An oil pump comprising: a rotor that is adapted to be rotationally driven by an engine; a tubular body that forms a pump chamber between the tubular body and an outer circumference side of the rotor; a casing that houses the rotor and the tubular body; a suction port and a discharge port that are formed in the casing; a pump mechanism that is adapted to cause oil suctioned into the pump chamber from the suction port to be discharged from the discharge port following rotation of the rotor; a capacity adjustment mechanism that is adapted to change a pump capacity by moving the tubular body in a tube radial direction relative to the rotor; a control valve that is adapted to convert oil pressure from the discharge port into control pressure; and a control oil passage that is capable of moving the tubular body in the tube radial direction by causing the control pressure from the control valve to act on the capacity adjustment mechanism, wherein the capacity adjustment mechanism has a configuration that moves the tubular body in a direction in which pump capacity decreases, as the control pressure increases, the control valve maintains the control oil passage in an open state, in a pressure region in which the oil pressure is less than a first control value and in a pressure region in which the oil pressure reaches a second control value that exceeds the first control value, the capacity adjustment mechanism, in a case where the control pressure is less than the first control value, increases an oil discharge amount at a first gradient following an increase in engine rotational speed by setting the pump capacity to maximum, and, in a case where the control pressure exceeds the first control value, increases the oil discharge amount at a second gradient that is less than the first gradient following an increase in engine rotational speed in a state where the pump capacity is reduced by moving the tubular body in the direction in which pump capacity decreases, the control valve, in a case where the oil pressure rises in a pressure region from the second control value up to a third control value that exceeds the second control value, operates to decrease the control pressure by narrowing the control oil passage as the oil pressure rises, and the capacity adjustment mechanism decreases the reduction in pump capacity by reducing or stopping movement of the tubular body in the direction in which pump capacity decreases, and increases the oil discharge amount at a third gradient that is greater than the second gradient following an increase in engine rotational speed. a rotor that is adapted to be rotationally driven by an engine; a tubular body that forms a pump chamber between the tubular body and an outer circumference side of the rotor; a casing that houses the rotor and the tubular body; a suction port and a discharge port that are formed in the casing; a pump mechanism that is adapted to cause oil suctioned into the pump chamber from the suction port to be discharged from the discharge port following rotation of the rotor; a capacity adjustment mechanism that is adapted to change a pump capacity by moving the tubular body in a tube radial direction relative to the rotor; a control valve that is adapted to convert oil pressure from the discharge port into control pressure; and a control oil passage that is capable of moving the tubular body in the tube radial direction by causing the control pressure from the control valve to act on the capacity adjustment mechanism, wherein the capacity adjustment mechanism has a configuration that moves the tubular body in a direction in which pump capacity decreases, as the control pressure increases, the control valve maintains the control oil passage in an open state, in a pressure region in which the oil pressure is less than a first control value and in a pressure region in which the oil pressure reaches a second control value that exceeds the first control value, the capacity adjustment mechanism, in a case where the control pressure is less than the first control value, increases an oil discharge amount at a first gradient following an increase in engine rotational speed by setting the pump capacity to maximum, and, in a case where the control pressure exceeds the first control value, increases the oil discharge amount at a second gradient that is less than the first gradient following an increase in engine rotational speed in a state where the pump capacity is reduced by moving the tubular body in the direction in which pump capacity decreases, the control valve, in a case where the oil pressure rises in a pressure region from the second control value up to a third control value that exceeds the second control value, operates to decrease the control pressure by narrowing the control oil passage as the oil pressure rises, and the capacity adjustment mechanism decreases the reduction in pump capacity by reducing or stopping movement of the tubular body in the direction in which pump capacity decreases, and increases the oil discharge amount at a third gradient that is greater than the second gradient following an increase in engine rotational speed.