FOUNDATIONAL ASSUMPTIONS OF CDM Roy Maartens Western Cape & - - PowerPoint PPT Presentation

FOUNDATIONAL ASSUMPTIONS OF ∧CDM

Roy Maartens Western Cape & Portsmouth

FOUNDATIONAL ASSUMPTION 1

COSMOLOGICAL PRINCIPLE The Universe is isotropic + homogeneous

FOUNDATIONAL ASSUMPTION 2

GENERAL RELATIVITY GR describes gravity at all (classical) scales and times

The Cosmological Principle is critical

§ to ∧CDM and dynamical DE models in GR § to all Modified Gravity models – and to testing GR with

cosmology

§ to deduce DE as a cause for acceleration

The Cosmological Principle is critical

§ to ∧CDM and dynamical DE models in GR § to all Modified Gravity models – and to testing GR with

cosmology

§ to deduce DE as a cause for acceleration

But what does the Cosmological Principle really mean? Universe is not exactly isotropic + homogeneous – it is at best I+H on average, on large enough scales:

§ How to average and coarse grain in nonlinear GR? § Exact I+H perturbative I+H ?

statistical I+H perturbative I+H ? (Rasanen 2009-) These are important unresolved issues for cosmology.

⇒ ⇒

Observational motivation?

The Copernican Principle

§ We cannot directly observe homogeneity – only isotropy. § We need the Copernican Principle to deduce homogeneity

from isotropy:

we are typical observers – not at a special position in the universe

What do isotropic CMB observations +

Copernican Principle tell us?

It seems obvious that we should get FLRW. But we have to show it using the general, fully nonlinear Einstein-Liouville equations. Nonlinear perturbations are not an option.

General intensity multipoles: Lowest ones Then Liouville becomes:

(Ellis, Treciokas, Matravers 1984; RM, Gebbie, Ellis 1999)

History: 1968 mathematical theorem by Ehlers, Geren, Sachs. Update: Generalized to include baryons, CDM and DE.

(EGS 1968; Ellis, Treciokas 1971; Stoeger, RM, Ellis 1995; Ferrando, Morales, Portilla 1999;

Clarkson, Barrett 1999; Clarkson, Coley 2001; Rasanen 2009; Clarkson, RM 2010)

__________________________________________ CMB isotropy + Copernican Principle + GR gives FLRW In a region, if * collisionless radiation is exactly isotropic, * the radiation 4-velocity is geodesic and expanding, * baryons and CDM are pressure-free, and DE has no anisotropic stress, then the region is FLRW __________________________________________

History: 1968 mathematical theorem by Ehlers, Geren, Sachs. Update: Generalized to include baryons, CDM and DE.

(EGS 1968; Ellis, Treciokas 1971; Stoeger, RM, Ellis 1995; Ferrando, Morales, Portilla 1999;

Clarkson, Barrett 1999; Clarkson, Coley 2001; Rasanen 2009; Clarkson, RM 2010)

__________________________________________ CMB isotropy + Copernican Principle + GR gives FLRW In a region, if * collisionless radiation is exactly isotropic, * the radiation 4-velocity is geodesic and expanding, * baryons and CDM are pressure-free, and DE has no anisotropic stress, then the region is FLRW __________________________________________

Note: It follows that matter and DE have the same 4-velocity as radiation

More powerful result: __________________________________________ We only need the first 3 multipoles to vanish! In a region, if * collisionless radiation has zero dipole, quadrupole and octupole, * the radiation 4-velocity is geodesic and expanding, * baryons and CDM are pressure-free, and DE has no anisotropic stress, then the region is FL _________________________________________

(Ellis, Treciokas, Matravers 1985, generalized in Clarkson, RM 2010)

= = =

abc ab a

F F F

More powerful result: __________________________________________ We only need the first 3 multipoles to vanish! In a region, if * collisionless radiation has zero dipole, quadrupole and octupole, * the radiation 4-velocity is geodesic and expanding, * baryons and CDM are pressure-free, and DE has no anisotropic stress, then the region is FL _________________________________________

(Ellis, Treciokas, Matravers 1985, generalized in Clarkson, RM 2010)

This is the best motivation we have for (exact) FLRW. = = =

abc ab a

F F F

More powerful result: __________________________________________ We only need the first 3 multipoles to vanish! In a region, if * collisionless radiation has zero dipole, quadrupole and octupole, * the radiation 4-velocity is geodesic and expanding, * baryons and CDM are pressure-free, and DE has no anisotropic stress, then the region is FL _________________________________________

(Ellis, Treciokas, Matravers 1985, generalized in Clarkson, RM 2010)

This is the best motivation we have for (exact) FLRW. Major open question: do these results lead to near-FLRW from near-isotropy? (Rasanen 2009+) = = =

abc ab a

F F F

Testing homogeneity and FLRW

We should test the foundations of FLRW:

§ If we find no violation – this strengthens the evidence

for homogeneity.

§ If we find one violation, this can disprove homogeneity.

Geometric consistency tests. 1

Luminosity distance in FLRW For any energy-momentum tensor, any gravity theory – this is a consistency condition for distance/ curvature in FLRW

DL = 1+ z H0 −ΩK 0 sin −ΩK 0 dz' H(z') / H0

z

∫

$ % & ' ( ) D ≡ H0DL(1+ z)−1 Then ΩK 0 = H(z)D'(z)

[ ]

2 −1

H0

2D(z)2

for all z

Differentiate with respect to z: ____________________________________________ Null test of homogeneity: K(z0) significantly differs from zero non-FLRW

____________________________________________

(Clarkson, Bassett, Lu 2009)

If K is consistent with 0 – then this strengthens support for homogeneity.

• Test can be applied using the same data that is needed

for determining the DE equation of state w(z).

• Future BAO data to get H(z) [radial BAO] and DA(z)

[transverse] DL(z)

⇒

K(z) ≡1+ H(z)2 D(z)D''(z)− D'(z)2 # $ % &+ H(z)H '(z)D(z)D'(z) = 0 for FLRW spacetime

⇒

Geometric consistency tests. 2

Distance sum rule Strong lensing + SNIa distances . zs (k=K/H0

2 )

. zl . z=0 Tests FLRW independent of energy-momentum tensor or gravity theory. (Rasanen, Bolejko, Finoguenov 2014)

⇒

Geometric consistency tests. 3

Cosmic parallax Using our motion relative to the CMB as a baseline + angular diameter distance Tests FLRW independent of energy-momentum tensor or gravity theory. (Quercillini, Quartin, Amendola 2008, Rasanen 2013)

⇒

Probe inside the lightcone – galaxies

Galaxies carry a fossil record of their star formation history in their spectra. This has been used to find H(z) in FLRW (‘cosmic clocks’). We can also use it to test FLRW. For each galaxy spectrum at (z,θ,Φ), get the age or star formation rate – then compare galaxies at the same lookback time. ____________________________________________ Null test of (statistical) homogeneity: Fossil record differs significantly non-FLRW universe ____________________________________________

(Heavens, Jimenez, RM 2011)

⇒

Use radial BAO to find lookback time

dt = −uµkµdv = −(1+ z)dv dz dv = (1+ z)2H|| (z) tLB = tO −tE = dz (1+ z)H|| (z)

zE

∫

r

BAO,||(z) =

Δz H|| (z)

ttLB ttform

Using stacked LRG data from SDSS: No evidence of inhomogeneity

(Hoyle et al 2012)

Probing inside the past lightcone – CMB Clusters act like giant mirrors that give us a glimpse

• f the last scattering surface inside our past lightcone.

This probes remote multipoles of the CMB.

Sunyaev-Zeldovich effect on CMB temperature Scattered CMB photons at distant clusters: * thermal SZ distorts the blackbody spectrum – a large effect if there is large anisotropy at the cluster

(Goodman 1995; Caldwell, Stebbins 2008)

* kinetic SZ probes bulk radial velocity

(Garcia-Bellido, Haugbolle 2008; Zhang, Stebbins 2011)

Non-perturbative SZ effect implies violation of FLRW.

r

v

Cluster polarization of CMB photons Scattered CMB photons at distant clusters are polarized: * by the CMB quadrupole – a large effect if there is a large quadrupole at the cluster * by the transverse velocity

• f the cluster

(Sunyaev, Zeldovich 1980; Kamionkowski, Loeb 1997)

Non-perturbative polarization implies violation of FLRW.

In principle we can evade the Copernican Principle!

* kSZ lets us see CMB as others see it

* if we see isotropy then the spacetime is FLRW Problem – need to detect double scatterings.

(Clifton, Clarkson, Bull 2011)

Consistency between galaxies & CMB In a FLRW universe, the dipole of the matter distribution should agree with the dipole of the CMB. NVSS all-sky radio survey shows consistency in direction (within very large error bars) but not amplitude. SKA angular correlation function (100’s millions galaxies) will be able to detect dipole within ~5o (Phase 1) and ~1o (Phase 2). (Schwarz et al 2014)

Consistency between galaxies & CMB In a FLRW universe, the dipole of the matter distribution should agree with the dipole of the CMB. NVSS all-sky radio survey shows consistency in direction (within very large error bars) but not amplitude. SKA angular correlation function (100’s millions galaxies) will be able to detect dipole within ~5o (Phase 1) and ~1o (Phase 2). Also – try to test whether the CMB large-scale anomalies are also in the matter distribution. (Schwarz et al 2014)

FOUNDATIONAL ASSUMPTION 2

GENERAL RELATIVITY GR describes gravity at all (classical) scales and times § Newtonian approximations to GR are essential to make computations manageable (and to build intuition). § But they can also sometimes lead to confusion, to errors and to missed opportunities – especially in the coming era

• f huge-volume galaxy surveys.

Redshift range and volume of different surveys (Bull 2014)

GR effects on the power spectrum

§ On scales near and above the Hubble radius,

different gauges for δ=δρ/ρ give different answers: Pδ(k) § Gauge-invariance does not solve the problem – any δGI defined on constant time-slices is not unique.

(Bonvin 2012)

§ We observe on the past lightcone, not t =const.

§ We must use the observed δobs * it is unique * it is automatically gauge-invariant § GR effects: redshift, lensing and volume distortions

(δθ, δϕ) = (θS −θO, ϕS −ϕO) δz 1+ z = n⋅v −Φ− dη(Φ'+ Ψ')

ηo

∫

(Yoo et al 2009, Yoo 2010, Bonvin, Durrer 2011,

Challinor, Lewis 2011, Bruni et al 2011, Jeong et al 2011, Bertacca et al 2013)

(Bonvin 2014)

(No integrated terms, flat-sky)

(Jeong et al 2011)

New information in the observed overdensity. Relativistic terms grow on very large scales – cosmic variance

(Yoo, Hamaus, Seljak, Zaldarriaga 2012)

Detectability of GR effects using multi-tracer method

(No integrated terms, flat-sky)

H k

H2 k2

Primordial non-Gaussianity in the galaxy distribution

• Primordial quantum fluctuations generated during

Inflation – may be non-Gaussian.

• Primordial non-Gaussianity is ‘frozen’ on large scales

during the expansion of the Universe.

• The effect of PNG of local type is to modify the bias of

galaxies relative to the underlying total matter distribution: Local PNG thus boosts the clustering of galaxies on large scales.

Δg = bΔm where b → b+ Δb, Δb∝ fNLk−2

HI power spectrum: Gaussian CDM (solid), non-Gaussian galaxy distribution (dashed) Green: z=0.4, Red: z=2.5

(Camera, Santos et al 2013)

dashed fNL =10

The GR corrections on very large scales are similar to the effect of PNG (Bruni et al 2011)

(Jeong et al 2011)

Therefore – in order to constrain PNG from the galaxy power spectrum, we should use the correct GR form for the power spectrum. There is another GR correction – nonlinear & primordial:

• The relativistic Poisson equation corrects the Newtonian

Poisson constraint at second order:

• This leads to an intrinsic PNG with
• The best-fit value from galaxy bias needs a GR correction:

(Camera, Santos, RM 2014)

δd = 2 3H2

d

r2φd h 1 2 ⇣ fNL 5 3 ⌘ φd i

f GR

NL = −5

3

(Bartolo,Matarrese, Riotto 2005, Verde, Matarrese 2009, Bruni et al 2013, Villa et al 2014)

• Simple slowroll inflation will therefore lead to
• Planck achieved

To test standard inflation via galaxies, we need

• In this regime, ignoring GR effects can lead to a

significant bias in the best-fit value:

fNL = −2.2

σ(fNL) = 7.5

σ(fNL) . 2

(Camera, RM, Santos 2014)

(Camera, RM, Santos 2014)

GR effects in lensing magnification (Bonvin 2008)

(Bolejko, Clarkson, RM et al 2012)

(Bacon et al 2014)