Logbook160121NsConstraints < Sandbox

---+ n_s constraints from 30m telescope

---++ Background
Questions to be answered: <br>
   * If we have a 30m telescope geared with a polarization sensitive receiver at Advanced ACTPol sensitivity (7-8 uK-arcmin at 90/150GHz, from arXiv: 1510.02809), we can measure really high ell E-modes. Does that improve the constraints on n_s?

Big picture: <br>
   * For the Starobinsky model (R^2 inflation) on fig. 12 of the Planck 2015 Constraints on Inflation paper (arXiv: 1502.0211), aside from further constraining r, tightening the n_s constraint will be more effective in ruling it out in the near future before constraints on r gets below 1e-3.
   * So the question is, how quickly would measuring more high ell E-modes tighten the constraint on n_s?

---++ Current constraints on n_s
From Planck's 2015 XIII. Cosmological parameters (arXiv: 1502.01589), XI. CMB power spectra, likelihoods, and robustness of parameters (arXiv: 1507.02704), and Planck's 2013 XVI Cosmological parameters paper ((arXiv: 1303.5076), the basic 6-parameter LCDM constraints are:
| *parameter* |  *Planck TT + lowP*  | *Planck TT+TE+EE+lowP* | *Planck TT only (2013) * |
| ombh2 | 0.02222 ± 0.00023 | 0.02225 ± 0.00016 | 0.02207 ± 0.00033 | 
| omch2 | 0.1197 ± 0.0022 | 0.1198 ± 0.0015 | 0.1196 ± 0.0031 | 
| 100theta | 1.04085 ± 0.00047 | 1.04077 ± 0.00032 | 1.04132 ± 0.00068 | 
| tau | 0.078 ± 0.019 | 0.079 ± 0.017 | 0.097 ± 0.038 |
| ln(1e10 A_s) | 3.089 ± 0.036 | 3.094 ± 0.034 | 3.103 ± 0.072 | 
| n_s | 0.9655 ± 0.0062 | 0.9645 ± 0.0049 | 0.9616 ± 0.0094 | 
|||| 
| H_0 | 67.31 ± 0.96 | 67.27 ± 0.66 | 67.4 ± 1.4 | 

I added the row for H_0  because I use H_0 instead of 100*theta in my fisher code.

---++ Cosmology and Inputs for the forecast
*fiducial cosmology*: <br>
The fiducial cosmology I use in this forecast based on the LCDM best fit from Planck's 2013 XVI cosmological parameter (arXiv: 1303.5076) as listed in the last column in the previous section. In addition, I assume a minimal neutrino mass of ~60meV (and r = 0)

*Inputs for the 30m telescope:* <br>
   * Beam = 0.33' ( 1 deg is about ell~200, lmax*0.33 = 200*60, lmax~36000)
   * Polarization noise (T noise is 1/sqrt(2) of this): 1, 3, 7.5, 10 uK-arcmin
   * lmaxEE = 4000, 10000, 20000; stepping through to see how much of a difference lmax makes
   * fsky = 0.0125, 0.125, 0.5; 0.0125 is 500 sq deg.
   * lmaxTT = 3000; any higher is contaminated by foregrounds
   * lmin various with fsky
   * use only the TT, TE, and EE lensed spectra -- not using phi (C^dd) 

*Planck prior inputs:* <br>
   * For the fsky = 500 sq deg, sample variance degrades the constraints by a significant amount, it is essential to have some large area survey prior (e.g. Planck)
   * We take the lower foreground freqs (100, 143, 217 GHz)
   * noise (T)  = [6.8,6.0,13.1] uK-arcmin
   * noise (P) = [10.9,11.4,26.7] uK-arcmin
   * beam = [ 9.5,7.1,5.0] arcmin
   * lmax = 2500
   * lmin = 30 
   * fsky = I tweak the fsky input so that the TT only constraints match the 2013 parameter constraints and the TT/TE/EE constraints match the 2015 parameter constraints to within about 10%; and fsky = 0.12

---++ sigma(n_s) vs fsky using lensed TT/TE/EE (no delensing)

     <img src="%ATTACHURLPATH%/ns_vs_fsky.png" alt="ns_vs_fsky.png" width=60% />



   * the solid lines are without Planck priors; the points at fsky=0.0125 has Planck prior
   * practically no difference going from ell=4000 to ell=20000. <strike>Perhaps expected given how smooth the peaks are beyond the 9th peak at ell~3000 (see e.g. fig 5 of SPTpol 100 sq deg EE spectrum paper arXiv: 1411.1042) </strike>
      * The peaks fading out should not matter. It is the overall tilt of the spectrum
   * Also, sensitivity isn't a big driver. A 10uK-arcmin experiment is almost as good as a 1uK-arcmin experiment. Also expected because E-modes are much brighter.
   * The improvement of adding high ell EE over the Planck prior for the fsky=0.0125 case is about 7%
   | parameter | Planck prior alone (TT+TE+EE) | Planck + 10uK-am, lmax=4000 | Planck + 10uK-am, lmax=10000 |
   | sigma(n_s) | 0.0051 | 0.00474 | 0.00474 | 


<strike> Next: one would expect that the peaks to be sharper in its unlensed (de-lensed) state. So perhaps one can extract more information/tighter constraints on n_s after delensing the E-modes. The ideal case would be fully delensed T and E-modes, i.e. with unlensed TT, TE, and EE. This will be the best case scenario on n_s constraint given CMB TT/TE/EE spectra. </strike>

*Next: Run* 
   1 a noiseless case 
   1 an extremely low noise case 1e-5 uK-arcmin, so that beam contribution will be present in N_ell = noise^2 exp (ell(ell+1)*beam^2/8log2)

---++ Compare noiseless/1e-5uK-arcmin/1uK-arcmin cases

     <img src="%ATTACHURLPATH%/ns_vs_fsky_noiseless_vs_1uk.png" alt="ns_vs_fsky_noiseless_vs_1uk.png" width=60% />

   * The 1e-5uK-arcmin lines completely overlaps with the noiseless case (see next section for explanation)
   * The 1uK-arcmin case with lmax = 4000, 10000, 24700 all overlap with the lmax=4000 case with noiseless/1e-5uK-arcmin case


----++ Beam cross-over
At 1 uK-arcmin, the cross-over of the noise due to beam with the EE spectrum is around ell~3000-4000. To make even higher ell measurements of the EE spectrum sensitive to the tilt of the spectrum. One needs to push the noise level lower.

     <img src="%ATTACHURLPATH%/beam_spectrum_lownoise_withEEsansprefactor.png" alt="beam_spectrum_lownoise_withEEsansprefactor.png" width=60% />


---++ References
   * Starobinsky model:
      * [[https://ncatlab.org/nlab/show/Starobinsky+model+of+cosmic+inflation][nLab: Starobinsky model of cosmic inflation]]
      * [[http://indico.ipmu.jp/indico/getFile.py/access?contribId=16&sessionId=7&resId=0&materialId=slides&confId=28][Sergei Ketov, PLANCK mission, Starobinsky inflation and its realization in old-minimal supergravity]]
   * [[http://arxiv.org/pdf/1402.0526v2.pdf][Linde, Inflationary Cosmology after Planck 2013?]]

-- %USERSIG{KimmyWu - 2016-01-21}%
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Topic attachments
I	Attachment	History	Action	Size	Date	Who
png	beam_spectrum_lownoise_withEEsansprefactor.png	r1	manage	51.3 K	2016-01-22 - 14:02	KimmyWu
png	ns_vs_fsky.png	r1	manage	59.5 K	2016-01-21 - 11:57	KimmyWu
png	ns_vs_fsky_noiseless_vs_1uk.png	r1	manage	100.6 K	2016-01-22 - 14:50	KimmyWu
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Topic revision: r2 - 2016-01-22 - KimmyWu