John D Barrow

S Cotsakis

A Tsokaros

A General Sudden Cosmological Singularity

We construct an asymptotic series for a general solution of the Einstein equations near a sudden singularity. The solution is quasi isotropic and contains nine independent arbitrary functions of the space coordinates as required by the structure of the initial value problem.

Introduction

The standard homogeneous and isotropic Friedmann-Lemaître cosmological models of general relativity are determined by the solution of two independent Einstein equations for three unknown time-dependent functions, the metric scale factor a(t), the fluid density, ρ(t), or the isotropic fluid pressure, p(t). If an equation of state p = f (ρ) is chosen, the system closes and the two remaining unknown functions are determined. Physically reasonable equations of state, like those for perfect fluids, produce well-behaved expansion factors with behaviours that offer simple Newtonian interpretations. However, the 1 Confidential: not for distribution. Submitted to IOP Publishing for peer review 16 April 2010 challenge of providing a compelling explanation for the observed acceleration of the universe in terms of a 'dark energy' fluid has led to an exploration of other less familiar equations of state [1], motivated by the form of bulk viscous stresses [2,3,[START_REF] Barrow | The Formation and Evolution of Cosmic Strings eds G Gibbons, S W Hawking and T Vaschaspati[END_REF], where dp/dρ is not everywhere continuous, or scenarios in which there is no equation of state at all. These less constrained material stresses introduce new possibilities for the evolution of Friedmann universes and can produce unexpected types of finite-time singularity which are far weaker than the curvature singularities that often accompany geodesic incompleteness [START_REF] Tipler | [END_REF]6,7,8,9,[START_REF] Hawking | The large scale structure of space-time[END_REF]. We found [START_REF] Barrow | [END_REF]12,13] a whole new class of pressure-driven singularities that keep the scale factor, a, the cosmological expansion rate, ȧ/a, and the density, ρ, finite while permitting the pressure, p, and the deceleration -ä/a, to blow up at a finite time despite the energy conditions ρ > 0 and ρ + 3p > 0 holding, and regardless of the three-curvature. There is no geodesic incompleteness associated with these singularities.

The stability of general relativistic solutions with sudden singularities in the presence of small scalar, vector, and tensor perturbations has been demonstrated in a gauge covariant formalism [14], and they have also been shown to be stable against quantum particle production processes as the singularities is approached [15]. They have been investigated in a number of different cosmological setting and their behaviour has been classified in the light of other types of finite-time singularity that can arise in isotropic and anisotropic cosmologies [16,17,18,19,20,21,22,23,24,25,26]. The formal definitions of these singularities have been discussed by Lake [27] in the case where infinities occur in second derivatives of a(t), but similar examples exist where the sudden singularity occurs in (arbitrarily) higher derivatives of a(t) and so no energy conditions are threatened [13,16]. Recently, we have also provided another formal construction procedure for these solutions in isotropic universes using fractional power series [28,29].

Sudden singularities have appeared in a wide range of cosmological investigations and our aim in this paper is to try to obtain information about the general cosmological behaviour in the neighbourhood of such a singularity by an asymptotic analysis of the Einstein equations without using simplifying assumptions of homogeneity or isotropy. We shall use a function-counting method, familiar from ref [START_REF] Landau | The Classical Theory of Fields 4th rev edn[END_REF]. Past uses of this method have failed to identify general behaviour near isotropic singularities [START_REF] Landau | The Classical Theory of Fields 4th rev edn[END_REF][START_REF] Khalatnikov | [END_REF]32] because of the ubiquity of chaotic behaviour near curvature singularities. However, although the full complement of independently arbitrary functions did not appear, the series expansions obtained provided useful asymptotic forms for particular cosmological models containing fluids and combinations of fluids [33,34]. The only use of this technique which identified a general behaviour in the Einstein cosmological equations is the study of the late-time behaviour of expanding universes with ρ > 0 and ρ + 3p > 0 in the presence of a positive cosmological constant . A series expansion around the de Sitter solution [35] produces a general complement of arbitrary functions in accord with the expectations of the cosmic no hair theorem [35,36,[START_REF] Barrow | The Very Early Universe[END_REF][START_REF] Jensen | [END_REF]39].

In this paper we are going to seek a series expansion around the particular FRW solution with a future sudden singularity introduced in ref [START_REF] Barrow | [END_REF]12]. In the limit that the sudden singularity is approached at finite future time we will determine the number of independently arbitrary functions of the three space variables that are necessary and sufficient to prescribe the solution locally on a spacelike hypersurface of constant comoving proper time.

Friedmann-Lemaître sudden singularities

Consider the most general form of sudden finite-time singularity of the sort discussed in [START_REF] Barrow | [END_REF]12] that can arise in an Friedmann expanding universe. In [START_REF] Barrow | [END_REF] we showed that singularities of this sort can arise from a divergence of the pressure, p, at finite time despite the scale factor, a(t), the material density, ρ, and the Hubble expansion rate, H = ȧ/a remaining finite. The pressure singularity is accompanied by a divergence in the acceleration of the universe, ä, at finite time. Remarkably, these singularities occur without violating the strong energy conditions ρ > 0 and ρ + 3p > 0. They can even prevent a closed Friedmann universe that obeys these energy conditions from attaining an expansion maximum [12]. In order to prevent the occurrence of these sudden singularities it is necessary to bound the pressure. A sufficient condition is to require dp/dρ to be continuous or p/ρ to be finite.

We will construct a quasi-isotropic ansatz for the general behaviour by first considering the Friedmann universe with curvature parameter k, zero cosmological constant, and units 8πG = c = 1. The Einstein equations reduce to

3H 2 = ρ - 3k a 2 , (1) ρ
+ 3H(ρ + p) = 0, (2) ä a = - ρ + 3p 6 . (3)
In [START_REF] Barrow | [END_REF] we constructed an explicit example by seeking, over the time interval 0 < t < t s , a solution for the scale factor a(t) of the form

a(t) = 1 + Bt q + C(t s -t) n , (4)
where B > 0, q > 0, C and n > 0 are free constants to be determined. If we fix the zero of time by requiring a(0) = 0, so Ct n s = -1, we have

a(t) = ( t t s ) q (a s -1) + 1 -(1 - t t s ) n , (5)
where a s ≡ a(t s ). Hence, as t → t s from below, we have

ä → q(q -1)Bt q-2 - n(n -1) t 2 s (1 -t ts ) 2-n → -∞, (6)
whenever 1 < n < 2 and 0 < q ≤ 1; the solution exists on the interval 0 < t < t s . Hence, as t → t s we have a → a s , H → H s and ρ → ρ s > 0 where a s , H s , and ρ s are all finite but p s → ∞. The infinite deceleration in ( 6) is why this sometimes also termed a 'big brake' singularity. By contrast, as t → 0 there is an initial all-encompassing strong-curvature singularity, with H → ∞, ρ → ∞ and p → ∞. From ( 6) and ( 3), we see that ρ and ρ + 3p remain positive throughout the evolution but, because ρ is finite asymptotically, the dominant-energy condition, |p| ≤ ρ, must always be violated as t → t s , [START_REF] Barrow | [END_REF]27,16].

Using the isotropic and homogeneous solution (5) as a guide we consider a series expansion of the metric in the neighbourhood of the sudden singularity in a general cosmological solution to the Einstein equations.

3 General series expansions near the singularity

Fractional power series

We use the Landau-Lifshitz [START_REF] Landau | The Classical Theory of Fields 4th rev edn[END_REF] notation: Latin indices for spacetime components, Greek for spatial components. The general form of the metric in synchronous coordinates is

ds 2 = dt 2 -γ αβ dx α dx β . (7)
On approach to the sudden singularity as t → t s , the solution (5) has the linear asymptotic form

a → a s + q(1 -a s )(1 - t t s ). (8)
Guided by this FRW example, (5) and its asymptotic form (8) we carry out a change of time coordinate to place the future sudden singularity at t = 0 and then consider a

general series expansion about it, with

γ αβ = a αβ + b αβ t + c αβ t n + ...,
where 1 < n < 2, and a αβ , b αβ , c αβ are functions of the space coordinates only; the inverse metric tensor is

γ αβ = a αβ -b αβ t -c αβ t n + • • • , so that γ αβ γ βγ = δ γ α .
We note that a αβ a βγ = δ γ α and the indices of b αβ , c αβ are raised by a αβ . The extrinsic curvature, its derivatives, and its contractions are, to leading order,

K αβ = ∂γ αβ ∂t = b αβ + c αβ n t n-1 + • • • , K α β = b α β + nc α β t n-1 + • • • , K = b + nc t n-1 + • • • , ∂K α β ∂t = n(n -1)c α β t n-2 + • • • , ∂K ∂t = n(n -1)c t n-2 + • • • , K β α K α β = b µν b µν + 2nb µν c µν t n-1 + • • • , KK α β = bb α β + n(cb α β + bc α β ) t n-1 + • • •
The components of the Ricci tensor are

R 0 0 = - 1 2 ∂K ∂t - 1 4 K β α K α β = - n(n -1)c 2 t n-2 - 1 4 b µν b µν - n 2 c µν b µν t n-1 + • • • , R 0 α = 1 2 (K β α ;β -K ;α ) = 1 2 (b β α;β -b ;α ) + n 2 (c β α;β -c ;α ) t n-1 + • • • , R α β = -Q α β - 1 4 KK α β - 1 2 ∂K α β ∂t = - n(n -1) 2 c α β t n-2 -[ 1 4 bb α β + P α β ] - n 4 [cb α β + bc α β ] t n-1 + • • • ,
where Q αβ is the Ricci tensor associated with γ αβ , and P αβ is the Ricci tensor associated with a αβ . In the synchronous reference frame

g 00 = 1, g 0α = 0.
The 4-velocity u i = (u 0 , u α ) has u 0 = u 0 and

1 = u i u i = u 2 0 -(a αβ + b αβ t + c αβ t n + • • • )u α u β .
The stress energy tensor for a perfect fluid

T i j = 8π[(ρ + p)u i u j -pδ i j ]
gives (to lowest order)

T 0 0 = (ρ + p)u 2 0 -p ≈ ρ, T 0 α = (ρ + p)u 0 u α ≈ (ρ + p)u α , T α β = (ρ + p)u α u β -pδ α β ≈ -pδ α β .
From the Einstein equations

R i j - 1 2 Rδ i j = 8πT i j ,
we use the 0 0 component to calculate the energy density and the trace of the α β equations to calculate the pressure. The 4-velocity can be calculated then from the 0 α component. Also we use the α β components to get restrictions on the arbitrary functions of a αβ , b αβ , c αβ . We have

16πρ = 1 2 (R 0 0 -R α α ) = P + b 2 -b µν b µν 4 - n 2 (b µν c µν -bc) t n-1 + • • • ,
while from the trace of the α β component we get

16πp = 1 2 (R 0 0 + 1 3 R α α ) = - 2n(n -1)c 3 t n-2 - 3b µν b µν + b 2 + 4P 12 - n 2 (b µν c µν + bc 3 ) t n-1 +• • • The Ricci scalar is R = R i i = -n(n -1)ct n-2 - b µν b µν + b 2 + 4P 4 - n 2 (b µν c µν + bc) t n-1 + • • • , the α β components of the Einstein equations give R α β = 1 2 (R -16πp)δ α β ,
and therefore

- n(n -1) 2 c α β t n-2 - 1 4 bb α β + P α β - n 4 (cb α β + bc α β ) t n-1 + • • • = - n(n -1)c 6 δ α β t n-2 -δ α β b 2 + 4P 12 -δ α β nbc 6 t n-1 + • • •
From the O(t n-2 ) terms, we get the constraints

c α β = c 3 δ α β or c αβ = c 3 a αβ , (9)
and so the components c αβ are all but one determined from a αβ .

Since γ αβ is to order t n , the extrinsic curvature will be O(t n-1 ) and the Ricci components R 0 0 , R α β will be O(t n-2 ). Hence this will be the only restriction on a αβ , b αβ , c αβ . Therefore, we have 6 + 6 + 1 = 13 independent functions. Subtracting the 4 coordinate covariances which may still be used to remove four functions, leaves 9 independent arbitrary functions of the three space coordinates on a surface of constant t time. This is the maximal number of independent arbitrary spatial functions expected in a local representation of part of the general solution of Einstein's equations near a sudden singularity. In general, we expect there will be 6 × g αβ and 6 × ġαβ , plus 3 free velocity components, u α ,and p and ρ, giving a total of 17 independent functions. We can remove four of these by using the R 0 a constraints and four more by using the general coordinate covariances. This leaves a total of 9 free functions expected in the general solution for the metric locally. If an equation of state had been assumed to relate the pressure to the density this number would have been reduced by one to 8.

We have found that the pressure is O(t n-2 ), so ρ + p = O(t n-2 ), and from

R 0 α = 8π(ρ + p)u 0 u α ≈ 8π(ρ + p)u α , since R 0 α ≈ O(t 0 ), we get that u α ≈ O(t -(n-2)
), and the full expansion is

u α = - 3(b β α;β -b ;α ) 2n(n -1)c t 2-n ≡ Φ α t 2-n , 0 < 2 -n < 1.
The equation for the stress energy conservation T ij ;i = 0 is split into time and spatial components as follows:

T i0 ;i = T α0 ;α + ∂ t T 00 + 1 2 (KT 00 + K αβ T αβ ) = 0, T iβ ;i = T αβ ;α + ∂ t T 0β + 1 2 KT 0β + K β α T 0α = 0,
where

T 00 = ρ, T 0α = (ρ + p)u α , T αβ = pγ αβ .
The time component gives

[(ρ + p)u α ] ;α + ∂ t ρ + 1 2 K(ρ + p) = 0. ( 10
)
To lowest order, the first term of eq. ( 10) is constant (O(t 0 )). The other two terms to lowest order are O(t n-2 ); therefore, to lowest order the equation is O(t n-2 ). Hence the

O(t n-2 ) term of ∂ t ρ must cancel the O(t n-2 ) term of 1 2 K(ρ + p). The first term is 16π(ρ + p)u α = -a αβ (b µ β;µ -b ;β ) -na αβ (c µ β;µ -c ;β )t n-1 + • • •
The second is

16π∂ t ρ = - n(n -1) 2 (b µν c µν -bc)t n-2 + • • • ,
and the third is

1 2 K(16πρ + 16πp) = - n(n -1)bc 3 t n-2 + • • • .
Therefore this equation will be satisfied to order O(t n-2 ) if

n(n -1) 2 (b µν c µν -bc) + n(n -1)bc 3 = 0, that is, by b µν c µν = 1 3 bc,
which is indeed satisfied because of the constraint eq. ( 9) on c αβ found earlier. Therefore, the time component of T ij ;i = 0 does not give any new constraints on the terms of the series.

Similarly, the spatial part of the stress-energy conservation is expressed as

γ αβ p ;α + ∂ t [(ρ + p)u β ] + 1 2 K(ρ + p)u β + K β α (ρ + p)u α = 0.
The lowest order of the first two terms is O(t n-2 ) while the lowest order of the last two terms is O(t 0 ) and so they are neglected. For the first term,

16πγ αβ p ;α = - 2n(n -1) 3 a αβ c ;α t n-2 + • • • , while for the second, 16π∂ t [(ρ + p)u β ] = -n(n -1)a βν (c µ ν;µ -c ;ν )t n-2 + • • • .
Hence, we must have

2n(n -1) 3 a αβ c ;α + n(n -1)a βν (c µ ν;µ -c ;ν ) = 0 or c β α;β = 1 3 c ;α
which again is satisfied by the restriction of eq. ( 9) on c αβ . Hence, both the temporal and the spatial components of the conservation of the stress energy tensor are satisfied to lowest order by the constraint on c α β that we have found already and there remain 9 independent functions in our series solution.

As an aside we note that he speed of sound is

v 2 s = dp dρ = ṗ ρ = 4(2 -n)c 3(bc -b αβ c αβ ) t -1
which goes to infinity as t → 0 which is unsurprising given the divergence of the pressure at the sudden singularity.

Puiseux and logarithmic series

We shall now discuss some other series expansions around the singularity. Consider first a Puiseux series of the form

a(t) = ∞ j=0 a j t j + t n [b 0 + t 1/q b 1 + t 2/q b 2 + • • • ]
as an asymptotic form for the Friedmann-Lemaître equations instead of eq. ( 5). This suggests an ansatz for the quasi-isotropic form of the general inhomogeneous metric of the form

γ αβ = a αβ + b αβ t + c αβ t n + ψ αβ t m + ω αβ t r + • • • , with 1 < n < m < r < 2.
If we perform the same analysis we will find again that a αβ and b αβ are arbitrary and

c αβ = c 3 a αβ , and
ψ αβ = ψ 3 a αβ , and
ω αβ = ω 3 a αβ ,
with c, ψ, ω again arbitrary. The pressure and the density will be

16πρ = 16πρ 0 + nbc 3 t n-1 + mbψ 3 t m-1 + rbω 3 t r-1 + • • • , (11) 16πp
= - 2n(n -1)c 3 t n-2 1 + m(m -1)ψ n(n -1)c t m-n + m(m -1)ω n(n -1)c t r-n + • • • , (12)
and

m -n > 0 r -n > 0.
Therefore we have the same phenomenon as with the series that was described in the previous section. If we keep adding terms of the form t s with 1 < s < 2 we will getting one more arbitrary function so if we add an infinite number of terms we will have an infinite number of functions. The reason that they are irrelevant is that they do not alter the nature of the singularity in its neighborhood as can be seen from the density and pressure eqs. ( 11)- (12). In other words, all three forms

γ αβ = a αβ + b αβ t + c αβ t n + • • • , γ αβ = a αβ + b αβ t + c αβ t n + ψ αβ t m + ω αβ t r + • • • , γ αβ = a αβ + b αβ t + c αβ t n + ψ αβ t m ln(t) • • • give lim t→0 ρ = ρ 0 , and 16π lim t→0 p = -lim t→0 2n(n -1)c 3 t n-2
and hence the same behaviour on approach to the sudden singularity as t → 0. Thus the number of relevant independent functions is 9 = 6 + 6 + 1 -4 which is the number required for a specification of the general solution.

Discussion

If we had sought a power-series expansion of the metric with time dependence dictated by the form of the radiation-dominated Friedmann-Lemaître universe on approach to an initial big bang curvature singularity as t → 0 then it would have taken the form of eq. ( 7) with

γ αβ = ta αβ + t 2 b αβ + ... (13)
with a αβ , and b αβ spatial functions [START_REF] Landau | The Classical Theory of Fields 4th rev edn[END_REF][START_REF] Khalatnikov | [END_REF]32] However, at most 3 arbitrary spatial functions in the series expansion are left free in this solution. A general solution of this form, with a radiation equation of state requires 8 independently arbitrary functions of space. The reason for the non-generic form of (13) for the radiation fluid is its quasiisotropic character. On approach to a general relativistic cosmological singularity the anisotropic shear and 3-curvatures dominate the dynamics and destroy the quasi isotropic character of a radiation-dominated universe.

By contrast, our results are quite different. We find that on approach to a latetime sudden singularity, where the density, expansion rate, and metric remain finite, it is possible for a quasi-isotropic solution to be part of the general solution of the Einstein equations. This solution has no equation of state and is characterised by nine independently arbitrary spatial functions. On approach to a sudden singularity the mean scale factor and the shear approach constant values, the chaotically anisotropic degrees of freedom which dominate at an initial curvature singularity are frozen out, and anisotropic velocities grind to a halt because of the divergent pressure and inertia. The situation is far simpler than for a initial vacuum or p < ρ dominated fluid singularity.

Comparable simplicity is achieved in the non-singular late-time approach to a quasiisotropic de Sitter universe in the presence of a cosmological constant or a p = -ρ fluid, shown by Starobinsky [35]. We have shown that there is an analogous 'no hair' behaviour on approach to a late-time sudden singularity and the general structure of such a finite-time singularity admits a simple quasi-isotropic form.


Johanna Riha

Catherine H Mercer

Kate Soldan
email: kate.soldan@hpa.org.uk

Clare E French

Mary Macintosh

Who is being tested by the English National Chlamydia Screening Programme? A comparison with national probability survey data

Keywords: Chlamydia trachomatis, screening, England, probability survey

Objectives:

We compare data collected by England's National Chlamydia Screening Programme (NCSP) with national probability survey data to examine demographic and behavioural differences that may be important in understanding who the NCSP is reaching and interpreting chlamydia positivity.

Methods: Data for 538,119 males and females aged 16-24 years who were screened in 2008 and data collected from 2,180 interviewees in Britain's second National Survey of Sexual Attitudes and Lifestyles 1999-2001 (Natsal-2), of whom 644 were tested for chlamydia, were compared using the Chi-square statistic and logistic regression.

Results: Compared with Natsal-2, the NCSP tested more women (67% vs. 49%). NCSP participants were more likely to be younger: 29% were 16-17 years vs. 16% of men and 15% of women in Natsal-2; from ethnic minority groups: 17% of men and 14% of women vs. 8% and 6%, respectively, in Natsal-2; not to have used condoms at last sex: 66% of men and 68% of women vs. 48% and 63%, respectively, in Natsal-2: and more likely to report 2+ partners in the last year: 62% of men and 47% of women vs. 47% and 30%, respectively, in Natsal-2. In multivariate analyses, higher adjusted odds ratios (AOR) of chlamydia positivity were found for those reporting non-use of condoms and for those reporting multiple partners, in both the NCSP and Natsal-2.

Conclusions:

The NCSP is testing young people at increased risk of chlamydia. The impact of this testing bias on the effectiveness of the programme should be evaluated.

INTRODUCTION

Genital Chlamydia trachomatis infection is the most commonly diagnosed bacterial sexually transmitted infection in the UK (1). It is commonly thought, based on experience in GUM clinics, that around 50% of infections in men and 60-70% of infections in women are asymptomatic, although some data put this higher at around 90% in men [START_REF] Mckay | Genital Chlamydia trachomatis infection in a subgroup of young men in the UK[END_REF]. If left untreated genital chlamydial infections can have serious long term consequences including pelvic inflammatory disease, ectopic pregnancy and tubal factor infertility in women and urethritis, epididymitis and Reiter's syndrome in men [START_REF] Westrom | Pelvic inflammatory disease and fertility[END_REF][START_REF] Hillis | Recurrent chlamydial infections increase the risks of hospitalization for ectopic pregnancy and pelvic inflammatory disease[END_REF][START_REF] Tay | Regular Review: Ectopic pregnancy[END_REF]. The National Chlamydia Screening Programme (NCSP) was launched in England in 2003 to control chlamydial infection through early detection and treatment of predominantly asymptomatic infections to prevent onward transmission and the development of sequelae (6). Unlike the cervical, breast or bowel screening programmes in England where medical registers are used to invite eligible subgroups of the population, the NCSP was not implemented as a call-recall register-based programme. Instead the NCSP was set up as an opportunistic programme offering testing and treatment to men and women under the age of 25 years in healthcare and non-healthcare based settings. The proportion of those testing positive through the programme has been higher than the prevalence observed in the general population [START_REF] Simms | The English National Chlamydia Screening Programme: variations in positivity in 2007/2008[END_REF][START_REF] Fenton | Sexual behaviour in Britain: reported sexually transmitted infections and prevalent genital Chlamydia Trachomatis infection[END_REF] and is likely to be influenced by factors associated with who is offered and who accepts chlamydia testing.

Between 1999 and 2001, Britain's second National Survey of Sexual Attitudes and Lifestyles (Natsal-2), a population-based probability sample survey, was carried out. In addition to collecting data on sexual behaviour and attitudes, the survey collected urine samples from a subset of sexually active adults and estimated the population-based prevalence of genital chlamydia. At the time of writing, the Natsal-2 data are the most up-to-date and wellcharacterised population-based chlamydia prevalence data available for Britain. These data therefore provide an opportunity to explore differences between those who are tested through the NCSP and a nationally representative general population sample. The aims of this study are to (i) compare the demographic and behavioural profiles of those participating in the NCSP relative to a general population sample (Natsal-2); (ii) examine whether the same demographic and behavioural risk factors for chlamydia are seen in both the NCSP and the general population; and (iii) consider how this comparison informs interpretation of NCSP positivity rates.

METHODS

Data sources

NCSP

The NCSP tests sexually active young men and women aged under 25 years for chlamydia outside of genitourinary medicine clinics. 542,052 tests were recorded in the NCSP database for 2008. Reported data do not necessarily include a unique patient identifier. 2,450 test records matched other records by sex, date of birth, ethnicity and postcode of residence. These were considered highly likely to be repeat tests of the same individual and were excluded from our analysis. The first test was retained unless a subsequent test was positive in which case the first positive test was retained. An additional 1,483 tests were excluded due to unknown sex. Data for 538,119 men and women aged 16 to 24 years (an estimated 8.6% of the English population aged 16-24 years) who were tested for chlamydia through the programme in 2008 entered analysis. The programme collects sociodemographic data (age, gender both 100% completeness, and ethnicity, with around 76% completeness) and sexual behavioural data (two or more sexual partners in the last year, with around 67% completeness) for each test conducted. Information about condom use at last sex is additionally requested of NCSP participants in less than 20% of programme areas. Where asked, this question is completed with a "yes" or "no" response for around 50% of participants.

Natsal-2

Natsal-2 involved a stratified probability sample of 11,161 male and female British residents aged 16 to 44 years interviewed between 1999 and 2001 based on addresses drawn using a multistage probability cluster design from the Postcode Address File. Because the prevalence of many HIV risk factors had been found to be higher in London relative to the rest of the country in the first Natsal survey conducted a decade earlier, addresses in Greater London were over sampled in Natsal-2 [START_REF] Johnson | Sexual attitudes and lifestyles[END_REF]. Information was collected by face-to-face Computer-Assisted Personal Interviews ('CAPI') with the more sensitive questions asked through Computer-Assisted Self-Interviews ('CASI'). In addition, a random half of sexually active respondents aged 18 to 44 years were asked to provide a urine sample to test for Chlamydia trachomatis using ligase chain reaction. Natsal-2 achieved a response rate of 65% for participation in the survey, which is in line with other major surveys conducted in Britain (10-12). The urine sample achieved a response rate of 71%, which did not vary significantly by age-group [START_REF] Mccadden | Chlamydia Trachomatis testing in the second British National Survey of Sexual Attitudes and Lifestyles: Respondent uptake and treatment outcomes[END_REF]. Further details of the survey methodology and questionnaire are published elsewhere [START_REF] Fenton | Sexual behaviour in Britain: reported sexually transmitted infections and prevalent genital Chlamydia Trachomatis infection[END_REF][START_REF] Mccadden | Chlamydia Trachomatis testing in the second British National Survey of Sexual Attitudes and Lifestyles: Respondent uptake and treatment outcomes[END_REF][START_REF] Johnson | Sexual behaviour in Britain: partnerships, practices, and HIV risk behaviours[END_REF][START_REF] Erens | National Survey of Sexual Attitudes and Lifestyles II: Technical Report[END_REF]. As described in other publications, [START_REF] Johnson | Sexual behaviour in Britain: partnerships, practices, and HIV risk behaviours[END_REF][START_REF] Erens | National Survey of Sexual Attitudes and Lifestyles II: Technical Report[END_REF] the Natsal-2 questionnaire data were weighted for differential selection and response probabilities and the urine sample data were given an additional weight to take account of non-response in providing a urine sample, calculated as the inverse probability of providing a urine sample [START_REF] Fenton | Sexual behaviour in Britain: reported sexually transmitted infections and prevalent genital Chlamydia Trachomatis infection[END_REF]. Details regarding the factors found to be associated with providing a urine sample are given in the paper by McCadden et al [START_REF] Mccadden | Chlamydia Trachomatis testing in the second British National Survey of Sexual Attitudes and Lifestyles: Respondent uptake and treatment outcomes[END_REF].

Data analyses

We compare data collected by the NCSP throughout 2008 and Natsal-2. Two analyses were conducted: Firstly, we compared the demographic and behavioural characteristics of those participating in the NCSP with those respondents aged 16-24 years who participated in the national probability survey with regards to five factors for which data were available from the two data sources: gender, age, ethnicity, condom use at last sex and having two or more sexual partners in the last year (expressed hereon for brevity as 'multiple partners'). Due to the large proportion of the NCSP participants for whom data on condom use was not available (see above), we compared the characteristics of NCSP participants with and without data for this variable to see whether there were any differences. We also compared the NCSP and Natsal-2 data limited to those participants with data for all five factors, as the multivariate analyses (described below) also focused on these sub-samples. In the bivariate analyses, the Chi-square statistic was used to determine whether any differences between the NCSP and the Natsal-2 data sources were statistically significant, which was considered as p<0.05 for all analyses.

Secondly, we investigated how testing positive for chlamydia is associated with the five factors for each of the two data sources. We used binary logistic regression to do this, calculating the odds ratio for testing positive for those in a particular category relative to the reference category, for example, among those aged 20-24 years relative to those aged 18-19 years. To take account of any confounding, we also used multivariate binary logistic regression to enable us to calculate the odds ratio adjusting for the other variables we had data for. The denominator for this second analysis was limited to those Natsal-2 participants aged 18-24 years who had met the sample selection criteria for providing a urine sample [START_REF] Fenton | Sexual behaviour in Britain: reported sexually transmitted infections and prevalent genital Chlamydia Trachomatis infection[END_REF], who had agreed to do so and whose result had been tested for chlamydia (n=644). Only those people for whom we had data on all five variables were included in this analysis. This corresponds to 627 of the eligible Natsal-2 participants (97% of 644) and 17,259 NCSP participants aged 18-24 (4%). We also ran this multivariate analysis excluding the condom use variable, thus including those participants who had data available for gender, age, ethnicity, and multiple partners (630 (98%) of eligible Natsal-2 participants; 234,034 (61%) of NCSP participants aged 18-24), to see if the results differed to the first multivariate model described.

To account for clustering, weighting and stratification of the Natsal-2 data, all analyses were performed in STATA 9.2 using the complex survey analysis functions [START_REF] Corp | STATA 9.2 2006[END_REF]. The NCSP data were considered as a simple sample, with no weighting (i.e. a weight of one).

RESULTS

Demographic and behavioural characteristics

Compared with Natsal-2, the NCSP tested larger proportions of women (67.2%, 95%CI 66.8% -67.2%; vs 49.3%, 95% CI 46.5%-52.1%) (data not shown in tables). Table 1 compares the demographic and behavioural characteristics of NCSP participants relative to the general population aged 16-24 years, according to Natsal-2, by gender. Relative to Natsal-2 respondents, men and women participating in the NCSP were more likely to be younger, specifically aged 16 to 17 years: 29% of men and women in the NCSP vs. 16% of men and 15% of women in Natsal-2. NCSP participants were also more likely to be from ethnic minority groups: 17% of men and 14% of women vs. 8% and 6%, respectively, in Natsal-2, particularly from black ethnic groups: 8% of men and 7% of women in contrast to 2% and 1%, respectively, among Natsal-2 participants. NCSP participants were also more likely to report not having used condoms at last sex than their counterparts in the general population: 66% of men and 68% of women vs. 48% and 63%, respectively in Natsal-2. NCSP participants were also more likely to report multiple partners in the last year: 62% of men and 47% of women vs. 47% and 30%, respectively, in Natsal-2.

Due to the large proportion of NCSP participants for whom data on condom use was not available (see above), we compared the characteristics of NCSP participants with and without data for this variable to see whether there were any differences. While statistically significant differences do exist, reflecting the size of these two groups, the magnitude of these differences are slim, typically within four percentage points (Web-table 1). One notable exception though is that the proportion of women from ethnic minority groups is larger among those for whom data on condom use were not collected: 15% in contrast to 5% of women for whom these data were collected. Given these differences, we also compared the NCSP and Natsal-2 participants limited to those with data for all four factors (Web-table 2). We found that the differences between the two data sources shown in Table 1 persisted, although there were smaller differences in the proportions of participants of non-white ethnicity.

Factors associated with chlamydia positivity

Overall, both men and women (aged 18-24 years) who tested in the NCSP had higher chlamydia positivity than those tested in the general population sample: Among men: 9.4% (95%CI:9.2%-9.5%) vs. 2.9% (95%CI:1.3%-5.2%), respectively, and among women: 8.5%

(95%CI:8.4%-8.6%) vs. 3.1% (95%CI:1.6%-5.7%), respectively. The corresponding numbers of positive chlamydia cases were 11,454 men and 21,316 women in the NCSP; 8 men and 16

women in the Natsal-2 sample.

Table 2 shows the factors associated with increased risk of chlamydia infection in each data source for men and women separately. Age was only significantly associated with having chlamydia among women in the NCSP, with lower odds for those aged 20-24 years relative to those aged 18-19 years (adjusted odds ratio: 0.84, 95%CI 0.74-0.96). Ethnicity appeared significantly associated for women in the NCSP (only), with lower odds for black women relative to white women (AOR: 0.34, 95%CI 0.15-0.76). However, NCSP women of black ethnicity who responded to the question regarding condom use (and were therefore included in our analysis) had a lower prevalence of chlamydia infection than all NCSP women of black ethnicity (3% vs 9.5%), which will have introduced bias into our analysis with respect to ethnicity. When the multivariate model was run for those participants with data for the other factors, excluding condom use (Web-tables 3A and 3B), the AOR for these women increased to 1.19 (95%CI 1.11-1.27) and a similar relationship was observed for men in the NCSP of black ethnicity: AOR:

1.42 (95%CI 1.30-1.54).

In terms of sexual risk behaviours, for both genders in both data sources the AOR was greater than one for participants who reported not using condoms at last sex: Among men, AORs of 1.53 for NCSP participants and 10.3 for men in Natsal-2; among women, AORs of 1.23 and 1.31, respectively. Similarly, NCSP participants who reported multiple partners in the year prior to testing for chlamydia were significantly more likely test positive: AORs of 1.62 and 1.81 for men and women, respectively. This relationship was also observed among Natsal-2 women but was not statistically significant (AOR 2.85, 95%CI 0.88-9.22). Among men in Natsal-2 sample, none of those reporting at most one partner in this time-frame tested positive for chlamydia, while prevalence was 6.7% (95%CI 3.0%-14.1%) among those reported multiple partners.

DISCUSSION

Statement of principal findings

This study has shown that the NCSP is testing individuals at higher risk of chlamydia than those in the general population. Similar behavioural risk factors for testing positive for chlamydia were identified for both populations. In line with previous published figures [START_REF] Hook | Comparative behavioral epidemiology of gonococcal and chlamydial infections among patients attending a Baltimore Maryland, sexually transmitted disease clinic[END_REF][START_REF] Van Duynhoven | Different demographic and sexual correlates for chlamydial infection and gonorrhoea in Rotterdam[END_REF], individuals reporting not having used a condom at last sex and those having multiple partners had higher odds of testing positive for chlamydia.

Strengths and weaknesses of the study

This study included a large sample size particularly for NCSP records, with over half a million records used in these analyses. The use of nationally representative data allowed us to investigate the disparities that exist between those testing through the NCSP in contrast to a random sample of the general population.

There are several limitations with this study. Firstly, the two data sources used different methods to collect genital samples and detect Chlamydia trachomatis. Natsal-2 used the most sensitive and specific Nucleic Acid Amplification Assay (NAAT) available at the time of that study, the ligase chain reaction (LCR) on participant-collected urines posted to the laboratory [START_REF] Fenton | Sexual behaviour in Britain: reported sexually transmitted infections and prevalent genital Chlamydia Trachomatis infection[END_REF]. NCSP testing in 2008 was by more recently developed NAAT assays conducted on a mixture of patient-collected and clinician collected samples. Natsal samples may have suffered more loss of sensitivity as a result of greater specimen transport delays [START_REF] Fenton | Sexual behaviour in Britain: reported sexually transmitted infections and prevalent genital Chlamydia Trachomatis infection[END_REF]. This may have increased the difference in observed prevalence of chlamydia in the two populations but should not affect the differences in behaviours or risk factors associated with the infection in each of the two populations.

The response rate for Natsal-2 was consistent with other major social surveys,(10-12) and the data were weighted to be broadly representative of the British general population with respect to the age, gender and regional distribution [START_REF] Johnson | Sexual behaviour in Britain: partnerships, practices, and HIV risk behaviours[END_REF][START_REF] Erens | National Survey of Sexual Attitudes and Lifestyles II: Technical Report[END_REF] and to account for non-response to the urine sample collection [START_REF] Mccadden | Chlamydia Trachomatis testing in the second British National Survey of Sexual Attitudes and Lifestyles: Respondent uptake and treatment outcomes[END_REF]. Nevertheless, response bias can not be completely eliminated and remains a limitation of these data. The possibility of response bias also applies to the NCSP data, since these data were also not 100% complete. Our inclusion of data on condom use at last sex appears to have introduced such a response bias in our analysis by ethnic group, rather than to have improved the adjustment for confounding that we hoped it would. Our analysis of risk without this variable, and associated response bias, finds an elevated risk for individuals with black ethnicity, as reported by a previous NCSP study [START_REF] Simms | The English National Chlamydia Screening Programme: variations in positivity in 2007/2008[END_REF].

We had limited power due to the relatively small Natsal-2 sample size for the age-group of interest, as noted above. There are also differences in terms of how the behavioural data were collected as Natsal-2 used CASI while the NCSP used a self-administered or clinician completed pen-and-paper questionnaire. However, it is worth noting that in a randomised comparison of administering the Natsal-2 questionnaire as a CASI versus pen-and-paper selfcompletion, no consistent evidence of an increased willingness to report sensitive behaviours was observed [START_REF] Johnson | Effect of computer-assisted self interviews on reporting of sexual HIV risk behaviours in a general population sample: a methodological experiment[END_REF].

It would have been interesting to examine whether the strength of the associations found in both populations differed in the NCSP relative to the general population. For example, it is possible that the relationship between a single, binary risk behaviour variable on the probability of infection is less discriminating in the NCSP than in the general population. Unfortunately there were too few chlamydia cases in the Natsal-2 (6 among men, 16 among women) and thus insufficient statistical power to test this hypothesis.

Finally, data for the two populations being compared were also collected during two different time periods, with the general population sample collected around eight years before the NCSP sample. At the time of writing however, Natsal-2 data are the most up-to-date data available on chlamydia prevalence in the general population. As has been previously reported [START_REF] Johnson | Sexual behaviour in Britain: partnerships, practices, and HIV risk behaviours[END_REF], it is possible that sexual behaviour in the British population has changed since the interviews for Natsal-2, which may account, at least in part, for the differences detected between those tested through the screening programme and Natsal-2. If the young British population has become more sexually active since Natsal-2 was conducted it is possible that the results from this study, which describe the NCSP population as being more at risk, reflect this change. A comparison of NCSP participants and participants in Britain's third National Survey of Sexual Attitudes and Lifestyles (Natsal-3), being conducted in 2010-12, should further our understanding of who is being tested by England's chlamydia screening programme.

Comparison with other studies

Others have compared certain characteristics of young people undergoing screening to those of young people more generally. For example, Sheringham et al have analysed residence-based Index of Multiple Deprivation scores (a variable we did not investigate) for young people screened through the NCSP in comparison to total population data, and found that in 2008 the NCSP tested a larger proportion of individuals who resided in deprived areas compared to those in more affluent areas [START_REF] Sheringham | Socioeconomic variations in chlamydia screening coverage and positivity in England: Analysis of the National Chlamydia Screening Programme in[END_REF]. We are not aware of any studies prior to ours that have compared behavioural data of young people assessing opportunistic chlamydia screening in England with a representative sample of the general population. This is probably because there are no population-based behavioural data with chlamydia status that are comparable to Natsal for this purpose.

Interpretation of the findings

Our findings suggests that the testing venues delivering opportunistic NCSP chlamydia screening, and/or testing offers, have been preferentially accessed, and/or accepted, by sexually active young adults at higher than average risk of infection. The reasons for this, whether selection by attendance at screening venues, by health professionals offering of tests or by acceptance of test offers, or a combination of these potential selection pressures, are a matter for separate research.

Implications of the findings

Models used to explore the effectiveness of chlamydia screening in England have assumed that those accepting testing are a random sample of the population eligible for chlamydia testing [START_REF] Turner | Modelling the effectiveness of chlamydia screening in England[END_REF]. However, the results from this study suggest that those participating in the screening programme are at higher risk than the general population. The consequent higher yield of infections detected (and treated) by screening is an important feature that should be factored in when evaluating the impact of chlamydia screening in England. This could be done by incorporating test-positivity and population prevalence into models as two separate variables. Assuming all other model assumptions were correct, this bias to higher risk participants in screening that we have observed in England suggests that the effectiveness of chlamydia screening in England has probably been higher than 'random sample' models would have predicted for the observed levels of coverage. Further analysis of this kind should be conducted as opportunities arise to monitor who is accessing screening and to inform modelling studies.

KEY MESSAGES

Table 1 : A comparison of the demographic and behavioural characteristics of NCSP participants relative to the general population aged 16-24 years, according to Natsal-2, by gender.

 1

	Gender:	Men

Table 1 :

 1 1. Weighted, unweighted denominators in Natsal-2. 2. Natsal-2 counts and percentages are weighted.

Table 2A : Percentages, crude and adjusted odds ratios for demographic and behavioural risk factors associated with testing positive for chlamydia, by data source: Males

 2A

	Data source:		Natsal-2					NCSP
	Demographic and behavioural risk factors:	Row % (95% CI) testing positive	Crude odds ratio (95% CI)	Adjusted odds ratio (95% CI)	3	Denominator	1, 2	Row % (95% CI) testing positive	Crude odds ratio (95% CI)	Adjusted odds ratio (95% CI)	3	Denominator	2
	All	3.0%	-	-		270, 342		9.8%	-	-
		(1.4%-6.6%)						(9.0%-10.6%)
	Age group (years)		p=0.986	p=0.744					p=0.118	p=0.175
	18-19	3.1%	1	1		68, 79		9.1%	1	1
		(0.7%-11.9%)						(8.0%-10.3%)
	20-24	3.0%	0.98	1.36		202, 263		10.4%	1.15	1.13
		(1.2%-7.6%)	(0.17-5.69)	(0.21-8.79)			(9.3%-11.5%)	(0.96-1.38)	(0.95-1.35)
	Ethnic group		p=0.535	p=0.056					p=0.333	p=0.123
	White	3.0%	1	1		243, 310		9.7%	1	1
		(1.3%-6.9%)						(8.9%-10.5%)
	Black	0.0%	N/A	N/A		11, 12		12.3%	1.31	1.47		285
								(9.0%-16.6%)	(0.91-1.89)	(1.02-2.13)
	Other minority	5.8%	2.03	9.13		16, 20		9.1%	0.94	1.00		263
	groups	(0.8%-32.9%)	(0.22-19.0)	(0.94-88.6)			(6.2%-13.3%)	(0.61-1.44)	(0.65-1.55)
	Condom use at last sex	p=0.033	p=0.070					p<0.001	p<0.001
	Yes	0.5%	1	1		125, 151		7.5%	1	1
		(<0.01%-3.5%)						(6.3%-8.9%)
	No	5.0%	10.4	10.3		145, 191		10.7%	1.49	1.53
		(2.2%-11.3%)	(1.21-89.3)	(0.82-127.8)			(9.8%-11.8%)	(1.20-1.84)	(1.23-1.89)
	≥ 2 sexual partners in last year							p<0.001	p<0.001
	No	0.0%	Not	Not		140, 187		7.8%	1	1
								(6.8%-8.9%)
	Yes	6.7%	applicable	applicable	130, 155		11.4%	1.53	1.62
		(3.0%-14.1%)						(10.3%-12.6%)	(1.27-1.84)	(1.34-1.96)

Table 2B : Percentages, crude and adjusted odds ratios for demographic and behavioural risk factors associated with testing positive for chlamydia, by data source: Females

 2B

	Data source:		Natsal-2					NCSP
	Demographic and behavioural risk factors:	Row % (95% CI) testing positive	Crude odds ratio (95% CI)	Adjusted odds ratio (95% CI)	2	Denominator	1, 2	Row % (95% CI) testing positive	Crude odds ratio (95% CI)	Adjusted odds ratio (95% CI)	2	Denominator	2
	All	2.8%				357, 337		9.0%				11768
		(1.6%-4.9%)						(8.5%-9.5%)
	Age group (years)		p=0.967	p=0.809					p=0.001	p=0.010
	18-19	2.9%	1	1		87, 85		10.0%	1	1		4773
		(0.9%-8.8%)						(9.2%-10.9%)
	20-24	2.8%	0.97	1.18		270, 252		8.2%	0.80	0.84		6995
		(1.5%-5.3%)	(0.25-3.74)	(0.30-4.73)			(7.6%-8.9%)	(0.71-0.91)	(0.74-0.96)
	Ethnic group		p=0.147	p=0.145					p=0.011	p=0.030
	White	2.5%	1	1		323, 309		9.1%	1	1		11129
		(1.3%-4.5%)						(8.6%-9.7%)
	Black	13.7%	6.31	5.85		11, 7		3.0%	0.30	0.34		203
		(2.6%-48.5%)	(0.95-41.8)	(0.96-35.8)			(1.3%-6.4%)	(0.13-0.69)	(0.15-0.76)
	Other minority	5.1%	2.12	2.01		23, 21		7.8%	0.84	0.91		436
	groups	(1.1%-20.6%)	(0.38-11.7)	(0.37-10.9)			(5.6%-10.7%)	(0.59-1.21)	(0.64-1.31)
	Condom use at last sex	p=0.759	p=0.034					p=0.016	p=0.004
	Yes	2.5%	1	1		110, 97		8.0%	1	1		3636
		(1.0%-6.1%)						(7.2%-8.9%)
	No	3.0%	1.20	1.31		247, 240		9.4%	1.19	1.23		8132
		(1.5%-5.8%)	(0.37-3.86)	(0.43-3.93)			(8.8%-10.0%)	(1.03-1.37)	(1.07-1.42)
	≥ 2 sexual partners in last year	p=0.068	p=0.079					p<0.001	p<0.001
	No	1.9%	1	1		244, 241		6.6%	1	1		6070
		(0.8%-4.6%)						(6.0%-7.2%)
	Yes	5.3%	2.92	2.85		113, 96		11.5%	1.83	1.81		5698
		(2.7%-10.1%)	(0.92-9.22)	(0.88-9.22)			(10.7%-12.3%)	(1.61-2.09)	(1.58-2.06)

• Condom non-use and having multiple sexual partners is strongly associated with testing positive for chlamydia, amongst both National Chlamydia Screening Programme (NCSP) participants and the general population.

• The NCSP is testing individuals at higher risk of chlamydia than those in the general population.

• As chlamydia screening becomes more widespread, further study of who is being tested would inform evaluation of screening effectiveness.

WORD COUNT: 3515 excluding tables, key messages, refs and statements below, 3155 also excluding title page and abstract.

Contributors:

JR and KS initiated the study. All authors contributed to determining the research questions and study design. JR and CM conducted the statistical analyses. JR wrote the first draft of the manuscript. All authors provided substantive contributions to the final version of the manuscript.

STATEMENT OF COMPETING INTEREST

JR, KS, CF, and MM are employed by the Health Protection Agency, and work for the National Chlamydia Screening Programme.

SOURCE OF FUNDING STATEMENT

This work was undertaken by the Health Protection Agency who received funding from the Department of Health; the views expressed in this publication are those of the authors and not necessarily those of the Department of Health.

LICENCE FOR PUBLICATION

The Corresponding Author has the right to grant on behalf of all authors and does grant on behalf of all authors, an exclusive licence (or non exclusive for government employees) on a worldwide basis to the BMJ Publishing Group Ltd to permit this article (if accepted) to be published in STI and any other BMJPGL products and sublicences such use and exploit all subsidiary rights, as set out in our licence http://group.bmj.com/products/journals/instructionsfor-authors/licence-forms.