Tolman Model Research Articles

Electromagnetic field on the complexity of minimally deformed compact stars

In the context of this endeavor, we establish a simple protocol for formulating interior stellar solutions that exhibit spherically symmetric configurations against the backdrop of relativistic gravitational decoupling through radial metric deformation (minimal geometric deformation scheme). In this pursuit, we make use of the vanishing complexity factor (Y~TF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{Y}_{TF}$$\\end{document}) condition, based on Herrera’s (Phys Rev D 97, 044010, 2018) innovative concept regarding the complexity of static or slowly evolving spherical matter configurations. The idea of a complexity factor emerges as the outcome of the orthogonal splitting of the Riemann–Christoffel tensor, which yields different scalar functions, known as structure scalars. The protocol is demonstrated by employing the Buchdahl and Tolman relativistic stellar ansatzes as isotropic seeds. Both of these ansatzes exhibit similar physical features, with a minor variation in their magnitudes in the case of Y~TF≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{Y}_{TF}\ e 0$$\\end{document}, where 0≤α<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0\\le \\alpha <1$$\\end{document}, and α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha $$\\end{document} represents a coupling parameter. However, when Y~TF=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\widetilde{Y}_{TF}=0$$\\end{document}, the Buchdahl stellar ansatz exhibits a uniform density matter configuration, while the Tolman model features an increasing pressure profile. The obtained relativistic stellar models satisfy the basic viability constraints required for the physically realistic configurations.

Role of Complexity on Self‐gravitating Compact Star by Gravitational Decoupling

AbstractIn this paper, we study novel classes of solutions characterizing the role of complexity on static and spherically symmetric self‐gravitating systems proposed by L. Herrera (Phys Rev D 97: 044010, 2018) in the gravitational decoupling background. We start by considering the minimal geometric deformation approach as a ground‐breaking tool for generating new physically viable models for anisotropic matter distributions by exploiting the Buchdahl and Tolman models. In both models, all solutions show similar results with a slight change in their magnitude for a non‐vanishing complexity factor i.e., , where β is a decoupling constant. However, under vanishing complexity factor i.e., , the minimally deformed anisotropic Buchdahl model yielded a constant density isotropic fluid distribution and anisotropic matter distribution becomes an isotropic fluid matter distribution without invoking any isotropy requirement. On the other hand, Tolman model possesses an increasing pressure when the complexity factor vanishes. Furthermore, we also extend our findings by calculating the mass‐complexity factor relationship for both presented models, revealing that the mass is larger for small values of the decoupling constant β and the complexity factor .

Improved Lemaitre–Tolman Model and the Mass and Turn-around Radius in Group of Galaxies. II. The Role of Dark Energy

Abstract In this paper, we extend our previous study on the Lemaitre–Tolman (LT) model showing how the prediction of the model changes when the equation of state (EoS) parameter (w) of dark energy (DE) is modified. In the previous study, it was considered that DE was merely constituted by the cosmological constant. In this paper, as in the previous study, we also took into account the effect of angular momentum and dynamical friction (J ηLT model) that modifies the evolution of a perturbation, initially moving with the Hubble flow. As a first step, solving the equations of motion, we calculated the relationship between mass, M, and the turn-around radius, R 0. If one knows the value of the turn-around radius R 0, it is possible to obtain the mass of the studied objects. As a second step, we build up, as in the previous paper, a relationship between the velocity, v, and radius, R. The relation was fitted to data of groups and clusters. Since the relationship v–R depends on the Hubble constant and the mass of the object, we obtained optimized values of the two parameters of the objects studied. The mass decreases of a factor of maximum 25% comparing the J ηLT results (for which w = −1) and the case w = −1/3, while the Hubble constant increases going from w = −1 to w = −1/3. Finally, the obtained values of the mass, M, and R 0 of the studied objects can put constraints on the DE EoS parameter, w.

Expansion of bundles of light rays in the lemaître-tolman models

The locus of $\theta \equiv {k^{\mu}};_{\mu} = 0$ for bundles of light rays emitted at noncentral points is investigated for Lema\^{\i}tre -- Tolman (L--T) models. The three loci that coincide for a central emission point: (1) maxima of $R$ along the rays, (2) $\theta = 0$, (3) $R = 2M$ are all different for a noncentral emitter. If an extremum of $R$ along a nonradial ray exists, then it must lie in the region $R > 2M$. In $2M < R \leq 3M$ it can only be a maximum; in $R > 3M$ both minima and maxima can exist. The intersection of (1) with the equatorial hypersurface (EHS) $\vartheta = \pi/2$ is numerically determined for an exemplary toy model (ETM), for two typical emitter locations. The equation of (2) is derived for a general L--T model, and its intersection with the EHS in the ETM is numerically determined for the same two emitter locations. Typically, $\theta$ has no zeros or two zeros along a ray, and becomes $+ \infty$ at the Big Crunch (BC). The only rays on which $\theta \to - \infty$ at the BC are the radial ones. Along rays on the boundaries between the no-zeros and the two-zeros regions $\theta$ has one zero, but still tends to $+ \infty$ at the BC. When the emitter is sufficiently close to the center, $\theta$ has 4 or 6 zeros along some rays (resp. 3 or 5 on the boundary rays). For noncentral emitters in a collapsing L--T model, $R = 2M$ is still the ultimate barrier behind which events become invisible from outside; loci (1) and (2) are not such barriers.

Improved Lemaitre–Tolman model and the mass and turn-around radius in group of galaxies

We extended the modified Lemaitre-Tolman model taking into account the effect of angular momentum and dynamical friction. The inclusion of these quantities in the equation of motion modifies the evolution of a perturbation, initially moving with the Hubble flow. Solving the equation of motions we got the relationships between mass, $M$, and the turn-around radius, $R_0$. Knowing $R_0$, the quoted relation allows the determination of the mass of the object studied. The relationships for the case in which also the angular momentum is taken into account gives a mass $\simeq 90$ \% larger than the standard Lemaitre-Tolman model, and two times the value of the standard Lemaitre-Tolman model, in the case also dynamical friction is taken into account. As a second step, we found relationships between the velocity, $v$, and radius, $R$, and fitted them to data of the Local Group, M81, NGC 253, IC342, CenA/M83, and to the Virgo clusters obtained by Ref.[New Astronomy 11(4):325, A&A 488(3):845]. This allowed us to find optimized values of the mass and Hubble constant of the objects studied. The fit gives values of the masses smaller with respect to the $M-R_0$ relationship method, but in any case 30-40\% larger than the $v-R$ relationship obtained from the standard Lemaitre-Tolman model. Differently from mass, the Hubble parameter becomes smaller with respect to the standard Lemaitre-Tolman model, when angular momentum, and dynamical friction are introduced. This is in agreement with Ref.[New Astronomy 11(4):325, A&A 488(3):845], who improved the standard Lemaitre-Tolman model taking into account the cosmological constant. Finally, we used the mass, $M$, and $R_0$ of the studied objects to put constraints to the dark energy equation of state parameter, $w$. Comparison with previous studies show different constraints on $w$.

Augmented Gibbs-Tolman Model for Surface Tension.

Gibbs developed the thermodynamics of a liquid-vapor system by introducing the idea of a "dividing surface" a hypothetical surface that separates the system into two homogeneous phases. The area and curvatures of a conveniently chosen dividing surface, the "surface of tension", are used to account for the effects of the smooth variation of properties across the actual transition layer between the phases. Tolman (1948, 1949) considered a more detailed model of the interfacial region and obtained expressions for surface tension (σ) and the location of the surface of tension. Based on qualitative arguments, Tolman's model introduced a surface of tension, such that the pressure (P) increases from its saturation value (Psat) to a maximum value (Pmax) as the surface is approached from the vapor side and decreases from (Psat) to its minimum value (Pmin) as the surface is approached from the liquid side. Assuming an exponential decay of (P) away from the surface, Tolman obtained an explicit expression for (σ) in terms of Psat, Pmax, Pmin, and two length scales. In the this work, the Gibbs-Tolman (GT) model is used along with the Lee and Kesler (1975) equation of state. The model is augmented to take into account the effect of the density gradient in the transition zone and a 4-parameter augmented model (AGT model) is proposed. The GT and AGT models are shown to fit the data for 152 pure liquids with an absolute average deviation (AAD) of 4.91% and 2.02%, respectively. The corresponding AAD values for 57 liquid mixtures are 4.2% and 3.0% respectively. Arguments are also presented to counter some of the fundamental concerns that have been raised about the GT approach. Although the model correlates the data very well, one of the length parameters turns out to be persistently negative, and the reason for this behavior is not clear.

Existence of blueshifts in quasispherical Szekeres spacetimes

In Lema\^{\i}tre -- Tolman (L--T) models, light rays emitted \textit{radially} at the Big Bang (BB) at such radial coordinates $r$ where the bang-time function $t_B(r)$ has $\dril {t_B} r \neq 0$ reach every observer with \textit{infinite blueshift}, $z = -1$. Consequently, there exist rays, emitted soon after the BB, that will reach later observers with finite blueshift ($-1 < z < 0$). But in spacetimes without symmetry there are no radial directions. The question thus arises whether blueshifts can exist at all in the Szekeres models that contain L--T as a limit, but in general have no symmetry. The aim of the present paper is to show that strong blueshifts can be generated in quasi-spherical Szekeres (QSS) models. It is shown that in an axially symmetric QSS model, infinite blueshift can appear only on axial rays, which intersect every space orthogonal to the dust flow on the symmetry axis. In an exemplary QSS model it is numerically shown that if such a ray is emitted from the Big Bang where $\dril {t_B} r \neq 0$, then indeed observers see it with $z \approx -1$. Rays emitted shortly after the BB and running close to the symmetry axis will reach the observers with a strong blueshift, too. Then, in a toy QSS model that has no symmetry, it was shown by numerical calculations that two null lines exist such that rays in their vicinity have redshift profiles similar to those in a vicinity of the axial rays in the axially symmetric case. This indicates that rays generating infinite blueshifts exist in general QSS spacetimes and are concentrated around two directions.

Ricci time in the Lemaître–Tolman model and the block universe

It is common to think of our universe according to the “block universe” concept, which says that spacetime consists of many “stacked” three-surfaces, labelled by some kind of proper time, $$\tau $$ . Standard ideas do not distinguish past and future, but Ellis’ “evolving block universe” tries to make a fundamental distinction. One proposal for this proper time is the proper time measured along the timelike Ricci eigenlines, starting from the big bang. This work investigates the shape of the “Ricci time” surfaces relative to the the null surfaces. We use the Lemaître–Tolman metric as our inhomogeneous spacetime model, and we find the necessary and sufficient conditions for these $$\{\tau = $$ constant $$\}$$ surfaces, $$S(\tau )$$ , to be spacelike or timelike. Furthermore, we look at the effect of strong gravity domains by determining the location of timelike S regions relative to apparent horizons. We find that constant Ricci time surfaces are always spacelike near the big bang, while at late times (near the crunch or the extreme far future), they are only timelike under special circumstances. At intermediate times, timelike S regions are common unless the variation of the bang time is restricted. The regions where these surfaces become timelike are often adjacent to apparent horizons, but always outside them, and in particular timelike S regions do not occur inside the horizons of black-hole-like models.

Mimicking acceleration in the constant-bang-time Lemaître-Tolman model: Shell crossings, density distributions, and light cones

The Lema\^{\i}tre--Tolman model with $\mathrm{\ensuremath{\Lambda}}=0$ and constant bang time that imitates the luminosity distance--redshift relation of the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ model using the energy function $E$ alone contains shell crossings. In this paper, the location in spacetime and the consequences of existence of the shell-crossing set (SCS) are investigated. The SCS would come into view of the central observer only at $t\ensuremath{\approx}1064T$ to the future from now, where $T$ is the present age of the Universe, but would not leave any recognizable trace in her observations. Light rays emitted near to the SCS are blueshifted at the initial points, but the blueshift is finite, and is overcompensated by later-induced redshifts if the observer is sufficiently far. The local blueshifts cause that $z$ along a light ray is not a monotonic function of the comoving radial coordinate $r$. As a consequence, the angular diameter distance ${D}_{A}$ and the luminosity distance ${D}_{L}$ from the central observer fail to be functions of $z$; the relations ${D}_{A}(z)$ and ${D}_{L}(z)$ are multiple-valued in a vicinity of the SCS. The following quantities are calculated and displayed: (1) The distribution of mass density on a few characteristic hypersurfaces of constant time; some of them intersect the SCS. (2) The distribution of density along the past light cone of the present central observer. (3) A few light cones intersecting the SCS at characteristic instants. (4) The redshift profiles along several light cones. (5) The extremum-redshift hypersurface. (6) The ${D}_{A}(z)$ and ${D}_{L}(z)$ relations. (7) The last scattering time and its comparison with the $\mathrm{\ensuremath{\Lambda}}\mathrm{CDM}$ last scattering epoch.

Accelerating expansion or inhomogeneity? II. Mimicking acceleration with the energy function in the Lemaître-Tolman model

This is a continuation of the paper published in {\it Phys. Rev.} {\bf D89}, 023520 (2014). It is investigated here how the luminosity distance -- redshift relation $D_L(z)$ of the $\Lambda$CDM model is duplicated in the Lema\^{\i}tre -- Tolman (L--T) model with $\Lambda = 0$, constant bang-time function $t_B$ and the energy function $E(r)$ mimicking accelerated expansion on the observer's past light cone ($r$ is a uniquely defined comoving radial coordinate). Numerical experiments show that $E > 0$ necessarily. The functions $z(r)$ and $E(r)$ are numerically calculated from the initial point at the observer's position; then backward from the initial point at the apparent horizon (AH). Reconciling the results of the two calculations allows one to determine the values of $E/r^2$ at $r = 0$ and at the AH. The problems connected with continuing the calculation through the AH are discussed in detail and solved. Then $z(r)$ and $E(r)$ are continued beyond the AH, up to the numerical crash that signals the contact of the light cone with the Big Bang. Similarly, the light cone of the L--T model is calculated by proceeding from the two initial points, and compared with the $\Lambda$CDM light cone. The model constructed here contains shell crossings, but they can be removed by matching the L--T region to a Friedmann background, without causing any conflict with the type Ia supernovae observations. The mechanism of imitating the accelerated expansion by the $E(r)$ function is explained in a descriptive way.

Inhomogeneous cosmological models and fine-tuning of the initial state

Inhomogeneous cosmological models are often reported to suffer from a fine-tuning problem because of the observer's location. We study if this is a generic feature in the Lemaître–Tolman (LT) models, by investigating if there are models with freedom in the initial state. In these cases, the present fine-tuned location would be evolved from a non-fine-tuned initial state thus vanishing the problem. In this paper, we show that this is not a generic problem and we give the condition when the LT models do not have fine-tuned initial state. The physical meaning of this condition, however, requires more investigation. We investigate if this condition can be found from a special case: homogeneous models with matter, dark and curvature density as parameters. We found that with any reasonable density values, these models do not satisfy this condition and thus do not have freedom in the initial state. We interpret this to be linked with the fine-tuning problem of the initial state of the homogeneous models, when the early time inflation is not included to them. We discuss the condition in the context of non-homogeneous models.

Dicyclohexyl(4-isopropylphenyl)phosphane selenide

In the title compund, C21H33PSe, the Se=P bond is part of a distorted tetra­hedral environment on the P atom. Both cyclo­hexyl groups adopt chair conformations. A cone angle of 170° was calculated using an adaptation of the Tolman model. Inter­molecular C—H⋯Se and C—H⋯Cg contacts are observed (Cg is the centroid of the benzene ring).

Formation of Gyrs old black holes in the centers of galaxies within the Lemaître–Tolman model

In this article we present a model of formation of a galaxy with a black hole in the center. It is based on the Lemaître–Tolman solution and is a refinement of an earlier model. The most important improvement is the choice of the interior geometry of the black hole allowing for the formation of Gyrs old black holes. Other refinements are the use of an arbitrary Friedmann model as the background (unperturbed) initial state and the adaptation of the model to an arbitrary density profile of the galaxy. Our main interest was the M87 galaxy (NGC 4486), which hosts a supermassive black hole of mass 3.2 × 109 M⊙. It is shown that for this particular galaxy, within the framework of our model and for the initial state being a perturbation of the ΛCDM model, the age of the black hole can be up to 12.7 Gyrs. The dependence of the model on the chosen parameters at the time of last scattering was also studied. The maximal age of the black hole as a function of the Ω m and ΩΛ parameters for the M87 galaxy can be 3.717 or 12.708 Gyr.

Imitating accelerated expansion of the Universe by matter inhomogeneities: corrections of some misunderstandings

A number of misunderstandings about modeling the apparent accelerated expansion of the Universe and about the ‘weak singularity’ are clarified: (1) Of the five definitions of the deceleration parameter given by Hirata and Seljak (HS), only q 1 is a correct invariant measure of acceleration/deceleration of expansion. The q 3 and q 4 are unrelated to acceleration in an inhomogeneous model. (2) The averaging over directions involved in the definition of q 4 does not correspond to what is done in observational astronomy. (3) HS’s equation (38) connecting q 4 to the flow invariants gives self-contradictory results when applied at the centre of symmetry of the Lemaître–Tolman (L–T) model. The intermediate equation (31) that determines q 3' is correct, but approximate, so it cannot be used for determining the sign of the deceleration parameter. Even so, at the centre of symmetry of the L–T model, it puts no limitation on the sign of q 3'(0). (4) The ‘weak singularity’ of Vanderveld et al. is a conical profile of mass density at the centre—a perfectly acceptable configuration. (5) The so-called ‘critical point’ in the equations of the ‘inverse problem’ for a central observer in an L–T model is a manifestation of the apparent horizon (AH)—a common property of the past light cones in zero-lambda L–T models, perfectly manageable if the equations are correctly integrated.

Volume averaging in the quasispherical Szekeres model

This paper considers volume averaging in the quasispherical Szekeres model. The volume averaging became of considerable interest after it was shown that the volume acceleration calculated within the averaging framework can be positive even when the local expansion rate decelerates. This issue was intensively studied within spherically symmetric models. However, since our Universe is not spherically symmetric similar analysis is needed in non-symmetrical models. This papers presents the averaging analysis within the quasispherical Szekeres model which is a non-symmetrical generalisation of the spherically symmetric Lemaître–Tolman family of models. In the quasispherical Szekeres model the distribution of mass over a surface of constant t and r has the form of a mass-dipole superposed on a monopole. This paper shows that when calculating the volume acceleration, ä, within the Szekeres model, the dipole does not contribute to the final result, hence ä only depends on a monopole configuration. Thus, the volume averaging within the Szekeres model leads to literally the same solutions as those obtained within the Lemaître–Tolman model.

Apparent and average accelerations of the Universe

In this paper we consider the relation between the volume decelerationparameter obtained within the Buchert averaging scheme and the decelerationparameter derived from supernova observation. This work was motivated byrecent findings that showed that there are models which despite havingΛ = 0 have volumedeceleration parameter qvol<0. This opens the possibility that back-reaction and averaging effects may be used as aninteresting alternative explanation to the dark energy phenomenon.We have calculated qvol in some Lemaître–Tolman models. For those models which arechosen to be realistic and which fit the supernova data, we find thatqvol>0, while those models which we have been able to find which exhibitqvol<0 turn out to be unrealistic. This indicates that care must be exercised in relating thedeceleration parameter to observations.

The Great Attractor and the Shapley Concentration

We develop a non-linear relativistic model of the Shapley Concentration (SC) and its environs, including the Great Attractor (GA) and the Local Group (LG). We take the Shapley Concentration as a major attractive centre, and we use the Lemaître–Tolman model. We constrain our model with present day observations, plus the requirement that it have a physically reasonable evolution from small perturbations at last scattering. We investigate possible mass and velocity distributions, and we find that the peculiar velocity maximum near the SC is ~800 km/s inwards, the density between GA and SC must be about ~0.9 times background, the mass of the GA is probably 4–6 × 1015 M⊙, the SC’s contribution to the LG motion is negligible, and the value of the cosmological constant is not significant on this scale.

Light propagation through large-scale inhomogeneities in the Universe and its impact on cosmological observations

This paper analyses cosmological observations within inhomogeneous and exact solutions of the Einstein equations. In some way the analyses presented here can be freed from assumptions such as small amplitude of the density contrast. The supernova observations are analysed using the Lemaitre–Tolman model and the CMB observations are analysed using the quasispherical Szekeres model. The results show that it is possible to fit the supernova data without the cosmological constant. However if inhomogeneities of sizes and amplitudes as observed in the local Universe are considered, their impact on cosmological observations is small.

Steric and electronic properties in bicyclic phosphines. Crystal and molecular structures of Se = Phoban-Q (Q = C 2, C 3Ph, Cy and Ph)

Reacting a series of the bicylic Phoban-Q (9-Q-9-phosphabicyclo[3.3.1]nonane and 9-Q-9-phosphabicyclo[4.2.1]nonane) derivatives (Q = alkyl, cyclo alkyl, aryl) with KSeCN results in the formation of the corresponding phosphine selenides. The first order phosphorus–selenium coupling constants, 1 J P–Se, ranges from 682 to 689 Hz for the [3.3.1] isomers and from 703 to 717 Hz for the [4.2.1] isomers indicating the former to be significantly more electron rich. The crystal structures of Se = Phoban[3.3.1]-Q (Q = CH 2CH 3, C 3H 6Ph, Cy, and Ph) and Se = Phoban[4.2.1]-Q (Q = Cy and Ph) are reported and reveal P Se bond distances ranging from 2.1090(9) to 2.1245(7) Å. For Q = Cy and Ph the two isomers ([3.3.1] and [4.2.1]) co-crystallise in the same crystal enabling the determination of the molecular structures for both from the same data collection. The cone angles for all ligand derivatives were determined according to the Tolman model but by using the actual P–Se bond distances and were found to be virtually identical ranging from 165° to 175°. Changes in the Q substituent have a minor effect on the overall steric and electronic properties of the Phoban family of ligands and can be used to manipulate physical properties without changing the chemical properties significantly.

Bicyclic phosphines as ligands for cobalt catalysed hydroformylation. Crystal structures of [Co(Phoban[3.3.1]-Q)(CO)3]2 (Q = C2H5, C5H11, C3H6NMe2, C6H11)

A range of tertiary bicyclic phosphine ligands derived from cis, cis-1,5-cyclooctadiene (Phoban family) was studied by batch autoclave reactions during the hydroformylation of a mixture of linear internal decenes using a cobalt catalyst system. Comparative runs were performed with PBu(3) as representative of standard trialkyl phosphine behaviour. The Phoban ligands comprise of a cyclooctyl bicycle with a mixture of the [3.3.1] and [4.2.1] isomers where the third substituent was systematically varied, Phoban-Q (Q = CH(2)CH(3), (CH(2))(4)CH(3), (CH(2))(9)CH(3), (CH(2))(19)CH(3), (CH(2))(3)N(CH(3))(2), C(6)H(11) and C(6)H(5)). An increase in ligand concentration resulted in a decrease in the reaction rate while the selectivity towards the n-alcohol product increased in accordance with a move from more unmodified catalysis to more modified catalysis. Alcohol yields of 77-85% were obtained at rates of 1.8-2.4 h(-1) for highly modified catalysis. Under highly modified conditions the linearity of the alcohol ranges in a narrow band from approximately 85-90% from Phoban-Ph to Phoban-Cy respectively. Hydrogenation of the alkene substrate varied from approximately 9-15% for Phoban-Ph and Phoban-Cy respectively the least and most electron donating derivatives. The two phosphine isomers were separated for Phoban-C(2) and the hydroformylation activity were re-evaluated for each isomer. The less electron donating [4.2.1] isomer required slightly higher ligand concentrations to achieve fully modified catalysis and gave rates and linearities comparable to the [3.3.1] isomer but giving slightly higher yields due to less hydrogenation of the olefin. In comparison, at fully modified conditions, PBu(3) gave a rate of 0.6 h(-1), alcohol yield of 77%, linearity of 81% and 17% hydrogenation. The crystal structures of the cobalt dimers [Co(CO)(3)(Phoban[3.3.1]-C(2))](2), [Co(CO)(3)(Phoban[3.3.1]-C(5))](2), [Co(CO)(3)(Phoban[3.3.1]-C(3)NMe(2))](2), and [Co(CO)(3)(Phoban[3.3.1]-Cy)](2) have been determined and indicated very similar geometries with Co-Co and Co-P bond distances ranging from 2.6526(10)-2.707(3) and 2.1963(8)-2.2074(9) A respectively. The cone angles of the Phoban ligands were calculated from the crystallographic data, according to the Tolman model, and ranges from 159-165 degrees.