1. High-redshift 21cm and redshift distortions
Antony Lewis
Institute of Astronomy, Cambridge
Work with:
Anthony Challinor (IoA, DAMTP); astro-ph/ Richard Shaw (IoA), arXiv:
Following work by Scott, Rees, Zaldarriaga, Loeb, Barkana, Bharadwaj, Naoz, Scoccimarro + many more …

2. Evolution of the universe
Opaque
Easy
Transparent
Dark ages
30<z<1000
Hard
Hu & White, Sci. Am., (2004)

3. CMB temperature

4. 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

5. 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 % at l=106 (k Mpc ~ 100)

6. 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 ~ Expected change in Pk ~ 10-3 per log k
- measure running to 5 significant figures!?
So worth thinking about… can we observe the baryons somehow?

7. How can light interact with the baryons (mostly neutral H + He)?
- 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
- Need transition at much lower energy * Essentially only candidate for hydrogen is the hyperfine spin-flip transition
triplet
singlet
Credit: Sigurdson
Define spin temperature Ts

8. What can we observe?
Spontaneous emission: n1 A10 photons per unit volume per unit proper time 1
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

9. 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=TCMB Later, 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)

11. What’s the power spectrum?
Use Boltzmann equation for change in CMB due to 21cm absorption:
Background:
Perturbation:
l >1 anisotropies in TCMB
Fluctuation in density of H atoms, + fluctuations in spin temperature
Doppler shift to gas rest frame
CMB dipole seen by H atoms: more absorption in direction of gas motion relative to CMB
+ reionization re-scattering terms

12. Solve Boltzmann equation in Newtonian gauge:
Redshift distortions
Main monopole source
Effect of local CMB anisotropy
Sachs-Wolfe, Doppler and ISW change to redshift
Tiny Reionization sources
For k >> aH good approximation is
(+re-scattering effects)

13. 21cm does indeed track baryons when Ts < TCMB
z=50
Kleban et al. hep-th/
So can indirectly observe baryon power spectrum at 30< z < via 21cm

14. Observable angular power spectrum:
Integrate over window in frequency
Small scales:
1/√N suppression within window
‘White noise’ from smaller scales
Baryon pressure support
baryon oscillations
z=50

15. What about large scales (Ha >~ k)?
Narrow redshift window
<~ 1% effect at l<50
Extra terms largely negligible

16. New large scale information? - potentials etc correlated with CMB
Dark ages ~2500Mpc
l ~ 10
Mpc
z=30
Opaque ages ~ 300Mpc
Comoving distance
z~1000

17. 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.

18. Also lensing Modified Bessel function Unlensed Lensed Wigner functions
Lensing potential power spectrum
Lewis, Challinor: astro-ph/
c.f. Madel & Zaldarriaga: astro-ph/

19. like convolution with deflection angle power spectrum; generally small effect as 21cm spectrum quite smooth
Cl(z=50,z=50)
Cl(z=50,z=52)
Lots of information in 3-D (Zahn & Zaldarriaga 2006)

20. 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/ , Peterson et al: astro-ph/
Current 21cm:
LOFAR, PAST, MWA: study reionization at z <20 SKA: still being planned, z< 25

22. 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/ , Kleban et al. hep-th/
Varying alpha: A10 ~ α13 (21cm frequency changed: different background and perturbations) Khatri & Wandelt: astro-ph/
Isocurvature modes (direct signature of baryons; distinguish CDM/baryon isocurvature) Barkana & Loeb: astro-ph/
CDM particle decays and annihilations (changes temperature evolution) Shchekinov & Vasiliev: astro-ph/ , Valdes et al: astro-ph/
Primordial non-Gaussianity (measure bispectrum etc of 21cm: limited by non-linear evolution) Cooray: astro-ph/ , Pillepich et al: astro-ph/
Lots of other things: e.g. cosmic strings, warm dark matter, neutrino masses, early dark energy/modified gravity….

23. 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 emission Ionizing radiation: - ionized regions have little hydrogen – regions with no 21cm signal Both 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.

24. 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

25. A closer look at redshift distortions
Real space
Redshift space y
y(z) x x

26. + 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

27. Linear-theory n Redshift-space distance: Actual distance:
Define 3D redshift-space co-ordinate
Transform using Jacobian: Redshift-space perturbation
Fourier transformed:

28. Linear power spectrum:
Messy astrophysics
Depends only on velocities > potentials
(n.k)4 component can be used for cosmology
Barkana & Loeb astro-ph/

29. Is linear-theory good enough?
RMS velocity
Radial displacement ~ MPc
Megaparsec
scale
Real Space
Redshift space
Looks like big non-linear effect!
M(x + dx) ≠ M(x) + M’(x) dx
BUT: bulk displacements unobservable. Need more detailed calculation.

30. Non-linear redshift distortions
Assume all fields Gaussian.
Shaw & Lewis, 2008 also Scoccimarro 2004
Power spectrum from:
Exact non-linear result (for Gaussian fields on flat sky):

31. Significant effect Depending on angle
Small scale power boosted
by superposition of lots of large-scale modes
z=10

32. More important at lower redshift. Not negligible even at high z.
Comparable to non-linear evolution
z=10

33. Similar effect on angular power spectrum:
(A ~ velocity covariance), u= (x-z,y-z’)
(sharp z=z’ window, z=10)

34. What does this mean for component separation?
Angular dependence now more complicated – all μ powers, and not clean.
Assuming
gives wrong answer…
z=10
Need more sophisticated separation methods to measure small scales.

35. Also complicates non-Gaussianity detection Redshift-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

36. Conclusions
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

37. Correlation function
z=10