Testing the Cosmological Principle Subir Sarkar
3bth Conf. on Recent Developments in High Energy Physics & C0smology, Thessaloniki, 16 June 2021 All we can ever learn about the universe is contained within our past light cone
We cannot move over cosmological distances and check if the universe looks the same from ‘over there’ … so need to assume that our position is not special “The Universe must appear to be the same to all observers wherever they are. This ‘cosmological principle’ …” Edward Arthur Milne, ’Kinematics, Dynamics & the Scale of Time’ (1936) THE COSMOLOGICAL PRINCIPLE BY D. E. LITTLEWOOD Volume 51, Issue 4, October 1955 , pp. 678-683
The ‘Perfect Cosmological Principle’ was abandoned following the discovery of the Cosmic Microwave Background in 1964 and the realisation that the universe does have a beginning, but the Cosmological Principle lived on …
… …
Steven Weinberg, Gravitation and Cosmology (1972) PHYS 652: Astrophysics 19
After straightforward yet tedious calculations2 (whichµ I re⌫legate to homework), we obtain the com- ds gµ⌫ dx dx ponents of the Ricci tensor: ⌘ = a2(⌘) d⌘2 dx¯2 0 a¨ R0 =3, a 0 ⇥ ⇤ R =0, 1 i R Rg + g Tµ⌫ = ⇢ fields gµ1⌫ µ⌫2 µ⌫ µ⌫ Ri = aa¨ +2˙a +22k !!. h i j a2 " ! "=8⇡GNTµ⌫ (93)
The t t component of the Einstein’s equation given in eq. (92) becomes ! 3¨a 1 =8"G (# + P )+ (# P ) , (94) a ⇢m #! k 2 ! $ ⇤ ⌦m 2 , ⌦k 2 2 , ⌦⇤ 2 (3H /8⇡GN) (3H a ) (3H or
2˙a2 2k a¨ =4"G(# P )a + , (98) ! ! a a which can be combined with eq. (95) to cancel out P dependence and yield
16"G#a 2k 2˙a2 =0, (99) 3 ! a ! a or 2 8"G 2 a˙ + k = #a . (100) 3 When combined with the eq. (62) derived in the context of conservation of energy-momentum tensor, and the equation of state, we obtain a closed system of Friedmann equations:
2 8"G 2 a˙ + k = #a , (101a) 3 $# a˙ +3(# + P ) =0, (101b) $t a P = P (#). (101c)
19 It is also a good rule not to put overmuch confidence in the observational results that are put forward until they are confirmed by theory – Arthur Eddington (1934) Interpreting Λ as vacuum energy also raises the ‘coincidence problem’:
Why is ΩΛ≈ Ωm today? An evolving ultralight scalar field (‘quintessence’) can display ‘tracking’ behaviour: this 1/4 -12 2 2 -42 requires V(φ) ~ 10 GeV but √d V/dφ ~ H0 ~10 GeV to ensure slow-roll … i.e. just as much fine-tuning as a bare cosmological constant A similar comment applies to models (e.g. ‘DGP brane-world’) wherein gravity is modified on the scale of the present Hubble radius 1/H0 so as to mimic vacuum energy … this scale is absent in a fundamental theory and must be put in by hand (there is similar fine-tuning in every proposal: massive gravity, chameleon fields, …) Therefore every attempt to explain the coincidence problem is equally fine-tuned
2 The only ‘natural’ option is if Λ ~ H always, but this is just a renormalisation of GN! 2 (recall: H = 8πGN/3 + Λ/3) ➙ ruled out by Big Bang nucleosynthesis which requires GN to be within 5% of lab. value … in any case this will not yield accelerated expansion
2 -42 Do we infer Λ ~ H0 from observations simply because H0 (~10 GeV) is the only scale in the F-R-L-W model … so this is the value imposed on Λ by construction? “Data from the Planck satellite show the universe to be highly isotropic” (Wikipedia) :A1,2014 571 Planck collaboration, A&A
We observe a ~statistically isotropic, ~gaussian random field of small temperature fluctuations (quantified by their 2-point correlations ➛ angular power spectrum) Standard model of structure formation : NASA Courtesey
The ~10-5 CMB temperature fluctuations are understood as due to scalar density perturbations with a ~scale-invariant spectrum which were generated during an early de Sitter phase of inflationary expansion … these perturbations have subsequently grown into the large-scale structure of galaxies observed today through gravitational instability in a sea of dark matter However the CMB is not isotropic There is a dipole with DT/T ~ 10-3 i.e. 100 times bigger than the small-scale anisotropy
T 1 2 T (✓)= 0 1 cos ✓ p 1967, Peebles & Wilkinson 1968 Wilkinson & 1967, Peebles Sciama
This is interpreted as due to our motion at 370 km/s wrt the ‘cosmic rest frame’ in which the CMB is truly isotropic … and in which the F-L-R-W equations hold
This motion is presumed to be due to local inhomogeneity in the matter distribution Its scale – beyond which we converge to the CMB frame – is supposedly of O(100) Mpc (Counts of galaxies in e.g. SDSS & WiggleZ surveys are said to scale as r3 on larger scales) Peculiar Velocity of the Sun and its Relation to the Cosmic Microwave Background
J. M. Stewart & D. W. Sciama
If the microwave blackbody radiation is both cosmological and isotropic, it will only be isotropic to :1349,2007 an observer who is at rest in the rest 79 frame of distant matter which last scattered the radiation. In this article an estimate is made of the velocity of the Sun relative to distant matter, from which a prediction can be made of the anisotropy to be expected in the microwave
Smoot, Rev. Mod. Phys. Mod. Phys. Smoot, Rev. radiation. It will soon be possible to compare this prediction with experimental results. NATURE 216:748,1967
The predicted CMB dipole was found soon afterwards … in broad agreement with expectations Structure within a cube extending extending cube a within Structure … We to appear movingbe towards Shapley supercluster the todue ‘Great a Attractor’ if so, our local ‘peculiarshould fall velocity’ off as rest frame’ in which the Friedmann ~ - 200 Lema Mpc î tre - Robertson from position our ~ 1/r as we converge to the `cosmic - Walker cosmology holds ( Supergal
Tully, Courtois, Hoffman, Pomarede, Nature 513:71,2014 actic Coord.) Theory of peculiar velocity fields (x)=[⇢(x) ⇢¯]/⇢¯ In linear perturbation theory, the growth of the density contrast
We are interested in the ‘growing mode’ solution – the density contrast grows self- similarly and so does the perturbation potential and its gradient … so the direction of the acceleration (and its integral – the peculiar velocity) remains unchanged.
The peculiar velocity field is related to the density contrast as: 2 x y v(x)= d3y (y), 3H x y 3
So the peculiar Hubble flow, �H(x) = HL(x) – H0 (⇒ trace of the shear tensor), is: x y H(x)= d3yv(y) W (x y), x y 2
H0 (2⇡)
h
~insensitive to shape of ~insensitive to shape of matter power spectrum matter power spectrum Hunt & S.S., MNRAS (Colin et al, MNRAS 414:264,2011) No convergence to the CMB frame out to ~260 Mpc
17 Need attractor mass of ~10 MSun at Feindt et al, A&A 560:A90,2013 ~300 Mpc to account for the flow Feindt : Ulrich Courtesey 6-degree Field Galaxy Survey confirms that bulk flow is higher than expected Largest single sample of 11,000 galaxy peculiar velocity measurements … using ‘Fundamental Plane’ distance estimates (2016) et al LCDM expectation for Gaussian window (90% CL) , Mould, Colless , Springbob , Magoulas
In the ‘Dark Sky’ simulations, <1% of Milky Way–like observers experience a bulk flow as large as is observed and extending out as far as is seen … so we are not ‘typical’ observers (Rameez et al, MNRAS 477:1772,2018, Mohayaee et al, arXiv:2003.10420) This distribution of the brightest radio sources at l ∼ 6 cm supposedly demonstrates the isotropy of the Universe Peebles, Peebles, Principles of Physical Cosmology Physical of Principles
Milky Way , 1993 ,
Unobserved regions
But if we are moving w.r.t. the cosmic rest frame, then distant sources should not be isotropic! If the dipole in the CMB is kinematic in origin, then cosmologically distant sources should exhibit a similar dipole in their sky distribution due to aberration
� � � = � [1 + 2 + � 1 + � cos(�)] !"# $ # �
Aberration Doppler boosting (Bradley 1727) S Rest frame Moving frame Power-law spectrum sin � tan � = � + S ∝ �--� γ ∗ ���� − � Differential flux Differential � � Frequency �
Integral flux distribution: N (>S) ∝ S-x Observer, velocity �
Ellis & Baldwin, ➙ Flux-limited catalog more sources in direction of motion MNRAS 206:377,1984 ⃗ Consider an all-sky catalogue of N � = � (�⃗!"#, �, α) + � (N) + � (D(z)) sources with redshift distribution D(z) from a directionally unbiased survey � → The ‘kinematic dipole’: independent of source distance, but depends on observer velocity, source spectrum, and source flux distribution D(z) � → The ‘random dipole’ ∝ 1/√� isotropically distributed 1 I � → The ‘clustering dipole’ due to the anisotropy in the source distribution redshift (significant for shallow surveys)
NVSS + SUMSS: 600,000 radio sources
Wide Field Infrared Survey Explorer: 1,200,000 galaxies,
Wide Field Infrared Survey Explorer: 1,360,000 quasars,
�#" Vary the direction of the Add up unit vectors hemispheres until maximum corresponding to directions in asymmetry is observed the sky for every source
�̂ "# # cos� �̂ "# # �! = � � sin�d�d� � = � � cos�sin�d�d� � "#$ #$ cos� � "#$ #$
However these estimators are biased (Rubart & Schwarz, A&A 555:A117,2013)
New: Unbiased Least-Squares Estimator Secrest et al, ApJL 908:L51,2021
(A1,1 /A0 , A1,2 /A0 , A1,3 /A0) where np denotes the number density of sources in sky pixel p, A0 is the mean density (monopole), A1j are the amplitudes of the three orthogonal dipole templates dj,p, and the sum is to be taken over all unmasked pixels The NRAO VLA Sky Survey (NVSS) + Sydney University Molonglo Sky Survey (SUMSS) (1.4 GHz survey down to Dec = -40.4o) (843 MHz survey at Dec < -30o) [Rescale the SUMSS fluxes by (843 MHz/1.4 GHz)-0.75 = 1.46 to match with NVSS]
To get rid of any ‘clustering dipole’: • Remove Galactic plane ±10o :1045,2017 (also Supergalactic plane) 471
• Remove nearby sources which are in common with 2MRS/LRS surveys
The direction is within 10° of CMB dipole, but velocity is ~ 1355 ± 174 km/s , Rameez & S.S., MNRAS Mohayaee
Confirms claim by Singal (ApJ 742:L23,2011) … however source redshifts are not Colin, directly measured (also the statistical significance is only 2.8� – by Monte Carlo) The CatWISE quasar catalogue (new)
0.8 Inferred from cross-
0.7 correlating with :L51,2021 0.6 SDSS DR16 Stripe 82: 908 3.4 μm (W1) 4.6 μm (W2) 4 0.5 ~10 eBoss redshifts PDF ApJL 9 > W1 > 16.4 0.4 W1 − W2 ≥ 0.8 Low-z AGNs excluded 0.3 by cross-correlation 0.2 with 2MASS XSC 0.1
2 0.0 2 , S.S. & Colin, 30 source deg° 90 0.00.51.0166.7 .52source.02 deg.53° .0369.8.5 redshift We now have a catalogue of 1.36 million quasars, with 99% at redshift > 0.1
Mohayaee 1
, 10
100
1 10°
Hausegger 2 10° PDF 3 10°
4 10°
5 10° , Rameez, Von , Von Rameez, 6 10° 1 0 1 2 10° 10 10 10 1 3 5 7 SW1 [mJy] ÆW1 Secrest The dipole can be compared to that expected, knowing the spectrum & flux distribution Our peculiar velocity wrt quasars ≠ peculiar velocity wrt the CMB Secrest , Rameez,Von, Galactic ± 60 ± = 30 b Hausegger ± 0 ` = 330± 300± 270± 240± 210± CatWISE CMB dipole , The direction of the quasar dipole is consistent with the CMB dipole - but not its amplitude Mohayaee , S.S. & Colin, Colin, & S.S. , ApJL 908 :L51,2021 The kinematic interpretation of the CMB dipole is rejected with p = 5 x 10-7 ⇒ 4.9� (Please check our results – all data on: https://doi.org/10.5281/zenodo.4431089) What impact does this have on cosmological analyses? Joint Lightcurve Analysis catalogue (740 SNe Ia) :A22,2014 568 , A&A et al et Betoule Spectral Adaptive Lightcurve Template (SALT2) used to make ‘stretch’ and ’colour’ corrections to the observed peak magnitude)
B-band
NB: The measured redshifts (in the heliocentric frame) have been ‘corrected’ to zCMB Cosmology Distance Luminosity modulus distance
So the µ-z data enables extraction of the parameter combination: ~ 0.8 WL – 0.6 Wm (NB: to determine H0 requires knowing the absolute magnitude M ➛ “distance ladder”)
Cosmography Acceleration is a kinematic quantity so data can also be analysed without assuming any dynamical model … by expanding the time variation of the scale factor in a Taylor series (e.g. Visser, CQG 21:2603,2004) ➙ good to <6% for JLA (extends to z ~ 1.2) q (¨aa)/a˙ 2 � ≡ (�⃛/a)(�̇/�) 0 ⌘ $ NB: Previous supernova analyses used the ‘constrained chi-squared’ method … 2 wherein sint is adjusted to get c of 1/d.o.f. for the fit to the assumed LCDM model!
We employ a Maximum Likelihood Estimator … and get rather different results Nielsen, Guffanti & S.S., Sci.Rep. 6:35596,2016 Well-approximated as Gaussian
‘Stretch’ corrections ‘Colour’ corrections
intrinsic distributions
cosmology SALT2 Jacques Colin et al.: Evidence for anisotropy of cosmic acceleration
Jacques Colin et al.: Evidence for anisotropy of cosmic acceleration
Fig. 1. The sky distribution of the 4 sub-samples of the JLA catalogue in Galactic coordinates: SDSS (red dots), SNLS (blue dots), low redshift (green dots) and HST (black dots). Note that the 4 big blue dots are clusters of many individual SNe Ia. The directions of the CMB dipole (star), the SMAC bulk flow (triangle), and the 2M++ bulk flow (inverted triangle) are also shown.
Fig. 1. The sky distribution of the 4 sub-samples of the JLA catalogue in Galactic coordinates: SDSS (red dots),Figure SNLS (blue 1 is a dots), Mollewide low redshift projection (green of dots) the directionsand HST (black of the dots). 740 SNeNote Ia that in the Galactic 4 big blue coordinates. dots are clusters of many individual SNe Ia. The directions of the CMB dipole (star), the SMAC bulk flow (triangle), andDue the to 2M the++ diversebulk flow survey (inverted strategies triangle) of are the also sub-samples shown. that make up the JLA catalogue, its sky coverage is patchy and anisotropic. While the low redshift objects are spread out unevenly across theFigure sky, the 1 is intermediate a Mollewide redshift projection ones of from the directions SDSS are of mainly the 740 confined SNe Ia in to Galactic a narrow coordinates. disk at low Due to the diverse survey strategies of the sub-samples that make up the JLA catalogue, its sky declination, while the high redshift ones from SNLS are clustered along 4 specific directions. coverage is patchy and anisotropic. While the low redshift objects are spread out unevenly across The JLA analysis (Betoule et al. 2014) corrects the observed redshifts zhel in the heliocentric the sky, the intermediate redshift ones from SDSS are mainly confined to a narrow disk at low frame in order to obtain the cosmological redshifts z after accounting for peculiar motions in declination, while the high redshift ones from SNLS areCMB clustered along 4 specific directions. the local Universe. These corrections are carried over unchanged from an earlier analysis (Conley If the CMB dipole is due to Theour JLAmotion analysis w.r.t. (Betoulethe CMB et al. 2014) frame corrects in which the observed the universe redshifts zhel in the heliocentric et al. 2011), which in turn cites an earlier method (Neill et al. 2007) and the peculiar velocity (supposedly) looks F-L-R-frameW, then in order the tomeasured obtain the cosmologicalredshift zhel redshiftsis relatedzCMB after to z accountingCMB ≡ z as: for peculiar motions in themodel local of Universe. Hudson Theseet al. corrections(Hudson et are al. 2004).carried It over is stated unchanged that the from inclusion an earlier of analysis these corrections (Conley etallow al. 2011), SNe Ia which with redshifts in turn cites down an to earlier 0.01 to method be included (Neill in the et al. cosmological 2007) and analysis, the peculiar in contrast velocity to
where z⊙ is the redshift inducedmodelearlier of analyses Hudsonby our (Riess etmotion al. et(Hudson al. w.r.t. 2006) et whichthe al. 2004). CMB employed Itand is stated onlyzSN is SNe that the Ia the downredshift inclusion to z = of0.023. these corrections due to the peculiarallow motionIn SNe Figure Ia of with 2supernova we redshifts scrutinise down host these to galaxy corrections0.01 to bein included the by exhibiting CMB in the frame the cosmological velocity parameter analysis, in, defined contrast as to C We find that the peculiarearlier velocity analyses ‘corrections’ (Riess et al. 2006) applied which to employed the JLA only catalogue SNe Ia down have to z = 0.023.
assumed that we converge to=In the[(1 Figure+ CMBzhel 2) we frame(1 scrutinise+ zCMB at )(1~ these150+ zd corrections)]Mpcc (contrary by exhibitingto observations) the velocity parameter , defined as(3) C ⇥ C
where= [(1 +z zheland) z(1 + arezCMB as)(1 tabulated+ zd)] byc JLA, while z is given by (Davis et al. 2011) (3) C hel CMB ⇥ d
where zhel1anduCMBzCMB are.nˆ/c as tabulated by JLA, while zd is given by (Davis et al. 2011) = , zd 1 :L13,2019 (4) 1 + uCMB .nˆ/c s
1 uCMB .nˆ/c 631 z = 1, (4) d 1 wheres1uCMB+ uCMBis 369.nˆ/c km s in the direction of the CMB dipole,(Kogut et al. 1993) andn ˆ is the , A&A unit vector in the direction1 of the supernova. It can be seen in Figure 2 that SNe Ia beyond z 0.06 where uCMB is 369 km s in the direction of the CMB dipole,(Kogut et al. 1993) andn ˆ ⇠is the have been assumed to be stationary w.r.t. the CMB rest frame,et al and corrections applied only to those unit vector in the direction of the supernova. It can be seen in Figure 2 that SNe Ia beyond z 0.06 ⇠ at lower redshifts. It is not clear how these corrections were made beyond z 0.04, which is the have been assumed to be stationary w.r.t. the CMB rest frame,Colin and corrections applied⇠ only to those atmaximum lower redshifts. extent to It whichis not clear the Streaming how these Motions corrections of Abell were Clusters made beyond (SMAC)z sample0.04, (Hudson which is et the al. ⇠ So we undid the correctionsmaximum to recover extent to whichthe original the Streaming data Motions in the ofheliocentric Abell Clustersframe (SMAC)Article sample number, (Hudson page 4 et of al. 12 … to check if the inferred acceleration of the expansion rate is indeed isotropic Article number, page 4 of 12 2 Cosmological analysis
We nowcompare the distance modulus (eq.1) obtained from the JLA sample with the apparent
magnitude (eq.2) using the Maximum Likelihood Estimator 25. For the luminosity distance we
use its kinematic Taylor series expansion up to the third term 40 since we wish to analyse the data
without making assumptions about the matter content or the dynamics:
2 cz 1 1 2 kc 2 dL(z)= 1+ [1 q0]z [1 q0 3q0 + j0 + 2 2 ]z (5) H0 ⇢ 2 6 H0 a0 where q aa/¨ a˙ 2 is the cosmic deceleration parameter in the Hubble flow frame, defined in terms ⌘
of the scale factor of the universe a and its derivatives w.r.t. proper time, j0 is the cosmic ‘jerk’
j = a/aH¨˙ 3, and kc2/(H2a2) is just ⌦ . Note that the last two appear together in the coefficient 0 0 k of the z3 term so cannot be determined separately. In the ⇤CDM model: q ⌦ /2 ⌦ . 0 ⌘ M ⇤
To look for a dipole in the deceleration parameter, we allow it to have a direction dependence: If we now do a cosmographic analysis allowing for a dipole in q0 , we find the MLE prefers one (x50 times the monopole!) … in the same direction as the CMB dipole
�� 1 2 �� � = 1 + 1 − �0 � + … , q0 (¨aa)/a˙ q⟾= qm + ~qd.nˆ (z,S) (6) �0 2 ⌘ F (Related work by: Cooke & Lynden-Bell 2010, Bernal et al. 2017, Javanmardi et al. 2018)
where qm and qd are the monopole and dipole components, while nˆ is the direction of the dipole
standard
:L13,2019 and (z,S) describes its scale dependence. We consider four representative functional forms: LCDM F acceleration
631 deceleration (a) No scale dependence: (z,S)=1independent of z, F (b) ‘Top hat’: (z,S)=1for z , Rameez & S.S., A&A (d) Exponential: (z,S)=exp( z/S). F 9 Mohayaee The significance of qo being negative has now decreased to only 1.4s Colin, This strongly suggests that cosmic acceleration is an artefact of our being located in a deep bulk flow (which includes most of the observed SNe Ia) … and not due to Λ T he 60he were were SNe High Ia mainly - studied by z SN SearchTeamSN in the direction the wherein Riess et al. al. et ⤺ • • dipole CMB (1998) and the 45 the and (1998) z z > 0.1> 0.1< the the apparent being due to flow due being bulk the acceleration is consistent with directions i.e. the inferred two littlethe between log The within 23 best The Supernova Cosmology Project Cosmology Supernova by by acceleration - Perlmutter likelihood changes very - fit fit direction of 0 of the CMB dipole. dipole. CMB the of p et al. al. et eaks q (1999) d . is is Colin, Mohayaee, Rameez & S.S., A&A 631:L13,2019 Do we infer acceleration although the expansion is actually decelerating … because we are embedded in a local ‘bulk flow’? (Tsagas 2010, 2011, 2012; Tsagas & Kadiltzoglou 2013, 2015, Tsagas et al. 2021) … if so, there should be a dipole asymmetry in the inferred deceleration parameter in the same direction – i.e. ~aligned with the CMB dipole ˜ a ˙ The patch A has mean peculiar velocity v drops below 1 and the comoving observer ‘measures’ negative deceleration parameter ~ Similar anisotropy (in H0) found in a sample of 500 X-ray clusters Redshift LX –T relation Luminosity distribution Migkas distribution : Kostas : Kostas Courtesey Migkas et al, A&A 636:A15,2020, amplitude 649:A151,2021 slope A ‘tilted’ Universe? ØThere is a dipole in the recession velocities of nearby galaxies and clusters ⇒ we are embedded in a ‘bulk flow’ which stretches out well beyond the scale at which the Universe supposedly becomes statistically homogeneous ØThe inference that the Hubble expansion rate is accelerating may be an artefact of this bulk flow – the acceleration is mainly a dipole aligned with the flow, and the monopole drops in significance to be consistent with zero [‘Cosmological fitting problem’ (Ellis & Stoeger 1987): use of heliocentric vs. CMB frame ⇒ different choices of corresponding 2-spheres in the ‘null fitting’ procedure] ØThe rest frame in which distant quasars are isotropic ≠ rest frame of the CMB The standard assumptions of isotropy and homogeneity are questionable (and so is the inference that the universe is dominated by dark energy) Need to reconstruct the standard cosmology … now starting with real data!