The Effect of a Scanning Flat Fold Mirror on a CMB B-Mode Experiment

arXiv:1105.3409v1 [astro-ph.IM] 17 May 2011 nvre oee,iflto xoeta expansion exponential - the first inflation of the within observations However, many explaining universe. in successful very nsae rae hn1 po- (CMB) than of background greater microwave pattern scales 23] cosmic on 22, “B-mode” the 20, or on 9, curl-like, larization [1, a waves produce gravity models of which Inflationary background a confirmation. predict strong lacks it indeed a scaatrzdb h esrt-clrratio, sig- tensor-to-scalar this the by of characterized amplitude is The nal 25]. [11, perturbations density orlto oavalue a to cross- tempera- correlation temperature-polarization the and of spectrum power measurements ture current by constrained is rmrilBmd inl rcntani for it the constrain detect or to need signal, we B-mode [2] primordial models inflationary on straints a fi speeta reasonable a at being, present sig- to B-mode is close the it detect are if to nal sensitivity or raw been, the with have of fielded (Planck experiments number balloon- satellite-based and 17]) A [15]), [4]) [14, Spider PolarBear [7], [10]. (EBEX [21], based required (QUIET is ground-based systematics of control nK. 10 fluctuations of polarization level the of at measurement requires This fplrzto rmrflcin.I eto I edescribe we III Section production In the reflections. review from and polarization system, of the describe we II tion sim- components much mount moving steerable of the pler. of mass design the the make in would reduction requirements, cryogenic large mirror. the the fold of a simplification with the receiver Besides however, stationary imagined, a not be combines can has which system accelerations A to sensitivity demonstrated. differ- the been at [8], stable re- angles are cryogenic tilt that ent Although demonstrated been tilts. have and ceivers re- notable with accelerations Scanning stability to a the spect on [5]. together; requirements experiment places optics cryostat Tenerife the and the receiver, cryostat being the exception the to respect scan with and stationary mitigating optics and their hold systematics reducing 1 instrumental strat- with against modulation stable, assist a and to include low, egy often have and (IP), to polarization designed all are h rviigΛD omlgclmdlhsbeen has model cosmological ΛCDM prevailing The omaueti inllvli hlegn,adgood and challenging, is level signal this measure To eivsiaesc ytmi hsppr nSec- In paper. this in system a such investigate We /f h ffc fasann a odmro naCBBmd experim B-mode CMB a on mirror fold flat scanning a of effect The ffc ntepwrsetu lutae n osbemeth possible a and sc illustrated realistic power-spectrum the a on of effect simulations Time-domain way. synchronous mn.A lmnu a-odmro sfudt add to found is mirror flat-fold aluminium An iment. os.Temjrt fteeexperiments these of majority The noise. eivsiaetepsiiiyo sn a-odba stee beam flat-fold a using of possibility the investigate We ∼ .INTRODUCTION I. 10 − 35 < r ◦ hc antb rdcdby produced be cannot which , a ayucrandetails; uncertain many has - s 0 . ila .Gane,CrsE ot,Ptr .R Ade R. A. Peter. North, E. Chris Grainger, F. William 0[2.T u ih con- tight put To [12]. 20 srnm ntuetto ru,Cri University. Cardiff Group, Instrumentation Astronomy r hs experiments These . Dtd oebr1,2018) 19, November (Dated: < r r which , 0 . 01. aea vrl oaiaino 0 of polarization overall an have 28 and etwv ihEfil aallt h boundary, the to parallel E-field with wave dent n so and o ea lt,w s h eainfrtecomplex the for relation the use refraction, we of plate, index metal a For hsa noaie nu eetdwith reflected input unpolarized an Thus lgigti noSelslwalw aclto of calculation allows law Snell’s into this Plugging a,fl irr lhuhtefltmro ol elarge; be single, would a mirror with flat output sky the ∼ the Although The to mirror. steered fold then flat, is horizon. CRA the the ground- of a beam observing in nominally placed (CRA), screen, antenna range compact and uny o odcnutr ecnassume can we conductor, good a For quency. σ and ihohrptnilpolm nScinIV. Section in problems along potential discussed, Results other and presented with pipeline. then are simulation pipeline and this from strategy scan fiducal a h rse qain ieterflcincecet for coefficients reflection angle the prin- incident polarizer”. give the “pile-of-plates equations is Fresnel the and The example, known for well is behind, effect ciple The of polarization. beam observations.a the allow to that sky such the tilted across moves and telescope tipped the then is mirror o eea ilcrc h conductivity the dielectric, general a For R ⊥ ∼ o hsivsiain ecnie onfdreceiver fed horn a consider we investigation, this For ffnra eetos scue yti odn,induce folding, this by caused as reflections, Off-normal . = 4 2 : .7%plrzto,wihvre nascan a in varies which polarization, 0.075% × × R ωǫ ⊥ 10 ,sc ytmcnb osdrd h flat The considered. be can system a such m, 4 nigptenaepromd n the and performed, are pattern anning ′′ o ueauiimpae 1 plate, aluminium pure a For . do orcinapplied. correction of od igmro o M -oeexper- B-mode CMB a for mirror ring ǫ − 0 9 where m efind we Ωm, θ i R R n rnmsinangle transmission and ( ⊥ k ω n I METHOD II. = = + 2 = n iκ 2 tan tan sin sin πf R ) = 2 k 2 2 2 2 = 2 0 = and ( ( ( ( ωǫ θ θ θ θ σ ǫ i i i i ′ 0 + + − − . + . 90and 9970 7%Ti goe the ignores This 075% f θ θ θ θ iǫ t t t t sterdainfre- radiation the is ) ) ) ) ′′ ent θ θ σ R t i /σ ⊥ 5would 45 = o ninci- an for sgvnby given is 0 = ǫ = ′ R ≪ . k 9985. ρ and (2) (4) (1) (3) R ǫ ′′ = k , 2 phase-delay from the reflections, which elliptically polar- izes the reflected beam, producing both circular polar- TABLE I. Simulation parameters. Detector positions are given in (az, el) in the telescope frame. ization and rotating the plane of polarization. Using a Drude-Lorentz model to calculate n and κ for aluminium Parameter Value at 90GHz[24], we calculate that the plane of polariza- Telescope location Atacama desert −9 tion is rotated by an additional 10 degrees when ◦ ∼ Patch centre 9h00m,-44 00’ the phase delay is included. Ignoring this is then valid ◦ Patch radius 10 as this is a much smaller than the effect from differen- tial reflection ( 0.2◦). The reflection is a total power to Azimuthal scan speed 0.5 deg/s polarization conversion∼ (T Q,U); in the context of a Nominal elevations 45, 60 deg CMB polarization measurement→ the total power is 2.7 K, Observing frequency 90 GHz and so the induced polarized signal is then 200 µK, Data rate 100 Hz ∼ −9 higher than even the EE polarization spectrum. How- Resitivity of aluminium 28.2 × 10 Ωm ever, the signal is changing consistently with the scan, Input r 0.1 and so is varying on different scales from the CMB sig- Detector 1 position (0,0) arcmin nals of interest and so could, in principle, be removed. Detector 2 position (14.8, 14.8) arcmin In the next section, we describe the pipeline with which Detector 3 position (29.6, 29.6) arcmin we investigate the effect on CMB reconstruction and Cℓ estimation. Detector 4 position (36.9, 7.4) arcmin We note in passing that it is possible to construct a Detector 5 position (51.7, 22.1) arcmin two mirror system that provides a fold with no net polarization. Consider a beam travelling along the x axis. The first mirror, mounted at 45◦ to the yz plane, then azimuthal scans at fixed elevation (nominally 45◦) as the reflects the beam so it is travelling along the y axis, and chosen sky patch rises for around 1 hour, continually ad- the second, mounted at 45◦ to the xy plane, reflects that justing the azimuthal center to achieve as circular a patch beam along the z. As the reflection axes have switched, as possible. Once the patch has risen sufficiently through both input Ex and Ey receive an attenuating factor of the scan, the elevation is increased, nominally to 60◦, ∼ RkR⊥. Any net polarization is then due to differences and the patch continues to rise for another 3 hours. The between the conductivity of the mirrors. If the two mir- same strategy is used as the patch sets, giving four dis- rors are fixed relative to each other, and allowed to rotate tinct data streams for a single patch during a 8-12-hour about the x axis, the beam can be steered. A second set observation period. of mirrors, in a similar configuration, can be added to al- The second step in the pipeline is to take the pointing low full beam steering around the sky. This is illustrated from the TOD data, determine the angle of the flat fold- in figure 1. 2 2 ing mirror, rotate the observed Ex and Ey to align with the plane of the mirror, apply the calculated Rk and R⊥, and then rotate back. The TOD is then naively binned III. SCAN STRATEGY AND PIPELINE to produce T, Q and U maps, following the map-making scheme outlined in [3], and then written out as a Healpix We simulate observations with the following four step format FITS files. We make maps at Nside = 512 so that the multipole range ℓ 100 to 1000 is retrievable, and pipeline. Firstly, time ordered data (TOD) is simulated ∼ with 100Hz sampling for four 3-hour observation periods, avoid holes in the maps by simulating a five pixel focal corresponding to a day of observations, and the sky lo- plane. The pixel angular positions from boresight are cation (RA, Dec), telescope pointing direction (az, el), listed in Table I. 2 2 Finally, the angular power spectrum (APS) coefficients and observed Ex and Ey recorded. The input CMB maps were produced by simulating a power spectrum Cℓ are estimated using the anafast Healpix routine. We ◦ with CAMB [16], and producing a realization with the apply a circular mask of radius 10 with a simple cosine- ◦ Healpix[Note1] [6] synfast routine. This input all-sky squared taper of width 2 at the edge. map was also convolved with a Gaussian beam with FWHM of 5.5 arcmin. Parameters used in the simulation are listed in Table I. Note that we include the monopole A. Determination of angles in this simulation, with a temperature of 2.7 K and so 2 2 E E =1.3 K. In order to determine the effect for a mirror at an ar- x ∼ y ∼ It is generally desirable for ground-based and sub- bitrary orientation, two angles are required; the angle of orbital experiments to scan the sky at constant eleva- incidence, θi and the angle, β, between the mirror and tion to minimize the effect of variable airmass. We fol- the horizon as seen by the incident photons. Consider low this and consider the proposed Clover scan strat- the the flat fold mirror to be at the origin, and the unit egy, the optimization of which is fully described else- vector a points to the dewar, and b points to the posi- where [North and Johnson]. The strategy is to perform tion on the sky being observed. This is is illustrated in 3

y z

R P1

R P2

P1 R R P1

R R P P2 2

FIG. 1. A pair of ﬂat-fold mirrors with no net polarization.

y b x θe

ϕ Ey Dewar a Ex b α a α a b θ θ i r MIRROR

FIG. 2. Fiducial geometry considered for calculating the relevant angles. Top left: the telescope beam from the dewar (direction a) to the sky (b), directed at the azimuthal polar coordinates (θe,φ). Bottom left: the angle α through which the beam is reﬂected by the mirror. Right: Ex and Ey, the two polarization vectors on the sky.

figure 2. α is the angle between a and b. In polar coordi- As the angles of incidence and reflection, θi and θr are ◦ ◦ nates, a = (1, 90 , 180 ) and b = (1,θe, φ) where 90 θe equal, we have and φ are the the elevation and azimuth of the sky− lo- θi =1/2α =1/2 arccos( sin θe cos φ)) (6) cation. Converting into Cartesian coordinates, we have − a b = ( 1, 0, 0) and = (sin θe cos φ, sin θe sin φ, cos θe). We define E to be parallel to the horizon, and E to be − a b x y To determine α, we take the dot-product of and and “upwards” towards the zenith. We require the angle β by get which Ex and Ey need to be rotated (around the line of sight) so that the rotated Ex is parallel to the mirror. We a.b = cos α = sin θe cos φ (5) define c, a unit vector that points in the same direction − 4

◦ ◦ as Ex, so, in polar, coordinates, c = (1, 90 , φ + 90 ). spectra all match the pipeline output as before. In order a b is parallel to the mirror. By taking the dot product to show problems more clearly, we also plot the difference of∧a b with c, we can determine the angle β as between the spectra produced from the maps with and ∧ without the flat fold mirror. These are shown against (a b).c = (a b) c cos β. (7) ℓ in figure 5. We find that the TT and EE spectra are ∧ | ∧ || | reconstructed well, with less than 1% difference between ℓ After some algebra, of 250 and 900. The BB spectra shows both fluctuation 2 of 1% and an overall offset of 2.5% over the same ℓ r = arccos(cos φ). (8) range.∼

Thus we have the angle of incidence, θi, required for the calculation above, and we have the angle, β, so that the B. Noise Rk and R⊥ factors can be applied to the rotated Ex and Ey. The shortest required observing time, for a generalized CMB experiment, would be achieved using a photon IV. RESULTS AND DISCUSSION noise limited telescope at a good site. For a instrument such as Clover, situated in the Atacama desert, the sys- A. Simulation Results tem NET is 175µK √s. With a 100Hz data stream, this noise level overwhelms the induced polarization signal from T Q,U mixing. However, simple binning by a fac- We use CAMB to simulate a sky, observe it with our → scanning strategy code, and follow the pipeline through tor of ten is sufficient to see the induced polarization sig- to calculate power spectra. Firstly we test our pipeline nal in the presence of white noise. If we add white noise at the level of 175 µK √s, with an ρ = 28.2 10−9 Ωm, by ignoring the effect of the fold mirror; resultant APS × are shown in figure 3. In the region of interest (ℓ 100– the best fitting resistivity returned with this procedure ∼ is 28.8 10−9 Ωm. 1000) the output spectra match the input spectra well for × TT and EE. The spectra from the masked pipeline maps Detector 1/f noise is assumed to be unimportant, as- for TT and EE are slightly supressed for larger ℓ; 5% at suming a detector-based modulation strategy is imple- at ℓ 900 when compared to the spectra made from the mented (e.g. using a rotating or stepped half-wave plate full sky∼ maps. However, as the spectra from the masked such as EBEX, or a fast detector switching such as full sky maps are also supressed, we conclude that this is QUIET). Atmospheric 1/f total power variations con- due to the effect of the partial sky mask and fairly naive tributes strongly however as the effect of the reflection apodization procedure. Similarly, the BB spectra from is a total power to polarization conversion. Thus the at- the masked map is higher than the all-sky map, but as mospheric 1/f becomes a polarization 1/f term which the pipeline BB spectra agrees with the masked all-sky does not get removed by any modulation strategy. A map BB spectra, we conclude that it is the mask rather straightforward differencing strategy, for example differ- than the map-making procedure which is increasing the encing successive samples during azimuthal scanning, or BB power. Although this level of analysis is not sufficient differencing two adjacent detectors, to remove this vary- for a full B-mode experiment, this level of analysis is suf- ing atmospheric contribution is likely to be the most ef- ficient for investigating the effect of the flat-fold mirror. ficient method, but would reduce the effective observing We then apply the effect of a flat fold mirror in the time by a factor of two. pipeline, using the resistivity of aluminium of 28.2 10−9 Ωm, and observations at 90GHz. A section of the× time stream is shown in figure 4, both the input and out- C. Temperature variations 2 put Ex and the azimuth of the observed sky location. The correlation between induced polarization with observed The resistivity of aluminium varies with temperature, sky azimuth is clear. and so the induced polarization will vary with temper- The induced polarization is big, and so it is reasonable ature. Given a resistivity at room temperature of ρ0 = to remove it without requiring the entire data set. By 28.2 10−9Ωm, the difference between the two polariza- applying the reverse of the observing algorthim, we can tion reflection× coefficients is 0.075%. We introduce a tem- −3 −1 determine a “pre-mirror” Ex and Ey for a given mirror perature coefficient of resistance, ρT , of 3.8 10 K , 2 2 × resitivity. We use Σ(Ex Ey ), summed over the day of with ρ = ρ0(1 + ρT ∆T ). Varying the temperature by observation for a particular− pixel, as a measure of the 20◦C gives a difference of 0.078% between the two po- amount of polarization, and minimize it by varying the larization reflection coefficients, meaning that the T resistivity, using the Nelder-Mead simplex direct search Q,U mixing varies by 7%. For a reasonably large flat→ [13], implemented in Matlab. This time-line can then fold mirror – in this case∼ the flat is 2m diameter – the be re-inserted into the pipeline, maps made and spectra temperature variation is likely to be∼ on the timescale of computed. The resultant APS are shown in figure 3. We hours, driven largely by the ambient temperature vari- see that in the range ℓ 100–1000, the TT, EE and BB ations. Such variations should be removable by fitting ∼ 5

FIG. 3. TT, EE, and BB spectra from various maps. dash-dot-dot-dot line is the input spectra, as simulated by CAMB. Dotted line is the all sky map, masked with the same mask used for the observed patches. solid is after our pipeline, but with no mirror. dashed is the spectra after having been observed with a ﬂat fold mirror. dot-dash is the spectra after observed, but with the ﬁtting algorthim (described in the text) applied.

the resistivity to the data over different timescales on a D. Oxide growth pixel-by-pixel basis. We note that, apart from the pixel on the optical axis, each pixel will see a section of the The polarizing effect of the fold mirror comes from mirror whose location varies in a predictable way with its resistivity. As such, effects that alter the resistivity azimuth (and elevation) and so fitting with respect to alter the polarizing effect. Aluminium grows an oxide mirror position might also be required. In this case, the layer very quickly, which then hydrates. The hydrated error on the fit can be reduced by fitting multiple pixel aluminium oxide can then be degraded chemically, or by data simulatiously to produce a mirror resistivity map. physical damage to the surface. Such a surface is thin (5- 15 nm) and so does not change the resistivity by a large amount. It is not unreasonable to assume that there will be changes to the surface during an extended observing campaign, both with oxide thickness and additional surface contamination. As with the temperature variation, 6

ﬁt out the eﬀect. As a result, we expect the resistivity variation from surface oxidation and contamination to be better characterised than the temperature variation.

E. Polarized foregrounds

The work so far has assumed that the light incident on the flat-fold mirror is polarized at a very low level, and that the majority of the polarized signal comes from the flat-fold mirror. This is the case when considering the CMB only, where the polarization signal has zero mean. These assumptions do not hold, however, when observing an arbitrary patch of sky, where both synchrotron FIG. 4. Upper plot: time stream of the input and observed Ex and dust from the galaxy is expected to be polarized at . × −9 for a noiseless simulation, with a 28 2 10 Ωm fold mirror. the 1 10% level, with a total intensity of 10–100 µK Lower plot: time stream of the telescope azimuth. close∼ to− the Galactic Plane, but down to . 1µK at high Galactic latitudes. As such, we would recommend this method only be used in regions of known low foreground emission, where the resulting error in polarization reconstruction would be small. Over sufficiently large areas of sky the magnitude of the effect could be modelled, and in regions with relatively little large-scale variation in the foreground emission, the effect would be a constant offset, similar to an additional absolute calibration error. The areas of sky observed by most B-mode experiments are selected to meet these requirements. In order to test this, we re-ran the simulation with some fiducial dust and synchrotron models added. After applying the same method as discussed above, the map- plane residuals are large; we see residuals of +20 to -40 µK in the Q map, in comparison with the signal level of 20µK across the map. We leave fully developing this ±method to work in the presence of significant polarized foregrounds for future work.

V. CONCLUSIONS FIG. 5. Fractional errors for the TT (dotted), EE (dashed) and BB (dot-dash) spectra. Calculated as the difference between the no-mirror case and the fitted mirror case, divided We have analyzed the effect of a flat-fold mirror on by the no-mirror case. polarization measurements of the CMB. In the case of no foregrounds and no noise with no corrections, we find that the E and B spectra are heavily contaminated by the contamination would almost certainly be localised, and mixing of total power to polarization signal. However, as so fitting resistivity with respect to mirror position would the CMB is expected to have a net polarization of zero, probably be required. However such changes would be the effect can be fitted out. In the case of having a net expected to be slow (timescales of days or more), or at polarization, such as in a real observation of the sky, the least sudden and stable, and so more data is available to reconstructed maps contain significant residuals.

[1] Abbott, L. F. and Wise, M. B. (1984). Constraints on Research. ArXiv Astrophysics e-prints. generalized inﬂationary cosmologies. Nuclear Physics B, [3] Brown, M. L., Challinor, A., North, C. E., Johnson, 244:541–548. B. R., O’Dea, D., and Sutton, D. (2009). Impact of [2] Bock, J., Church, S., Devlin, M., Hinshaw, G., Lange, modulation on CMB B-mode polarization experiments. A., Lee, A., Page, L., Partridge, B., Ruhl, J., Tegmark, MNRAS, 397:634–656. M., Timbie, P., Weiss, R., Winstein, B., and Zaldarriaga, [4] Crill, B. P., Ade, P. A. R., Battistelli, E. S., Benton, S., M. (2006). Task Force on Cosmic Microwave Background Bihary, R., Bock, J. J., Bond, J. R., Brevik, J., Bryan, S., 7

Contaldi, C. R., Doré, O., Farhang, M., Fissel, L., Gol- mization, 9:112–147. wala, S. R., Halpern, M., Hilton, G., Holmes, W., Hris- [14] Lamarre, J. M., Puget, J. L., Bouchet, F., Ade, P. A. R., tov, V. V., Irwin, K., Jones, W. C., Kuo, C. L., Lange, Benoit, A., Bernard, J. P., Bock, J., de Bernardis, P., A. E., Lawrie, C., MacTavish, C. J., Martin, T. G., Charra, J., Couchot, F., Delabrouille, J., Efstathiou, G., Mason, P., Montroy, T. E., Netterfield, C. B., Pascale, Giard, M., Guyot, G., Lange, A., Maffei, B., Murphy, A., E., Riley, D., Ruhl, J. E., Runyan, M. C., Trangsrud, Pajot, F., Piat, M., Ristorcelli, I., Santos, D., Sudiwala, A., Tucker, C., Turner, A., Viero, M., and Wiebe, D. R., Sygnet, J. F., Torre, J. P., Yurchenko, V., and Yvon, (2008). SPIDER: a balloon-borne large-scale CMB po- D. (2003). The Planck High Frequency Instrument, a larimeter. In Society of Photo-Optical Instrumentation third generation CMB experiment, and a full sky submil- Engineers (SPIE) Conference Series, volume 7010 of So- limeter survey. New Astronomy Review, 47:1017–1024. ciety of Photo-Optical Instrumentation Engineers (SPIE) [15] Lee, A. T., Tran, H., Ade, P., Arnold, K., Borrill, J., Conference Series. Dobbs, M. A., Errard, J., Halverson, N., Holzapfel, [5] Davies, R. D., Gutierrez, C. M., Hopkins, J., Mel- W. L., Howard, J., Jaffe, A., Keating, B., Kermish, Z., huish, S. J., Watson, R. A., Hoyland, R. J., Rebolo, R., Linder, E., Miller, N., Myers, M., Niarchou, A., Paar, Lasenby, A. N., and Hancock, S. (1996). Studies of cos- H., Reichardt, C., Spieler, H., Steinbach, B., Stompor, mic microwave background structure at Dec.=+40 deg - R., Tucker, C., Quealy, E., Richards, P. L., and Zahn, I. The performance of the Tenerife experiments. MNRAS, O. (2008). POLARBEAR: Ultra-high Energy Physics 278:883–896. with Measurements of CMB Polarization. In H. Kodama [6] Górski, K. M., Hivon, E., Banday, A. J., Wandelt, B. D., & K. Ioka, editor, American Institute of Physics Confer- Hansen, F. K., Reinecke, M., and Bartelmann, M. (2005). ence Series, volume 1040 of American Institute of Physics HEALPix: A Framework for High-Resolution Discretiza- Conference Series, pages 66–77. tion and Fast Analysis of Data Distributed on the Sphere. [16] Lewis, A., Challinor, A., and Lasenby, A. (2000). Effi- ApJ, 622:759–771. cient computation of CMB anisotropies in closed FRW [7] Grainger, W., Aboobaker, A. M., Ade, P., Aubin, F., models. Astrophys. J., 538:473–476. Baccigalupi, C., Bissonnette, E.,´ Borrill, J., Dobbs, M., [17] Mandolesi, N., Bersanelli, M., Butler, R. C., Artal, E., Hanany, S., Hogen-Chin, C., Hubmayr, J., Jaffe, A., Baccigalupi, C., Balbi, A., Banday, A. J., Barreiro, R. B., Johnson, B., Jones, T., Klein, J., Korotkov, A., Leach, S., Bartelmann, M., Bennett, K., Bhandari, P., Bonaldi, A., Lee, A., Levinson, L., Limon, M., Macaluso, J., MacDer- Borrill, J., Bremer, M., Burigana, C., Bowman, R. C., mid, K., Matsumura, T., Meng, X., Miller, A., Milligan, Cabella, P., Cantalupo, C., Cappellini, B., Courvoisier, M., Pascale, E., Polsgrove, D., Ponthieu, N., Reichborn- T., Crone, G., Cuttaia, F., Danese, L., D’Arcangelo, O., Kjennerud, B., Renbarger, T., Sagiv, I., Stivoli, F., Stom- Davies, R. D., Davis, R. J., De Angelis, L., de Gasperis, por, R., Tran, H., Tucker, G., Vinokurov, J., Zaldarriaga, G., De Rosa, A., De Troia, G., de Zotti, G., Dick, J., M., and Zilic, K. (2008). EBEX: the E and B Experi- Dickinson, C., Diego, J. M., Donzelli, S., Dörl, U., Dupac, ment. In Society of Photo-Optical Instrumentation En- X., Enßlin, T. A., Eriksen, H. K., Falvella, M. C., Finelli, gineers (SPIE) Conference Series, volume 7020 of Soci- F., Frailis, M., Franceschi, E., Gaier, T., Galeotta, S., ety of Photo-Optical Instrumentation Engineers (SPIE) Gasparo, F., Giardino, G., Gomez, F., Gonzalez-Nuevo, Conference Series. J., Górski, K. M., Gregorio, A., Gruppuso, A., Hansen, [8] Grainger, W. F., Ade, P. A. R., Hargrave, P., Chase, F., Hell, R., Herranz, D., Herreros, J. M., Hildebrandt, S., Kiernan, B., and North, C. E. (2010). Thermal ar- S., Hovest, W., Hoyland, R., Huffenberger, K., Janssen, chitecture of the Clover cryostats. In Society of Photo- M., Jaffe, T., Keihänen, E., Keskitalo, R., Kisner, T., Optical Instrumentation Engineers (SPIE) Conference Kurki-Suonio, H., Lähteenmäki, A., Lawrence, C. R., Series, volume 7739 of Society of Photo-Optical Instru- Leach, S. M., Leahy, J. P., Leonardi, R., Levin, S., Lilje, mentation Engineers (SPIE) Conference Series. P. B., López-Caniego, M., Lowe, S. R., Lubin, P. M., [9] Grishchuk, L. P. (1974). Amplification of gravitational Maino, D., Malaspina, M., Maris, M., Marti-Canales, waves in an isotropic universe. Soviet Journal of Experi- J., Martinez-Gonzalez, E., Massardi, M., Matarrese, S., mental and Theoretical Physics, 40:409–+. Matthai, F., Meinhold, P., Melchiorri, A., Mendes, L., [10] Hu, W., Hedman, M. M., and Zaldarriaga, M. (2003). Mennella, A., Morgante, G., Morigi, G., Morisset, N., Benchmark parameters for CMB polarization experi- Moss, A., Nash, A., Natoli, P., Nesti, R., Paine, C., Par- ments. Phys. Rev. D, 67(4):043004–+. tridge, B., Pasian, F., Passvogel, T., Pearson, D., Pérez- [11] Kamionkowski, M., Kosowsky, A., and Stebbins, A. Cuevas, L., Perrotta, F., Polenta, G., Popa, L. A., Pouta- (1997). A Probe of Primordial Gravity Waves and Vor- nen, T., Prezeau, G., Prina, M., Rachen, J. P., Rebolo, ticity. Physical Review Letters, 78:2058–2061. R., Reinecke, M., Ricciardi, S., Riller, T., Rocha, G., [12] Komatsu, E., Smith, K. M., Dunkley, J., Bennett, C. L., Roddis, N., Rohlfs, R., Rubiño-Martin, J. A., Salerno, Gold, B., Hinshaw, G., Jarosik, N., Larson, D., Nolta, E., Sandri, M., Scott, D., Seiffert, M., Silk, J., Simon- M. R., Page, L., Spergel, D. N., Halpern, M., Hill, R. S., etto, A., Smoot, G. F., Sozzi, C., Sternberg, J., Stivoli, Kogut, A., Limon, M., Meyer, S. S., Odegard, N., Tucker, F., Stringhetti, L., Tauber, J., Terenzi, L., Tomasi, G. S., Weiland, J. L., Wollack, E., and Wright, E. L. M., Tuovinen, J., Türler, M., Valenziano, L., Varis, J., (2010). Seven-Year Wilkinson Microwave Anisotropy Vielva, P., Villa, F., Vittorio, N., Wade, L., White, M., Probe (WMAP) Observations: Cosmological Interpreta- White, S., Wilkinson, A., Zacchei, A., and Zonca, A. tion. ArXiv e-prints. (2010). Planck pre-launch status: the Planck-LFI pro- [13] Lagarias, J., Reeds, J., Wright, M., and Wright, P. gramme. ArXiv e-prints. (1998). Convergence properties of the nelder-mead sim- [North and Johnson] North, C. and Johnson, B. Clover scan plex method in low dimensions. SIAM Journal of Opti- strategy, in prep. in prep. [Note1] Note1. http://healpix.jpl.nasa.gov. 8

[20] Rubakov, V. A., Sazhin, M. V., and Veryaskin, A. V. [24] Ung, B. and Sheng, Y. (2007). Interference of surface (1982). Graviton creation in the inflationary universe and waves in a metallic nanoslit. Opt. Express, 15(3):1182– the grand unification scale. Physics Letters B, 115:189– 1190. 192. [25] Zaldarriaga, M. and Seljak, U. (1997). All-sky analysis [21] Samtleben, D. and for the QUIET collaboration (2008). of polarization in the microwave background. Phys. Rev. QUIET - Measuring the CMB polarization with coherent D, 55:1830–1840. detector arrays. ArXiv e-prints. [22] Starobinskii, A. A. (1983). The Perturbation Spectrum Evolving from a Nonsingular Initially De-Sitter Cosmol- VI. ACKNOWLEDGEMENTS ogy and the Microwave Background Anisotropy. Soviet Astronomy Letters, 9:302–+. [23] Starobinsky, A. A. (1982). Dynamics of phase transition We thank Enzo Pascale and Giorgio Savini for useful in the new inflationary universe scenario and generation discussions during this work. Some of the results in this of perturbations. Physics Letters B, 117:175–178. paper have been derived using the HEALPix[6].