text: The dark side of gravity: Modified theories of gravity Francisco S. N. LobcB Centro de Astronomia e Astrofisica da Universidade de Lisboa, Campo Grande, Ed. C8 1749-016 Lisboa, Portugal and Institute of Gravitation & Cosmology, University of Portsmouth, Portsmouth P01 2EG, UK (Dated: July 10, 2008) Modern astrophysical and cosmological models are faced with two severe theoretical difficulties, that can be summarized as the dark energy and the dark matter problems. Relative to the former, it has been stated that cosmology has entered a 'golden age', in which high-precision observational data have confirmed with startling evidence that the Universe is undergoing a phase of accelerated expansion. Several candidates, responsible for this expansion, have been proposed in the literature, in particular, dark energy models and modified gravity, amongst others. One is liable to ask: What is the so-called 'dark energy' that is driving the acceleration of the universe? Is it a vacuum energy or a dynamical field ("quintessence")? Or is the acceleration due to infra-red modifications of Einstein's theory of General Relativity? In the context of dark matter, two observations, namely, the behavior of the galactic rotation curves and the mass discrepancy in galactic clusters, suggest the existence of a (non or weakly interacting) form of dark matter at galactic and extra-galactic scales. It has also been proposed that modified gravity can explain the galactic dynamics without the need of introducing dark matter. We briefly review some of the modified theories of gravity that address these two intriguing and exciting problems facing modern physics. I. INTRODUCTION CT. u . bJO. > o r^ Cosmology is said to be thriving in a golden age, where a central theme is the perplexing fact that the Universe is undergoing an accelerating expansion [l|. The latter, one of the most important and challenging current problems in cosmology, represents a new imbalance in the governing gravitational equations. Historically, physics has addressed such imbalances by either identifying sources that were previously unaccounted for, or by altering the governing equations. The cause of this acceleration still remains an open and tantalizing question. The standard model of cosmology has favored the first route to addressing the imbalance, namely, a missing stress- y/S ■ energy component. In particular, the dark energy models arc fundamental candidates responsible for the cosmic expansion (see Refs. 2] for a review and references therein). A simple way to parameterize the dark energy is by an equation of state of the form ui = p/p, where p is the spatially homogeneous pressure and p the energy density of the dark energy. A value of uj < —1/3 is required for cosmic expansion, as dictated by the Friedmann equation a J a — ~AttG{p + p/3), and to = — 1 corresponds to a cosmological constant. A possibility that has been widely ^^ ■ explored, is that of quintessence, a cosmic scalar field cj) that has not yet reached the minimum of its potential V{4>) [3|]. A common example is the energy of a slowly evolving scalar field with positive potential energy, similar to the inflaton field used to describe the inflationary phase of the Universe. In quintessence models the parameter range is — 1 < lo < —1/3, and the dark energy decreases with a scale factor a(t) as pq ex a _3< - 1+w ^ J4j. A specific exotic form of dark energy denoted phantom energy, with w < — 1, has also been proposed [5(, and possesses peculiar properties, such as the violation of the energy conditions and an infinitely increasing energy density. However, recent fits to supernovae, cosmic microwave background radiation (CMBR) and weak gravitational lensing data indicate that an evolving equation_of state crossing the phantom divide, is mildly favored, and several models have been proposed in the literature [6, 7]. In particular, models considering a redshift dependent equation of state, possibly provide better fits to the most recent and reliable SN la supernovae Gold dataset. It is also interesting to test specific models that are motivated by particle physics against the SN data, rather than trying to fit the phenomenological equations of state. In a cosmological setting, it has also been shown that the transition into the phantom regime, for a single field is probably physically implausible [6j, so that a mixture of various interacting non-ideal fluids is necessary. If confirmed in the future, this behavior has important implications for theoretical models of dark energy. For instance, this implies that dark energy is dynamical and excludes the cosmological constant and the models with a constant parameter as possible candidates for dark energy. An alternative model to dark energy is that of the generalized Chaplygin gas (GCG), based on a negative pressure fluid, which is 'Electronic address: firstname.lastname@example.com inversely proportional to the energy density [8||, i.e., p c h = —A/p" h , where A and a are positive constants. An attractive feature of this model, is that at early times, the energy density behaves as matter, p^ ~ a~ 3 , where a is the scale factor, and as a cosmological constant at a later stage, p c h = const. This dual behavior is responsible for the interpretation that the GCG model is a candidate of a unified model of dark matter and dark energy [9], and probably contains some of the key ingredients in the dynamics of the Universe for early and late times. All of these models present an extremely fascinating aspect for future experiments focussing on supernovae, CMBR and weak gravitational lensing and for future theoretical research. One may also explore the alternative viewpoint, namely, through a modified gravity approach. A very promis- ing way to explain these major problems is to assume that at large scales Einstein's theory of General Relativity breaks down, and a more general action describes the gravitational field. The Einstein field equation of General Relativity was first derived from an action principle by Hilbert, by adopting a linear function of the scalar curva- ture, R, in the gravitational Lagrangian density. However, there are no a priori reasons to restrict the gravitational Lagrangian to this form, and indeed several generalizations of the Einstein-Hilbert Lagrangian have been proposed, including "quadratic Lagrangians" . involving second order curvature invariants such as i? 2 , R^R^, R a f3 l j,vR al3 ^ ', e al3 ^ v Rap-ysKly, Capfj.uC ' 11 ^ 1 ' , etc [1J|. The physical motivations for these modifications of gravity were related to the possibility of a more realistic representation of the gravitational fields near curvature singularities and to create some first order approximation for the quantum theory of gravitational fields. In this context, a more general modifi- cation of the Einstein-Hilbert gravitational Lagrangian density involving an arbitrary function of the scalar invariant, f(R), was considered in [ll| . and further developed in [12, [Da, LLJ] • Recently, a renaissance of f(R) modified theories of gravity has been verified in an attempt to explain the late-time accelerated expansion of the Universe. Earlier interest in f(R) theories was motivated by inflationary scenarios as for instance, in the Starobinsky model, where f(R) = R — A + aR 2 was consider ed fl5l| . In particular, it was shown that cosmic acceleration can be indeed explained with the context of f(R) gravity [la ], and the conditions of viable cosmological models have also been derived [171 ]. In the context of the Solar System regime, severe weak field constraints seem to rule out most of the models proposed so far [H, [l9| , although viable models do exist [20] • In the context of dark matter, the possibility that the galactic dynamics of massive test particles may be understood without the need for dark matter was also considered in the framework of f(R) gravity models [2l|, [22|, [23, [2J, |2a |. and connections with MOND and the pioneer anomaly further explored by considering an explicit coupling of an arbitrary function of R with the matter Lagrangian density [261 |27| . The issue of dark matter is a long outstanding problem in modern astrophysics. Two observational aspects, namely, the behavior of the galactic rotation curves and the mass discrepancy in clusters of galaxies led to the necessity of considering the existence of dark matter at a galactic and extra-galactic scale. The rotation curves of spiral galaxies show that the rotational velocities increase from the center of the galaxy and then attain an approximately constant value, Vt goo ~ 200 — 300 km/s, within a distance r from the center of the galaxy [28[ . In these regions the mass increases linearly with the radius, even where very little luminous matter can be detected. Relatively to the mass discrepancy in clusters of galaxies, the total mass of a cluster can be estimated in two ways. First, by taking into account the motions of its member galaxies, the virial theorem provides an estimate, My. Second, the total baryonic mass M may be estimated by considering the total sum of each individual member's mass. The mass discrepancy arises as one generally verifies that My is considerably greater than M, with typical values of My jM ~ 20 — 30 [28[. This is usually explained by postulating the existence of a dark matter, assumed to be a cold pressure-less medium distributed in a spherical halo around the galaxies. Still in the context of modified gravity, an interesting possibility is the existence of extra dimensions. It is widely believed that string theory is moving towards a viable quantum gravity theory, and one of the key predictions of string theory is precisely the existence of extra spatial dimensions. In the brane-world scenario, motivated by recent developments in string theory, the observed 3-dimensional universe is embedded in a higher-dimensional spacetime [29j . Most brane-world models, including those of the Randall- Sundrum type [30j, produce ultra-violet modifications to General Relativity, with extra-dimensional gravity dominating at high energies. However it is also possible for extra- dimensional gravity to dominate at low energies, leading to infra-red modifications of General Relativity. New features emerge in the brane scenario that may be more successful in providing a covariant infra-red modification of General Relativity, where it is possible for extra-dimensional gravity to dominate at low energies. The Dvali-Gabadadze-Porrati (DGP) models [3l| achieve this via a brane induced gravity effect. The generalization of the DGP models to cosmology lead to late-accelerating cosmologies [32| , even in the absence of a dark energy field 33]. This exciting feature of "self acceleration" may help towards a new resolution to the dark energy problem, although this model deserves further investigation as a viable cosmological model [34[. While the DGP braneworld offers an alternative explanation to the standard cosmological model, for the expansion history of the universe, it offers a paradigm for nature fundamentally distinct from dark energy models of cosmic acceleration, even those that perfectly mimic the same expansion history. It is also fundamental to understand how one may differentiate this modified theory of gravity from dark energy models. The DGP braneworld theory also alters the gravitational interaction itself, yielding unexpected phcnomcnological extensions beyond the expansion history. Tests from the solar system, large scale structure, lensing all offer a window into understanding the perplexing nature of the cosmic acceleration and, perhaps, of gravity itself [34]]. The structure formation (35[ and the inclusion of inflation are also important requirements of DGP gravity, if it is to be a realistic alternative to the standard cosmological model. Generalizations of the DGP model with the inclusion of a Gauss-Bonnet (GB) term have also been explored [33], the global structure of the DGP cosmologies have been analyzed [36J , and research into other brane- world approaches to dark energy, such as the supersymmetric large extra dimensions (SLED) model have been undertaken [371 ]. In this work, we review several modified theories of gravity exploring some of their interesting properties and characteristics, in particular, focussing mainly on the late-time cosmic acceleration. We refer the reader to excellent reviews, for instance, on dark energy and modified gravity in Ref. Q, f(R) gravity in Ref. [38|, an introduction to modified gravity as an alternative for dark energy in Ref. [39(, and the late-time acceleration in braneworlds in Ref. [4fJ. This paper is outlined in the following manner. In Section [HI we review f(R) modified theories of gravity, for instance, focussing on the scalar-tensor representation of f(R) and on the late-time acceleration; we further analyze an interesting extension to f{R) gravity by considering an R— matter coupling, which yields some intriguing properties; we also discuss Gauss-Bonnet gravity and the late-time acceleration, and briefly consider modified Gauss-Bonnet gravity; the DGP brane-world model is also briefly reviewed. In Section [TTl] we consider the possibility of dark matter being a geometric effect in f(R) gravity, focussing on the f(R) generalized virial theorem, and its applications to galactic cluster observations. Finally in Section [IV! we conclude with a summary and discussions. II. MODIFIED THEORIES OF GRAVITY: LATE-TIME COSMIC ACCELERATION A. f(R) modified theories of gravity A promising avenue that has been extensively investigated recently are the f(R) modified theories of gravity, where the standard Einstein- Hilbert action is replaced by an arbitrary function of the Ricci scalar R J4ll |. In this work, we use the metric formalism, which consists in varying the action with respect to q^", although other alternative approaches have been considered in the literature, namely, the Palatini formalism [42|, [43|, where the metric and the connections are treated as separate variables; and the metric-affine formalism, where the matter part of the action now depends and is varied with respect to the connection [431 ] . 1. Action and field equations The action for the f(R) modified theories of gravity is given by S=^Jd i xV^f(R) + S M (9^^), (1) where k = 8ttG. Sm(<7' w ') ip) is the matter action, defined as Sm = / d^x^J—g C m (g^ u ,tlj), where C m is the matter Lagrangian density, in which matter is minimally coupled to the metric g^ v and ip collectively denotes the matter fields. Using the metric approach, by varying the action with respect to g 1 *", provides the following field equation FR»» - \f g"u - V M V"F + g^DF = k t£?> , (2) where F = df/dR. The matter stress-energy tensor, T^™, is defined as "" V~9 S(g^) ■ l > Now, considering the contraction of Eq. ^\j, provides the following relationship FR-2f + 3DF= kT, (4) which shows that the Ricci scalar is a fully dynamical degree of freedom. Note that the field equation, Eq. ([5]), may be written as 1 cff (j>l/ = Rp,U — nRd^U = K ^ fill J (5) where the effective stress energy tensor is given by T°^ = TJfJ + Tfi? ■ The components TJu? and the curvature stress-energy tensor, T^V , are defined as fjZ? = TW/F, (6) T$ = -^ V IM V"F-~g tiV (RF + nF + K T) (7) respectiveiy. It is also interesting to consider the conservation law for the above curvature stress-energy tensor. Taking into account the Bianchi identities, V M G M " = 0, and the diffeomorphism invariance of the matter part of the action, which yields V M T^™ = 0, we verify that the effective Einstein field equation provides the following conservation law V^t( c ) = rHycp (r\ 2. Scalar-tensor representation for f(R) gravity f(R) gravity may be written as a scalar-tensor theory, by introducing a Legendre transformation {R, /} — > {(f>, V} defined as cj> = F(R), V(<f>) = R(<P)F-f(R(d>)). (9) In this representation the field equations of f(R) gravity can be derived from a Brans-Dicke type action with parameter u> = 0, given by S=^~ f[<f>R- V{4>) + L m ] JS d 4 x. (10) 2k J The only requirement for the gravitational field equations to be expressed in the form of a Brans-Dicke theory is that F(R) be invertible, that is, R(F) exists p9{ . Thus, the field equations of f(R) gravity can be reformulated as 1 G Rixv - -^g^R = 8tt—t^ + 0^ v , (n) where V = ~\v [4>) 9nu + -r (V M V, - 9ltv U) cj>. (12) Using these variables the trace equation, i.e., Eq. ([5]), takes the form 3D<£ + 2V{6) - (f>-rr = kT. (13) dep The modification of the standard Einstein-Hilbcrt action leads to the appearance of an effective gravitational constant G c g = G/(j> in the field equations. Note the presence of a new effective source term for the gravitational field, given by the tensor 6^. 3. R— matter couplings in f(R) gravity Recently, in the context of f(R) theories of modified gravity, it was shown that a function of R— matter coupling induces a non-vanishing covariant derivative of the stress-energy, V ^F" ^ 0. This potentially leads to a deviation from geodesic motion, and consequently the appearance of an extra force [261 ]. Implications, for instance, for stellar equilibrium have been studied in Ref. [27[ ■ The equivalence with scalar-tensor theories with two scalar fields has been considered in Ref. [4J], and a viability stability criterion was also analyzed in Ref. [45|. It is interesting to note that nonlinear couplings of matter with gravity were analyzed in the context of the accelerated expansion of the Universe [46J , and in the study of the cosmological constant problem [47 1 . The action for R— matter couplings, in f(R) modified theories of gravity [261 ] . takes the following form 1 S = -f 1 (R) + [l + Xf 2 (R)]C 7 -g d x (14) where fi(R) (with i — 1,2) are arbitrary functions of the curvature scalar R. For notational simplicity we consider k — 1 throughout this subsection. Varying the action with respect to the metric g^ yields the field equations, given by FiR" v ~ \h9y.v ~ V M V, Fx + g^UFx = -2XF 2 C m R^ + 2A(V"V" - g^U)C m F 2 + (1 + Xf 2 )T^ , (15) where we have denoted Fi(R) = fl(R), and the prime represents the derivative with respect to the scalar curvature. Now, taking into account the generalized Bianchi identities [26|, [48| , one deduces the following corrected conservation equation V M T (m) = XF2 ^ l + A/ a a r - T( m ) VR, (16) where the coupling between the matter and the higher derivative curvature terms describes an exchange of energy and momentum between both. In the following, consider the equation of state for a perfect fluid T^ = {p + p)U,U" + pg^u (17) where p is the energy density and p, the pressure, respectively. The four- velocity, U^, satisfies the conditions U^U^ = -1 and U^U^u = 0. Introducing the projection operator /i M " = g M " + U^U V , gives rise to non-geodesic motion governed by the following equation of motion for a fluid element: dU 1 * jds + T^plf^U 13 — / M , where the extra force, / M , is given by f" P XF 2 1 + A/j {Z m -p)VuR + V v p h^ (18) In a recent paper 49], the authors argued that a "natural choice" for the matter Lagrangian density for perfect fluids is C m = p, based on Refs. [5Ctl5l|. where p is the pressure. This choice has a particularly interesting application in the analysis of the R— matter coupling for perfect fluids, which implies in the vanishing of the extra force [261 ]. However, it is important to point out that despite the fact that C m — p does indeed reproduce the perfect fluid equation of state, it is not unique [52J ]. Other choices include, for instance, C m = — p [5X|. l53||. where p is the energy density, or C m — ~na, were n is the particle number density, and a is the physical free energy defined as a = p/n — Ts, with T being the temperature and s the entropy per particle (see Ref. [5ll.l52j] for details). Hence, it is clear that no immediate conclusion may be extracted regarding the additional force imposed by the non-minimum coupling of curvature to matter, given the different available choices for the Lagrangian density. One may conjecture that there is a deeper principle or symmetry that provides a unique Lagrangian density for a perfect fluid [53]. This has not been given due attention in the literature, as arbitrary gravitational field equations depending on the matter Lagrangian have not always been the object of close analysis. See Ref. [52] for more details. 4. Late-time cosmic acceleration In this subsection, we show that f(R) gravity may lead to an effective dark energy, without the need to introduce a negative pressure ideal fluid. Consider the FLRW metric ds l -dt 2 + a 2 (t) dr 2 1 — kr 2 + r 2 (d6 2 + sin 2 9d(f> 2 ) (19) Taking into account the perfect fluid description for matter given by Eq. (|17p . we verify that the gravitational field equation, Eq. ([5]), provides the generalised Friedmann equations in the following form [54ll55|: 1 2F(R) {a ^{llf(K)-mR)}-Z^)RF>(R)} = Ip, 1 -RF'{R) + RF'(R) + R 2 F"(R)--[f(R)-RF{R:)}} = --(p + 3p) 6 (20) (21) These modified Friedmann field equations may be rewritten in a more familiar form, as a \ k a J 6 -) = ~(ptot+3ptot), (23) a J b where p tot — p + pt c \ and p tot = p + Pm, and the curvature stress-energy components, pt c \ and pt c )i are defined as P (c) = J^^im-RFm-S^RF'iR)} , (24) P {c) = ^y J2 Q RF'(R) + RF'(R) + R 2 F'\R) - \ [f(R) RF(R)]j , (25) respectively. The late-time cosmic acceleration is achieved if the condition p to t + 3ptot < is obeyed, which follows fromEq. pg|) . For simplicity, consider the absence of matter, p = p = 0. Now, taking into account the equation of state w e ff — P(c)/p(c)i with f(R) oc R n and a generic power law a(t) = ao(£/£o) a [54j ]. the parameters w g and a are given by 6n 2 - In - 1 -2n 2 + 3n - 1 Wpff = ^ , a = , (26 6n 2 - 9n + 3 ' n - 2 ' V ; respectively, for n 7^ 1. Note that a suitable choice of n can lead to the desired value of u> e g < —1/3, achieving the late-time cosmic acceleration. Another example is a model of the form f(R) = R — /i 2 ( n+1 ) / ' R n [lq . Choosing once again a generic power law for the scale factor, the parameter can be written as " C " = - 1 + _&±H> , 27) 6(2n + l)(n + 1) and once again a desired value of w e ff < — 1/3 may be attained, by appropriately choosing the value of the parameter n. Note that as n — * 00 the spacetime is approximately de Sitter. Other forms of f(R) have also been considered in the literature, for instance those involving logarithmic terms, such as /(i?) = R + a\n{R/n 2 )+0R m or f(R) = R + jR- n [\n(R/p 2 )] m [H, HI] . These models also yield acceptable values for the effective equation of state parameter, resulting in the late-time cosmic acceleration. B. Gauss-Bonnet gravity and cosmic acceleration 1. Gauss-Bonnet gravity In considering alternative higher-order gravity theories, one is liable to be motivated in pursuing models consistent and inspired by several candidates of a fundamental theory of quantum gravity. In this context, it may be possible that unusual gravity-matter couplings predicted by string/M-theory may become important at the recent low-curvature Universe. For instance, one may couple a scalar field not only with the curvature scalar, as in scalar-tensor theories, but also with higher order curvature invariants. Indeed, motivations from string/M-theory predict that scalar field couplings with the Gauss-Bonnet invariant Q are important in the appearance of non-singular early time cosmologies. It is also possible to apply these motivations to the late-time Universe in an effective Gauss-Bonnet dark energy model m- Consider the action of Gauss-Bonnet gravity given by j4 / ^ - \d^d^ - V(4>) + f{<t>)G ■SuisT, ■>!>), (28) where A = +1 is defined for a canonical scalar field, and A = — 1 for a phantom field, respectively. The Gauss-Bonnet invariant in given by Q = R 2 — AR^R^ + R^apR^ 01 ^ ' ■ It is also important to note that in the matter action, matter is minimally coupled to the metric and not to the scalar field, making Gauss-Bonnet gravity a metric theory. Thus, using the diffeomorphism invariance of Sm (q^ j V 1 ) yields the covariant conservation of the stress-energy tensor, V' J TJ'™ ) = 0. Varying the action with respect to 0, provides the equation of motion for the scalar field, given by \V 2 0-V'( ( f>)+f'(0)g = Q. (29) The gravitational field equations are given by varying the action with respect to the metric g pv ', and provides the following relationship [55|, [57| -G^ - \gTf{<t>)Q + 2f(cj>)RR^ - 2WV v [f(cf>)R} + 2g pv V 2 [f{4>)R] K 2 ~8f(cf>)R%R»? + 4V p VTO)iH + iVpVmW"] - 4V 2 [/(0)i?H -Ag^V p V a [fWR P l + 2f{$)R ppaT R% aT - 4V P V CT [/(<£) R^ v ] = T pv + T$ v , (30) where Tj? v is given by Id^Q^-lg^d^dpA-l, TT = A ( tt^ d u 4> - -y v d p ^ d p <p ) - -g^V(4>) . (31) One may use the following identities obtained from the Bianchi identity [55l . 157] : VRpr^v = V"R vr ~ V"iV , Vi^ M = - V M i? , V /i V"i2'"' = -OR , V p V a R ppl/a = V 2 R^ - ^VViJ + R ppva R pa - R p p R vp , VpV (M^)P = v^V^-R - RWRpg + R p p R vp , in the Gauss-Bonnet gravitational field equation, which may then be formally simplified to K 2 +2f{cj>)R ppaT R l/ pcTT - 4f(0)R pp ™R pa = T pv + T$ v + T? v , (32) -G pv - ^g pv I{cp)Q + 2f(4>)RR^ + Af{<t>)R» p R v P where T pv is defined as T pv = 2[V p V v J{(t>)]R-2g pu [V 2 f{4>)]R-A[V p V p f{4>)]R vp - A[V p V u f{4>)]R pp +4{V 2 f(cb)}R p » + 4 5 ^[V p V ff /(0)]i?" CT - 4[V p V a f(4>)}R pp ™ . (33) Using the FLRW metric, the modified Friedmann equations for Gauss-Bonnet gravity reduce to the following relationships a \ k , k a J 6 3 d (35) A0 2 - 1/(0) + l2H^f{cj>) + 12- (ff 2 /) and the equation of motion for the scalar field is given by A U + ZH^j + V'{4>) - 2Af'{<t>)H 2 [H + H 2 ) = , (36) where the expression for the Gauss-Bonnet invariant, Q = 24H 2 (H + H 2 ), is used. Note that in the absence of matter, the Gauss-Bonnet gravitational field equations may be written as p GB - -H 2 , pgb = -- (SH 2 + 2H) , (37) where the Gauss-Bonnet curvature stress-energy tensor components are defined as Pgb = \tf + V{ct>) - 240/'(0)ff 3 , (38) Pgb = ^ 2 -y(^)+8^(i/ 2 /)+16ff 3 0/'(0). (39) These relationships are particularly interesting as one may now define an effective equation of state given by PGB 1 2H Pgb (40) Consider the choices of an exponential scalar-GB coupling and exponential scalar potential given by [57| V(cf>) = Voe-W* , f{4>) = foe 2 ^ . (41) The scale factor and the scalar field are respectively chosen in the following form a(t) a t h °, for ho > (quintessence), ao(t s — t) h ° , for ho < (phantom), and m .In for fto > 0, ^o In ( %-* ) , for /i < 0, where ti is an arbitrary constant [57|. Using the above choices, and considering the absence of matter, then taking into account Eqs. finally reorganizing the terms leads to the following equations Vot\ 48/ feg 1 k(1 + M 6 k(1 + /io) 3ftg(l - fto) A0qk(1 — 5/iq) /jo A0q« (42) (43) and ([3"6")l , and (44) (45) The effective equation of state parameter, Eq. (|4"0")) . takes the following form Weff = -1 - — . 3/l (46) Note that if /iq < then w e ff < — 1, reflecting an effective phantom regime; and if ho > then u> e & > — 1, which reflects an effective quintessence regime. However, it has been shown that the case of ho < is always stable, while the case of a non-phantom ho > cosmology is always unstable [57} • An interesting example is the case of Vo = 0, which from Eq. (|4~4"|) imposes the following condition [57[: </>g = — 6/Iq(1 — h )/[An(l — 5h )]. In order for cf> to be real, in the case of the a canonical scalar, A = 1, one finds that 1/5 < ho < 1; for a phantom field, A = — 1, then /io < 1/5 or /io > 1. To achieve an effective equation of state parameter value that mimics dark energy is to consider, for instance, ho = —80/3, which imposes the w c g = —1.025. We refer the reader to Ref. [57| for more examples and details. It is also interesting to note that observational constraints [58| , such as CMBR, galaxy distribution, large scale structure and supernovae seem to favour the Gauss- Bonnet coupling. 2. Modified Gauss-Bonnet gravity and the late-time acceleration An interesting alternative gravitational theory is modified Gauss-Bonnet gravity, which is given by the following action: S = -9 £ + w SmGT,^). (47) This theory has been extensively analyzed in the literature [39|, [59|, |6CJ , and rather than review all of its intricate details here, we note that it is a subset of Gauss-Bonnet gravity given by the action (j2"5|) . To see this, we follow closely the approach outlined in Ref. [5a, [60]. By introducing two auxiliary scalar fields A and B, the gravitational part of the action ([47]) . may be rewritten as S = j4 / £- + B(g-A) + f(A) (48) Now varying with respect to B, one obtains A = Q, so that the action (f48|) is recovered. Varying with respect to A, one obtains B = f'(A), and substituting in (|4"5)l leads to S = Id Xy/—g With the following definitions <p = A , and one finally ends up with the following action R R 2k + f'(A)(g-A) + f(A) V{<j>) = Af (A) - /(A) d xy—g 2 k V(^) + /($G s M (ar,ii>) (49) (50) (51) which is simply the action for Gauss-Bonnet gravity given by Eq. (j2"8|) with the absence of the kinetic term. Thus, modified Gauss-Bonnet theory, given by (|4"T|) is dynamically equivalent to Gauss-Bonnet gravity with A = |55l . l6(j | . We refer the reader to Refs. [39j, [59j for more details. C. DGP brane gravity and self-acceleration One of the key predictions of string theory is the existence of extra spatial dimensions. In the brane-world scenario, motivated by recent developments in string theory, the observed 3-dimensional universe is embedded in a higher- dimensional spacetime [29]. One of the simplest covariant models is the Dvali-Gabadadze-Porrati (DGP) braneworld model, in which gravity leaks off the AD Minkowski brane into the 51? bulk at large scales. The generalization of the DGP models to cosmology lead to late-accelerating cosmologies [32|], even in the absence of a dark energy field [33j |. While the DGP braneworld offers an alternative explanation to the standard cosmological model, for the expansion history of the universe, it offers a paradigm for nature fundamentally distinct from dark energy models of cosmic acceleration, even those that perfectly mimic the same expansion history. The 5D action describing the DGP model is given by S 1 2k 5 cPxy/=g w R (5), 1 2^4 d A x, Mr- d X\f— 7 C r (52) where k$ = 8nG5 and K4 = %-kG^. The first term in the action is the Einstein- Hilbert action in five dimensions for a five-dimensional bulk metric, gAB, with a five-dimensional Ricci scalar ^R; the second term is the induced Einstein-Hilbert term on the brane, with a four-dimensional induced metric 7 on the brane; and C m represents the matter Lagrangian density confined to the brane. The transition from AD to hD behavior is governed by a cross-over scale, r c , given by ^5 2K4 (53) Gravity manifests itself as a 4-dimensional theory for characteristic scales much smaller than r c ; for large distances compared to r c , one verifies a leakage of gravity into the bulk, consequently making the higher dimensional effects important. Thus, the leakage of gravity at late times initiates acceleration, due to the weakening of gravity on the brane. For weak fields, the gravitational potential behaves as $ for for r < r c , r > r c . (54) by Taking into account the FRW metric, and considering a flat geometry, the modified Friedmann equation is given H 2 -H lirG -p. (55) where e — ±1, and the energy density satisfies the standard conservation equation, i.e., p + 3H(p + p) = 0. For scales H^ 1 <C r c , the second term is negligible, and Eq. ([55)1 reduces to the general relativistic Friedmann equation, 10 i.e., H 2 = 8irGp/3. The second term becomes significant for scales comparable to the cross-over scale, H~ 1 > r c . Self- acceleration occurs for the branch e = +1, and the modified Friedmann equation shows that at late times in a CDM universe characterized by a scale factor p ex a~ 3 , the universe approaches a de Sitter solution H^H oc = - (56) r c Thus, one may achieve late-time acceleration if the Hq is of the order of ffoo. Note that the late-time acceleration in the DGP model is not due to the presence of a negative pressure, but simply due to the weakening of gravity on the brane as a consequence of gravity leakage at late times. Although the weak-field gravitational DGP behaves as AD on scales smaller than r c , linearized DGP gravity is not described by General Relativity [4CJ, [61j . It is also interesting to note that despite the fact that the expansion history of the DGP model and General Relativity are the same, the structure formation in both are essentially different [62]. Combining these features provides the possibility of distinguishing the DGP model from dark energy models in General Relativity. We should also emphasize that while the DGP model is not ruled out by current observations, the ACDM model fits the data comfortably [63|] . Another interesting aspect of this model is that the self-accelerating branch in the DGP model contains a ghost at the linearized level [6l|, [64]]. The presence of the ghost implies a negative sign for the kinetic term, resulting in negative energy densities, consequently leading to the instability of the spacetime. However, in a recent paper it was claimed that a higher codimension generalization of the DGP scenario is free of ghost instabilities [63 , and further work along these lines is currently underway. We refer the reader to Ref. (401. l6lj. and references therein, for more details on the DGP model. III. DARK MATTER AS A GEOMETRIC EFFECT OF MODIFIED GRAVITY The issue of dark matter is a long outstanding problem in modern astrophysics. Two observational aspects, namely, the behavior of the galactic rotation curves and the mass discrepancy in clusters of galaxies led to the necessity of considering the existence of dark matter at a galactic and extra-galactic scale. The galactic rotation curves of spiral galaxies 28] are probably the most striking evidences for the possible failure of Newtonian gravity and of General Relativity on galactic and intergalactic scales. In these galaxies, neutral hydrogen clouds are observed at large distances from the center, much beyond the extent of the luminous matter. As these clouds are moving in circular orbits with nearly constant tangential velocity v tg , such orbits are maintained by the balance between the centrifugal acceleration v 2 /r and the gravitational attraction GM(r)/r 2 of the total mass M(r) contained within the radius r. This yields an expression for the galactic mass profile of the form M(r) = rv 2 /G, with the mass increasing linearly with r, even at large distances, where very little luminous matter has been detected [281 ] . This peculiar behavior of the rotation curves is usually explained by postulating the existence of dark matter, assumed to be a cold and pressureless medium, distributed in a spherical halo around the galaxies. There are many possible candidates for dark matter, the most popular ones being the weakly interacting massive particles (WIMP) [69 ]. One cannot also a priori exclude the possibility that Einstein's (and Newton's) theory of gravity breaks down at galactic scales. In this context, several theoretical models, based on a modification of Newton's law or of General Relativity, have been proposed to explain the behavior of the galactic rotation curves [671 ] . A promising avenue that has been extensively investigated recently are the f(R) modified theories of gravity. In this context, early work in explaining dark matter using f(R) gravity using models of the form f{R) ex R n found large values for n [2l|, [22j, [23j • In these papers, a power law modified Newtonian potential of the form <f>(r) = — ^?[1 + {r/r c )P] was considered, to describe the observed behavior of the galactic rotation curves, where m is the mass of the particle, r c a constant and the coefficient (3 depends on the 'slope' parameter n in the modified action. Using this modified Newtonian potential, it was found that the best fit to 15 low luminosity rotation curves in R n gravity is obtained for n — 3.5 |2l| (somewhat lower values, in particular, n — 2.2, were obtained in [22, UM)- These results seem to suggest that a strong modification of standard general relativity is required to explain the observed behavior of the galactic rotation curves. Note that these large values of n are in gross violation with the Solar System tests. However, recently is was found that only slight deviations from General Relativity are needed, i.e., n = 1 + e with e <C 1 [2J, |2J|. This discrepancy of values can be traced back to the correction term of the modified Newtonian potential. It was shown in Ref. [24| that the correct modified term to the Newtonian potential in the "dark matter" dominated region, where the rotation curves are strictly flat, must have a logarithmic dependence on the radial coordinate r, of the form $Ar(r) = — ^^ + v 2 g ln(r/ro), where ro is an arbitrary constant of integration (we refer the reader to Ref. 24] for details). These differences in the Newtonian limit in the two models result in different values of the parameter n in the power-law modified action of the gravity. In the following sections, we shall analyze the 'dark matter' problem by considering a generalized version of the virial theorem in the framework of f(R) modified theories of gravity [25[. Recall that due to its generality and wide 11 range of applications, the virial theorem plays an important role in astrophysics. Assuming steady state, one of the important results which can be obtained with the use of the virial theorem is to deduce the mean density of astrophysical objects such as galaxies, clusters and super clusters, by observing the velocities of test particles rotating around them. Hence the virial theorem can be used to predict the total mass of the clusters of galaxies. The generalized virial theorem, in the context of f(R) gravity is obtained by using a method based on the collisionless Boltzmann equation [25| . The additional geometric terms present in the modified gravitational field equations provide an effective contribution to the gravitational energy, which at the galactic/extra-galactic level acts as an effective mass, playing the role of the 'dark matter'. The total virial mass of the galactic clusters is mainly determined by the effective mass associated to the new geometrical term, the geometrical mass. It is important to note that the latter term may account for the well-known virial theorem mass discrepancy in clusters of galaxies. A. The generalized virial theorem in f(R) gravity Consider an isolated and spherically symmetric cluster described by a static and spherically symmetric metric ds 2 = -e^dt 2 + e x ^dr 2 + r 2 {dd 2 + sin 2 ^ 2 ) . (57) The galaxies, treated as identical and collisionless point particles, are described by a distribution function fg, which obeys the general relativistic Boltzmann equation. In terms of the distribution function the stress-energy tensor can be written as T^ v = J f B mu^u^ du, where m is the mass of the particle (galaxy) [68j |. u^ is the four- velocity of the galaxy and du = du r duedu v / u t is the invariant volume element of the velocity space. Thus, the stress-energy tensor of the matter in a cluster of galaxies can be (r) represented in terms of an effective density p c g and of an effective anisotropic pressure, with radial p y c g and tangential p cS components, given by Pcti = P (u 2 ) , p^ = p (u 2 r ) , p[^ ] = p (uf) = p (u 2 ,) , (58) where, at each point, (w 2 ) is the average value of u 2 , etc, and p is the mass density [69J | . In what follows, we use this form of the stress-energy tensor, and for convenience take into account the scalar- tensor representation of f(R) gravity outlined in Section Hi A 21 As we are interested in astrophysical applications at the extra-galactic level, we may assume that the deviations from standard General Relativity (corresponding to the background value <j> = 1) are small. Therefore we may represent as = 1 4- eg'(R), where e is a small quantity, and g'(R) describes the modifications of the geometry due to the presence of the tensor 9^ v [19|], so that 1/0 ~ 1 — eg'(R). Now adding up the non-zero components of the gravitational field equation Eq. (fTTj) . and taking into account the above approximations (see Ref. [25j for details), one obtains the following relationship / a ii i i/'\'\ er x 1^— + — + -- — J ~ AnGp (u 2 ) + ^G P4n (59) where (u 2 ) = (u 2 ) + (u 2 ) + (w 2 ,) + (u 2 ), and the useful quantity p^ is defined as -ep(u 2 )g'(R) + ' 4ttG \v {(j))+ i(2V t V* + D) (60) b=l+eg'(R) which may be interpreted as the geometric energy density. It is convenient to introduce some approximations that apply to test particles in stable circular motion around galaxies, and to the galactic clusters. First of all, we assume that v and A are slowly varying (i.e. v' and A' small), so that in Eq. ([5"9"]l the quadratic terms can be neglected. Secondly, we assume that the galaxies have non-relativistic velocities, so that (u 2 ) « (u 2 ,) « (u§) <C (uf,) w 1. Thus, Eq. (|55j) becomes ±1 ( r ^ 2r 2 dr \ dr AnGp + 4irG P4> . (61) In order to derive the virial theorem for galaxy clusters, one uses the relativistic Boltzmann equation, which provides the following relationship (see Ref. [25| for details) 2K-- 4Trr 3 p^-dr = 0, with K = 2np [(uj) + (uj) + (u 2 )] r 2 dr . (62) 2 Jo " r Jo 12 K is the total kinetic energy of the galaxies, and the total mass of the system is given by M = /" dM(r) = L 4wpr 2 dr. The main contribution to M is due to the baryonic mass of the intra-cluster gas and of the stars, but other particles, such as massive neutrinos, may also contribute significantly to M. Now, multiplying Eq. (|6ip by r 2 and integrating from to r we obtain GM{r) = -r 2 -^-~GM^(r), with Af (r) = 4tt / P4> {r')r n dr' . (63) The useful quantity M^ is denoted as the geometric mass of the cluster. By multiplying Eq. (f6"3"|) with dM(r), followed by an integration one deduces the relationship n = n^--j i^p-^dr, (64) with the following definitions n=-[ R ^n dM{r) , and n^^^MdMir), (65) o r where the quantity fl is the usual gravitational potential energy of the system. Finally, with the use of Eq. (I62|) . we obtain the generalization of the virial theorem, in f(R) modified theories of gravity, which takes the form IK + ft - ft = 0. (66) Note that the generalized virial theorem, given by Eq. (166|) , can be written in an alternative form if we introduce the radii Ry and R^ defined by Ry=M 2 / fMWdMM, and R+ = Mil C ^^ dM(r) , (67) respectively. We denote i?^ as the geometric radius of the cluster of galaxies. Thus, the quantities f2 and Q<p are finally given by O = —GM 2 /Ry and Cl^ = GMl/R^, respectively. The virial mass My is defined as 2K = ^^ . (68) Rv After substitution into the virial theorem, given by Eq. (|66[) , we obtain M + M 2 R 4> • [ > If My/M > 3, a condition which is valid for most of the observed galactic clusters, then Eq. (j6"5|) provides the virial mass in f(R) gravity, which can be approximated by Mv ^^L. (70) M R^ V ' B. Geometric mass and geometric radius from galactic cluster observations An interesting application of the generalized virial theorem can be inferred from the galaxy cluster observations. According to the modified f(R) gravity model, the total mass of the cluster consists of the sum of the baryonic mass (mainly the intra-cluster gas), and the geometric mass, so that Af tot (r) = in J" (p g + p$) r 2 dr. Hence it follows that Mtot(f) satisfies the following mass continuity equation dM 2 (r) =^r 2 p g (r)+^r 2 PlP {r). (71) 13 Note that most of the baryonic mass in the clusters of galaxies is in the form of the intra-cluster gas. The gas mass density p g distribution can be fitted with the observational data by using the following expression for the radial baryonic mass (gas) distribution [70] / 2 \ -3/3/2 Pg(r) = Po ( 1 + ^J , (72) where r c is the core radius, and po and_5 are (cluster-dependent) constants. Using the Jeans equation [28J|, one may obtain the total mass distribution [7(J, [71( s so that taking into account the density profile of the gas given by Eq. (|72[) , the total mass profile inside the cluster is given by M tot (r)= 3kBPT J J* , (73) pm p G r£ + r* where ks is Boltzmann's constant, T g is the gas temperature, p rs 0.61 is the mean atomic weight of the particles in the cluster gas, and m p is the proton mass [701 ]. Using the mass continuity equation, Eq. ([TTj) . then Eqs. (172p and f73"|) provide the expression of the geometric density term inside the cluster, given by 3k B 0T g (r 2 +3r 2 c ) _ AnGpo 47rp0 (r) In the limit r > ^ we obtain for p^ the simple relation ^^ - AnGporfr 2 -^} ' and the geometric mass in the limit r » r c , may be approximated as ~3k B (3T g inGpor^r 2 - 3 ! 3 ' 2 GM^ (r) (75) (76) prrip 3(1-/3) One may assume that the contribution of the gas density and mass to the geometric density and geometric mass, respectively, can be neglected. The latter approximations are very well supported by astrophysical observations, which show that the gas represents only a small fraction of the total mass [70ll72]. Therefore, we obtain «A r " 2 , and GM.ir)*^* AnGp^r) « ^JllLJL r ~\ and GM<t> (r) a ^JULJL r> (77) respectively. One may also estimate an upper bound for the cutoff of the geometric mass. The idea is to consider the point at which the decaying density profile of the geometric density associated to the galaxy cluster becomes smaller than the average energy density of the Universe. Let the value of the coordinate radius at the point where the two densities are equal to be nf r . Then at this point p,p{R? r ) = p un iv, where p un iv is the mean energy density of the universe. By assuming p un iv = Pc = 3H 2 /8irG — 4.6975 x 10 _30 /i| g/cm~ 3 , where H = 5O/150 km/Mpc/s J7CJ, we obtain The total geometric mass corresponding to this value is M™ = M, « } ) = 4.83 x 10 16 /5 3 / 2 (|gf ^ K$M®. (79) This value of the mass is consistent with the observations of the mass distribution in the clusters of galaxies. However, according to f(R) modified theories of gravity, we predict that the geometric mass and its effects extends beyond the virial radius of the clusters, which is of the order of only a few Mpc. By assuming that R^ ~ -Rl , we obtain the following relation between the virial and the baryonic mass of the cluster For a cluster with gas temperature T g — 5x 10 7 K, (3 = 1/2 and Ry = 2 Mpc we obtain My ~ 32M, a relation which is consistent with the astronomical observations 1701 . 14 IV. SUMMARY AND DISCUSSION Cosmology has entered a 'golden age', in which the rapid development of increasingly high-precision data has turned it from a speculative to an observationally based science. Recent experiments call upon state of the art technology to provide detailed information about the contents and history of the Universe. These experiments include the Hubble Space Telescope, the NASA WMAP satellite instrument, that measures the temperature and polarisation of the CMBR, and the Sloan Digital Sky Survey (SDSS), that is automatically mapping the properties and distribution of 1 million galaxies. High-precision cosmology has allowed us to tie down the parameters that describe our Universe with growing accuracy. The standard model of cosmology is remarkably successful in accounting for the observed features of the Universe. However, there remain a number of fundamental open questions at the foundation of the standard model. In particular, we lack a fundamental understanding of the acceleration of the late universe. Recent observations of supernovae, together with the WMAP and SDSS data, lead to the remarkable conclusion that our universe is not just expanding, but has begun to accelerate [l|. What is the so-called 'dark energy' that is driving the acceleration of the universe? Is it a vacuum energy or a dynamical held ("quintessence")? Or is the acceleration due to infra-red modifications of Einstein's theory of General Relativity? How is structure formation affected in these alternative scenarios? What will the outcome be of this acceleration for the future fate of the universe? The aspects of these fundamental questions whose resolution is so important for theoretical cosmology, need to look beyond the standard theory of gravity. It is clear that these questions involve not only gravity, but also particle physics. String theory provides a synthesis of these two parts of physics and is widely believed to be moving towards a viable quantum gravity theory. One of the key predictions of string theory is the existence of extra spatial dimensions. In the brane-world scenario, motivated by recent developments in string theory, the observed 3-dimensional universe is embedded in a higher-dimensional spacetime [231 . The generalization of the Dvali-Gabadadze-Porrati (DGP) brane models [3l| lead to late-accelerating cosmologies |32|], even in the absence of a dark energy field. This exciting feature of "self acceleration" may help towards a new resolution to the dark energy problem, although this model deserves further investigation as a viable cosmological model [73|]. It will be interesting to generalize the DGP model with the inclusion of a Gauss-Bonnet (GB) [33[, and it will also be important to investigate the effects of the GB term relatively to the issues of strong coupling and ghosts in the DGP models. Infra-red modifications to General Relativity, where the consistency of various candidate models, including 4-dimensional modifications to the Einstein-Hilbert action, especially GB modifications with a scalar field coupling, have also be analyzed [57] . In this context, a more general modification of the Einstein-Hilbert gravitational Lagrangian density in the form of L — f(R) has recently been extensively analyzed. Relatively to the construction of "quintessential" dark energy models, recent fits to observational data indicate that an evolving equation of state crossing the phantom divide is mildly favored [7J, UM- ^ n a cosmological setting, it has also been shown that the transition into the phantom regime, a mixture of various interacting non-ideal fluids is necessary [6] , with important implications to the model construction of dark energy. If confirmed in the future, this behaviour holds important implications to the model construction of dark energy. The latter models, considering a redshift dependent equation of state, possibly provide better fits to the most recent and reliable SN la supernovae Gold dataset. Deciding between these possible sources of the cosmic acceleration will be one of the major objectives in cosmology in the next decade with several surveys and experiments to address the nature of dark energy. One may mention new several major SNIa supernovae projects, such as the SuperNova Legacy Survey (SNLS), SDSS-II, Destiny, the Large Synoptic Survey Telescope (LSST), Dark Energy Survey (DES) and the SuperNova Acceleration Probe (SNAP). Other dark energy probes include the Dark UNiverse Explorer (DUNE); the Wide Field Multi-Object Spectrograph (WFMOS), which will perform surveys to measure dark energy and the history of our Galaxy; and the Panoramic Survey Telescope and Rapid Response System (PanSTARRS), amongst others. All of these aspects present an extremely fascinating aspect for the above-mentioned experiments and for future theoretical research. Acknowledgments I thank Orfeu Bertolami, Christian Boehmer, Tiberiu Harko, Kazuya Koyama, Roy Maartens and Antonios Papa- zoglou for helpful comments, and acknowledge funding by Fundagao para a Ciencia e a Tecnologia (FCT)-Portugal through the grant SFRH/BPD/26269/2006. 15 [1] Perlmutter, S. et al. 1999, Astrophys. J., 517, 565; Riess, A. 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subset: commoncrawl


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Have Medical Questions? Ask a Doctor Online!
Hi my name is***** will reply to you in just a moment, allow me to read the question.
I'm glad the CT scan and blood tests were normal, that's very reassuring. However the symptoms you are suffering from sound miserable since they have been going on for a month.
A virus can hang on for quite some time, even though most patients only have it for a few days, some will have it for months.
If it's improving then that's really important, your body is likely encountering a very strong virus and building an immune response.
The dryness could be from the inflammation of the sinuses and nasal tract. Medications can cause it too but you would know because the dryness would be temporary. I assume you are no longer having any high fevers or severe body aches?
Please let me know if this information helped.
It can last another 1-2 weeks but should improve soon. The tingling and numbness is from peripheral vasoconstriction, it will take some time for that to resolve. The best thing you can do is hydrate aggressively. Get plenty of rest, I know we say this a lot but it's vital.
Cytomegalovirus is another common virus that can cause this. Also benign bacterial infections such as lysteria can cause this.
Quite possible and common.
Please let me know if you have any other questions about it, I'm happy to answer them.
If you have no other questions you can click on "Rating" to complete this session.
It was my pleasure chatting with you.
Dr. Ashori
How are you doing today? Any improvement?
It might take time for it to resolve, but I'm convinced that the sensation will fully go back to normal in time.
Hope my answers were helpful.If you are done with this case please click on "Rating" to complete it.Thank youDr Ashori
If you have any other question please let me know, otherwise please click on "Rating" to close this visit.
On average 2 weeks.
Some patients get relief as early as 10 days.
Anything else I can help you out with?
Please click on "Rating" in order to close this case and complete the transaction. Dr Ashori
No, it wouldn't affect the vessel function.
Bacteria can do that but no virus of that kind.


text:
Conditions with symptom: 'pubis'
Search Results
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Excess Fat on Pubis
The Mons Pubis is the area of the body that sits just above the pubic bone and consists mostly of fat. Read more →
I am 6 months postpartum and just noticed a knot/lump the size odňal dime on my mons pubis then after looking a little more that I have about 10 tiny bumps spread out in between the mons pubis and the clitoris. The big one is painful. The littlebones are painful as well but itch morebthan anything, Breast and cervical cancer are very strong in ky family so I'm pretty terrified right now. Please help!!!
I am 15/16 and have a fat pubis how can I get rid of it ? :)
you poor thing Robbie...are you male or female? I have lived my whole life with this embarassing problem. It has really held me back in relationships. I suggest you ask your doctor and not let it rule you life. You are lucky to have the internet sites and TV shows like this for help. AAll the best to you.
I have a large mons pubis since putting on weight 6 years ago, is there anything i can go as it,s embarassing. Fay
i'v been watching the embarrassing fat bodies and most of them lose weight or get surgery.
Im 20 and have had an enlarged pubis since as long as I can remember, I'm not overweight it's just always been that way. Is there help available for this on the nhs? It rules my life and I am getting to the point where I can't cope any more.
I'm a slim build, and i have a large mons pubis and more fat ontop than I would like. This exsits in the female population, its just the medias perception or what we've come to accept as an acceptable a female body (but its not true). You are still a woman,capable of having sex/a family/a good life. its just the way our pelvic bone sits in out body (the angle changes between person to person)
I've had a red spot/lump on the mons pubis just above the clitoris, it's sore to touch and appears quite red, I'm quite concerned and really hoping it isn't a STD, any ideas please?
I have found it difficult to find any clinics which specialise in the removal of fat and excess skin from the pubis area.Please could you point me in the right direction. Thanks
Hi. They dont seem to advertise mons pubis reduction.. I found it difficult to gather info online. I contacted lots of popular clinics and enquired about a reduction.. most of the surgeons I met said the usually conbine the pubic lift with a tummy tuck.. I said thats no good as I want to have children in the future.. so I opted for liposuction to get rid of the fat...
My mons pubis area is very fat and sticks out alot and shows through clothes. I hate it and am so embarrassed by it that I have never had sex and I don't think I ever will if it stays the same. Is this normal and is there any way to get rid of it (possibly avoiding surgery)?
I have the same problem. I have done my research and it is very difficult to lose the fat by healthy eating and exercise. It is usually genetic. The only answer I have came across is liposuction. NHS won't fund it so it is very expensive. I am currently saving for liposuction. Hate not being able to wear normal clothes and pretty bikini's. Having a fatty mons pubis is very common so you're not alone x
Hi i feel like a freak too my mons is huge and im not exagerating. I cant afford lipoand was turned down on nhs. Id rather be dead than have this. I dont ho out coz cant wear anything nice.i wont have sexir smear tests etci feel so ashamed
i have excess skin on my stomach and pubis area ive only had this since my c sections i have had 5 ive been to doctors told them how depressed i am how pain full it is it gets sore and sweaty ive been refused treatment on nhs I dont know what to do
Hi thats ashame.. the nhs dont like to do cosmetic surgery anymore. . Try your local cosmetic sugery and look into a tummy tuck.. its expensive but can be done on finance..
I have just asked a question about enlarged mon pubis...or veneris mons and have found out through a link on this site it is possible to have liposuction of that area. Is there anywhere in the UK that does this and is safe? As I have only found clinics abroad which will make it more expensive
I have the same problem. Not sure if you will see this now because you posted this so long ago. Harley Medical Group does it. I am currently saving for liposuction.
Hi I am 39 years old and I have had 4 sections and I'm left with a hangover and it as got worse since i have lost 4 stone and it as also left me with a fatty pubis to :( I have been to my gp and he as said there is no exacise that will get rid of it and there is nothing they can do surgically as he said it is classed as cosmetic procedure I don't know what else to do and it making me so depressed.please could you help!
Hi, I am also a 17 years old female, also have some kind of spots in between my legs and in my pubis area, other thing that I have is a brown mass growing and sticking to my pubis area. I don't know what it's and it's scaring me ! Can you help find out what it is and what solution do I have for it ?? Thank you
I have been on a large amount of morphine for the last 3 years now after inreasing slowly but have put on a lot of weight and have a fat pubis and a large stomach, i also do not get erections as often as before. What can i do to lose weight and get rid of the fat pubis, as dieting does not seem to be working.
Look into slimming world. Not just for women! It is not a diet but a way of life. Steroids will always make you gain some sort ooptimising your food will help to keep weight gain via food will slow even help lose.
I have this 'problem' and also have quite a large and pronounced mons pubis. These two things together make me extremely self conscious. I am 19 and a virgin and can't imagine letting anyone see my vagina because I hate it so much. I am aware that many people have enlarged labia and some people don't mind it or even like it but what about those of us who don't? Surely the most natural thing in the world is to have sex and get pregnant? So if you find it too stressful to do that and something like surgery can be done to help you why is it not covered by the NHS? There's no way in the world I could raise £3000 for this, I just think I'll live the rest of my life a virgin. Also, does anyone know of anything that can be done about the mons pubis area as this can be very noticeable through clothes?
Hi, i 13 and my pubis are blonde is this normal
Unless its green and purple and smells bad you really don't need to worry what colour your pub's are
Please please help me!! I'm 25 and have been with my partner since we were 14. Since giving birth to my son who is now 5 years old, I've had a constant problem with my pubis area. Stretch marks have obviously helped gravity take its toal on my body- but also on the skin from my pubis area. Because of this my whole vagina has dropped and is fat and saggy. (I'm not overweight).I am effected by this day to day. I cannot wear underwear at all as I'm sore. I cannot walk for too long as my vagina rubs inbetween my legs- I constantly have thrush as a result of this. I can't wear jeans or skirts or smart trousers, I live in leggings as they are the comfiest. Not only that, but because the 'sag' is so much my vaginal skin splits and is extremely sore all of the time. I also have not had an orgasm in 5 years and I think all these problems are connected. I really need to know if I can get any type of help with this, it's destroying my life. Is there any type of surgery available that will hopefully lift my skin to where it used to be?!
Hi, I have a fatty pubis and I hate it. What can I do? Any advise would be much appreciated.
You will learn to execpt that not all women look like the girls in adds with flat mons pubis', some people have a more prominent pubic bone and therefore need more fat to protect the bone. The more you're proud of it , and accept that it's there and it's part of you the more other girls will feel they can too and not hide it under clothes. The issue eases with time, I promise. Don't save up for plastic surgery you're made the way you are for a reason.
Really wanted to see this but video kept going off after only a minute...I need to know what to do after examining my mons pubis...help
I'm 13 and I have a spot around my pubis area. It is a bit big and it doesn't have a head on it, it's very sore though. Do you know what it is?
This is called the 'mons pubis' and is perfectly normal - look on wikipedia!
Hi, For a few months now I have experienced severe itching at the top of my vulva (on the pubis I believe it is called and above my clit) also on the side of my leg near the vulva. It usually happens at night. I have never had sexual intercourse so STI's can't be an issue. I have been given a tablet for thrush last month and it seemed to stop for a little bit (Well it still itched just not as severe) I have noticed it starts getting worse nearer my period. I have very sensitive skin too so not sure if that may be playing a part. I have completely shaved the area to see what might be going on and there seems to be a red patch of skin (where I have scratched) around the pubis and an extremely sore area above the clit. I assume this could be dermatitis/exzema. (Could someone suggest what else it could be?) I want to know whether there are any creams to use? I have used canesten and vagisil (I believe it is called) and it does not help. I have just now resorted to using savlon cream to sooth the soreness. I have been to the drs countless times but they seem to just 'brush it under the carpet' and say it'll clear over time. It is getting to the point now where it is so sore to walk or wear any type of clothing below.
Sounds a bit like you have created some exzema, and the scratching and shaving has made it worse. Rather than go down the pharmaceutical path use a mix of castor and coconut oil, both are very soothing and anti fungal/antibacterial. Apply twice a day. Clip your pubic hair rather than shave, or buy an electric shaver; razors remove the top layers of skin, shavers don't. Also use cornflower as powder, NOT TALC as talc has shown up in ovarian cancer. Also, don't use soap on your vulva, just plain water, keeps the natural oils there to protect her.
Hi im diabitic 14 years old and my mons pubis looks long and big,I masturbate a lot like twice a day can you help me please and im a virgin
You do not say in what respects you need help, Zoey. What is wrong with having a long and large pubis ? As regards masturbation, there is nothing wrong. The old theory that it is harmful has been discredited. It is actually beneficial, according to latest opinion. It keeps the heart healthy. Women who do not masturbate are a tiny minority.
hello everyone im a 15 year old girl i have a fat mons pubis and i cannot wear anything tight because it just looks like a big bump!! it gets me down and really upsets me, its there anything i can do for this? thank you so much
My name is claire and I'm 31 at my heaviest I was 27stone 8 pound I'm now 9 stone 11 but my doctor without viewing me under my clothes has said that I need to lose another stone and half to be put forward for any kind of excess skin removal surgery. I want to know if this is true because I believe the excess skin that surrounds my arms, stomach, thighs and especially pubis would have quite a considerable weight and am already displease with the way my ribs and collar bones pertrude. What are the guidelines for GPS to put forward a patient for skin removal surgery. My excess skin is making it difficult to exercise as my skin slaps so violently, I sweat so bad and get irritated in the folds between my thighs and pubis, when I go to the toilet I wee on myself as the excess skin gets in the way. Due to excessive washing to combat the many abscesses I have had on my upper inner thighs I suffer with Bv regularly. Losing the weight has been a loss in my self asteem and confidence. Should my Gp be helping me? Desperate
Firstly ccongratulationsfor losing your weight. They should be supporting you and realise that this is affecting your mental health and hindering your further weight loss because of the sores and embarracement you feel. Unfortunately you may have to bare all to your dr so they can see the sores. Good luck and congrats again for your loss.
hello im a 15 year old girl and i have a fat mons pubis i cant wear anything tight and it really upsets me, im not over weight im actually silm please help :(
i have red lumps/spots on my pubis, around my vagina and on my buttocks. the can rub and be irritating. its really embarrassing i have been to my GP and got anti-bioticts but they never worked. help!
I sometimes get this and i was told it could be ingrown hairs
Hey, I have a large,fat mons pubis.It protrudes in clothes and gives me an embarrassing camel toe.Would losing weight help this.PLEASE HELPPPPP!
hello everyone im a 15 year old female and i have a fat mons pubis it sticks out so much i cant wear leggings tight clothes can somebody find out how or what i can do to make this go away or redcuce the size:(
Thank you for visiting our web site and for discussing your problem so openly. If you would like the opportunity to be seen by one of the Embarrassing Bodies Doctors and have your problems assessed and treated by a specialist, then why not apply for the chance to take part in the programme. Please visit the following link and click 'Apply to be on the show' where you can fill in an application form: http://www.channel4embarrassingillnesses.com/features/be-on-embarrassing-bodies/
Ohmygosh same! I am 14 and started my period when i was 13, and i am a normal weight. However i have always had a "chubby mons pubis" and am afraid to show anyone it. I have got close to boys but fear intimacy because of that? HELP!
Okay, well most of the teenagers suffers from these problems! FirSt thing you have to do is wash your clothes with boiled, water if these are crabs(phthirus pubis), use antiinsectants lotion and shampoos!
Hello Sophie, Yes this is called "mons pubis". This in english means pubic mound or 'mountain or ,hill. This generally is considerd to be georgeous by men, so don't be selfconcious.
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How Do You Know If You Are Getting the Best Quality Human Hair Weaves?
(comments: 1)
Written by Deborah Moffett
How Do You Know If You Are Getting the Best Quality Human Hair Weaves?
Shopping for hair can be a challenge! With prices continuing to increase due to high demand, you need to be selective of what you purchase. So many companies claim to sell human hair. Unfortunately many of these shops sell hair that is human hair mixed with animal hair. A good indicator is when you see very low prices from the norm. This usually indicates that the hair may not be 100% human. Here are some industry tips that will help as you navigate through the many hair choices today.
Remy/Virgin Remy
Many stores now use the word Remy for their product. It is important to note that Remy means specifically that all the hair cuticles have not been stripped. This process allows for the hair to lie in its natural position making for the most natural looking hair extension. The hair is expensive and is collected as a ponytail, donated from a woman's head, so it is intact, from one donor, and full length. This explains why the hair does vary slightly from bundle to bundle. Virgin Remy is hair that is untouched or unprocessed. To maintain the high quality of this hair it should only be steam processed without the use of any harsh chemicals.
Brazilian, Malaysian and Peruvian
These descriptions are common in the hair industry but they do not necessarily indicate this is where they were collect. Brazilian hair is more representative of the texture, a thick and wavy texture, holds color and curls very well and is soft and smooth. Our International Collection is representative of this hair. Malaysian Hair is shiny, the curls and waves are steam set so they will last. Malaysian hair is thick loves a leave-in conditioner. Our New York Remi Collection is representative of this hair. Peruvian hair is more coarse and thicker than the Brazilian and Malaysian hair so will blend well with Caucasian hair textures and African-American relaxed hair textures. Our Pure Remi Collection is representative of this hair.
Online/ Corner Stores
Do your homework when shopping online or at the corner store. Check the stores ratings. Ask how long they have been in business and check to see their knowledge of the product. It is a good indicator that you can trust shopping with a company that has been in the business for an extensive amount of time. Understand that you get what you pay for and high quality hair is in high demand so can be very expensive.
Human hair is expensive so by mindful of what you are purchasing. Real human hair weaves will last longer when cared for properly and look nicer than those hair extensions that are mixed with synthetic or animal hair. That is not to say that you cannot get a great look with lesser quality or synthetic hair. Just understand that you will need to spend more time on upkeep as well as replace the hair sooner than with a higher quality product.
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Pretty easy, overall.
Course Overview:
Going in to the communication disorders field, learning about special education is very interesting and informative. I will definitely be utilizing the information I have received in this class.
Course highlights:
I learned about the IEP, and how to (theoretically) work well and efficiently with children with disabilities.
Hours per week:
0-2 hours
Advice for students:
Go to class, always participate. Don't be afraid to ask questions, and be respectful! It is a large lecture so anything you do is very noticeable. You are required to have an electronic device, but don't go on social media!


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will headers increase horse power on my 06 ford focus zx3 2.0L?
i already have a manga flow exhaust and a cold air intake, but i want to know if i should even buy headers. they
have some for my car but i don't know if it will really make a difference . could any engine wizz help me out?
3 Answers
Headers might give you just a little, very little more HP, mostly people install them for sound, and if you live in Calif, don't do it, and some other States
a header will give you more power than your catback....
i had a OBX header on mine and felt a huge difference! The stock header restricts airflow and hp, once i installed the OBX 4-1 race header my car actually pulled hard in 4th and 5th gear. I recommend it.
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New addition
Three new ladies add cool factor to capital FM.
Capital FM has just introduced three new faces, or rather, voices to its station. there's the chatty duo pamela and Janice to brighten up the mornings on Capital Breakfast (6am-10am on weekdays) and to accompany you in the evening rush hour is isabel on The Jam Break (4pm-8pm, with co-host Non).
Newcomers to radio, Janice, pamela and isabel bring a new level of wholesomeness to Capital FM. Each one has a background that enables them to relate to the women of today.
"to connect with the diverse, complex and colourful listeners of the station, the lingua franca adopted at Capital is that of adventure, Compassion and Humility," says general manager lynette Ow.
"these are qualities pamela, Janice and isabel have in spades. they are able to relate and be relatable to our listeners."
On Capital Breakfast, Janice and pamela will bring real issues faced by Malaysian females to the forefront. is the glass ceiling still present in maledominated industries?
Do workplace policies ensure sexually-hostile environments cease to exist in Malaysia, or is this merely on paper to comply with labour laws? Janice and pamela have witnessed enough to know that women with real issues need a place where things can be discussed.
Stressed out because you are stuck in the evening traffic jam? let isabel and Non help de-stress you!
the two announcers are like chalk and cheese where they both offer multiple perspectives that mirror the listeners point of view on the day's topics, ranging from the serious to the irreverent and sometimes just plain "i didn't know that!".
isabel shares: "listeners can look forward to fun chats with Non and i about interesting stories we've found from across the globe, insightful how-to guides and random fun facts, all with a good dose of cheeky banter!"
although new to the radio industry, isabel is fitting in just fine. Non finds his new co-host a pleasant to work with.
"isabel is one of the most energetic, composed people i have ever met. it sounds like a contradiction, i know, but if you tune in to The Jam Break you'll hear what i mean," Non says.
"She's easily the most mature 'young' person i've had the pleasure of working with. and respectful too; she always let's me cross the road first."
He said, she said: non and Isabel offer multiple perspectives on the day's topics, ranging from the serious to the irreverent on TheJambreak.


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Store's Best Items
|NewYork Mania
|Seller's Address
Basic Information of Product
|Product Status/Sales Type
|Product No.
|Weight (kg)
|1 kg
|Country of Origin
|Shipping Method
Courier Service
|GST Applicable
|Not Applicable (Non-GST seller)
|Tax Invoice
|After Sale Service
11street receives report on products to protect buyers' right. For further information on safe trade, please contact us.Reportgo
For order related and other issues, kindly click on E-mail Enquiry here.E-mail Enquiry
Product Name
Vitscost 123 Gluten Free Southern Style Corn Bread Mix -- 13.23 oz
Product Code: PROD1540017242
Product Information
123 Gluten Free Southern Style Corn Bread Mix Description Easy to Make, Can**t Beat The TasteYields 16 Large SquaresGluten, Wheat, Dairy, Casein, Peanut, Tree Nut, Sugar, Egg, Soy, Rice & Potato FREEKosher Enjoy the corn bread hot out of the oven or use this to make a wonderful stuffing! Micah**s Mouthwatering Corn Bread Mix can also be used to make jalapeno cheddar cornbread, which is great with chili. 123 Gluten Free has a wonderful corn bread that is so enjoyable you can eat it alone! This mouthwatering recipe yields 16 large squares of corn bread goodness. There is an egg free recipe and a dairy free recipe on the box. With no trans fat and only 80 calories per serving you have a whole lot of loving southern style. DirectionsTo Make Corn Bread: 1 package Micah**s Mouthwatering Corn Bread Mix™ 3 Tbsp. sugar (optional) 2 large eggs, room temperature 2/3 cup buttermilk 2/3 cup milk or milk substitute 2 Tbsp. unsalted butter or unsalted margarine, melted or oil. Preheat oven to 425 degrees F. Generously grease 9** square pan with non-stick spray.Place 1-2-3 Gluten Free® Micah**s Mouthwatering Corn Bread Mix™ in bowl. Add sugar, eggs, buttermilk and milk and stir until combined. Stir in melted butter/oil.Pour batter into pre-greased pan.Bake for approximately 20 minutes, until top is golden brown and sides begin to pull away from pan. Baking times may vary by appliance.Cut corn bread into squares. Serve warm. Makes 16 squares.Free OfGluten, wheat, dairy, casein, peanut, tree nut, sugar, egg, soy, rice, and potato. Disclaimer These statements have not been evaluated by the FDA. These products are not intended to diagnose, treat, cure, or prevent any disease.
Calories 80
Calories from Fat 0
Total Fat 0 g 0%
Saturated Fat 0 g 0%
Trans Fat 0 g
Cholesterol 0 g 0%
Sodium 300 mg 13%
Total Carbohydrate 20 g 7%
Dietary Fiber less than 1 g 2%
Sugars 0 g
Protein less than 1 g
Vitamin A0%
Vitamin C0%
One Color
Seller Information
|NewYork Mania
|Ship-From Address
|NJ , SUITE 6 Clifton
|Return/Exchange Address
|Notice on Return/Exchange
Return / Exchange Policy
- Request for cancellation if product has not been dispatched after payment completion.
- Request for return within 7 calendar days following delivery date.
- Check seller's approval on buyer's request.
- Buyer sends product in question to seller.
- Seller receives and checks returned product
- Seller approves return and return is completed.
- Request for exchange within 7 calendar days following receipt of delivery.
- Check seller's approval on buyer's request.
- Buyer sends product in question to seller.
- Seller receives and checks returned product
- Seller send new product to buyer, and buyer receives it, and exchange is completed.
How to Request for Cancellation/Return/Exchange
You can request for cancellation after completing payment and if product has not been dispatched by then, cancellation request is approved and you can receive refund.
But, in case seller has already dispatched product, cancellation request can be rejected.
You can request for return after obtaining product, you should send the product in question to seller.
After seller checks if returned product is flawed, and approves your request, return is completed and payment is refunded.
You can request for exchange if wrong product is shipped, you should send the product in question to seller.
After seller checks if returned product is flawed, seller approves your request, and re-sends new product.
- You can request for cancellation after completing payment and if product has not been dispatched by then, cancellation request is approved and you can receive refund.
Criteria for Return/Exchange
You can request for return/exchange within 7 calendar days following delivery date. But return/exchange request cannot be made in following cases:
- When request due to change of mind is made after 7 calendar days following delivery date
- When product is used, destroyed or damaged
- When tag attached to product is removed or package of product is opened and product value is damaged
- When sealed package is opened or packaging materials are lost
- When too much time has passed and product value has so decreased that its re-sale is not possible
- When return/exchange request is made for customized product such as hand-made shoes or accessories
- When components of product (including free gift) have been used or lost
- When buyer did not follow instructions included in product
- As e-voucher products are immediately sent to your e-mail, returns/exchanges are not available. Please take note of this before purchase, and input your e-mail address correctly.
- As Mobile Top-up products are directly reloaded on your cell phone number, returns/exchanges are not available. Please take note of this before purchase, and enter your cell phone number correctly.
For any inconvenience regarding Cancellation/Return/Exchange, please contact Customer Service at 03-2720 0000 and we will do our best to help you.
- Please note that, direct transaction with the seller without using 11street payment service can cause problems such as; not receiving the product, receiving different product etc.
- For sales products and contents registered by the seller, 11street run by Celcom Planet has no responsibility on any of the registered contents.


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Professionally Treated Lawns - Tree and Shrub Care
Lawn Rx is the first local business to donate services in support of our non-profit community activities. Lawn Rx is the exclusive turf treatment company for the Loyalhanna Soccer Park. Our fields benefit from the five treatment program provided by Lawn Rx.
We encourage all members to utilize Lawn Rx for their personal home lawns, trees and shrubs.
Phone: (724) 539-1003
Fax: (866) 597-5296
website: www.lawnrxinc.com
"Your Prescription For A Healthy, Beautiful Lawn"


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Slim and lightweight construction that provides the maximum protection for your Galaxy S6 Edge by using Air Cushion Technology.
When it comes to your smartphone, it's not just about the protection. It's also about your style, your personality, your life. With OtterBox Symmetry Series you don't have to choose between a protective case and a stylish one. You can have it all.
Reduced price!
The Samsung Galaxy S6 Edge Clear Protective Cover is the perfect way to not only minimise damage but also show your style.
The Samsung Galaxy S6 Edge Protective Cover is the perfect way to not only minimise damage but also show your style.
The Samsung Galaxy S6 Edge Clear View Cover will protect your Ga;laxy S6 Edge from scratches, damage and bumps. The transparent design allows you keep track of the time, get message alerts and answer calls without opening the cover. A suitably intelligent fit for your Samsung Galaxy S6 Edge.
Reduced price!
The Samsung Galaxy S6 Edge Wallet Flip Cover is designed to provide great protection as well as an attractive look, with a range of attractive colours to choose from. It also includes a handy card slot that you can use to store your credit card on the go.
The Muvit MyFrame case is a combination of a transparent shell showing the back of the Samsung Galaxy S6 Edge, and rubber finish around the edges providing drop protection.
It's quite the show off. This protective, two-piece Naked Tough case for Samsung Galaxy S6 Edge keeps it cool with its smooth transparent finish and ultra slim design.
Reduced price!
The Case-Mate Sheer Glam case for the Samsung Galaxy S6 Edge is inspired by dazzling fine jewelry that shimmers.
Inspired by military armor, UAG (Urban Armor Gear) brings lightweight cases with military-grade shock protection for your Samsung Galaxy S6 Edge, without adding the extra bulk.
Not afraid to get your hands dirty? Neither is Tough Jacket™. It's ready to handle whatever life throws your way. With reinforced corner protection and raised lips, we dare you.
Our thinnest case, yet. Shell adds just 1.4 mm in thickness. Sleek, one-piece design shields your device from the indignities of the outside world even as it preserves the look and feel of your phone.
For the purist and minimalist. It offers a slim profile with quality details and protection.