Reaching small scales with low frequency imaging: applications to the Dark Ages
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Subject Areas: observational astronomy, cosmology
Keywords: cosmology, interferometry, highresolution imaging, Dark Ages
Author for correspondence:
Leah Morabitoe-mail:[email protected]
Reaching small scales withlow frequency imaging:applications to the Dark Ages
L. K. Morabito and J. Silk , Centre for Extragalactic Astronomy, Department ofPhysics, Durham University, Durham, DH1 3LE, UK Institut d’Astrophysique de Paris, UMR 7095CNRS,Sorbonne Universités, 98 bis, boulevard Arago,F-75014, Paris, France Beecroft Institute of Particle Astrophysics andCosmology, University of Oxford, Oxford OX1 3RH, UK
The initial conditions for the density perturbationsin the early Universe, which dictate the large scalestructure and distribution of galaxies we see today, areset during inflation. Measurements of primordial non-Gaussianity are crucial for distinguishing betweendifferent inflationary models. Current measurementsof the matter power spectrum from the CMB onlyconstrain this on scales up to k ∼ . Mpc − . Reachingsmaller angular scales (higher values of k ) can providenew constraints on non-Gaussianity. A powerful wayto do this is by measuring the HI matter powerspectrum at z (cid:38) . In this paper, we investigate whatvalues of k can be reached for the LOw FrequencyARray (LOFAR), which can achieve (cid:46) (cid:48)(cid:48) resolution at ∼
50 MHz. Combining this with a technique to isolatethe spectrally smooth foregrounds to a wedge in k (cid:107) - k ⊥ space, we demonstrate what values of k we canfeasibly reach within observational constraints. Wefind that LOFAR is ∼
1. Introduction
The end of Inflation set the density fluctuations whichdictated the growth of large-scale structure and thedistribution of galaxies we see today. Observations ofthe Cosmic Microwave Background (CMB) and galaxysurveys which trace the large scale structure havebeen used to constrain the matter power spectrum onlarge scales (co-moving wavenumbers k (cid:46) . Mpc − [1,2]) while observations of galaxy clustering, weakgravitational lensing and the Lyman- α forest extend thisof c (cid:13) The Authors. Published by the Royal Society under the terms of theCreative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author andsource are credited. a r X i v : . [ a s t r o - ph . C O ] J u l r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. to the range of . (cid:46) k (cid:46) Mpc − (e.g., Fig 19 of [3]). Probing higher co-moving wavenumbersprovides new information, and places powerful constraints on the distribution of matter in theearly Universe. Using the spin-flip transition of neutral hydrogen allows us to access the matterdistribution in the Dark Ages (after the formation of the first stars, but before reionisation). Athigh enough redshifts, the temperature of the Universe was coupled with the CMB temperaturethrough Compton scattering, but after z ∼ the Universe began to cool more rapidly thanthe CMB. This provides the ideal conditions for detecting the spin-flip in absorption. At lowredshifts, the small scales (high k ) are heavily influenced by the complex astrophysical processesof galaxy formation and reionization. Reaching above z ∼ allows us to avoid many of thesecomplications.Measuring the 21 cm power spectrum in the early Universe is the goal of many low-frequencyinstruments, which usually focus on the lower redshifts of the Epoch of Reionisation [4–10]. Someexperiments focus on achieving a global detection, which measures the average temperaturebrightness across the sky of the 21 cm line, T , as a function of redshift. This is an inexpensiveway to make a detection, as in principle it can be done with a single antenna and enoughintegration time. However, astrophysical foregrounds have to be removed or simultaneouslyfitted for in the data to extract the 21 cm signal, and it can be hard to interpret the results (seenrecently with the claimed detection from EDGES [11]).Although more resource intensive, using interferometers to measure 21 cm fluctuations canprovide more stringent constraints. With resolution comes the possibility of isolating foregroundsto a ‘wedge’ in k space [12–16]. In the 2D power spectrum, foregrounds will occupy low values of k (cid:107) (i.e., along the line of sight). The imperfect sampling of smoothly varying foreground spectraresults in an upscattering of the foreground power in this parameter space, but the foregroundscan still be isolated to a particular region. By cleanly isolating the foregrounds, we can make themost of the data which probes the rest of the k (cid:107) - k ⊥ parameter space.Angular resolution offers more than just foreground isolation: it allows us to probe highervalues of k ⊥ , which is crucial for extracting new information from the data to place constraintson the 21 cm power spectrum [17,18]. By measuring the small-scale matter distribution in theDark Ages, we can place completely new constraints on models of inflation, and simultaneouslyhelp probe open questions on the more nearby Universe. For example, measuring the small-scalepower spectrum can inform us on whether or not primordial black holes were large enough toexplain the rapid growth of the most massive super-massive black holes at z (cid:38) [19,20]. It is alsopossible that primordial black holes can increase our chances of detecting the 21 cm signal: theirpresence can deepen the absorption at z (cid:46) when both the product of Eddington ratio and mass,as well as their source density, are large [21]. This enhancement in the absorption is expected tohappen at k (cid:38) − - − Mpc − , and increase for larger values of k .To reach the relevant values of k ⊥ , it is necessary to use an interferometer with high resolutionand a wide field of view. The LOw Frequency ARray (LOFAR) [6] is the only low-frequency radiointerferometer capable of the necessary sub-arcsecond resolution, and simultaneously offers a 5deg field of view. With stations currently spread across 7 countries (the Netherlands, Germany,UK, France, Sweden, Ireland, Poland) and growing (Latvia and Italy are building stations, andthere are future plans for expansion to other countries), its capabilities will remain unique evenin the era of the Square Kilometre Array (SKA). As such, it is interesting to ask the questionwhether LOFAR could be used to detect the small-scale 21 cm power spectrum, which is thefocus of this speculative study. We intend this to be informative only, with an eye towards thefuture construction of a low-frequency lunar array. Such an array would remove the ionosphere(opening up the regime below ∼
10 MHz, where the ionosphere becomes opaque) and mitigateradio frequency interference [22]. Currently LOFAR’s longest baseline is about two-thirds thatof the diameter of the moon, and offers a real-world example of the type of long-baseline low-frequency array that could be built.The paper is organised as follows: in Section 2 we discuss using LOFAR as a cosmologicalinstrument, with particular attention to its high-resolution imaging capability. In Section 3 we r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. address foreground isolation. Section 4 presents our sensitivity predictions, with discussion inSection 5 and conclusions and future work in Section 6.Throughout this paper we have assumed cosmological parameters from the Planck
2. The Low Frequency Array as a cosmological instrument
The LOw Frequency ARray (LOFAR) is a phased array comprised of fixed dipoles, with phasedelays introduced into the signal paths to electronically ‘point’ the telescope. Groups of 96 dipolesform stations, which are correlated in the same way as a traditional interferometer. LOFAR hastwo frequency bands: the Low Band Antenna (LBA) which operates between 10 – 90 MHz, and theHigh Band Antenna (HBA) which operates between 110 and 240 MHz. There is a hard limit of 96MHz of available bandwidth, which is typically split into 2 ×
48 MHz to conduct two simultaneousobservations. In practice, this bandwidth is adjusted be centred on the sharply peaked maximumsensitivity of the LBA at ∼
55 MHz, or to avoid the strong bandpass roll-off below 120 MHz forthe HBA. The bandwidth is divided into 244 subbands, which can each be divided further intochannels. The finest frequency resolution possible in a standard observing mode is 0.76 kHz (256channels per subband), which corresponds to higher values of k (cid:107) .While default LOFAR operations use only the the stations in the Netherlands, baselinesup to 1989 km (Ireland to Poland) provide sub-arcsecond imaging at MHz frequencies. Thisunique capability is competitive with higher-frequency instruments like e -MERLIN, as LOFARcan achieve sub-arcseocnd imaging across a ∼ field of view in a single pointing. Thishigh-resolution capability has been exploited to achieve scientific results on individual objectsfor both the HBA [24] and LBA [25]. The combination of high resolution and wide field of viewalso makes LOFAR an attractive survey instrument, and it is the only instrument current capableof sub-arcsecond resolution at MHz frequencies, where the Dark Ages signal can be reached. Thisprovides access to higher values of k ⊥ .Figure 1 shows what values of k ⊥ and k (cid:107) are currently available to LOFAR. Horizontalshaded regions indicate the redshift ranges which LOFAR cannot access because of its frequencycoverage. This is based on the default observational setup and current array configuration. Thisincludes 13 international stations, which provide k ⊥ ∼ . - . Mpc − for the HBA and k ⊥ ∼ . - . Mpc − for the LBA. The default frequency setup is 30 – 78 MHz for theLBA and 120 – 168 MHz for the HBA. The LOFAR EoR project [26,27] uses a slightly differentfrequency range to access higher redshifts, but has to deal with the increasing radio frequencyinterference (RFI) at the top of the HBA band (the RFI rapidly increases above ∼
168 MHz). TheLBA is clearly more interesting for studying the Dark Ages, and in fact [28] have placed the firstlimits on the power spectrum for z = 19 . − . . They use only the Dutch array, which limits k ⊥ to < . Mpc − , and places a 2 σ limit of ∆ < ( ∼ mK ) at k = 0 . Mpc − from twosimultaneously observed fields. They centre their bandwidth on the most sensitive portion ofthe LBA band, setting frequency limits of 39 - 72 MHz. Increasing the redshift upper limit to z > is technically possible by shifting the usable bandwidth down to start at the hard limitof 10 MHz, but this comes at the cost of greatly reduced sensitivity. It is worth noting that finerfrequency increments provide diminishing returns in reaching higher values of k (cid:107) . The limitsare k (cid:107) ∼ − . - . Mpc − for the HBA and k (cid:107) ∼ − . - . Mpc − for the LBA. Thesecalculations assume that there are no foregrounds to remove, which is addressed in the nextsection.
3. Foregrounds
Although LOFAR can theoretically access high- k modes, this will be limited by foregroundcontamination. Setting aside the problem of calibrating for ionospheric distortions as the radiowaves pass through the atmosphere (which would not be a problem for a lunar array), theforegrounds consist of Galactic emission and extragalactic sources. The Galactic foreground in r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. R ed s h i ft LBAHBAHorizon
CoreDutchInternational10 − − − k [Mpc -1 ] R ed s h i ft Δν =48 MHz Δν =0.76 kHz LBAHBA − − − k [Mpc -1 ] Figure 1.
Translating LOFAR’s angular and spectral resolution into k ⊥ and k (cid:107) . Left:
The shaded regions represent thevalues of k ⊥ which can be reached by the Dutch (blue) and International (green) stations. The gap in between themarises from the fact that there is a large geographical distance between the outermost Dutch stations and the closestinternational stations. The gray shaded region on the left shows the horizon limit. Right:
The gray lines show the limitson k (cid:107) based on the total bandwidth (48 MHz) and the smallest frequency resolution possible in a standard observingmode (0.76 kHz). The frequency spacing will be channelised, and the colour scale runs from wider channels (blue), whichmeans more infrequent coverage in k (cid:107) space, to to narrower channels (orange), which means more frequent coverage in k (cid:107) space. This is plotted as a continuous shaded region, but in practice will be discrete. particular has a steep power law spectrum, with S ν ∝ ν − . . Extragalactic sources are alsocontaminants, and can range in shape and size from compact galaxies up to bright radio relicsin galaxy clusters. Modelling and subtracting these forergrounds is perhaps the most obviousapproach, but in practice this becomes extremely complex and difficult to do with high accuracy.Another approach is to isolate the foregrounds rather than subtract them, as described in theintroduction. This technique uses the 2D power spectrum in k (cid:107) - k ⊥ space to identify and remove aforeground ‘wedge’. Although we have made substantial progress in the last year on calibratingthe full international LOFAR telescope, there is not yet an appropriate data set with which tostart investigating the foreground wedge specifically for large wavenumbers using LBA data. Weexpect this situation to change within the next year, but here we show the theoretical foregroundwedge (Eqn. 5 in [29]) in Figure 2, for the international stations of LOFAR (the green band inthe left-hand panel of Figure 1). The ranges of k ⊥ and k (cid:107) which LOFAR will probe are redshiftdependent, although it is clear from Figure 2 that this does not change significantly within theallowable bandwidth of the LBA. The theoretical foreground wedge is shown, calculated for thespecific redshift examples in the Figure, and assuming a 5 deg field of view. This is the field ofview of a single LOFAR pointing which has been calibrated including the international stations.Sources outside this field of view are typically subtracted, but residual power may still leak infrom imperfect source subtraction. We therefore also plot the ‘worst case’ wedge limit as dottedlines, assuming the full 20 deg field of view of LOFAR calibrated with only the Dutch stations.The wedge will clearly limit the range of k (cid:107) values. In the best case scenario (no foregroundleakage; solid lines in Figure 2), we can reach the full range of k ⊥ values by setting a lower limitof k (cid:107) (cid:38) . Mpc − which corresponds to a maximum allowable bandwidth of ∼
98 kHz. r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. z=30z=35z=40z=45 Foregroundwedge k [Mpc -1 ] k [ M p c - ] Figure 2.
The shaded rectangles show the portion of the k ⊥ - k (cid:107) plane which the angular and spectral resolution ofLOFAR can access using the international stations, for four representative redshifts. The theoretical foreground wedgeis shown as the gray shaded region in the bottom right hand corner of the plot. The dotted lines show the theoreticalforeground wedge assuming the full LOFAR field of view using just the Dutch stations. The coloured rectangles and linesare for z = 45 (light) to z = 30 (dark).
4. Sensitivity predictions
From the proceeding two sections, we see that we are limited to a narrow range in k ⊥ and k (cid:107) that depends on the foreground wedge. This translates to a limitation in baseline length andbandwidth. In this section, we will calculate the sensitivity of LOFAR to these angular and spectralscales. While we do not expect this to be sufficient for a detection of the 21 cm absorption fromthe Dark Ages, it is instructive to see how far off we are with this particular array design.The sensitivity of an an array depends on many factors. In general, the sensitivity for aninterferometer is defined as: ∆S = S sys (cid:112) N ( N − δνδt (4.1)where S sys is the System Equivalent Flux Density (SEFD), N is the number of stations, and δν , δt are the bandwidth and integration time, respectively. This equation is valid if all elements of theinterferometer are the same, which is not the case for LOFAR. We must account for the differentSEFDs (related to the collecting area) of the core, remote, and international stations, which are alldifferent. Expanding Equation 4.1 for LOFAR, we get: ∆S = W (cid:114) δνδt ) (cid:104) N c ( N c − S c + N c N r S c S r + N c N i S c S i + N r ( N r − S r + N r N i S r S i + N i ( N i − S i (cid:105) (4.2) r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. where W is the imaging weight (assumed here to be unity), δν , δt , and N are as above, with thesubscripts c , r , and i denoting core, remote, and international stations respectively. The systemsensitivity, S , depends on the effective collecting area of each station and the system temperature: S sys = 2 ηk B A eff T sys (4.3)where η is an efficiency factor (assumed here to be unity), k B is the Boltzmann constant, A eff is theeffective collecting area of a station, and the system temperature T sys is the summation of the skytemperature T sky and the instrument temperature, T sys . We use tabulated values of T sys for eachtype of station, and T sky = 60 λ . to calculate the senstivity of each type of station. This followsthe methods of [30] should the reader desire a more in-depth discussion.Here we adapt the sensitivity calculations to the specific idea we want to investigate. For agiven redshift, k ⊥ and k (cid:107) can be translated into values of spatial resolution and bandwidth. Weuse the range of the spatial resolutions to specify which LOFAR stations will contribute to thesensitivity on those spatial scales, and only these are used in Equation 4.2. Note that the shorterbaselines will still contribute to the overall sensitivity at smaller scales: a perfect point source willbe detected on both short and long baselines. The converse is not true – if a source is extended,it may be detected on short baselines but not long baselines. Once the stations are selected, wecalculate the array sensitivity as in Equation 4.2 using only those stations. Finally, we convert thesensitivity to brightness temperature in Kelvin: T b = Sc ln (2)2 ν k B πθ (4.4)where S is the array sensitivity from Equation 4.2, c is the speed of light, ν is the observingfrequency, k B is the Boltzmann constant, and θ is the FWHM of a circular resolution element.From this we can see that T b depends inversely on the resolution, and thus for smaller resolutionelements (higher k ⊥ values) we are sensitive to higher values of T b . It is useful to keep in mindthat Equation 4.2 depends on the individual stations’ effective collecting area, the bandwidth,and the integration time. An example of the sensitivity for z = 30 and z = 45 is given in Figure 3,for observation times of 800 hours and 10 years. Values of T b for four different redshift slices aregiven in Table 1, for the 10 year observation. Table 1. T b sensitivity values for 10 year observation time, for four different redshift slices. z k ⊥ range θ range k (cid:107) range ∆ν range T b, min T b, max T b, median [Mpc − ] [arcsec] [Mpc − ] [kHz] [K] [K] [K]30 . - . . - . . . . . - . . - . . . . . - . . - . . . . . - . . - . . . .
5. Discussions
The 21 cm signal from the Dark Ages is expected to be (cid:46) mK [31], but could be as smallas ∼ mK at the most pessimistic, in the standard cosmology. Assuming an optimistic case of ∼ mK, with 10 years of observations and assuming no data loss or corruption from calibration,Figure 3 indicates the best LOFAR can do is ∼ K. Stacking in redshift slices can help alleviatethis: the most one can stack is the entire 48 MHz bandwidth. For the limited allowed values ofbandwidth, this would increase the signal to noise by a factor in the range of - . However, ifthe goal is to detect individual sources in absorption to construct the 3-point correlation function, r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. Figure 3.
The log of the temperature brightness to which LOFAR is sensitive, as a function of k ⊥ and k (cid:107) at z = 30 (leftcolumn) and z = 45 (right column) for an 800 hour observation (top row) and a 10 year observation (bottom row). stacking in redshift will not help. We therefore conclude that we are approximately five orders ofmagnitude away from the desired sensitivity.As mentioned in the introduction, this signal could be boosted by primordial black holes,which could narrow the gap between expected and achievable sensitivity by 2 - 3 ordersof magnitude depending on the number and combined Eddington ratio of primordial blackholes. However, other scenarios have also been proposed. The EDGES absorption profile [11],although not yet independently confirmed, is more than twice as deep as expected from standardcosmology. This necessitates either an increase in the radio background, or cooling in the IGM (orboth), which would also impact the Dark Ages signal (see, e.g. Fig. 1 in [32] for a demonstrationof models invoking extra cooling). New exotic physics have been proposed to explain thegap between the EDGES detection and standard cosmology. For example, baryon-dark matterscattering can increase the global signal and its r.m.s fluctuations at the relevant Dark Agesredshifts by over an order of magnitude [33,34]. Another option for cooling baryons is for a smallfraction of dark matter to have a mini-charge [35,36] which allows coupling with photons. Moreexotic options for explaining the stronger than expected EDGES detection include (but are notlimited to): interacting dark energy [37], axions [38,39], and neutrino decay [40]. Not all of thesepredict significant increases for the redshift ranges consider here, and thus any detection wouldhelp place constraints on these models. r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A ..................................................................
6. Conclusion and Future Work
In this paper we have considered the question of probing the small-scale 21 cm power spectrumfrom the Dark Ages, using LOFAR, which is the only currently operational telescope capableof reaching sub-arcsecond scales at the relevant low frequencies. This resolution is interestingfor probing high values of k ⊥ , which can help place new constraints on models of inflation.Specifically, it is interesting to test whether the scalar spectral index measured by Plank for k (cid:46) Mpc − holds at higher values of k or if it there are deviations from primordial non-Gaussianity, as discussed in [41–43]. This can have far-reaching implications, as an amplificationof power in the smaller angular scales (higher k ) is necessary for the formation of primordialblack holes of sufficient size to help explain the presence of super-massive black holes in galaxiesat high redshifts.We have shown that the LOFAR LBA is sensitive to k ⊥ ∼ . - . Mpc − based onthe international stations which provide its angular resolution; and k (cid:107) ∼ − . - . Mpc − based on its flexible spectral resolution. These values of k ⊥ and k (cid:107) overlap with the theoreticalexpectation of where the ‘foreground wedge’ is located, which places a lower limit on k (cid:107) of ∼ Mpc − . Using these ranges of k ⊥ and k (cid:107) , we have estimated the LOFAR sensitivity in termsof temperature brightness for four redshift slices accessible from the LBA frequency range, for 10years of integration time. We find that we can reach ∼ K at best without stacking in redshiftto improve signal to noise, in the absence of any calibration errors or incomplete foregroundisolation. This is five orders of magnitude above the expected strength of the 21 cm signal fromthe Dark Ages, and LOFAR clearly does not have sufficient collecting area.The SKA-LOW is expected to be an order of magnitude more sensitive than LOFAR, but thedesign baseline does not include the long baselines to match LOFAR’s unique spatial resolution,and it will not extend to frequencies below 50 MHz. Although LOFAR cannot achieve the desiredsensitivity, it is useful as a testbed to quantify the foreground isolation technique for high valuesof k ⊥ by testing this on real data. At the time of writing, no fully calibrated LBA dataset with theinternational baselines exists, but we expect this to change within the next year.Further work should include an assessment of improving the collecting area of a low-frequency array with long baselines. For example, when designing a lunar array, the sensitivityof the individual dipoles must be improved over current existing hardware, then their stationconfiguration must optimised to increase collecting area, and finally the array configuration mustbe taken into account to maximise the sensitivity to specific angular scales. Investigating theforeground wedge at high values of k (cid:107) , i.e., for long baselines, will be most useful for optimisingthe array configuration.Data Accessibility. This analysis was carried out using R and all scripts are available at https://github.com/lmorabit/cosmic_dawn . Authors’ Contributions.
LKM carried out the calculations and authored the manuscript. JS conceived theproject and provided guidance for relevant content.
Competing Interests.
The authors declare that they have no competing interests.
Acknowledgements.
The authors would like to thank the anonymous referees for their helpful comments.The authors acknowledge useful conversations with Ian Harrison. LKM gratefully acknowledges theinstructive and well-commented code from https://gitlab.com/radio-fisher , which is describedin [44].
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