaxies observed in any direction and at any redshiftwhere we are able to determine the lookback time. Conversely, any significant difference for even onecomparison would indicate a breakdown of homogeneity:SE′(tO − tE)− SE(tE) significantly different from 0 ⇒ violation of homogeneity. (3.4)A more ambitious approach is to obtain the lookback time also from stellar ages of the stellarpopulations of galaxies. If the Universe is homogeneous, then in patches of the sky that are sufficientlylarge to measure global properties, S should be be the same at the same time from the beginning ofstar formation in galaxies. We thus need both absolute ages and relative ones for the different timebins in which the galaxy lookback time has been split.We obtain the above determination of the specific star formation rate and lookback time for twogalaxies, our own O and E . We also measure the redshift zE . Let us consider the different possible– 4 –Figure 2. Schematic of star formation histories for 2 galaxies showing how the time difference from the lastperiod of activity can be used to probe homogeneity.outcomes. Suppose that galaxy O has an absolute age T . If we measure that galaxy E has an age> T then we conclude that the universe is inhomogeneous, as it is otherwise impossible to explainhow the distant galaxy with positive lookback time is older.The other outcome is that the absolute age of galaxy E is smaller than T . Then we need to lookat the specific star formation rate per unit of mass for both galaxies. In a homogeneous Universe,these should be the same at the same time from t=0. Comparing the specific star formation rates,we look at the time difference from the last event when stars where formed at galaxy E and galaxyO (see Fig. 2). Because we know the redshift, we can now compare with the lookback time (3.3)determined via the BAO.This method would not work if the specific star formation rate is an exact power law, but this isnever the case [20, 21]. In order to perform this test in practice we can use the currently determinedMOPED/VESPA [20, 22] star formation histories. For an individual galaxy the time resolution iscoarse (about 30% of the Hubble time), but we can improve the spatial resolution of the test ofhomogeneity by binning the spectra (cf. [24]), thus improving the time resolution of the recoveredstar formation histories. At some point the method is limited by the accuracy of current models, but10% resolution should be possible with current data and models, allowing a test of homogeneity onscales of a few hundred Mpc. We are currently working on such a test (Hoyle et al., in preparation).4 ConclusionsAlthough we cannot directly observe homogeneity, we can test the Copernican Principle at the foun-dation of homogeneity, using observations that carry information from inside our past lightcone.Previously this has been discussed via consistency relations between distances and expansion rates,using supernova and BAO data, and via Sunyaev-Zeldovich effects from galaxy clusters on the CMBtemperature and polarization. Here we have proposed a new test of homogeneity based on the fossilrecord of distant galaxies. We make observations at constant time slices within our past light cone bycomputing the lookback time to a distant galaxy and adding it to the proper time along the galaxypast worldline. The lookback time is computed without assuming Friedmann geometry, using stan-dard rulers (the radial BAO); the proper time on the galaxy worldline is effectively computed usingthe fossil record, dependent on the physics of stellar evolution but not on a Friedmann geometry.As remarked, this test has the advantage that it allows one to look inside our past light cone, inprinciple everywhere inside. Naturally, we cannot make any global statements about homogeneity, aswe are inevitably restricted to the region on or within our past light cone. The method is interestingin the sense that once the lookback time has been computed, then the fossil record comparisons shouldbe limited only by observational error and sample variance, and if fossil record observations are not– 5 –consistent in different locations then homogeneity does not apply. The converse is that any failure tofind inhomogeneity strengthens the case for the Copernican Principle.Acknowledgments: We thank Chris Clarkson, Catherine Cress, George Ellis, Licia Verde andChristian Wagner for discussions. RM is supported by a South African SKA Research Chair, bythe UK Science & Technology Facilities Council (grant no. ST/H002774/1) and by a NRF (SouthAfrica)/ Royal Society (UK) exchange grant.References[1] R. Maartens, arXiv:1104.1300[2] C. Clarkson and R. Maartens, Class. Quant. Grav. 27 (2010) 124008 [arXiv:1005.2165].[3] G. F. R. Ellis, arXiv:1103.2335.[4] D. Stern, R. Jimenez, L. Verde, M. Kamionkowski and S. A. Stanford, JCAP 1002 (2010) 008[arXiv:0907.3149].[5] S. M. Crawford, A. L. Ratsimbazafy, C. M. Cress, E. A. Olivier, S. L. Blyth and K. J. van der Heyden,arXiv:1004.2378.[6] M. Moresco, R. Jimenez, A. Cimatti and L. Pozzetti, JCAP 1103 (2011) 045 [arXiv:1010.0831].[7] D. P. Carson, R. C. Nichol, Mon. Not. Roy. Astron. Soc. 408 (2010) 213. [arXiv:1006.2830].[8] W. B. Bonnor and G. F. R. Ellis, Mon. Not. Roy. Astron. Soc. 218 (1986) 605.[9] C. Clarkson, B. Bassett and T. H.-C. Lu, Phys. Rev. Lett. 101 (2008) 011301 [arXiv:0712.3457].[10] A. Shafieloo and C. Clarkson, Phys. Rev. D 81 (2010) 083537 [arXiv:0911.4858 [astro-ph.CO]].[11] C. Clarkson, M. Regis, JCAP 1102 (2011) 013. [arXiv:1007.3443].[12] J. Goodman, Phys. Rev. D 52 (1995) 1821 [arXiv:astro-ph/9506068].[13] R. R. Caldwell and A. Stebbins, Phys. Rev. Lett. 100 (2008) 191302 [arXiv:0711.3459].[14] P. Zhang, A. Stebbins, Phys. Rev. Lett. 107 (2011) 041301. [arXiv:1009.3967].[15] J. P. Zibin, A. Moss, Class. Quant. Grav. 28 (2011) 164005. [arXiv:1105.0909].[16] J.-P. Uzan, C. Clarkson and G.F.R. Ellis, Phys. Rev. Lett. 100 (2008) 191303 [arXiv:0801.0068].[17] J. Jia and H. B. Zhang, JCAP 0812 (2008) 002 [arXiv:0809.2597].[18] H. J. Seo and D. J. Eisenstein, Astrophys. J. 665 (2007) 14 [arXiv:astro-ph/0701079].[19] C. Wagner, private communication.[20] A. Heavens, B. Panter, R. Jimenez and J. Dunlop, Nature 428 (2004) 625 [arXiv:astro-ph/0403293].[21] B. Panter, R. Jimenez, A. F. Heavens and S. Charlot, Mon. Not. Roy. Astron. Soc. 378 (2007) 1550[arXiv:astro-ph/0608531].[22] R. Tojeiro, A. F. Heavens, R. Jimenez R. and B. Panter, Mon. Not. Roy. Astron. Soc. 381 (2007) 1252[arXiv:0704.0941].[23] R. Tojeiro, S. Wilkins, A. F. Heavens, B. Panter and R. Jimenez, Astrophys. J. Suppl. 185 (2009) 1[arXiv:0904.1001].[24] R. Tojeiro, W. J. Percival, A. F. Heavens, R. Jimenez,[arXiv:1011.2346 [astro-ph.CO]].– 6 –

Testing homogeneity with the fossil record of galaxies

Alan F Heavens

Raul Jimenez

Roy Maartens

Crossref

Heavens, A.

Jimenez, R.

Maartens, Roy

Name not available

Prepared for submission to JCAP
Testing homogeneity with the fossil
record of galaxies
Alan F. Heavens,a Raul Jimenezb and Roy Maartensc,d
aSUPA, Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh, EH9 3HJ, UK
bICREA & ICC, University of Barcelona (IEEC-UB), Barcelona 08024, Spain
cDepartment of Physics, University of Western Cape, Cape Town 7535, South Africa
dInstitute of Cosmology & Gravitation, University of Portsmouth, Portsmouth PO1 3FX, UK
E-mail: afh@roe.ac.uk, raul.jimenez@icc.ub.edu, roy.maartens@port.ac.uk
Abstract. The standard Friedmann model of cosmology is based on the Copernican Principle, i.e.
the assumption of a homogeneous background on which structure forms via perturbations. Homo-
geneity underpins both general relativistic and modified gravity models and is central to the way in
which we interpret observations of the CMB and the galaxy distribution. It is therefore important to
probe homogeneity via observations. We describe a test based on the fossil record of distant galaxies:
if we can reconstruct key intrinsic properties of galaxies as functions of proper time along their world-
lines, we can compare such properties at the same proper time for our galaxy and others. We achieve
this by computing the lookback time using radial Baryon Acoustic Oscillations, and the time along
galaxy world line using stellar physics, allowing us to probe homogeneity, in principle anywhere inside
the past light cone. Agreement in the results would be an important consistency test – although it
would not in itself prove homogeneity. Any significant deviation in the results however would signal
a breakdown of homogeneity.
Keywords: Early Universe; Gravityar
X
iv
:1
10
7.
59
10
v2
  [
as
tro
-p
h.C
O]
  1
5 S
ep
 20
11
Contents
1 Introduction 1
2 Lookback time in a general universe 2
3 Probing homogeneity with the fossil record 4
4 Conclusions 5
1 Introduction
The standard model of the Universe is homogeneous, with structure formation described via pertur-
bations on a fixed background. Within general relativity (GR), it then follows that the acceleration
of the Universe is driven by dark energy. The homogeneous standard model is a simple, predictive
model that successfully accommodates all observations up to now. However, we should probe the
foundations of this model as far as possible. This is further motivated by the fact that there is still
no satisfactory description of the dark energy that is central to the model. We can probe the assump-
tion that GR holds on cosmological scales, by investigating modified gravity theories and by devising
consistency tests of GR. This probe is only effective if we assume homogeneity. Alternatively, we can
probe the assumption of homogeneity itself [1–3].
Homogeneity is not established by observations of the CMB and the galaxy distribution – we
cannot directly observe homogeneity, since we observe down the past lightcone, recording properties
on 2-spheres of constant redshift and not on spatial surfaces that intersect that lightcone. What
these observations can directly probe is isotropy. In order to link isotropy to homogeneity, we have
to assume the Copernican Principle, i.e. that we are not at a special position in the Universe. The
Copernican Principle is not observationally based; it is an expression of the intrinsic limitation of
observations from one spacetime location.
We cannot observationally prove homogeneity, but we can develop consistency tests that would
uncover deviations from homogeneity if these deviations existed. In other words, we can test for
consistency of the Copernican Principle and for violations of the Copernican Principle. If we find no
violation, then the indirect evidence for homogeneity is strengthened. However, any single significant
violation of the Copernican Principle would disprove homogeneity.
Here we propose a new test of homogeneity, based on intrinsic properties of galaxy evolution.
Homogeneity means that fundamental properties, such as star formation histories, should be uniform
– this is the ‘fossil record’ implicitly carried by galaxies via their evolution. By observing distant
galaxies, and then tracing our own galactic history back to the proper time of emission, we can check
for consistency between properties of our galaxy and distant galaxies.
This is closely related to the ‘cosmic chronometer’ project [4–6], which uses galaxy spectral prop-
erties to find H(z) in a Friedmann model (see also [7]). Here we do not assume Friedmann geometry
but rather test for it. It is also related to earlier work which attempted to prove homogeneity from
uniformity of the thermal history of galaxies [8]. However it turns out that there are inhomogeneous
models with uniform thermal histories [8], and the proposal fails as a proof of homogeneity. It does
survive however as a probe of homogeneity – i.e., violations of uniformity would indicate a break-
down of the Copernican Principle and thus of homogeneity. The question is how to apply the idea in
practice.
Before describing our proposal, we briefly review other tests of the Copernican Principle and
homogeneity.
• The geometric null test of [9] is based on the tight relation between curvature and expansion
history in a Friedmann spacetime. This relation is independent of dark energy and other con-
tributions to the energy-momentum tensor (and also independent of the field equations). The
– 1 –
quantity
K(z) := H20 (1 + z)4 +H2(z)
[
(1 + z)2
{
DL(z)D
′′
L(z)−D′2L (z)
}
+D2L(z)
]
+ (1 + z)H(z)H ′(z)DL(z) [(1 + z)D′L(z)−DL(z)] (1.1)
is identically zero at all redshifts in a Friedmann spacetime with arbitrary content and curvature.
It follows that
K(z) significantly different from 0 ⇒ violation of homogeneity. (1.2)
(How to implement the test is discussed in [10].)
• The baryon acoustic oscillation (BAO) feature in the galaxy distribution provides a further
geometric test of homogeneity [1, 9, 11]. Future large-volume surveys will allow the detection of
the BAO scale in both radial and transverse directions. These scales are equal in a Friedmann
background, so that
radial BAO
transverse BAO
− 1 significantly different from 0 ⇒ violation of homogeneity. (1.3)
• Galaxy clusters with their hot ionized intra-cluster gas, act via scattering of CMB photons like
giant mirrors that carry information about the last scattering surface seen by the cluster – and
therefore serve as indirect probes inside our past lightcone. The Sunyaev-Zeldovich (SZ) effect
on the CMB temperature anisotropies probes the monopole (thermal SZ) and dipole (kinetic
SZ) seen by the cluster in a perturbed Friedmann model. Thus [1, 12]
Non-perturbative thermal or kinetic SZ temperature effect ⇒ violation of homogeneity.
(1.4)
(In related work, the SZ effect has been used to place constraints on inhomogeneous void models
[13–15].)
The SZ effect on CMB polarization provides a new and different probe of the monopole and
dipole, and in addition probes the quadrupole and octupole seen by the cluster, in a perturbed
Friedmann model. Thus [1]:
Non-perturbative SZ polarization effects ⇒ violation of homogeneity. (1.5)
Other tests that may become possible with future observations include: a geometric null test based on
the time drift of the cosmological redshift [16]; massive neutrinos propagate inside our past lightcone
and provide in principle a probe of deviations from homogeneity [17], if the cosmic neutrino background
can be observed.
We now turn to our proposed probe of the Copernican Principle, based on the fossil record
carried by galaxies, which is also an indirect probe of conditions inside our past lightcone. First we
must deal with the problem of lookback time.
2 Lookback time in a general universe
A distant galaxy, with worldline E , emits photons at event E that we observe with redshift zE at
event O on our galaxy worldline O (see Fig. 1). In order to compare the intrinsic properties of E and
O at the same proper time, we need to compute the lookback time tLB = tO − tE , where t denotes
proper time along galaxy worldlines. This is straightforward in a Friedmann model – but we cannot
assume the geometry of spacetime if our aim is to test for homogeneity. So we need to compute the
lookback time in a covariant way, valid in a general spacetime.
The galaxy 4-velocity field is uµ = dxµ/dt. The past-pointing photon 4-momentum is kµ =
dxµ/dv, where v is the null affine parameter with v = 0 at O. Then
1 + z = uµk
µ, kµ = (1 + z)(−uµ + nµ), uµnµ = 0, nµnµ = 1, (2.1)
– 2 –
Figure 1. Lookback time in a general spacetime.
where nµ is a unit vector along the line of sight. For observers comoving with the matter, an increment
dv in null affine parameter corresponds to a time increment dt, where
dt = −uµkµdv = −(1 + z)dv. (2.2)
We need to relate v to z: by (2.1),
dz
dv
= kν∇ν(uµkµ) = kµkν∇µuν , (2.3)
where the last equality follows since kµ is geodesic. The covariant derivative is split as
∇µuν = 1
3
Θhµν + σµν + ωµν − uµu˙ν , (2.4)
where hµν = gµν +uµuν projects into the galaxy instantaneous rest space, Θ is the volume expansion
rate (Θ = 3H in a Friedmann model), σµν is the shear, ωµν is the vorticity and u˙µ is the acceleration.
Since matter is pressure-free, u˙µ = 0. Putting everything together, we get
dz
dv
= (1 + z)2
[1
3
Θ + σµνn
µnν
]
. (2.5)
Now we integrate along the lightray from O to E, using (2.2) and (2.5):
tO − tE =
∫ zE
0
dz
(1 + z)
[
Θ(z)/3 + σµν(z)nµnν
] . (2.6)
This will give us the lookback time – provided that we can uniquely relate the time intervals along
galaxy worldlines that cross the light ray to a time interval along our worldline O. In order to do this,
we need the existence of spatial 3-surfaces that are everywhere orthogonal to uµ; these will then be
– 3 –
surfaces of constant proper time. The necessary and sufficient condition for these surfaces to exists is
an irrotational flow:
ωµν = 0. (2.7)
Then we can uniquely identify the event E′ where the constant proper time surface t = tE through
E intersects O. (For rotating matter, it is not clear whether we can consistently define a lookback
time.)
3 Probing homogeneity with the fossil record
In order to compare the two galaxies at the same proper time, we need a method to evaluate the
lookback time (2.6) from some observable. The observed expansion rate along the line of sight is [2]
Hobs,los(z) = n
µnν∇µuν = 1
3
Θ(z) + σµν(z)n
µnν . (3.1)
This leads to a covariant expression for the radial part of the Alcock-Paczynski formula, valid in any
spacetime; in particular, we get a covariant expression for the comoving radial BAO scale, set by the
maximum comoving distance a sound wave travels, rbao,los, without assuming Friedmann geometry
[1]:
rbao,los =
δz(z)
Hobs,los(z)
=
δz(z)
Θ(z)/3 + σµν(z)nµnν
. (3.2)
In principle, Hobs,los can be found from observations of the extent of the baryon ruler in redshift,
δz(z), for a range of redshifts along the line of sight to the emitting galaxy. Then the lookback time
is simply
tO − tE =
∫ zE
0
dz
(1 + z)Hobs,los(z)
. (3.3)
We can estimate how many independent patches there are in the sky where the BAO feature
can be measured, using the Friedmann formalism in [18]. Note that radial BAO observations give
us Hobs,los times the rbao,los; we assume the latter is computed to sufficient accuracy (∼ 1% in the
standard model) that the main error source is from Hobs,los. The angular size of a box with redshift
depth of δz is δθ = (1 + z)DA/rbao,trans, where for this estimate we assume rbao,los = rbao,trans. Then
taking redshift bins of, for example, width 0.2, a 10 (20)% measurement of Hobs,los(z) requires boxes
with size ∼ 650(450)Mpch−1. We ignore complications from bias and assume that the survey will be
large enough such that shot noise is subdominant. This estimate is supported by looking directly into
a suite of dark matter simulations and determining the BAO feature [19]. Therefore, there are ∼ 50
independent patches in the sky where the homogeneity test can be performed.
For the galaxy fossil record, we assume that from the integrated spectrum of a galaxy E observed
at redshift zE , we can obtain its specific star formation rate S at the event E of emission, corresponding
to E ’s proper time tE (see Fig. 1). Then we use a model of star formation to compute S for our
galaxy in the past, at the event E′, corresponding to proper time t = tO − tE , which we determine
from the radial BAO data via (3.3). If the universe is homogeneous, then these rates should agree
within the errors – and this should be true for galaxies observed in any direction and at any redshift
where we are able to determine the lookback time. Conversely, any significant difference for even one
comparison would indicate a breakdown of homogeneity:
SE′(tO − tE)− SE(tE) significantly different from 0 ⇒ violation of homogeneity. (3.4)
A more ambitious approach is to obtain the lookback time also from stellar ages of the stellar
populations of galaxies. If the Universe is homogeneous, then in patches of the sky that are sufficiently
large to measure global properties, S should be be the same at the same time from the beginning of
star formation in galaxies. We thus need both absolute ages and relative ones for the different time
bins in which the galaxy lookback time has been split.
We obtain the above determination of the specific star formation rate and lookback time for two
galaxies, our own O and E . We also measure the redshift zE . Let us consider the different possible
– 4 –
Figure 2. Schematic of star formation histories for 2 galaxies showing how the time difference from the last
period of activity can be used to probe homogeneity.
outcomes. Suppose that galaxy O has an absolute age T . If we measure that galaxy E has an age
> T then we conclude that the universe is inhomogeneous, as it is otherwise impossible to explain
how the distant galaxy with positive lookback time is older.
The other outcome is that the absolute age of galaxy E is smaller than T . Then we need to look
at the specific star formation rate per unit of mass for both galaxies. In a homogeneous Universe,
these should be the same at the same time from t=0. Comparing the specific star formation rates,
we look at the time difference from the last event when stars where formed at galaxy E and galaxy
O (see Fig. 2). Because we know the redshift, we can now compare with the lookback time (3.3)
determined via the BAO.
This method would not work if the specific star formation rate is an exact power law, but this is
never the case [20, 21]. In order to perform this test in practice we can use the currently determined
MOPED/VESPA [20, 22] star formation histories. For an individual galaxy the time resolution is
coarse (about 30% of the Hubble time), but we can improve the spatial resolution of the test of
homogeneity by binning the spectra (cf. [24]), thus improving the time resolution of the recovered
star formation histories. At some point the method is limited by the accuracy of current models, but
10% resolution should be possible with current data and models, allowing a test of homogeneity on
scales of a few hundred Mpc. We are currently working on such a test (Hoyle et al., in preparation).
4 Conclusions
Although we cannot directly observe homogeneity, we can test the Copernican Principle at the foun-
dation of homogeneity, using observations that carry information from inside our past lightcone.
Previously this has been discussed via consistency relations between distances and expansion rates,
using supernova and BAO data, and via Sunyaev-Zeldovich effects from galaxy clusters on the CMB
temperature and polarization. Here we have proposed a new test of homogeneity based on the fossil
record of distant galaxies. We make observations at constant time slices within our past light cone by
computing the lookback time to a distant galaxy and adding it to the proper time along the galaxy
past worldline. The lookback time is computed without assuming Friedmann geometry, using stan-
dard rulers (the radial BAO); the proper time on the galaxy worldline is effectively computed using
the fossil record, dependent on the physics of stellar evolution but not on a Friedmann geometry.
As remarked, this test has the advantage that it allows one to look inside our past light cone, in
principle everywhere inside. Naturally, we cannot make any global statements about homogeneity, as
we are inevitably restricted to the region on or within our past light cone. The method is interesting
in the sense that once the lookback time has been computed, then the fossil record comparisons should
be limited only by observational error and sample variance, and if fossil record observations are not
– 5 –
consistent in different locations then homogeneity does not apply. The converse is that any failure to
find inhomogeneity strengthens the case for the Copernican Principle.
Acknowledgments: We thank Chris Clarkson, Catherine Cress, George Ellis, Licia Verde and
Christian Wagner for discussions. RM is supported by a South African SKA Research Chair, by
the UK Science & Technology Facilities Council (grant no. ST/H002774/1) and by a NRF (South
Africa)/ Royal Society (UK) exchange grant.
References
[1] R. Maartens, arXiv:1104.1300
[2] C. Clarkson and R. Maartens, Class. Quant. Grav. 27 (2010) 124008 [arXiv:1005.2165].
[3] G. F. R. Ellis, arXiv:1103.2335.
[4] D. Stern, R. Jimenez, L. Verde, M. Kamionkowski and S. A. Stanford, JCAP 1002 (2010) 008
[arXiv:0907.3149].
[5] S. M. Crawford, A. L. Ratsimbazafy, C. M. Cress, E. A. Olivier, S. L. Blyth and K. J. van der Heyden,
arXiv:1004.2378.
[6] M. Moresco, R. Jimenez, A. Cimatti and L. Pozzetti, JCAP 1103 (2011) 045 [arXiv:1010.0831].
[7] D. P. Carson, R. C. Nichol, Mon. Not. Roy. Astron. Soc. 408 (2010) 213. [arXiv:1006.2830].
[8] W. B. Bonnor and G. F. R. Ellis, Mon. Not. Roy. Astron. Soc. 218 (1986) 605.
[9] C. Clarkson, B. Bassett and T. H.-C. Lu, Phys. Rev. Lett. 101 (2008) 011301 [arXiv:0712.3457].
[10] A. Shafieloo and C. Clarkson, Phys. Rev. D 81 (2010) 083537 [arXiv:0911.4858 [astro-ph.CO]].
[11] C. Clarkson, M. Regis, JCAP 1102 (2011) 013. [arXiv:1007.3443].
[12] J. Goodman, Phys. Rev. D 52 (1995) 1821 [arXiv:astro-ph/9506068].
[13] R. R. Caldwell and A. Stebbins, Phys. Rev. Lett. 100 (2008) 191302 [arXiv:0711.3459].
[14] P. Zhang, A. Stebbins, Phys. Rev. Lett. 107 (2011) 041301. [arXiv:1009.3967].
[15] J. P. Zibin, A. Moss, Class. Quant. Grav. 28 (2011) 164005. [arXiv:1105.0909].
[16] J.-P. Uzan, C. Clarkson and G.F.R. Ellis, Phys. Rev. Lett. 100 (2008) 191303 [arXiv:0801.0068].
[17] J. Jia and H. B. Zhang, JCAP 0812 (2008) 002 [arXiv:0809.2597].
[18] H. J. Seo and D. J. Eisenstein, Astrophys. J. 665 (2007) 14 [arXiv:astro-ph/0701079].
[19] C. Wagner, private communication.
[20] A. Heavens, B. Panter, R. Jimenez and J. Dunlop, Nature 428 (2004) 625 [arXiv:astro-ph/0403293].
[21] B. Panter, R. Jimenez, A. F. Heavens and S. Charlot, Mon. Not. Roy. Astron. Soc. 378 (2007) 1550
[arXiv:astro-ph/0608531].
[22] R. Tojeiro, A. F. Heavens, R. Jimenez R. and B. Panter, Mon. Not. Roy. Astron. Soc. 381 (2007) 1252
[arXiv:0704.0941].
[23] R. Tojeiro, S. Wilkins, A. F. Heavens, B. Panter and R. Jimenez, Astrophys. J. Suppl. 185 (2009) 1
[arXiv:0904.1001].
[24] R. Tojeiro, W. J. Percival, A. F. Heavens, R. Jimenez,
[arXiv:1011.2346 [astro-ph.CO]].
– 6 –


https://researchportal.port.ac.uk/portal/services/downloadRegister/118503/1107.5910v2.pdf

Testing homogeneity with the fossil record of galaxies

Abstract

Similar works

Full text

Available Versions

Spiral - Imperial College Digital Repository

Crossref

Name not available