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High-redshift 21cm and redshift distortions. Antony Lewis Institute of Astronomy, Cambridge http://cosmologist.info/. Work with:. Anthony Challinor (IoA, DAMTP); astro-ph/0702600 Richard Shaw (IoA ), arXiv:0808.1724 .

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high redshift 21cm and redshift distortions
High-redshift 21cm and redshift distortions

Antony Lewis

Institute of Astronomy, Cambridge


Work with:

Anthony Challinor (IoA, DAMTP); astro-ph/0702600Richard Shaw (IoA), arXiv:0808.1724

Following work by Scott, Rees, Zaldarriaga, Loeb, Barkana, Bharadwaj, Naoz, Scoccimarro + many more …


Evolution of the universe




Dark ages



Hu & White, Sci. Am., 290 44 (2004)


Why the CMB temperature (and polarization) is great

  • Probes scalar, vector and tensor mode perturbations
  • The earliest possible observation (bar future neutrino anisotropy surveys etc…)- Includes super-horizon scales, probing the largest observable perturbations- Observable now

Why it is bad

- Only one sky, so cosmic variance limited on large scales

- Diffusion damping and line-of-sight averaging: all information on small scales destroyed! (l>~2500)- Only a 2D surface (+reionization), no 3D information

if only we could observe the cdm perturbations
If only we could observe the CDM perturbations…

- not erased by diffusion damping (if cold): power on all scales

- full 3D distribution of perturbations

What about the baryons?

  • fall into CDM potential wells; also power on small scales
  • full 3D distribution
  • but baryon pressure non-zero: very small scales still erased

How does the information content compare with the CMB?

CMB temperature, 1<l<~2000: - about 106 modes - can measure Pk to about 0.1% at l=2000 (k Mpc~ 0.1)

Dark age baryons at one redshift, 1< l < 106: - about 1012 modes

- measure Pk to about 0.0001% at l=106 (k Mpc ~ 100)


What about different redshifts?

  • About 104 independent redshift shells at l=106
  • - total of 1016 modes - measure Pk to an error of 10-8 at 0.05 Mpc scales

e.g. running of spectral index:

If ns = 0.96 maybe expect running ~ (1-ns)2 ~ 10-3Expected change in Pk ~ 10-3 per log k

- measure running to 5 significant figures!?

So worth thinking about… can we observe the baryons somehow?


- after recombination, Hydrogen atoms in ground state and CMB photons have hν << Lyman-alpha frequency* high-frequency tail of CMB spectrum exponentially suppressed * essentially no Lyman-alpha interactions * atoms in ground state: no higher level transitions either

  • How can light interact with the baryons (mostly neutral H + He)?

- Need transition at much lower energy* Essentially only candidate for hydrogen is the hyperfine spin-flip transition



Credit: Sigurdson

Define spin temperature Ts


What can we observe?

Spontaneous emission: n1 A10 photons per unit volume per unit proper time


h v = E21

Rate: A10 = 2.869x10-15 /s


Stimulated emission: net photons (n1 B10 – n0 B01)Iv

Total net number of photons:

In terms of spin temperature:

Net emission or absorption if levels not in equilibrium with photon distribution

- observe baryons in 21cm emission or absorption if Ts <> TCMB


What determines the spin temperature?

  • Interaction with CMB photons: drives Ts towards TCMB
  • Collisions between atoms: drives Ts towards gas temperature Tg

TCMB = 2.726K/a

At recombination, Compton scattering makes Tg=TCMBLater, once few free electrons, gas cools: Tg ~ mv2/kB ~ 1/a2

Spin temperature driven below TCMB by collisions:

- atoms have net absorption of 21cm CMB photons

  • (+Interaction with Lyman-alpha photons - not important during dark ages)

What’s the power spectrum?

Use Boltzmann equation for change in CMB due to 21cm absorption:



l >1 anisotropies in TCMB

Fluctuation in density of H atoms,+ fluctuations in spin temperature

Doppler shiftto gas rest frame

CMB dipole seen by H atoms:more absorption in direction of gas motion relative to CMB

+ reionization re-scattering terms


Solve Boltzmann equation in Newtonian gauge:

Redshift distortions

Main monopolesource

Effect of localCMB anisotropy

Sachs-Wolfe, Doppler and ISW change to redshift

Tiny Reionization sources

For k >> aH good approximation is

(+re-scattering effects)


21cm does indeed track baryons when Ts < TCMB


Kleban et al. hep-th/0703215

So can indirectly observe baryon power spectrum at 30< z < 100-1000 via 21cm


Observable angular power spectrum:

Integrate over window in frequency

Small scales:

1/√N suppressionwithin window

‘White noise’from smaller scales


baryon oscillations



What about large scales (Ha >~ k)?

Narrow redshift window

<~ 1% effect at l<50

Extra terms largely negligible


New large scaleinformation?- potentials etccorrelated with CMB

Dark ages~2500Mpc

l ~ 10

14 000 Mpc


Opaque ages ~ 300Mpc

Comoving distance


non linear evolution
Non-linear evolution

Small scales see build up of power from many larger scale modes - important

But probably accurately modelled by 3rd order perturbation theory:

On small scales non-linear effects many percent even at z ~ 50

+ redshift distortions, see later.


Also lensing

Modified Bessel function



Wigner functions

Lensing potential power spectrum

Lewis, Challinor: astro-ph/0601594

c.f. Madel & Zaldarriaga: astro-ph/0512218


like convolution with deflection angle power spectrum;generally small effect as 21cm spectrum quite smooth



Lots of information in 3-D (Zahn & Zaldarriaga 2006)

observational prospects
Observational prospects

No time soon…

- (1+z)21cm wavelengths: ~ 10 meters for z=50- atmosphere opaque for z>~ 70: go to the moon?- fluctuations in ionosphere: phase errors: go to moon?- interferences with terrestrial radio: far side of the moon?- foregrounds: large! use signal decorrelation with frequency

But: large wavelength -> crude reflectors OK

See e.g. Carilli et al: astro-ph/0702070, Peterson et al: astro-ph/0606104

Current 21cm:

LOFAR, PAST, MWA: study reionization at z <20SKA: still being planned, z< 25

things you could do with precision dark age 21cm
Things you could do with precision dark age 21cm
  • High-precision on small-scale primordial power spectrum(ns, running, features [wide range of k], etc.)e.g. Loeb & Zaldarriaga: astro-ph/0312134,Kleban et al. hep-th/0703215
  • Varying alpha: A10 ~ α13(21cm frequency changed: different background and perturbations)Khatri & Wandelt: astro-ph/0701752
  • Isocurvature modes(direct signature of baryons; distinguish CDM/baryon isocurvature)Barkana & Loeb: astro-ph/0502083
  • CDM particle decays and annihilations(changes temperature evolution)Shchekinov & Vasiliev: astro-ph/0604231, Valdes et al: astro-ph/0701301
  • Primordial non-Gaussianity(measure bispectrum etc of 21cm: limited by non-linear evolution)Cooray: astro-ph/0610257, Pillepich et al: astro-ph/0611126
  • Lots of other things: e.g. cosmic strings, warm dark matter, neutrino masses, early dark energy/modified gravity….
back to reality after the dark ages
Back to reality – after the dark ages?
  • First stars and other objects form
  • Lyman-alpha and ionizing radiation present:Wouthuysen-Field (WF) effect: - Lyman-alpha couples Ts to Tg- Photoheating makes gas hot at late times so signal in emissionIonizing radiation: - ionized regions have little hydrogen – regions with no 21cm signalBoth highly non-linear: very complicated physics
  • Lower redshift, so less long wavelengths:- much easier to observe! GMRT (z<10), MWA, LOFAR (z<20), SKA (z<25)….
  • Discrete sources: lensing, galaxy counts (~109 in SKA), etc.

How do we get cosmology from this mess?

Would like to measure dark-matter power on nearly-linear scales.

Want to observe potentials not baryons

1. Do gravitational lensing: measure source shears - probes line-of-sight transverse potential gradients (independently of what the sources are)

2. Measure the velocities induced by falling into potentials - probe line-of-sight velocity, depends on line-of-sight potential gradients

Redshift distortions


A closer look at redshift distortions

Real space

Redshift space






Density perturbed too. In redshift-space see


Both linear (+higher) effects – same order of magnitude. Note correlated.

More power in line-of-sight direction -> distinguish velocity effect




Redshift-space distance:

Actual distance:

Define 3D redshift-space co-ordinate

Transform using Jacobian: Redshift-space perturbation

Fourier transformed:


Linear power spectrum:

Messy astrophysics

Depends only on velocities -> potentials

(n.k)4 component can be used for cosmology

Barkana & Loeb astro-ph/0409572

is linear theory good enough
Is linear-theory good enough?

Radial displacement ~ 0.1-1 MPc

RMS velocity 10-4-10-3



Redshift space

Real Space

Looks like big non-linear effect!

M(x + dx) ≠ M(x) + M’(x) dx

BUT: bulk displacements unobservable. Need more detailed calculation.

non linear redshift distortions
Non-linear redshift distortions

Shaw & Lewis, 2008 also Scoccimarro 2004

Assume all fields Gaussian.

Power spectrum from:

Exact non-linear result (for Gaussian fields on flat sky):


Significant effectDepending on angle

Small scale power boosted

by superposition of lots oflarge-scale modes



More important at lower redshift.Not negligible even at high z.

Comparable to non-linear evolution



Similar effect on angular power spectrum:

(A ~ velocity covariance), u= (x-z,y-z’)

(sharp z=z’ window, z=10)

what does this mean for component separation
What does this mean for component separation?

Angular dependence now more complicated – all μ powers, and not clean.


gives wrong answer…


Need more sophisticated separation methods to measure small scales.

also complicates non gaussianity detection redshift distortion bispectrum
Also complicates non-Gaussianity detectionRedshift-distortion bispectrum
  • Mapping redshift space -> real space nonlinear, so non-Gaussian

Linear-theory source

Just lots of Gaussian integrals (approximating sources as Gaussian)..

Zero if all k orthogonal to line of sight.

Can do exactly, or leading terms are:

Also Scoccimarro et al 1998, Hivon et al 1995

Not attempted numerics as yet

  • Huge amount of information in dark age perturbation spectrum- could constrain early universe parameters to many significant figures
  • Dark age baryon perturbations in principle observable at 30<z< 500 up to l<107 via observations of CMB at (1+z)21cm wavelengths.
  • Dark ages very challenging to observe (e.g. far side of the moon)
  • Easier to observe at lower z, but complicated astrophysics
  • Redshift-distortions probe matter density – ideally measure cosmology separately from astrophysics by using angular dependence
  • BUT: non-linear effects important on small scales - more sophisticated non-linear separation methods may be required