Cristina Ballesteros
email: cristina.ballesteros@uclm.es

Joaquín Vicente

Ricardo Carrasco-García

Rafael Mateo

José Fuente

José De La Fuente

Christian Gortázar

Specificity and success of oral-bait delivery to Eurasian wild boar in Mediterranean woodland habitats

Keywords: Bait, Bovine tuberculosis, Iophenoxic acid, Vaccine, Wild boar

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Eurasian wild boar (Sus scrofa) are key wildlife reservoir hosts for diseases affecting livestock, such as classical swine fever [START_REF] Kaden | Oral immunisation of wild boar against classical swine fever in Baden-Württenberg: development of the seroprevalence based on the hunting bag[END_REF], and Aujeszky's disease [START_REF] Ruiz-Fons | Epidemiological risk factors of Aujeszky's disease in wild boar (Sus scrofa) and domestic pigs in Spain[END_REF], as well as for several zoonotic diseases including bovine tuberculosis (bTB; [START_REF] Naranjo | Evidence of the role of European wild boar as a reservoir of Mycobacterium tuberculosis complex[END_REF]. The total eradication of a disease is very difficult if a wildlife host is able to serve as a natural reservoir of the pathogen [START_REF] Gortázar | Diseases shared between wildlife and livestock: a European perspective[END_REF]). Hence, the eradication of diseases shared between livestock and wildlife may require the development of control strategies that reduce pathogen transmission between wildlife and domestic animals using vaccination or by reducing host density [START_REF] Brauer | Oral immunisation of wild boar against classical swine fever: uptake studies of new baits and investigations on the stability of lyophilised C-strain vaccine[END_REF][START_REF] Ballesteros | Recent developments in oral bait vaccines for wildlife[END_REF][START_REF] Cross | The potential of oral vaccines for disease control in wildlife species[END_REF].

Along with hunting, live-trapping, and habitat management, control of both wild boar and feral pigs may be achieved through baiting with baits containing poison [START_REF] Mcilroy | Effectiveness of a warfarin-poisoning campaign against feral pigs, Sus scrofa, in Namadgi National Park, A.C.T[END_REF][START_REF] Saunders | Evaluation of a warfarin poisoning programme for feral pigs (Sus scrofa)[END_REF][START_REF] Cowled | Efficacy of manufactured PIGOUT® baits for localised control of feral pigs in the semi-arid Queensland rangelands[END_REF][START_REF] Twigg | Bait consumption by, and 1080based control of, feral pigs in the Mediterranean climatic region of south-western Australia[END_REF] or contraceptives [START_REF] Linhart | A review of baits and bait delivery systems for free ranging carnivores and ungulates[END_REF]. Baiting with oral vaccines is also used for wildlife disease control [START_REF] Baer | The oral rabies immunization of foxes and dogs with sausage baits[END_REF][START_REF] Cross | The potential of oral vaccines for disease control in wildlife species[END_REF]). However, the effective and efficient field vaccination of wildlife species requires the development of stable and speciesspecific baits as delivery vehicles for oral vaccines [START_REF] Brauer | Oral immunisation of wild boar against classical swine fever: uptake studies of new baits and investigations on the stability of lyophilised C-strain vaccine[END_REF][START_REF] Ballesteros | Recent developments in oral bait vaccines for wildlife[END_REF]) and designing appropriate baiting strategies [START_REF] Vos | Feasibility of oral rabies vaccination campaigns of young foxes (Vulpes vulpes) against rabies in summer[END_REF]. In Spain, controlled vaccination trials are being conducted to eventually use wildlife vaccination as a disease management tool for Aujeszky's disease [START_REF] Ruiz-Fons | Epidemiological risk factors of Aujeszky's disease in wild boar (Sus scrofa) and domestic pigs in Spain[END_REF] and bTB (Ballesteros et al. 2009c). Additionally, we developed oral baits suitable to deliver pharmaceuticals to free-living wild boar, and developed techniques to improve specificity and uptake rate in overabundant wild boar populations (Ballesteros et al. 2009a). These baits could be used for the oral immunization of 2-4-month-old wild boar piglets (Ballesteros et al. 2009b).

Chemical markers are incorporated into baits to enable identification of individuals consuming baits. Substances like rhodamine B, tetracycline, and iophenoxic acid (IPA) have been used as markers in bait delivery experiments [START_REF] Fletcher | A field trial to determine the feasibility of delivering oral vaccines to wild swine[END_REF][START_REF] Farry | Efficacy of bait distributional strategies to deliver canine rabies vaccines to coyotes in Southern Texas[END_REF][START_REF] Fleming | Aerial baiting of feral pigs (Sus scrofa) for exotic disease control in semi-arid rangelands of New South Wales[END_REF]. Iophenoxic acid (α-ethyl-2-hydroxy-2,4,6-triiodebenzenepropanoic acid; hereafter Et-IPA) was clinically employed as a cholecystographic medium in 1950s due to its low systemic toxicity and low level of unpleasant side reactions [START_REF] Shapiro | A preliminary report on Teridax, a new cholecystographic medium[END_REF]). However, it was proven to cause prolonged elevation of serum iodine levels so it was withdrawn from clinical use in 1957 because of its long persistence in plasma. The plasma half-life in humans is of the order of 2.5 years [START_REF] Astwood | Occurrence in the sera of certain patients of large amounts of a newly isolated iodine compound[END_REF].

Recently, Et-IPA has been used successfully to investigate wild boar baits and baiting strategies to deliver oral vaccines, contraceptives, and toxicants [START_REF] Fletcher | A field trial to determine the feasibility of delivering oral vaccines to wild swine[END_REF][START_REF] Mitchell | The effectiveness of aerial baiting for control of feral pigs (Sus scrofa) in North Queensland[END_REF][START_REF] Fleming | Aerial baiting of feral pigs (Sus scrofa) for exotic disease control in semi-arid rangelands of New South Wales[END_REF][START_REF] Campbell | Using baits to deliver pharmaceuticals to feral swine in Southern Texas[END_REF][START_REF] Cowled | Vaccination of feral pigs (Sus scrofa) using iophenoxic acid as a simulated vaccine[END_REF][START_REF] Massei | Iophenoxic acid as a long-term marker for wild boar[END_REF]). Et-IPA can be incorporated into baits as a chemical marker because it binds to proteins in the blood plasma and elevates the protein-bound iodine in animals that consume Et-IPA-marked baits. Percentages of wild boar and feral pigs sampled in the sites where Et-IPA-marked baits have been distributed vary from 31% (in one of the three study sites of [START_REF] Fleming | Aerial baiting of feral pigs (Sus scrofa) for exotic disease control in semi-arid rangelands of New South Wales[END_REF]) to 95% [START_REF] Fletcher | A field trial to determine the feasibility of delivering oral vaccines to wild swine[END_REF]. [START_REF] Mitchell | The effectiveness of aerial baiting for control of feral pigs (Sus scrofa) in North Queensland[END_REF] found 63% of adult feral pigs eating marked baits whereas [START_REF] Campbell | Using baits to deliver pharmaceuticals to feral swine in Southern Texas[END_REF] and [START_REF] Cowled | Efficacy of manufactured PIGOUT® baits for localised control of feral pigs in the semi-arid Queensland rangelands[END_REF] reached 74% and 73% of the feral pigs sampled, respectively.

Determination of the proportion of wild boar and nontarget animals consuming baits is crucial to optimize and evaluate the cost and success of baiting campaigns. Moreover, we hypothesized that age-specific delivery could be achieved by deploying baits in special cages or under heavy stones. The aim of this study was to determine the proportion of young and adult wild boar and non-target animals consuming two kinds of marked baits (one directed to adult wild boar and another targeting wild boar up to 8 months old) in a bait delivery trial conducted at four different sites in the Mediterranean region of Spain where wild boar are abundant.

Materials and methods

Study sites

Experiments were conducted in four hunting estates in the provinces of Toledo: Quintos de Mora estate (403926 E; 4361344 N; 68.6 km 2 ; site 1); Ciudad Real: Riofrio (371401 E; 4326485 N; 32.7 km 2 ; site 2) and Rosario (378481 E; 4326095 N; 20.9 km 2 ; site 3) estates; and Guadalajara: CM estate (581715 E; 4517113 N; 11.0 km 2 ; site 4), Castilla-La Mancha, central Spain. Sites 1 and 4 are fenced, thus making them almost impermeable for ungulate immigration or emigration. Sites 2 and 3 are quite permeable so movements of animals are more likely.

The study sites were selected to be representative of the wide range of wild boar abundances that can be found in South Central Spain. A population of 400 wild boar was estimated according to manager interviews in site 1 whereas the population of wild boar estimated at sites 2, 3, and 4 was of 200 individuals for each site (Table 1). These estimates were in agreement with our own independent estimates of wild boar relative abundance based on the dropping frequency index. Briefly, each count consisted of n=40 transects of 100 m, divided into ten sectors of 10 m in length. Dropping frequency was defined as the average of the number of 10-m sectors with wild boar droppings in each transect of 100 m (DF=∑Di/n; where "D" is the number of dropping-positive sectors and ranges from zero to ten, and "n" is the number of 100-m transects, usually 40 per site; [START_REF] Acevedo | Estimation of European wild boar relative abundance and aggregation: a novel method in epidemiological risk assessment[END_REF]; see Table 1). Artificial selective feeders were evenly distributed across the sites 1, 2, and 3. Site 4 had its own feeders since artificial feeding is routinely practiced in this estate. Feeder density ranged from 4.4 to 6.7 per 10 km 2 and from 19.3 to 98.0 per wild boar [START_REF] Acevedo | Estimation of European wild boar relative abundance and aggregation: a novel method in epidemiological risk assessment[END_REF]; Table 1). Feeders were placed close to water holes (which is a limiting resource in summer in the study areas) or using already existing feeding points (site 4).

Artificial feeders consisted of a 1-cm-ø metal-grid cage with an opening (15-cm wide) to allow only access of piglets younger than 5 months (Fig. 1). A green mesh that provides shade covered the cage. Each feeder (triangular shape, side=1 m) was pre-baited with pelleted piglet feed and maize to accustom animals to visit them. There was the possibility to combine two grids per side to get an extra 30-cm-wide opening, which allowed access to wild boar up to 8 months.

Heavy pavel stones were placed outside the feeders. Pavel stone density ranged from 12.5 to 45.4 per 10 km 2 and from 73.5 to 330 per wild boar abundance index (Table 1). Each stone weighed 7 kg and measured 34×9×20 cm. The stones had a hollowed shape so feed could be placed under them. The stones were slightly buried in the ground in order to restrict access to all but adult wild boar, which could turn them and access the feed and baits placed below.

Half of the total artificial feeders in each site were monitored with digital game cameras with infrared illumination (Leaf River Outdoor Products, Taylorsville, MS, USA). Cameras were set on either native vegetation (tree trunks or branches) or artificial structures (e.g., fence posts).

Although it has been previously reported that metal-grid selective feeders improve age-related bait specificity and uptake rate in wild boar populations accustomed to being fed by managers (Ballesteros et al. 2009a, b), in the present study, we targeted <8-month-old wild boar (approximately 30-kg weight in the study area) because at the time the trial was performed, the largest proportion of the last wild boar cohort was between 4 and 8 months old. Sites 1 and 4 were pre-baited for 5 weeks, and sites 2 and 3 were pre-baited for 3 weeks. Every 3 days, we checked them to record feed consumption and pictures were examined to determine if young wild boar entered the feeders (Ballesteros et al. 2009a) and if adult ones turned the stones (unpublished observations). The fact that adult wild boar turned the stones did not mean that they consumed all the baits below them since other animals like young wild boar could gain access to the baits after adult ones turned the stones.

Baits

Baits used in the experiment had a hemispherical shape (ø 3.4×1.6 cm, 10 g). They consisted of piglet feed, Fig. 1 Percent of boars by age and other species per study site a outside and b inside the selective feeders. Animal presence was determined from examining pictures taken with digital game cameras with infrared illumination on each site. In addition, red fox inside selective feeders at sites 1 and 3 and adult wild boar (sites 1 and 3), red deer (site 1), and red fox (sites 1 and 3) outside selective feeders represented 1% of the animals detected per site and were not included in graph. Other ungulates in the study sited included fallow deer, roe deer, and mouflon paraffin, sucrose, and cinnamon-truffle powder attractant (Ballesteros et al. 2009b). A 0.2-mL polyethylene capsule filled with water was placed inside the baits to simulate the vaccine formulation.

IPA was diluted in ethanol to a concentration of 80 mg/ml. Then, 0.5 ml of the dilution was added to the surface of the recently prepared bait. The final concentration of IPA was 40 mg/bait.

Baiting field trial

The field trial was conducted in summer 2008 (first baiting started on 4 August, last baiting on 30 September). Baits were set in two different places: baits directed to wild boar piglets were set inside selective feeders while baits directed to adult wild boar were set under pavel stones around the feeders (<10-m apart). The number of baits laid at each site was chosen based on the number of wild boar estimated by the managers and our independent abundance indexes, to result in a broad range of baiting intensities (Tables 1 and2). According with the size of each site our baiting densities varied from 6.41 at site 1 to 27.3 per km 2 at site 4 (in the case of the baits directed to wild boar piglets) whereas the baiting densities of baits directed to adult ones varied from 2.51 at site 1 to 9.09 per km 2 at site 4. Wild boar abundance indexes were estimated for each site as explained above [START_REF] Acevedo | Estimation of European wild boar relative abundance and aggregation: a novel method in epidemiological risk assessment[END_REF]). Indexes of abundance of 0.17, 0.05, 0.02, and 0.28 were obtained for sites 1, 2, 3, and 4, respectively.

Two different markers were used to distinguish bait consumption inside or outside piglet feeders. Baits placed inside selective feeders contained ethyl-iophenoxic acid (Et-IPA) whereas baits placed under pavel stones contained the methyl analog of Et-IPA (Mt-IPA).

Two cycles of bait deployment were conducted at each site. In the first cycle, we placed 20 Et-IPA baits inside each selective feeder; whereas, three or four pavel stones (according to the density of wild boar) were placed next to each selective feeder, except at site 4 where due to the high density of wild boar we placed 50 Et-IPA baits inside each feeder and eight to nine stones per feeder. Two Mt-IPA baits were placed under each stone. Baits were delivered together with piglet feed in order to attract the animals and favor bait consumption. Four days after baiting we revisited the selective feeders and pavel stones to record complete or incomplete bait consumption and the number of intact and chewed capsules found. All digital cameras were checked and pictures downloaded for further analysis. Three days after the first cycle examination finished, the procedure was repeated (second cycle of bait deployment).

Animal samples from the baiting field trial Hunter-harvested wild boar and non-target animals (red deer (Cervus elaphus) and mouflon (Ovis aries musimon)) from the sites in which baits were placed were blood sampled. Also, wild boar from an estate next to site 1 were blood sampled. In this estate, no marked baits were delivered. The objective was to check whether emigration could happen between closed estates.

Animals were legally hunted from October 2008 to February 2009. Blood samples were obtained post-mortem from the thoracic cavity. These were centrifuged to separate serum. Sera were placed in labeled tubes and stored at -20°C until IPA detection by LC/ESI-MS as described by [START_REF] Jones | High-performance liquid chromatographic determination of iophenoxic acid in serum[END_REF] and [START_REF] Wiles | High-performance liquid chromatographic determination of iophenoxic acid in serum[END_REF] with some modifications. Briefly, wild boar serum (0.1 ml) was mixed with 0.63 ml of acetonitrile and 20 μl of buthyl-IPA as internal standard (10 ng/μl in acetonitrile). The mixture was vortexed during 5-10 s, and then 0.2 ml of a sodium tungstate dihydrate solution (10%, w/v in water) and 0.2 ml of sulphuric acid 0.33 M were added. The resultant mixture was vortexed during 10 min at a high speed (2,500 rpm) and cooled for 20 min at -20°C until two phases were separated. Then it was centrifuged at 5,000×g at 4°C for 10 min and 0.6 ml of the upper acetonitrile phase was transferred to a glass vial for LC/MS analysis [START_REF] Ballesteros | Analysis by LC/ESI-MS of iophenoxic acid derivatives and evaluation as markers of oral baits to deliver pharmaceuticals to wildlife[END_REF].

Statistical analysis

Contact rates were compared using χ 2 tests. We designed logistic models to test the statistical effect of sex (categorical predictor) and age class (categorical predictor, young vs. adult) on the probability of presence of any marker (Et-IPA and Mt-IPA as dependent variables, respectively; absence=0, presence=1) and its combinations (combination of Et-IPA and Mt-IPA and of Et-IPA or Mt-IPA as dependent variables, respectively; absence=0, presence=1). These models were separately built for study sites 1, 2, and 3, since sampling size and composition did not allow for comparison in site 4. The significant level was set at p<0.05. We used SPSS 17.0 statistical software.

Results

Bait consumption and targeted species

After examinations of feeders and pavel stones we recorded the number of baits that had disappeared (consumed) and the number of capsules found as well as their state (chewed or intact). Intact capsules had water inside them. The number of baits (both Et-IPA and Mt-IPA treated baits) placed and the number of baits consumed at each site is summarized in Table 2. Two sites were found where the proportion of consumed baits was high both inside feeders and under pavel stones (>90%, sites 1 and 3). Site 2 presented lower proportions of consumed baits both inside the feeder and under pavel stones. Finally, site 4 presented moderate rates of bait consumption inside the feeders (averaging 74.3%) and a total success when baits were placed under pavel stones (100%). The percentages of consumption were similar when comparing the first and the second cycle trials for a given site and way of bait disposal (Table 2).

The number of intact and chewed capsules found at each site is recorded in Table 3. It was likely that capsules found inside selective feeders would correspond to ethyl-IPA-marked baits (baits targeted to wild boar piglets). Except for capsules found inside feeders in site 4, over 70% of the capsules were chewed, being remarkable that over 90% of them (both inside and outside feeders) were chewed in site 1 (Table 3).

The relative frequency of selective feeder visits by different animal species was determined at each site (Fig. 1). Inside selective feeders, young wild boar (<8 months of age, as established from picture examination and maximum grid width of 30 cm) were detected more frequently than adult (over 8 months old) wild boar (χ 2 =2,394, χ 2 =630, χ 2 =3,338, and χ 2 =156 in sites 1 to 4, respectively; p<0.05). Adult wild boar (only detected in sites 1 and 3) and red deer (only found in site 1) detected inside the feeders represented <1% of the visits per site. This finding was due to accidental circumstances when the shadowing cover or the metal grid was damaged. Birds and other animal species such as the Eurasian badger (Meles meles) were detected and were more abundant in sites 1 and 4, respectively. At site 4, young wild boar were detected inside three of six feeders, while badgers were detected in all six feeders. Timing of feeder visits by both species was similar. Wild boar detection declined while badger detection increased between the start and the end of the bait deployment trials (data not shown).

Outside feeders, a wider range of species such as wild boar, red deer, small-sized birds, red fox (Vulpes vulpes), roe deer (Capreolus capreolus), fallow deer (Dama dama), mouflon, and badger were detected (Fig. 1). In sites 1 and 3, wild boar (both adult and young) were detected more frequently (p<0.05) when compared to red deer, red fox, and roe deer. In site 2, no significant differences were found between the detection frequency of wild boar and red deer (χ 2 =2.88, p=0.09). In site 4, the presence of wild boar was significantly lower than that of red deer (χ 2 =1,131, p<0.05) whereas it was significantly higher than that of red fox, mouflon, roe deer, and badger (χ 2 =3,830, χ 2 =744, χ 2 =2,246 and χ 2 =3,827, respectively; p<0.05). Adult and young wild boar were found in similar frequencies outside selective feeders in sites 1 and 2, but young wild boar were significantly more frequent in sites 3 and 4 (χ 2 =1971.0 and χ 2 =17.74, respectively; p<0.05).

Overall, artificial stones were turned in 115 cases. In 71.3% of the cases, adult wild boar turned the stones, 24.4% were turned by young wild boar (all of them close to 6 months of age), and only 4.3% were turned by red deer. Piglets and juveniles were often pictured roaming close to adult females that turned the stones.

Presence of methyl-and ethyl-IPA in animal serum

The presence of methyl-and ethyl-IPA was determined in the serum of sampled wild boar (Fig. 2). A total of 39, 34, 26, and 17 wild boar were sampled at sites 1, 2, 3, and 4, respectively. The number and age class of animals sampled depended on the number of animals hunted at each estate so only low percentages of the total population could be sampled (between 8.5% and 17.0%). The proportion of wild boar showing methyl-or/and ethyl-IPA markers in serum ranged from 11.5% in site 3 to 56.4% in site 1. When attending to age classes, 72.7% of young individuals (estimated to be <8 months when baits were disposed) from study site 1 presented any or both markers, and this rate decreased in sites 4 (40.0%), 2 (31.6%), and 3 (12.6%). It is remarkable that the predominant marker in young wild boar from site 1 was Et-IPA whereas in site 4 it was Mt-IPA. For adults, a similar trend was found with sites 1 and 4 presenting the highest proportion of wild boar showing methyl-or/and ethyl-IPA markers in serum (up to 40.0%). Again, Mt-IPA predominated in the study site 4. Agerelated differences in the presence of serum markers were found in site 2 only where the proportion of adult wild boar with serum markers was less than that of young animals (χ 2 =4.24; p=0.04). However, wild boar of all ages and sexes appeared equally likely to consume baits containing IPA since logistic models for sites 1, 2, and 3 did not reveal sex or age effects on the probability of presence of any marker (and combinations).

A total of 43 non-target animals were sampled at the four sites to determine species-specific consumption of baits. Twenty-six red deer from site 1 were analyzed, two of them showing low concentrations of Mt-IPA in serum (<0.1 ng/μl). Five and ten red deer were sampled at sites 2 and 3, respectively and none of them showed methyl-or ethyl-IPA in serum. Finally, two mouflons were sampled at site 4 and did not show any marker in serum. Thirty-five wild boar were sampled from an estate next to site 1 where marked baits were not delivered. None of them showed any marker in its serum.

Discussion

The proportion of free-living wild boar consuming two types of marked baits directed to adult and juvenile animals was determined. Variable oral-bait delivery success to wild boar was obtained among the four study sites where the baiting trials were conducted, ranging from a minimum of 12.6% to a promising 72.7% of juveniles having eaten marked baits, most of them (Fig. 2a) directed to wild boar piglets inside selective feeders. The examinations conducted at each site after bait deployment showed that the proportion of consumed baits in two sites was high (>90%) both inside feeders and under the heavy pavel stones (sites 1 and 3; Table 2). However, in site 4 only moderate rates of bait consumption were obtained inside the feeders (Table 2). In this site, badgers predominated inside feeders, probably preventing young wild boar from getting inside feeders. This observation merits further research, since our results suggest that feeding wild boar may actually benefit badgers, with potential consequences regarding TB epidemiology in multi-host systems. Inside selective feeders, young wild boar were more frequently detected than adult wild boar. This result confirms the efficacy of selective feeders to deliver baits to young wild boar (Ballesteros et al. 2009a). Birds, highly abundant inside feeders in site 1, have been demonstrated to have no effects on the integrity and persistence of baits (Ballesteros et al. 2009a), which was also confirmed in the present study. Outside feeders, a wider range of species was detected, being the wild boar (both adult and young) and red deer the most frequently found in accordance with the distribution and abundance of both species (Fig. 1; [START_REF] Acevedo | Relationship between bronchopulmonary nematode larvae and relative abundances of Spanish ibex (Capra pyrenaica hispanica) from Castilla-La Mancha, Spain[END_REF][START_REF] Acevedo | Estimation of European wild boar relative abundance and aggregation: a novel method in epidemiological risk assessment[END_REF]. For instance, red deer locally presented even higher presence figures than wild boar, which may relate to higher densities of deer in the study areas [START_REF] Acevedo | Estimating red deer abundance in a wide range of management situations in Mediterranean habitats[END_REF].

Pavel stones were tested as a bait delivery method to adult wild boar. In most cases, stones were turned by adult wild boar and less than 5% were due to non-target species such as red deer. In these cases, we observed that stones probably had not been appropriately buried in the ground due to the hard soil. In fact, in these cases animals displaced rather than turned the stones. Therefore, this method may be optimized to selectively deliver baits to adult wild boar.

From camera pictures it was not possible to visualize the individual baits since cameras only provide a punctual image, which do not always capture the moment when the animal takes the bait. So, we could not assess which animals consumed them. This information was only used to characterize the activity around the bait release points.

We observed in previous experiments that wild boar can get immunized by having oral mucosal contact with vaccines (Ballesteros et al. 2009b, c). So, it is not necessary that animals swallow the capsules to produce an immune response. A high proportion of chewed capsules were found (Table 3), thus confirming previous results on bait performance as a vehicle for liquid-based formulations targeting the oral mucosa (Ballesteros et al. 2009a, b). Nonetheless, only a low proportion of capsules found inside feeders in site 4 were chewed. This result was probably due to the predominant use of the selective feeders at this site by badgers rather than by young wild boar. It is almost impossible to avoid badgers entering the feeders. However, apparently, badgers do not chew these capsules as efficiently as wild boar do (Ballesteros et al. 2009a) so it is unlikely that badgers could have contact with the vaccine. These results confirmed the selectivity of the delivery methods for wild boar, although badgers could interfere locally with the wild boar bait uptake rate. However, in Spain, badgers present usually low densities in areas where bTB has become a problem that involves wild boar as the main reservoir [START_REF] Naranjo | Evidence of the role of European wild boar as a reservoir of Mycobacterium tuberculosis complex[END_REF][START_REF] Sobrino | Bovine tuberculosis in a badger (Meles meles) in Spain[END_REF].

Biomarker detection rate estimations were based on low sample sizes, given by the limited number of wild boar harvested in the four study sites (<10% of the estimated total population in all cases). Thus, results obtained in this study need to be considered with caution. A 72.7% of young wild boar in study site 1 presented IPA marker in their serum, being Et-IPA (disposed in baits inside feeders) the predominant marker whereas 40% of young wild boar in site 4 presented only Mt-IPA (disposed in baits under stones; see Fig. 2). Similar findings were found for adult wild boar in sites 1 and 4, although with lower proportions of animals with serum markers. These results could be explained by the fact that young wild boar can forage around feeders together with adult females, gaining access to the baits placed below the stones after they are turned by adults. Conversely, some adult wild boar could gain access to baits placed inside feeders. A few cases of adult wild boar introducing their heads inside the metal grid were observed in pictures taken at site 1. So, these animals could consume some of the baits directed to young wild boar. In addition, young wild boar could move some baits or bait fragments outside the feeder, a detail that is difficult to evidence with camera trapping. These facts indicate that selective feeders would need to be improved to avoid adult wild boar accessing the baits placed inside them, for example increasing their side size from 1 to 2 m or changing their triangular shape to a square one.

The low concentrations of IPA found in two red deer can be due to contact with remains of bait but not to whole bait consumption.

Bait delivery studies have used concentrations of IPA between 20 and 40 mg/bait [START_REF] Fletcher | A field trial to determine the feasibility of delivering oral vaccines to wild swine[END_REF][START_REF] Mitchell | The effectiveness of aerial baiting for control of feral pigs (Sus scrofa) in North Queensland[END_REF][START_REF] Fleming | Aerial baiting of feral pigs (Sus scrofa) for exotic disease control in semi-arid rangelands of New South Wales[END_REF][START_REF] Campbell | Using baits to deliver pharmaceuticals to feral swine in Southern Texas[END_REF]. Twenty milligrams of IPA per bait may be insufficient to detect bait consumption in adult animals or when they consume one bait only [START_REF] Cowled | Vaccination of feral pigs (Sus scrofa) using iophenoxic acid as a simulated vaccine[END_REF]. Therefore, higher amounts of IPA (40 mg/bait) as used herein may give more reliable estimations of bait uptake [START_REF] Cowled | Vaccination of feral pigs (Sus scrofa) using iophenoxic acid as a simulated vaccine[END_REF]. The proportion of wild boar (young or adult) showing any marker in the serum or the combination of both strongly varied among study sites. The percentages of wild boar or feral pigs showing serum markers in previous bait delivery experiments varied from 31% to 95% depending on the study site, density of target animals and baiting density (baits per square kilometer; [START_REF] Fletcher | A field trial to determine the feasibility of delivering oral vaccines to wild swine[END_REF][START_REF] Mitchell | The effectiveness of aerial baiting for control of feral pigs (Sus scrofa) in North Queensland[END_REF][START_REF] Fleming | Aerial baiting of feral pigs (Sus scrofa) for exotic disease control in semi-arid rangelands of New South Wales[END_REF][START_REF] Campbell | Using baits to deliver pharmaceuticals to feral swine in Southern Texas[END_REF]). In the present study, baiting densities varied from 2.51 Mt-IPA-marked baits/km 2 in site 1 to 27.3 Et-IPA-marked baits/km 2 in site 4. These baiting densities were lower than those used in previous studies in other countries (e.g., 68 baits/km 2 in [START_REF] Campbell | Using baits to deliver pharmaceuticals to feral swine in Southern Texas[END_REF]), where feral pig densities are lower than wild boar densities in Mediterranean Spain [START_REF] Naranjo | Evidence of the role of European wild boar as a reservoir of Mycobacterium tuberculosis complex[END_REF]. Therefore, in future experiments it would be desirable to use higher baiting densities to target a higher percentage of the wild boar population.

Additionally, other factors that could affect bait consumption such as the density of feeding points, the wild boar abundance in relation to the density of feeding points, the number of disposed baits at each feeder, the interference/competence with other bait-consuming species, and the population variations in bait acceptance and habituation to feeders should be considered in future experiments. In fact, in both sites where the pre-baiting period lasted longer (≥5 weeks), the results of bait consumption by wild boar were better. This may indicate that wild boar were more accustomed to use feeders in these sites. [START_REF] O'brien | Factors influencing the intake of sodium monofluoroacetate (compound 1080) by free-ranging feral pigs[END_REF] and [START_REF] Saunders | Evaluation of a warfarin poisoning programme for feral pigs (Sus scrofa)[END_REF] emphasized the need for a sufficient pre-feeding regime prior to a trapping or poisoning program.

Finally, the spatial-temporal population dynamics of wild boar in the period between bait delivery and sampling, that is, immigration-emigration, could introduce some bias in the access to the population actually present at delivery time. Nonetheless the only estate where hunting occurred during the study period, potentially causing increased wild boar movements [START_REF] Keuling | How does hunting influence activity and spatial usage in wild boar Sus scrofa L.?[END_REF], was site 1, where the highest marker presence was detected. The regular hunting season started at least 1.5 months after bait deployment in the remaining study sites, and therefore we do not suspect much movement due to these circumstances. This was further supported by analyzing samples from wild boar captured in an estate contiguous to site 1, finding no IPA-marked individuals. Site 4 was also impermeable for any ungulate immigration. Therefore, wild boar populations in sites 1 and 4 were highly fixed spatially. In contrast, the fences in sites 2 and 3 were quite permeable, suggesting that wild boar immigration-emigration movements were more likely, limiting apparent bait deployment success (due to immigration) or maker detectability (due to emigration).

In conclusion, the results obtained in these experiments showed that bait delivery systems used for young and adult wild boar were effective in targeting the desired species but not as effective in targeting the desired age groups since in a few cases adults got baits placed inside the selective feeders and young wild boar got baits placed under pavel stones after adult ones turned them. Variable results were obtained in the percentage of animals that ingested the baits (Table 2, Fig. 2). Future experiments should test higher baiting densities and longer pre-baiting periods to increase baiting success and thus the efficacy of wild boar baiting strategies as a population or disease management tool in Spanish Mediterranean woodlands.

Fig. 2

 2 Fig. 2 Numbers of wild boar sampled at each study site and percentages of them showing only Et-IPA, only Mt-IPA, or both markers in serum according to the age (a refers to young wild boar whereas b refers to adult animals). The boxes on the top of the bars indicate the percentage of wild boar that showed any marker in serum (Et-IPA, Mt-IPA, or both)

Table 1

 1 Estimated population size and density of feeders and pavel stones used to deploy the baits at each site

	Site	Area (km 2 )	Pre-baiting	No. of wild boar	Abundance	No. and density of feeders	No. and density of pavel stones
			period		index
	1	68.6	5 weeks	400	0.17	30	86
						4.4 a /25.9 b	12.5 a /73.5 b
	2	32.7	3 weeks	200	0.05	16	54
						4.9 a /98.0 b	16.5 a /330 b
	3	20.9	3 weeks	200	0.20	14	48
						6.7 a /33.5 b	23.0 a /115 b
	4	11.0	>5 weeks	200	0.28	6	50
						5.4 a /19.3 b	45.4 a /162 b
	a Feeders or pavel stone density per 10 km 2
	b Feeders or pavel stone density per wild boar, calculated as [feeders or pavel stone density per 10 km 2 ]÷[wild boar abundance index]

Table 2

 2 Number of baits placed and consumed (consumption percentage) for each cycle

		No. Et-IPA baits placed	No. Et-IPA baits consumed	No. Mt-IPA baits placed	No. Mt-IPA baits consumed
	Site a	Cycle 1	Cycle 2	Cycle 1	Cycle 2	Cycle 1	Cycle 2	Cycle 1	Cycle 2
	1	440 b	540	400 (90.9%)	486 (90.0%)	172 b	204	158 (91.2%)	193 (94.6%)
		6.41 c /37.7 d	7.87 c /46.3 d			2.51 c /14.7 d	2.97 c /17.5 d
	2	320	320	174 (54.4%)	116 (36.2%)	108	108	78 (72.2%)	59 (54.6%)
		9.79/195.8	9.79/195.8			3.30/66.1	3.30/66.1
	3	280	280	268 (95.7%)	280 (100%)	96	96	83 (86.4%)	96 (100%)
		13.4/67	13.4/67			4.6/23.0	4.6/23.0
	4	300	300	223 (74.3%)	207 (69.0%)	100	100	100 (100%)	100 (100%)
		27.3/97.5	27.3/97.5			9.09/32.5	9.09/32.5

a Twenty and 50 baits were disposed in sites 1-3 and 4, respectively for each cycle b In the first cycle, five selective feeders were not visited by animals during the pre-baiting period. Therefore, no baits were placed inside them c Bait density per square kilometer d Bait density per wild boar, calculated as [baits density per square kilometer]÷[wild boar abundance index]

Eur J Wildl Res (2011) 57:749-757

Acknowledgements

We thank many colleagues from IREC for their technical assistance. Caterina Falconi, Luca Carrel, Jesús Carrasco, and Mauricio Durán helped in the field. Pablo R. Camarero helped in the IPA determination. We thank Bernardo Herrera for his help during field work. We also thank the managers of Quintos de Mora, Rosario, and Riofrio public estates for allowing access to the study areas, and the guards for helping during field work. This study was supported by grants from Instituto Nacional de Investigación y Tecnología Agraria y Alimentaria (INIA) (Project FAU 2006-00017-C03-01), Consejería de Educación y Ciencia, Junta de Comunidades de Castilla-La Mancha (JCCM) (projects PAI 06-0046-5285 and 07-0062-6611), EU FP7 Grant TB-STEP 212414; the Grupo Santander and the Fundación Marcelino Botín, Spain.


F Roth

M Huth

High-κ field-effect transistor with copper-phthalocyanine

Keywords: numbers: 85.30.Tv, 77.55.-g, 81.05.Fb

The use of SrTiO 3 dielectrics as high-permittivity insulator in organic thin film field effect transistors (FET) is evaluated. Field-effect transistors with sputtered SrTiO 3 and copper-phthalocyanine (CuPc) as semiconducting layer were fabricated. The device preparation was performed in-situ in an ultra high vacuum chamber system. The dielectric in the transistors had a permittivity of up to 200 which led to low driving voltages of 3 V. The field effect transistors were p-type and reached mobilities of about µ = 1.5 • 10 -3 cm 2 /Vs and an on/off ratio of 10 3 . These properties are compared to devices based on other dielectric materials.

Introduction

Over the past 15 years remarkable progress has been made in the development of thin film transistors (TFTs) based on organic semiconductors. In several key figures of merit, such as e. g. the on-to-off current ratio, the field effect mobility, and the threshold voltage, the best organic TFTs now can compete with commercial amorphous silicon TFTs. The gate dielectric plays a crucial role in this regard. It is responsible for the reliability of the device, governs the required driving voltages and also limits the polarizations which can be achieved at the interface. In organic thin film transistors the most common dielectric is SiO 2 . It is well known and characterized from its use in the conventional semiconductor industry, and has the advantages of chemical stability and excellent insulation with low leakage currents. The required silicon wavers with an oxide dielectric layer are commercially available and the source and drain electrodes can be easily fabricated by lithography or by deposition through a shadow mask. Various groups have in this way prepared transistors with different organic layers such as pentacene or copper-phthalocyanine [1,2]. This approach allows for concentrating on the growth of the active organic layers, but limits the experiments to the regime of low charge carrier concentrations. Many organic materials show electronic interactions which are strongly influenced by the charge carrier concentration. Examples are charge transfer salts based on organic molecules such as tetrathiofulvalene-tetracyanoquinodimethane (TTF)-(TCNQ) [3,4,5] or K + TCNQ - [6] which show the behavior of a Peierls or Mott insulator [START_REF] Mott | Metal-Insulator Transitions[END_REF]. Large changes in the charge carrier concentration may even induce a Mott metal-insulator transition, allowing for a so called Mott transition field-effect transistor (MTFET) [6,[START_REF] Newns | [END_REF]. It is therefore interesting to grow these materials in transistor structures where high polarizations can be reached. But also for other organic semiconductors, where no Mott metal-insulator transition occurs, the utilization of highκ dielectrics is interesting, because the high polarizations result in transistors with low driving voltages. The charge carrier concentration at the insulator surface which can be induced in fieldeffect experiments is the product of breakdown field and dielectric constant evaluated at the breakdown field [9]. It therefore depends on the intrinsic properties of the material. With SiO 2 as dielectric a breakdown field of E BD = 10 MV cm -1 and polarizations of up to 3 µC cm -2 are possible at best, while with complex oxide dielectrics with a perovskite structure polarizations in the range from 10 -40 µC cm -2 can be achieved under carefully optimized conditions [10].

Materials

From the perovskite class SrTiO 3 is one of the best characterized materials. It has a cubic crystal structure with a lattice constant of a = 3.905 Å and the advantage of a high dielectric constant of r = 300 for bulk material at room temperature. Thin films of SrTiO 3 have been grown with various techniques, such as RF sputtering [11], pulsed laser deposition [12] or molecular beam epitaxy [13]. The dielectric constant and breakdown voltage in thin film samples is in general lower than in bulk samples most likely due to defects and inhomogeneities of the electric field caused by surface roughness [11]. Furthermore, SrTiO 3 can be doped with niobium, which results in a conductive crystalline substrate material, ideally suited for the growth of an epitaxial dielectric SrTiO 3 layer [14]. In this work, to demonstrate the capability of SrTiO 3 as high-κ dielectric for organic thin film field-effect transistors, the well-known organic p-type semiconductor copper phthalocyanine was chosen as the active layer. CuPc (CuN 8 C 32 H 16 ) is a crystalline synthetic blue pigment from the group of phthalocyanine dyes. It has been used by various groups in field-effect experiments, usually with SiO 2 as the dielectric when deposited as thin films, but also already with high-permittivity insulators [1,15].

Device preparation

An ultra high vacuum (UHV) system was employed for the preparation of the devices. It consisted of distinct chambers used for the individual process steps. The growth of the dielectric layer, active layer and the contact preparation was studied separately before they were combined to fabricate field effect transistors completely in-situ.

SrTiO 3 sputtering

For the growth of the SrTiO 3 layers RF-sputtering was used. The growth of sputtered SrTiO 3 was characterized with X-ray diffraction employing Cu-K α radiation in parallel beam mode and optical microscopy and optimized with respect to high crystallinity, a smooth surface and a sufficient growth rate. The sputtering target had a diameter of two inches and was at a distance of about 15 cm from the substrate.

For the final FET structure preparation SrTiO 3 layers had to be grown on Nb-doped SrTiO 3 (100) substrates. Since by X-ray diffraction thin film and substrate related Bragg reflexions can in this case hardly be discriminated, the dielectric layer growth was optimized on MgO (100) substrates. The thus obtained sputtering parameters were then used for FET preparation on Nb-doped SrTiO 3 (100) employing an argon/oxygen mixture of 2:1 at a chamber pressure of 0.025 mbar, a substrate temperature of 720 • C and a forward sputter power of 100 W. The films were post annealed inside the chamber at 890 • C for 1 hour in an oxygen atmosphere of 0.01 mbar. The deposition time was 90 minutes for field-effect transistor preparation which resulted in a nominal thickness of about 450 nm. During the optimization on MgO the SrTiO 3 films grew preferentially in the (100) direction. The Bragg peaks of the sputtered films were shifted to lower 2θ values as compared to bulk SrTiO 3 . This was probably related to the larger lattice constant of MgO (a = 4.216 Å) as compared to SrTiO 3 (a = 3.905 Å) which caused a tensile stress due to the misfit [16]. Post annealing further improved the crystalline quality of the SrTiO 3 films. This was demonstrated ex-situ in an oven at 800 24 h, so that the sample properties could be compared before and after annealing. The SrTiO 3 films showed an increase in the height of the Bragg peak and a shift in position in direction of bulk SrTiO 3 , as is exemplarily shown in Fig. 1.

CuPc growth

The organic semiconductor as active layer was prepared by organic molecular beam deposition in an UHV chamber with base pressure of 10 -10 mbar. The source material was commercial CuPc and it was used without further purification. The deposition rates were kept low, between 0.4 and 0.02 nm/ min. CuPc growth was studied at different substrate temperatures on α-Al 2 O 3 (100) and SrTiO 3 (100). No difference in growth behavior between these different substrates could be observed. In contrast to this the substrate temperature had a strong impact on the growth, as observed by X-ray diffraction. The temperature influence was studied in the range from room temperature to 140 • C. The films changed with increasing substrate temperature from smooth, with small angle oscillations and only a small Bragg peak to higher crystallinity, with a larger Bragg peak and a rougher surface with few or no small angle oscillations. A selection of Bragg scans is shown in Fig. 2 for films grown on α-Al 2 O 3 (100). The Bragg peaks for all films and substrates were detected at an angle of 2θ = 6.8 • which corresponds to a distance between molecular planes of 1.3 nm. The Bragg peak is in agreement with the (200) plane of the α-phase of CuPc. The α-phase is frequently reported for vacuum deposition of CuPc at moderate substrate temperatures [17], while the β-phase can be found at high substrate temperatures or deposition rates [18]. In the α-phase the molecules are oriented close to edge-on with respect to the substrate. For the field-effect transistors prepared in this study an intermediate substrate temperature of 85 chosen. This temperature was selected to balance the somewhat conflicting requirements of high crystallinity and layer smoothness. Field-effect transistors in thin film geometry with CuPc as active layer and SrTiO 3 as dielectric were prepared in-situ on (100) Nb-doped SrTiO 3 substrates in bottom gate geometry. The source and drain electrodes were made of a 30 nm gold layer by evaporation through shadow masks. Transistors were prepared in bottom contact (contact preparation before active layer) and top contact (contact preparation after active layer) geometry as sketched in Fig. 3. The channel length was 20 µm and the channel width was 1 mm. The shadow masks had a four-segment arrangement, so that there were always four (identical) transistors prepared, one in each sector of the substrate.

Results

When measured, the four FETs on one substrate showed comparable results. The dielectric layer was characterized and I -V and transfer characteristics were measured ex-situ at room temperature.

The capacitance was routinely measured with an LCR meter at a test frequency of 100 Hz. To further characterize the dielectric, the capacitance was also measured with frequencies up to 100 kHz. Up to 10 kHz the dielectric frequency response was almost constant with a loss of about 1%. At 100 kHz the measured capacitance dropped by about 5% as compared to the low frequency value. The dielectric constant of the dielectric layer was calculated from the capacitance of the electrodes. The values obtained for κ from the transistors with SrTiO 3 as the dielectric layer were typically around 180 to 200 and the dielectric could usually withstand voltages of about 3 V which corresponds to a breakdown voltage of about 700 kV cm -1 and a polarization of 1.3 µC cm -2 . To calculate mobilities from the FET characteristics the capacitance per unit area was required, which was calculated from the measured capacitance and the area of the electrodes to about 413 nF cm -2 . All CuPc transistors showed p-type semiconducting behavior. For all transistors only low voltages were necessary to achieve a pronounced modulation of the source drain current. The response characteristics of the transistors were measured multiple times with increasing voltages. The voltages were ramped up and down slowly and often a hysteresis was observed. The top contact transistors showed a better performance (i.e. higher mobility), possibly due to an annealing effect of the active layer during contact preparation, i. e. Au evaporation through a shadow mask. The response curve of a top contact FET is shown in Fig. 4. The transfer curves were measured for different fixed source drain voltages. The gate voltage was ramped up and down and also a hysteretic behavior was observed. A hysteresis behavior in organic FETs is not uncommon and usually related to trapped charges [19].

For positive gate voltages the conductivity was suppressed and the drain current decreased below the zero gate voltage current, which can be interpreted as a depletion behavior. Saturation was hardly obtained for drain currents exceeding the gate voltage. This indicates a negative build in zero (threshold) voltage which biased the transistor. This FET had an on/off ratio of about 10 3 , which is comparable to literature results

-3 -2.5 -2 -1.5 -1 -0.5 0 U DS [V] -3e-07 -2.5e-07 -2e-07 -1.5e-07 -1e-07 -5e-08 0 I DS [A] V G =-3V -2V -1V 0V 3V 2V 1V Figure 4.
Current voltage characteristic of a CuPc FET with SrTiO 3 as dielectric layer. The "on" current at -3 V source drain and -3 V gate voltage was 2.4 • 10 -7 A.

The "off" current at -3 V source drain voltage and 0 V gate voltage was 2.5 • 10 -8 A and was further suppressed to 3.5 • 10 -10 A with a gate voltage of 3 V. The on/off ratio was about 10 3 .

[15]. The mobility of the transistors can be obtained in different ways. From the current voltage curve the mobility can be calculated in the saturation regime by [20,21]

µ = ∂ √ I D ∂V G 2 • 2L C I W , (1)
where L is the channel length, W is the channel width, C I is the capacitance per unit area of the gate insulator, I D is the drain current and V G is the gate voltage. In the linear regime the mobility is obtained from the transfer curve by [20,21]

µ = ∂I D ∂V G • L W C I V D (2)
or from the slope of a plot, where the slopes of the response curves in the linear regime at constant gate voltage are plotted over gate voltage by [20,21]

µ = ∂I D ∂V D • L W C I (V G -V T ) , (3)
where V D is the drain and V T is the threshold voltage. The mobility obtained in the linear regime was derived from transfer curves to µ = 1.8 • 10 -3 cm 2 /Vs. Additionally, the mobility was determined from the slopes of the response curves for small drain voltages. A plot of the derived slopes is presented in Fig. 5. From a fit to theses values a mobility of µ = 1.5 • 10 -3 cm 2 /Vs and a threshold voltage of V T = 0.27 V was calculated.

The mobility was also extracted from the saturation current as depicted in Fig. 6. The currents at V D = -3 V were obtained from the response curves from Fig. 4. The square roots of these currents were then plotted over the gate voltage and from the slope the mobility was calculated to µ = 1.1 • 10 -3 cm 2 /Vs and the threshold voltage to V T = 0.98 V. The plot showed the typical increase in slope towards higher gate voltages [21].

Conclusions

In summary, FETs with SrTiO 3 as dielectric and CuPc as semiconductor were prepared in-situ. The values derived for the mobility in the linear and the saturation regime were in good agreement and amounted to µ = 1.5 • 10 -3 cm 2 /Vs. These mobilities were obtained without optimization of the CuPc layer with respect to transport properties. In the literature the highest mobilities of about µ = 1.5 to 2.0 • 10 -2 cm 2 /Vs were reported for a substrate temperature of 125 insulator. Both groups report a dramatic field effect mobility increase with substrate temperature, in conjunction with higher crystalline order. At even higher substrate temperatures, the field effect mobilities decreased again, which was attributed to gaps between large non-space-filling crystals. Our field-effect mobility is well in agreement with the mobilities obtained by Bao et al. on SiO 2 at a substrate temperature of 85 • C. Further evidence that our lower mobilities can be attributed to the growth of the CuPc and not, for instance, to the roughness of the dielectric is the low SrTiO 3 surface roughness as evident from the large number of Kiessig fringes observed in the thickness measurements of the SrTiO 3 films (comp. Fig. 1). Furthermore, Okuda et al. [15] reported very similar mobilities for devices prepared on SiO 2 and PZT although the PZT films had a 20 times larger surface roughness as compared to the SiO 2 layer. We therefore conclude, that mobilities comparable to other substrates can be reached on SrTiO 3 by optimizing the growth of the CuPc layer. The on/off ratio in our devices (10 3 ) is comparable to the PZT devices, but lower than the best SiO 2 FETs (10 5 ). The driving voltages (3 V) are also comparable to the PZT devices (2 V) and much lower than the SiO 2 FETs (100 V). These results show that organic transistors with low driving voltages can be realized with a SrTiO 3 dielectric. Further optimization of the SrTiO 3 insulator is required to fully utilize its potential to achieve high polarizations and enable organic MTFETs, as well as basic research on the effects of large induced interface charges in organic charge transfer materials [22,23].

Figure 3 .

 3 Figure 3. Schematic drawing of a top contact (a) and bottom contact (b) thin film transistor with bottom gate geometry. The bottom gate is realized by a conducting substrate, here Nb-doped SrTiO 3 , with the dielectric layer grown on top. With the use of shadow masks both top and bottom contact devices can be realized by altering the deposition order of contact and active layer deposition.

Figure 5 .Figure 6 .

 56 Figure5. This plot is used to determine the mobility in the linear regime from the response curve. The slopes of the drain current over the drain voltage dI D /dV D as a function of the gate voltage V G >> V D (linear regime) are plotted. From the linear fit the threshold voltage was estimated by the x-axis intercept to V T = 0.27 V and the mobility was derived from the slope to µ = 1.5 • 10 -3 cm 2 /Vs.

  (Color online) X-ray scan with the (200) Bragg peak of a sputtered SrTiO 3 film on a MgO substrate before and after annealing in air. The height of the peak increased and the center shifted toward the literature value for bulk SrTiO 3 (2θ = 46.472 • ). Similar behavior was observed for several films. The inset shows the corresponding thickness measurement of the SrTiO 3 film and a simulated curve to determine the film thickness (52 nm).

	Intensity (a.u.)	1e+04 1e+06 1e+08	MgO (200) Sub.	Intensity (a.u.)	1e-06 1e-04 1e-02	1	2 2Θ (degrees) 3 4 film SrTiO 3 (200) Fit: thickness 52 nm 5 As measured
				after annealing
				before annealing
		42	43	44	45	46	47	48
						2Θ (degrees)
	Figure 1.

• C for

  Figure 2. (Color online) Small angle X-ray diffraction pattern of the CuPc thin films prepared in the OMBD chamber on α-Al 2 O 3 substrates. The films were deposited on substrates at different temperatures. The film prepared at 50 • C showed pronounced Kiessig fringes but only a small Bragg peak. It was observed that this changed with higher substrate temperature -the Kiessig fringes disappeared in favor of Laue oscillations. The inset shows the corresponding rocking curves. The width (FWHM) of the rocking curve from film 009 was 0.038. Deposition time was 2 h for all films. Effusion cell temperatures were 355 • C for film 007, 350 • C for film 008 and 360 • C for film 009.

	Intensity (a. u.)	1e+02 1e+04 1e+06	Intensity (a. u.)	1e+03 1e+04 1e+05	3.2 3.4 3.6 3.8 Θ (degrees)
		T=50°C	-007 -52nm
		T=120°C -008 -23nm
		T=85°C	-009 -42nm
		1	2	3	4	5	6	7	8	9
					2Θ (degrees)

• C was

Acknowledgments

The authors are grateful for financial support through Transregio SFB TR 49 (Frankfurt, Mainz, Kaiserslautern)


Boudewijn F Roukema

Vincent Blanloeil

A measure on the set of compact Friedmann-Lemaître-Robertson-Walker models

Keywords: numbers: 98.80.Jk, 04.20.Gz, 02.40.-k

Compact, flat Friedmann-Lemaître-Robertson-Walker (FLRW) models have recently regained interest as a good fit to the observed cosmic microwave background temperature fluctuations. However, it is generally thought that a globally, exactly-flat FLRW model is theoretically improbable. Here, in order to obtain a probability space on the set F of compact, comoving, 3-spatial sections of FLRW models, a physically motivated hypothesis is proposed, using the density parameter Ω as a derived rather than fundamental parameter. We assume that the processes that select the 3-manifold also select a global mass-energy and a Hubble parameter. The requirement that the local and global values of Ω are equal implies a range in Ω that consists of a single real value for any 3-manifold. Thus, the obvious measure over F is the discrete measure. Hence, if the global mass-energy and Hubble parameter are a function of 3-manifold choice among compact FLRW models, then probability spaces parametrised by Ω do not, in general, give a zero probability of a flat model. Alternatively, parametrisation by a spatial size parameter, the injectivity radius r inj , suggests the Lebesgue measure. In this case, the probability space over the injectivity radius implies that flat models occur almost surely (a.s.), in the sense of probability theory, and non-flat models a.s. do not occur.

Introduction

Recent empirical analyses regarding the shape of the comoving spatial section of the Universe, assuming a perturbed Friedman-Lemaître-Robertson-Walker (perturbed-FLRW) model, have mostly focussed on the Wilkinson Microwave Anisotropy Probe (WMAP) all-sky maps of the cosmic microwave background. The results are presently inconclusive regarding both curvature and topology. The curvature has been found to be "nearly" zero, i.e., comoving space could be hyperbolic, flat or spherical. Writing the curvature as an effective density parameter Ω k := 1 -Ω, where Ω is the total density parameter, ‡ none of the three possible curvatures k ∈ {0, ±1} have been observationally excluded to high statistical significance. It is true that the precision in upper limits to |Ω k | has improved by about one a half orders of magnitude in the last decade and a half, and that the accuracy in |Ω k | upper limits has similarly improved, but these limits do not exclude any of k ∈ {0, ±1}.

Results regarding topology, i.e. the fundamental group Γ of holonomy transformations of the comoving spatial section, are also inconclusive.

Several groups have found that the Poincaré dodecahedral space S 3 /I * , with covering space M = S 3 and fundamental group Γ = I * , is preferred over simply connected infinite flat space [START_REF] Luminet | [END_REF]Aurich et al., 2005a,b;Gundermann, 2005;[START_REF] Caillerie | [END_REF]Roukema, Buliński, Szaniewska and Gaudin, 2008;Roukema, Buliński and Gaudin, 2008) in order to fit the WMAP data, while others exclude detectable cosmic topology (Cornish et al., 2004;Key et al., 2007), or prefer simply connected infinite flat space (Niarchou and Jaffe, 2007;Lew and Roukema, 2008).

While a well-established theoretical basis for cosmic curvature exists (the Einstein field equations match the spatial curvature of the Universe to its matter-energy density), there are at present only hints as to what might constitute a theory of cosmic topology. Various quantum gravity aspects of cosmic topology include the decay from pure quantum to mixed states (e.g., Hawking, 1984), smooth topology evolution (e.g., Dowker and Surya, 1998), and work that could contribute to a physical motivation for deciding which 3-manifold should be favoured by a theory of quantum cosmology (Seriu, 1996;Anderson et al., 2004). A recent heuristic result is that of the dynamical effect of cosmic topology in the presence of density perturbations. A residual weak-limit gravitational effect in the presence of a density perturbation selects well-proportioned spaces (Weeks et al., 2004) in general, and the Poincaré dodecahedral space S 3 /I * in particular, to be special in the sense of being "better balanced" (Roukema et al., 2007;Roukema and Różański, 2009). The agreement between the residual gravity effect and many of the empirical analyses in choosing the same space, the Poincaré space, suggests that this effect might have been important in the early universe in selecting the comoving ‡ The total density parameter is defined Ω := ρ tot /ρ crit , where ρ tot and ρ crit are the total density and the critical density, respectively. The latter is ρ crit := 3H 2 /(8πG) where H is the Hubble parameter and G the gravitational constant.

3-manifold in which we live.

On the other hand, Aurich et al. (Aurich et al., 2007;Aurich, 2008;Aurich et al., 2008Aurich et al., , 2010) ) have recently carried out several studies showing that a flat compact model (specifically, T 3 ) generally provides a better fit to the WMAP data than infinite flat models. This work provides one of the few ways of potentially showing that the Universe is flat in the sense of being a flat 3-manifold (apart from perturbations), if that is indeed the true shape of the Universe, rather than a Poincaré space. Approaches that ignore topology sometimes argue in favour of flatness while simultaneously stating (correctly) that they cannot determine the curvature. For example, Vardanyan et al. (2009) calculate the odds "in favour of a flat Universe", but also state that "the [curvature] of the Universe is not knowable" if log 10 |Ω k | < -4, and make no claim that log 10 |Ω k | ≥ -4. That is, they assert that either the curvature has not yet been determined, or it is unknowable. In contrast, use of the global properties of the comoving spatial section potentially offers a way to determine the curvature, even if the curvature is zero, as suggested by Aurich et al.'s work, and even in the (realistic) presence of perturbations.

Although flat, compact space models have been frequently modelled, they have generally been disliked for fine-tuning reasons, especially in relation to inflationary motivations and scenarios. It is frequently stated that the probability of the spatial section of the Universe, according to the exact-FLRW model, being exactly flat is zero. For example, in a discussion about common misconceptions regarding "standard universe models" and inflation, Ellis (2006) states that being exactly flat "requires infinite fine tuning of initial conditions; if say the two millionth digit in the value of Ω k is non-zero at any time, then the universe is not spatially flat" and that "although the scale-free k = 0 exponential case is the model underlying the way many people approach the problem, it is highly exceptional -it is of zero measure within the space of all inflationary FL models" (Sect. 2.8.1, 2.7.1, resp., Ellis, 2006). That is, if the measure µ on the space F of all exact-FLRW models is a probability measure, i.e. if µ(F ) = 1, then the probability of exact flatness is zero:

P (k = 0) = µ(k = 0) = 0.
Since inflationary scenarios only favour near flatness, not exact flatness, the subclass of inflationary scenarios I ⊂ F can validly have µ(I) > 0, i.e. a positive probability, according to this implicit definition of a measure space (F, Σ, µ).

However, is being exactly flat necessarily a physical possibility that has zero measure? [START_REF] Ellis | NATO ASIC Proc. 541: Theoretical and Observational Cosmology[END_REF] argue that finding a natural, plausible, measure in the "full space of cosmological space-times" is an open problem. Previous suggestions of measures (e.g., Gibbons et al., 1987;Coule, 1995;Evrard and Coles, 1995;Kirchner and Ellis, 2003;Gibbons and Turok, 2008) have typically only considered the curvature of the spatial section and not the fact that it is a 3-manifold that has both a curvature and a topology. In other words, the set F of exact-FLRW models is considered to be equivalent to F Γ=0 := {Ω ∈ R : Ω > 0}, i.e. it is parametrised only by the density parameter Ω. The measure is implicitly assumed to be a simple function of the Lebesgue measure on F Γ=0 that gives µ(Ω = 1) = 0 and a normalised measure P (F Γ=0 ) = 1, so that P (Ω = 1) = 0.

The fact that an FLRW model has both a curvature and a topology is not a new insight. Both curvature and topology have been mentioned quite clearly by several of the founders of relativistic cosmology (de Sitter, 1917;[START_REF] Friedmann | Mir kak prostranstvo i vremya (The Universe as Space and Time) Leningrad[END_REF][START_REF] Friedmann | [END_REF]Lemaître, 1931;Robertson, 1935). The curvature and topology of FLRW models have usually been thought to be independent of one another, except that the set of possible 3-manifolds is divided between hyperbolic, flat, and spherical 3-manifolds. Some work that does consider topology includes that of Seriu (1996) and Anderson et al. (2004). Seriu (1996) defined a "spectral distance" function over pairs of universes with (in general) different spatial sections, but did not try extending this to a measure. Anderson et al. (2004) applied a Hartle-Hawking "no boundary" path integral approach to the spatial section that constitutes a boundary to a global space-time. Alternatively, a Bayesian approach to measure spaces and the flatness problem, using Jaynes' principle, finds that the flatness problem is not a problem (Evrard and Coles, 1995;Kirchner and Ellis, 2003).

Here, a more elementary approach to obtaining a measure on the set of exact-FLRW models, considering both the curvature and topology of the models, is presented. In Sect. 2.1, a physically motivated hypothesis is proposed. This hypothesis is used to determine the set of compact, 3-spatial sections of exact-FLRW models where the density parameter Ω is a derived rather than fundamental physical parameter. Only compact models are considered, with the motivation that this is physically more reasonable than infinite models. § The resulting set and the natural measure on it are presented in Sect. 2.4. An alternative definition of the set, using the injectivity radius r inj (half the length of the shortest closed comoving spatial geodesic), and the natural measure on it are discussed in Sect. 3. Conclusions are given in Sect. 4.

For early references to cosmic topology see Schwarzschild (1900) ;de Sitter (1917);[START_REF] Friedmann | Mir kak prostranstvo i vremya (The Universe as Space and Time) Leningrad[END_REF][START_REF] Friedmann | [END_REF]; Lemaître (1931) and Robertson (1935). For a short, modern introduction see Roukema (2000). For reviews, see Lachièze-Rey and Luminet (1995); Luminet (1998); Starkman (1998); [START_REF] Luminet | NATO ASIC Proc. 541: Theoretical and Observational Cosmology[END_REF]; [START_REF] Blanloeil | Blanloeil & Roukema[END_REF]; Rebouças and Gomero (2004).

Parametrisation by Ω

Hypothesis

The implicit assumption that the space of exact-FLRW models consists of F Γ=0 := {Ω ∈ R : Ω > 0} is a statement that Ω is a free parameter, unconstrained by prior physics. One way of introducing some physical motivation for a theory of a set of possible universes is to extrapolate from known physics. § For hyperbolic models, replacing the condition "compact" by "finite volume", i.e. allowing finite volume models containing infinite spatial geodesics, should not modify the results in this paper.

English translation: Stewart et al. (1998).

Local geometry is a fundamental and experimentally well-established part of modern physics, via the Einstein field equations. It would be physically reasonable that as a result of quantum gravity processes or during (mostly) smooth early-universe topology evolution as explored by Dowker and Surya (1998), some global physical properties of the Universe in the FLRW approximation (not necessarily the density Ω) could be determined by a global geometrical property. The most obvious global geometrical property is geometria situs [START_REF] Euler | [END_REF], known today as topology. For exact-FLRW models, the topological characteristic that is usually discussed is equivalent to the choice of constant curvature 3-manifold.

What property or properties could most reasonably be hypothesised to be determined by the choice of topology? Although density (in an exact-FLRW model) can be evaluated globally, a more fundamental property in a classical, pre-relativistic sense, is mass. In pre-relativistic physics, conservation of mass in any closed system is a fundamental principle. In FLRW cosmological models, let us define the global (by volume), total (by component) non-relativistic plus relativistic mass-energy in the 3-spatial section as m := ρ tot V , where ρ tot is the total (by component) mass-energy density, and V is the volume of the spatial section in physical units, if k = 0, and as m := ρ tot V R 3 C , where R C is the curvature radius in physical units and V is the volume of the spatial section in units of R 3 C , if k = 0. Since, in general, m is not conserved with time, due to the change in frequency of (in particular) photons with respect to locally comoving observers, this should be evaluated at a given epoch.

Another parameter that could reasonably depend on the choice of 3manifold is the Hubble parameter H. Thus, along with the above definition of m, the following hypothesis is proposed.

Hypothesis 1 The topological evolution processes in the early universe that determine the comoving spatial 3-manifold M of an FLRW universe lead to, at a given post-topology-evolution epoch t t , a global, total ¶ matterenergy m(M) := m(M)| tt , and a rate of expansion, i.e. the Hubble parameter H(M

) := H(M)| tt = (ȧ/a)(M)| tt .
That is, m and H at the epoch t t are functions from the space of exact-FLRW models F to R, written m(M), H(M) to denote this dependence. The total mass-energy m(M) is determined in the comoving frame. The manifold M is assumed to be compact and orientable, and to be fixed for all epochs at t > t t .

The epoch t t is thought of here to be pre-inflationary if an inflationary scenario is invoked, but neither requires nor rejects an inflationary epoch at t > t t . To investigate ¶ The term "global" refers to the whole of comoving space, while "total" refers to all of the matterenergy density components, such as baryons, non-baryonic dark matter, neutrinos, radiation, and dark energy.

possible measure spaces using Hypothesis 1, spatially curved and flat models need to be considered separately.

Curved models

For curved models, first let us consider the density parameter Ω as a local parameter, i.e. as a limit towards a space-time point. In order to have a real, positive curvature radius R C for both negative (k = -1) and positive (k = +1) curvature, let us define

R C ∈ R such that R C > 0 and the Friedmann equation is Ω = 1 ± c H(M)R C 2 , k = ±1.
(1)

In a compact, exact-FLRW model, Ω also has a global meaning. Let us write the volume of a compact 3-manifold M of constant, non-zero curvature and (real) curvature radius R C > 0 as V R 3 C , i.e. V is dimensionless. For example, the lens space L(p, q) for p, q ∈ Z, 0 < q ≤ p/2, with p, q relatively prime (e.g., Gausmann et al., 2001), has volume 2π 2 R 3 C /p, so that V [L(p, q)] = 2π 2 /p. In the hyperbolic case, the uniqueness of V for a given fundamental group Γ was less obvious than for the spherical case, but follows from Mostow (1968)'s rigidity theorem. + The definitions of the critical density and the density parameter give

Ω := ρ ρ crit = m(M) V R 3 C 8πG 3 [H(M)] 2 .
(

) 2
What values of Ω simultaneously satisfy (1) and (2)? Clearly, the local and global values of the density parameter must be equal in an exact-FLRW model. Hence,

1 ± c H(M)R C 2 = m(M) V R 3 C 8πG 3 [H(M)] 2 , (3)
where ± correspond to curvatures k = ±1, respectively, as above. The range of R C values that satisfy this equation, and in turn, the values of Ω that satisfy both (1) and

(2) for a given pair [m(M), H(M)], are presented in Sect. 2.4.

Zero spatial curvature

The equivalent of (1) for a flat, compact, exact-FLRW model is

Ω = 1. (4)
Hence, the equivalent of (2) is not needed in order to determine the valid range of Ω for flat models.

+ Extended to the non-compact, finite volume case by Prasad. 2) is shown for V = 2π 2 ("×" symbols; this is for the largest spherical 3-manifold S 3 ) and V = π 2 , π 2 /10, and π 2 /60. The unphysical range R C ≤ 0 yields no solutions.

2.4. The set F Ω m,H of FLRW models parametrised by Ω

The set F of compact, exact-FLRW models is considered here to be the set of compact, comoving spatial sections that have an FLRW metric. A superscript Ω is used to indicate the parameter over which a measure space is to be defined. The subscripts m, H indicate that Hypothesis 1 is assumed.

Equation (3) is a cubic equation in R C . The discriminant is ∆ = -324V 2 H(M) 2 (±c 6 V 2 + 48π 2 G 2 m(M) 2 H(M) 2 ). ( 5
)
For positive curvature, ∆ < 0, so there is only one real root for R C . For negative curvature, there can be up to three distinct, real roots, but from ( 1) and ( 2), it is clear that only one of these is positive. This is illustrated in Figures 1 and2. The multiple root or the two additional roots that can occur in the hyperbolic case (Figure 1) occur for R C < 0, which is unphysical given the definition of R C used here. *

For illustrative purposes, we can consider the case where m(M) and H(M) are equal for all 3-manifolds. Figure 1 shows that in this case, for hyperbolic 3-manifolds, there is a maximum possible curvature radius R C , i.e. a maximum possible density parameter Ω, at t t . These maxima are attained for the smallestvolume hyperbolic 3-manifold. If we consider only orientable, hyperbolic 3-manifolds, then the Weeks-Matveev-Fomenko space [m003(-3, 1) in the SnapPea census ] of volume V ≈ 0.943 is the smallest-volume orientable hyperbolic 3-manifold (Weeks, 1985;Matveev and Fomenko, 1988;Gabai et al., 2009), shown here by "×" symbols. Larger volume hyperbolic 3-manifolds give the continuous curves in the figure [shown for V ≈ 3.0, e.g. m117(-5, 2) in the SnapPea census, and for V = 10.0, 100.0], which successively give lower R C and Ω. In Figures 1 and2, not only are m(M) and H(M) fixed, but the displayed ranges of numerical values of R C and Ω have been chosen for convenience only, i.e. with the units c = H(M) = 8π G m(M)/3 = 1. Elsewhere, these parameters are considered to retain their physical units.

Similarly, Figure 2 shows that for the spherical case, fixed values of m(M) and H(M) independent of M imply a minimum possible curvature radius R C as the choice of 3-manifold (i.e. Γ) is varied, which corresponds to a maximum possible density parameter Ω. These are attained for the largest-volume spherical 3-manifold, i.e. S 3 itself, of volume V = 2π 2 . While both signs of the curvature give upper limits to Ω for fixed m(M) and H(M) as Γ varies, the hyperbolic case prevents Ω from approaching the flat case, while the spherical case allows Ω to approach arbitrarily close to flatness as the volume decreases. The volume V = π 2 /60 is that of the Poincaré dodecahedral space.

More generally, i.e. without fixing m(M) and H(M), the unique physical * An alternative definition could use imaginary values of R C for the hyperbolic case.

The file /usr/share/snappea/ClosedCensusData/ ClosedCensusInvariants.txt of version 3.0d3-20 of SnapPea is referred to in this paper.

solution for R C in both cases is R C± (M) = √ ±c 6 V 2 + 48π 2 m 2 G 2 H 2 + 4 √ 3πmGH 2 3 ∓ c 2 V 2 3 √ 3HV 1 3 √ ±c 6 V 2 + 48π 2 m 2 G 2 H 2 + 4 √ 3πmGH 1 3 , (6)
where the dependence of R C on the manifold M occurs through the dependences V (M), m(M), and H(M).

The density parameter for a curved FLRW model can be written using either (1) or (2). The former gives

Ω ± (M) = 1 ± c H(M)R C± (M) 2 , ( 7
)
where M = H 3 /Γ or M = S 3 /Γ, and R C± is given in (6). As noted above (4), the solution in the flat case is

Ω k=0 (M) = 1. (8)
Together, these give the following result. For any given manifold M, Hypothesis 1 implies a single value of Ω rather than a continuous interval of possible values of Ω. In other words, there is no freedom in choosing Ω once M has been selected. Since the set of constant curvature manifolds {M} is countable (e.g., Lachièze-Rey and Luminet, 1995), we can write F Ω m,H as the countable (infinite) set

F Ω m,H = {Ω(M) : M ∈ F }, (9)
where Ω(M) is given by ( 7) and (8).

Measure spaces and probability spaces

Let us choose the obvious σ-algebra, 2 F Ω m,H , i.e. the set of all subsets of F Ω m,H , and any discrete measure µ on R satisfying

µ(Ω) > 0, Ω ∈ F Ω m,H µ(Ω) = 0, Ω ∈ F Ω m,H . (10)
Then (F Ω m,H , 2 F Ω m,H , µ) is a measure space over the density parameter Ω of exact-FLRW models of the Universe. It is clear that there is no physical requirement that µ(Ω = 1) be zero, i.e. there is no requirement that the measure of the case of a flat FLRW model be zero.

For a probability space (F Ω m,H , 2 F Ω m,H , µ) to be made starting from the measure µ, it would be required that µ(F Ω m,H ) = 1. Let the manifolds be enumerated {M i } ∞ i=1 and Ω(M) as given by ( 7) and (8) be enumerated Ω j , where a non-bijective function j(i) is provided so that j 1 = j 2 ⇒ Ω j 1 = Ω j 2 i.e. manifolds of equal Ω are included as a single case with respect to the measure. (Otherwise, µ(Ω j ) would be multiply defined for some values of Ω j , e.g. Ω j = 1.) Any convergent series {x j } ∞ j=1 with Σ j x j = 1 now satisfies

µ[Ω j (M i )] = x j , M i ∈ F µ(Ω) = 0, Ω ∈ F Ω m,H , (11)
giving a probability space (F Ω m,H , 2 F Ω m,H , µ). For example, this would be satisfied by

j = 1, x j = 1/e, 1 ≤ i ≤ 6 (12) j ≥ 2, x j = 1/[(j -1)! e], i ≥ 7 M i = E i , 1 ≤ i ≤ 6, ( 13
)
where E i are the six compact, flat, orientable 3-manifolds as labelled in Table I of Riazuelo et al. (2004), and the non-flat, compact, orientable 3-manifolds are enumerated by i ≥ 7, j(i) ≥ 2. This gives

µ(Ω = 1) = µ(Ω 1 ) = x 1 = 1/e ≈ 0.37 > 0 (14)
That is, given this example of {x j }, the probability of a flat space would be about 37%, i.e. strictly greater than zero. No physical motivation is suggested for this particular choice of {x j } and enumeration {M i } of the 3-manifolds. This choice of {x i } and corresponding (partial) enumeration {M i } are presented only as a numerical example of how P (Ω = 1) > 0 is possible, with P (Ω = 1) ≡ µ(Ω = 1) = 1/e. It would remain possible to choose {x j } in a way that gives P (Ω = 1) = 0, but it would be not be necessary to do so. Additional physical assumptions would be required to determine what the normalised discrete measure µ should be. Alternatively, Ω might not be a physically useful choice for defining a measure space. The construction in Sect. 3 suggests one alternative.

How can Ω = 1 not be a case of measure zero?

Fine-tuning arguments have played a role in cosmology for (at least) several decades. It may seem strongly counterintuitive that exactly Ω = 1 models do not necessarily constitute a case of measure zero, since the number 1 is a single real number on the continuous real number line. How does the topology of spatial sections invalidate this argument? The reason is that while topology in some sense might appear to give "additional" parameter freedom to the FLRW models, curvature can be thought of as a type of rigidity that together with topology reduces a continuous interval in the possible range of Ω to a set of discrete points. Thus, the hypothesis presented here leads to a physical link between curvature and topology. This link is represented algebraically in (3). This equation represents a physical relation between curvature and topology implied by Hypothesis 1. For a given compact manifold M, the relativistic mass-energy m(M) must be distributed uniformly throughout the space, expanding at the given rate H(M) = (ȧ/a)(M), in a way that simultaneously gives the curvature implied locally at each point in comoving space by the Friedmann equation and fills the volume implied by the global shape of the manifold. The local and global roles of the density Ω do not have the freedom to be unrelated to one another if m and H are determined by the choice of 3-manifold.

The full mathematics that underlies this physical argument has only become known in the second half of the twentieth century, and some details even more recently. The complete classification of spherical 3-manifolds and Mostow (1968)'s rigidity theorem for compact, hyperbolic 3-manifolds play a role by implying a unique volume V for any given curved, compact 3-manifold. A proof that the Weeks-Matveev-Fomenko manifold is the smallest-volume orientable hyperbolic 3-manifold was only published in 2009 (Gabai et al., 2009).

It is this physical relation between Ω being both local and global, which is a result of the rigidity of (constant curvature) curved 3-manifolds, that reduces the apparent continuum of values of Ω in the curved case to a discrete set. The set of values of Ω in the flat case is already discrete, since it contains just one member, Ω = 1. Hence, the full set of allowed values of Ω is discrete, and the natural measure is a discrete measure µ, as indicated in (10), rather than the Lebesgue measure. This is why Ω = 1 need not be a case of measure zero.

Parametrisation by a one-dimensional size parameter

Moreover, while Hypothesis 1 reduces the freedom of Ω in curved FLRW models, it implies more parameter freedom in flat FLRW models for other parameters.

In the 3-torus model T 3 , there are (at least) three parameters required to define the fundamental domain. Following Lehoucq et al. (1996), we can write the size of the domain as L a L e L u . For orthogonal side-lengths of the fundamental parallelepiped of a T 3 model, the definition of the critical density gives

L a L e L u = 8π G m(T 3 ) 3[H(T 3 )] 2 , ( 15
)
as for (2). This equation was ignored in Sect. 2.3, since it provides no constraint on Ω. The Friedmann equation in this case is (4), which provides no constraint on the global parameters L a , L e , L u . Hence, two of these parameters are free. In general, the angles between the faces of the fundamental domain constitute additional free geometrical parameters, but for L a , L e , L u defined orthogonally, these angles do not affect the volume of the manifold. The parameter freedom for compact, flat FLRW models has frequently been thought of as an empirically undesirable property, since it makes the models easier to fit to observations, reducing the models' falsifiability. This freedom is not total. In the presence of an inhomogeneity, the residual gravity effect (Roukema et al., 2007;Roukema and Różański, 2009), can be invoked as a motivation for L a ≈ L e ≈ L u .

The set F

2r inj m,H of FLRW models parametrised by 2r inj However, without invoking the residual gravity effect, the parametrisation of compact, flat FLRW models can be made in a way that also applies to compact, curved FLRW models, by using a size parameter. Here, we propose twice the injectivity radius, i.e., the length of the shortest closed comoving spatial geodesic, written 2r inj R C , where 2r inj is dimensionless, in the curved case, and 2r inj , including a length dimension, in the flat case. This underestimates the full parameter freedom of the flat case, providing a conservative approach to constructing a measure space.

Equation (15) implies an upper limit to 2r inj . This occurs for the regular T 3 model, in which L a = L e = L u , so that

2r inj = min(L a , L e , L u ) = 8π G m(T 3 ) 3[H(T 3 )] 2 1/3 . ( 16
)
Increasing any of the dimensions decreases at least one of the other dimensions, forcing 2r inj to decrease. A reasonable empirical lower bound for 2r inj can also be set, e.g. a comoving scale of ∼ 10h -1 Gpc. Aurich (2008) estimate 2r inj = 11.5 ± 0.3h -1 Gpc from applying the cross-correlation Monte Carlo Markov chain method (Roukema, Buliński, Szaniewska and Gaudin, 2008;Roukema, Buliński and Gaudin, 2008) to the 5-year WMAP data for a T 3 model. Hence, for the T 3 and related compact models with rectangular-prism fundamental domains

10a(t t )h -1 Gpc < ∼ 2r inj = min(L a , L e , L u ) ≤ 8π G 3 1/3 max i=1,2,3 m(E i ) [H(E i )] 2 1/3 , ( 17
)
where E 1 , E 2 , E 3 follow Table I of Riazuelo et al. (2004). The compact flat models with hexagonal-prism fundamental domains, E 4 and E 5 , have the corresponding limit

10a(t t )h -1 Gpc < ∼ 2r inj ≤ 2 1/3 3 -1/6 8π G 3 1/3 max i=4,5 m(E i ) [H(E i )] 2 1/3 (18)
(Appendix A.1). The other compact, orientable, flat model, i.e. the Hantzsche-Wendt space, E 6 , has a smaller injectivity radius, maximised in the regular case, giving

10a(t t )h -1 Gpc < ∼ 2r inj ≤ 2 -1/3 8π G 3 1/3 m(E 6 ) [H(E 6 )] 2 1/3 (19) (Appendix A.2).
Hence, over all curvatures, we can classify the full set of compact FLRW models by 2r inj for the flat case and 2r inj R C for the curved cases. F 2r inj m,H can now be written

F 2r inj m,H = 2r inj : 10a(t t )h -1 Gpc < 2r inj ≤ 8π G 3 1/3 η 2r inj (M) R C (M) : M ∈ F, M ∈ {H 3 , S 3 } , (20)
where

η := max m(E i ) [H(E i )] 2 1/3 , i = 1, 2, 3 2 1/3 3 -1/6 m(E i ) [H(E i )] 2 1/3 , i = 4, 5 2 -1/3 m(E 6 ) [H(E 6 )] 2 1/3 . ( 21
)
As in the case of F Ω m,H (9), a single value of 2r inj or 2r inj R C can correspond to more than one exact-FLRW model.

Measure spaces and probability spaces

Let us set Σ to be the smallest σ-algebra that contains all open sets on F Ω m,H induced by the usual topology on R. Then (F 2r inj m,H , Σ, λ), where λ is the Lebesgue measure, is a measure space over 2r inj , twice the injectivity radius of exact-FLRW models of the Universe. Let us normalise this by defining

λ = λ 8πG 3 1/3 η -10a(t t )h -1 Gpc , ( 22
)
where η is given by ( 21), yielding the probability space (F 2r inj m,H , Σ, λ). Hence, under the obvious probability space over the size of the 3-spatial section of FLRW models, the probability of a flat FLRW model is unity and the probability of a curved FLRW model is zero:

λ (2r inj | Ω=1 ) = 1, λ (2r inj | Ω =1 ) = 0. ( 23
)
This follows from ( 20) and ( 22), since the Lebesgue measure on an interval on the real line is the length of that interval, and the Lebesgue measure on a set of discrete, isolated points is zero.

An event that occurs with a probability of one is referred to as being almost sure (a.s.) in the Kolmogorov construction of probability spaces. This should not be confused with certainty. For example, an event of a uniform random process that selects a value from the interval 0 ≤ x ≤ 1 on the real line results in a specific value x * , despite the fact that the process a.s. † † does not choose that specific value x * . Thus, an a.s. outcome is not certain, even though its probability is one. In the case of interest here, if the Lebesgue measure is adopted and normalised, then a flat model a.s. occurs and a non-flat model a.s. does not occur. † † Almost sure and almost surely are both abbreviated as a.s.

How can Ω = 1 occur almost surely?

The probability space over Ω, i.e. (F Ω m,H , 2 F Ω m,H , µ), follows from the link between local and global definitions of the density parameter, i.e. between curvature and topology, implied by Hypothesis 1. This relation can be described in terms of the curvature radius R C , as shown above. Thus, equality of the local and global definitions of Ω discretises it. In the flat case, the curvature is zero and R C is undefined. The mathematical nature of flat, constant curvature 3-manifolds allows them a continuous range of fundamental domain shapes. The physical processes that are presumed to select a 3-manifold from among those mathematically available are not constrained to choose fundamental domain size parameters from a discrete set. Unless a physical constraint is added, a continuous, finite range in 2r inj , the size parameter adopted here, is allowed.

On the other hand, constant-curvature, curved 3-manifolds do not allow 2r inj to vary. The same rigidity that suggests the discrete measure in (F Ω m,H , 2 F Ω m,H , µ) leads even further, to the Lebesgue measure in (F 2r inj m,H , Σ, λ). Another way of saying this is that Hypothesis 1 not only links together the curvature and topology of exact-FLRW models, so that the rigidity of curved 3-manifolds constrains them to a discrete parameter space, but it also frees up flat 3-manifolds to occupy a continuous parameter space, if the parameter chosen is, for example, the injectivity radius. This is why, given Hypothesis 1, the natural choice of measure leads to a probability space where a flat model a.s. occurs (P ≡ λ = 1) and a non-flat model a.s. does not occur (P ≡ λ = 0).

Conclusions

Cosmic topology has been referred to in the cosmological context in the pre-relativistic era (Schwarzschild, 1900) and by de Sitter (1917[START_REF] Friedmann | Mir kak prostranstvo i vremya (The Universe as Space and Time) Leningrad[END_REF][START_REF] Friedmann | [END_REF], Lemaître (1931) and Robertson (1935) during the founding of relativistic cosmology. When the topology of the FLRW models is taken into account, the popular idea that Ω = 1 FLRW models constitute a class of measure zero among the full set of exact-FLRW models is no longer self-evident. This has been shown here by proposing that the parameters determined at a given epoch t t following topology evolution, i.e. as the results of the processes of primordial spatial 3-manifold evolution (quantum or otherwise), are the global (by volume), total (by component) mass-energy m(M)| tt and the Hubble parameter H(M)| tt , i.e. these are functions of the 3-manifold M selected by those processes, while the density Ω is relegated to a derived parameter (Hypothesis 1).

This hypothesis leads to quite different measure spaces and associated probability spaces than those in which Ω = 1 FLRW models constitute a class of measure zero.

When Ω is used to parametrise the FLRW models, the hypothesis leads to the discrete measure as the obvious choice. In this case, there is no obvious constraint requiring µ(Ω = 1) = 0. Moreover, use of the injectivity radius r inj to parametrise FLRW models, rather than Ω, suggests the Lebesgue measure as the natural measure. In this case, the measure of the class of non-flat models is zero. Since the measure is normalisable, the probability of a flat FLRW model in the corresponding probability space is unity. Hence, in this case, a flat model occurs a.s. and a non-flat model a.s. does not occur, in the Kolmogorov probability sense of these terms. This is the reverse of what has been thought to be the case when topology is ignored.

How does this happen? For exact-FLRW models with compact spatial sections, the density parameter has both a local and a global physical meaning. The requirement of equality between the two definitions of the density parameter, and the rigidity of curved, constant-curvature 3-manifolds, reduce the parameter freedom of the non-flat models. In contrast, the lack of rigidity of the flat models allows them a continuum of possible sizes. Hence, the flat models become much more probable. This approach suggests a motivation independent of inflationary scenarios for the recent work finding that a T 3 FLRW model provides a good fit to the WMAP sky maps (Spergel et al., 2003;Aurich et al., 2007;Aurich, 2008;Aurich et al., 2008Aurich et al., , 2010)). Moreover, a physical relation between the curvature and topology of comoving space is implied.

Figure 1 .

 1 Figure 1. Example of local and global dependence of the (total) density parameter Ω on the curvature parameter R C in hyperbolic, compact FLRW models, given Hypothesis 1, in the special case where m(M ) and H(M ) are constants independent of M . The local definition of the density parameter (1) gives the pair of curves that are symmetric around R C = 0 and decrease without bound as R C → 0 (solid curves). The global definition of the density parameter (2) is shown for dimensionless volume V = 0.943 ("×" symbols; this is for the smallestvolume orientable hyperbolic 3-manifold, the Weeks-Matveev-Fomenko manifold) and for V = 3.0, 10.0, and 100.0. The unphysical range R C ≤ 0 is shown for algebraic completeness, in order to show the cases where one or more unphysical solutions with R C < 0 occur. The units are chosen so that c = H(M ) = 8π G m(M )/3 = 1, i.e. R C is in units of c/H(M ).

Figure 2 .

 2 Figure 2. As for Figure 1, for spherical models. The local definition of the density parameter (1) gives the pair of curves symmetric around R C = 0 that increase without bound as R C → 0 (solid curves). The global definition of the density parameter (2) is shown for V = 2π 2 ("×" symbols; this is for the largest spherical 3-manifold S 3 ) and V = π 2 , π 2 /10, and π 2 /60. The unphysical range R C ≤ 0 yields no solutions.

  Figure 3. Orthogonal projection in z direction of one copy of the fundamental domain (FD) of a hexagonal-prism (third-turn or sixth-turn) space illustrating its volume, given that the injectivity radius (horizontal and vertical shortest closed geodesic) is L. The hexagonal area can be cut and paste into an L × √ 3L/2 rectangle.

Acknowledgments

Thank you to Karolina Zawada, Bartosz Lew and anonymous referees for several useful comments. Some of this work was carried out within the framework of the European Associated Laboratory "Astrophysics Poland-France". Use was made of the computer algebra program maxima, the GNU Octave command-line, high-level numerical computation software (http://www.gnu.org/software/octave), and the Centre de Données astronomiques de Strasbourg (http://cdsads.u-strasbg.fr).

Appendix A. Injectivity radii Appendix A.1. Injectivity radius of hexagonal-prism spaces

Figure 3 shows a projection of the fundamental domain (FD) of a hexagonalprism space, with dimensions that maximise the injectivity radius for a fixed volume, i.e. the "vertical" and "horizontal" shortest closed geodesics are both of length L. The volume of the FD is Since the shift in the x direction is identical for all points in the FD, 2r inj cannot be smaller than L/2. The point p 1 = (L/4, L/4, 0) in Fig. 4 is mapped to g(p 1 ) = (3L/4, L/4, 0), i.e. by a distance of L/2. Thus, the L/2 lower bound is attained. Hence, .4)