Marita Andersson Gronlund
email: marita.gronlund@neuro.gu.se

MD, PhD Jovanna Dahlgren

CO, PhD Eva Aring

MD Maria Kraemer

Ann Hellstrom

MD, PhD Marita Andersson Grönlund

MD, PhD A Hellström

Ophthalmological findings in children and adolescents with Silver Russell Syndrome

Keywords: visual function, ocular findings, Silver Russell Syndrome (SRS), prenatal programming, intrauterine growth-retarded (IUGR)

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

INTRODUCTION

Silver Russell Syndrome (SRS) is a rare syndrome characterized by pre-and postnatal growth retardation, a triangular face, micrognathia, lateral asymmetry, and clinodactylia (Fig. 1A). [START_REF] Rossignol | Silver-Russell syndrome and its genetic origins[END_REF] So far, little is known about the cause of the disease, although several genetic abnormalities have been described involving chromosomes 1, 7, 8, 11, 15, 17, and 18.[2-3] Today we know that a large proportion of children with SRS have either hypomethylation at the imprinting control region of chromosome 11p15 (30-65%) with a more severe phenotype [4] or maternal uniparental disomy of chromosome 7 (5-10%) with a milder phenotype. [4][START_REF] Preece | Maternal uniparental disomy 7 in Silver-Russell syndrome[END_REF] The candidate genes are those imprinted in the regions of chromosomes 7 and 11, such as insulin-like growth factor 2 (IGF2), IGF2 receptor (IGF2R), and growth factor receptor-bound protein 10 (Grb 10).

Different ophthalmological findings, such as ptosis, epicanthal folds, hypertelorism, long eyelashes, eyebrows meeting in the midline, protruding eyeballs, microphthalmia, strabismus, blue sclera, absence of the lacrimal duct, microcornea, heterochromia, myopia, subnormal visual acuity (VA), cataract, asymmetry and central excavation of the optic discs, pigmentary retinopathy, retinal detachment, decreased full-field electroretinogram (ERG) potential, and congenital glaucoma, have previously been described in separate case reports in patients with SRS. [START_REF] Kass | Russell-Silver dwarfism[END_REF][START_REF] Siegel | Optic nerve asymmetry in a child with Russell-Silver syndrome[END_REF][START_REF] Kato | Congenital glaucoma and Silver-Russell phenotype associated with partial trisomy 7q and monosomy 15q[END_REF] However, to our knowledge, no detailed ophthalmological investigation in a larger number of patients with SRS has previously been reported. Therefore, the purpose of this study was to prospectively evaluate visual function and ocular findings in patients with SRS, a group born extremely small for gestational age (SGA).

MATERIALS AND METHODS

Patients

Eighteen children and adolescents (8 girls, 10 boys; mean age 11.6 years, range 3.4-18.1 years) fulfilling all criteria for the diagnosis of SRS were referred from across Sweden and examined by a national multidisciplinary team at the Queen Silvia Children's Hospital, Gothenburg, Sweden. Table 1 shows auxological data at birth and proportion of preterm births in the children with SRS. Birth weight and birth length were converted into standard deviation scores (SDSs) based on Swedish reference values. [START_REF] Wikland | Swedish population-based longitudinal reference values from birth to 18 years of age for height, weight and head circumference[END_REF] The mean birth weight SDS was -3.1 (range -1.2 --4.5) and the mean birth length SDS was -3.2 (range 0 --5.6); all of these children were SGA by birth weight, birth length, or both. At the time of the investigation, 16 of the 18 children had been treated with growth hormone (GH) for a mean period of 6.6 years (range 2-14 years).

Methods

A detailed ophthalmological evaluation was performed, including:

Determination of best corrected visual acuity for near and distant fixation

Best corrected VA (BCVA) was tested with a linear KM-Boks chart. [START_REF] Moutakis | Using the KM visual acuity chart for more reliable evaluation of amblyopia compared to the HVOT method[END_REF] For children who could not read the KM-Boks chart, we used the HOTV chart. Distance VA was tested monocularly at a distance of 3 m and near vision was tested binocularly at a distance of 0.33 m.

Refraction under cycloplegia

Refraction tests were performed with an autorefractor (Topcon A6 300, Topcon Corporation, Tokyo, Japan) after a single instillation of a mixture of cyclopentolate (0.85%) and phenylephrine (1.5%). Significant refractive errors were defined as the spherical equivalent (SE) of ≥0.5 dioptre (D) for myopia or ≥2.0 D for hyperopia. Astigmatism was assessed at a level of ≥0.75 D SE, and anisometropia at ≥1.0 D SE.

Investigation of strabismus and ocular motility

Heterotropia, defined as intermittent or constant, near (0.33 m) or at distance (3 m), was diagnosed with cover-uncover tests. Heterophoria was diagnosed with alternate cover tests, and deviations were quantified using alternate prism cover tests. Exophoria was defined as values below the 5 th percentile in the control group (negative values), and esophoria as values above the 95 th percentile (positive values). Thus, the cut-off values defining significant heterophoria were between <-2 and >0 prism dioptres (pD) for distance and between <-10 and >0 pD for near vision. Motility was assessed with a penlight and described in terms of overor under-functioning.

Testing of stereo acuity

Stereo acuity was tested with the TNO random dot stereo test, the Lang I stereo card, or the Titmus test, as appropriate. Subnormal stereo acuity was defined as >60 seconds of arc.

Near point of convergence

Near point of convergence (NPC), in cm, was measured three times with a Royal Air Force ruler and the mean value of the measurements was recorded.

Fixation recordings

The simultaneous horizontal and vertical positions of the right and left eye were recorded using the Orbit infrared (IR) system (IOTA Inc, Timrå, Sweden). In this IR device, pulsed infrared light, emitted inside a pair of goggles, is reflected against the ocular surface and detected by eight detectors. Eye position signals are conducted via a sound card to a computer, where they are recorded. The investigation is described in detail elsewhere. [START_REF] Aring | Visual fixation development in children[END_REF] 7.

Assessment of ocular dimensions

The medial intercanthal distance (ICD) and right and left palpebral fissure lengths (PFLs), in mm, were measured with a ruler. Total axial length (TAL) was measured by ultrasound biometry (Paxis, version 2.01; BIOVISION, Clermont-Ferrand, France).

Examination of the anterior segment, media, and ocular fundus

Examination of the anterior segment of the eye was performed with a slit lamp and the ocular fundus was examined by indirect ophthalmoscopy.

Electroretinogram

A full-field ERG of one eye was recorded using a bipolar contact lens in a Nicolet Analysis System (Nicolet Biomedical Instruments, Madison, WI, USA) as described elsewhere, [START_REF] Andréasson | Phenotypes in three Swedish families with x-linked retinitis pigmentosa caused by different mutations in the RPGR gene[END_REF] according to International Society for Clinical Electrophysiology of Vision standards. [START_REF] Marmor | Standard for clinical electroretinography (1999 update)[END_REF] In some cases, skin electrode ERGs were registered. This involved placing a silver-silver chloride electrode just below the central lower lid margin. A single flash, using a red filter, with a Grass flashlight stimulus of supramaximal intensity (<100 µs, ~0.5 J) was projected at a distance of approximately 20 cm to the child's eye.

Photography of the ocular fundus for quantitative digital image analysis

Ocular fundus photographs taken in cycloplegia were analysed with a specially designed computer-assisted digital mapping system [START_REF] Strömland | Morphometry of the optic nerve and retinal vessels by computer-assisted image analysis[END_REF] in regard to the optic disc area (ODA), optic cup area, neuroretinal rim area, tortuosity of veins and arteries, and number of branching points.

Statistical analysis

Means, standard deviations (SDs), medians. and ranges were calculated for descriptive purposes. For a comparison between two groups, Mann-Whitney's U-test was used for ordered and continuous variables; for dichotomous variables, Fisher's exact test was used.

Test results were considered to be significant at values of p<0.05. The reference group for this study was selected individually by minimizing the maximal t-values over the variables age and sex between the group of children with SRS and a reference group of 143 healthy, Swedish school-aged children. [START_REF] Grönlund | Ophthalmological findings in a sample of Swedish children aged 4-15 years[END_REF]

Ethical approval

The study was approved by the Ethical Committee at the Medical Faculty, Sahlgrenska

Academy at the University of Gothenburg, Sweden. Informed consent was obtained from the parents of all the children participating in the study.

RESULTS

Altogether, 17/18 children with SRS had ophthalmologic abnormalities. Table 2 summarizes the ophthalmological findings in each of the 18 children and youths with SRS.

Fixation

No significant difference between the eyes or between the children with SRS and the controls was found regarding fixation in primary position, i.e. fixation time, number of intruding saccades, drifts, or blinks recorded by the Orbit IR system.

Ocular dimensions

The mean and SDs of ICD, PFL, and TAL of right and left eyes for the children with SRS as well as for the reference group are shown in Table 4. No significant difference between right and left eyes was found regarding PFL and TAL in the children with SRS. An ICD of ≥5-10

Examination of the anterior segment, media, and ocular fundus

Abnormal growth of the eyebrows was recognized in four children with SRS and four had long, full prominent lashes bilaterally. Epicanthal folds and unilateral ptosis were seen in one child (Table 2).

Electroretinogram

Electroretinogram was performed in 16 children with SRS, in 11 of whom skin-electrode ERG was performed, while 5 underwent full-field ERG. The retinal conditions showed normal activity in all children.

Digital image analysis

Ocular fundus photographs of 16 children were analysed with regard to the ODA, cup area, and neuroretinal rim area. Tortuosity of veins and arteries and number of branching points were analysed in 13 children.

Optic disc morphology

Of 16 children, 3 had an ODA and 3 had a neuroretinal rim area that was smaller than the 5 th percentile for the controls (<1.82 mm 2 and <1.52 mm 2 , respectively) (Table 2; Fig. 1B). Three children had a cup area larger than the 95 th percentile for the controls (>0.79 mm 2 ) (Table 2).

There was no significant difference in optic disc, cup, or rim area between the two groups.

However, there was a tendency towards larger cups in children with SRS (median 0.56; range 0.0-1.20) than in controls (median 0.34; range 0.0-1.61) (p=0.056). No significant difference between right and left eyes regarding optic disc, cup, or rim area was found.

not previously been reported in children with SRS. In addition, three children (cases no 3, 5, and 8) have an astigmatism with ≥1.0 D difference between the eyes (Table 2). Knowledge of an asymmetry of refraction is important for the development of the child's visual function and for preventing anisometric amblyopia.

Growth hormone and insulin-like growth factor 1 (IGF-1) are involved in ocular growth by influencing the synthesis of the extracellular matrix of the sclera and by inducing angiogenesis. [START_REF] Parentin | Refractive evaluation in children with growth defect[END_REF] An association between optic nerve hypoplasia and reduced retinal vascularization, as well as a mean hyperopic defect related to shorter axial length and an increased central corneal thickness, is documented in individuals with GH deficiency. [START_REF] Parentin | Refractive evaluation in children with growth defect[END_REF][START_REF] Bourla | Insulinlike growth factor I affects ocular development: a study of untreated and treated patients with Laron syndrome[END_REF][START_REF] Parentin | Central corneal thickness in children with growth hormone deficiency[END_REF] At the time of this investigation, 16 children with SRS were being treated with GH, three of whom had small optic discs as measured by the digital analysis system. Interestingly, despite being treated with GH, these children still had significantly shorter TALs than the controls.

Overall, however, their emmetropization seemed to process normally. [START_REF] Parentin | Congenital growth hormone deficiency and eye refraction: a longitudinal study[END_REF] hypothesized, in their study on the effect of GH therapy on refraction, that correct and well-timed substitutive GH therapy could permit normal emmetropization. [START_REF] Parentin | Congenital growth hormone deficiency and eye refraction: a longitudinal study[END_REF] It can only be speculated whether the change in refraction is related to the GH-induced somatic growth per se, or whether it is a direct effect of GH and/or IGF-1. Further studies are needed regarding the effects of GH replacement therapy on the development of the visual and ocular system.

Abnormal optic nerve and retinal vascular morphology, i.e. reduced rim area and decreased number of central branching points, have previously been reported in young adults who were IUGR infants. [START_REF] Hellström | Abnormal retinal vascular morphology in young adults following intrauterine growth restriction[END_REF][START_REF] Ley | Abnormal retinal optic nerve morphology in young adults after intrauterine growth restriction[END_REF] In the present study, 3/16 children had a reduced rim area and 5/13 children had a decreased number of branching points. A decrease in the neuroretinal rim area reflects either a decrease in the number of axons or a reduction in axonal volume in the optic disc. The underlying mechanisms and the importance of the decreased number of central branching points are not known.

It is known that humans who have been undernourished during gestation become less healthy later in life. [START_REF] Barker | Fetal programming of coronary heart disease[END_REF] Several studies have shown that prenatal exposure to famine is associated with cardiovascular risk factors and coronary heart disease, brain abnormalities, affective disorders, and antisocial behaviour. [START_REF] Neugebauer | Prenatal exposure to wartime famine and development of antisocial personality disorder in early adulthood[END_REF][START_REF] Brown | Further evidence of relation between prenatal famine and major affective disorder[END_REF] Children with SRS are a heterogeneous group with different genetic backgrounds. [START_REF] Abu-Amero | The genetic aetiology of Silver-Russell syndrome[END_REF] Though they exhibit different severity, however, they all share the same general phenotype and have all had both in utero and postnatal growth retardation. Whether it is the genetic background or the foetal milieu per se which has an adverse impact in these children is not known. Children with SRS are found to have impaired growth early in gestation, and this would probably affect several organs, including the brain and the eyes. Besides the ophthalmological impact, as described here, we hypothesize that perception and brain function may also be impaired. It is therefore of great interest to evaluate further the visual perception and other cognitive functions of children with SRS.

In conclusion

On the basis of our findings, we recommend that an ophthalmological examination be performed in children born with severe IUGR, and especially in children with SRS, to optimize the visual and ocular development of these children.

Table 1 .

 1

		SRS group	Reference group
		(n=18*)	(n=99*)
	Birth weight (g)
	Mean (range)	1,880 (1,164-2,890)	3,590 (1,540-4,890)
	Birth length (cm)
	Mean (range)	42 (34-48)	50 (40-57)
		(n=17)
	Head circumference (cm)
	Mean (range)	32 (28.5-34)	33.6 (31-38.5)
		(n=11)	(n=98)
	Gestational age (wks)
	Mean (range)	37 (33-41)	39.6 (35-42)
	Preterm birth (<37 wks GA)
	n (%)	9 (50.0%)	3 (3.0%)

Table 2

 2

	.
	Case/sex/	Visual	Refraction	Aniso-	Strabismus/	Stereo	Anterior segment/
	age (yrs)	acuity	Sph/cyl	metropia	ocular	Acuity ('')/	media/fundus
	and if	logMAR	OD	>1D SE	motility	NPC (cm)
	treated with	OD/OS	OS
	GH (+/-)
	1/M/17/+	0.1/0.1 † +6.50/-0.50	-	NA	NA	Small ODA
			+6.25/-0.5
	2/F/3/-	0.2/0.2* +2.25/-0.25	+	Orthophoria	240''	‡, abnormal lashes
			+3.50/-0.25			6 cm
	3/M/9/+	0.05/0.2	+0.25/-0.75	-	Esotropia	1,200''	‡, ptosis, epicanthal
		†	+0.75/-1.75			6 cm	folds, abnormal
							eyebrows, large cup,
							small rim area
	4/F/8/+	0.1/0.1 † +1.0/-0.50	-	Exophoria	60''	Large cup,
			+1.25/-0.50			6 cm	tortuosity of retinal
							vessels
	5/F/14/+	0.0/0.0 † -0.5/-1.25	-	Exophoria,	30''
			-1.0/-0.25		NPC >6 cm	6 cm
	6/M/12/+	0.0/0.0 † +2.75/-1.25	+	Exotropia	60''	‡, abnormal
			+4.50/-1.75			20 cm	eyebrows and
							lashes, small ODA,
							small rim area
	7/F/15/+	-	+1.0/-0.25	-	Esophoria	60''	‡, abnormal lashes
		0.1/0.0 †	+1.25/-0.50			6 cm
	8/M/8/+	0.2/0.3	+1.5/-0.5	-	Orthophoria,	60''	‡, abnormal lashes,
			+1.25/-1.50		underaction	6 cm	ODAs >0.8 mm diff
					of both LR
	9/M/18/+	0.0/-	-1.0/-1.75	-	Orthophoria,	60''	Abnormal eyebrows
		0.1 †	-1.0/-1.0		NPC >6 cm	10 cm
	10/M/12/+	0.0/0.0	+0.25/-0.75	-	Exophoria	30''	Small rim area
			+0.50/-0.25			6 cm
	11/M/13/+	0.0/0.0 † -3.50/-1.50	-	Orthophoria	60''	Large cup
			-3.50/-1.50			6 cm
	12/M/13/+	0.0/0.0	+1.5/-0.25	+	Orthophoria	30''	Abnormal
			+0.5/-0.25			6 cm	eyebrows, small
							ODA
	13/F/10/+	0.1/0.1	+1.0/-0.50	-	Orthophoria	60''	Tortuosity of retinal
			+0.75/-0.25			6 cm	vessels
	14/F/15/-	0.05/0.2 -1.5	-	Esophoria	240''
			-1.0/-0.75			6 cm
	15/M/9/+	0.3/0.3 † +0.50/-0.25	-	Esotropia,	100''	‡, tortuosity of
			+0.50/-0.50		overaction of	neg	retinal vessels
					both IOs
	16/F/10/+	0.1/0.1	+0.25	-	Exophoria	60''
			+.25/-0.25			6 cm
	17/M/12/+	0.0/0.0	+-0/-0.50	-	NA	NA	ODAs >0.8 mm diff
			+0.25/-0.50
	18/F/12/+	0.0/0.0	+1.5/-0.25	-	Orthophoria	15''	‡, tortuosity of
			+1.75/-0.5			6 cm	retinal vessels

In total, 11/18 children with SRS had refractive errors (ref 33/99) (p=0.05). The median SE in the study cohort was +0.56 (range -4.25-+6.25) for right eyes and +0.31 (range -4.25-+6.125) for left eyes. No statistical difference between right and left eyes was found regarding VA, refraction in SE, or astigmatism in the children with SRS.

Table 3 .

 3

	Ophthalmological	SRS group	Ref group	p-value
	findings	(n=18*)	(n=99*)
	n (%)
	Visual acuity (logMAR)
	best eye at distance
	≥0.3	1 (5.6%)	0 (0%)	n.s.
	0.2-0.1	6 (33.3%)	1 (1.0%)	<0.0001
	<0.1	11 (61.1%)	98 (99.1%)	<0.0001
	Visual acuity (logMAR)
	binocular at near range
	≥0.3	1 (6.7%); n=15	0 (0%)	n.s.
	0.2-0.1	2 (13.3%); n=15	0 (0%)	0.04
	<0.1	12 (80.0%); n=15	99 (100%)	0.004
	Refraction
	myopia (≥0.5 D SE)	4 (22.2%)	9 (9.1%)	n.s.
	hyperopia (≥2.0 D SE)	3 (16.7%)	5 (5.1%)	n.s.
	astigmatism (≥0.75 D)	8 (44.4%)	25 (25.3%)	n.s.
	anisometropia (≥1.0 D SE)	3 (16.7%)	3 (3.0%)	0.046
	wearing glasses	9 (50.0%)	6 (6.1%)	<0.001
	Strabismus
	heterotropia	3 (18.8%); n=16	3 (3.0%)	n.s.
	significant heterophoria
	at distance	1 (6.3%); n=16	5 (5.1%)	n.s.
	significant heterophoria
	at near	2 (12.5%); n=16	2 (20.2%)	n.s.
	Stereo acuity (>60'') †
		2 (12.5%); n=16	0 (0%)	0.02
	NPC >6 cm †
		2 (12.5%); n=16	0 (0%)	0.02
	*Where numbers differ from the total number of children in the group, they are given separately for each
	category; †without having heterotropia.
	D = dioptre; logMAR = log of the minimal angle of resolution; NPC = near point of convergence; n.s. = not
	significant; SE = spherical equivalent.

  PFL was recorded in 7/15 children (ref 10/99) (p=0.002).

	Table 4.
	Ocular	SRS	Ref	p-value
	dimensions	(n=18)*	(n=99)*
		Mean (SD)	Mean (SD)
	ICD (mm)	29.40 (3.36)	28.45 (2.23)	n.s.
		(n=15)	(n=88)
	PFL (mm) OD	25.38 (2.55)	26.32 (1.89)	n.s.
		(n=16)	(n=88)
	PFL (mm) OS	25.25 (2.59)	26.48 (1.84)	0.03
		(n=16)	(n=88)
	TAL (mm) OD	22.09 (1.07)	22.71 (0.68)	0.006
		(n=12)	(n=75)
	TAL (mm) OS	21.96 (1.11)	22.76 (0.70)	0.001
		(n=12)	(n=75)
	*Where numbers differ from the number of children in the group, they are given separately for each category.
	ICD = medial intercanthal distance; n.s. = non-significant; OD = right eye; OS = left eye; PFL = palpebral
	fissure length; SD = standard deviation; SRS = Silver Russell Syndrome; TAL = total axial length.

mm ≥

ACKNOWLEDGEMENTS

This study was supported by the Gothenburg Medical Society, the W & M Lundgrens Vetenskapsfond II, the Sven Jerring Foundation, the Swedish Research Society (grant #10863 and grant 522-2005-7238), and Research and Development of Region Västra Götaland (grant 2002-2004). Our study results have been presented in part at the 11th Nordic Paediatric Ophthalmology Congress held in Uppsala, Sweden, in September 2003 and at the ARVO annual meeting held in Fort Lauderdale, FL, USA, in April 2004.

The authors would like to thank Birgitta Melander, Jonna Johansson, and Eva Rudholm for their technical assistance. We would also like to thank Professor Kerstin Albertsson-Wikland, Birgit Lidwall, and the staff at Endocrine Ward 335, who took care of the children with SRS.

Emma Jaensson and Per Ekman of the Statistiska Konsultgruppen, Gothenburg, Sweden, are gratefully acknowledged for their help with the statistical analyses.

Licence for publications: The corresponding author has the right to, and does, grant on behalf of all the authors an exclusive licence on a worldwide basis to the BMJ Publishing Group Ltd and its licencees, to permit this article to be published in the British Journal of Ophthalmology and any other BMJ Group products and to exploit all subsidiary rights, as set out in the BJO's licensing agreement (http://bjo.bmjjournals.com//ifora/licence.pdf).

Competing interests: The authors, Andersson Grönlund M, Dahlgren J, Aring E, Kraemer M, Hellström A, have no competing interests to declare.

Retinal vessel morphology

Two out of 13 children with SRS had an increased tortuosity of arteries (ITA), and four children had an increased tortuosity of veins (ITV) above the 95 th percentile for the controls (>1.17 and >1.09, respectively) (Table 2; Fig. 1C). Five children with SRS had a decreased number of central vascular branching points below the 5 th percentile for the controls (<21) (see Fig. 1C). However, no significant difference was found in ITA, ITV, or number of branching points between the two groups.

DISCUSSION

The children and adolescents with SRS in this prospective study showed a high number of visual and ocular abnormalities, such as subnormal VA, refractive errors, anisometropia, subnormal stereo acuity, and subnormal NPC, as well as short eyes compared with controls.

In addition, ptosis, epicanthal folds, long eyelashes, eyebrows meeting in the midline, hypertelorism, small optic discs, large cups, and increased tortuosity of retinal vessels were seen in some of the children.

Children with SRS commonly show a lateral facial and/or skeletal asymmetry. In an interesting case report presented by [START_REF] Siegel | Optic nerve asymmetry in a child with Russell-Silver syndrome[END_REF], an asymmetry of the optic discs was demonstrated in a child with SRS. [START_REF] Siegel | Optic nerve asymmetry in a child with Russell-Silver syndrome[END_REF] A 0.6 mm difference between the optic discs was measured biomicroscopically using a 78 D-ruled indirect lens. Two of the children in our study (cases 8 and 17) also showed a difference of optic disc size (0.88 mm), as measured by the digital analysing system used in this study (Table 2). However, for the group as a whole, no significant difference between the sizes of the right and left optic discs could be found. On the other hand, we found that anisometropia (≥1.0 D SE), reflecting an asymmetric disorder, was more common in children with SRS than in controls (Tables 2 and3). This finding has

LEGENDS Table 1

Auxological data at birth and number of preterm births in 18 children with Silver Russell Syndrome (SRS) as well as in an age-and sex-matched reference group (n=99).

.

Table 2

Ophthalmological findings in the study population, children and adolescents with Silver Russell Syndrome (SRS).

Table 3

Ophthalmological findings, expressed as number (n) and percentage (%), in children with Silver Russell Syndrome (SRS) compared with the reference group.

Table 4

Ocular dimensions, expressed as mean (standard deviation, SD), in children with Silver Russell Syndrome (SRS) and the reference group, respectively.

Figure 1A-C

A. Silver Russell Syndrome (SRS). Photograph showing a 3-year-old girl with SRS, born in gestational week 36 at birth weight 1,550 g (-4.2 standard deviation score, SDS). The prominent forehead, triangular face, micrognathia, small lips, full prominent eyelashes, and the longer intercanthal distance compared with palpebral fissure length are noticeable. At examination her visual acuity was 0.2 logMAR in both eyes and she was hyperopic as well as anisometropic. Her stereo acuity was 240''.


C Deil

W Domainko

G Hermann

A.-C Clapson

A Förster

C Van Eldik

W Hofmann

Capability of Cherenkov Telescopes

Keywords: Ultra-fast optical photometry, Cherenkov telescope PACS: 95.55.Ka, 95.55.Qf, 95.75.Wx, 97.80.Jp

Introduction

At optical wavelengths, the sky has been surveyed in deep exposures typically lasting minutes to hours. Much faster optical flares may have escaped detection, since the dynamic sky at sub-second timescales has rarely been explored in the past [1]. The availability of high quantumefficiency, fast photodetectors (photomultiplier tubes (PMTs), avalanche photodiodes or charge-coupled devices (CCDs)) has lead a few groups to construct high-timeresolution photometers, imagers or spectrographs and operate them on medium to large optical telescopes [2][3][4][5][6].

The very large light collecting areas (> 100 m 2 ) of reflectors of imaging Cherenkov telescopes such as H.E.S.S. [7], MAGIC [8] and VERITAS [9] provide an interesting alternative to normal optical telescopes [1]. Up to now, the only optical observations performed using the reflector of a Cherenkov telescope concerned the measurement of the optical pulsed emission from the Crab pulsar [10,11].

Compact objects like neutron stars and stellar-mass black holes are prime candidates as sources of fast optical flares. The fastest expected time variability of the emission from these systems is given by the light travel time through the immediate vicinity of the compact object, 10 -100 μs for a typical size of tens of kilometers [1]. Such short flares can be produced by the accretion of lumps of matter, for instance in X-ray binary systems from the stellar companion [12]. Relativistic outflows can further shorten the observed timescales by Doppler boosting (for exceptional examples in the case of super-massive black holes see [13][14][15]). Optical variability in the millisecond range has been observed for pulsars [16] and X-ray binaries [17][18][19] and has been searched for in objects known to exhibit flares in other wavelength regions, for instance at radio frequencies from rotating radio transients (RRATs) [20,21]. No optical variability on timescales shorter than 100 μs has been detected from any astronomical target so far.

When trying to detect astronomical flares, natural and artificial flaring events occurring in the earth's atmosphere or orbit have to be considered. Known producers of such terrestial optical flares on sub-second timescales are airplanes, satellites, lightning and shooting stars [22]. Additionally, cosmic-ray induced air showers produce light flashes lasting a few nanoseconds [23].

This paper reports on 43 h of high-time-resolution optical photometry of X-ray binaries and one RRAT using the reflector of a H.E.S.S. Cherenkov telescope during moonshine. A comparison of the performance of H.E.S.S. with Preprint submitted to Elsevier 30 December 2008 regular optical telescopes for such fast photometry observations is presented in Sec. 2. The custom-built detector that was used to record continuous light curves at microsecond time resolution is described in Sec. 3. An overview of the observations and a description of the flare-finding algorithm that was used to search for flares on timescales of 2 μs -100 ms is given in Sec. 4. The events that were detected are presented in Sec. 5, and Sec. 6 gives an outlook on how this new technique might be improved in the future.

Comparison with Regular Optical Telescopes

The main difference in optical performance between Cherenkov and regular optical telescopes is that Cherenkov telescopes have huge reflectors but poor angular resolution, whereas optical telescopes have smaller reflectors and good angular resolution (see also [24]). Thus we assume here that the sensitivity of a Cherenkov telescope is limited by the shot noise from night-sky background (NSB) photons, whereas the optical telescope is considered completely background-free and photon-limited.

Consider a flare of known duration τ occuring at a known point in time. This scenario might happen when a flare is detected in X-ray or radio observations and searched for in the optical band. For an optical telescope of diameter D O , the minimum detectable flux φ O [ ph m -2 s -1 ] is given by the requirement that it has to collect at least one photon,

φ O ∼ 4 π 1 D 2 O τ . ( 1
)
The detection limit for a Cherenkov telescope is given by the condition that the number of photons detected from the source S = φ C π 4 D 2 C τ must be at least as large as the fluctuations in the number of photons from the NSB during the flare. The photodetector aperture σ C [rad] must be chosen to be approximately equal to the width of the point spread function (PSF) of the telescope. For an isotropic night-sky surface brightness ψ NSB [ ph m -2 s -1 sr -1 ], telescope diameter D C and flare duration τ , the average number of NSB photons during the flare will be

B = ψ NSB ( π 4 ) 2 D 2 C σ 2 C τ and the Poisson noise N = √ B. Solving the equation S/N ∼ 1 for φ C gives the minimum detectable flux φ C ∼ σ C D C √ ψ NSB √ τ , (2)
and hence

φ C φ O ∼ π 4 σ C D C D 2 O ψ NSB τ . ( 3
)
The main result from Eqn. (3) is that the relative sensitivity φ C /φ O scales with √ τ , i.e. that Cherenkov telescopes perform better relative to optical telescopes for the detection of shorter flares. Inserting the values for H.E.S.S. into Eqn. (3), an absolute comparison with regular optical telescopes can be made. Figure 1 shows that even under with a normal optical telescope for high-time-resolution photometry. For a given flare duration τ , the diameter of a background-free optical telescope was calculated to achieve the same sensitivity as H.E.S.S. using Eqn. (3) and the condition φ C φ O = 1 with the following parameters: H.E.S.S. telescope diameter D C ∼ 11 m, PSF width σ C ∼ 2.5 arcmin and typical NSB brightness without moon at the H.E.S.S. site ψ NSB ∼ 2 × 10 12 ph m -2 s -1 sr -1 in the wavelength range 300 -650 nm [25]. For typical NSB levels during moonshine see Fig. 4. moderate moonshine conditions Cherenkov telescopes are competitive with meter-class optical telescopes in the submillisecond time domain, whereas for the search of longer flares optical telescopes perform better.

For observations of high-mass X-ray binary systems with a bright companion optical telescopes will be backgroundlimited as well. Then, the relative sensitivity of Cherenkov and optical telescopes no longer depends on the flare duration, but is given by

φ C φ O ∼ D O D C R C,Background R O,Background , (4)
where R Background are photon detection background rates. These rates are the sum of the rates caused by light from a companion star, light from the night sky, or potentially a dark rate in the detector.

The Camera System

For the optical high-time-resolution observations, a 7pixel camera was custom-built and mounted on the lid of the Cherenkov camera of a H.E.S.S. telescope using a plane secondary mirror to put it into focus (see Fig. 2). The central pixel was used to continuously record the light curve of the astronomical target, while a ring of six 'outer' pixels was used both to monitor the NSB level and as a veto system to reject terrestial background events occurring in the atmosphere (see Fig. 3).

For all seven pixels, Photonis PMTs of type XP2960 were used as light detectors. For this setup, the total system response mainly is in the wavelength range 350 -550 nm, roughly matching the canonical astronomical B band (effective wavelength 445 nm, effective width 94 nm). In this range the average transmissivity of the atmosphere is known to be ∼ 70% at 15 • zenith angle [26]. The primary and secondary mirror reflectivities (both on average ∼ 80%) and PMT quantum efficiency (on average ∼ 20%) have been measured in the lab as a function of wavelength. Assuming a flat spectrum in this range, an observed photoelectron rate can be converted into an actual flux density in Jansky. The PMTs were equipped with hexagonal light funnels of 10 arcmin width [27], and the central pixel had a circular aperture of 5 arcmin (see Fig. 3). This diameter was chosen such that it contains more than 90% of the light from the target, but blocks ∼ 3/4 of the NSB compared to the outer pixels. Note that the H.E.S.S. optical PSF has a full width at half maximum (FWHM) of 2.5 arcmin. The advantage of this pixel layout is that there are no gaps in the veto ring and that no part of the signal from the astronomical target will enter one of the veto pixels. However, the NSB level in the outer pixels is a factor ∼ 4 higher compared to the central pixel, making them 2 times less sensitive to detect similar flares.

Pointing corrections for atmospheric refraction, for bending of the masts holding the Cherenkov camera, and for the offset of the central pixel relative to the optical axis of the telescope were applied. The pointing accuracy was checked frequently by observing a bright star near the target. This was done by closing the pneumatic lid of the 7-pixel camera and taking a picture of the image of the star on that lid with a CCD located between mirror segments in the center of the H.E.S.S. reflector. The remaining mispointing was below 0.5 arcmin, compared to 5 arcmin diameter of the central pixel, resulting only in a negligible signal loss.

The analog signals from the PMTs were transmitted via 50 m coaxial cables to an electronics container. There, they were amplified and passed through an analog low-pass filter with a cutoff near 0.5 μs. Then, the signals were continuously digitized with a sampling period of 10 ns, and 256 consecutive samples were averaged to reduce the data rate to 6 megabyte/s, resulting in a 2.56 μs sampling. The absolute time of each sample is known with a precision of ∼ 1 μs from a GPS clock. An additional coaxial cable was connected to a termination resistor in the 7-pixel camera and its signal was used to monitor possible electronic interference (for details see [28]).

Observations and Data Analysis

In May 2007 high-time-resolution observations of several X-ray binaries containing a black hole or a neutron star (Sco X-1, GX 339-4, MXB 1735-44, V4641 Sgr and SS 433) and of the isolated neutron star RRAT J1317-5759 were performed during moonshine. After removing runs where clouds entered the field of view, 43 h (∼ 1 terabyte) of good quality data remain. The level of the NSB depends mainly on the brightness of the moon, the altitude of the target and the moon-target angular separation. Our observations range from new moon to more than 3/4 moon, altitudes of 30 • -90 • and moon-target separations of 30 • -120 • . The measured NSB level increases up to a factor of ∼ 20 relative to no moon (see Fig. 4). Note that the shot noise from the NSB, which determines the sensitivity for detecting flares, varies by a factor of ∼ √ 20 for these conditions. Non-astronomical events are expected to occur, producing a background of events that has to be rejected to find flares from the astronomical target. Airplanes, satellites and shooting stars [10] passing through the field of view of the whole camera and also the central pixel are expected to produce time-shifted flares with the following pattern: first in one or two of the outer pixels, then in the central pixel and then in one or two of the outer pixels on the opposite side (see Fig. 5 top). Lightning at the horizon being scattered in the atmosphere will illuminate all pixels in the same way, making it easy to veto (see Fig. 5 bottom). ph m -2 s -1 sr -1 in the wavelength range 300 -650 nm [25]). Note that about 50% of the observations occured at NSB levels less than five times the level without moon.

Cosmic-ray induced air showers typically show an elliptical light distribution on the sky with diameters of ∼ 30 arcmin and last for a few nanoseconds. Such flashes will register as temporally unresolved events in our detector. Given the ∼ 30 arcmin field of view of the 7-pixel camera, cosmic rays seen in the central pixel should be seen simultaneously by at least one veto pixel.

The data from all seven pixels and the electronics interference monitoring channel were searched for events that stand out above the noise. The challenge of finding weak signals in a noise-dominated time series is similar to the one faced by gravitational wave detectors, and a large set of literature exists on the topic (see [29] and references therein). The complexity of the methods and their performances depend to a large degree on the assumptions that can be made both on the noise in the series and the characteristics of the signal.

Of particular interest for its simplicity is the matched filter, where a particular waveform is assumed for the signal. A generic flare search algorithm called Peak Correlator (PC) was derived from this principle (presented in [30], implemented in [START_REF]BuL-Virgo Burst Software Library[END_REF]), using a bank of Gaussian templates exp(-t 2 /2τ 2 ) as assumed signal shapes. Gaussians roughly fit any structureless flare, and using several widths τ i makes it possible to cover a range [τ Min , τ Max ] of timescales. The template spacing is chosen such that the loss of detection significance for an actual Gaussian shaped flare does not decrease by more than some for any width τ in the specified range.

The PC algorithm calculates the statistical significance of flares, assuming a Gaussian waveform, by taking the noise in the series into account, estimated by the power spectral density (PSD) of the data. In practice, the time series is split into chunks of length T Chunk and each chunk into a number of windows N Windows such that the length of a window T Window = T Chunk /N Window is large compared to τ Max . The PSD of a chunk is the average of the PSDs of its windows. The underlying assumption is that the noise level is stable on timescales longer than T Chunk .

As can be seen in Fig. 6, above ∼ 100 Hz (corresponding to timescales shorter than ∼ 10 ms), the noise is approximately flat, which is characteristic of shot noise from random photon arrival times. This background is usually dominated by the NSB and sometimes by a bright star in the field of view. It changes slowly with zenith angle on timescales of many minutes while the telescope tracks the target. Additionally, due to field rotation (the H.E.S.S. telescope has an altitude-azimuth mount), a bright star sometimes rotates around the center and moves from one outer pixel to the next, changing their background levels on timescales of many minutes. Below 100 Hz (corresponding to timescales above 10 ms) the signal is dominated by electronic noise (1/f noise). Its level changes mainly with temperature, again on timescales much longer than T Chunk . The PSD contains strong lines at 50 Hz and multiples up to ∼ 1 kHz, resulting from electrical interference from the power supply. The PC algorithm correctly takes these noise components into account and prevents false event detections on the corresponding timescales. Fig. 6. A typical PSD calculated from one chunk of data containing no detected events (see text for explanations).

The output of the PC filter for a given template width τ i is a time series of signal-to-noise ratio (SNR) values at the same sampling as the original light curve (t s = 2.56 μs). For an input time series of Gaussian-distributed noise, the output is also Gaussian. Thus a simple amplitude threshold A Thr,Det on the SNR series can be used to trigger a detection in the central pixel, and A Thr,Veto to trigger a detection in one of the six outer pixels or in the electronics interference monitoring channel. For actual parameter values used in this analysis see Tab. 1. When consecutive samples are above threshold, the first and last ones define the start t Start and stop time t Stop of a so called micro-event. The SNR of the micro-event is given by the maximum SNR value in this time range, and the micro-event time is determined by the time of this maximum SNR. A flare will trigger on several nearby timescales τ i and might contain substructures, leading to several closeby micro-events. Micro-events are clustered if their clustering windows [t Startnτ i , t Stop + nτ i ] overlap, where n is the clustering factor of Tab. 1. The time, timescale and SNR of the clustered event is defined by the micro-event with the highest SNR.

The following simple coincidence veto algorithm was applied to every event detected in the central pixel: if any of the six outer pixels had detected an event on any of the tested timescales within ΔT Veto of the central pixel event time, the flare was vetoed. The longest observed time shifts are 50 ms for some shooting stars, thus a veto time window ΔT Veto = 100 ms was chosen conservatively. The veto threshold A Thr,Veto = 7 was chosen such that even the shortest timescales do not trigger on statistical fluctuations due to the large number of trial factors (∼ 10 11 ). Electrical interference will also influence the signal from the PMTs, although with a much lower significance, since their time series is dominated by photon shot noise, which is absent from the resistor channel. Whenever the electronics interference monitoring channel detected an interference signal within ΔT Veto of the central pixel event time, this event was rejected. In Sec. 5 only events where no electronic interference occured are discussed. When applying this veto algorithm, a total of 4% of the observation time is vetoed.

Results

Applying this flare search and veto algorithm to the data, with the parameters shown in Tab. 1, we find the events shown in Fig. 7. On timescales above 500 μs, a large number of 1175 events was detected. Most of these events are probably background from shooting stars, since a visual inspection of their light curves shows that most have a temporal pattern similar to the pattern of the event shown in Fig. 5 top. 21% of these events are not vetoed, especially those of low SNR. This is understandable as some shooting stars might travel close to head-on towards the telescope. These events may remain in an outer pixel for some time and give a flat-topped flare that is not very Gaussian-like.

Events with τ < 3 μs (1538 events, 51% not vetoed) must be considered as temporally unresolved, since they last only one or two samples, which would also be the case for much shorter flares because of the analog shaping of 0.5 μs. These events might be due to air showers of a few nanoseconds duration. In that case many low-SNR events might not be detected by the outer pixels because air showers exhibit large intensity variations within the shower on scales of few arcminutes.

In the time range 3 -500 μs, where the fastest flares from stellar-mass black holes and neutron stars are expected to occur, there is considerably less backgound: only eight events were detected. The five vetoed events at τ ∼ 120 μs are isolated events that occured in the nights of May 11 and 12. They show the same structureless waveform in all pixels and are most likely caused by single lightning flashes (Fig. 5 bottom shows the one with the highest SNR as an example).

The three events at τ ∼ 20 μs (light curves shown in Fig. 8) were not vetoed by any of the outer pixels. They are Fig. 7. SNR and timescale τ for all events in 43 h of observations. Every circle represents a flare detected in the central pixel. Red events were not vetoed, blue events were. Note that blue events were shifted to the right by multiplying their τ by a factor 1.1 such that they can be seen more clearly. Events vetoed by the electronics interference monitoring channel are not shown at all. separated by 68.608 (3) ms from each other and thus only form one exceptional event. Using the method explained in Sec. 3, the peak flux of the brightest flare (SNR = 15 at τ = 22μs) was determined as 0.2 Jy. The flares were recorded while observing the X-ray binary black hole candidate V4641 Sgr on May 14, 2007 at 3:47 UTC. Observing conditions were very good (altitude 63 • , 2.6 days before new moon, moon-target separation 101 • ). A careful analysis of the data shows, however, that the flares were not caused by the astronomical target, but very likely by a piece of space debris crossing the field of view. A flare separation of 68.6 ms corresponds to a rotation frequency of 14.6 Hz. One shiny surface would produce a beam of reflected sunlight of diameter 0.5 • , the diameter of the sun on the sky. A telescope placed in this rotating beam would see flares of approximately rectangular shape and 0.5 • /360 • × 68.6 ms = 95 μs duration. Given that the telescope might not be located in the center of the 0.5 • circular beam but towards the edge, any flare duration shorter than but of the same order as 95 μs is possible, in rough agreement with the observed τ = 20 μs (corresponding flare FWHM = 2.4τ = 50 μs). One consequence of this interpretation would be that at a separation of a few times 68.6 ms further flares should be present in one of the outer pixels when the debris passed through their field of view. As can be seen in Fig. 9, this is actually the case. Three events at SNR ∼ 4 occurred in outer pixel #3 right before the events in the central pixel, and five events at SNR ∼ 3 right after in the opposite outer pixel #6. This shows that indeed an object was crossing the field of view of the whole camera and that the flares cannot be of astrophysical origin. Assuming a height of ∼ 1000 km (the object had to be above 600 km to be outside the earth's shadow and almost all space debris is at a height of less that 2000 km), the angular speed of 0.6 • /s corresponds to a speed of 10 km/s, a typical value for space debris orbits [START_REF]Technical Report on Space Debris[END_REF][START_REF][END_REF]. An airplane with strobe lights could in principle produce the observed signal, but neither the 15 Hz repetition rate, nor the 20 μs duration are used in airplane strobes, and the Namibian air control confirmed that there were no airplanes within 100 km of the H.E.S.S. site at the time the flares occurred. Satellites do not have strobe lights and do not rotate as fast as 15 Hz and can thus also be excluded as the origin of the flares.

Outlook

In this paper it is demonstrated that the huge optical reflector of Cherenkov telescopes can be used to observe faint optical flashes in the sub-millisecond time domain. Flares with a duration of ∼ 20 μs as faint as 0.1 Jy have been detected with high statistical significance using a H.E.S.S. telescope equipped with a dedicated detector during moderate moonshine.

Compact objects like neutron stars and black holes are known to produce bright and short optical flares (e.g. For the study of bright objects, the night-sky background faced by optical observations with a Cherenkov telescope is less critical and a large reflector is especially important. In this case Cherenkov telescopes could be used to monitor such systems, since e.g. H.E.S.S. is not operated during moonshine. Significant observing time (∼ 40 h/month) is available without interfering with normal gamma-ray observations, whereas time on large optical telescopes is expensive.

The observations presented here show that a considerable background of optical flashes with origin likely in the earth's atmosphere is present on timescales below 3 μs and above 0.5 ms. A ring of veto pixels has been shown to be effective, but not perfect to reject these events. A more efficient veto system could consist of a second telescope at a distance of several hundred kilometers observing the same target at the same time. Any flare from an astronomical target must occur coincident with the same waveform in both telescopes. Light sources at smaller distances will be seen by only one system if the parallax of the light emitting object is larger than the angular diameter of the central pixel. A separation of the two telescopes of 700 km with an angular diameter of the central pixel of 0.1 • will place any coincident event at a distance of at least 400 000 km and thus outside the earth-moon system.

The H.E.S.S. array will be extended with a fifth, much larger telescope. This H.E.S.S. II telescope will feature a light collecting area of 600 m 2 , allowing more sensitive searches for ultra-fast optical transients. Continuing hightime-resolution optical photometry observations with an improved setup (higher quantum efficiency, smaller pixels, faster readout) in conjunction with a second telescope (optical or Cherenkov) at a large distance seems promising.

Fig. 1 .

 1 Fig.1. Comparison of a H.E.S.S. Cherenkov telescope (100 m 2 ) with a normal optical telescope for high-time-resolution photometry. For a given flare duration τ , the diameter of a background-free optical telescope was calculated to achieve the same sensitivity as H.E.S.S. using Eqn. (3) and the condition φ C

Fig. 2 .

 2 Fig. 2. The 7-pixel camera mounted on the lid of the H.E.S.S. Cherenkov camera.

Fig. 3 .

 3 Fig. 3. Left: Front view of the 7-pixel camera. Right: Comparison of the dimensions of the H.E.S.S. optical PSF full width at half maximum (FWHM= 2.5 arcmin), central pixel aperture diameter (5.0 arcmin) and hexagonal outer pixel flat-to-flat width (10 arcmin). The telescope focal length is 15 m resulting in a plate scale of 4.4 mm/arcmin.

Fig. 4 .

 4 Fig.4. NSB levels during the 43 hours of good quality observations in May 2 -29, 2007, roughly one moon period. This integral plot shows the fraction of the observing time the NSB was below a given level, relative to the NSB level without moon (ψ NSB ∼ 2 × 10 12 ph m -2 s -1 sr -1 in the wavelength range 300 -650 nm[25]). Note that about 50% of the observations occured at NSB levels less than five times the level without moon.

Fig. 5 .

 5 Fig. 5. Example background events very likely caused by a shooting star (top) and lightning (bottom). The intensities were arbitrarily offset such that the light curves from different channels do not overlap.

Fig. 9 .

 9 Fig.9. Temporal signature of the space debris events. This plot shows the SNR (for τ = 22 μs) at multiples of 68.608 ms before and after the SNR = 15 event in the central pixel.

Fig. 8 .

 8 Fig.8. Light curve of the three optical flares from the direction of V4641 Sgr onMay 14, 2007. Panel (a) shows one second of the raw light curve with a binning of 38.4 μs (15 samples). The time is given relative to 3:47:28 UTC. (Due to technical problems with the time synchronization of the measurement computer, the absolute time of the event might be offset by at most 10 seconds. For the relative timing between samples, the GPS signal was used directly by the analog-to-digital converter and is thus correct.) Panels (b), (c) and (d) each show a zoom-in of one millisecond of observations at the original binning of 2.56 μs, with the time axis centered on the flare peaks. The red curve is a fit of a Gaussian (width and amplitude as free parameters).

  12

Table 1

 1 Flare search parameters for the PC and veto algorithm (see text for explanations).

	Parameter	Symbol	Value
	Timescale range	[τ Min , τ Max ]	[ 2 μs, 100 ms]
	Maximum significance loss		1%
	Chunk duration	T Chunk	21 s
	Windows per chunk	N Windows	16
	Detection SNR threshold	A Thr,Det	8
	Veto SNR threshold	A Thr,Veto	7
	Clustering factor	n	4
	Veto window	ΔT Veto	100 ms

Acknowledgments

The authors would like to thank the H.E.S.S. collaboration for permission to use a H.E.S.S. telescope for these observations. We thank Thomas Kihm for help with the design and implementation of the data acquisition and Ste-

ACCEPTED MANUSCRIPT

fan Schmidt and Christian Neureuther for constructing the 7-pixel camera. Toni Hanke and Eben Tjingaete were very helpful with the installation and operation of the instrument in Namibia.


J L Cortés

J Induráin

An extension of the cosmological standard model with a bounded Hubble expansion rate

The possibility of having an extension of the cosmological standard model with a Hubble expansion rate H constrained to a finite interval is considered. Two periods of accelerated expansion arise naturally when the Hubble expansion rate approaches to the two limiting values. The new description of the history of the universe is confronted with cosmological data and with several theoretical ideas going beyond the standard cosmological model.

INTRODUCTION

According to General Relativity, if the universe is filled with the particles of the Standard Model of particle physics, gravity should lead to a deceleration of the expansion of the universe. However, in 1998 two independent evidences of present accelerated expansion were presented [START_REF] Riess | [END_REF]2] and later confirmed by different observations [3][4][5]. On the other hand, measurements of large scale structure [6] and CMB anisotropy [3]

also indicate that the universe evolved through a period of early accelerated expansion (inflation).

There is no compelling explanation for any of these cosmic accelerations, but many intriguing ideas are being explored. In the case of inflation, the origin of the accelerated expansion can be either a modification of gravity at small scales [7] or a coupling of the expansion of the universe to the progress of phase transitions [8]. In the case of the present accelerated expansion these ideas can be classified into three main groups: new exotic sources of the gravitational field with large negative pressure [9] (Dark Energy), modifications of gravity at large scales [10] and rejection of the spatial homogeneity as a good approximation in the description of the present universe [11].

Different models (none of them compelling) of the source responsible for each of the two periods of accelerated expansion have been considered. Einstein equations admit a cosmological constant Λ, which can be realized as the stress-energy tensor of empty space. This Λ, together with Cold Dark Matter, Standard Model particles and General Relativity, form the current cosmological model (ΛCDM). However, quantum field theory predicts a value of Λ which is 120 orders of magnitude higher than observed. Supersymmetry can lower this value 60 orders of magnitude, which is still ridiculous [12]. In order to solve this paradox, dynamical Dark Energy models have been proposed.

This has also lead to explore the possibility that cosmic acceleration arises from new gravitational physics. Also here several alternative modifications of the Einstein-Hilbert action at large and small curvatures [13][14][15][16][17][18][19], or even higher dimensional models [20,21], producing an accelerated expansion have been identified. All these analy-sis include an ad hoc restriction to actions involving simple functions of the scalar curvature and/or the Gauss-Bonnet tensor. This discussion is sufficient to establish the point that cosmic acceleration can be made compatible with a standard source for the gravitational field, but it is convenient to consider a more general framework in order to make a systematic analysis of the cosmological effects of a modification of general relativity.

In this paper we parametrize the evolution of the universe (considered isotropic and homogeneous) with the Hubble parameter H. One finds that it is possible to restrict the domain of H to a bounded interval. This restriction naturally produces an accelerated expansion when the Hubble expansion rate approaches any of the two edges of the interval. Therefore, we find a new way to incorporate two periods of cosmic acceleration produced by a modification of general relativity. But the dependence on the two limiting values of H can be chosen independently. One could even consider a Hubble expansion rate constrained to a semi-infinite interval with a unique period of accelerated expansion. In this sense the aim to have a unified explanation of both periods of accelerated expansion is only partially achieved. This simple phenomenological approach to the problem of accelerated expansion in cosmology proves to be equivalent at the homogeneous level to other descriptions based on modifications of the Einstein-Hilbert action or the introduction of exotic components in the matter Lagrangian.

In the next section we will review how a general modification of the gravitational action leads to a generalized first Friedman equation at the homogeneous level. In the third section we will present a specific model (ACM) based on the simplest way to implement a bounded interval of H. In the fourth section we will contrast the predictions of ACM for the present acceleration with astrophysical observations. In the fifth section we will show that it is always possible to find modified gravitational actions which lead to a given generalized first Friedman equation, and present some simple examples. In the sixth section we will show that it is also always possible to find Dark Energy models which are equivalent to a given generalized first Friedman equation, and present some simple examples. The last section is devoted to summary and conclusions.

ACTION OF THE COSMOLOGICAL STANDARD MODEL EXTENSION

The spatial homogeneity and isotropy allow to reduce the gravitational system to a mechanical system with two variables a(t), N (t) which parametrize the Robertson-Walker geometry

ds 2 = N 2 (t)dt 2 -a 2 (t)∆ ij dx i dx j (1)
∆ ij = δ ij + kx i x j 1 -kx 2 . ( 2
)
Invariance under parameterizations of the time variable imply that the invariant time differential N (t)dt must be used. Also a rescaling of the spatial variables x i → λx i together with a(t) → λ -1 a(t) and k → λ -2 k is a symmetry that must be kept in the Lagrangian. The action of the reduced homogeneous gravitational system can be then written as δI g δN (t) ( 4)

I g = dtN L k a 2 , H, 1 N dH dt , 1 N d dt 1 N dH dt , ... , (3)
-(8πG N ) p = - 1 N a 2 δI g δa(t) . ( 5
)
It is possible to choose a new time coordinate t ′ such that

dt ′ dt = N (t) . (6)
This is equivalent to set N = 1 in the action (3) of the gravitational system and the evolution equations of the cosmological model reduce to a set of equations for the scale factor a(t). If we introduce the notation

H (i) = d dt i H (7)
(δ i L) (j) = d dt j a 3 ∂ i L , (8)
with ∂ i L denoting the partial derivatives of the Lagrangian as a function of the variables (k/a 2 , H, H (1) , H (2) , ...) we have 1) .

8πG N 3 ρ = -L + ∞ i=2 i-2 j=0 (-1) i-j a 3 H (j) (δ i L) (i-j-2) (9)
-(8πG N ) p = -3L + 2k a 2 ∂ 1 L + ∞ i=2 (-1) i a 3 (δ i L) (i-
(10) In the homogeneous and isotropic approximation, the vanishing of the covariant divergence of the energymomentum tensor leads to the continuity equation

d dt ρa 3 = -p d dt a 3 . ( 11
)
In the radiation dominated era one has

ρ = 3p = σ a 4 , ( 12
)
where σ is a constant parameterizing the general solution of the continuity equation. In a period dominated by matter one has a pressureless fluid and then

p = 0 , ρ = η a 3 , (13)
with constant η. When these expressions for the energy density and pressure are plugged in (9-10), one ends up with two compatible differential equations for the scale factor a(t) which describe the evolution of the universe.

From now on we will use the more common notation dH dt ≡ Ḣ,...

THE ASYMPTOTIC COSMOLOGICAL MODEL

Let us assume that the Lagrangian L of the gravitational system is such that the evolution equations (9-10) admit a solution such that a(t) > 0 (absence of singularities), ȧ > 0 (perpetual expansion) and Ḣ < 0. In that case, one has a different value of the scale factor a and the Hubble rate H at each time and then one has a one to one correspondence between the scale factor and the Hubble rate. Since the continuity equation (together with the equation of state) gives a relation between the scale factor and the density, one can describe a solution of the evolution equations of the generalized cosmological model through a relation between the energy density and the Hubble rate, i.e. through a generalized first Friedman equation

8πG N 3 ρ = g(H) , (14)
with g a smooth function which parametrizes the different algebraic relations corresponding to different solutions of different cosmological models. Each choice for the function g(H) defines a phenomenological description of a cosmological model. Then one can take it as a starting point trying to translate any observation into a partial information on the function g(H) which parametrizes the cosmological model. Eq. ( 14) is all one needs in order to reconstruct the evolution of the universe at the homogeneous level. The generalized second Friedman equation is obtained by using the continuity equation (11) and the expression for ρ as a function of H. One has

-(8πG N ) p = 3g(H) + g ′ (H) H Ḣ . ( 15
)
In the matter dominated era one has Ḣ

H 2 = -3 g(H)
Hg ′ (H) (16) and then the assumed properties of the solution for the evolution equations require the consistency conditions

g(H) > 0 , g ′ (H) > 0 . ( 17
)
In the period dominated by radiation one has Ḣ

H 2 = -4 g(H) Hg ′ (H) ( 18
)
instead of ( 16) and the same consistency conditions (17) for the function g(H) which defines the generalized first Friedman equation.

We introduce now a phenomenological cosmological model defined by the condition that the Hubble rate has an upper bound H + and a lower bound H -. This can be implemented through a function g(H) going to infinity when H approaches H + and going to zero when H approaches H -. We will also assume that there is an interval of H in which the behavior of the energy density with the Hubble parameter is, to a good approximation, scale-free i.e. g(H) ∝ H 2 . The source of the gravitational field will be a homogeneous and isotropic fluid composed of relativistic and non-relativistic particles. The Cosmological Standard Model without curvature is recovered in the limit H -/H → 0 and H + /H → ∞ which is a good approximation for the period of decelerated expansion.

Notice that this interpretation is independent of the underlying theory of gravitation. The Hubble parameter can be used to parametrize the history of universe as long as Ḣ = 0 ∀ t. The total density can be thus expressed as a function of H. If H is bounded, then ρ(H) will have a pole at H = H + and a zero at H = H -. Far from these scales, the behavior of ρ(H) can be assumed to be approximately scale-free.

Under these conditions, we can parametrize the dependence of the cosmological model on the lower bound H - by

g(H) = H 2 h - H 2 - H 2 (19)
and similarly for the dependence on the upper bound

H + g(H) = H 2 h + H 2 H 2 + , (20)
where the two functions h ± satisfy the conditions

lim x→0 h ± (x) = β ±1 , lim x→1 h ± (x) = 0 . ( 21
)
β is a constant allowed in principle by dimensional arguments. If β = 1 then it can be moved to the lhs of the Friedman equation, turning G N → βG N , and can be interpreted as the ratio between an effective cosmological value of the gravitational coupling and the value measured with local tests. But β = 1 would be in conflict with Nucleosynthesis, through the relic abundances of 4 He and other heavy elements (for 3 neutrino species) [START_REF] Mukhanov | Physical Foundations of Cosmology[END_REF], so we set β = 1 . Therefore

lim x→0 h ± (x) = 1 , lim x→1 h ± (x) = 0 . ( 22
)
The consistency conditions ( 17) result in

h ± (x) > 0 , h ± (x) > xh ′ ± (x) ( 23
)
for the two functions h ± defined in the interval 0 < x < 1. Thus, we can divide the cosmic evolution history into three periods. In the earliest, relativistic particles dominate the energy density of the universe and the generalized first Friedman equation shows a dependence on the upper bound H + . There is also a transition period in which the effect of the bounds can be neglected and the rhs of the first Friedman equation is scale-free; this period includes the transition from a radiation dominated universe to a matter dominated universe. In the third present period, non-relativistic particles dominate the energy density of the universe but the dependence on the lower bound H -must be accounted for in the generalized first Friedman equation.

From the definition of the Hubble parameter, one has Ḣ

H 2 = -1 + aä ȧ2 . (24)
Then, in order to see if there is an accelerated or decelerated expansion, one has to determine whether Ḣ/H 2 is greater or smaller than -1.

In the period dominated by radiation one has Ḣ

H 2 = -2 1 1 - xh ′ + (x) h+(x) , (25)
with x = H 2 /H 2 + , where we have used (20) assuming that only the dependence on the upper bound (H + ) of the Hubble parameter is relevant. A very simple choice for this dependence is given by

g(H) = H 2 1 -H 2 H 2 + α+ , (26)
with α + a (positive) exponent which parametrizes the departure from the standard cosmological model when the Hubble rate approaches its upper bound. With this simple choice one has a transition from an accelerated expansion for

H 2 > H 2 + /(1 + α + ) into a decelerated ex- pansion when H 2 < H 2 + /(1 + α + ),

which includes the domain of validity of the standard cosmological model (H ≪ H + ).

In the period dominated by matter (which corresponds to lower values of the Hubble rate) we assume that only the dependence on the lower bound (H -) of the Hubble rate is relevant. Then one has Ḣ

H 2 = - 3 2 1 1 - xh ′ -(x) h-(x) , ( 27
)
with x = H 2 -/H 2 .
We can also consider a dependence on H -parametrized simply by an exponent α -

g(H) = H 2 1 - H 2 - H 2 α- . ( 28
)
With this choice one has a transition from a decelerated expansion for

H 2 > H 2 -(1 + α -/2
) leaving the domain of validity of the standard cosmological model and entering into an accelerated expansion when the Hubble rate approaches its lower bound for

H 2 < H 2 -(1 + α -/2
). The possibility of describing ρ as a function of H is independent of the existence of spatial curvature k. However, in the kinematics of observables in the expanding universe we do need to specify the value of k. In the rest of the paper we will assume that the universe is flat (k = 0), although the same analysis could be done for arbitrary k.

The properties of the expansion obtained in this simple example (a period of decelerated expansion separating two periods of accelerated expansion) are general to the class of phenomenological models with a generalized first Friedman equation (14) with g(H) satisfying the consistency conditions (17) and a Hubble rate constrained to a finite interval. The specific part of the example defined by (26,28) is the simple dependence on the Hubble rate bounds and the values of the Hubble rate at the transitions between the three periods of expansion. From now on we will name this description the Asymptotic Cosmological Model (ACM). With respect to the late accelerated expansion, ACM can be seen as a generalization of ΛCDM, which can be recovered by setting α -= 1. It also includes an early period of exponential expansion which can be seen as a phenomenological description of the evolution of the universe at inflation in the homogeneous approximation.

Horizon problem

One can see that the horizon of a radiation-dominated universe can be made arbitrarily large as a consequence of an upper bound on the Hubble parameter and in this way one can understand the observed isotropy of the cosmic microwave background at large angular scales.

Let us consider the effect of the modification of the cosmological model on the calculation of the distance d h (t f , t i ) of a source of a light signal emitted at time t i and observed at time

t f d h (t f , t i ) = c a(t f ) t f ti dt a(t) . ( 29
)
We have

dt a = da a 2 H = 8πG N σ 3 -1/4 g ′ (H)dH 4Hg(H) 3/4 , ( 30
)
where in the first step we have used the definition of the Hubble expansion rate H and in the second step we have used the relation between the scale factor a and H as given by (12)(13)(14). We are considering both times t i and t f in the radiation dominated period.

The distance d h is then given by

d h (H f , H i ) = 1 4g(H f ) 1/4 Hi H f dHg ′ (H) Hg(H) 3/4 . ( 31
)
If H i is very close to H + (i.e. if we choose the time t i when the light signal is emitted well inside the period of accelerated expansion) then the integral is dominated by the region around H i which is very close to H + . Then one can approximate in the integrand

g(H) ≈ H 2 + 1 -H 2 H 2 + α+ . ( 32
)
On the other hand if the observation is made at a time t f within the domain of validity of the cosmological standard model (H 2 -≪ H 2 f ≪ H 2 + ) then the factor g(H f ) 1/2 in front of the integral can be approximated by H f and then one has

d h (H f , H i ) ≈ α + 4 H f H i B H 2 i H 2 + (1/2, -α/4) , (33)
where B z (m, n) is the incomplete Beta function, and it can be made arbitrarily large by choosing H i sufficiently close to H + . In this way we see that a cosmological model with a finite interval of variation for H solves the horizon problem.

CONSTRAINTS OF ACM BY OBSERVATIONS

In this section we will carry out a more technical analysis about how the astrophysical observations constrain the parameter space of ACM in the matter dominance period. This analysis is based on the use of (assumed) standard candles, basically Type Ia Supernovae [START_REF] Riess | [END_REF] and CMB [25]. These observations constrain the parameter space to confidence regions in which the combination Ω m ≡ (1 -H 2 -/H 2 0 ) α-is constrained to be around one quarter. The consideration of both Type Ia SNe and CMB together favor α -> 1.5. The results of this analysis can be seen in figures (FIG. 123). A reader not interested in technical details might well skip this section.

We center our discussion of the experimental tests of the Asymptotic Cosmological Model in the late accelerated expansion produced when H approaches its lower bound H -. The vast amount of supernovae data collected by [START_REF] Riess | [END_REF] and [24], the data from the SDSS Baryon Acoustic Oscillation [26], the mismatch between total energy density and total matter energy density seen at CMB anisotropies [25] and the measurements of present local mass density by 2dF and SDSS [5,27] compared with the measurements of H 0 from the HST Cepheids [28] show that the universe undergoes a surprising accelerated expansion at the present time.

We will firstly confront the model with the Supernovae Ia data from Riess et al. and SNLS collaboration. The usefulness of the Supernovae data as a test of Dark Energy models relies on the assumption that Type Ia SNe behave as standard candles, i.e., they have a well defined environment-independent luminosity L and spectrum. Therefore, we can use measured bolometric flux (34)

F = L
The computed value must be compared with the one obtained experimentally from the measured extinctioncorrected distance moduli (µ 0 = 5log 10 ( dL 1Mpc ) + 25) for each SN. The SNe data have been compiled in references [START_REF] Riess | [END_REF], and we have limited the lowest redshift at cz < 7000km/s in order to avoid a possible "Hubble Bubble" [29,30]. Therefore our sample consists on 182 SNe. We will determine the likelihood of the parameters from a χ 2 statistic,

χ 2 (H 0 , H -, α -) = i (µ p,i (z i ; H 0 , H -, α -) -µ 0,i ) 2 σ 2 µ0,i + σ 2 v , (35)
where σ v is the dispersion in supernova redshift due to peculiar velocities (we adopt v p = 400km/s in units of distance moduli), σ µ0,i is the uncertainty in the individual measured distance moduli µ 0,i , and µ p,i is the value of µ 0 at z i computed with a certain value of the set of parameters H 0 , H -, α -. This χ 2 has been marginalized over the nuisance parameter H 0 using the adaptive method in reference [31]. The resulting likelihood distribution function e -χ 2 /2 has been explored using Monte FIG. 1: Confidence regions in parameter space of the Asymptotic Cosmological Model (ACM) with (bottom) and without (top) priors from CMB and estimations of present matter energy density at 1σ (green), 2σ (red) and 3σ (blue). The ΛCDM is inside the 1σ region if we consider no priors, but is moved outside the 2σ region when we confront the SNe data with CMB and estimations from present matter energy density. In this case a value α-> 1.5 is favored.

Carlo Markov Chains. We get a best fit of ACM at α -= 0.36 and H- H0 = 0.95, for which χ 2 = 157.7. In contrast, fixing α -= 1, we get Ω Λ ≡ We can add new constraints for the model coming from measurements of the present local matter energy density from the combination of 2dF and SDSS with HST Cepheids, rendering Ω m = 0.28 ± 0.03; and the distance to the last scattering surface from WMAP, which leads H0dz ′ H(z ′ ) = 1.70 ± 0.03 [START_REF] Riess | [END_REF]. Including these priors we get a best fit of ACM at α -= 3.54 and H- H0 = 0.56 (our simulation explored the region with α -< 3.6), for which χ 2 = 164.1. In contrast, we get Ω Λ = 0.73 and χ 2 = 169.5 for the best fit ΛCDM, which is outside the 2σ confidence region shown in Fig. 1 (bottom). The fits of the best fit ΛCDM and ACM taking into account the priors to the SNe data are compared in Fig. 2. The information which can be extracted from the data is limited. This can be seen in Fig. 3, in which it is explicit that the data constrain mainly the value of the present matter density, Ω m ≡ ρm ρC = (1 -

H 2 - H 2 0
) α-.

GENERALIZED FIRST FRIEDMAN EQUATION AND f (R) GRAVITY

The extension of the cosmological model considered here could be compared with recent works on a modification of gravity at large or very short distances [13][14][15][16][17][18][19]. It has been shown that, by considering a correction to the Einstein Hilbert action including positive and negative powers of the scalar curvature, it is possible to reproduce an accelerated expansion at large and small values of the curvature in the cosmological model. Some difficulties to make these modifications of the gravitational action compatible with the solar system tests of general relativity have lead to consider a more general gravitational action, including the possible scalars that one can construct with the Riemann curvature tensor [32], although the Gauss-Bonnet scalar is the only combination which is free from ghosts and other pathologies. In fact, there is no clear reason to restrict the extension of general relativity in this way. Once one goes beyond the derivative expansion, one should consider scalars that can be constructed with more than two derivatives of the metric and then one does not have a good justification to restrict in this way the modification of the gravitational theory. It does not seem difficult to find an appropriate function of the scalar curvature or the Gauss-Bonnet scalar, which leads to a cosmology with a bounded Hubble expansion rate.

One may ask if a set of metric f(R) theories which include ACM as an homogeneous and isotropic solution exists. The answer is that a bi-parametric family of f(R) actions which lead to an ACM solution exists. In the following section we will derive them and we will discuss some examples. The derivation follows the same steps of modified f(R)-gravity reconstruction from any FRW cosmology [33].

We start from the action

S = 1 16πG N d 4 x √ -g[f (R) + 16πGL m ] , (36)
which leads to the "generalized first Friedman equation"

-18(H Ḧ + 4H 2 Ḣ)f ′′ (R) -3( Ḣ + H 2 )f ′ (R) -1 2 f (R) = (8πG N ) ρ (37) R = -6( Ḣ + 2H 2 ) . ( 38
)
If the energy density of the universe is mainly due to matter (as in the late accelerated expansion), we can use ( 16) and ( 14) to express ρ, R, Ḣ and Ḧ as functions of H. Thus we get

[H 3 gg ′2 -3H 2 g 2 g ′ + 3H 3 g 2 g ′′ ]f ′′ (R) + [3Hgg ′2 -H 2 g ′3 ] 18 f ′ (R) -g ′3 108 f (R) = gg ′3 18 (39) R(H) = 18Hg(H)/g ′ (H) -12H 2 . ( 40
)
H(R) can be obtained from ( 40) and set into (39); then we get an inhomogeneous second order linear differential equation with non-constant coefficients. Therefore, there will always be a bi-parametric family of f (R) actions which present ACM as their homogeneous and isotropic solution. The difference between these actions will appear in the behavior of perturbations, which is not fixed by (14). Some of these actions are particularly easy to solve. If g(H) = H 2 then R = -3H 2 and the differential equation becomes

6R 2 f ′′ (R) -Rf ′ (R) -f (R) + 2R = 0 . ( 41
)
Its general solution is

f (R) = R + c 1 R 1 12 (7- √ 73) + c 2 R 1 12 (7+ √ 73) , ( 42
)
which will give ( 14) as First Friedman equation as long as the radiation energy density can be neglected. In general the differential equation (39) will not be solvable analytically, and only approximate solutions can be found as power series around a certain singular point R 0 (a value of R such that H(R) cancels out the coefficient of f ′′ (R) in ( 39)). These solutions will be of the form

f (R) = f p (R) + c 1 f 1 (R) + c 2 f 2 (R) with f i (R) R 0 = ∞ m=0 a (i) m ( R -R 0 R 0 ) si+m , ( 43
)
where i = p, 1, 2, a (1) 0
= a

(2) 0 = 1 and the series will converge inside a certain radius of convergence. An interesting choice of R 0 is R 0 = -12H 2 -≡ R -, which is the value of R at H -. One can also find the approximate solution of the differential equation for | R |≫| R -|, which will be of the form

f i (R) R = ∞ m=0 a (i) m ( R - R ) si+m . ( 44
)
One can in principle assume that the action (44) could be considered as valid also in the region in which radiation begins to dominate, but this action reproduces (14) only if matter dominates. However, some of the new terms appearing in (44) will be negligible against R when radiation begins to dominate. The others can be canceled out by setting to zero the appropriate integration constant and therefore the Cosmological Standard Model will be recovered as a good approximation when radiation begins to dominate. f (R)-theories are not the only modified gravity theories studied in the literature; f (G)-theories [34] are also a popular field of research. In these theories, the Einstein-Hilbert action is supplemented by a function of the Gauss-Bonnet scalar G = R 2 -4R µν R µν + R µνρσ R µνρσ (not to be confused with the gravitational coupling G N ).The same approach can be used to answer what f (G) actions are able to reproduce ACM homogeneous evolution, once more in analogy with the modified f(G)gravity reconstruction of a FRW cosmology [35]. The form of the action will be

S = 1 16πG N d 4 x √ -g[R + f (G) + 16πG N L m ] , (45)
from which we derive the modified Friedman equation

(8πG N ) ρ = 3H 2 - 1 2 Gf ′ (G) + 1 2 f (G) + 12f ′′ (G) ĠH 3 .
(46) Following the same procedure as in the case of f (R)theories we arrive to the differential equation

3g(H) = 3H 2 -1 2 Gf ′ (G) + 1 2 f (G) + 864H 6 g g ′3 [9gg ′ -Hg ′2 -3Hgg ′′ ]f ′′ (G) (47)
G = 24H 2 (H 2 -3Hg/g ′ ) . ( 48
)
For a given g(H), one can use (48) to get H(G), then set it into (47) and solve the second order linear differential equation. As in the previous case, there will be a biparametric family of solutions for f (G) which will have (14) as their homogeneous isotropic solution as long as matter dominates. The parameters will need to be fixed in order to make the contribution of undesired terms in the Friedman equation to be negligible when radiation dominates. Again g(H) = H 2 is an example which can be solved analytically. The differential equation turns to be

12G 2 f ′′ (G) -Gf ′ (G) + f (G) = 0 , (49)
which has the solution

f (G) = c 1 G + c 2 G 1/12 . (50)
In most cases this procedure will not admit an analytical solution and an approximate solution will need to be found numerically.

A similar discussion can be made to explain the early accelerated expansion (inflation) as a result of an f (R) or f (G) action, taking into account that in this case the dominant contribution to the energy density is radiation instead of matter. The analogue to (44) will be now a solution of the form

f i (R) R = ∞ n=0 a (i) n ( R R + ) ri+n , (51)
for

|R| ≪ |R + | with R + = -12H 2 + .
Terms in the action which dominate over R when R becomes small enough should be eliminated in order to recover the Standard Cosmological model before matter starts to dominate.

Both accelerated expansions can be described together in the homogeneous limit by an f (R) action with terms ( R- R ) si+m with s i > 0 coming from (44) and terms ( R R+ ) ri+n with r i > 0 coming from (51). The action will contain a term, the Einstein-Hilbert action R, which will be dominant for H -≪ H ≪ H + including the period when matter and radiation have comparable energy densities.

ALTERNATIVE DESCRIPTIONS OF THE ACM

In general, by virtue of the gravitational field equations, it is always possible to convert a modification in the gravitational term of the action to a modification in the matter content of the universe. In particular, at the homogeneous level, it is possible to convert a generalized first Friedman equation of the type (14) to an equation in which, apart from the usual matter term, there is a dark energy component with an unusual equation of state p = p(ρ) and in which General Relativity is not modified.

The trivial procedure is the following. Assume that the source of the gravitational field in the modified gravity theory behaves as p d = ωρ d (matter or radiation). This is the case when the modification of gravity is relevant. We can then define a dark energy or effective gravitational energy density as

ρ g = 3 8πG N H 2 -g(H) . ( 52
)
The continuity equation ( 11) allows to define a pressure for this dark energy component as p g = -ρ g -ρg /3H. The equation of state of the dominant content of the universe leads to

Ḣ = -3(1 + ω) Hg(H) g ′ (H) , (53)
which can be used to express ρg as a function of H, and one gets the final expression for p g ,

p g = -3 8πG N H 2 -2(1 + ω)H g(H) g ′ (H) + ωg(H) . ( 54
)
Then, for given g(H) we can use (52) to get H = H(ρ g ) and substitute in (54) to get an expression of p g (ρ g ; H ± , ω) which can be interpreted as the equation of state of a dark energy component. In the simple case of a matter dominated universe with ACM, α -= 1 gives obviously a dark energy component verifying p g = -ρ g . Another simple example is α -= 2, for which

8πG N 3H 2 - p g = 2 8πGN 3H 2 - ρ g -3 . ( 55
)
The inverse procedure is also straightforward. Suppose we have a universe filled with a standard component p d = ωρ d and a dark energy fluid p g = p g (ρ g ). Using the continuity equation of both fluids one can express their energy densities as a function of the scale factor a and then use this relation to express ρ g as a function of ρ d . Then the Friedman equation reads

H 2 = 8πG N 3 (ρ d + ρ g (ρ d )) , (56)
which, solving for ρ d , is trivially equivalent to (14). This argument could be applied to reformulate any cosmological model based on a modification of the equation of state of the dark energy component [36] as a generalized first Friedman equation (14). Until now we have considered descriptions in which there is an exotic constituent of the universe besides a standard component (pressureless matter or radiation). In these cases, pressureless matter includes both baryons and Dark Matter. However, there are also descriptions in which Dark Matter is unified with Dark Energy in a single constituent of the universe. One of these examples is the Chaplygin gas p g = -1/ρ g [37,38]. The use of the continuity equation leads to

ρ g = A + Ba -6 , ( 57
)
where A and B are integration constants. The previous method can be used to find the generalized first Friedman equation for the baryon density in this model,

8πG N 3 ρ b = H 2 k 1 + k(1 - H 4 - H 4 ) -1 , (58)
where H -= 8πG 3 A 1/4 , k = Bρ -2 b0 -1, and ρ b0 is the present value of ρ b . In the H ≫ H -limit this model can be interpreted as a universe filled with baryons and dark matter or as a universe filled with baryons and with a higher effective value of G N . Equation (58) does not fulfill [START_REF] Mukhanov | Physical Foundations of Cosmology[END_REF] because it describes the behavior of just baryon density. In the H > ∼ H -limit, the model can be interpreted as a universe filled with baryons and a cosmological constant or as an ACM model with α -= 1 and filled only with baryons.

A similar example in the early period of accelerated expansion would be a universe filled with a fluid which behaves as a fluid of ultra-relativistic particles if the energy density is low enough but whose density has an upper bound

ρ X = σ a 4 + C . ( 59
)
This dependence for the energy density in the scale factor follows from the equation of state

p X = 1 3 ρ X - 4C 3σ ρ 2 X . (60)
This model turns out to be equivalent to a universe filled with radiation following eq. ( 26) with α + = 1 and

H 2 + = ( 8πGN
3 ) σ C . Another equivalent description would be to consider that the universe is also filled with some self interacting scalar field ϕ which accounts for the discrepancy between the standard energy-momentum and GR Einstein tensors. Given an arbitrary modified Friedman equation ( 14) a potential V (ϕ) can be found such that the cosmologies described by both models are the same. The procedure is similar to the one used to reconstruct a potential from a given cosmology [39]. From the point of view of the scalar field, the cosmology is defined by a set of three coupled differential equations

φ + 3H φ + V ′ (ϕ) = 0 (61) 8πG N 3 (ρ d + φ2 2 + V (ϕ)) = H 2 (62) -3(1 + ω)Hρ d = ρd . ( 63
)
The solution of these equations for a certain V (ϕ) will give H(t) and ρ d (t), and therefore ρ d (H) which is g(H) up to a factor 8πGN 3 . In this way one finds the generalized first Friedman equation associated with the introduction of a self interacting scalar field. Alternatively, given a function g(H) in a generalized first Friedman equation ( 14) for the density ρ d , one can find a scalar field theory leading to the same cosmology in the homogeneous limit. By considering the time derivative of (62) and using ( 61) and (63) one gets φ2

= -(1 + ω)ρ d - Ḣ 4πG N , (64)
where we can use ( 53) and ( 14) to get φ2 as a function of H. Setting this on (62) we get V (ϕ) as a function of H. On the other hand, ϕ ′ (H) = φ/ Ḣ, so

8πG N 3 (ϕ ′ (H)) 2 = (2H -g ′ (H))g ′ (H) 9(1 + ω)H 2 g(H) (65)
and

8πG N 3 V (ϕ(H)) = H 2 -g(H)+ (1 + ω)(2H -g ′ (H))g(H) 2g ′ (H) .
(66) If g(H) is such that the rhs of (65) is positive definite, it can be solved and the solution ϕ(H) inverted and substituted into (66) in order to get V (ϕ). This is the case of a function of the type (28) with α -> 1. For the particular case of an ACM expansion with α -= 1, the solution is a flat potential V = V 0 and a constant value of ϕ = ϕ 0 .

If g(H) is such that the rhs of (65) is negative definite, as it happens in (28) with α -< 1 or in (26), the problem can be solved by changing the sign of the kinetic term in (62). The result is a phantom quintessence in the case of (28) with α -< 1. The case ( 26) is more complicated because the energy density of the associated inflaton turns out to be negative. Moreover, it is of the same order as the energy density of ultra-relativistic particles during the whole period of accelerated expansion. Therefore, it seems that this model is inequivalent to other inflation scenarios previously studied.

In summary, there are many equivalent ways to describe the discrepancy between observed matter content of the universe and Einstein's General Relativity. At the homogeneous level, it is trivial to find relations among them. The possibility to establish the equivalence of different descriptions is not a peculiarity of the description of this discrepancy in terms of a generalized first Friedman equation (14). The same relations can be found among generalized equations of state, scalar-tensor theories and f(R) modified gravity [40].

SUMMARY AND DISCUSSION

It may be interesting to go beyond ΛCDM in the description of the history of the universe in order to identify the origin of the two periods of accelerated expansion. We have proposed to use the expression of the energy density as a function of the Hubble parameter as the best candidate to describe the history of the universe. In this context the late time period of accelerated expansion and the early time inflation period can be easily parametrized.

We have considered a simple modification of the cosmological equations characterized by the appearance of an upper and a lower bound on the Hubble expansion rate. A better fit of the experimental data can be obtained with this extended cosmological model as compared with the ΛCDM fit. Once more precise data are available, it will be possible to identify the behavior of the energy density as a function of the Hubble parameter and then look for a theoretical derivation of such behavior.

We plan to continue with a systematic analysis of different alternatives incorporating the main features of the example considered in this work. Although the details of the departures from the standard cosmology can change, we expect a general pattern of the effects induced by the presence of the two bounds on H. We also plan to go further, considering the evolution of inhomogeneities looking for new consequences of the bounds on H.

The discussion presented in this work, which is based on a new description of the periods of accelerated expansion of the universe, can open a new way to explore either modifications of the theory of gravity or new components in the universe homogeneous fluid. Lacking theoretical criteria to select among the possible ways to go beyond ΛCDM, we think that the phenomenological approach proposed in this work is justified.

We are grateful to Paola Arias and Justo López-Sarrión for discussions in the first stages of this work. We also acknowledge discussions with Julio Fabris, Antonio Seguí, Roberto Emparán, Sergei Odintsov, Aurelio Grillo and Fernando Méndez. This work has been partially supported by CICYT (grant FPA2006-02315) and DGIID-DGA (grant2008-E24/2). J.I. acknowledges a FPU grant from MEC.

  with H = 1 aN da dt . If we keep the standard definition of the gravitational coupling and the density (ρ) and pressure (p) of a cosmological homogeneous and isotropic fluid as a source of the gravitational field, we have the equations of the reduced system 8πG N 3 ρ = -1 a 3

4πd 2 L

 2 and frequency to determine luminosity distance d L and redshift z. The luminosity distance is given now by d L (z) = c(1 + z) Hg 1/3 (H0)g 2/3 (H) .

2

 2 = 159.1 for the best fit ΛCDM. The confidence regions are shown in Fig. 1 (top).

FIG. 2 :

 2 FIG. 2: Distance Moduli vs. Redshift comparison between the best fits of ACM (H0 = 66 Km/sM P c, red) and ΛCDM (H0 = 65 Km/sM P c, blue) to the SNe data with priors. ACM fits clearly better the medium redshift SNe.