Gravitational Lensing of Cosmological 21 cm Emission

Transcription

1 Mon. Not. R. Astron. Soc. 000, (0000) Printed 22 January 2015 (MN ATEX stye fie v2.2) Gravitationa ensing of Cosmoogica 21 cm Emission A. Pourtsidou 1,2 & R. Benton Metcaf 1 1 Dipartimento di Fisica e Astronomia, Universitá di Boogna, viae B. Pichat 6/2, 40127, Boogna, Itay 2 Institute of Cosmoogy & Gravitation, University of Portsmouth, Burnaby Road, Portsmouth, PO1 3FX, United Kingdom The 21cm radiation provides an exceent source for gravitationa ensing studies. Earier work (Zahn & Zadarriaga 2005; Metcaf & White 2009) had shown that if the EoR is at redshift z 8 or ater, a arge radio array such as SKA coud measure the ensing convergence power spectrum and constrain the standard cosmoogica parameters however, these studies assumed a much arger survey area than is currenty panned for the argest such experiment, SKA ow (see Section 3 for detais). The authors extended the Fourier-space quadratic estimator technique, which was first deveoped by (Hu 2001) for CMB ensing observations to three dimensiona observabes, i.e. the 21 cm intensity fied I(θ, z). These studies did not consider 21 cm observaarxiv: v2 [astro-ph.co] 21 Jan January INTRODUCTION 21cm cosmoogy is a new and exciting area of research with a great dea of potentia. Future radio teescopes ike the SKA 1 (Square Kiometer Array) wi give us access to previousy unexpored epochs of the Universe, such as the Epoch of Reionization (EoR), the end of the Dark Ages and the beginning of the Cosmic Dawn (Dewdney 2013). Another important deveopment in the fied has been the HI intensity mapping technique in which the distribution of gaaxies can be measured without detecting individua gaaxies. Instead, the 21cm emission is treated as a continuous (unresoved) background, much ike the Cosmic Microwave Background (CMB), but extended in the frequency (redshift) dimension. Severa groups are panning to use this technique to measure the Baryon Acoustic Osciations (BAO) scae at redshifts of order unity (Chang et a. 2008, 2010; Seo et a. 2010; Masui et a. 2010; Ansari et a. 2012; Battye et a. 2012; Chen 2012; Pober et a. 2013; Smoot & Debono 2014; Bu et a. 2014), an important probe of dark energy. By combining resuts from different kinds of surveys (CMB, gaaxies, 21cm), which probe compimentary redshift 1 ABSTRACT We investigate the feasibiity of measuring weak gravitationa ensing using 21 cm intensity mapping with specia emphasis on the performance of the panned Square Kiometer Array (SKA). We find that the current design for SKA Mid shoud be abe to measure the evoution of the ensing power spectrum at z 2 3 using this technique. This wi be a probe of the expansion history of the universe and gravity at a unique range in redshift. The signa-to-noise is found to be highy dependent on evoution of the neutra hydrogen fraction in the universe with a higher HI density resuting in stronger signa. With reaistic modes for this, SKA Phase 1 shoud be capabe of measuring the ensing power spectrum and its evoution. The signa-tonoise s dependance on the area and diameter of the array is quantified. We further demonstrate the appications of this technique by appying it to two specific couped dark energy modes that woud be difficut to observationay distinguish without information from this range of redshift. We aso investigate measuring the ensing signa with 21 cm emission from the Epoch of Reionization (EoR) using SKA and find that it is unikey to constrain cosmoogica parameters because of the sma survey size, but coud provide a map of the dark matter within a sma region of the sky. Key words: cosmoogy: theory arge-scae structure of the universe gravitationa ensing: weak dark energy ranges, we wi be abe to span a very arge observationa voume and more precisey investigate the acceeration of the Universe. Of particuar reevance to this paper is the possibiity of extending the cosmoogica probes to redshifts between those accessibe with gaaxy redshift surveys and the high redshift of the CMB. By studying the evoution of expansion and structure formation across a wide range of redshift we can investigate whether dark energy or modified gravity effects are present at a higher redshift than the standard cosmoogica mode predicts (Copeand, Sami & Tsuikawa 2006; Cifton et a. 2012).

2 2 Pourtsidou & Metcaf tions from redshifts after reionization when the average HI density in the universe is much smaer. In Pourtsidou & Metcaf (2014) we extended the aforementioned studies to redshifts after reionization, but before those probed by gaaxy surveys in the visibe bands and showed that ensing can be measured using the HI intensity mapping technique. Here we deveop this concept further. In Section 2 we present a preiminary overview of our study, which spans two distinct regimes (EoR and z 2 5), and give the forma form of the ensing estimator and ensing reconstruction noise. In Section 3 we study how we the current SKA design wi be abe to map the ensing convergence at typica EoR redshifts. In Section 4 we investigate weak ensing intensity mapping with a SKA-ike interferometer array at redshifts z 2 3. Different modes for the HI mass function and different array configurations are considered. We concentrate on the abiity of different SKA phases to probe the ensing signa and cacuate the corresponding signa-to-noise predictions. In Section 5 we show that these measurements can be used to differentiate between some specific and nove interacting dark energy modes. We concude in Section 6. Throughout the paper we adopt a fat ΛCDM Universe with the PANCK cosmoogica parameter vaues (Ade et a. 2013), uness otherwise stated. 2 PREIMINARIES We wi investigate the ensing of 21 cm emission in two distinct regimes. The first is from the EoR and the second is from z 2 5. In the first case, the neutra hydrogen fraction is high and HI gas is not restricted to individua gaaxies. In this case the HI distribution wi be approximated as a Gaussian random fied that roughy foows the distribution of matter. More sophisticated modes are under investigation, but wi not be considered here. The biggest source of noise for 21 cm observations at these high redshifts comes from foreground contamination. SKA ow is being panned to probe this regime. Current pans ca for it to survey a 5 5 sky area with a frequency range of MHz corresponding to redshifts z 5 35 (Dewdney 2013). At ower redshift the HI fraction is much ower and essentiay a the HI gas is in discrete gaaxies. In this regime we treat the HI distribution as consisting of discrete sources that are custered according to the standard CDM paradigm, but aso exhibit random Poisson fuctuations. This is the standard way of modeing the distribution of gaaxies in redshift surveys. We find that this Poisson or shot noise contribution to the gaaxy custering is important for measuring ensing from 21 cm after (and perhaps during) reionization. The evoution in the HI fraction as a function of redshift is a matter of some debate and specuation and has an important effect on the expected ensing signa to noise. We wi address this uncertainty in Section 4 by adopting severa different modes. The quadratic ensing estimator for the CMB or a singe redshift sice of the 21 cm emission works by essentiay measuring differences in the power spectrum in different regions of the sky which resuts in correations in Fourier (or spherica harmonic) modes that woud not exist otherwise. For the CMB the oca power-spectrum becomes anisotropic because of shearing and the acoustic peaks are sighty shifted to arger (smaer) anguar scaes by gravitationa magnification (demagnification). The effect of ensing can be divided into a shearing which can make the oca power spectrum anisotropic and magnification which isotropicay scaes the oca power spectrum. It can be shown that an isotropic magnification cannot be measured (with a quadratic estimator) if the powerspectrum is scae free (C 2 ). It can aso be shown the the shear cannot be measured if the power-spectrum is constant (see Bucher et a. (2012) for a nice demonstration of this). As we wi see, the proected matter power-spectrum in the CDM mode is approximatey a constant at arge scaes (sma ) and C 2 at sma scaes so to the extent that the HI distribution foows the dark matter distribution, the quadratic estimator picks up the magnification on arge scaes and the shear on sma scaes and some combinations in between. After reionization the remaining HI resides in discrete gaaxies. The gaaxies are custered in the same way as matter on arge scaes (moduo a bias factor), but on sma scaes their discreteness enters into the power-spectrum as shot noise or Poisson noise. The power-spectrum of this component is fat (for equa uminosity sources, C = 1/η where η is the number density on the sky) so ony the magnification can be measured. Magnifying (demagnifying) a region of the sky reduces (increases) the number density of sources. The resut is that the power-spectrum wi have features on the scae of the ensing. Unike the CMB, the 21 cm emission wi be observabe at many different redshifts. The 3D ensing estimator effectivey stacks redshift sices. Since the ensing is coherent for different sices at the same anguar position, but the 21 cm emission is statisticay independent, the residua effect can be attributed to ensing. This wi be made more expicit in the next section. 2.1 The quadratic ensing estimator Here we wi give a more expicit description of the ensing estimator and noise. The advantage of 21cm ensing is that one is abe to combine information from mutipe redshift sices, and that the 21cm signa extends to far smaer anguar scaes than CMB fuctuations (oeb 2004), which are suppressed by Sik damping at > The intensity fuctuations, I(θ, z), are expressed in discrete Fourier space in the radia direction owing to the finite band width (wave vector k = 2π, where is the depth of the observed voume) and in continuous Fourier space perpendicuar to the ine of sight (wave vector k = /D where D the anguar diameter distance to the source redshift and is the dua of the anguar coordinate on the sky). See Appendix A for the connection between estimators in discrete and continuous Fourier space. Considering modes with different independent, an optima estimator can be found by combining the individua estimators for different modes without mixing them. The quadratic estimator for the ensing potentia Ψ is

3 Gravitationa ensing of Cosmoogica 21 cm Emission 3 thus of the form ˆΨ() = g(,, ) Ĩ,Ĩ,, (1) where Ĩ, is the discrete Fourier transform of the ensed intensity fied Ĩ(θ, x ). The form of the fiter g(,, ) and the expected noise in the estimator depends on whether the source is discrete or continuous. In appendices A through C we derive the cases that are used in the foowing sections. The mean observed brightness temperature at redshift z due to the average HI density is T (z) = 180 Ω HI(z) h (1+z)2 H(z)/H 0 mk (Battye et a. 2012). The anguar power spectrum of the dispacement fied (the dispacement of points on the sky caused by gravitationa ensing) for sources at a redshift z s can be cacuated with C δθδθ = 9Ω2 mh0 3 zs [W (z)]2 dz P (k = /D(z), z) ( + 1)c 3 0 a 2 E(z), (2) (Kaiser 1992) in the weak fied imit where W (z) = (D(z s) D(z))/D(z s), E(z) = H(z)/H 0 and P (k) is the power spectrum of matter. The convergence power spectrum is simpy. Note that the imber approximation (imber 1953) has been used in deriving Eq. (2). This approximation is quite accurate even for arge scaes 10, as the resuting formua invoves an integra over the ine of sight which effectivey averages out the effect of the approximation, see (Schmidt et a. 2008) for a detaied discussion. The ensing reconstruction noise incudes the therma noise of the array which -assuming a uniform teescope distribution- is cacuated using the formua C κκ = ( 2 /4) C δθδθ C N = (2π) 3 T 2 sys Bt obs f 2 cover max(ν) 2, (3) where T sys is the system temperature, B is the chosen frequency window, t obs the tota observation time, D te the diameter (maximum baseine) of the core array, max(λ) = 2πD te /λ is the highest mutipoe that can be measured by the array at frequency ν (waveength λ), and f cover is the tota coecting area of the core array A co divided by π(d te /2) 2 (Furanetto et a. 2006). Assuming that the temperature (i.e. the 21cm emissivity) is Gaussian distributed we find that the ensing reconstruction noise N() is (see (Zahn & Zadarriaga 2005) and Appendix A) 3 ENSING STUDIES USING 21CM RADIATION FROM THE EPOCH OF REIONIZATION Observing the EoR through 21 cm has been one of the primary science goas for severa arge radio teescope proects incuding the SKA. The coecting area and resoution of OFAR 2 or MWA 3 are unikey to be high enough to observe gravitationa ensing as we wi see. For this reason we concentrate on the SKA ow instrument here which is the part of SKA designed for observing the EoR and the cosmic dawn. The current SKA pans ca for a 25 square degree survey 4 with SKA ow (Dewdney 2013). The convergence power spectrum measured in this size fied wi be dominated by sampe variance. This wi make it difficut to measure the cosmoogica parameters through inear growth in the matter power spectrum at a competitive eve with these observations even if the signa-to-noise in the ensing map is very high. This is not true of the SKA Mid at ower redshift where the survey area wi be much arger see Section 4. However, it might be possibe to map the ensing convergence within the 25 square degree EoR survey area with high fideity. This woud aow us to actuay see the distribution of dark matter in a typica region of the sky, something that is ony possibe with gaaxy ensing around very atypica, arge gaaxy custers. For the EoR, the convergence (or, equivaenty, the dispacement fied) estimator and the corresponding ensing reconstruction noise are cacuated assuming that the temperature (brightness) distribution is Gaussian, and consider the reionization fraction f HI = 1 unti the Universe is rapidy and uniformy reionized at redshift z s. Before reionization this is probaby a very good approximation. Reionization is not expected to be homogeneous and may extend over a significant redshift range so for some period there wi be ionized regions that grow and intersect unti they fi amost a of space when reionization is compete. Because the contrast in the brightness temperature is arger during this period, detecting and mapping the 21 cm emission wi be easier, but it wi change the power spectrum and increase the fourth order moments of the brightness temperature (Metcaf & White 2009). It is not yet cear how this wi affect the detectabiity of ensing. To assess this wi require numerica simuations which we pan to present in a future pubication. Note that a semi-anaytic mode used to describe the patchy regime in (Zahn & Zadarriaga 2005) showed an increase in noise due to a decrease of the fuctuation eve of the 21cm signa on the smaest resoved scaes and the con- N(, ν) = [ max =1 1 4 ] 1 d 2 [ C, + ( )C, ] (2π) 2 2C tot, Ctot, (4) where C, tot = C, + C N and C, = [ T (z)] 2 P, with P, the underying dark matter power spectrum. One can aso consider stacking mutipe bands ν and then the noise is reduced as N = 1/ [N(, ν)] 1. The Gaussian approximation formaism wi be used in the next Section to investigate ν the possibiity of measuring the ensing signa with 21 cm emission from the Epoch of Reionization Here we note that suggestions for a three-tier SKA ow survey have been recenty put forward (eon Koopmans & Jonathan Pritchard, private communication). These incude an additiona 200 square degree survey in hrs as we as a 2000 square degree survey in hrs. Note that these numbers assume 1 beam ony, and a muti-beam request (N beam = 5) has aso been put forward. This woud further increase the sky area surveyed in a given amount of time and woud greaty enhance the SKA ow science output, adding the exciting possibiity of measuring the cosmoogica parameters (see (McQuinn et a. 2006) and (Metcaf & White 2009)).

4 4 Pourtsidou & Metcaf tribution of the connected four-point function which acts as a sampe variance term correating different k-modes. Of particuar importance is the noninear structure effect during the EoR. We wi be abe to account for these effects once more detais about the reionization process are known, but under the assumption that these features appear in higher resoution than that of our considered interferometers there is no need for the connected four-point contribution to the variance of the quadratic ensing estimator to be taken into account (Kovetz & Kamionkowski 2013). At redshift z s 8, we assume the SKA1 Baseine Design (Dewdney 2013) parameters of A co 0.3 km 2 with maximum baseine D te = 4 km, whie for SKA2 we can consider A co 1.2 km 2. The estimated ensing noise is shown in Figure 1 aong with the estimated signa. Here C δθδθ is the convergence fied power spectrum at z s = 8 and N the ensing reconstruction noise assuming a reionization fraction f HI = 1. Note that at such high redshifts the system temperature T sys is dominated by gaactic synchrotron radiation and can be approximated by (Dewdney 2013) ( ν ) 2.55 T sys = 60 K. (5) 300 MHz For SKA ow we can consider a 1, 000 hr observation time and mutipe stacked bands ν with B = 8 MHz 5. Here we note that the noise N(, ν) converges quicky with (as going to higher vaues of the signa decays quicky beow the therma noise eve), so that max 20 at z 8. It can be seen in Fig. 1 that the noise for at east SKA2 shoud be we beow the power-spectrum over a arge range in anguar scaes. This means that typica modes wi be measurabe with high signa-to-noise and a high fideity map is possibe. The noise crosses the power-spectrum at 800 for SKA2 which corresponds to an anguar resoution of 10 arcmin. This measurement greaty benefits from the arger coecting area that wi come with Phase 2 of the SKA. Such a high fideity map coud be cross-correated with many other observabes from ower redshift to determine how gas and stars foow dark matter as a function of redshift or to test the predictions of genera reativity. 4 WEAK ENSING WITH HI INTENSITY MAPPING AFTER REIONIZATION After reionization HI gas exists ony in discrete gaaxies. This introduces some differences in the cacuation of the ensing estimator. As wi be seen, the noise per mode wi increase from the EoR case, but because panned 21 cm intensity mapping surveys wi cover a much arger fraction of the sky the sampe variance is much ower and a very good measurement of the convergence power-spectrum and its redshift evoution shoud be possibe. The signa-to-noise is criticay dependent on the array parameters, the area surveyed and the evoution of the HI mass fraction with redshift. 5 As discussed in (Metcaf & White 2009), the quadratic ensing estimator is optimized for the case where the statistica properties of the 21cm radiation signa and noise are constant within a band, and an observation bandwidth of a few MHz is sma enough so that this assumption is ustified. 1 C Θ Θ 2Π SKA1 SKA Figure 1. The ensing dispacement fied power spectrum, C δθδθ, for sources at z = 8 is shown as a soid back ine and ensing reconstruction noise N as dashed/dotted ines. The bue dotted curve is for the SKA1 Baseine Design with 10 8 MHz frequency bins around z = 8 spanning the redshift range z The red dashed ine is for SKA2 and the same frequency bins. The vertica ine is approximatey the owest accessibe with a 5- by-5 degree fied. Where the noise curves are beow C, typica fuctuations in the ensing defection shoud be recoverabe in a map. 4.1 Measuring ensing with the SKA In previous work (Pourtsidou & Metcaf 2014) we extended the 21 cm intensity mapping ensing method further, taking into account the discreteness and custering of gaaxies, and investigated the possibiity of measuring ensing at intermediate redshifts without resoving (in anguar resoution) or even identifying individua sources. Here we perform an improved anaysis of the signa-to-noise expected from an SKA-ike interferometer, and demonstrate the performance of SKA Mid Phases 1 and 2. A forma treatment of how a quadratic ensing estimator can be derived for discrete sources is incuded in appendices B and C. The expected measurement error in the power spectrum C δθδθ in a band of width is given by C δθδθ = 2 ( ) C δθδθ + N. (6) (2 + 1) f sky Here, N is the derived δθ estimate reconstruction noise (see Appendix C for the detaied derivation and formuas for this case) which invoves the underying dark matter power spectrum, the HI density Ω HI(z) as we as the HI mass moments up to 4th order and, of course, the therma noise of the array C N, equation (3). The system temperature in this case is the sum of the receiver and sky noise (Dewdney 2013) [ ( ν ) ] 2.55 T sys = K. (7) 300 MHz The moments of the HI mass function (or 21 cm uminosity function) enter into the estimator s noise expressions (C15) through (C20). Assuming that the HI mass function foows the Schechter function (which is an exceent fit to the oca Universe data see, for exampe, the HIPASS survey resuts in (Zwaan et a. 2003)) we can write dn(m, z) dm dm = φ (z) ( M M (z) ) α [ exp M M (z) ] dm M (z). (8)

5 Gravitationa ensing of Cosmoogica 21 cm Emission 5 Damped yman-α systems observations are used to measure Ω HI(z) up to z 5, whie no measurements of φ (z) and M (z) are avaiabe other than in the oca Universe. However, for this form of the mass function we have the reation SKA0-1 7 SKA2 Ω HI(z) = φ (z) M (z) φ (z = 0) M (z = 0). (9) At redshift z = 0 we use the vaues α = 1.3, M = 3.47h M, φ = h 3 Mpc 3 reported from the HIPASS survey (Zwaan et a. 2003). We wi initiay consider the most conservative scenario, i.e. a no evoution mode that we wi ca Mode A. Mode A was aso used in (Pourtsidou & Metcaf 2014). Here we wi aso show resuts using a few HI evoution modes. et us first concentrate on source redshift z s 2. In the recenty pubished SKA1 Baseine Design (Dewdney 2013), SKA Mid operates in the frequency range MHz divided in three bands. Our chosen z s corresponds to frequency ν s = 473 MHz (Band 1). Foowing the proposed parameters for SKA Mid we consider a 2 yr observation time, f sky 0.7, and we further choose B = 20 MHz and = 36. From Eq. (B6) we see that the measurement errors increase as 1/ f sky and the fact that SKA Mid wi cover a arge fraction of the sky contributes to the high S/N significance we predict. An increase in the observation time woud reduce the therma noise of the interferometer and woud aso ead to an increase of the overa S/N eves. A simiar effect comes from increasing the bandwidth B of the observation however, note that we avoid choosing very wide bands since in that case there woud be non-negigibe correations within the band. It is not cear whether these correations woud ust ead to an increase of the noise ony or if they woud aso increase the ensing signa. We pan to investigate this issue in a future pubication. Since the gaaxies are being treated as point sources we wish to incude ony frequency modes that do not resove the interna structure of the gaaxies. Assuming a typica veocity dispersion for a gaaxy at z = 2 to be around 200 km/sec, we find max = 63. However the noise has converged at 40, which is the vaue we use in our numerica simuations (see Fig. C1 for a demonstration of the convergence with ). Keeping these vaues constant, we present a signa-to-noise contour pot (Figure 2) at mutipoe = 100 for z s = 2 on the (A co, D te ) parameter space, and show three SKA Mid performance cases: SKA Mid Phase 1 with 50% sensitivity (SKA0), SKA Mid Phase 1 (SKA1), and SKA Mid Phase 2 (SKA2). We aso present the dispacement fied power spectrum and measurement errors for z s = 2 and z s = 3 using the SKA2 specifications in Figure 3. These resuts are not sensitive to the exact vaue of min. For exampe, the S/N vaues remain practicay unchanged if we use a very sma min = 10 or the one corresponding to the current SKA1 fied of view ( 1 deg), min = 180. This technique shoud enabe us to measure the ensing power spectrum at source redshifts we beyond those accessibe with more traditiona weak ensing surveys based on the shearing of individua gaaxy images. The noise is expected to be sma enough that evoution in the ensing power spectrum between z s = 2 and 3 wi be ceary detectabe as iustrated in Figure 3. D te km A co km 2 Figure 2. The signa-to-noise at = 100 in a = 36 for various array configurations. Sources are at z s = 2. The contour ines are abeed with the (S/N) vaues. The area under the thick bue soid ine is excuded, since it corresponds to f cover > 1. SKA0, SKA1 and SKA2 are shown. The no-evoution in Ω HI (Mode A) scenario is used. C δθδθ (+1)/2π 1e-07 1e-08 1e SKA2 Figure 3. Dispacement fied power spectrum for z s = 2 (soid back ine) and z s = 3 (dashed magenta ine) and the corresponding measurement errors using the SKA2 specifications and Mode A for the HI mass function. 4.2 Importance of the redshift evoution in the HI mass function It is usefu to iustrate how our resuts depend on a possibe evoution of the HI mass function with redshift. We wi foow Zhang & Pen (2006). The authors construct three different evoution modes for the HI mass function. Observations of damped yman-α systems and yman-α imit systems measure Ω HI from z = 0 to z 5 (see, for exampe, (Peroux et a. 2003)).These observations found that Ω HI increases by a factor of 5 toward z 3 and then decreases toward higher

6 6 Pourtsidou & Metcaf e SKA2 - Mode B HI z C δθδθ (+1)/2π 1e-08 1e Figure 5. The evoution of Ω HI (z) with redshift in (Zhang & Pen 2006) Figure 4. Dispacement fied power spectrum for z s = 2 (soid back ine) and z s = 3 (dashed magenta ine) and the corresponding measurement errors using the SKA2 specifications and Mode B for the HI mass function. redshift. Combining Eq. (9) and these observations, one can put constraints on the product of φ and M. Either an increase in φ or M increases the detectabiity of ensing, so the no evoution mode is considered a conservative choice. Foowing (Zhang & Pen 2006), the evoution of Ω HI wi be approximated as g(z) ΩHI(z) Ω HI(z = 0) = (1 + z)2.9 exp( z/1.3). (10) This function is potted in Fig. 5 and the constraint for the redshift evoution of the product φ M is φ (z)m (z) = φ (z = 0)M (z = 0)g(z). (11) Here we wi investigate the foowing evoution scenarios (Zhang & Pen 2006): Mode (B) No evoution in M (z). φ (z) = φ (z = 0)g(z). Mode (C) No evoution in φ (z). M (z) = M (z = 0)g(z). Mode (D) φ (z)/φ (z = 0) = M (z)/m (z = 0) = g(z) 1/2. We aso note that HI mass function modeing has been performed by other authors (Abdaa, Bake & Rawings 2010), where the evoution of the Ω HI(z) function agrees with (Zhang & Pen 2006) unti z 3 and so woud make very itte difference to our predictions Mode B Mode B assumes no evoution in M. The shot noise terms in the estimator noise are changed in this case. The dispacement fied power spectrum and measurement errors for mode B are shown in Fig. 4, and the signa-to-noise contour at = 100 is shown in Fig. 6 as a function of array parameters. Notice the improvement of the S/N across a of the D te km SKA SKA A co km 2 Figure 6. The signa-to-noise at = 100 in a = 36 for various array configurations. Mode B for the evoution of the HI mass function is used. Sources are at z s = 2. The contour ines are abeed with the (S/N) vaues. The area under the thick bue soid ine is excuded, since it corresponds to f cover > 1. SKA0, SKA1 and SKA2 are shown. parameter space from the no-evoution mode the ensing signa can be detected even with SKA Mode C Mode C assumes no evoution in φ. Ω HI wi increase, but the shot noise terms wi stay the same as the no-evoution ones (Mode A), because the integras do not get affected by M. Hence, the signa-to-noise is higher than Mode A by a factor 1.5 at = 100 using the SKA2 specifications, but ower than mode B by a factor 0.5 at = 100 using the SKA2 specifications Mode D Mode D assumes that both φ and M are evoving as φ (z)/φ (z = 0) = M (z)/m (z = 0) = g(z) 1/2. The signa-to-noise is higher than mode C by a factor 1.5 at = 100 using the SKA2 specifications, but ower than 36

7 Gravitationa ensing of Cosmoogica 21 cm Emission 7 mode B by a factor 0.7 at = 100 using the SKA2 specifications. We see that for a modes where the HI density increases with redshift the ensing signa-to-noise is greater than in the no-evoution mode. Furthermore, we earned that there is a stronger dependence on the evoution of φ than M. This is aso expected, since an increase in φ increases the detectabiity of ensing at this point we shoud stress that in our case the contributions of the Poisson moments contribute both to the signa as we as the noise of our estimator, so the dependence on φ is crucia. Mode B, in which φ is soey responsibe for the increase in Ω HI is the most optimistic scenario, but the most conservative no-evoution Mode A is probaby ess reaistic. Previousy, we used the signa-to-noise cacuations at a singe mutipoe number ( = 100) to demonstrate the capabiities of various arrays (see Fig. 2 and 6). A more comprehensive expression woud invove contributions from many mutipoes. We therefore define a new (S/N) for the detection of the ensing signa as C 2 b S/N = C 2 b b 1/2, (12) where the sum is over defection fied power spectrum band powers (using = 36 as before) and C b is the forecasted error on each band power. We cacuate (S/N) as a function of coecting area and maximum baseine, with bmin = 18 and bmax = 990. We show the signa-to-noise contour pots for Mode A and mode B for the HI mass functions in Figures 7 and 8, respectivey. As we have aready stressed, Mode A is the most conservative, no-evoution scenario one can consider for modeing the HI mass function. From Figures 3 and 7 we can deduce that in order to get a good measurement of the ensing signa up to mutipoe number 1000 we need a S/N of about 25. Then from Fig. 8 we can see that SKA1 assuming Mode B does amost as we as SKA2 assuming no evoution (sources at z s = 2). This shows how crucia the HI mass function evoution is for our method and the SKA s science goas. In Fig. 9 we show the resuts of our cacuations using the SKA1 specifications and Mode B as expected from the above discussion, the measurement errors are simiar to the SKA2 no-evoution case. 5 CONSTRAINING INTERACTING DARK ENERGY MODES In this Section we demonstrate the constraining power of 21cm ensing measurements using the intensity mapping method with an instrument ike the SKA. The probems of the nature and evoution of dark energy and dark matter are arguaby the most important ones in modern cosmoogy, and ambitious future missions ike EUCID 6 are dedicated to the exporation and mapping of the dark Universe by investigating the evoution of cosmic structures out to redshifts 1.5 2, a period over which dark energy is thought to dominate the Universe s expansion. 6 D te km SKA A co km 2 SKA2 Figure 7. The signa-to-noise from Eq. (12) for various array configurations. Sources are at z s = 2 and we use Mode A for the HI mass function. The contour ines are abeed with the (S/N) vaues. The area under the thick bue soid ine is excuded, since it corresponds to f cover > 1. SKA0, SKA1 and SKA2 are shown. D te km SKA0-1 SKA A co km 2 Figure 8. The signa-to-noise from Eq. (12) for various array configurations. Sources are at z s = 2 and we use Mode B for the HI mass function. The contour ines are abeed with the (S/N) vaues. The area under the thick bue soid ine is excuded, since it corresponds to f cover > 1. SKA0, SKA1 and SKA2 are shown. An important question is whether dark energy is indeed a cosmoogica constant, ike a the avaiabe data from Cosmic Microwave Background measurements (Ade et a. 2013), the Hubbe constant (Riess et a. 2009) uminosity and distance at high redshift with supernovae Ia (Kowaski et a. 2008) and Baryon Acoustic Osciation surveys (ampeit et a. 2009) seem to suggest (however this success comes with the price of the cosmoogica constant and coincidence probems), an evoving scaar fied ike quintessence, or a modification of gravity. To differentiate between the pethora of avaiabe modes, we might have to go even deeper in redshift 31 39

8 8 Pourtsidou & Metcaf C δθδθ (+1)/2π 1e-07 1e-08 SKA1 - Mode B 1e Figure 9. Dispacement fied power spectrum for z s = 2 and the corresponding measurement errors using the SKA1 specifications and Mode B for the HI mass function. space z > 1.5, and this is where 21 cm radiation becomes a unique probe of cosmoogy. Interactions in the dark sector, such as a nongravitationa couping between dark matter and dark energy (for exampe a couping between a quintessence fied φ paying the roe of dark energy and the matter sector, see (Amendoa 2000)), can cause modifications to the background evoution of the matter density and the Hubbe parameter, as we as changes in the evoution of structure growth with respect to ΛCDM. However, as we wi demonstrate beow, the background effects are very difficut to probe as aternative dark energy and modified gravity modes can be very successfu in mimicking the ΛCDM background. Therefore, we need to study inear perturbations and quantify the effects of the various modes in the CMB and matter power spectra. Modifications in the evoution of structure growth in interacting dark energy modes eave distinctive signatures in the weak ensing signa, and measurements such as those presented in Fig. 3, coud be used to constrain such modes. In a recent study Pourtsidou, Skordis & Copeand (2013) constructed three genera casses (Types) of modes of dark energy in the form of a scaar fied φ couped to cod dark matter. The first cass, Type 1, is a generaization of the couped quintessence (CQ) mode suggested by (Amendoa 2000). In such modes, the CDM energy density is not conserved separatey (ike in the uncouped case) but there is a non-zero couping current which represents the decay of DM to DE or vice versa. In the specific (Amendoa 2000) mode the evoution of the background CDM density ρ c is found to be ρ c = ρ c,0a 3 e αφ. (13) We therefore see that the CDM density evoves differenty than the uncouped quintessence case and ΛCDM, which means that for positive couping parameter α (decay of dark matter to dark energy) the CDM density has to be higher in the past in order to reach the same vaue today. This mode and its variants have been extensivey studied in the iterature, and constraints on the strength of the couping parameter (roughy < 0.1 assuming it is constant) have been derived using its effects on the CMB and matter power spectra, the growth of structure and the weak ensing signa (e.g. (Tarrant et a. 2012; Xia 2013; Cadera-Cabra, Maartens & Schaefer 2009; De Bernardis et a. 2011; Amendoa et a. 2011) and references therein). Here we wi investigate the weak ensing convergence signatures of modes beonging to the new casses of couped DE theories, namey Type 2 and Type 3, constructed in Pourtsidou, Skordis & Copeand (2013). Type 2 modes invove both energy and momentum transfer between the two components of the dark sector (i.e. ike Type 1 modes but with a distinctivey different couping mechanism), whie Type 3 is a pure momentum transfer theory. The dispacement fied power spectrum for these modes is cacuated using (Kaiser 1992) C δθδθ 9 = ( + 1)c 3 zs 0 dz [Ω m(z)] 2 P (k = /D, z)[w (z)] 2 (1 + z) 4 [H(z)] 3, (14) which simpifies to Eq. (2) for ΛCDM by setting Ωm(z) Ω m = H 2 0 a 3 [H(z)] Type 2 modes of CDM couped to DE Type 2 modes are cassified via the agrangian function (Pourtsidou, Skordis & Copeand 2013) (n, Y, Z, φ) = F (Y, φ) + f(n, Z), (15) where n is the partice number density, Y = 1 2 µφ µ φ is used to construct a kinetic term for the quintessence fied φ, and Z = u µ µφ pays the roe of a direct couping of the fuid veocity to the gradient of the scaar fied. We can proceed by making further assumptions. First of a, we want to consider couped quintessence modes, hence we can write F = Y + V (φ). Secondy, since we choose the scaar fied to be couped to CDM, the function f is separabe, i.e. f = n h(z). The detais of the fied and fuid equations for the Type-2 cass of theories under consideration can be found in (Pourtsidou, Skordis & Copeand 2013). An important quantity that appears in the cosmoogica equations is the function K(Z) = h z/(h Zh z), with h z = dh/dz. We therefore see that we cannot simpy choose h to be proportiona to Z, i.e. h(z) = βz with β the (constant) couping parameter, as in that case K(Z) diverges. Choosing h(z) = Z β we find that the background CDM density ρ c evoves as ρ c = ρ c,0a 3 Zβ, (16) where Z = φ/a (note that the dot denotes derivatives with respect to conforma time). The fact that the CDM density depends on the time derivative φ instead of φ itsef is a notabe difference from the Type 1 (Amendoa) case. Note aso that the specific form of the couping function Z β means that in order to have meaningfu soutions Z has to be positive throughout the cosmoogica evoution. In (Pourtsidou, Skordis & Copeand 2013) the authors derive the background

9 Gravitationa ensing of Cosmoogica 21 cm Emission CDM Moresco et a. (2012) Type 3, γ = 0.05 Type 2, β = CDM WMAP9 Type 2, β = 0.02 Type 3, γ = H(z) (km/s/mpc) (+1)C /2π [µκ 2 ] z Figure 10. The Hubbe parameter versus redshift for ΛCDM (soid back ine) and the Type 2, 3 modes under consideration (red dashed ine for Type 2, bue dotted dashed ine for Type 3), together with expansion history measurements from (Moresco et a. 2012), which combine the observationa constraints on the Hubbe parameter from (Simon et a. 2005; Stern et a. 2010; Moresco et a. 2012b). and perturbed Kein-Gordon equations for the evoution of the quintessence fied, as we as the perturbed CDM equations for the evoution of the density contrasts and veocities. With these equations at hand, and considering a singe exponentia potentia for the quintessence fied, we use a modified version of the CAMB code (ewis, Chainor & asenby 2000) to study the observationa signatures of the chosen couped mode. In Figures 10 and 11 we show the Hubbe parameter evoution and the CMB TT power spectra for these modes together with ΛCDM. We aso construct the dispacement fied power spectrum (Fig. 12) and compare it with the ΛCDM prediction in Fig. 3. Note that the cosmoogy of each mode evoves to the PANCK cosmoogica parameter vaues at z = 0 (Ade et a. 2013) and the matter power spectra P (k) have been normaized to CMB fuctuations. From Fig. 12 we see that the Type 2 mode with a couping parameter β = 0.02 woud be excuded by the proposed ensing / intensity mapping observations. That is because in this case there is energy transfer from dark matter to dark energy making the dark matter density arger in the past compared to the non-interacting case for fixed Ω m today, the growth is increased, the gravitationa potentia is higher and the convergence power spectrum is enhanced with respect to ΛCDM. Another important point demonstrated in Fig. 13 is that the Type 2 mode inear growth factor is strongy scae dependent, and the difference between the growth of the two scaes persists up to the source redshift z = 2. Note that there is aso an effect on the Hubbe parameter evoution (see Fig. 10) and the CMB power spectrum (see Fig. 11), but subter than the effect on the ensing signa. Figure 11. The CMB TT spectra for ΛCDM (soid back ine) and the Type 2, 3 modes under consideration (red dashed ine for Type 2, bue dotted dashed ine for Type 3), together with WMAP9 measurements. 5.2 Type 3 modes of CDM couped to DE Type 3 modes are cassified via the agrangian function (Pourtsidou, Skordis & Copeand 2013) (n, Y, Z, φ) = F (Y, Z, φ) + f(n). (17) We can consider a couped quintessence function of the form F = Y + V (φ) + h(z), and choose h(z) = γz 2. Note that this mode is physicay acceptabe for γ < 1/2, otherwise it suffers from ghost and strong couping probems (Pourtsidou, Skordis & Copeand 2013). As we have aready mentioned, Type 3 is a pure momentum transfer theory. The background CDM density ρ c evoves as in the uncouped case and the energy conservation equation remains uncouped even at the inear eve. Foowing the same procedure, using the equations derived in (Pourtsidou, Skordis & Copeand 2013) and the modified version of CAMB we construct the dispacement fied power spectrum and compare it with the ΛCDM prediction in Fig. 3. From Fig. 12 we see that the Type 3 mode with a couping parameter γ = 0.05 woud be excuded. The signature of this mode is growth suppression and hence a decrease in the ensing signa. Again, we note that there is aso an effect on the Hubbe parameter evoution (see Fig. 10) and the CMB power spectrum (see Fig. 11), but subter than the effect on the ensing signa. 6 CONCUSIONS HI intensity mapping is an innovative technique which can be utiized to map the arge scae structure of the Universe in three dimensions. In this paper, we investigated the potentia of a SKA-ike interferometer to perform accurate measurements of the ensing signa over a wide range of redshifts using the intensity mapping method. Extending the work in

10 10 Pourtsidou & Metcaf C δθδθ (+1)/2π 1e-07 1e-08 1e Type 3, γ = 0.05 Type 2, β = 0.02 ΛCDM Figure 12. Dispacement fied power spectrum for ΛCDM (soid back ine) compared with Type 2 (red dashed ine) and Type 3 (bue dotted dashed ine) interacting dark energy modes. Sources are at z s = 2. The circe points with back error bars are the same as in Figure 3, i.e. using the most conservative scenario (no evoution in the HI mass function), Mode A. Assuming a more optimistic scenario ike Mode B (magenta error bars) woud further improve the constraints. P k,z P CDM k,z k Mpc 1 k 0.05 Mpc 1 CDM Our cacuations showed that the ensing convergence can be measured with high fideity at redshifts z 2 3, hence it can be used for tomographic studies aong many redshift (or frequency) bins in order to map, for exampe, the evoution of the growth function at intermediate redshifts, i.e. higher than those of gaaxy shear surveys. We aso extended our cacuations to incude a possibe evoution of the HI mass function, showing that the ess ikey no-evoution scenario is the most conservative one and our caim that a high S/N measurement can be achieved using this method is robust. To quantify the power of this technique we performed an optimization study using a SKA-ike interferometer and showed that SKA wi be abe to deiver very good measurements of the ensing signa and its redshift evoution. We aso demonstrated the constraining power of our technique by using it to distinguish specific interacting dark energy modes. Our resuts confirm that the 21cm technique can be used to compete (and compement) with future gaaxy surveys, and SKA in particuar can be used to investigate the evoution of dark energy and dark matter in order to test the standard ΛCDM paradigm against exotic modes with possibe observationa signatures in higher redshifts than the ones probed by gaaxy ensing surveys. Another exciting prospect is the study of the reionization epoch we showed that the ow frequency SKA instrument can map the ensing convergence and aow us to see the distribution of dark matter in a typica region of the sky, something that is ony possibe with gaaxy ensing around very atypica, arge gaaxy custers. This woud provide a great opportunity to correate visibe obects with mass and test the dark matter paradigm. Acknowedgments This research is part of the proect GENCO, funded under the Seventh Framework Programme, Ideas, Grant Agreement n The authors woud ike to thank eon Koopmans, Jonathan Pritchard and Mario Santos for usefu discussions Figure 13. The ratio of the inear matter power spectrum in the Type 2 couped DE mode to the one in ΛCDM for two characteristic scaes as a function of redshift. (Pourtsidou & Metcaf 2014), we presented our technique for measuring gravitationa ensing in HI observations after reionization. Other methods have been proposed based on counting the number of detected gaaxies in 21 cm emission (Zhang & Pen 2005, 2006; Zhang & Yang 2011). In such methods the custering of gaaxies and the shot noise from Poisson fuctuations in the number counts contribute purey to the noise in the ensing estimator. The corresponding errors can be controed to 10% 20% eve for intermediate source redshifts z 1, but for z s 2 the reconstruction is not successfu (Zhang & Yang 2011). Our estimator takes into account both the custering and discreetness of gaaxies, which contribute to both the noise and to improving the signa resuting in a significant improvement in performance. REFERENCES Abdaa, F. B., Bake, C., Rawings, S., 2010, MNRAS, 401, 743 Ade, P. et a. (Panck Coaboration), 2013, Amendoa,., 2000, Phys. Rev. D, 62, Amendoa,., Pettorino, V., Querceini, C., Vomer, A., 2012, Phys. Rev. D, 85, Ansari, R., Campagne, J. E., Coom, P., Magnevie, C., Martin, J. M., Moniez, M., Rich, J., Yeche, C., 2012, C. R. Phys., 13, 46 Battye, R. A., Browne, I. W. A., Dickinson, C., Heron, G., Maffei, B., Pourtsidou, A., 2013, MNRAS, 434, 1239 Bucher, M., Carvaho, C. S., Moodey, K., Remazeies, M., 2012, Phys. Rev D, 85, Bu, P., Ferreira, P. G., Pate, P., Santos, M. G., arxiv: [astro-ph.co]. Cadera-Cabra, G., Maartens, R., Schaefer, B. M., 2009, JCAP, 0907, 027 Chang, T., Pen, U.-., Bandura, K., Peterson, J. B., Mc- Donad, P., 2008, Phys. Rev. ett., 100,

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12 12 Pourtsidou & Metcaf APPENDIX A: ENSING ESTIMATOR FOR A CONTINUOUS SOURCE Here we derive the east squares, or minimum variance, quadratic estimator on a discrete grid for a source that can be approximated as a Gaussian random fied. This wi be the discrete version of the estimator found by Zahn & Zadarriaga (2005). The discrete version is necessary for cacuations on a finite grid and usefu in the deviation of the discrete source case (see Appendix B). This appendix wi aso serve to introduce some notation. We write the discrete Fourier transform (DFT) of the intensity fied I(x) as I k = Ωs e ik x I(x), (A1) N N where k = (, ), x = (θ, z) and Ω s = Θ s Θ s for a square survey geometry. We aso have I(x) = 1 e ik x I k. Ω s N is the number of ces in the direction parae to the ine-of-sight and N is the number of ces on a pane perpendicuar to the ine-of-sight. We use a fat sky or sma ange approximation. The correation between discrete modes is x k (A2) I k I k = Ω2 sδ K kk P k (A3) where P k is the discrete power spectrum. The Kronecker deta impies statistica homogeneity. The discrete power spectrum is reated to the continuous one, P (k), by P k = P (k) V s = P (k) Ω sd 2 = 1 Ω s C,. (A4) Here D is the comoving anguar size distance to the source voume from the observer. The anguar Fourier coordinate is = Dk and denotes the DFT coordinate in the radia direction. The comoving ength of the source voume in the radia direction is. This serves to define the anguar power spectrum in ange and radia coordinates C,. So in terms of the anguar power spectrum I k I k = Ωs C, δ K, δk. (A5) Gravitationa ensing causes the the observed emission fied to be inhomogeneous over a region of the sky with coherent defection. The observed intensity after ensing is Ĩ(θ, x ) = I ( ) θ α(θ), x (A6) I ( ) θ, x α(θ) θ I ( ) θ, x (A7) I ( ) θ, x + θ Ψ(θ) θ I ( ) θ, x (A8) where α(θ) is the defection caused by ensing and Ψ(θ) is the ensing potentia. The defection fied is a potentia fied to very good approximation. In Fourier space this becomes Ĩ, I, + 1 Ω s ( ) I, Ψ (A9) From this we can find the correation between modes to first order Ĩ,Ĩ, = ΩsC,δ K,0δ K + δk [ C, + ( )C, ]Ψ. We seek to construct a quadratic estimator of the form ˆΨ() = g(,, ) Ĩ,Ĩ,. Putting (A10) into (A11) and requiring the estimator to be unbiased gives the constraint g(,, )[ C, + ( )C, ] = 1. The variance of this estimator is V = g(,, )g (,, ) I, I,I, I, = 2Ω 2 s g 2 (,, )C, tot C,, tot where in cacuating the fourth order correations the resut for a Gaussian random fied has been used and (A10) (A11) (A12) (A13) C tot, = C, + C N, (A14)